Technische Universit¨atDresden Faculty for Forestry, Geosciences and Hydrosciences
Systems Analysis in Water Management
Peter-Wolfgang Gr¨aber
Summer Semester 2010 This lecture material is only internal to use for the study courses of the Department
Hydro Sciences of the TU Dresden. This books are subject to the copyright and is only for the intern use in connection of the education inside of the Technische Universitaet
Dresden.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval, without permission in writing form the publisher.
Editing and layout: Prof. Dr.-Ing. habil. Peter-Wolfgang Gr¨aber
Technische Universit¨atDresden
Faculty of Forest, Geosciences and
Hydrosciences
Institute of Waste Management and
Contaminated Site Treatment
Phone: +49 (0) 3501 530029
Fax: +49 (0) 3501 530022
e-mail: [email protected]
Internet: http://www.tu-dresden.de/fghhiaa
Issue: 28.02.2009
II Author presentation
The lecture Systems Analysis in Water Management is based on foundation courses such as mathematics, physics and informatics. And the chapters, which play a role in water management task solutions, will be specially discussed in the lecture, such as vector analysis, solution of equation systems, matrix calculation, and solution of differential equations as well as numerical integration.
The following chapters are revisions of basic knowledge and will only be shown with corresponding key points. Self study is strongly recommended during the revision.
The further chapters go beyond basic knowledge and indicate mathematical methods, which are related to the water management practice.
The teaching contents of the subject Systems Analysis in Water Management require an advanced mathematical knowledge, including abstraction ability. In the ex- ercise and computer courses, some problem will be discussed combined with practically relevant cases in order to develop a deeper understanding of this lecture.
III Table of contents
Author presentation ...... III
List of tables ...... XIV
List of Figures ...... XXI
Abbreviations and Formula Symbols ...... XXII
I Mathematical fundamentals 1
1 Algebra fundamentals ...... 5
1.1 Exponential and logarithmic expressions ...... 6
1.1.1 Exercises ...... 8
1.2 Matrices ...... 9
1.2.1 Fundamentals ...... 9
1.2.2 Calculation rules ...... 14
1.2.3 determinant of a matrix ...... 19
1.2.4 Exercises ...... 21
1.3 Linear Equation Systems ...... 22
1.3.1 Total step method ...... 24
1.3.1.1 Gauss elimination ...... 24
1.3.1.2 Cramer’s rule ...... 28
1.3.1.3 Construction of the inverse matrix ...... 31
IV 1.3.1.4 LU decomposition ...... 33
1.3.1.5 Cholesky decomposition ...... 38
1.3.2 Iterative methods ...... 48
1.3.3 Overdetermined equation systems (m > n) ...... 49
1.3.4 Exercises ...... 50
2 Vector algebra and vector calculus ...... 51
2.1 Unit vectors ...... 53
2.2 Algorithms ...... 55
2.3 Examples of vector calculus ...... 62
2.4 Exercises ...... 66
3 Interpolation methods ...... 69
3.1 Polynomial interpolation ...... 74
3.1.1 Analytical power function ...... 75
3.1.2 Lagrange interpolation formula ...... 78
3.1.3 Newton interpolation formula ...... 81
3.1.3.1 Arbitrary supporting points ...... 81
3.1.3.2 Equidistant supporting point distribution ...... 84
3.1.3.3 Examples for the Newton method ...... 87
3.2 Spline interpolation ...... 91
3.3 Kriging method ...... 100
3.4 Exercises ...... 105
V 4 Optimisation problems...... 107
4.1 Analytical solution of extreme value problems ...... 108
4.2 Iterative optimum search ...... 108
4.3 Least squares method (MKQ method) ...... 108
4.4 Search strategies ...... 109
4.4.1 Jones spiral method ...... 111
5 Ordinary differential equations...... 113
5.1 Setting up equations ...... 116
5.1.1 Examples of setting up differential equations: ...... 117
5.1.2 Exercises for setting up ODE’s: ...... 121
5.2 Analytical solution methods ...... 125
5.2.1 First order ordinary differential equations ...... 125
5.2.1.1 Solution of homogeneous differential equations . . . . . 125
5.2.1.2 Solution of the inhomogeneous first order ODE . . . . 128
5.2.1.3 Exercises ...... 133
5.2.2 Ordinary differential equations of higher order ...... 136
5.2.2.1 OED of type a ...... 136
5.2.2.2 ODE of type b ...... 137
5.2.2.3 ODE of type c ...... 142
5.2.2.4 Exercises ...... 143
5.3 Integral transforms ...... 144
VI 5.3.1 Time- and Frequency domain ...... 144
5.3.2 Laplace Transform ...... 147
5.3.2.1 Forward transformation ...... 147
5.3.2.2 Important calculation rules ...... 148
5.3.2.3 Inverse transformation ...... 150
5.3.2.4 Correspondence table ...... 151
5.3.3 Solution of differential equations by means of
Laplace transform ...... 153
5.3.3.1 Algorithm ...... 153
5.3.3.2 Examples ...... 153
5.3.3.3 Example of ODE systems ...... 156
5.3.3.4 Exercises to the integral transforms ...... 158
5.4 Methods for Numerical Integration ...... 161
5.4.1 Integration ...... 161
5.4.1.1 Rectangle rule ...... 161
5.4.1.2 Trapezoidal rule ...... 164
5.4.1.3 Simpson’ rule ...... 165
5.4.1.4 Newton formula ...... 165
5.4.1.5 Examples for application of numerical integration . . . 166
5.4.1.6 Exercises for the application of the numerical
integration ...... 169
5.4.2 Numerical Solutions of Differential Equations ...... 171
VII 5.4.2.1 Euler method ...... 172
5.4.2.2 Runge Kutta method ...... 175
5.4.2.3 Predictor Corrector Method ...... 178
5.4.2.4 Exercises to numerical solutions of ODE ...... 181
II Partial differential equations of underground processes 183
6 Overview ...... 185
6.1 One dimensional flow equation ...... 188
6.2 Horizontal plane groundwater flow equation ...... 189
6.3 One dimensional material transfer ...... 190
6.4 Multiphase flow ...... 190
7 Horizontal plane Groundwater flow equation ...... 193
7.1 Dupuit assumption and ballance equation ...... 194
7.2 Potential illustration ...... 197
7.3 Boundary conditions ...... 201
7.3.1 Initial conditions ...... 201
7.3.2 Randbedingungen ...... 202
8 Analytical Solutions ...... 205
8.1 Theis well equation (rotationally symmetrical flow) ...... 206
8.1.1 General solution ...... 206
8.1.2 Consideration of special effects ...... 214
8.1.2.1 Imperfect well ...... 214
VIII 8.1.2.2 Multi-well plants ...... 216
8.1.2.3 Variable conveying curve of a well ...... 217
8.1.2.4 Limitations ...... 222
8.1.2.5 Multilateral boundary ...... 228
8.1.3 Supply from neighbouring layers ...... 231
8.2 Exercises to analytical solutions ...... 234
9 Numerical methods ...... 241
9.1 Methods of local quantization ...... 244
9.1.1 Finite Differencen Method ...... 246
9.1.1.1 Balance equation ...... 247
9.1.1.2 Consideration of boundary conditions ...... 260
9.1.2 Finite Element Method ...... 266
9.2 Time quantization methods ...... 270
9.2.1 Backward difference - Implicit methods ...... 272
9.2.2 Mixed methods ...... 277
9.2.3 Extrapolation method ...... 279
9.3 Exercises of numerical calculation ...... 281
10 Simulation programm system ASM ...... 285
III System theory and Modelling 291
11 Fundamentals ...... 293
11.1 Model classification ...... 294
IX 11.2 Methods of process analysis ...... 300
11.2.1 Theoretical process analysis ...... 300
11.2.2 Experimental process analysis ...... 301
11.3 Signal representation ...... 303
11.3.1 Basic signal forms ...... 304
11.3.2 Application of selected test signals ...... 305
11.3.2.1 Impulse function ...... 305
11.3.2.2 Step function ...... 306
11.3.2.3 Ramp function ...... 307
11.3.3 Signal syntheses ...... 309
11.3.4 Signal analysis ...... 310
11.3.5 Quantization ...... 312
11.4 Transmission systems ...... 314
11.4.1 Mathematical description ...... 315
11.4.2 Basic transient characteristics ...... 318
11.4.2.1 Proportional characteristic =⇒ P element ...... 318
11.4.2.2 Integral characteristic =⇒ I element ...... 320
11.4.2.3 Differential characteristic =⇒ D element ...... 324
11.4.2.4 First order delay =⇒ PT1 element ...... 325
11.4.2.5 Second order delay =⇒ PT2 element ...... 328
11.4.2.6 Duration behaviour =⇒ PTL element ...... 331
11.4.2.7 Overview of basic transient characteristic ...... 332
X 11.4.3 Combined transient characteristic ...... 335
12 Modell regulation based on parameters ...... 343
12.1 Transient characteristic with 1st order delay ...... 344
12.1.1 Mathematical description ...... 344
12.1.2 Time constant from integer multiples ...... 347
12.1.3 Time constant from slope ...... 348
12.2 Transient characteristic with 2nd order delay ...... 349
12.2.1 Mathematical description ...... 349
12.2.2 Unit step as input signal(transfer function h(t)) ...... 351
12.2.2.1 model type I ...... 354
12.2.2.2 model type II ...... 355
12.2.3 Dirac impuls as input signal (weight function g(t)) ...... 355
12.2.4 Exercises to experimental process analysis ...... 358
12.3 Arbitrary transient characteristic and arbitrary input signals ...... 363
12.3.1 Introductory ...... 363
12.3.2 decompositions of arbitrary input functions
(signal analysis) ...... 364
12.3.3 Composition of output functions (signal synthesis) ...... 366
12.3.4 Determination of weighting function g(t) for general case . . . . 370
12.3.5 Prognosemodelle ...... 373
12.3.6 Exercises and applications for the faltung integral ...... 375
XI IV Indirect parameter identification 377
13 Estimation procedures ...... 381
14 Flow parameters...... 385
14.1 Pumping test evaluation ...... 386
14.1.1 Fundamentals ...... 386
14.1.2 Practical realisation ...... 388
14.2 Pumping test simulator ...... 393
15 Suction power distribution ...... 397
BIBLIOGRAPY ...... 399
XII List of tables
2.1 coordinate systems for description of vectors ...... 53
2.2 Description of the NABLA operator in different coordinate systems . . 58
2.3 differential operators in the different coordinate systems ...... 61
4.1 Iterative estimation methods(according to WERNSTEDT) ...... 110
5.1 Representation of differential equations ...... 114
5.2 identification of differential equations ...... 115
5.3 Special cases in continuous integral transforms ...... 145
5.4 Special cases for discrete transformations ...... 146
5.5 Correspondence table Laplace transform ...... 151
5.5 Correspondence table - continuation ...... 152
8.1 Well function W (σ) for the range 1 · 10−12 ≤ σ ≤ 9 ...... 210
8.2 Error of COOPER and JACOB formula as a function of σ ...... 213
9.1 Einteilung von Netzformen ...... 244
9.2 equation system for two dimensional aquifer quantization ...... 251
9.3 equation system in diagonal shape ...... 252
9.4 continuation 1 ...... 253
9.5 cotinuation 2 ...... 254
9.6 continuation 3 ...... 255
XIII 11.1 comparison of theoretical and experimental process analysis ...... 302
11.2 classification of information parameter and signal carrier ...... 303
11.3 relationship between different basic signals ...... 307
11.4 Measuring instruments and methods and signal forms ...... 313
11.5 relationship between different functions of transient characteristic . . . 317
11.6 transient characteristics ...... 318
11.7 basic transient characteristic with mathematical description ...... 334
11.8 composite transfer elements ...... 336
12.1 classification of model type according STREJC ...... 353
12.2 time constant ratio dependent on step response ...... 355
12.3 parameter estimation for model type II ...... 356
12.4 variables of impulse response function for 2nd order delay ...... 357
12.5 comparison of applied abbreviations ...... 364
14.1 difference between pumping test evaluation and laboratory method . . . 386
14.2 Realisierte Programmversionen mit geohydraulischem Schema . . . . . 389
XIV List of figures
2.1 vector representation in Cartesian coordinates ...... 53
2.2 vector representation in spherical coordinates ...... 54
2.3 vector representation in two-dimensional space ...... 54
3.1 representation of the discontinuous measured data acquisition . . . . . 71
3.2 representation of an interpolation problem ...... 72
3.3 Representation of the measured interpolated values ...... 90
3.4 Representation of linear spline curves ...... 91
3.5 Representation of a spline curve with cubic polynomials ...... 92
3.6 Spline interpolation function ...... 99
3.7 Illustration of a Kriging problem ...... 101
4.1 Iterative model adjustment ...... 109
4.2 Spiral algorithm according to JONES ...... 112
5.1 Diagram of the pressure ratios ...... 117
5.2 Illustration of an filament heated industrial furnace ...... 119
5.3 Refilling of an old surface mine with constant volume flow rate . . . . . 121
5.4 Refilling of an old surface mine with variable volume flow rate . . . . . 122
5.5 Linked storage cascade ...... 122
5.6 Representation of the groundwater level with a block diagram . . . . . 123
XV 5.7 Water level regulation of a feeder ...... 123
5.8 Pollutant transport out of the soil into the water ...... 124
5.9 Schematic representation of the groundwater level ...... 134
5.10 Refilling procedure of a remainder hole ...... 135
5.11 Connection between original and complex variable domain ...... 144
5.12 Schematic representation of the groundwater level ...... 159
5.13 filling procedure of a remainder hole ...... 160
5.14 creation from rectangles to the numerical integration ...... 162
5.15 Numerical integration by means of trapezoidal rule ...... 165
5.16 groundwater level as a function of the radius ...... 170
5.17 Computation of the function value y(b) from the initial value y(a) . . . 172
5.18 iterative solution of the differential equation ...... 173
5.19 result development with the Euler method ...... 175
5.20 filling procedure of a remainder hole ...... 182
7.1 effect of boundary conditions on an aquifer ...... 204
8.1 Coordinate System for a rotationally symmetrical well ...... 206
8.2 Infinitively expanded aquifer ...... 208
8.3 imperfect well with filter in the
a) upper, b) lower, c) middle part of the aquifer ...... 215
8.4 multi-well plant (Br = well) ...... 217
8.5 superposition of the drawdown potentials ...... 218
XVI 8.6 Virtual conveying flows with time-dependent conveying curve ...... 219
8.7 composite conveying curve ...... 219
8.8 drawdown potential ...... 220
8.9 groundwater rising ...... 221
8.10 approximation of a continuous conveying curve ...... 221
8.11 1st type boundary condition modelling by a virtual well ...... 223
8.12 consideration of one-sided 1st type boundary condition ...... 223
8.13 2nd type boundary condition modelling by a virtual well ...... 224
8.14 consideration of one-sided 2nd type boundary condition ...... 224
8.15 consideration of one-sided 3rd type boundary condition ...... 227
8.16 dependence of the auxiliary length on the standardized river width . . . 228
8.17 multilateral boundary conditions with corresponding virtual wells . . . 230
8.18 function r/B dependent on σ ...... 233
8.19 infinitely expanded aquifer with well (Br) and GWOW (GWBR) . . . . 234
8.20 aquifer with imperfect river, well and foundation pit ...... 235
8.21 aquifer with well and foundation pit ...... 235
8.22 groundwater level dependent on radius ...... 236
8.23 aquifer with river, well and influx ...... 237
8.24 Infinitely expanded aquifer with a conveying well ...... 238
8.25 real river with pumping well (artificial groundwater recharge plant) . . 238
8.26 groundwater plant with well and sheet pile wall ...... 239
8.27 induced recharge plant with well and river ...... 239
XVII 9.1 network configurations ...... 245
9.2 transition between continuum and resistance network ...... 248
9.3 balance for node n, m ...... 249
9.4 4 x 4 grid net ...... 250
9.5 allocation of hydraulic parameter to compensatory conductivity . . . . 256
9.6 consideration of 2nd type boundary condition ...... 261
9.7 consideration of a 3rd type boundary condition ...... 262
9.8 outside nodes lying boundary condition ...... 264
9.9 Relationship from tangent to secant with a typical drawdown procedure 270
9.10 Argument association in time quantization of a 1-D-field problem . . . 271
9.11 consideration of time quantization ...... 273
9.12 Time quantized computed drawdown of a ditch flow ...... 275
9.13 Time quantized computed groundwater rise of a ditch flow ...... 275
9.14 Dependence of the time quantization error on the time increment . . . . 276
9.15 equivalent circuit diagram of local point Pn,m ...... 279
9.16 stratified aquifer with steady flow regime ...... 281
9.17 stratified aquifer with steady flow regime ...... 281
9.18 tunnel construction in an aquifer ...... 282
9.19 dyke construction and core seal ...... 282
9.20 aquifer with river and well ...... 283
9.21 influence of river and hanging inflow on the aquifer ...... 284
10.1 aquifer with well and foundation pit ...... 287
XVIII 10.2 stratified aquifer with steady flow regime ...... 287
10.3 stratified aquifer with unsteady flow regime ...... 288
10.4 tunnel construction in an aquifer ...... 288
10.5 dyke construction with core seal ...... 289
11.1 coupling of migration- and simulator model ...... 295
11.2 model classification by Krug ...... 296
11.3 devices and technical programming realization ...... 298
11.4 process character of modelling and simulation ...... 299
11.5 basic signal forms ...... 304
11.6 test signals ...... 308
11.7 signal syntheses ...... 309
11.8 approximation arbitrary signals by bar signals ...... 311
11.9 representation forms of signals ...... 312
11.10 system with its connection to environment ...... 314
11.11 transfer element ...... 315
11.12 relationship between time and complex variable domain ...... 316
11.13 Filling procedure with constant flow rate ...... 321
11.14 Filling procedure with variable flow rate ...... 326
11.15 coupled storage cascade ...... 328
11.16 basic forms of transient characteristic ...... 333
11.17 interconnection of linear transfer elements ...... 335
11.18 two series connected PT1 elements ...... 337
XIX 11.19 realisation of a PI-element ...... 338
11.20 realisation of a PID-element ...... 340
11.21 feedback system for water level control ...... 341
12.1 equivalent circuit diagram of a transfer element with 1st order delay . . 344
12.2 transient characteristic and circuit of a PT1 element ...... 346
12.3 time constant determination from its multiple ...... 347
12.4 tangent intersection and time constant ...... 348
12.5 equivalent circuit diagram of a transfer element with 2nd order delay . . 349
12.6 Step response function and equivalent circuit diagram
of a PT2TL element ...... 352
12.7 Kenngr¨oßender Sprungantwort ...... 353
12.8 determination of parameters T1 and T2 ...... 354
nd 12.9 2 order transfer element (PT2TL) ...... 356
12.10 impulse response function g(t) for a 2nd order delay element ...... 357
12.11 impulse response of a column flow test ...... 360
12.12 groundwater level dependent on radius ...... 361
12.13 approximation of a function by impulse ...... 365
12.14 impulse response function g(t) for a 2nd order delay element ...... 365
12.15 overlay of individual step response function ...... 367
12.16 formation of discontinuous response signals from measured values . . . 372
12.17 groundwater level dependent on radius ...... 375
XX 13.1 iterative model adjustment ...... 383
14.1 iterative model adjustment in a pumping test ...... 387
14.2 programme system for pumping test evaluation according
to Beims/Graber¨ ...... 389
14.3 quality mountain in a pumping test evaluation with search procedure of
different start points ...... 390
14.4 Iteration performance in the pumping test evaluation ...... 392
14.5 structure scheme of pumping test simulator (Beims/Mucha/Graber¨ ) 394
XXI Abbreviations and symbols
Symbol Unit English German
A m2 Area Fl¨ache
geohydraulic geohydraulsiche a s · m−2 time constant Zeitkonstante
Lage des a m position of the aquitarde Grundwasserstauers
B m leakage faktor Speisungsfaktor
b m width Breite
C g · l−1; % concentration Konzentration
DGL differential equation Differentialgleichung
D m flow trough thickness durchstr¨omte M¨achtigkeit
d m distance Schichtdicke, Abstand
Elektrodenpotential EV voltage source Spannungsquelle
XXII Symbol Unit English German
Elektron
e− C electron elektr. Elementarladung
e− =−1, 60210 · 10−19C
Faradaysche Konstante F C Faraday constant F = 96491, 6C
F Fourier transform Fourier-Transformierte
GS electrical conductance elektrischer Leitwert
Erdbeschleunigung
−2 g m · s gravitational acceleration g ≈9, 832m · s−2 (Pol)
g ≈ 9, 780m · s−2 (Aquator)¨
g weight function Gewichtsfunktion
GF quality function G¨utefunktion
GW groundwater Grundwasser
groundwater observation Grundwasser- GWBR well beobachtungsrohr
GWL aquifer Grundwasserleiter, Aquifer
GWU¨ gruoundwater monitoring Grundwasser¨uberwachung
GWPN groundwater sample Grundwasserprobenahme
XXIII Symbol Unit English German
GWR water level Grundwasserressource
H m water level, general Wasserstand, allgemein
H m2 equivalent potential Ersatzpotential
Grundwasserstand h m groundwater level ¨uberstauter Wasserstand
h m, s increment Schrittweite
h transit function Ubergangsfunktion¨
IA electric current elektrischer Strom
kf -Wert k m · s−1 intrinsic permeability Durchl¨assigkeitsbeiwert
L − Laplace transform Laplace-Transformierte
LGS linear equation system Lineares Gleichungssystem
l m dinstance L¨ange
elektrische Leitf¨ahigkeit LF S · cm−1 electrical conductivity von Fl¨ussigkeiten
M − frequency H¨aufigkeit
XXIV Symbol Unit English German
ordinary differential Gew¨ohnliche ODE equation Differentialgleichung
partial differential Partielle PDE equation Differentialgleichung
p N · m−2 pressure Druck
QC electric charge elektrische Ladung
Q m3 · s−1 volume flow Volumenstrom
R Ω electric resistance elektrischer Widerstand
R plant Regelstrecke
Re − Reynolds number Reynolds-Zahl
representative Repr¨asentatives REV unit volume Einheitsvolumen
S − Speicherkoeffizient
S plant Steuerstrecke
s m distance Weg
Temperatur TK temperature T [oC] = T [K]−273, 15K
XXV Symbol Unit English German
T m2 · s−1 transmissibility Transmissibilit¨at
t s time Zeit
UV electric potential elektrische Spannung
V m3 volume Volumen
V˙ m3 · s−1 volume flow Volumenstrom
v m · s−1 velocity Geschwindigkeit
Theisschen Brunnenformel W (σ) Theis well formula Potenzreihe
w − command variable F¨uhrungsgr¨oße
x m spatial coordinate Ortskoordinate
x − general signal allgemeine Signalgr¨oße
x − controlled variable Regelgr¨oße,Steuergr¨oße
y m spatial coordinate Ortskoordinate
y − input value Stellgr¨oße
Potentialdifferenz Z m2 potential difference (Grundwasser)
XXVI Symbol Unit English German
z m spatial coordinate, heigth Ortskoordinate, H¨ohe
z − pertubation St¨orgr¨oße
α Grad degree (angle) Winkelgrad
Euler-Konstante γ − Euler constant γ ≈ 0, 5772156649
Temperatur δ oC temperature δ[oC] = T [K] − 273, 15K
δ − Dirac impulse Dirac-Impuls
ε F · m−1 permittivity Dielektrizit¨atskonstante
Dielektrizit¨atskonstante
−1 ε0 F · m vacuum permittivity im Vakuum
−12 −1 ε0 =8, 855 · 10 F · m
ε − general error allgemeine Fehlergr¨oße
λ m wavelength Wellenl¨ange
effective boundary wirksame Randbeding- λ∗ m condition distance ungsentfernung
µ H · m−1 permeability Permeabilit¨at
XXVII Symbol Unit English German
Permeabilit¨atdes
−1 µ0 H · m permeability of vakuum Vakuums
−6 −1 µ0 =1, 2566 · 10 H · m
ρ g · m−3 density Dichte
specific electrical spezifischer elektrischer ρ Ω · m−1 resistance Widerstand
argument of the Theis Argument der Theisschen σ − well formula Brunnenformel W (σ)
τ s delay Verz¨ogerungszeit
Girinskij-Potential Φ m2 Girinskij potential (Grundwasser)
ϕ V electrostatic potential elektrisches Potential
spezifische elektrische κ S · m2 · m−1 electrical conductance Leitf¨ahigkeit
κ S · cm−1 electrical conductivity elektrische Leitf¨ahigkeit
1 uni step Einheitssprung
XXVIII Part I
Mathematical fundamentals
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management
The part mathematical fundamentals is directly based on the elementary learn- ing of mathematics and actually shows some solutions of selected problems, which are important in the water management subjects. After this review special, advanced topics will be discussed.
3 4 Chapter 1
Algebra fundamentals
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 1.1 Exponential and logarithmic expressions
The most important transformations of exponential expressions are:
xa · xb = xa+b
xa a−b xb = x
(xa)b = xa·b (1.1) √ b a xa = x b
x0 = 1
If there are exponential expressions with more than on basis, the identity
x = aloga(x) (1.2) helps to transform the expressions to the same basis. Specially:
x = eln(x) (1.3)
The most important transformations of logarithmic expressions are:
log(a · b) = log(a) + log(b)
a log( b ) = log(a) − log(b)
log(xb) = b · log(x)
√ 1 1 b b log( x) = log(x ) = b log(x) (1.4) xloga y = yloga x
loga(x) = logb(a) · logb(x)
logx(1) = 0
logx(x) = 1
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 1.1. Exponential and logarithmic expressions
Specially:
log10(x) = lg(x) with lg(10) = 1
loge(x) = ln(x) with ln(e) = 1
ln(eb) = b · ln(e) = b
lg(x) = ln(10) · ln(x) (1.5)
ln(x) = lg(e) · lg(x)
ln(10) = 2, 302585093 ...
lg(e) = 0, 434294482 ...
7 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.1.1 Exercises
Exercises to 1.1:
1. Simplify the following expressions:
(18a2x)4 (15ax2)4 0, 004 · 102 · 0, 23 (a) · (b) (27ax2)5 (20a3x)2 0, 2 · 10−3 · 16 √ √ √ x · 3 x2 · 10 x (c) √ (d) 2 log x3 − 3 log y2 5 x3 10 10
(e) (ln u + 4 ln v)
2. Transform the following formulas according to t:
t t (a) I = I0 exp − T (b) I = I0 1 + 9 log9 T
µt (c) I = I0 (e − 1)
3. Given the linear regression s, compute the variable T in semilogarithmic scale from the gradient of the linear regression.
V˙ 2, 25 · t · T s = · ln (V˙ is given) 4 · π · T r2 · S
4. Given the linear regression s (exercise 3). Compute the specific capacity S in the
case s = 0 for t0 and a given r.
8 1.2 Matrices
1.2.1 Fundamentals
In the following the most important calculation rules for matrices are to be specified.
• General matrix A system of m times n elements (e.g. numbers, functions), which is arranged in m rows and n columns, is called matrix of type (m; n): a a ··· a ··· a 11 12 1j 1n a a ··· a ··· a 21 22 2j 2n ...... ...... A = (aij) = (1.6) a a ··· a ··· a ←− i-th row i1 i2 ij in . . . . ...... . . ··· . . am1 am2 ··· amj ··· amn
↑
j-th column
• Square matrix If the number of rows m and columns n of a matrix are same, i.e. m = n, the
matrix is square. The elements a11, a22, a33 . . . ann form the main diagonal and a1n, a2n−1, a3n−2 . . . an1 is the secondary diagonal. • Transpose of a matrix A resultant matrix AT will be obtained by interchanging the rows and columns of matrix a A, which is called transpose of A. In general it can be written as:
A = (ajk) (1.7)
T T A = (ajk) = (akj)
specially: (ajj) = (akk)
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 1. ALGEBRA FUNDAMENTALS
The transpose of a square matrix is its reflection through the main diagonal. The transpose of a symmetric matrix is the same like the original matrix.
T As = As (1.8) In a skew symmetric or antimetric matrix:
T Aa = −Aa (1.9)
(ajk) = − (akj)
(ajj) = 0
Each square matrix (Aq) can be decomposed in form of the sum of a symmetric (As) and an antimetric (Aa) matrix.
Aq = As + Aa (1.10) 1 A = · A +AT s 2 q q 1 A = · A −AT a 2 q q Examples of transpose matrix calculation: 1. 2 −3 2 1 4 T A = ⇒ A = 1 0 −3 0 2 4 2 1 4 5 1 7 8 T A = 7 2 6 ⇒ A = 4 2 9 8 9 3 5 6 3
2. 1 4 5 T Asym = 4 2 6 = Asym 5 6 3
10 1.2. Matrices
3. 3 −1 4 3 5 −4 T A = 5 7 8 ⇒ A = −1 7 0 −4 0 5 4 8 5
A = As + Aa 3 −1 4 3 5 −4 3 2 0 1 As = · 5 7 8 + −1 7 0 = 2 7 4 2 −4 0 5 4 8 5 0 4 5
1 A = · (A −AT ) a 2 q q 3 −1 4 3 5 −4 0 −3 4 1 As = · 5 7 8 − −1 7 0 = 3 0 4 2 −4 0 5 4 8 5 −4 −4 0
• Unit matrix A square matrix, in which all elements of the main diagonal are equal to one and all other elements are zero, is called unit matrix and designated as E. 1 0 ··· 0 0 0 1 ··· 0 0 . . . . . E = ...... (1.11) 0 0 ··· 1 0 0 0 ··· 0 1
A · E = E · A = A (1.12)
En = E
11 CHAPTER 1. ALGEBRA FUNDAMENTALS
• Diagonal matrix A diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero.
Thus, a matrix A = (aij) with n columns and n rows is diagonal if and only if aij = 0 for all i 6= j. a 0 ··· 0 11 . . 0 a .. . 22 . . . . .. .. 0 0 ··· 0 ann Any diagonal matrix is also a symmetric matrix. • Tridiagonal matrix A tridiagonal matrix is a square matrix in which the entries outside the main diagonal and outside the first diagonal below and outside the first diagonal above the main diagonal are zero. The entries in the three middle diagonals may or may not be zero.
This means: aij = 0 for |i − j| > 1. a a 0 ··· 0 11 12 . . a a a .. . 21 22 23 . . 0 a .. .. 0 32 . . . . ...... a n−1 n 0 ··· 0 an n−1 ann
• Band matrix A band matrix contains a large number of zero elements. A band matrix is a square matrix in which all entries outside the main diagonal and some selected parallel diagonals are zero. The entries in this selected diagonals may or may not be zero. Is p the number of the lower diagonals with nonzero entries and q the number of
the upper diagonals with nonzero entries bij of a band matrix B, so the number l = p + q + 1 is called the band width of B.
Thus: bij = 0 for j + p < i and i + q < j.
12 1.2. Matrices
Examples for band matrices: 1. The unit matrix is a band matrix with band width l = 0 + 0 + 1 = 1. 2. A tridiagonal matrix is a band matrix with band width l = 1 + 1 + 1 = 3 3. a a 0 ··· a 0 0 11 12 1 k a a a ··· 0 a 0 21 22 23 2 (k+1) 0 a a a ··· 0 a 32 33 34 3 (k+2) ...... A = ...... a 0 0 a a a 0 k 1 i (j−1) ij i j+1 ...... 0 a(k+1) 2 0 0 a(m−1) n 0 0 a(k+2) 3 0 ··· am (n−1) amn
This matrix has the band width (k − 1) + (k − 1) + 1 = k + k − 1. 4. a ··· a 0 ··· 0 11 1 k . . . 0 .. .. . . . . . ...... 0 A = . . . . .. .. a (n−k) n . . . . ...... . . 0 ········· 0 amn
This matrix has the band width (k − 1) + 1 = k.
13 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.2.2 Calculation rules
• Addition and subtraction of matrices
If A = (ajk) and B = (bjk) have the same order, e.g. the same number of rows and the same number of columns, then:
A ± B = (ajk) ± (bjk) (1.13)
Example of matrix addition: 4 6 3 −15 A = B = 6 9 −2 10
4 + 3 6 − 15 7 −9 A + B = = 6 − 2 9 + 10 4 19
• Matrix multiplication and division
If A = (ajk) is a m × p matrix and B = (bjk) is a p × n matrix, we define the product of A·B as a matrix C = (cjk) with order m×n. It means that matrix C has the same number of rows as the matrix A and the same number of columns as the matrix B. The product A · B is only defined, if the number of columns of A (p of matrix m × p) is the same as the number of rows of B (p of matrix p × n). We speak about multiplication as rows time columns. The rows of A will be multiplied by the columns of B. According to Falk’s scheme: P C = (cjk) = (ajl) · (blk) with j = 1 ··· m for k = 1 ··· n The multiplication B · A forms a matrix C with order of p × p What should be noticed is that the commutative law is not applicable (A · B 6= B · A). In case of square matrix A, B and C have the same order m×m. The division is reverse procedure of multiplication, as the inverse matrix is built.
14 1.2. Matrices
Examples of matrix multiplication: 2 1 1 2 4 Given: A = and B = 0 3 −1 0 3 −1 2
Find: A · B and B · A
1 · 2 + 2 · 0 + (4 · (−1)) 1 · 1 + 2 · 3 + 4 · 2 A · B = (−1 · 2) + 0 · 0 + (3 · (−1)) (−1 · 1) + 0 · 3 + 3 · 2
−2 15 = −5 5
2 · 1 + (1 · (−1)) 2 · 2 + 1 · 0 2 · 4 + 1 · 3 B · A = 0 · 1 + (3 · (−1)) 0 · 2 + 3 · 0 0 · 4 + 3 · 3 ((−1) · 1) + (2 · (−1)) ((−1) · 2) + 2 · 0 ((−1) · 4) + (2 · 3)
1 4 11 = −3 0 9 6= A · B −3 −2 2
15 CHAPTER 1. ALGEBRA FUNDAMENTALS
4 6 3 −15 Given: A = and B = 6 9 −2 10
Find: A · B
4 6 3 −15 A · B = · 6 9 −2 10 (3 · 4 − 2 · 6) (4 · (−15) + 6 · 10) 0 0 = = (3 · 6 − 2 · 9) (6 · (−15) + 9 · 10) 0 0
Attention: In contrast to the algebra of real numbers the commutative law is not appli- cable in the matrix multiplication. The associative law A·(B · C = (A · B) · C and the distributive law A·(B + C) = A · B + A · C on the other hand are available for the matrices A square matrix A can be multiplied by itself. A2 = A · A. Then we get an exponentiation of matrices. • Inverse matrix The inverse of a square matrix A is a matrix A−1, such that:
A · A−1 = A−1 · A = E (1.14)
The inverse matrix A−1 can be exactly built if the determinant D = det A (chapter 1.2.3 determinant of a matrix, page 19ff) of the matrix A is unequal −1 to zero (D 6= 0). The inverse matrix A is formed by the minor Uij for the element aij. With the determinant Uij pertaining to element aji:
U a = (−1)i+j ji (1.15) ij D The change between row and column must be considered. The matrix of minor is transposed. Furthermore a change of sign takes place as a function of the
distance to main diagonals aii of the matrix, i.e., when the sum i and j is odd number, the minor should be multiplied by -1.
16 1.2. Matrices
The construction of the inverse matrix is demonstrated on the basis of a two- and a three-row matrix. For a two-row matrix: a a 11 12 A = a21 a22 a22 a12 − a22 −a12 (1.16) −1 D D 1 A = = D a21 a11 − D D −a21 a11
D = det A = a11 · a22 − a21 · a12
For a three-row matrix: a a a 11 12 13 A = a a a 21 22 23 a31 a32 a33 U11 − U21 U31 U −U U D D D 11 21 31 −1 1 A = − U12 U22 − U32 = −U U −U D D D D 12 22 32 U13 U23 U33 D − D D U13 −U23 U33
D = a11a22a33 + a12a23a31 + a13a21a32 − a31a22a13 − a32a23a11 − a33a21a12
(1.17)
Examples of inverse matrix calculation:
5 4 1. If A = , then D = 5 · 2 − 2 · 4 = 2 and 2 2
2 −4 1 −2 −1 1 A = = 2 −2 5 −1 2, 5
17 CHAPTER 1. ALGEBRA FUNDAMENTALS
2. Find the inverse matrix of 2 1 −1 A = 5 2 0 1 1 −2
The determinant A can be developed for example according to the second row:
1 −1 2 −1 2 1
D = −5 + 2 − 0
1 −2 1 −2 1 1
= −5(−2 + 1) + 2(−4 + 1) = −1
The minors are:
2 0 5 0 5 2
U11 = = −4 U12 = = −10 U13 = = 3
1 −2 1 −2 1 1
1 −1 2 −1 2 1
U21 = = −1 U22 = = −3 U23 = = 1
1 −2 1 −2 1 1
1 −1 −2 1 2 1
U31 = = 2 U32 = = 5 U33 = = −1
2 0 5 0 5 2
Then: 4 −1 −2 −1 A = −10 3 5 −3 1 1
18 1.2. Matrices
1.2.3 determinant of a matrix
A number can be assigned to a square matrix A, whose value is the determinant D = det A. The n-th order determinant development can be defined recursively with help of the Laplace expansion theorem.
a a ··· a 11 12 1n
a a ··· a 21 22 2n D = det A = (1.18) ......
an1 an2 ··· ann The development can take place according to the elements of rows or columns.
1. Development according to the elements of i-th row n X D = det A = aiνAiν i fixed (1.19) ν=1 2. Development according to the elements of k-th column n X D = det A = aµkAµk k fixed (1.20) µ=1
Here Aik means the adjunct belonging to the element aik,e.g. the minor Uik of the i+k element aik multiplied by the factor (−1) . We get the minor Uik of the element aik from the n-th order determinant by deleting the i-th row and k-th column; it has order n-1. Thus the rank of the minor is always one order lower than the associated determinant. The rank of a matrix is determined by the highest order.
For a three-row matrix A: a a a 11 12 13 A = a a a (1.21) 21 22 23 a31 a32 a33 then for example the minor for element a11:
a a 22 23 U11 = = a22a33 − a23a32 (1.22)
a a 32 33
19 CHAPTER 1. ALGEBRA FUNDAMENTALS
For three-row determinant the Sarrus’ rule can be also applied. The first two columns are written fictitiously right beside the determinant and afterwards the sum of the diagonal products of the ”minor diagonal” are subtracted from that of the ”main diagonals”.
a a a a a 11 12 13 11 12
A = det A = a a a a a 21 22 23 21 22
a a a a a 31 32 33 31 32
= a11a22a33 + a12a23a31 + a13a21a32 − a13a22a31 − a11a23a32 − a12a21a33
Tips: For the development of the determinants it is always favourable to select the row or column with most zero elements. If the matrix consists of only one elementA = (a), then det A = a.
20 1.2. Matrices
1.2.4 Exercises
Exercises to 1.2: 2 1 −1 1 1 4 1. Given: A = , B = , C = 4 3 2 −4 −2 −1
Find:
(a) 2A + 3B (b) A − 2B − 3C (c) A · B
(d) (A · B)T (e) B · A (f) (A · B) · C
(g) (B · A) · C (h) BT AT (i) AT + BT
2. Find the matrix C = A · BT (a) A = 2 3 1 1 B = 2 −1 1 3
1 0 1 3 (b) A = B = 2 −3 −2 1 3 −8
3. Find the inverse matrix A−1: 1 1 −1 3 5 (a) A = (b) A = 1 −1 1 2 7 −1 1 1
2 2 3 (c) A = −4 −2 3 4 3 2
21 1.3 Linear Equation Systems
Linear equation systems play an important role in the mathematical treatment of one-, two-, or three dimensional field problems as well as in hydrology and hydrogeology. Such equation systems with a large number of unknown quantities and equations, whose magnitudes of order may reach million, originate from quantization methods. The continuous field problems are decomposed in generally potential fields and discon- tinuous partial processes. And these can be described by linear, non-linear or linearised equations. The equation systems have the following figure:
a11x1 +a12x2 +a13x3 + ··· a1jxj ··· + a1nxn = r1
a21x1 +a22x2 +a23x3 + ··· a2jxj ··· + a2nxn = r2 ...... (1.23)
ai1x1 +ai2x2 +ai3x3 + ··· aijxj ··· + ainxn = ri ......
am1x1 +am2x2 +am3x3 + ··· amjxj ··· + amnxn = rm
In this equation system there are xj, with j = 1 . . . n, n unknown quantities and ri, with i = 1 . . . m, m known quantities at the right side. And aij with i = 1 . . . m and j = 1 . . . n are characterized as coefficients. So we have n unknown quantities and m equations.
If n = m, the equation system is definitely solvable, and it is determined. For n < m it is so called over determined equation system , and mostly there are approximate solutions which fulfil all equations. In the case n > m several solutions exist, i.e. the equation system is not clearly solvable, and it is undetermined. By the introduction of the matrix notation we can write them for short and the rules of the matrix calculus can also be used at the same time.
Thus the above equation systems can be noted in the following form:
A · X = R (1.24)
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 1.3. Linear Equation Systems
A is the coefficient matrix, X the solution vector and R on the right side a column vector as follows: a a ··· a x r 11 12 1n 1 1 a a ··· a x r 21 22 2n 2 2 A = X = R = . . . . . . . . .. . . . am1 am2 ··· amn xn rn
Different methods are used for the solution of such matrix equations or equation sys- tems. The relatively simple direct solution methods are mostly not treatable, since the construction of the inverse coefficient matrix proves itself complicated with higher rank. We differentiate the solution methods between the direct total step methods and iterative equation solutions. The methods are just as their names imply. By the to- tal step methods the equation system is separated based on algebraic transformations until the equation with only one unknown quantity remaining. In the following some representative examples of both methods are indicated.
23 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.3.1 Total step method
1.3.1.1 Gauss elimination
In the Gauss elimination we try to get an equation with one unknown by successive substitution.
The result of the elimination is the equation system:
A · X = R
Step by step we transform the equation system in to an equation system
A0 · X = R0 with an upper triangle matrix A0 and R0 a transformed right side.
a a a ··· a ··· a r 11 12 13 1j 1n 1 0 a0 a0 ··· a0 ··· a0 r0 22 23 2j 2n 2 0 0 a0 a0 a0 ··· a0 r0 33 34 3j 3n 3 0 . . . . . 0 . A = 0 0 ...... R = . (1.25) 0 0 0 0 a0 a0 a0 r0 ij ij+1 in j . . 0 0 0 0 .. .. a0 r0 m−1n n−1 0 0 0 0 0 0 0 0 amn rn
The supreme row, or the supreme equation remains unchanged. The (j − 1) equations are multiplied by factor fakj and subtracted from the j-th equation. For the second row or the second equation: ai2 fak2 = (1.26) ai1 or in general 0 aij fakj = 0 (1.27) aij−1 The lowest row or equation can be solved. This solution is inserted backwards into all
24 1.3. Linear Equation Systems
other equations. This back substitution generally yields the value xi:
0 rn xn = 0 (1.28) amn n ! 1 X x = r0 − a0 x i a0 i ij j ii j=i+1
Thus the computation of vector X can be achieved.
Example for using the Gauss elimination: Find the solution of the following system:
2x − 3y + 4z = 19
4x − 4y + 3z = 22
−6x − y + 5z = 7
The coefficients and absolute terms are shown in the following scheme:
x y z 1
2 −3 4 19
4 −4 3 22
−6 −1 5 7
The row which is used for the elimination (in this example the first row) is marked by the letter E. A variable should be eliminated with the help of E i.e. how many multiples of E should be added on other rows. The added multiple of E can be marked beside the corresponding rows:
x y z 1
E 2 −3 4 19
−2 4 − 4 −4 + 6 3 − 8 22 − 38
3 −6 + 6 −1 − 9 5 + 12 7 + 57
25 CHAPTER 1. ALGEBRA FUNDAMENTALS
In such way the 2nd and 3rd row are similarly changed:
x y z 1
0 2 −5 −16
0 −10 17 64
x y z 1
E 0 2 −5 −16
5 0 −10 + 10 17 − 25 64 − 80
The E-row and the last row contain the coefficients and absolute terms of the new system: 2 −3 4 x 19 0 2 −5 · y = −16 0 0 −8 z −16
2x − 3y + 4z = 19
2y − 5z = −16
8z = 16
The results of successive insertion
z = 2, y = −3, x = 1
In general each result should be checked, if it solves the equation system.
2x − 3y + 4z = 19
4x − 4y + 3z = 22
−6x − y + 5z = 7
26 1.3. Linear Equation Systems
2 · 1 − 3 · (−3) + 4 · 2 =? 19
19 = 19
4 · 1 − 4 · (−3) + 3 · 2 =? 22
22 = 22
− 6 · 1 − (−3) + 5 · 2 =? 7
7 = 7
Thus the result solves the equation system.
27 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.3.1.2 Cramer’s rule
According to the Cramer’s rule the solution of matrix equation is achieved by the determinant computations. The elements of the solution vector X:
D x = i (1.29) i D
D = det A is the determinant of the matrix A. Di is the Cramer’s determinant. It is developed from D i consequence of replacing the i-th column by the right side, the vector R. This method only has practical meaning for small matrices which contain many zeros. Example for using the Cramer’s rule:
Find the solution of the linear equation system with the Cramer’s rule:
2x + y − z = 0
5x + 2y = 8
x + y − 2z = −5
The linear equation system is written in matrix form as follows: 2 1 −1 x 0 5 2 0 · y = 8 1 1 −2 z −5 or A · X = R with 2 1 −1 A = 5 2 0 1 1 −2
The determinant D = det A of the matrix A can be developed according to second
28 1.3. Linear Equation Systems row:
2 1 −1
D = 5 2 0
1 1 −2
1 −1 2 −1 2 1
= −5 + 2 − 0
1 −2 1 −2 1 1
= −5(−2 + 1) + 2(−4 + 1)
D = −1
The Cramer’s determinants are:
0 1 −1
8 0 8 2
Dx = 8 2 0 = −1 − 1 = −2
−5 −2 −5 1
−5 1 −2
2 0 −1
8 0 5 8
Dy = 5 8 0 = 2 − 1 = 1
−5 −2 1 −5
1 −5 −2
2 1 0
2 8 5 8
Dz = 5 2 8 = 2 − 1 = −3
1 −5 1 −5
1 1 −5
Thus the solution is:
D D D x = x y = y z = z D D D x = 2 y = −1 z = 3
29 CHAPTER 1. ALGEBRA FUNDAMENTALS
Verifying the solution
2x + y − z = 0
5x + 2y = 8
x + y − 2z = −5
2 · 2 + (−1) − 3 =? 0
0 = 0
5 · 2 + 2 · (−1) =? 8
8 = 8
2 + (−1) − 2 · 3 =? −5
−5 = −5
Thus the solution is right.
30 1.3. Linear Equation Systems
1.3.1.3 Construction of the inverse matrix
The solution vector X of a linear equation system is noted:
A · X = R
A−1 · A · X = A−1 · R
X = A−1 · R (1.30)
with A−1 · A = E
Example for construction of the inverse matrix A−1: Find solution of the same linear equation system (see former example page 28) with help of inverse matrix.
2x + y − z = 0
5x + 2y = 8
x + y − 2z = −5
Written in matrix form: 2 1 −1 x 0 5 2 0 · y = 8 1 1 −2 z −5 or A · X = R with 2 1 −1 A = 5 2 0 1 1 −2
31 CHAPTER 1. ALGEBRA FUNDAMENTALS
The determinant D = det A of the matrix A can be developed with the second row:
2 1 −1
D = 5 2 0
1 1 −2
1 −1 2 −1 2 1
= −5 + 2 − 0
1 −2 1 −2 1 1
= −5(−2 + 1) + 2(−4 + 1)
D = −1
The inverse matrix of A is (see example - construction of inverse matrix, page 17): 4 −1 −2 −1 A = −10 3 5 −3 1 1
Then: X = (A−1 · R)
4 −1 −2 0 2 X = −10 3 5 · 8 = −1 −3 1 1 −5 3
x = 2, y = −1, z = 3
This result is the same like for the Cramer’s rule, so we skip the verifying of the result.
32 1.3. Linear Equation Systems
1.3.1.4 LU decomposition
The so called LU decomposition assumes that a regular, invertible square matrix A of an equation system A · X = R can be written as the product of the matrices L and U: L · U = A (1.31) Here L is a lower triangle matrix and U an upper triangle matrix. l 0 ··· 0 u u ··· u a a ··· a 11 11 12 1n 11 12 1n l l ··· 0 0 u ··· u a a ··· a 21 22 22 2n 21 22 2n · = . . . . . . . . . . . . .. . 0 0 .. . . . .. . lm1 lm2 ··· lmn 0 0 ··· umn am1 am2 ··· amn Under the permission that the matrix A is square, e.g. m = n the equation transforms to: A · X = (L · U) · X = L · (U · X) = L · Y = R (1.32) So the original equation can be decomposed into two equations, which contain the simple solvable triangle matrices. First the vector Y will be calculated. This solution vector Y helps to calculate the solution vector X. Y is the right side for the equation for calculating X. L · Y = R (1.33) U · X = Y For the calculating of Y the forward substitution is used:
r1 y1 = (1.34) l11 i−1 ! 1 X y = y − l y with i = 2, 3, .....n i l i ij j ii j=1 For the calculating of X thr backward substitution is used:
yn xn = (1.35) umn n ! 1 X x = y − u x with i = n − 1, n − 2,...., 1 i u i ij j ii j=i+1
33 CHAPTER 1. ALGEBRA FUNDAMENTALS
Out of the definition of the equation system we get:
L · U = A or (1.36)
l 0 ··· 0 u u ··· u a a ··· a 11 11 12 1n 11 12 1n l l ··· 0 0 u ··· u a a ··· a 21 22 22 2n 21 22 2n · = (1.37) . . . . . . . . . . . . .. . 0 0 .. . . . .. . lm1 lm2 ··· lmn 0 0 ··· umn am1 am2 ··· amn
If these two matrices are multiplied and an element comparison is made, a complicated equation system with m · n unknown quantities appears. The difficulty can be avoided if the main diagonal elements lii of the L matrix are set to one.
(lii) = 1 (1.38)
Then we get the following simple calculation scheme for the elements: 1 0 ··· 0 u11 u12 ··· u1n a11 a12 ··· a1n l21 1 ··· 0 0 u22 ··· u2n a21 a22 ··· a2n · = . . .. . .. . . . .. . . . . . 0 0 . . . . . . lm1 lm2 ··· 1 0 0 ··· umn am1 am2 ··· amn (1.39) 1 · u11 1 · u12 ··· 1 · u1n a11 a12 ··· a1n l21 · u11 l21 · u12 + 1 · u22 ··· l21 · u1n + 1 · u2n a21 a22 ··· a2n = . . .. . . . .. . . . . . . . . . lm1 · u11 lm1 · u12 + lm2 · u22 ··· lm1 · u1n + ··· am1 am2 ··· amn (1.40)
34 1.3. Linear Equation Systems
Example of solving an equation system with the LU decomposition:
−3x +2x −3x = 6 −3 2 −3 1 2 3 9x −2x +10x = −10 =⇒ A = 9 −2 10 1 2 3 6x1 +8x2 +14x3 = 22 6 8 14
with: lii = 1
1 0 0 u u u 11 12 13 L = l 1 0 U = 0 u u 21 22 23 l31 l32 1 0 0 u33
L · U = A
We know from the element comparison:
1 · u11+ 0 · 0+ 0 · 0 = −3 =⇒ u11 = −3
l21 · u11+ 1 · 0+ 0 · 0 = 9 =⇒ l21 = −3
l31 · u11+ l32 · 0+ 1 · 0 = 6 =⇒ l31 = −2
1 · u12+ 0 · u22+ 0 · 0 = 2 =⇒ u12 = 2
l21 · u12+ 1 · u22+ 0 · 0 = −2 =⇒ u22 = 4
l31 · u12+ l32 · u22+ 1 · 0 = 8 =⇒ l32 = 3
By the same way we find:
1 · u13+ 0 · u23+ 0 · u33 = −3 =⇒ u13 = −3
l21 · u13+ 1 · u23+ 0 · u33 = 10 =⇒ u23 = 1
l31 · u13+ l32 · u23+ 1 · u33 = 14 =⇒ u33 = 5 | {z } 6
35 CHAPTER 1. ALGEBRA FUNDAMENTALS
Then the triangle matrices have the following structure: 1 0 0 −3 2 −3 L = −3 1 0 U = 0 4 1 −2 3 1 0 0 5
Then the solution vector X can be computed with the help of the vector Y. In general:
A · X = R
A = L · U y 1 L · U · X = R with Y = y | {z } 2 Y y3
L · Y = R
1 · y1+ 0 · y2+ 0 · y3 = 6
−3 · y1+ 1 · y2+ 0 · y3 = −10
−2 · y1+ 3 · y2+ 1 · y3 = 22 y = 6 6 1 y = 8 Y = 8 2 y3 = 10 10
U · X = Y
−3 · x1+ 2 · x2− 3 · x3 = 6
0 · x1+ 4 · x2+ 1 · x3 = 8
0 · x1+ 0 · x2+ 5 · x3 = 10
36 1.3. Linear Equation Systems
x = −3 −3 1 x = 3 X = 3 2 2 2 x3 = 2 2 Try out:
−3x1+ 2x2− 3x3 = 6
9x1− 2x2+ 10x3 = −10
6x1+ 8x2+ 14x3 = 22
3 ? −3 · (−3)+ 2 · 2 − 3 · 2 = 6 ⇒ 6 = 6
3 ? 9 · (−3)− 2 · 2 + 10 · 2 = −10 ⇒ −10 = −10
3 ? 6 · (−3)+ 8 · 2 + 14 · 2 = 22 ⇒ 22 = 22
37 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.3.1.5 Cholesky decomposition
With the Cholesky decomposition the solution of the matrix equation in the special case of a symmetric coefficient matrix can be traced back by the solution of two sub- systems, as the coefficient matrix is decomposed into an upper and a lower triangle matrix.
The Cholesky decomposition is not generally applicable, it presupposes that the coefficient matrix A of an equation system
A · X = R is symmetric , i.e. A = AT , and positive definite.
A matrix is symmetric, if: A = AT . This is directly clear.
If a symmetric matrix is positive definite has to be computed. For a symmetric (n × n) matrix A the following propositions are equivalent:
• A is positive definite
• All eigenvalues λi of A are greater than zero (λi > 0 for i = 1, . . . , n) • For all vectors ~x ∈ Rn is true: xT · A · x > 0 • All leading principal minors of A are strictly positive
An example for the calculation of the leading principal minors will be shown now:
A minor of a matrix A figures out by deletion of columns and rows of the matrix A. The principal leading minors are the minors which figures out by the deletion of the corresponding k last columns and rows. Therefor a (n × n) matrix A has n principal leading minors/principal minors.
Example: Given the symmetric (3 × 3) matrix 2 −1 −3 A = −1 2 4 −3 4 9
Obviously A is symmetric.
38 1.3. Linear Equation Systems
The principal leading minor A0 is the minor which figures out by deletion of the last 0 columns and rows, so A0 equals det(A).
2 −1 −3
A0 = det A = −1 2 4 = 2 · (18 − 16) − (−1) · (−9 + 12) + (−3) · (−4 + 6) = 1
−3 4 9
The principal leading minor A1 is the minor which figures out by deletion of the last column and row.
2 −1
A1 = = 4 − 2 = 2
−1 2
The principal leading minor A2 is the minor which figures out by deletion of the last 2 columns and rows, so A2 equals a11.
A2 = a11 = 2
So the given matrix A is positive definite.
In the most cases the matrices for the quantization methods in simulations of ground- water flow are symmetric and positive definite (FDM,FEM or FVM). In the Cholesky decomposition the symmetric matrix A of an equation system is written as the product of two matrices, a lower triangle matrix B and an upper triangle matrix BT , which is equal to the transpose of the lower,
B · BT = A (1.41)
Here B is an upper triangle matrix, whose elements bik = 0 for i > k. So the equation system is:
A · X = R
B · BT ·X = R
We write: BT ·X = Y (1.42)
39 CHAPTER 1. ALGEBRA FUNDAMENTALS and determine the elements of B by element comparison
B · BT = A so Y can be computed: B · Y = R (1.43) The solution X results from the backward calculation according to equation 1.42. Gen- erally the following algorithm can be indicated for the computation of the elements of B:
k−1 P 1 akj − bljblk f¨ur k + 1 < j < n, j = 2 to n l=1 bkk bkj = (1.44) 0 f¨ur k > j v u j−1 u X 2 bjj = tajj − blj f¨ur j = 1 to n l=1 j−1 ! X 1 yj = rj − bljyl f¨ur j = 1 to n bjj l=1 The Cholesky decomposition has some advantages compared to the Gauss elimi- nation. E.g. the method is characterised by the fact that it works numerically stable, since the dominance of the main diagonals is strengthened by extraction of the square root from very small elements. If the coefficient matrix A possesses a band structure, this is also transferred to the triangle matrix. The algorithm is independent of the val- ues on the right side. Thus the solution of the equation system can be repeated with small expenditure for different values on the right side (boundary and initial values), which makes variant calculations very efficient.
For an equation system with three equations and three variables:
a11x1 + a12x2 + a13x3 = r1
a21x1 + a22x2 + a23x3 = r2
a31x1 + a32x2 + a33x3 = r3
A · X = R
40 1.3. Linear Equation Systems
a a a x r 11 12 13 1 1 a a a · x = r 21 22 23 2 2 a31 a32 a33 x3 r3
For using the Cholesky decomposition we must check whether the conditions sym- metric and positive definite are given for the matrix A. The equation system can be written in the following form: a a a x r 11 12 13 1 1 a a a · x = r 12 22 23 2 2 a13 a23 a33 x3 r3
According to the CHOLESKY decomposition the triangle matrix B and the corre- sponding transpose BT can be written as follows: b b b 11 12 13 T B = 0 b b (1.45) 22 23 0 0 b33 b 0 0 b 0 0 11 11 B = b b 0 = b b 0 12 22 21 22 b13 b23 b33 b31 b32 b33
According to equation 1.41: a a a b 0 0 b b b 11 12 13 11 11 12 13 a a a = b b 0 · 0 b b 12 22 23 12 22 22 23 a13 a23 a33 b13 b23 b33 0 0 b33
The elements of the matrix B are determined by multiplication of B with BT via
41 CHAPTER 1. ALGEBRA FUNDAMENTALS element comparison with the matrix A:
b11 · b11 +0 · 0 +0 · 0 = a11
b11 · b12 +0 · b12 +0 · 0 = a12
b11 · b13 +0 · b23 +0 · b33 = a13
b12 · b11 +b22 · 0 +0 · 0 = a12
b12 · b12 +b22 · b22 +0 · 0 = a22
b12 · b13 +b22 · b23 +0 · b33 = a23
b13 · b11 +b23 · 0 +b33 · 0 = a13
b23 · b12 +b23 · b22 +b33 · 0 = a23
b33 · b13 +b23 · b23 +b33 · b33 = a33
We recognize that some equations are redundant in the developed equation system due to the symmetry of the coefficient matrix. Thus only six of these equations are needed for the determination of the matrix B.
√ a11 = b11 · b11 =⇒ b11 = a11
a12 a12 a12 = b11 · b12 =⇒ b12 = = √ b11 a11 a13 a13 a13 = b11 · b13 =⇒ b13 = = √ b11 a11 r 2 2 2 p 2 a12 a22 = b12 + b22 =⇒ b22 = a22 − b12 = a22 − a11
a12 · a13 a23 − (1.46) a23 − b12 · b13 a a = b · b + b · b =⇒ b = = 11 23 12 13 22 23 23 r 2 b22 a a − 12 22 a 11 b = pa − b2 − b2 33 33 13 23 v u a · a 2 a = b2 + b2 + b2 =⇒ u a − 12 13 33 13 23 33 u a2 23 a = ua − 13 − 11 u 33 2 t a11 a12 a22 − a11
42 1.3. Linear Equation Systems
After determination of the matrices B and BT the matrix Y in equation 1.43 can be computed: B · Y = R b 0 0 y r 11 1 1 b b 0 · y = r 12 22 2 2 b13 b23 b33 y3 r3
r1 b11 · y1 = r1 =⇒ y1 = b11 r2 − b12 · y1 b12 · y1 + b22 · y2 = r2 =⇒ y2 = (1.47) b22 r3 − b13 · y1 − b23 · y2 b13 · y1 + b23 · y2 + b33 · y3 = r3 =⇒ y3 = b33 With this computed matrices B and Y the solution of the equation system X can be determined with help of equation 1.42:
BT ·X = Y b b b x y 11 12 13 1 1 0 b b · x = y 22 23 2 2 0 0 b33 x3 y3
y3 b33 · x3 = y3 =⇒ x3 = b33 y2 − b23 · x3 b22 · x2 + b23 · x3 = y2 =⇒ x2 = (1.48) b22 y1 − b12 · x2 − b13 · x3 b11 · x1 + b12 · x2 + b13 · x3 = y1 =⇒ x1 = b11 Attention: The equations 1.3.1.5 to 1.3.1.5 can be accordingly applied for the deter- mination of the elements of the matrix B, the matrix Y and the solution vector X in all equation systems with three rows and three variables, if they fulfil the conditions for the CHOLESKY decomposition. For each case the elements of the coefficient matrix and those on the right side must be accordingly used. These algorithms can be extended easily to any size of equation system. notice: If the matrix X is symmetric but not positive definite, the matrix B can have complex elements. It is still possible to use the Cholesky decomposition, but it can be harder to calculate the solution Most of the computer algebra systems (i.e. Maple, Mathlab etc.) don’t compute the
43 CHAPTER 1. ALGEBRA FUNDAMENTALS
Cholesky decompositon, if the matrix X isn’t positive definite. Example for the Cholesky decomposition:
1. With the same equation system as in the examples before follows:
2x + y − z = 0
5x + 2y = 8
x + y − 2z = −5
This equation system is not symmetric, so we can not use the Cholesky de- composition.
2. Given the following equation system:
9x1 + 2x2 + 3x3 = 6
2x1 + 8x2 + 4x3 = −10
3x1 + 4x2 + 10x3 = 22
We use the alternative form A · X= R with: 9 2 3 x 6 2 8 4 · y = −10 3 4 10 z 22
We see the matrix A is symmetric: 9 2 3 T A = 2 8 4 = A 3 4 10
44 1.3. Linear Equation Systems
To prove that A is positive definite we have to calculate all leading principal minors of A.
A11 = det(a11) = 9 > 0 a a 9 2 11 12 A22 = det = det = 9 · 8 − 2 · 2 = 68 a21 a22 2 8
A33 = det A = 9 · 8 · 10 + 2 · 4 · 3 + 3 · 2 · 4
−3 · 8 · 3 − 9 · 4 · 4 − 2 · 2 · 10
= 512
Thus the equation system fulfils the conditions for the Cholesky decomposition. According to equation 1.41 follows:
9 2 3 b 0 0 b b b 11 11 12 13 2 8 4 = b b 0 · 0 b b 12 22 22 23 3 4 10 b13 b23 b33 0 0 b33
To determine the elements of the matrix B and BT we use element comparison from the multiplication of B · BT with A. Since only six unknown elements must be determined, only six of these equations are needed:
9 = b11 · b11 =⇒ b11 = 3 2 2 = b · b =⇒ b = 11 12 12 3
3 = b11 · b13 =⇒ b13 = 1 √ r 4 68 8 = b2 + b2 =⇒ b = 8 − = = 2, 7487 12 22 22 9 3 2 4 − 3 · 1 10 4 = b12 · b13 + b22 · b23 =⇒ b23 = √ = √ = 1, 2127 68 68 3 r 100 10 = b2 + b2 + b2 =⇒ b = 10 − 1 − = 2, 744 13 23 33 33 68
45 CHAPTER 1. ALGEBRA FUNDAMENTALS
After the matrix B and its transpose BT were determined, the matrix Y can be computed according to equation 1.43:
B · Y = R 3 0 0 y 6 1 0, 6667 2, 7487 0 · y = −10 2 1 1, 2127 2, 744 y3 22
3 · y1 = 6
0, 6667 · y1 +2, 7484 · y2 = −10
1 · y1 +1, 2117 · y2 +2, 744 · y3 = 22 6 y = = 2 1 3 =⇒ −10 − 0, 6667 · 2 y2 = = −4, 123 2, 7484 22 − 1 · 2 − 1, 2117 · (−3, 638) y = = 9, 111 3 2, 744
With the known matrix B and the matrix Y the solution of the equation system X can be computed now by means of equation 1.42:
BT · X = Y 3 0, 6667 1 x 2 1 0 2, 7487 1, 2127 · x = −4, 123 2 0 0 2, 744 x3 9, 111
2, 744 · x3 = 9, 111
2, 748 · x2 + 1, 2127 · x3 = −4, 123 2 3 · x + · x + 1 · x = 2 1 3 2 3
46 1.3. Linear Equation Systems
8, 896 x = = 3, 3203 3 2, 744 =⇒ −3, 638 − 1, 2127 · 8, 896 x2 = = −2, 9648 2, 748 2 − 1 · 8, 896 − 0, 6667 · −3, 638 x = = 0, 2188 1 3 Thus the solution is:
x1 = 0, 2188; x2 = −2, 9648; x3 = 3, 3203
47 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.3.2 Iterative methods
For the iterative methods first an approximate solution is assumed for the equation system. This solution will be inserted into the system, and by means of optimisation methods the components of the solution vector are best adapted. After n iterations the approximation will approach to the accurate solution with a residue. The iterative methods play a dominating role for large equation systems, since they are usually substantially faster than the directly solving strategies.
Common applications are the CG methods(Conjugate Gradient method) and the multigrid methods. These methods are particularly subject to the further de- velopment in connection with applications of simulation in field problems. For the CG methods the use of additional preconditioning became generally accepted in recent years, with which the search strategy of the iterative optimisation steps is specified. Generally the kind of the preconditioning substantially determines the optimisation speed or the number of optimisation steps. We assume the general matrix notation of a linear equation system as below; X is designated as the unknown solution vector, A · X = R (1.49) So we can insert a known approximated solution X+δX into this equation. This yields an unknown δR on the right side, which deviates from the given value. A · (X + δX) = R + δR (1.50) If we subtract equation 1.49 from equation 1.50, then: A · δX = δR (1.51) or with above equation 1.50: A · δX = A · (X + δX) − R (1.52) The right side of this equation is known, since X + δX is the approximation solution. The goal now is to make the right side equal to zero by finding a new δX. This can be done via solving the matrices equation (see subsection 1.3.1, page 24) or by means of purposeful optimisation, e.g. with the CG method. Generally the CG method can be well used for linear, square, symmetric matrices (m = n). The basic idea is to minimize a function 1 f(x) = X · A · X − b · X 2 The function possesses a minimum if the gradient (see section 2.2) is equal to zero: ∇ f (x) = A · X − R ⇒ 0 (1.53)
This minimum can be found, if we formulate a function f(xk + αkpk) by means of a search direction pk. The index k is the number of continuous search loop.
48 1.3. Linear Equation Systems
1.3.3 Overdetermined equation systems (m > n)
Contrary to the so far described methods, solutions for overdetermined equation system are ton be demonstrated here. In this case there are more equations than variables , i.e. m is greater than n (m > n). This occurs when mathematical models are to be adapted to measured data. A typical case is thereby the application of the quantized (discrete) convolution integral (german: Faltung ;see section 12.3 convolution integral ).
A usual method is to regard this task as optimisation, and to try to adapt the free parameters , i.e. the solution vector X, to the measured values. Thereby most meth- ods differ in the conditioning of the optimisation problem and in the choice of the optimisation strategy.
The SVD method (Singular Value Decomposition) proceeds in the following way. Given the matrix equation:
A · X = R or (aij) · (xi) = (bi) m > n
In this case the coefficient matrix A can be decomposed into
T A = U · (wii) · V (1.54) whereby the matrix U has the same structure as A, wii is a square diagonal matrix with rank n and Vt is a transpose with rank n. This decomposition used in the equation above and computing the solution vector, shows that:
1 X = V · diag · UT · R (1.55) wjj
This equation can be solved with the Householder routine.
49 CHAPTER 1. ALGEBRA FUNDAMENTALS
1.3.4 Exercises
Exercises to 1.3:
Determine the solution vector X of the equation systems in five ways (A · X = R):
(a) with Gauss’ elimination (b) with the Cramer rule (c) with direct computation of A−1 · R (d) by using the LU decomposition (e) with the Cholesky method
3x + y − z = 2 x−2 y+2 x−2y 3 − 2 = 5 1. 2. 2x − y + 4z = 0 x−y 3y+2 x−2(y−1) 6 + 4 = 3 x + 5y − 2z = 1
2 0 −1 x 1 x + y + z = 3 3. 2 4 −1 · y = 1 4. 2x + 4y + 8z = 13 −1 8 3 z 2 3x + 9y + 27z = 34
2x − 4y + 9w = 25 2x + y − 2z = 1 3x − 3y − 3z + 11w = 27 5. 6. x + y + z = 6 4x + 6y − 15z + 5w = −5 −2x + y − 2z = −3 3x + y − 4z + 12w = 32
50 Chapter 2
Vector algebra and vector calculus
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
On the basis of simple, well-known representations of vector calculus the basic rules of the vector algebra are specified. Subsequently, the rules of vector differentiation with descriptive examples are discussed.
52 2.1 Unit vectors
Different unit vectors of vector representations are dependent on the use of coordinate system. Then the vector ~a can be expressed as the sum of multiples of the unit vectors. The unit vectors have the length (magnitude) one |e| = 1, and are always parallel to the coordinate system axis. For the practical work in water management three coordinate systems, the Cartesian, the cylindri- cal and the spherical, are generally used. In table 2.1 the vector ~a is described in each of this three coordinate systems (also see figures 2.2 and 2.1).
Table 2.1: coordinate systems for description of vectors coordinate unit vector~a system vectors → → → → → → → Cartesian i , j , k a = ax i + ay j + az k → → → → → → → cylindrical r , φ, z a = ar r + aφ φ + az z → → → → → → → sphericalrisch r , θ , φ a = ar r + aθ θ + aδ φ
Figure 2.1: vector representation in Cartesian coordinates
In two-dimensional space the polar coordinate system will be used (see Figure 2.3).
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
Figure 2.2: vector representation in spherical coordinates
Figure 2.3: vector representation in two-dimensional space
Since the vector ~a is independent of the used coordinate system, the following conver- sion is applicable between the Cartesian and the polar coordinate system:
p 2 2 ar = ax + ay = |~a| ay aα = arctan ax (2.1)
ax = cot (aα) · ay
ax = cos (aα) · |~a|
54 2.2 Algorithms
In the following some important basic arithmetic rules for vectors are to be demon- strated by examples in the Cartesian coordinate system.
• Addition The arguments of the Cartesian unit vectors are respectively added in the vector addition: ~ ~ ~a + b = (ax + bx)~i + (ay + by)~j + (az + bz)k (2.2)
Notice: This relationship applies only to the Cartesian coordinate system and can not be transferred to other coordinate systems. In the vector algebra the following laws apply:
commutative law A~ + B~ = B~ + A~ (2.3)
distributive law m(nA~) = (mn)A~ = n(mA~) (2.4)
distributive law (m + n)A~ = mA~ + nA~ (2.5)
distributive law m(A~ + B~ ) = mA~ + mB~ (2.6)
associative law A~ + (B~ + C~ ) = (A~ + B~ ) + C~ (2.7)
• Magnitude The magnitude of a vector is equal to its length and thus a scalar, which is direction-independent. q 2 2 2 |~a| = ax + ay + az (2.8) In paticular it applies that the magnitude of the unit vector is equal to one.
|~i| = |~j| = |~k| = |~r| = |~α| = |~δ| = 1 (2.9)
• Product There are two kinds of vector products with respect to the vector algebra, the scalar product (dot product) and the vector product(cross product). The scalar product between two vectors is defined:
~a ·~b = |~a| · |~b| · cos(~a,~b) (2.10)
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
Hence the scalar product between two vectors is equal to zero, if they stand perpendicularly to each other. In particular it applies that the scalar product of a vector with itself, i.e. the square, is equal to the square of the magnitude: ~ 0 ~a⊥b ~ ~ |~a| · |b| ~a b ~a ·~b = (2.11) ~ ~ −|~a| · |b| ~a b ~ ~ |~a| · |b| · cos ~a, b generally
Particularly for the unit vectors:
~i · ~j = 0; ~i · ~k = 0; ~j · ~k = 0; ~r · ~α = 0; ~r · ~z = 0; (2.12) ~i ·~i = 1; ~j · ~j = 1; ~k · ~k = 1; ~r · ~r = 1; ~α · ~α = 1; ~z · ~z = 1;
According to the above algorithms we compute the scalar product as follows:
~ ~ ~ ~a · b = (ax~i + ay~j + azk) · (bx~i + by~j + bzk) (2.13)
= axbx + ayby + azbz
For computing the angle between two vectors we use the equation:
a b + a b + a b cos(~a,~b) = x x y y z z (2.14) p 2 2 2p 2 2 2 ax + ay + az bx + by + bz
The cross product between two vectors yields a vector:
~a ×~b = ~v (2.15)
its magnitude is equal to the positive area of the parallelogram having ~a and ~b as sides |~v| = |~a ×~b| = |~a| · |~b| · sin(~a,~b)
and its direction is perpendicularly to ~a and ~b:
~v⊥~a and ~v⊥~b
56 2.2. Algorithms
Generally: ~ 0 ~akb ~ ~ |~a| · |b| ~a⊥b |~a ×~b| = (2.16) −|~a| · |~b| ~b⊥~a ~ ~ |~a| · |b| · sin(~a, b) generally For the Cartesian coordinate system applies:
~i ~j ~k
~a ×~b = a a a (2.17) x y z
b b b x y z Especially for unit vectors:
~ ~ ~ ~ ~ ~ i × j = 1 i × k = 1 j × k = 1 |~r × ~α| = 1 |~r × ~z| = 1 ~i × ~j = ~k ~i × ~k = ~j ~j × ~k =~i ~r × ~α = ~z ~r × ~z = ~α
~ ~ ~ ~ ~ ~ i × i = 0 j × j = 0 k × k = 0 |~r × ~r| = 0 |~α × ~α| = 0 |~z × ~z| = 0 (2.18)
Notice: For the vector product the commutative law is not applicable, but the anticom- mutative: ~a ×~b = −~b × ~a (2.19) The vector product is still distributive:
~a × (~b + ~c) = ~a ×~b + ~a × ~c (2.20)
• Differentiation In vector analysis we speak of three different kinds of differentiation, the gradient (grad), the divergence (div) and the curl or rotation (rot) of a vector. For all three methods a uniform differential vector, the Nabla operator ∇ applies (see table 2.2). Table 2.3 shows the ways of writing of the different kinds of differentiation in the overview as a function of the used coordinate system. For
57 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
further simplification the Laplace differential operator 4 will be used. This is equal to the double application of the Nabla operator: ∆ = ∇ · ∇ (2.21)
Table 2.2: Description of the NABLA operator in different coordinate systems
coordinate system
∂ ∂ ∂ Cartesian 5 = ~i + ~j + ~k ∂x ∂y ∂z
∂ 1 ∂ ∂ cylindrical 5 = ~r + · −→ϕ + ~z ∂r r ∂ϕ ∂z
∂ 1 ∂ 1 ∂ −→ spherical 5 = ~r + · −→ϕ + · θ ∂r r · sin θ ∂ϕ r ∂θ
For the gradient
∇ϕ = grad ϕ (2.22)
scalar ϕ =⇒ vector ∇ϕ
∂ −→ ∂ −→ ∂ −→ ∂ϕ−→ ∂ϕ−→ ∂ϕ−→ ∇ϕ = i + j + k ϕ = i + j + k ∂x ∂y ∂z ∂x ∂y ∂z the Nabla operator is applied to a scalar potential field ϕ. The result for this is a vector. The gradient can be regarded as the formal multiplication of the Nabla operator with a scalar. In the field of the hydrogeology this quantity can be the groundwater level h, temperature fields T , concentration distributions C, evaporation or groundwater regeneration rates vN and others. There scalars (potentials) are non- directional and have thereby no vector character. However they are location dependent. The most important application of the gradient is the Darcy law for the computation of the groundwater flow velocity (see section 7.1 ). ~v = −k grad h (2.23)
Example for the gradient: The groundwater level of an aquifer is indicated by the function: h = 2xy − 3x + 2
58 2.2. Algorithms
We compute the groundwater flow velocity for the case that the permeability coefficient of the aquifer is k = 2 · 10−3m · s−1. It applies:
~v = −k grad(h)
∂ (2xy − 3x + 2) ∂ (2xy − 3x + 2) ∂ (2xy − 3x + 2) m = −2 · 10−3 ~i + ~j + ~k ∂x ∂y ∂z s m m = (6 − 4y) 10−3 ·~i − 4 · x · 10−3 · ~j s s It is to be recognized, that:
1. there is no vertical stream 2. the velocity is dependent on the coordinates. Thus the current in the aquifer is not constant.
The divergence is the application of the Nabla operator on a vector:
∇~v = div ~v (2.24)
vector ~v =⇒ scalar
∂ ∂ ∂ ∂v ∂v ∂v ∇~v = ~i + ~j + ~k v ~i + v ~j + v ~k = x + y + z ∂x ∂y ∂z x y z ∂x ∂y ∂z The result of divergence formation is a scalar quantity. The divergence can be regarded as the formal application of the scalar product between the Nabla operator and a vector. According to the rule of scalar product formation the divergence of a vector is a scalar quantity. The divergence, also noted as productivity of an area G, indicates whether sources or sinks are in this area. If the divergence of a vector field is equal to zero (∇~v = div ~v = 0), the area has neither a source nor a sink. According to Gauss’ law the entire source and sink activity of an area G can be computed by the volume integral of the divergence. At the same time it is known from the balance laws that the difference between the source and sink activities, i.e. the flow rates, have to discharge through the surface: ZZZ II div ~vdV = ~v · ~ndS (2.25)
G G For the two-dimensional area follows similarly: ZZ I div ~vdA = ~v · ~ndL (2.26)
A L
59 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
~n is a normal (perpendicularly standing) unit vector to the surface or to the circumfer- ence. With Gauss’ theorem the volume integral can be converted into an integral over the surface and an area integral can be converted into an integral over the bound. Also the divergence plays a fundamental role in the hydrogeology, since all processes must be balance in the mathematical description. In particular a large number of further derivatives is based on the following relation:
div ~v = div(−k grad h) = q (2.27)
Example of divergence calculation: We compute the divergence of the velocity vector ~v for the previous example:
∂ ∂ ∇~v = (3 − 2y)2 · 10−3 + (−4 · 10−3x) = 0 ∂x ∂y
Thus this area is neither source nor sink.
The curl/rotation is the rotation of a vector field and is the cross product of the Nabla operator with a vector:
rot ~v = ∇ × ~v (2.28)
~i ~j ~k
∂vz ∂vy ∂vz ∂vx ∂vy ∂vx rot ~v = ∂ ∂ ∂ = − ~i − − ~j + − ~k ∂x ∂y ∂z ∂y ∂z ∂x ∂z ∂x ∂y
v v v x y z
The result is a vector. If rot ~v = 0, we say it is an irrotational field. We can also deduce from it, that rot grad ϕ = 0 always is applicable for irrotational potential fields j (a conservative vector field).
Further rules of computation in connection with the vectorial differentiation yiel as a result of application of the vector rules and the extended rules for the differentiation of products:
∇(ϕ1 · ϕ2) = ϕ1∇ϕ2 + ϕ2∇ϕ1 = ϕ1 grad ϕ2 + ϕ2 grad ϕ1 (2.29) ∇ · (ϕ(~a)) = ϕ∇~a + ~a · ∇ϕ = ϕ div(~a) + ~a · grad(ϕ) (2.30) ∇ × (ϕ~a) = ϕ∇ × ~a + ~a × ∇ϕ = ϕrot(~a) + ~a × grad(ϕ) (2.31)
If we examine the source and sink activity of an aquifer, we can write the Darcy law
60 2.2. Algorithms as follows:
div(~v) = div(−k · grad h) = q (2.32)
∇(~v) = ∇(−k · ∇ h) = q
∇(~v) = ∇(−k) · ∇ h − k · ∇(∇h) = q
∇(~v) = ∇(−k) · ∇ h − k · ∆h = q
div(~v) = grad(−k) · grad(h) − k · div(grad(h)) = q
Only for the homogeneous isotropic aquifer may be set grad(−k) = 0 and then the aquifer equation is: div(~v) = −k div(grad h) = q (2.33)
Table 2.3: differential operators in the different coordinate systems
coordinate system name/operator Cartesian cylindric NABLA operator ∂ ~ ∂ ~ ∂ ~ ∇ = ∂ ~r + 1 ∂ ~ϕ + ∂ ~z ∇ = ∂x i + ∂y j + ∂z k ∂r r ∂ϕ ∂z ∇ gradient ∂h~ ∂h~ ∂h~ ∇ h = ∂h~r + 1 ∂h ~ϕ + ∂h~z ∇ h = ∂x i + ∂y j + ∂z k ∂r r ∂ϕ ∂z grad h = ∇h divergence 1 ∂vϕ ∇ · v = ∂vx + ∂vy + ∂vz ∇ · ~v = ∂(r vr) + 1 + ∂vz ∂x ∂y ∂z r ∂r r ∂ϕ ∂z div ~v = ∇ · ~v LAPLACE operator ∂2h ∂2h ∂2h ∂(r ∂h ) 2 2 ∆ h = + + 1 ∂r 1 ∂ h ∂ h ∂x2 ∂y2 ∂z2 ∆h = r ∂r + r2 ∂ϕ2 + ∂z2 ∆h = div(grad h) = ∇2h
∂v ∂vz y ~ 1 ∂vz ∂vϕ curl/rotor ∇ × ~v = ∂y − ∂z i ∇ × ~v = r ∂ϕ − ∂z ~r ∂vx ∂vz ~ ∂vr ∂vz −→ + ∂z − ∂x j + ∂z − ∂r ϕ ∂(r v ) rot~v = ∇ × ~v ∂vy ∂vx ~ 1 ϕ 1 ∂vr + ∂x − ∂y k + r ∂r − r ∂ϕ ~z volume dV = dx · dy · dz dV = r · dr · dϕ · dz dV
61 2.3 Examples of vector calculus
3 −1 1. The filter velocity ~v consists of the components vx = 3 · 10 m · s , vz = −5 · −4 −1 10 m · s and vy = 0. Outline and compute the filter velocity and indicate the magnitude and the angle.
The magnitude of a vector is:
q 2 2 2 |~v| = vx + vy + vz
For the given vector this is:
m |~v| = p(3 · 10−3)2 + (−5 · 10−4)2 = 3, 04 · 10−3 s
The angle is calculated by the slope, which is equal to the tangent of the angle:
−4 vz −5 · 10 α = arctan = arctan −3 vx 3 · 10
= arctan (−0, 166) = 350, 52 o = 6, 12 rad
Thus the vector in Cartesian an polar coordinates is:
m m m m ~v = 3 · 10−3 ·~i + 5 · 10−4 · ~k = 3, 04 · 10−3 · ~r + 6, 12 · ~α s s s s
2. A pollutant particle moves by convection (~vkonv) and by the hydrodynamic dis- persion (~vdisp). Plot and compute the covered way and the end point, if the particel is transported from the origin with the following velocities:
m m ~v = 1 · 10−4 ~i + 10−3 ~j and konv s s m ~v = 3 · 10−10 ~r + 0, 785~α disp s
In this task two different coordinate representations are used. Since the natural
processes are independent of the type of representation, the task can be solved with the use of the Cartesian coordinate representation or by means of polar coordinates. In both cases a conversion between the two systems is necessary.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 2.3. Examples of vector calculus
In this case for the two-dimensional case the following relations are available:
~a = ax~i + ay~j
~a = ar~r + aα~α q 2 2 ar = ax + ay = |~a|
ay aα = arctan ax
ax = cos (aα) · ar
ay = sin (aα) · ar
ax = tan (aα) · ay
It is important that aα usually is indicated in radian measure and the following relation applies: _ α α◦ = 2 π 360◦ With the given numerical values we find:
~v = ~vkonv + ~vdisp
According to the definition:
q m v = v2 + v2 = p(10−4)2 + (10−3)2 = 10−7 r konv x y s vy ◦ αkonv = arctan = 84, 29 vx
3. Plot and compute the end point of a pollutant particle after one day, if it moves from the origin x = 0m, y = 0m by convection due to a potential gradient of ∆h = 1m between the points x = 0m, y = 0m and x = 30m, y = 40m with a k value k = 5 · 10−4m/s. The Basis of the convection is the filtration velocity ~v. Accurate the field velocity
has to be used, however not in this example. The mean transit velocity ~va is
63 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
equated thereby the pore velocity. ~v ~v = ~v filtration velocity, n seep through porosity a n0
~v = −k grad h (Darcy law)
dh ∆h v = k ⇒ v ≈ k r dr r ∆r
p 2 2 p 2 2 ∆r = (x1 − x2) + (y1 − y2) = (30 m) + (40 m) = 50 m m 1 m m v = 5 · 10−4 · = 10−5 r s 50 m s
−5 m Distance s: s = vr · t = 10 s · 86400 s = 0, 864 m
s = px2 + y2 Position: x2 = s2 − y2 From the linear equation: y = m x + n or the two points equation of the straight line follows: y − y y − y 1 0 = 0 x1 − x0 x − x0 With x0 = y0 = 0 and x1 = 30m and y1 = 40m we have the results: y 40 4 4 = = or y = x = m · x x 30 3 3
y2 = m2 · x2 = m2 · (s2 − y2)
m2 · s2 y2 = (1 + m2) s s m2 · s2 1, 3332(0, 864 m)2 y = = = p0, 4777m = 0, 689m (1 + m2) (1 + 1, 7778) y x = = 0, 517m m We can insert y immediately into the equation of the length s: 4 y = x 3 p s = x2 + y2
r 4 r 16 5 s = x2 + ( x)2 = x2 + x2 = x 3 9 3
64 2.3. Examples of vector calculus
With s = 0, 864m we get the value:
3s x = = 0, 6 · 0, 864m = 0, 518m 5 4 y = x = 0, 69m 3
65 2.4 Exercises
Exercises to 2:
1. The vectors ~a,~b,~c are given in the coordinates:
ax = 5 bx = 3 cx = −6
ay = 7 by = −4 cy = −9
az = 8 bz = 6 cz = −5 Determine the length of vector d~ = ~a +~b + ~c.
2. Given the vectors ~a = 2~i − 3~j + 5~k and ~b = 3~i − w~j + 2~k. Compute w such that the two vectors are perpendicular to each other.
3. Given ϕ = xy + yz + zx and A~ = x2y~i + y2z~j + z2x~k. Calculate: (a)A~ · ∇ϕ (b)ϕ∇ · A~ (c) (∇ϕ) × A~
4. A particle moves along a space curve in the coordinates x = t3 + 2t, y = −3e−2t, z = 2 sin 5t. Compute the velocity and the acceleration of the particle at any time t. Indicate the distances for t = 0 and t = 1.
5. Plot and compute the end point of a pollutant particle after one day, if it moves from the origin x = 0m, y = 0m by a convection due to a potential gradient of 4h = 1m between the points x = 0m, y = 0m and x = 30m, y = 40m with a k value k = 5 · 10−4m · s−1. 6. Given the scalar conservative vector field in a filter h = xy + yz + xz. a) Determine the filter velocity (vector and magnitude) b) Is the activity source or sink in the filter? c) Is the flow in the filter irrational? Given are k = 10−4ms−1 and grad(−k) = 0.
7. A pollutant plume spreads in the underground. The distribution of the pollutant varies in the range of the values x ::= 0 to 10 and y ::= 0 to 10 with the following geometry: C(x, y) = 50 − ((x − 5)2 + (y − 5)2)
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 2.4. Exercises
a) Plot the equipotential lines for the concentration values in range of C(x, y) = 0mg to 50mg with an increment δC(x, y) = 10. b) Compute the gradient at the point P (3, 4) and determine the magnitude and the direction angle.
8. A pollutant plume spreads in the underground. The distribution of the pollutant varies in the range of the values x ::= 0 to 10 and y ::= 0 to 10 with the following geometry: C(x, y) = 125 − ((2x − 10)2 + (y − 5)2) a) Plot the equipotential lines for the concentration values in range of C(x, y) = 0mg to 125mg with an increment δC(x, y) = 25. b) Compute the gradient at the point P (5, 10) and determine the magnitude and the direction angle.
9. The groundwater level of an aquifer which one side is limited by a barrier and a well has the following geometry:
1 (y − 10)2 z = R 2 x
a) Plot the hydroisohypses in the range of zR = 1m to zR = 5m with an increment δzR = 1m for the coordinate 0 ≤ x ≤ 10 b) Compute the filtration velocity with k = 0, 0001ms−1 at the point P (5, 5); determine the magnitude and the direction angle α. c) Is this field source or sink?
10. The ground water level of a rift inflow in an aquifer is given by the geometrical function:
zR = y + 3x
a) Plot the hydroisohypses for the range zR = 1m to 5m with an increment δzR = 1m. b) Compute the filtration velocity with k = 0, 0001ms−1 at the point P (5, 5); determine the magnitude and the direction angle. c) Is this field source or sink?
67 CHAPTER 2. VECTOR ALGEBRA AND VECTOR CALCULUS
68 Chapter 3
Interpolation methods
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 3. INTERPOLATION METHODS
Problem:
Some measured values (dependent variable) are dependent on independent variables in one -, two -, three- or four-dimensional space measurement, and generally they are represented by the three space coordinates (depending upon coordinate system e.g. xn, yn, zn or rn, αn, zn or rn, αn, ϑn (see 2 vector analysis)) and the time tn. We have a discontinuous value tables in this case. For the one dimensional case e.g.:
independent dependend
variable variable
x0 y0 = f(x0)
x1 y1 = f(x1) . . . .
xn yn = f(xn)
The points x0, x1, ..., xn are called supporting points, and the points y0, y1, ..., yn are the basic or supporting values.
If we are looking for function values, whose arguments lie within the range (x0, xn), we name it interpolation In contrast if the function values we are looking for lie outside the range (x0, xn) we call this extrapolation. By interpolation or extrapolation we try to find a continuous function w = p(x), which reflects the original function yn = f(xn) as exactly as possible (see figure 3.1). It is always assumed that the interpolation function only matches the original function on the supporting points. The accuracy for the interval in between, e.g. the matching of both functions, depends on the number and the distribution of the supporting points. According to the sampling theorem the quantisation error increases proportionally to the rise of the function.
Note:
No interpolation algorithm can be used as replacement for an enlargement ot the measured value density. By means of an interpolation algorithm one receives in each case approximated values.
70 Figure 3.1: representation of the discontinuous measured data acquisition
71 CHAPTER 3. INTERPOLATION METHODS
Example for application of interpolation: The pollutant concentration C(x), which runs from a refuse dump, is measured at the points x0, x1, x2 (see figure 3.2). The pollutant concentration at the point xF l, which cause the danger by flowing in the river, is to be estimated by interpolation. A conclusion is to be given whether this value exceeds the limiting value.
Figure 3.2: representation of an interpolation problem
x0 C0 = f(x0)
x1 C1 = f(x1)
xF l ?
x2 C2 = f(x2)
For the solution of this problem an interpolation function w = p(x) is to seek for as
”replacement” for the function Cn = f(xn). This function should fulfil the following conditions:
wi = p(xi) = Ci∀i = 1, . . . , n (3.1)
72 i.e.
w0 = p(x0) = C0
w1 = p(x1) = C1 (3.2) . .
wn = p(xn) = Cn
Then it is supposed that the intermediate values of the function w = p(x) are a good approximation of the intermediate values of the function Cn = f(xn). For the determination of the function w = p(x) different interpolation methods can be used. We differentiate thereby one- and multi-dimensional procedures. The mul- tidimensional methods play an important role in connection with the geographical information systems (GIS) and are also often applied in connection with geostatis- tics.
In the following some methods will be introduced in connection with water economical questions.
• Polynomial interpolation • Spline interpolation (peace wise polynomial interpolation) • Kriging method
73 3.1 Polynomial interpolation
In this method p(x) has the form of an algebraic polynomial of order n:
2 n w = p(x) = a0 + a1x + a2x + ... + anx (3.3)
The advantage of this method is that the intermediate values can be computed as easily as possible.
Based on a value table with n+1 pairs of variates maximally an n-th order polynomial can be exactly determined: n X k y := p(x) = ak · x (3.4) k=0 with the property: n X k y(xi) ≈ p(xi) = ak · xi = wi (3.5) k=0 This polynomial is the interpolation polynomial to the given system of interpolation supporting points.
Normally we look for polynomials of lower order (n ≤ 3), which fit together the pairs of values at least by pairs:
p(x) = a0 + a1x linear interpolation
2 p(x) = a0 + a1x + a2x quadratic interpolation
2 3 p(x) = a0 + a1x + a2x + a3x cubic interpolation
The application of polynomials with higher order makes the calculations more difficult and leads to very large fluctuations.
From the different display formats for the polynomials follow different interpolation methods for the determination of the coefficients ai of an n-th order polynomial. All this different methods lead to the same polynomial.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 3.1. Polynomial interpolation
This interpolation methods are:
• analytical power function
• Lagrange
• Aiken
• Newton
3.1.1 Analytical power function
This method assumes that for each supporting point the polynomial w = p(x) fulfils the condition y(xi) = p(xi). In this case we get for the n+1 supporting points a system of n + 1 equations with n + 1 unknown coefficients a0, . . . , an.
2 n a0 + a1x0 + a2x0 + ... + anx0 = y0 (3.6)
2 n a0 + a1x1 + a2x1 + ... + anx1 = y1 . .
2 n a0 + a1xn + a2xn + ... + anxn = yn
This equation system can be written as a matrix equation:
X · A = Y with the matrices: 1 x x2 ··· xn a y 0 0 0 0 0 1 x x2 ··· xn a y 1 1 1 1 1 X = A = Y = . . . . . . . ...... . . 2 n 1 xn xn ··· xn an yn
75 CHAPTER 3. INTERPOLATION METHODS
The matrix X and the vector Y on the right side represent the known coefficients, whereby A is the solution vector. The linear equation system can be solved by all the known methods (see section 1.3, solution of equations systems, page 22) The determinant of this linear equation system is: (x − x ) · (x − x ) · (x − x ) · ... · (x − x )· 1 0 2 0 3 0 n 0 1 x ··· xn ·(x − x ) · (x − x ) · ... · (x − x )· 0 0 2 1 3 1 n 1 1 x ··· xn ·(x − x ) · ... · (x − x )· 1 1 3 2 n 2 D = = (3.7) . . . . . . . .. . . 1 x ··· xn ·(x − x ) · (x − x )· n n n−1 n−2 n n−2 ·(xn − xn−1) and is named the Vandermond determinant.
Since all supporting points are (must be) pairwise different, is D 6= 0 and the linear equation system is explicit solvable.
There is only one polynomial of the order n which fulfils the property yi = f(xi) = p(xi) with the coefficients (see section 1.2.3 determinants, page 19):
D D D a = a0 , a = a1 , ··· , a = an (3.8) 0 D 1 D n D With this coefficients the interpolation polynomial is:
2 n y(x) ≈ p(x) = a0 + a1 · x + a2 · x + ··· + an · x
The interpolation value at the place xP is:
2 n y(xP ) ≈ p(xP ) = a0 + a1 · xP + a2 · xP + ··· + an · xP
Although the beginning of this method is very simple, the final determination of the interpolation polynomial requires a relative large computation, particularly if a great number of supporting points are to be taken in account.
76 3.1. Polynomial interpolation
Example for the interpolation with the analytical power function method: Find a quadratic polynomial by using the values of the following table and calculate 1 1 the value y = f( 2 ) at the place x = 2 . x 0 1 2
y 0 1 0 Since only three supporting points are given, the polynomial can only be a second order polynomial. A quadratic polynomial has the form
2 p(x) = a0 + a1x + a2x
It must be:
yi = p(xi)
2 yi = a0 + a1xi + a2xi
p(0) = 0 ⇒ a0 + a1 · 0 + a2 · 0 = 0 ⇒ a0 = 0
2 p(1) = 1 ⇒ a0 + a1 · 1 + a2 · 1 = 1 ⇒ a1 + a2 = 1
2 p(2) = 0 ⇒ a0 + a1 · 2 + a2 · 2 = 0 ⇒ 2a1 + 4a2 = 0 From this three equations follows:
a0 = 0
a1 = 2
a2 = −1
Thus the interpolation polynomial is:
p(x) = 2x − x2
1 With this function the value at the place x = 2 can be computed:
1 1 1 12 3 f ≈ p = 2 · − = 2 2 2 2 4
77 CHAPTER 3. INTERPOLATION METHODS
3.1.2 Lagrange interpolation formula
Lagrange wrote the interpolation in the following form:
y(xP ) ≈ p(xP ) = L0(xP ) · y0 + L1(xP ) · y1 + ... + Ln(xP ) · yn (3.9)
With the Lagrange interpolation no closed analytical functions are computed, but only single values p(xP ) for each interpolation point xP . Thereby the coefficients Li(x) of the interpolation values yi (for i = 0, 1, . . . , n) are n-th order polynomials of xP . These are computed from the supporting points xi and are called the Lagrange polynomials. The Lagrange polynomials of n-th order are:
(xP − x1)(xP − x2) ··· (xP − xn) L0(x) = (x0 − x1)(x0 − x2) ··· (x0 − xn) (xP − x0)(xP − x2) ··· (xP − xn) L1(x) = (x1 − x0)(x1 − x2) ··· (x1 − xn) . . (3.10) (xP − x0)(xP − x1) ··· (xP − xi−1)(xP − xi+1) ··· (xP − xn) Li(x) = (xi − x0)(xi − x1) ··· (xi − xi−1)(xi − xi+1) ··· (xi − xn) . .
(xP − x0)(xP − x1)(xP − x2) ··· (xP − xn−1) Ln(x) = (xn − x0)(xn − x1) ··· (xn − xn−1) Thus the Lagrange interpolation polynomial is:
y = f(xP ) ≈ p(xP ) = L0(xP )y0 + L1(xP )y1 + ... + Ln(xP )yn
(xP − x1)(xP − x2) ··· (xP − xn) y = y0 (x0 − x1)(x0 − x2) ··· (x0 − xn) (xP − x0)(xP − x2) ··· (xP − xn) + y1 (3.11) (x1 − x0)(x1 − x2) ··· (x1 − xn) . .
(xP − x0)(xP − x1)(xP − x2) ··· (xP − xn−1) + yn (xn − x0)(xn − x1) ··· (xn − xn−1)
If we insert for xP one of the points x0, x1, ..., xn−1, xn, there is always a factor in the numerator which is equal to zero. For the i-th Lagrange polynomial the denominator is also equal to zero, so the limes of the i-th Lagrange polynomial is equal to 1. Since all other Lagrange polynomials are zero, the Lagrange interpolation polynomial is:
yi = y(xi) = f(xi) ≈ p(xi) = 1 · yi
78 3.1. Polynomial interpolation
A disadvantage of the Lagrange method is that the computation of the Lagrange interpolation polynomials must be accomplished again when an increase of the support- ing point number should be taken into account, which is identical with the increase of the order of the interpolation polynomial. This is clearly to be seen in the following example. Note:
• The weights (coefficients) Li(xi) in the Langrange interpolation formula always have to be computed again, if the supporting points number changes.
• The sum of the weights is always equal to one (important for checking the results)
X Li(xi) = 1
Example for Lagrange interpolation:
For the function yn = f(xn) the values at the equidistant points xn = x0 + 2nh for n = −1, 0, 1, 2 (see table) are given:
n −1 0 1 2
xn x0 − 2h x0 x0 + 2h x0 + 4h
f(xn) y−1 y0 y1 y2
1 Find an approximate value w = f(x) = f(x0 + h) for x = 2 . According to the rules of the polynomial interpolation in the case with four supporting points maximally a 3rd order polynomial can be developed. It is also possible to accomplish a piecewise interpolation. This has the advantage that we can reduce computation work. The accuracy is however declined. In this case we try to find an optimum between the required accuracy and the cost of computation. The supporting points used in the piecewise interpolation are this next to the interpolation point.
1. Linear interpolation 1 The interpolation function at the point x = 2 is written with the help of the Lagrange interpolation formula as (see equation 3.1.2):
w 1 = L0(x 1 )y0 + L1(x 1 )y1 2 2 2
1 Here the supporting points x = 0 and x = 1 are used, where the point x = 2 is
79 CHAPTER 3. INTERPOLATION METHODS
in between. The weights L0 and L1 are (see equation 3.10):
x 1 − x1 2 x0 + h − x0 − 2h 1 L0(x 1 ) = = = 2 x0 − x1 x0 − x0 − 2h 2 x 1 − x0 2 x0 + h − x0 1 L1(x 1 ) = = = 2 x1 − x0 x0 + 2h − x0 2 Then the searched value is: 1 w 1 = (y0 + y1) 2 2 The result of the linear interpolation is thereby equal to the arithmetic middle. 2. Quadratic interpolation In this case the interpolation function is (see equation 3.1.2):
w 1 = L0(x 1 )y0 + L1(x 1 )y1 + L2(x 1 )y2 2 2 2 2 The corresponding coefficients are (see equation 3.10):
(x 1 − x1)(x 1 − x2) 2 2 3 L0(x 1 ) = = 2 (x0 − x1)(x0 − x2) 8
(x 1 − x0)(x 1 − x2) 2 2 3 L1(x 1 ) = = 2 (x1 − x0)(x1 − x2) 4
(x 1 − x0)(x 1 − x1) 2 2 1 L2(x 1 ) = = − 2 (x2 − x0)(x2 − x1) 8 and the result is: 3 3 1 w 1 = y0 + y1 − y2 2 8 4 8 3. cubic interpolation In the same way follows (see equation 3.1.2):
w 1 = L−1(x 1 )y−1 + L0(x 1 )y0 + L1(x 1 )y1 + L2(x 1 )y2 2 2 2 2 2 We get the following Lagrange factors (see equation 3.10):
(x 1 − x0)(x 1 − x1)(x 1 − x2) 2 2 2 1 L−1(x 1 ) = = − 2 (x−1 − x0)(x−1 − x1)(x−1 − x2) 16
(x 1 − x−1)(x 1 − x1)(x 1 − x2) 2 2 2 9 L0(x 1 ) = = 2 (x0 − x−1)(x0 − x1)(x0 − x2) 16
(x 1 − x−1)(x 1 − x0)(x 1 − x2) 2 2 2 9 L1(x 1 ) = = 2 (x1 − x−1)(x1 − x0)(x1 − x2) 16
(x 1 − x−1)(x 1 − x0)(x 1 − x1) 2 2 2 1 L2(x 1 ) = = − 2 (x2 − x−1)(x2 − x0)(x2 − x1) 16 Thus the result is: 1 9 9 1 w 1 = − y−1 + y0 + y1 − y2 2 16 16 16 16
80 3.1. Polynomial interpolation
3.1.3 Newton interpolation formula
3.1.3.1 Arbitrary supporting points
The disadvantage of the Lagrange method is that the Lagrange polynomials must be computed again and again, which can be avoided in the Newton method. With the Newton method only one auxiliary item should be added when further supporting points are taken into account. The method begins with the following formula:
p(x) = b0 + b1(x − x0)
+ b2(x − x0)(x − x1)
+ b3(x − x0)(x − x1)(x − x2) (3.12) . .
+ bn(x − x0)(x − x1) ··· (x − xn−1)
If we want to find a certain interpolation value p(xP ), x will be replaced by xP in the polynomial. The coefficients are determined again in such a way that the interpolation polynomial at the supporting point matches exactly with the interpolation values (xn, yn). If we respectively replace xP by x0, . . . , xn in the Newton formula, we get an equation system with n equations and n variables. Since in each case the corresponding factors
(xP − xi = 0) are equal to zero, the polynomial items will be omitted. Then we know the basic value yi from the polynomial value p(xi).
y0 = b0 + b1(x0 − x0) + ··· | {z } =0
y1 = b0 + b1(x1 − x0) + b2(x1 − x0)(x1 − x1) + ··· | {z } =0
y2 = b0 + b1(x2 − x0) + b2(x2 − x0)(x2 − x1) (3.13) . .
yn = b0 + b1(xn − x0) + b2(xn − x0)(xn − x1) + ···
+bn(xn − x0)(xn − x1) ··· (xn − xn−1)
81 CHAPTER 3. INTERPOLATION METHODS
The equation system can be solved step by step with b0, b1, b2 . . . , bn. By inserting the first equation into the second we get b1. Once again inserting this into the third equation it yields b2. After n + 1 steps we insert b0, . . . , bn−1 in to the n + 1-th equation and this yield bn.
b0 = y0
(y1 − y0) b1 = = [x1x0] (x1 − x0) (y1−y0) (y2 − y0) − (x2 − x0) (x1−x0) b2 = (x2 − x0)(x2 − x1) (y − y ) + (y − y ) − [x x ](x − x ) = 2 1 1 0 1 0 2 0 (x2 − x0)(x2 − x1) (y − y ) (y − y ) [x x ](x − x ) = 2 1 + 1 0 − 1 0 2 0 (3.14) (x2 − x0)(x2 − x1) (x2 − x0)(x2 − x1) (x2 − x0)(x2 − x1) [x x ] [x x ](x − x ) [x x ](x − x ) = 2 1 + 1 0 1 0 − 1 0 2 0 (x2 − x0) (x2 − x0)(x2 − x1) (x2 − x0)(x2 − x1) [x x ] [x x ](x − x ) − [x x ](x − x ) = 2 1 + 1 0 1 0 1 0 2 0 (x2 − x0) (x2 − x0)(x2 − x1) [x x ] = 2 1 + −[x1x0](x2−x1) (x2−x0)(x2−x1) (x2 − x0) [x2x1] − [x1x0] b2 = = [x2x1x0] (x2 − x0) Hereby we use the abbreviated notation, which is called divided differences of first and higher order:
yk − yi [xkxi] := xk − xi [xlxk] − [xkxi] [xlxkxi] := (xl − xi) [xmxlxk] − [xlxkxi] [xmxlxkxi] := (3.15) (xm − xi) . .
[xnxn−1 ··· x1] − [xn−1 ··· x1x0] [xnxn−1 ··· x1x0] := (xn − x0)
82 3.1. Polynomial interpolation
Thus the coefficients results to:
b0 = y0
(y1 − y0) b1 = = [x1x0] (x1 − x0) [x x ] − [x x ] b = 2 1 1 0 = [x x x ] 2 (x − x ) 2 1 0 2 0 (3.16) [x3x2x1] − [x2x1x0] b3 = = [x3x2x1x0] (x3 − x0) . .
[xnxn−1 ··· x1] − [xn−1 ··· x1x0] bn = = [xnxn−1 ··· x1x0] (xn − x0)
Particularly the coefficients can be determined conveniently according to the following computation scheme (example for 5 supporting points):
y0 x (y1−y0) 0 = [x1x0] (x1−x0) = b0 [x2x1x0] = b1 [x3x2x1x0] x1 y1 = b2 [x4x3x2x1x0] (y2−y1) = [x2x1] = b3 (x2−x1) x2 y2 [x3x2x1] = b4 (y3−y2) = [x3x2] [x4x3x2x1] (x3−x2) x3 y3 [x4x3x2] (y4−y3) = [x4x3] (x4−x3) x4 y4 (3.17) According to equation 3.12 the value y at the place x be interpolated:
y(x) ≈ p(x) = b0 + b1(x − x0)
+ b2(x − x0)(x − x1) (3.18)
+ b3(x − x0)(x − x1)(x − x2)
+ b4(x − x0)(x − x1)(x − x2)(x − x3)
This equation also can be used, in order to compute the interpolation function w = p(x) distribution and possibly to plot the function.
83 CHAPTER 3. INTERPOLATION METHODS
3.1.3.2 Equidistant supporting point distribution
The equidistant supporting point distribution x0, x1 = x0 + h, . . . , xn = x0 + nh (h is the increment) is given, then the interpolation by the Newton method is:
∆y ∆2y ∆ny p(x) = y + 0 (x−x )+ 0 (x−x )(x−x )+...+ 0 (x−x ) ... (x−x ) (3.19) 0 h 0 2! · h2 0 1 n! · hn 0 n−1
2 n The elements ∆y0, ∆ y0,..., ∆ y0 are called finite differences. The exponent does not represent exponentiation, but step by step differences. By comparing the equations 3.19 and 3.12 on page 81 we get:
b0 l y0
y1 − y0 ∆y0 b1 = l (3.20) x1 − x0 h 2 [x2x1] − [x1x0] ∆ y0 b2 = l 2 (x2 − x0) 2! · h
84 3.1. Polynomial interpolation
These differences are computed according to the following scheme: 0 y 1 − n ∆ − 1 y 1 − n = ∆ 0 y n ∆ ··· 2 − n y ∆ 0 1 − y y 1 ∆ ∆ − n . . . − − y 1 2 y y = ∆ 2 = ∆ = ∆ − 0 1 n y y y 2 2 2 ∆ ∆ ∆ 2 − 1 n − y n y − 0 1 1 − y y − n n . . . − − y y 1 2 y y = = 2 1 = = − − 0 1 n n y y y y ∆ ∆ ∆ ∆ 1 − 0 1 2 n n y y y . . . y y 1 − 0 1 2 n n x x x . . . x x
85 CHAPTER 3. INTERPOLATION METHODS
The scheme for an example with n = 4: 0 y 3 ∆ − 1 y 3 = ∆ 0 y 4 ∆ 0 1 y y 2 2 ∆ ∆ − − 1 2 y y 2 2 = ∆ = ∆ 0 1 y y 3 3 ∆ ∆ 0 1 2 y y y ∆ ∆ ∆ − − − 1 2 3 y y y = ∆ = ∆ = ∆ 0 1 2 y y y 2 2 2 ∆ ∆ ∆ 0 1 2 3 y y y y − − − − 1 2 3 4 y y y y = = = = 0 1 2 3 y y y y ∆ ∆ ∆ ∆ 0 1 2 3 4 y y y y y 0 1 2 3 4 x x x x x
By back substitution we see, that each finite difference is a combination of the y-values of the first column, e.g.: 3 ∆ y0 = y3 − 3y2 + 3y1 − y0 (3.21)
86 3.1. Polynomial interpolation
3.1.3.3 Examples for the Newton method
1. For the function yn = f(xn) the values at equidistant points xn = x0 + 2nh for n = −1, 0, 1, 2 are given (see table):
n −1 0 1 2
xn x0 − 2h x0 x0 + 2h x0 + 4h
f(xn) y−1 y0 y1 y2 1 Find an approximate value x = for y 1 = f(x0 + h). 2 2 Solve this example with the Newton method and compare the results with those from the Lagrange interpolation formula. a) linear interpolation:
∆y0 p(x 1 ) = y0 + (x 1 − x0) 2 ∆x 2 In this example is ∆x = 2h
y1 − y0 p(x 1 ) = y0 + (x0 + h − x0) 2 2h 1 = (y + y ) 2 0 1 b) Quadratic interpolation:
2 ∆y0 ∆ y0 p(x 1 ) = y0 + (x 1 − x0) + (x 1 − x0)(x 1 − x1) 2 ∆x 2 2!∆x2 2 2 In this example is ∆x = 2h
1 y2 − 2y1 + y0 p(x 1 ) = (y0 + y1) + (x0 + h − x0)(x0 + h − x0 − 2h) 2 2 (2h)22 3 3 1 = y + y − y 8 0 4 1 8 2 It applies:
2 ∆ y0 = ∆y1 − ∆y0 = y2 − y1 − (y1 − y0) = y2 − 2y1 + y0 Note:
The advantage of the Newton method is that the polynomial Li(x) does not change if the number of supporting points is changed, i.e. each time we only need to calculate the additional part of the interpolation function.
87 CHAPTER 3. INTERPOLATION METHODS
2. Given the following value table:
x 0 1 2 3 4 5
y 1 2 4 8 15 26
Determine the value y = f(2, 5). Choose a polynomial with a suitable order. How large is the deviation if the order of the polynomial is changed? Since the given supporting points are equidistant (h = 1), the Newton method is applicable to calculate the polynomials with different orders. First the finite differences are computed:
x0 = 0 y0 = 1 ∆y0 = 1 2 x1 = 1 y1 = 2 ∆ y0 = 1 3 ∆y1 = 2 ∆ y0 = 1 2 4 x2 = 2 y2 = 4 ∆ y1 = 2 ∆ y0 = 0 3 5 ∆y2 = 4 ∆ y1 = 1 ∆ y0 = 0 2 4 x3 = 3 y3 = 8 ∆ y2 = 3 ∆ y1 = 0 3 ∆y3 = 7 ∆ y2 = 1 2 x4 = 4 y4 = 15 ∆ y3 = 4 ∆y4 = 11 x5 = 5 y5 = 26
It is evident that the maximal interpolation polynomial order is third. a) Linear interpolation
The searched value x = 2, 5 lies between x2 = 2 and x3 = 3. Therefore the linear interpolation is accomplished only between this tow values
∆y p(x) = y + 2 (x − x ) 2 h 2 4 = 4 + (2, 5 − 2) 1
p(2, 5) = 6
88 3.1. Polynomial interpolation b) Quadratic interpolation Since the searched value is x = 2, 5 the quadratic interpolation can be
stretched among x1, x2 and x3.
∆y ∆2y p(x) = y + 2 (x − x ) + 2 (x − x )(x − x ) 2 h 2 2!h2 2 3 4 3 = 4 + (2, 5 − 2) + (2, 5 − 2)(2, 5 − 3) 1 2 · 1 0, 75 = 4 + 2 − 2
p(2, 5) = 5, 625 c) cubic interpolation The cubic interpolation formula requires three supporting places. In this
case both of the triple x1, x2 and x3 or the triple x2, x3 and x4 can be used. For the first case:
∆y ∆2y p(x) = y + 1 (x − x ) + 1 (x − x )(x − x ) 1 h 1 2!h2 1 2 ∆3y + 1 (x − x )(x − x )(x − x ) 3!h3 1 2 3 2 2 = 2 + (2, 5 − 1) + (2, 5 − 1)(2, 5 − 2) 1 2 · 1 1 + (2, 5 − 1)(2, 5 − 2)(2, 5 − 3) 6 · 1 = 2 +3 + 0, 75 − 0, 0625
p(2, 5) = 5, 6875 For the second triple:
∆y ∆2y p(x) = y + 2 (x − x ) + 2 (x − x )(x − x ) 2 h 2 2!h2 2 3 ∆3y + 2 (x − x )(x − x )(x − x ) 3!h3 2 3 4 4 3 = 4 + (2, 5 − 2) + (2, 5 − 2)(2, 5 − 3) 1 2 · 1 1 + (2, 5 − 2)(2, 5 − 3)(2, 5 − 4) 6 · 1 0, 75 = 4 +2 − + 0, 0625 2 p(2, 5) = 5, 6875
89 CHAPTER 3. INTERPOLATION METHODS
The deviation between the linear and the quadratic result is:
5, 625 − 6 = 6, 7%, 5, 625 While the deviation between the square and the cubic result is only:
5, 6875 − 5, 625 = 1, 1% 5, 6875
In order to estimate the results, the given points can be plotted (see figure 3.3). The diagram shows that: Actually the value should lie between 5
Figure 3.3: Representation of the measured interpolated values
and 6. Obviously the linear interpolation can not yield good results in this case. For this reason it is meaningful to plot given points and estimate the searched value. In a practical work it is important to have enough points in order to get a good approximation of the function. This can be ascertained that, the form of the function substantially does not change when additional points are taken into account.
90 3.2 Spline interpolation
To describe a given function in a certain interval we can link sections that consist of several lower degree polynomials instead of only one polynomial with high degree. The classical examples are line segments in subintervals (see figure 3.4). It is assumed that the function between two supporting points is nearly linear. This can be applied, if the supporting points are close enough to each other.
Figure 3.4: Representation of linear spline curves
Such approximations are continuous, however the first derivative is discontinuous, and i.e. vertices appear at the transition part from one interval to another. In the following the spline interpolation method will be described, in which cubic polynomials are built up such that the vertices are smooth, so that the first and second derivatives of the approximation are continuous. Polynomials with higher degree are not used since they oscillate strongly.
A given interval I = (a, b) is divided in n subintervals by the x-values x0 = a < x1 < x2 < . . . < xn = b. The cubic polynomial pieces will be fitted in the subintervals such that the y-values yi at the points xi match exactly. The first and second derivatives of the functions for the left and for the right subinterval at the data points must be equal
(see figure 3.5). The data points (xi, yi) are called the nodes of the spline (originally the term spline comes from an flexible spline devices).
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 3. INTERPOLATION METHODS
A cubic polynomial has four coefficients. Generally it can be written:
2 3 pi(x) = c0i + c1ix + c2ix + c3ix (3.22)
Figure 3.5: Representation of a spline curve with cubic polynomials
The Spline function is defined as follows:
1. The function S(x) is two times continuous differentiable on the interval [a, b].
2. S(x) is given by a cubic polynomial for each interval [xi, xi+1]. I.e.:
P S(x) := pi(x) (3.23) 2 3 pi(x) := ai + bi(x − xi) + ci(x − xi) + di(x − xi)
3. S(x) fulfils the interpolation constraints S(xi) = yi for all i = 1, . . . , n in the interval [a, b]. 4. Depending upon the form of connection constraints we get different kinds of spline functions. The following table shows special cubic spline functions:
92 3.2. Spline interpolation
Connecting conditions Description Comments
00 S(x0) = S (x0) = 0 S(x0) and S(xn) is the tangent natural 00 S(xn) = S (xn) = 0 function at the graph of S(x)
00 00 S (x0) = αS (xn) = generally
0 0 S (x0) = αS (xn) = β given first derivative at the boundary
000 000 S (x0) = αS (xn) = β given third derivative at the boundary
S(x0) = S(xn)
0 0 S (x0) = S (xn) periodic
00 00 S (x0) = S (xn)
p0(x) = pl(x) 000 000 not-a-knot S (xl) and S (xn) are continuous pn−2(x) = pn−1(x)
In the case of n intervals it yields to 4n coefficients and then 4n constraints are expected to compute the spline. At each node (xi, yi)(i = 1, . . . , n) we have 4 constraints (the y value and the correlation of the derivatives). This yields 4n − 4 constraints. At the boundary points a and b the y values are given and so we have 4n − 2 constraints. The spline is not completely defined. Two degrees of freedom remain.
pi(xi) = yi i = 0, . . . , n interpolation constraint pi(xi) = pi−1(xi) connecting constraints of (3.24) 0 0 p (xi) = p (xi) i = 1, . . . , n − 1 i i−1 the polynomials pi and pi−1 00 00 pi (xi) = pi−1(xi)
We can set the second derivative at the boundary points equal to zero and get a natural spline.
pn(xn) = an S(x0) and S(xn) are tangents (3.25) 00 pn(xn) = 2cn on the graph S(x)
93 CHAPTER 3. INTERPOLATION METHODS
Alternatively the first derivative at the boundary points can be given, in order to approximate a function. Thus it yields an equation system with 4n equations for 4n + 2 unknown quantities. The two missing equations are covered by the boundary conditions.
00 p0(x0) = 0 boundary conditions (3.26) 00 pn(xn) = 0
This equation system can be solved according to known methods. Usually the solution of this equation system is complex, so not only one-step but also iterative methods (see section 1.3 solution methods of equation system) must be used. As is shown below, a tridiagonal equation system can be generated by a certain algorithm, so then it can be solved with a little operating expense.
94 3.2. Spline interpolation
Algorithm
Given n + 1 data points xi (i = 0, . . . , n) with an increment hi = xi+1 − xi and n + 1 data values yi (e.g. as a list of measurement), so the following algorithm (see equations 3.24 to 3.26)can be used to compute the cubic spline function.
n−1 P S(x) := pi(x) i=0 (3.27) 2 3 pi(x) = ai + bi(x − xi) + ci(x − xi) + di(x − xi)
The cubic spline consist of n functions for the intervals xi ≤ x ≤ xi+1.
Step Calculation Validity Comment from equation 3.24 interpolation constraint 1 ai = yi i = 0, . . . , n pi(xi) = yi for all data points for natural splines: 00 00 S (x0) = S (xn) = 0 00 2 c0 = cn = 0 p (x) = 2ci + 6di(x − xi) 00 p (x0) = 2c0 = 0 00 p (xn) = 2cn = 0 from equation 3.24 ` ´ hi−1ci−1 + 2ci hi−1 + hi + hici+1 3 ` ´ 3 connection constraints 3 = ai+1 − ai − (ai − ai−1) i = 1; ··· n − 1 hi hi−1 pi+1(xi+1) = pi(xi+1) 1 hi 0 0 4 bi = (ai+1 − ai) − (ci+1 − 2ci) i = 0, . . . , n − 1 p (x ) = p (x ) hi 3 i+1 i+1 i i+1 5 d = 1 (c − c ) i = 0, . . . , n − 1 p00 (x ) = p00(x ) i 3hi i+1 i i+1 i+1 i i+1 The equation in the third step of the algorithm represents a linear equation system of n − 1 equations for the variables c1, . . . , cn−1. It can be written in the matrix form:
95 CHAPTER 3. INTERPOLATION METHODS
A · C = R (3.28)
2(h + h ) h 0 1 1 h 2(h + h ) h 1 1 2 2 . . . A = ...... (3.29) h 2(h + h ) h n−3 n−3 n−2 n−2 hn−2 2(hn−2 + hn−1)
3 3 (a − a ) − (a − a ) h 2 1 h 1 0 1 0 3 3 (a − a ) − (a − a ) 3 2 2 1 h2 h1 R = . (3.30) . 3 3 (an−1 − an−2) − (an−2 − an−3) hn−2 hn−3 3 3 (an − an−1) − (an−1 − an−2) hn−1 hn−2
c 1 c 2 . C = . (3.31) c n−2 cn−1
The matrix A is tridiagonally, symmetrically, diagonally dominant, positively definite and all entries are positive. Thus this matrix is always invertible and definitely solvable. As solution method the Gauss algorithm for tridiagonal matrices can be used (see section 1.3.1 solutions of equation system, Gauss algorithm).
96 3.2. Spline interpolation
Example spline interpolation:
Given the following measured data points and values:
i 0 1 2 3 4
xi −1 −0, 5 0 0, 5 1
yi 0, 5 0, 8 1 0, 8 0, 5
For these 5 pairs find a natural cubic spline. According to the definition of the spline function, 4 functions
2 3 pi(x) = ai + bi(x − xi) + ci(x − xi) + di(x − xi)
(i = 1,..., 4) defined respectively on the intervals xi ≤ x ≤ xi+1 are searched.
Using the algorithm the following steps are implemented:
Step Computation Result
a0 = 0, 5
a1 = 0, 8
1 ai = yi a2 = 1, 0
a3 = 0, 8
a4 = 0, 5
c0 = 0 2 c0 = c4 = 0 c4 = 0
97 CHAPTER 3. INTERPOLATION METHODS
Step Computation Result 2 (h + h ) h 0 c 0 1 1 1 h 2 (h + h ) h · c 1 1 2 2 2 0 h2 2 (h2 + h3) c3 3 3 (a − a ) − (a − a ) h 2 1 h 1 0 1 0 3 3 = (a − a ) − (a − a ) 3 2 2 1 h2 h1 3 3 c1 = 0 (a4 − a3) − (a3 − a2) h3 h2 3 c2 = −1, 2 2 (0, 5 + 0, 5) 0, 5 0 c 1 c3 = 0 0, 5 2 (0, 5 + 0, 5) 0, 5 · c 2 0 0, 5 2 (0, 5 + 0, 5) c3 3 3 (1, 0 − 0, 8) − (0, 8 − 0, 5) 0, 5 0, 5 3 3 = (0, 8 − 1, 0) − (1, 0 − 0, 8) 0, 5 0, 5 3 3 (0, 5 − 0, 8) − (0, 8 − 1, 0) 0, 5 0, 5
b0 = 0, 6
b1 = 0, 6 1 hi 4 bi = (ai+1 − ai) − (ci+1 − 2ci) hi 3 b2 = 0
b3 = −0, 6
d0 = 0
1 d1 = −0, 8 5 di = (ci+1 − ci) 3hi d2 = 0, 8
d3 = 0
98 3.2. Spline interpolation
This yields to the following functions according to equation 3.23:
2 3 pi(x) := ai + bi(x − xi) + ci(x − xi) + di(x − xi)
Function Interval
p0(x) = 0, 5 + 0, 6(x + 1) −1 ≤ x ≤ −0, 5
3 p1(x) = 0, 8 + 0, 6(x + 0, 5) − 0, 8(x + 0, 5) −0, 5 ≤ x ≤ 0
2 3 p2(x) = 1, 0 − 1, 2x + 0, 8x 0 ≤ x ≤ 0, 5
p3(x) = 0, 8 − 0, 6(x − 0, 5) 0, 5 ≤ x ≤ 1
The diagram of the spline is shown in figure 3.6:
Figure 3.6: Spline interpolation function
We recognize that the spline simulates the original analytic function 1 y = x2 + 1 very well. The maximum deviation of analytic solution amounts to 0.010244; which corresponds to 1.68%.
99 3.3 Kriging method
A family of special interpolation methods is called Kriging, which tries to handle with the following problem: The sampling at a place gives informations for certain spatial points. However it is unknown which values are available for the data points and values between these points. Kriging is a method, which makes possible, to compute the value of an intermediate point or the average over an entire block. The different special methods are all based on the creation of weighted mean values of the spatial variables. Block estimations are predominantly necessary in the mining industry, while point estimations are used for map representations, which is described know.
The individual Kriging methods differ either in the kind of the goal sizes which can be estimated or in their methodical extension for the inclusion of additional information.
Additional information about the spatial behaviour of a location dependent variable ex- ists in the cognition of other measurements, which relates to the observed variables. In hydrogeological practice for instance correlated dissolved matter or temporal repetition measurements of ground- water pressure head are common.
In a word Kriging methods have the following advantages compared to other interpo- lation procedures:
• Kriging yields the ”best” estimated value. • Kriging involves the information of the spatial structure of the variable and the variogram into the estimation. • The individual spatial arrangement of the measuring point net is considered with reference to the interpolation grid. • The reliability of the results is indicated in form of Kriging error for each esti- mated point.
Note: Also in the Kriging method it must be paid attention that no information gain can be achieved by the mathematical procedures. Only the information content of the measured values (basic values) is processed. Interpolation results might contradict physical laws (e.g. ground water contour line in receiving streams). If we want to get physically correct interpolations, a fine quantized simulation by means of physical models (e.g. ground-water flow models) is necessary and meaningful. Therefore such simulation programs offer internal diagram routines for creation of isoline.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 3.3. Kriging method
In order to understand the Kriging procedures, the following terms from the geostatis- tics must be known:
n 1 P Mean value m = Za n a=1 R z · p(z)dz = m Expected value E[Z] = with p(z) the density function
Variance var(Z) = σ2 = E [(Z − E[Z])2] = E [(Z − m)2]
Covariance of two cov(Zi,Zj) = E [(Zi − mi)(Zj − mj)] = σij
random variables Zi, Zj
correlations σij q ρij = σ2σ2 coefficient i j ~ 1 2 γ(h) = E Z(~x + ~h) − Z(~x) Variogramm 2
Z is a place dependent random variable with n measured values Za. The density function p(z) the probability that Z becomes the value zi. By computation of the inequality of two values, the variogram shows the variability of a random function, which correspond to points with distance to the vector ~h. Then the Kriging problem can be represented according to figure 3.7:
Figure 3.7: Illustration of a Kriging problem
101 CHAPTER 3. INTERPOLATION METHODS
We have a number of measured values Z(~xa), whereby Z is a random variable and ~xa is a measuring point of the range D.
We assume then that Z(~xa) is a subset of the random function Z(~x), which has the following characteristics: It is a second order stationary function, i.e.:
1. The expected value is constant over the range D: E[Z(~x + ~h)] = E[Z(~x)] 2. The covariance between two points depends only on the vector ~h: cov[Z(~x + ~h),Z(~x)] = C(~h)
Due to these assumptions we want to compute a weighted mean, in order to get an estimated value for the place ~x0.
∗ The Kriging estimator Z (~x0) represents a linear combination of weighted sample values
Zi and n of neighbouring points:
n ∗ X Z (~x0) = λiZ(~xi) (3.32) i=1
∗ The weights λi are determined in such a way that the estimated value Z (~x0) of the unknown true value fulfils the following conditions:
∗ ∗ ∗ 1. Z [~x0) is unbiased, i.e.: E [Z (~x0) − Z(~x0)] = 0 ∗ 2 2. The mean square value E[Z (~x0) − Z(~x0)] is minimal.
Assume the stationarity is the expected value E[Z(~xi)] = m and Z( ~x0) = m. Then the condition 1. yields:
" n # n n X X X E λiZ(~xi) − Z( ~x0) = λim − m = m( λi − 1) = 0 (3.33) i=1 i=1 i=1 From this follows that the sum of the weights must be one.
With the help of the variogram the expected value of the square error can be expressed:
∗ 2 ∗ E [Z (~x0) − Z(~x0)] = var (Z (~x0) − Z(~x0)) n n n P P P = 2 λiγ(~xi − ~x0) − λiλj(~xi − ~xj) − γ( ~x0 − ~x0) i=1 i=1 j=1
(3.34)
102 3.3. Kriging method
n P In order to minimize the error variance of the side condition 1 ( λi = 1), the La- i=1 grange multiplier µ will be introduced. Then the following function is minimized:
n ! ∗ X ϕ = var(Z (~x0) − Z(~x0)) − 2µ λi − 1 i=1 ∂φ ∂φ We get the minimum by setting of the partial derivative ,(i = 1, . . . , n) and ∂λ ∂µ zero. i These yield a linear Kriging equation system (KES) with n + 1 equations:
n X λjγ(~xi − ~xj) + µ = γ(~xi − ~x0) for i = 1, . . . , n j=1 n X λj = 1 j=1
In matrix form the KES is written as follows: γ( ~x − ~x ) γ( ~x − ~x ) . . . γ( ~x − ~x ) 1 λ γ( ~x − ~x ) 1 1 1 2 1 n 1 1 0 γ( ~x − ~x ) γ( ~x − ~x ) . . . γ( ~x − ~x ) 1 λ γ( ~x − ~x ) 2 1 2 2 2 n 2 2 0 . . . . . . . . .. . · . = . γ( ~xn − ~x ) γ( ~x − ~x ) . . . γ( ~x − ~x ) 1 λ γ( ~x − ~x ) 1 n 2 n n n n 0 1 1 ... 1 0 µ 1
In the case of point estimation γ(~xi − ~xi) = γ(0) = 0, i.e. the diagonal entries are zero. Since in the steady case the relationship of γ(~h) = C(0) − C(~h), γ(~h) in the KES can be substituted by the covariance C(~h). Thus the diagonal of the matrix emerges large elements. For numerical aspects this is preferable and therefore implemented in most programs.
2 The Kriging estimate variance σK for point estimation results from above equations is:
n 2 ∗ X σK = var(Z ( ~x0) − Z( ~x0)) = µ + λiγ(~xi − ~x0) (3.35) i=1 In a special case, in which no spatial dependence of the data exists, we get the weights 1 λ = . Then the Kriging estimator is the simple arithmetic means of the neighbouring i n samples. The following characteristics distinguish the Kriging estimator:
103 CHAPTER 3. INTERPOLATION METHODS
• The KES is only solvable if the determinant of the matrix (γij) 6= 0. Practically this means that a sample can not appear twice (i.e. with identical coordinates). • Kriging yields an accurate interpolator. • The KES depends only on γ(~h) or C(~h), however not on the values of the variable
Z in the points of sample xi. With identical data configuration the KES only need to be solved once. • Confidential limits of the estimation can be indicated under the help of the esti-
mation error σK
In practice a series of Kriging procedures were developed and applied, which regard more complex situations, e.g. intermittent variable, space time dependence etc.
104 3.4 Exercises
Exercises to 3: Interpolate by means of
• analytical power function • Lagrange interpolation formula • Newton interpolation formula • spline interpolation
−x2/2 1. Given some values of the normal distribution function y(x) = e√ : 2π
x 1, 00 1, 20 1, 40 1, 60 1, 80 2, 00
−x2/2 y(x) = e√ 0, 2420 0, 1942 0, 1497 0, 1109 0, 0790 0, 0540 2π
Find the value of y(x) for x = 1, 5.
√ 2. Interpolate the function y= x for the values x = 1, 03 and x = 1, 26 with help of the following basic values:
x 1, 00 1, 05 1, 10 1, 15 1, 20 1, 25 1, 30 √ y = x 1, 00000 1, 02470 1, 04881 1, 07238 1, 09544 1, 11803 1, 14017
3. Find a rational function with the degree as low as possible through the supporting points (1, −2), (2, 3) and (3, 1). How does this interpolation function change, if another supporting point (4, 4) is taken into account?
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 3. INTERPOLATION METHODS
106 Chapter 4
Optimisation problems
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 4.1 Analytical solution of extreme value problems
4.2 Iterative optimum search
4.3 Least squares method (MKQ method)
In water management practice the experimental process analysis (see section 11.1model classification) is used for the parameter determination of underground systems, e.g. k-, S- and T -values of soils, degradation rates, transportation parameters. The mathemat- ical model structure is specified by a theoretical process analysis. We try to transfer this model structure into easily solvable representations. The parameters can be deter- mined by the solving conditional equations or by a parameter approximation problem. So the task is, on the basis of structure knowledge or assuming such a
• the characteristics of the system which reflect reality as exact as necessary, and • eliminate the superposed influence of noise and errors as good as possible
To fulfil these demands we compare the output value with the original function of the inputs or an independent variable (time or place) with the model parameters. In the result a change of the parameters is to be made or the model change itself until the deviation reaches a minimum. The changes can be carried out according to a certain strategy (search algorithms, optimisation programs), statistically (random number gen- erator) or empirically. Also the visual comparison between the two diagrams (original and model output signal) is possible.
This task is also called parameter estimation. Particularly the procedure which is introduced here is classified as iterative estimation.
By the algorithmic model adjustment (see figure 4.1) we try to let the input vector and the manipulated vector y work in the process as well as in the model. With a 1 first parameter set, the starting parameters, the output vector of the model xM can be computed by first approximation. The deviation of this vector from the output vector x of the process (xi −xMi) is named as quality of the adjustment of the model. In water management applications the square evaluation will be carried out. The aim of the P 2 transformation of the parameters is minimizing the value Q = (xi − xMi) =⇒ Min.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber Figure 4.1: Iterative model adjustment
4.4 Search strategies
In these optimisation tasks it is very crucial in which processing time the minimum is found. The processing time Tv depends on the basic computing time Tn for the numeric analysis of the model and the number of iteration steps n. The number of the solution procedures is mainly determined by four influences:
• the number of the parameters which are looked for; it corresponds to the number of search directions and shrinks exponentially
• the formation of the quality mountains, i.e. the slope and the number of sub minimum,
• the search strategy, whereby the correct direction, the search step length and cognition of sub minimum are important
• the starting parameters, which crucially prevent unnecessary search steps.
The formation of the quality mountains and the search strategy can not be regarded independently. Generally it must be noted that there is not a ”best” search program, but for certain classes of quality mountains appropriate procedures are particularly suitable.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 4. OPTIMISATION PROBLEMS
There is a series of procedures to solve optimization problems. We divide these search methods according to their search strategy into non-gradient, gradient and coincidence search methods.
In table 4.1 the most important procedures for iterative processes of estimation (ac- cording to Wernstedt) are compiled.
Table 4.1: Iterative estimation methods(according to WERNSTEDT)
Solution equation for sˆ Method Examples sˆ i+1=ˆs + ∆ˆs i+1 i+1 ∆ˆs = a(i) • ∆ˆs POWELL gradient free > < if a(i) = 0 then Q(i) = Q(i − 1) ROSENBROCK methods < > ∆ˆs given vector FIBONACCI steepest gradient NEWTON i+1 ∆ˆs = −g(i + 1) • R(i + 1)∇Q(i) NEWTON gradient g(i) step length vector RAPHSON methods R(i) direction matrix MARQUART ∇Q(i) gradient for the i-th step LEVENBERG POWELL JONES i+1 ∆ˆs = b(i + 1) • R(i + 1)∆ˆs BROOKS random 0 f¨ur Q(i + 1) Q(i) b(i + 1) = = RASTRIGIN processes 1 f¨ur Q(i + 1) < Q(i) WHITE ∆ˆs random vector SCHWEFEL
110 4.4. Search strategies
4.4.1 Jones spiral method
The method of the non-linear regression, which is best suitable for pumping test evaluations, tries to adapt the model functions xM (ˆs) to the given values x(s) by choosing the parameterss ˆ. The deviations between xM (ˆs) and x(s) are the weight factors W . We assume the case that n samples (measured values) are available from the process and the model is determined by k independent parameters.
Starting from the initial valuess ˆ0 the goal function
n X 2 Q(ˆs) = (Wi(x(s)i − xM (ˆs)i) ) (4.1) i=1 is to be minimized concerning to the parameterss ˆi (i = 1, . . . , k). This is the method of least squares
An iteration consists of solving a linear equation system:
k X Ai,j · Tj = Gi (4.2) j=1 with dQi Gi = (i = 1, . . . , k) (4.3) dsˆi n X dxMi G = W · (x(s) − x (ˆs ) ) (4.4) j ii i M 0 i dsˆ i=1 j and n X dxMk dxMk Ai,j = Wkk · · (4.5) dsˆi dsˆj k=1
Tj represents the change of j-th parameter. We can get the linear equation system by developing the objective function to a Taylor series at the place s0. If we set the partial derivatives of this function equal to zero, we get an equation system as above. To check whether the linear approximation is adequate, the inequality must be fulfilled.
Q(ˆs0 + T ) < Q(ˆs0)
It is not always like this case in practical. According to Jones we find a better goal function value by a vector manipulation between the Taylor direction T and the negative gradient direction G (see figure 4.2). The minimum function value within the iteration is expected in the places ˆ0 + T . On the other hand the goal function
111 CHAPTER 4. OPTIMISATION PROBLEMS decreases in the direction of negative gradients. Thus it is sure that there is a better ∗ ∗ goal function value within the triangles ˆ0 − (ˆs0 + T ) − (ˆs0 + G ). G has the direction of G and the modulus of T . We act on the assumption of Taylor step in this search. An iteration is terminated, if a better goal function value were found. If the Taylor
Figure 4.2: Spiral algorithm according to JONES step is not successful, points will be calculated at the first spiral, given by:
∗ S = A1 · (µG + (1 − µ)T ) (4.6)
The different s-values are found by change of the µ value. µ begins with 0, 1 and is computed by the following relationship (Z ≥ 2):
Z · mn mn+1 = (4.7) 1 + (Z − 1) · mn
If µ > 0, 9 whe search will stop on the current spiral. If Q(ˆs0 + S) ≥ Q(ˆs0), the vector T is halved and it will be searched in the next spiral The larger Z is, the fewer points on a spiral are computed. If possible we interpolate either on the spiral or in Taylor direction. If no better value is found even along the last spiral, the search will be carried out in negative gradient direction with smaller increment.
112 Chapter 5
Ordinary differential equations
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Ordinary differential equations (ODE) are characterized by the fact that the searched function is dependent on a variable, while in a partial differential equation (PDE) more arguments and their appropriate derivatives appear as in the following examples:
dx + t2 · x = 2t2 Ordinary differential equation (ODE) dt ∂f ∂f + = 0 Partial differential equation (PDE) ∂x ∂y
Table 5 illustrates the characteristics:
Table 5.1: Representation of differential equations
ODE PDE one variable several variables Number (independent parameter) (independent parameters) Variables x, y, z or t x, y, z and/or t ~v = k · grad(h) ds Example v = ∂h ∂h ∂h dt ~v = k · ~i + ~j + ~k ∂x ∂y ∂z
1-D function with Vertex with arbitrary tangent function curved surface Illustration
In the following derivations and examples with the ordinary differential equations it is assumed that ”x” stands for the function value and ”t” for the argument. Of course all propositions can also be made with other arguments, and for the dependent func- tions arbitrary variable names can be used. The particular use of the letter x as a symbol of variable name appears in many mathematic teaching materials and in the signal theory (See Graeber: Lehrmaterial zur Automatisierung- stechnik bzw. Grundwassermesstechnik). In partial differential equations x, y and z are used as independent local coordinates.
114 The general form of an ODE is:
dx(t) d2x(t) dnx(t) F t, x(t), , , ··· , = g(t) (5.1) dt dt2 dtn
These differential equations are identified according to 5, if correspondent conditions are fulfilled.
Table 5.2: identification of differential equations
Description Condition Example dnx dn−1x dn−2x Order of the ODE n + + = 0 dtn dtn−1 dtn−2 dx inhomogeneous g(t) 6= 0 + t2 · x = 2t2 dt dx homogeneous g(t) = 0 + t2 · x = 0 dt dnx d2x explicit = 0 = 0 dtn dt2 dx dnx dx implicit F t, x, , = 0 + t2 · x − 2t2 = 0 dt dtn dt dx linear a (t) 6= a (x, t) t + t2x = 2t2 1 1 dt dx non-linear a (t) = a (x, t) x + t2x = 2t2 1 1 dt
115 5.1 Setting up equations
In the further sections solving differential equations is based on the mathematical description of natural processes. The derivatives of mathematical equations as trans- formation of natural processes is noted as modelling and as the transformation of mathematical model. The development of such mathematical models is the subject of chapter 11.2.1theoretical/experimental process analysis. The method described here is only how the mathematical models to be completed according to the physical or chemical basic laws and their effects. This way of the theoretical process analysis, also designated as mathematical modelling, is generally preferred by natural scientists.
In the theoretical process analysis the reciprocal effects of the process variables, state variables are formulated as mathematical model equations with their neighbourhoods. The most substantially reciprocal effects between the system and its neighbourhood are divided into causes and effects. The causes and the effects are called input and output variables. The description by means of the physical or chemical basic law is usually in formation of the balance equation, particularly the formation of the energy and mass balance equations.
Mostly the energy balance equations lead to the force equilibrium law and the flux laws. Generally we can speak of the transformation from potential to kinetic energy. Such energy transformations take place on so called flow resistances. A kinetic energy in form of material or mass flow results from different potential energies at the in- or outflow resistance (e.g. conduit, aquifer, electrical resistance), which act as driving force. We also say that flow resistances corresponding potential energy, also called as potential, is abolished, ”drops” (e.g. pressure difference, voltage drop). The mass balance equation assumes that mass is neither created nor destroyed within a regarded system (e.g. container, representative unit volume). The mass balance of a system can only be changed by outside sources or sinks. If dynamic systems are considered, the storage effect must be included likewise the mass balance. This means that all mass flows, which affect a system, must be zero in sum (junction law). Mathematically this circumstance can be also described by the divergence of a flow vector, which must be zero in this case (div ~v = 0).
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 5.1. Setting up equations
5.1.1 Examples of setting up differential equations:
1. Find the relationship between the flow rate V˙ , which flows from a pipe with free gradient, and the water level if the pipe is connected to a reservoir (see figure 5.1). According to the equilibrium of forces follows:
Figure 5.1: Diagram of the pressure ratios
patm + phydr1 = patm + phydr2
The hydrostatic water pressure amounts to:
phydr = h · ρ · g
The systems strives to reach a steady state
phydr = 0
h · ρ · g = 0
h = 0
This is the steady state. To study the non-steady-state a appropriate differential equation must be setted up. These we get by setting up the balance equation
V = h · A
dV d (h · A) = dt dt dh dA = A + h dt dt
117 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Since in the current reservoir the walls are perpendicular, e.g. A = const., the change of the areal A over the time is equal to zero. dV dh = A (5.2) dt dt Then again is the flow rate dependent form the difference of the pressure ratios and of the hydraulic resistance of the pipe. Along the pipe appears a decrease of the pressure which is among others proportional to the length of the pipe and inversely proportional to the profile of the pipe. This is collected as the hy- draulic resistance. Then the decrease of pressure is proportional to the hydraulic resistance.
(p + p ) − (p + p ) V˙ = atm hydr1 atm hydr2 Rhydr p − p = hydr1 hydr2 Rhydr h − 0 = ρ · g Rhydr ρ · g · h V˙ = Rhydr The discharge is equal to the balance equation 5.2:
dh ρ · g · h A = dt Rhydr A dh − h = 0 Rhydr · ρ · g dt The water level in the container is described by a homogeneous ordinary differ- ential equation of first order. To solve such equations see section 5.2.1 (ODE of first order), page 125.
Under the initial condition that the initial water level ht=0 = hAnf is, we get the solution:
t − h = hAnf · e T A with T = Rhydr · ρ · g
2. Find the differential equation to determinate the temperature of an industrial furnace with an filament heating (see figure 5.2). To simplify the theoretical
118 5.1. Setting up equations
Figure 5.2: Illustration of an filament heated industrial furnace process analysis we assume that the furnace temperature is not spatial dependent and the leakage heat flow is only controlled by the difference of the furnace temperature and the outside temperature. The overall mass m has the specific heat capacity c. The energy balance for the furnace is:
dϑ m · c i = P − Q (5.3) dt el v According to the Newton Law for heat transfer follows:
Qv = α · A(ϑi − ϑa)
Inserted in equation 5.3 we optain:
dϑ m · c i = P − α · A(ϑ − ϑ ) dt el i a m · c dϑ P i + ϑ = el + ϑ α · A dt i α · A a For the temperature profile inside the furnace we get an inhomogeneous ODE of first order. For the solution of such equations see section 5.2.1 ODE of first order, page 125 and section 5.2.1.2 solution of inhomogeneous ODE, page 128. Pel Here under the condition the right side is α·A + ϑa = const. and the initial
119 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
temperature inside the furnace is ϑi,t=0 = ϑi,Anf the solution is:
t Pel − ϑ = + ϑ 1 − e T + ϑ i α · A a i,Anf m · c mit T = α · A T is called the time constant and is a measure for the temperature alteration
speed ϑi inside the furnace.
120 5.1. Setting up equations
5.1.2 Exercises for setting up ODE’s:
Exercises to 5.1
1. The ground water level reincrease and therewith the refill of the former surface mines under natural conditions lasts to long. Therefore one tries to accelerate the refilling process.
Find a differential equation for the refilling process h(1,2)(t) without considering the aquifer and eventually groundwater recharge rates. In all cases the initial
condition ht=0(1,2) = 0 is given. a) constant volume flow rate (see figure 5.3) b) variable volume flow rate (see figure 5.4) c) linked storage cascade (see figure 5.5)
d) V1 + V2 = const
Figure 5.3: Refilling of an old surface mine with constant volume flow rate
2. For the given hydraulic diagram (see figure 5.6) with the corresponding block diagram set up the differential equation. Assume linear conditions and a homogeneous, isotropic aquifer with the param- −4 m eters k = 5 · 10 s , n0 = 0.2, zRmittel = 20m, l = 50m. 3. For the water level regulation of a feeder a float regulation is used (see figure 5.7). Set up the differential equation for the calculation of the water level H. The areal of the reservoir is A. The volume flow rate V˙ depends on the water level H. ˙ ˙ V = K · Vmax · (Hmax − H)
121 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Figure 5.4: Refilling of an old surface mine with variable volume flow rate
Figure 5.5: Linked storage cascade
4. For a steady experiment a soil sample with the volume VB is given into a beaker mg with water VH20. The pollutant concentration in the soil shall be CB = 125 l , in mg the water CH20t=0 = 0 l at the time t = 0. In a first approximation a pollutant transport out of the soil into the water will occur (see figure 5.8). Set up the differential equation for the concentration regime in the water. The
water solid ratio (W/F = 1) will be one. The diffusion resistance RDiff is given.
mqu = 125 mg (pollutant in the soil), VB = VH2O = 1l, RDiff
5. Two objects with different temperatures(T1 und T2) get connected at the time t = 0. The heat transfer is determined by the thermical conductivity Rtherm and
by the heat capacities(W1 und W2) of both objects. Draw a thermical block diagram and set up the corresponding differential equa-
tion for the variationZeichnen of the temperature T2(t).
122 5.1. Setting up equations
Figure 5.6: Representation of the groundwater level with a block diagram
Figure 5.7: Water level regulation of a feeder
123 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Figure 5.8: Pollutant transport out of the soil into the water
124 5.2 Analytical solution methods
5.2.1 First order ordinary differential equations
A solution for the following first order ODE should be found: dx a (t) + a (t)x = g(t) 1 dt 0 The following notations are often used:
dx dy =x ˙ = y0 dt dx First the inhomogeneous ODE will be transferred to a homogeneous ODE: dx a (t) h + a (t)x = 0 (5.4) 1 dt 0 h There are several methods to solve homogeneous ODE, and the separation of vari- ables and the ansatz method will be described now.
5.2.1.1 Solution of homogeneous differential equations
An arbitrary homogeneous first order ODE has the notation: dx a (t) h + a (t)x = 0 (5.5) 1 dt 0 h
For simplification the functions a0 and a1 are regarded as constants.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
• Separation of the variables
The separation of variables is a method to transform the ODE to an equation where on both sides is a total differential in one variable. Then both sides are integrable and the solution can be found.
dx a h + a x = 0 1 dt 0 h dxh a0 = − xh dt a1 dx a h = − 0 dt xh a1 R 1 a0 R dxh = − dt (5.6) xh a1 a0 a0 ln xh + C1 = − · t ln xh − ln C2 = − · t a1 a1 a0 a0 ln xh = − · t + C1 or ln xh = ln C2 − · t a1 a1
“ a0 ” “ a0 ” − a ·t+C1 − a ·t xh = e 1 xh = C2 · e 1
Both solutions are possible according to the logarithm laws (see section 1.1, page 6) and transferable to each other. E.g.:
C1 C2 = e or C1 = − ln C2 (5.7)
Since the integration constants C1 and C2 are still indefinite as well as the logarithms and exponential functions of the constants, both solutions are equal. Given some inital or final conditions the constants can be derived, e.g. C1 and C2 at t = 0 with the initial condition xh0 :
C1 = − ln xh0 C2 = xh0 (5.8)
For both variants the solution of the homogeneous first order ODE is:
a0 −( a ·t) xh = xh0 · e 1 (5.9)
Example for using the separation of variables:
Find the solution of the ODE dx + t2 · x = 0 with: x = 3 dt t=0
126 5.2. Analytical solution methods
According to the algorithm we try to separate the total differentials (dx and dt), each on one side of the equation.
dx = −t2x dt 1 dx = −t2dt x Z 1 Z dx = −t2dt x 1 ln x = − t3 + C 3 1
− 1 t3+C x = e 3 1
− 1 t3 x = C2e 3
Using the initial condition xt=0 = 3 in the general solution the constant will be C2 = 3, 1 − t3 so the solution is x = 3e 3 .
• Ansatz method
The basic idea of the ansatz method is to find a possible solution by using functions which solve similar problems. The most popular ansatz functions are combinations of exponential functions and/or sine/cosine functions as well as general power series. The advantage is that no longer difficult integrations have to be done, the ansatz functions have to be derivated, which is often more simply to realize:
dxh Differential equation: a1 · dt + a0xh = 0
λt Ansatz: xh = K · e
dxh λt Derivation of the ansatz: dt = λ · K · e
λt λt Inserting into the homogeneous ODE: a1 · λ · K · e + a0 · K · e = 0
λ = − a0 a1
(5.10)
127 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
The reciprocal value of the constant λ is the time. It is denoted as time T or τ. Then the solution is: a − 0 t t t a − − xh = K · e 1 = K · e T = K · e τ (5.11)
As in the separation of variables the constants are determined from the initial or final conditions. I.e. K can be determined at t = 0 with xh0 :
K = xh0 (5.12)
This yields the solution: a0 −( a t) xh = xh0 · e 1 (5.13) This is the same solution for the ODE like in the other method.
5.2.1.2 Solution of the inhomogeneous first order ODE
A general inhomogeneous first order ODE can be written in the form: dx a + a x = g(t) (5.14) 1 dt 0
Its general solution results from adding the homogeneous solution xh(t) (for the ODE with g(t) = 0) to a particular solution xp(t), i.e.
x(t) = xp(t) + xh(t) (5.15)
A method to get the particular solution xp(t) in this case is the variation of constants. After finding the homogeneous solution xh(t) in this method we assume the constant is dependent on the arguments.
128 5.2. Analytical solution methods
• Variation of constants
The solution will be carried out in four steps.
dx ODE: a + a x = g(t) 1 dt 0 dx First step homogeneous equation: a h + a x = 0 1 dt 0 h dx Second step separation of variables: = − a0 dt a1 xh a − 0 t a solution of the homogeneous equation: xh = Ce 1
a − 0 t a Third step variation of constants: xp = C(t)e 1
According to the rule of product differentiation:
a dxp dC − a0 t a0 − 0 t = · e a1 + C · − · e a1 (5.16) dt dt a1
dx Fourth step insertion into the original ODE: a p + a x = g(t) 1 dt 0 p
a a a dC − 0 t a − 0 t − 0 t a 0 a a a1 · · e 1 + C · − · e 1 + a0 · C · e 1 = g(t) dt a1 a a a dC − 0 t a − 0 t − 0 t a 0 a a a1 · · e 1 + a1 · C · − · e 1 + a0 · C · e 1 = g(t) dt a1 a dC − 0 t a · · e a1 = g(t) 1 dt a dC g(t) + 0 t = · e a1 (5.17) dt a1 This homogeneous first order ODE can be solved by the above methods, i.e. the separation of variables:
a g(t) + 0 t dC = · e a1 dt a1 a R R g(t) + 0 t dC = · e a1 dt (5.18) a1 a R g(t) + 0 t C(t) = · e a1 dt a1
129 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Thus the general solution of the differential equation is:
x(t) = xp(t) + xh(t) (5.19)
a Z a a − 0 t g(t) 0 t − 0 t x = Ce a1 + · e a1 dt e a1 (5.20) a1
The constants are determined by initial or final conditions, as in the method of variables separation, e.g. C at t = 0 with xh0 is:
Z g(t) a0t a C = x0 − · e 1 dt (5.21) a1 t=0
Then the solution is: Z a a Z a a g(t) + 0 t − 0 t g(t) + 0 t − 0 t a a a a x(t) = x0 − · e 1 dt · e 1 + · e 1 dt · e 1 (5.22) a1 t=0 a1 The solvability depends on the integrability of the pertubation function g(t).
Note: The product integration is just for some functions possible. Especialy if a function is the integral or the derivative of a function the product integration is used: Z Z u · dv = u · v − v · du (5.23)
That what we should keep in mind is not the final formula, but the way to the formula:
1. Transfer the equation into a homogeneous differential equation 2. Separation of variables 3. Variation of the constants or ansatz method 4. Insertion into the differential equation
Remarks on the variation of constants method:
1. It can be only used for linear differential equations. 2. The general solution is linearly dependent on the constants. 3. The general solution has a part without constants, which is a particular solution of the inhomogeneous differential equation. 4. It frequently occurs that a non-linear differential equation is transferred into a linear one by a simple substitution.
130 5.2. Analytical solution methods
Examples for solving inhomogeneous differential equations:
1. Find the solution of the ODE
tx˙ − x = t2 cos t
Solution:
ODE: tx˙ − x = t2 cos t
1. Find the homogeneous solution: tx˙ − x = 0
dx dt 2. Separation of variables: = x t
⇒ xh = Ct
3. Variation of constants: xp = C(t)t
⇒ x˙ = Ct˙ + C
4. Insertion into the ODE: t(Ct˙ + C) − Ct = t2 cos t
⇒ t2C˙ = t2 cos t
⇒ C˙ = cos t
⇒ C = sin t + C1
General Solution: x = (C1 + sin t)t = t sin t + C1t
131 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
2. Find the solution of the ODE:
2tx · x˙ + t2 − x2 + 1 = 0
Solution: By the substitution z = x2 andz ˙ = 2x · x˙ the ODE is transferd to the equation tz˙−z = −1−t2, Which is a linear ODE of the unknown function z. This equation can be solved with the ansatz method:
ODE: 2txx˙ + t2 − x2 + 1 = 0
Substitution: z = x2, z˙ = 2x · x˙ Substituted ODE: tz˙ − z = −1 − t2
1. Find the homogeneous solution: tz˙ − z = 0
dz dt 2. Separation of variables: = z t ⇒ zh = Ct
3. Variation of constants: zp = C(t)t ⇒ z˙ = Ct˙ + C
4. Insertion into the ODE: t(Ct˙ + C) − Ct = −1 − t2 ⇒ t2C˙ = −1 − t2 1 ⇒ C˙ = − − 1 t2 1 ⇒ C = − t + C t 1
2 General solution for z: z = 1 − t + C1t 2 2 Back substitution: x = 1 − t + C1t
p 2 General solution for x: x = 1 − t + C1t
132 5.2. Analytical solution methods
5.2.1.3 Exercises
Exercises to 5.2.1:
1. Find the solutions of the following ODE’s: a) y0 = (y − 3) cos x b) y0 = ex+y c) y0 sin x = y ln y 0 y2 d) 2xy + x = 0 e) y0 + y + ex = 0 0 y f) y + x = sin x dx g) + t2 · x = 2t2 dt h) y0 = −xy2 with y(0) = 2 dx i) + t2x = 0 with x(0) = 3 dt dx j) t − x = t2 cos t with x(π/2) = π dt
2. The differential equation for a system with simple memory effect is:
T x˙ a + xa = K xe
(xa output value, xe input value, T time constant, K proportional transfer func- tion)
How does the output value xa change in dependency on the time t if:
xe = c · t (c = const.)
3. Find the general and the special solution for the given initial conditions:
a) y0 = xy + 2x with y(0) = 2
b) y0 + x2y = x2 with y(2) = 1
4. For a given hydraulic scheme (see figure 5.9) with the associated block diagram applies the differential equation: dz h = R · C R + z F l dt R
133 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Linearised conditions and a homogeneous, isotropic aquifer with the following parameters are assumed: −4 m k = 5 · 10 s , n0 = 0.2, zRmiddle = zRm = 20m, l = 50m .
Figure 5.9: Schematic representation of the groundwater level
Compute the change of the groundwater level, if the river surface as a first ap- proximation changes as follows:
a) erratic (hF l = hFL · 1(t))
b) sinusoidal (hF l = hF Lm sin(ω · t) + hFL0, with ω = 2π/τ and τ = 7d)
5. For the concentration C given by sorption of pollutants to the soil matrix applies the ODE: ˙ T1C + C = K −1 mg T1 is a time constant and K a constant.( T1 = 1d , K = 100 l ) Given is the concentration C(t = 0) = 0. a) Solve the ODE with help of the analytical methods and compute the con- centration change for t = 1d. b) Draw the concentration change in a diagram dependent on the time.
6. The padding to the remainder holes of the former brown coal open pit caused by the rise of groundwater under natural conditions will last too long time. There-
fore external supply is introduced to the filling procedure (ht=0 = 0) for acceler- ation.(see figure 5.10) dh A = V˙ dt Zustr
Solve the ODE with help of the analytical methods.
134 5.2. Analytical solution methods
Figure 5.10: Refilling procedure of a remainder hole
7. The following ODE is given: dh + k · h = g dt
−1 −1 with ht=0 = 0, g = 0.015m · s and k = 0.01s Solve the ODE with help of the analytical methods.
135 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
5.2.2 Ordinary differential equations of higher order
A general solution of a differential equation with n-th order has n constants and repre- sents geometrically a n-parametric curve family. For determination of a single solution from this family we need n initial or boundary conditions.
Example of a2. order equation: y2y0 +y2 −1 = 0 is the movement of a particle. The general solution of these differential equation is (x − C)2 + y2 = 1. This equation stands for all circles of the radius r = 1 with the centre on the x axis. The initial condition y(0) = 1 yields the constant C = 0; then the single solution is x2 +y2 = 1. The particle moves around the circle with radius r = 1 whose centre is on the origin of the coordinate system.
Different types of higher order differential equation can be solved with different meth- ods:
5.2.2.1 OED of type a
d2y dy λ · + = 0 (5.24) dt2 dt
The degrees of higher order differential equation can be reduced by means of the fol- lowing substitution: dy z = (5.25) dt The derivative is: dz d2y = (5.26) dt dt2 These two equations are inserted into the ODE:
dz λ · + z = 0 (5.27) dt The solution of this first order homogeneous ODE is given in the previous section (see section 5.2.1.1, page 125) by: t − z = k1e λ (5.28) or t dz k1 − = − e λ (5.29) dt λ
136 5.2. Analytical solution methods
Due to the substitution condition we get:
dy z = dt (5.30) t dy − = k e λ dt 1 This differential equation can be solved again with the method of the separation of the variables:
t dy − = k e λ dt 1 t R R − dy = k1e λ dt (5.31)
t − y = −λ · k1 · e λ + k2
Since here two constants exists, two conditions equations must be found. For e−t functions the values t = 0 and t = ∞ are helpful, since the values for this exponential function in this case are simple (1 and 0):
k2 = y(∞) y(0) − y(∞) k = − 1 λ Then the solution is: t − y(t) = (y(0) − y(∞)) · e λ + y(∞) (5.32) Remark: This solution method can be applied also for the differential equation d2y dy a + b = g(t) (5.33) dt2 dt dy The substitution z = leads to a linear ODE, which can be solved by the method of dt the variation of the constants.
5.2.2.2 ODE of type b
d2y dy a + b + cy = 0 (5.34) dt2 dt This differential equation is to be solved according to the Ansatz method. Sense and purpose of the substitution method are to avoid complicated operations of the inte- gration of the differential equation and only carry out substantially simple operations
137 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS of differentiation. A popular substitution is used here, which looks promising from the experience. At the beginning all derivatives are developed, which appear in the differential equation.
Given the following Ansatz and derivatives:
y = C · eλt
dy λt (5.35) dt = C · λ · e
d2y 2 λt dt2 = C · λ · e
Inserted into the differential equation this yields:
aλ2 + bλ + c · C · eλt = 0 (5.36)
For t 6= −∞ we can divide by eλt with the result:
aλ2 + bλ + c = 0 (5.37)
The standard form of this quadratic equation is: b c λ2 + λ + = 0 or λ2 + dλ + f = 0 (5.38) a a b c whereby d = and f = . a a This equation is also designated as the characteristic equation of an ODE. The general solution of this characteristic equation is: r d d2 λ = − ± − f (5.39) 1,2 2 4 Depending upon the coefficients d and f there are three different cases:
d2 d2 1. The first case is, if − f > 0 =⇒ > f or b2 > 2 · c · a. 4 4 Then λ1 6= λ2 are real numbers, r d d2 λ = − ± − f (5.40) 1,2 2 4 The solution is then: λ1t λ2t y = C1e + C2e (5.41) The solution of this case is an asymptotic curve, which converges to a steady
state. This steady state is out of a real interval, if λ1 and λ2 are negetive values
138 5.2. Analytical solution methods
d2 d2 2. The second case is, if − f < 0 =⇒ < f, or b2 < 2 · c · a. 4 4 Then λ1,2 are complex numbers, since the radiant is negative.
d q d2 λ1,2 = − 2 ± 4 − f q d d2 λ1,2 = − 2 ± (−1) f − 4 (5.42) q d p d2 λ1,2 = − 2 ± (−1) · f − 4
d λ1,2 = − 2 ± j · β Inserting this solution of the characteristic equation into the substitution function follows:
(− d +j·β)·t (− d −j·β)·t y = C1 · e 2 + C2 · e 2 (5.43) − d ·t +j·β·t −j·β·t y = e 2 C1 · e + C2 · e
According to the calculation rules for exponential functions the sum of the ex- ponents can be decomposed into a product of two exponential functions. At the same time we can consider that the exponential functions with imaginary expo- nent can be transformed into trigonometric functions. Thus the solution is:
− d ·t y = e 2 (C1 cos βt + C2 sin βt) (5.44) This function represents the general form of the oscillation equation. For special
cases we get sinusoidal oscillations. This is the case, if C1 or C2 are identically equal to zero. With d = 0 we get an undamped oscillation, i.e. the amplitude is constant, if d < 0 a damped, with which the amplitude goes to zero, and if d > 0 a swing oscillation. d2 d2 3. The third case is, if − f = 0 =⇒ = f or b2 = 2 · c · a 4 4 In this case the radiant is zero and we get two identically solution: d b λ = λ = λ = − = − (5.45) 1 2 2 2a Thus the solution is no longer unique! We have two different functions, which solve the ODE:
λt y1 = Ce (5.46) λt λt y2 = C1te + C2e
139 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Examples of second order differential equations:
1. find the solution for the ODE: y00 − y = 0.
ODE : y00 − y = 0
Ansatz : y = eλt
: y0 = λeλt
: y00 = λ2eλt
Insertion into the ODE : λ2 − 1 eλt = 0
Characteristic equation : λ2 − 1 = 0
Solution of the characteristic equ. : λ1,2 = ±1
t −t general solution : y = C1e + C2e
2. find the solution for the ODE:y ¨ + y = 0
·· · ODE : y + y = 0y
Ansatz : y = eλt
· : y = λeλt
·· : y = λ2eλt
Insertion into the ODE : λ2 + 1 eλt = 0
Characteristic equation : λ2 + 1 = 0
Solution of the characteristic equ. : λ1,2 = ±j
: e±jt = cos t ± j sin t
general solution : y = C1 cos t + C2 sin t
140 5.2. Analytical solution methods
·· · 3. find the solution for the ODE: y + 2y + y = 0
·· ODE : y + 2y0 + y = 0
Ansatz : y = eλt
· : y = λeλt
·· : y = λ2eλt
Insertion into the ODE : (λ2 + 2λ + 1)eλt = 0
Characteristic equation : λ2 + 2λ + 1 = 0
Solution of the characteristic equ. : λ1,2 = −1
−t −t general solution : y = C1te + C2e
Remarks: This solution method can be likewise used for differential equation of higher order (n ≥ 3) with the appropriate ansatz of higher order algebraic equations.
Example of 3rd order ODE
··· · 1. Find the solution of the ODE: y + y = 0
··· · ODE : y + y = 0
Ansatz : y = eλt
· ·· ··· : y = λeλt , y = λ2eλt , y = λ3eλt
Insertion into the ODE : λ3 + λ eλt = 0
Characteristic equation : λ3 + λ = λ(λ2 + 1) = 0
Solution of the characteristic equation : λ1 = +j, λ2 = −j, λ3 = 0
General solution : y = C1 + C2 cos x + C3 sin x
141 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
5.2.2.3 ODE of type c
d2y 1 dy + + y = 0 (5.47) dt2 t dt These ODE we can solve again with the Ansatz method. The ansatz and the deriva- tives are:
2 3 4 y = 1 + a2t + a3t + a4t dy = 2a t + 3a t2 + 4a t3... dt 2 3 4 d2y = 2a + 2 · 3a t + 4 · 3a t2.. dt2 2 3 4 If these equations are inserted into the ODE and if the equation is arranged according to powers of t:
0 1 2 3 (1 + 2 · 2a2) · t + 3 · 3a3 · t + (a2 + 4 · 4a4) · t + (a3 + 5 · 5a5) · t (5.48) 2 n +...... + (an + (n + 2) · an+2) · t = 0
A solution of this equation, which applies to all t-values, is that the factors of the power series members are equal to zero. In this case:
a3 = a5 = a7 = ...... a2n+1..... = 0 1 a = − 2 22 a 1 a = − 2 = 4 42 2242 a 1 a = − 4 = − 6 62 224262
a2n−2 2n 1 2n 1 a2n = − = (−1) 2 = (−1) 2 2 2 2 2 2 n 2 (2n) 2 · 4 · 6 ···· (2n) Πk=1 (2k) If we set these coefficients into the solution, then we receive the solution of differential equation, which are called zero order Bessel function:
2 4 6 2n t t t 2n t y = 1 − + − + ..... + (−1) 2 + ..... = I (t) (5.49) 22 2242 224262 n 2 0 Πk=1 (2k)
142 5.2. Analytical solution methods
5.2.2.4 Exercises
Exercises to 5.2.2:
Find the solutions for the following ODE’s:
1. yy00 = y02 2. y00 − y0 = ex 00 0 3. y + 4y + a0y = 0 f¨ur a0 = 3, 4, 5
143 5.3 Integral transforms
5.3.1 Time- and Frequency domain
An integral transform is a method to solve differential equations over a detour. We distinguish two ranges in the transformations:
• the original or time domain and • the complex variable or frequency domain.
The integration according to the arguments within the original range is transformed into a multiplication in the complex variable domain. The difficult integration proce- dures can be bypassed. The relations between the ranges and their special character- istics are represented in the following scheme (see figure 5.11). The most well-known
Figure 5.11: Connection between original and complex variable domain transformations are the Laplace, the Laplace Carson, the Fourier, the Lau- rent and the Z transform. The theories of most these transformations can be gleaned in the multifaceted literature. Therefore here we only deal with the substantial criteria and disadvantages, which are against general application.
The following transformations are represented on the basis of time as argument, since these are most frequent applications of engineers, albeit the transformations are appli-
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 5.3. Integral transforms cable to all arguments, i.e. also to space variables.
The group of the integral transforms can be divided into the continuous and discrete transformations. The continuous integral transforms can be generally written:
2t Z F (f(t)) = k(t, f(t))f(t)dt (5.50)
1t Hereby k(t, f(t)) is designated as the kernel of the transform. To keep it simple only one argument (i.e. t) is regarded.
As special cases the relations specified in table 5.3 are known.
Table 5.3: Special cases in continuous integral transforms
transform lower integration upper integration kernel bound bound description k(t, f(t)) t1 t2
Laplace- e−pt 0(−∞) ∞ transform
Laplace- pe−pt 0(−∞) ∞ Carson- transform
Fourier- e−jωt 0(−∞) ∞ transform
The connection between the three mentioned integral transforms can be represented in the following descriptive form. According to the definition it will be characterized as complex frequency
p = σ + jω with s = real part and j = imaginary part (5.51)
If the real part of the complex frequency p converges to zero, the Laplace transform changes into the Fourier transform. It means that arbitrary (theoretical) time pro- cedure can be treated by means of the Laplace transform, and only sinusoidal one by means of the Fourier transform. The Laplace transform is particularly suit- able for application to deadbeat procedures, like e.g. bar signals. Nevertheless the Fourier transform has a large advantage as it is simpler in practice. Each periodic or
145 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS periodization function can be decomposed into a sum of sinusoidal oscillations by the Fourier series analysis. This decomposition of the excitation functions and overlay of the response functions are certainly only permitted in linear systems. The well-known complex computing methods of electro-technology for sinusoidal alternating current results from the Fourier transform. In the Fourier transformation the density of such oscillations, the so called spectrum will be analysed and treated by the rule of alternating current theory with only one sinusoidal oscillation, i.e. only one frequency. The discrete transformations are in contrast represented by a sum formula: X F (f(t)) = k(tn, f(tn)) (5.52) n
Here we get for the certain k(tn, f(tn)) some well-known special cases (see table 5.4).
Table 5.4: Special cases for discrete transformations
Transformation kernel Summation bounds description k(tn, f(tn)) discrete e−pn 0 ≤ n < ∞ Laplace transform
1 −∞ < n < ∞ Laurent transform zn 1 0 ≤ n < ∞ Z transform zn
We can also explain the connection between Laplace and Z transform in the following way. Replace in the Laplace integral for continuous functions ∞ Z F (p) = f(t)e−ptdt (5.53)
0
p = σ + jω the function f(t) by the function value series f(nT ), and the corresponding integral by an infinite sum and e−pt by e−pnT , then we get: ∞ X F (p) = T f(nT )e−npT (5.54) n=0 with ept = z follows: ∞ X −n FT (z) = T f(nT )z = T · F (z) (5.55) n=0
146 5.3. Integral transforms
5.3.2 Laplace Transform
5.3.2.1 Forward transformation
The following symbols are used in the Laplace transform:
L {f (t)} = F (p) Laplace transform of the function f(t)
L−1 {F (p)} = L−1 {L {f (t)}} Laplace inverse transform
= f (t)
The transform of time or original domain into the Laplace domain takes place by means of integral relationship stated above.
∞ Z F (p) = L {f (t)} = f(t)e−ptdt (5.56)
0
p = σ + jω
Examples for the application of the transform to functions:
Example 1:
f(t) = 0 (5.57)
∞ Z F (p) = L {0} = 0 · e−ptdt = 0
0 Example 2:
f(t) = 1 (5.58)
∞ Z F (p) = L {1} = 1 · e−ptdt
0 1 1 = − · e−pt∞ = − · (−1) p 0 p 1 = p
147 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Example 3:
f(t) = t (5.59)
∞ Z F (p) = L {t} = t · e−ptdt
0 e−pt ∞ = − 2 (pt + 1) p 0 1 = p2
There are tabular compositions of the Laplace transform for further basic functions (see table 5.5, page 151).
5.3.2.2 Important calculation rules
• Addition theorem
L {f1(t) + f2(t)} = L {f1(t)} + L {f2(t)} This addition theorem will be proved exemplarily for the other calculation rules. Hereby we use the definition of the Laplace transform, which is the integral of the product of the exponential function with the function to transform:
∞ Z −pt L {f1(t) + f2(t)} = e (f1(t) + f2(t))dt 0 ∞ Z −pt −pt = (e f1(t) + e f2(t))dt (5.60) 0 ∞ ∞ Z Z −pt −pt = e f1(t)dt + e f2(t)dt 0 0 According to the definition of the Laplace transform follows:
L {f1(t) + f2(t)} = L {f1(t)} + L {f2(t)}
General form of the addition theorem
L {λ1f1(t) + ... + λnfn(t)} = λ1F1(p) + ... + λnFn(p) (5.61)
148 5.3. Integral transforms
• Similarity theorem 1 p L {f(at)} = F (5.62) a a • Theorem of damping L e−atf(t) = F (p + a) (5.63)
• Shift theorem to right, L {f(t − a)} = e−apF (p) positive in the future a to left, L {f(t + a)} = eap F (p) − R e−ptf(t)dt 0 negative in the history Laplace transform with F (p) = L{f(t)} with out shift (5.64) • Differentiation The differentiation rules are the basis for the application of the Laplace trans- form to differential equations and their solutions.
L {f 0(t)} = pF (p) − f(0)
L {f 00(t)} = p2F (p) − f(0)p − f 0(0)
L f (n)(t) = pnF (p) − f(0)pn−1 − f 0(0)pn−2 − · · · − f (n−2)(0)p − f (n−1)(0)
(5.65)
• Integration t Z 1 L f(τ)dτ = F (p) (5.66) p 0 • Convolution theorem The convolution operation is important for transmission systems (see section 12.3): t Z L f1(t − τ)f2(τ)dτ = L {f1 (t)}· L {f2 (τ)} = F1(p) · F2(p) (5.67) 0
149 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
5.3.2.3 Inverse transformation
For the inverse transformation we use the so called L−1 transform.
L−1 {F (p)} = L−1 {L {f(t)}} = f(t) Laplace inverse transform (5.68)
In principle the following are possibilities can be used:
• Integral formula a+j∞ 1 Z f(t) = L {f(t)} eptdp (5.69) 2πj a−j∞
• Residue formula(decomposition into partial fractions)
∞ X f(t) = Res L {f(t)} ept (5.70) p=p n=1 n
Hereby pn are singularities in the left, complex half plane and (p − pn) yields the corresponding poles. • Series expansion
∞ X −σ0t f(t) = anBn (σ0t) with Bn(σot) = e Ln (2σ0t) (5.71) n=0
Because of this possibility the residue formula is always applicable and easy to handle for the technical problems.
150 5.3. Integral transforms
5.3.2.4 Correspondence table
Since these integrals are relatively complicated and different functions are very often repeated, arithmetic rules and correspondence tables are set up, from which the forward transformation and their inverse transformations are easy for reading (see table 5.5).
Table 5.5: Correspondence table Laplace transform
Nr. F(p) = L{f(t)} f(t) = L−1{F(p)}
1 0 0
1 2 p 1
1 tn−1 3 pn (n−1)!
1 tn−1 αt 4 (p−α)n (n−1)! e
1 eβt−eαt 5 (p−α)(p−β) β−α
p eβt−αeαt 6 (p−α)(p−β) β β−α
α 7 p2+α2 sin αt
α cos β+p sin β 8 p2+α2 sin(αt + β)
p 9 p2+α2 cos αt
p cos β−α sin β 10 p2+α2 cos(αt + β)
α 11 p2−α2 sinh αt
p 12 p2−α2 cosh αt
151 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Table 5.5: Correspondence table - continuation
Nr. F(p) = L{f(t)} f(t) = L−1{F(p)}
p2+2α2 2 13 p(p2+4α2) cos αt
2α2 2 14 p(p2+4α2) sin αt
p2−2α2 2 15 p(p2−4α2) sinh αt
2α2p 16 p4+4α4 sin αt sinh αt
α(p2+2α2) 17 p4+4α4 sin αt cosh αt
2αp 18 (p2+α2) t sin αt
p2−α2 19 (p2+α2)2 t cos αt
2αp 20 (p2−α2)2 t sinh αt
21 √1 √1 p πt
√1 √1 22 p p 2 t π
1 23 √ J0(αt)(Bessel function of order 0) p2+α2
1 24 √ I0 (αt) (modified Bessel function of order 0) p2−α2
α sin(αt) 25 arctan p t
2αp 2 26 arctan p2−α2+β2 t sin(αt) · cos(βt)
152 5.3. Integral transforms
5.3.3 Solution of differential equations by means of Laplace transform
5.3.3.1 Algorithm
This solution method consists of three sub steps:
• Application of the LAPLACE transformation to the differential equation (or the differential equation system) with consideration of initial conditions • Solution of the resulting algebraic equation (or equation system) with F (p) as unknown quantity • Inverse transformation of F (p) and determination of the searched function, i.e. solution function of the differential equation.
5.3.3.2 Examples
Example 1:
Find the solution of the ODEy ¨(t) + y(t) = 1 with the initial conditions y(0) = 1 and y˙(0) = 0:
• application of the Laplace transform:
ODE :y ¨(t) + y(t) = 1
Laplace transform : L {y¨(t) + y(t)} = L {1}
Addition theorem : L {y¨(t)} + L {y(t)} = L {1}
1 Transforming : p2F (p) − f(0)p − f 0(0) + F (p) = p 1 Initial conditions : p2F (p) − p + F (p) = p
153 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
• Solution of the resulting algebraic equation with F (p):
1 p2 + 1 F (p) = + p p p2 + 1 p2 + 1 F (p) = p 1 F (p) = p
• Inverse transformation and determination of y(t) by means of the correspondence table (see table 5.5, Page 151, row 2):
1 y(t) = L−1 {F (p)} = L−1 = 1 p
Example 2:
Find the solution of the ODEy ¨− 3y ˙ + 2y = 2e−t by applying the Laplace transform. The initial conditions are y(0) = 2 andy ˙(0) = −1.
• applications of the Laplace transform:
·· · ODE : y − 3y + 2y = 2e−t
n·· · o Laplce transform : L y − 3y + 2y = L 2e−t
n·· o n · o Addition theorem : L y (t) + L −3y + L {2y (t)} = L 2e−t
Transforming : p2F (p) − f (0) p − f 0 (0)
2 : − 3pF (p) + 3f (0) + 2F (p) = p + 1 2 Initial conditions : p2F (p) − 2p + 1 − 3pF (p) + 6 + 2F (p) = p + 1
• Solving the algebraic equation according to F (p):
2 p2 − 3p + 2 F (p) = + 2p − 7 p + 1 2 + (2p − 7) (p + 1) F (p) = (p2 − 3p + 2) (p + 1)
154 5.3. Integral transforms
• Inverse transform and determination of y(t): The inverse transform is achieved in this case via expansion into partial fractions. The zero positions of the denominator polynomial are searched and expressed as sum product. In the case under consideration the denominator is equal to zero, if:
p + 1 = 0 =⇒ p1 = −1 s −3 −32 p2 − 3p + 2 = 0 =⇒ p = − ± − 2 2,3 2 2
p2 = +1, p3 = +2
The equation for F (p) can be written: 2p2 − 5p − 5 F (p) = (p + 1)(p − 1)(p − 2) Then the partial fraction is: 2p2 − 5p − 5 A B C = + + (p + 1) (p − 1) (p − 2) p + 1 p − 1 p − 2
A common denominator (p + 1)(p − 1)(p − 2) should be used to determine the factors A, B and C: 2p2 − 5p − 5 A(p − 1)(p − 2) + B(p + 1)(p − 2) + C(p + 1)(p − 1) = (p + 1) (p − 1) (p − 2) (p + 1)(p − 1)(p − 2) This equation is fulfilled only when apart from the denominators the numerators are also same, i.e.:
2p2 − 5p − 5 = A(p − 1)(p − 2) + B(p + 1)(p − 2) + C(p + 1)(p − 1)
2p2 − 5p − 5 = (A + B + C)p2 + (−3A − B)p + (2A − 2B − C)
It must be an identity and valid for all p values. That means the coefficients of the power series of p are identical in each case. And we get follows:
p2 2 = A + B + C
p1 −5 = −3A − B
p0 −5 = 2A − 2B − C
155 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
This LES can be solved by the known methods, so the solution is:
1 7 A = ,B = 4,C = − 3 3 Then F (p) can be written in the following way:
1 1 1 7 1 F (p) = · + 4 · − · 3 p + 1 p − 1 3 p − 2
The inverse transform is to be read from the correspondence table (see table 5.5, page 151, line 4) It is:
1 tn−1 L−1 = eat for n = 1 (p − a)n (n − 1)! 1 L−1 = eat (p − a)1
for a1 = −1, a2 = 1, a3 = 2 gilt 1 7 L−1 {F (p)} = y (t) = · ea1t + 4 · ea2t − · ea3t 3 3 1 7 y (t) = e−t + 4et − e2t 3 3
Remark: The same procedure can also be used for the solution of linear ODE systems with constant coefficients.
5.3.3.3 Example of ODE systems
Find the solution of the ODE system:
·· · x = −y
·· · y = x with the initial conditions x(0) = 0,x ˙(0) = 1, y(0) = 0,y ˙(0) = 0.
• application of the Laplace transform: With F (p) = L {x (t)} and G (p) = L {y (t)} the application of the Laplace
156 5.3. Integral transforms
transform to the system yields (under consideration of the initial conditions):
p2F (p) − 1 = −pG (p)
p2G (p) = pF (p)
• Solving the linear equation according to F (p), G(p): According to the known rules or simple transformation, e.g. from the 2nd. equa- tion:
1 G (p) = F (p) p
Insert in the equation : p2F (p) − 1 = −F (p)
1 Resolve to F (p): F (p) = p2 + 1 1 1 G (p) = F (p) = p p (p2 + 1)
• Inverse transform and derivation of x(t), y(t) with help of the correspondence table (see table 5.5, page 151, row 3 and 7)
1 x (t) = L−1 {F (p)} = L−1 = sin t p2 + 1 1 1 p y (t) = L−1 {G (p)} = L−1 = L−1 − = 1 − cos t p (p2 + 1) p (p2 + 1)
157 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
5.3.3.4 Exercises to the integral transforms
Exercises to 5.3:
1. Solve the following ODE’s with the Laplace transform: y(x) = f(0) · e−x (a) y0(x) + y = 0 with f(0) = 1 → initial condition
y(0) = 1 (b) y00(t) − 3y0(t) + 2y(t) = 4 with y0(0) = 0
y(0) = 3 (c) y00(t) + 16y(t) = 32t with y0(0) = −2
y(0) = 3 (d) y00(t) + 4y0(t) + 4y(t) = 6e−2t with y0(0) = 8
y(0) = 1
(e) y000(t) + y0(t) = t + 1 with y0(0) = 1
y00(0) = 1
2. Solve the following equation systems with the Laplace transform: y0(t) + x(t) = 0 x(0) = 0 (a) with x0(t) + y(t) = 1 y(0) = 0
2x(t) − y(t) − y0(t) = 4(1 − e−t) x(0) = 0 (b) with 2x0(t) + y(t) = 2(1 + 3e−2t) y(0) = 0
3. For the following hydraulic scheme (see figure 5.12) and the corresponding block
158 5.3. Integral transforms
diagram applies the ODE dz h = R · C R + z F l dt R Linearized conditions and a homogeneous, isotropic aquifer are assumed and the following parameters are given: −4 m k = 5 · 10 s , n0 = 0, 2, zRmittel = 20m, l = 50m Compute the change of the water level with the Laplace transform, if the river
Figure 5.12: Schematic representation of the groundwater level
surface changes as a first approximation as follows:
a) erratic (hfl = hfl0 · 1(t))
b) sinusoidal (hfl = hfla sin(ω · t), with ω = 2π/τ and τ = 7 days). (only the difference to the steady state shall be taken into account!)
4. For the concentration C in sorption of pollutants at the sail matrix applies the ODE: ˙ T1C + C = K −1 Hereby T1 is a time constant and K a constant. T1 = 1d , K = 100 The concentration change at time t = 0 shall be C(0) = 0. a) Solve the ODE with the Laplace transform and compute the concentration change at time t = 1d. b) Sketch the principle time process of the concentration change.
5. The padding to the remainder holes of the former brown coal open pit caused by the rise of groundwater under natural conditions will last too long time. Therefore
external supply is introduced to the filling procedure (ht=0 = 0) for acceleration (see figure 5.13). The corresponding differential equation is: dh A = V˙ dt Zustr
159 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Transfer this differential equation by means of the Laplace transform in the image domain and solve this equation.
Figure 5.13: filling procedure of a remainder hole
6. Given the following ODE: dh + k · h = g dt −1 −1 ht=0 = 0, g = 0, 015m · s and k = 0, 01s Solve this ODE with the Laplace transform.
160 5.4 Methods for Numerical Integration
5.4.1 Integration
The numerical integration always yields the result of a certain integral between an upper and a lower limit. In a variable is inserted at the upper limit of the integral, the certain integral changes into a function, which is determined by the lower limit and the variable at upper border. The integral of a function, which can be also displayed as the area between the upper and lower limit and the abscissa, can be approximated by a simplified area computation. The numerical integration procedures differ with each other in the method of area computation. In most procedures it is assumed that total area between the upper and lower limit is divided into individual sub-area and the summation of these sub-areas yields the integral. The accuracy strongly depends on the method of sub-area creation and the quantization width of the abscissa. The approximation by summation of sub-areas will be worst with rectangle method and will be best with Predictor Corrector procedure or with higher order Runge Kutta procedure with the same quantization increment. The advantage of the trivial procedures exists in the simple, fast and stable computation of the sub-areas also in complicated, e.g. discontinuous function.
5.4.1.1 Rectangle rule
The rectangle rule as the simplest method assumes the creation of rectangles as sub- area (see figure 5.14). The area of the rectangles results from the multiplication of the function value (yn) on the left supporting place (xn) with the quantization increment ∆x = |xn − xn+1|. A substantial advantage of the rectangle method is no equidistant quantization neces- sary for the abscissa:
b m−1 Z X Fleft = y(x)dx ≈ (|xn+1 − xn|)yn (for m supintervals) (5.72) a n=0
We can use the function value on the right side instead of on the left. Then the area is computed by:
b m−1 Z X Fright = y(x)dx ≈ (|xn+1 − xn|)yn+1 (5.73) a n=0
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Figure 5.14: creation from rectangles to the numerical integration
The correct value of the integral has to be in between Fleft and Fright.
Examples for application of rectangle rule:
Z 2 1 1. We calculate the integral dx according to the table by using the rectangle 1 x rule (left and right) and compare the result with the analytical result:
Z 2 1 dx = ln 2 = 0, 693 1 x
x 1,0 1,2 1,4 1,6 1,8 2,0
1 1,000 0,833 0,714 0,625 0,556 0,500 x
The increment is regarded as constant, h = ∆x = 0, 2.
Flinks = 0, 2 (1, 000 + 0, 833 + 0, 714 + 0, 625 + 0, 556) = 0, 746
Frechts = 0, 2 (0, 833 + 0, 714 + 0, 625 + 0, 556 + 0, 500) = 0, 646
1 f (x) = is a monotonically increasing function, so F > F > F . The x links anal rechts
162 5.4. Methods for Numerical Integration
average of the two results is:
0, 746 + 0, 646 F = = 0, 696 average 2
Fanalytic = 0, 693
This value approaches to the actual analytical value. Remark:
The increment plays an important role for exact determination of the integral. The smaller it is, the more approaches the numerical value to the analytical, i.e. the numerical value converges. This is not only valid for rectangle rule, but for all numerical methods. Z 2 1 2. We calculate the integral dx with an increment h = 0.1 and the values in 1 x table and compare the results with example 1.
1 x Fleft Fright x
1, 0 1, 000 1, 000
1, 1 0, 909 0, 909 0, 909
1, 2 0, 833 0, 833 0, 833
1, 3 0, 769 0, 769 0, 769
1, 4 0, 714 0, 714 0, 714
1, 5 0, 667 0, 667 0, 667
1, 6 0, 625 0, 625 0, 625
1, 7 0, 588 0, 588 0, 588
1, 8 0, 556 0, 556 0, 556
1, 9 0, 526 0, 526 0, 526
2, 0 0, 500 0, 500
sum 7, 187 6, 669
163 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Flinks = 0, 1 · (7, 187) = 0, 719
Frechts = 0, 1 · (6, 669) = 0, 667
Fmittel = 0, 694
It is clear to notice that all the three values, which are calculated with increment
0.1 , lie closer to the analytical value Fanalytic = 0.693 than in example 1 with an increment of 0.2 .
5.4.1.2 Trapezoidal rule
The approximation by polynomials plays an important role in a multitude of proce- R b dures. The basic idea is that, if p(x) is an approximation for y(x), then a p(x)dx ≈ R b a y(x)dx. Different situations are dependent on the selected approximation.
With the trapezoidal rule the function between the supporting places xn and xn+1 is linear interpolated (see figure 5.15). Thus the wanted area is divided into trapezoid areas, which are calculated geometrically: a + b y + y I = h oder I = |x − x | n n−1 (5.74) 2 n n−1 2 By summation of the sub-areas:
b Z m X yn + yn+1 F = y (x) dx ≈ (|x − x |) (for m subintervals) n n+1 2 a n=0 (5.75) In the case of equidistant division the computation of ∆x can be simplified:
b m m−1 Z X X ∆x F = y (x) dx ≈ F = ∆x · (y ) + (y + y ) (5.76) n n+1 2 o m a n=0 n=0 In the trapezoidal rule we have another simple possibility to take in account irregular increment, i.e. not equidistant quantization.
164 5.4. Methods for Numerical Integration
Figure 5.15: Numerical integration by means of trapezoidal rule
5.4.1.3 Simpson’ rule
The Simpson’ rule is:
x2k Z h F = f(x)dx = (y + 4y + 2y + 4y + ... + 2y + 4y + y ) (5.77) 3 0 1 2 3 2k−2 2k−1 2k x0
It is likewise a compound formula as parabolic arcs are used instead of y(x).
Remark:
• the supporting places must be equidistant (constant increment h).
• the number of supporting places xn must be odd (n = 0 ... 2k).
5.4.1.4 Newton formula
In this method the Newton interpolation function (also see 3.1.3, page 81) will be integrated with following results:
165 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
number of interpolation Equation supporting method points
R x1 h p (x) dx = (y0 + y1) 2 linear x0 2
R x2 h p (x) dx = (y0 + 4y1 + y2) 3 quadratic x0 3
R x3 3h p (x) dx = (y0 + 3y1 + 3y2 + y3) 4 cubic x0 8
5.4.1.5 Examples for application of numerical integration
R π/2 We use trapezoidal rule and Simpson rule in order to determine the integral 0 sin x· dx from the following table. Compare the results with the analytical value Ianalytic = 1.
π 2π 3π 4π 5π π x 0 12 12 12 12 12 2
sin x 0 0, 259 0, 500 0, 707 0, 866 0, 966 1, 000
Trapezoidal rule π I = (0 + 0, 259 + 0, 5 + 0, 707 + 0, 866 + 0, 966 + 0, 5) = 0, 994 tr 12
Simpson rule π I = (0 + 4 · 0, 259 + 2 · 0, 5 + 4 · 0, 707 + 2 · 0, 866 + 4 · 0, 966 + 1) = 1, 000 s 3 · 12 Obviously the adjustment of quadratic polynomials yields one up to three decimal places exact result.
166 5.4. Methods for Numerical Integration
Newton interpolation function
1. linear (0 + 0, 259)+ (0, 259 + 0, 500)+ (0, 500 + 0, 707)+ π INl = 2 · 12 (0, 707 + 0, 866)+ (0, 866 + 0, 966)+ (0, 966 + 1, 000)
= 0, 994
2. quadratic (0 + 4 ∗ 0, 259 + 0, 500) + π INq = (0, 500 + 4 ∗ 0, 707 + 0, 866) + 3 · 12 (0, 866 + 4 ∗ 0, 966 + 1, 000)
= 1, 00003
3. cubic (0 + 3 ∗ 0, 259 + 3 ∗ 0, 500 + 0, 707) + 3 · π INk = 8 · 12 (0, 707 + 3 ∗ 0, 866 + 3 ∗ 0, 966 + 1, 000)
= 1, 00006
Apparently the adjustment of quadratic polynomials yields one up to three decimal places exact result.
Remark:
The accuracy of numerical methods must be always relating to the significant number of computed and represented places. If we solve e.g. the same problem with seven
167 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS digits of significant number, then the Simpson’s rule yields I = 1.000003 , which does not match the analytical value of Ianalytic = 1 .
168 5.4. Methods for Numerical Integration
5.4.1.6 Exercises for the application of the numerical integration
Exercises to 5.4.1:
1. Compute the given integral by using the trapezoidal rule with increment h = 0.1 and h = 0.01. 1 Z dx I = 1 + x 0 2. Calculate the following integrals. Use at least two numerical methods and two different increments and then compare the results:
1 2 x Z 2 Z e a) e(−x) dx b) dx x −1 1
3. Calculate the integral 1,3 Z √ I = xdx
1 with the rectangular rule, trapezoidal rule and the three Newton formulae and compare the results. 4. Calculate the integral 10 Z 1 I = dx x 1 by approximation. Choose h = 1. Use the Simpson’s rule for the interval [1, 9] and the trapezoidal rule for the interval [9, 10].
5. A measurement series of the specific heat c of Al2O3 depending on the tempera- ture T is listed in the table. Determine the amount of heat
1000 Z Q = c(T) dT
−200
o which must be supplied to a gram Al2O3 , in order to warm it up from −200 C to 1000oC. The integration is to be accomplished numerically according to
169 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
a) trapezoidal rule b) Simpson’s rule with the increment h = 200oC.
T [oC] −260 −200 −100 0 100 200 300 400 600 800 1000
c c[ g·K ] 0 0, 04 0, 012 0, 18 0, 22 0, 24 0, 25 0, 26 0, 27 0, 275 0, 28
6. In a pumping test the groundwater level were measured (see figure 5.16). Calculate the water deficit (volume) of the sinking funnel, if the aquifer is of following characteristic values: −1 −1 hn = 16m, M = 10m, k = 0, 001m · s , S0 = 0, 0001m , n0 = 0, 20 Use the methods of numerical integration.
Figure 5.16: groundwater level as a function of the radius
170 5.4. Methods for Numerical Integration
5.4.2 Numerical Solutions of Differential Equations
While the solutions of definite integrals were in the foreground in the former sections, the Euler’s method, the Runge Kutta method and the Predictor Corrector method are used now to solve ordinary differential equations. In contrast to analytical methods numerical methods always assumes boundary conditions, i.e. the initial and boundary conditions. Particularly in 1st. order differential equation the initial values are supposed, which leads to the concept of initial value problems.
The 1 st. order differential equation dy = f(x, y) (5.78) dx with the initial condition at the point x = a and the function value y(x=a) = ya leads by integration in the range x = a bis x = b to:
b b Z dy Z dx = f(x, y)dx (5.79) dx a a b Z b [y]a = f(x, y)dx a b Z yb = ya + f(x, y)dx a
Thus we obtain the wanted function value yb in the place x = b from the function value at the beginning point plus the definite integral of function y (see figure 5.17). The problem now is the function y is unknown. For this reason approximation solutions must be again used for the integral as described in the former section. The following methods differ from the application of approximation methods.
These methods can be improved if this approximation is only applied in sections and then iteratively expanded to the whole integration interval (see figure 5.18).
171 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Figure 5.17: Computation of the function value y(b) from the initial value y(a)
x Z n y1 ≈ ya + f(x, y)dx (5.80) a x Zn+1 yn+1 ≈ yn + f(x, y)dx (5.81)
xn b Z yb ≈ ym + f(x, y)dx (5.82)
xm
We recognize that the writing ways of integration limits could be synonymous: lower limit: x = a or x = xn upper limit: x = b or x = xn+1 The subscript is usually used for the intermediate intervals and the advantage is that it can be easily converted into programming language.
5.4.2.1 Euler method
The Euler method is the simplest method and actually the integral is approximately determined by means of the rectangle formula. The greater the distance between a and b, i.e. the increment h or ∆x, the worse the approximation is.
172 5.4. Methods for Numerical Integration
Figure 5.18: iterative solution of the differential equation
h = b − a
∆xn = xn+1 − xn
Thus we get the following solution
b Z yb = ya + f(x, y)dx ≈ ya + f(a, ya) · (b − a) (5.83) a
≈ ya + f(a, ya) · h (5.84) x Zn+1 yn+1 = yn + f(x, y)dx ≈ yn + f(xn, yn) · (xn+1 − xn) (5.85)
xn
≈ yn + f(xn, yn) · ∆xn (5.86)
In this method it is possible to work with different increments, i.e. with a not equidis- tant division. It is also suitable for an automatic increment control, because the error, which results from the approximation, is dependent on the slope of the function y and
173 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS on the increment (b − a) = ∆x. Inserting it into equation above, we get:
b Z dy yb = ya + f(x, y)dx ≈ ya + · (b − a) (5.87) dx x=a a
Example for application of the Euler’s method (also see figure 5.19):
An approximation solution for the differential equation y0 = xy1/3 with y(1) = 1 is looked for.
The Euler formula can be also written in the form:
0 yn+1 = yn + h · yn
1/3 yn+1 = yn + ∆xn · xn · yn
1/3 yn+1 = yn + (xn+1 − xn) · xn · yn
2 The stop error O(h ), which is produced in the interval xn to xn+1, is rather larger in the Euler method (i.e. proportinal to h2), so that for a high accuracy very small increments h are necessary. E.g. for h = 0, 01:
y1 ≈ 1 + (0, 01) 1 = 1, 0100
y2 ≈ 1, 0100 + (0, 01) (1, 01) (1, 0033) ≈ 1, 0201
y3 ≈ 1, 0201 + (0, 01) (1, 02) (1, 0067) ≈ 1, 0304
The stop error in each interval is about 0.00007. The fourth decimal place should be considered with caution. If we want a higher accuracy, a smaller increment h is necessary. The analytical values are
y1 = 1, 01007
y2 = 1, 02027
y3 = 1, 03060
174 5.4. Methods for Numerical Integration
Figure 5.19: result development with the Euler method
5.4.2.2 Runge Kutta method
The Runge Kutta method assumes the same approach, the approximation of the integral by area calculation, as Euler method. The difference lies in the degree of the approximation function for area calculation, which is linear in the Euler method. Here with the Runge Kutta method a higher order polynomial is used according to Taylor series expansion.
2 3 4 0 00 h 000 h 0000 h y = y + y · h + y · + y · + y · + ... (5.88) b a a a 2! a 3! a 4!
with h = |b − a|
Depending upon degrees of the considered derivative in the Taylor series we distin- guish Runge Kutta method in different n-th orders.
In the following subscript way of writing is used, as the entire integration interval (a to b) is mostly decomposed into subintervals and additionally this way will be converted in programming technique in practice.
yn+1 = yn + kn (5.89)
175 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
The Runge Kutta methods differ in the way of kn determination. In this classification the Euler method can be arranged:
kn = ∆xn · f (xn, yn)
∆xn = |xn+1 − xn|
The simplest procedure, which differs from Euler method in respect of accuracy, is the 2nd order Runge Kutta method:
yn+1 = yn + k2 (5.90)
with k1 = ∆xn · f (xn, yn) (5.91) 1 1 k = ∆x · f x + h, y + k (5.92) 2 n n 2 n 2 1
∆xn = |xn+1 − xn| The error of this method grows proportionally with h power 3 (0(h3)) and is one power better than Euler method (0(h2)). The 4th order Runge Kutta method is frequently used, which is a good compromise between accuracy and numerical expenditure.
For the general form:
yb = ya + k (5.93) we write: 1 k = · (k + 2 k + 2 k + k ) (5.94) 6 1 2 3 4
k1 = h · f (a, ya) h k k = h · f a + , y + 1 2 2 a 2 h k k = h · f a + , y + 2 3 2 a 2
k4 = h · f (a + h, ya + k3)
mit h = |b − a|
176 5.4. Methods for Numerical Integration
The error of this procedure is of 5th order (0(h5)). Here also for improvement of the accuracy we can divide the total interval of a to b into subintervals xn with yn and iteratively solve yb. Since we cannot change the increment within the subintervals, it is possible to control increment as a function of gradients: 1 y = y + (k + 2k + 2k + k ) (5.95) 2 a 6 1,1 2,1 3,1 4,1 . .
1 y = y + (k + 2k + 2k + k ) n+1 n 6 1,n 2,n 3,n 4,n . .
1 y = y + (k + 2k + 2k + k ) b b−∆xn 6 1,b−∆xn 2,b−∆xn 3,b−∆xn 4,b−∆xn If n = 1, it yields
∆xn = |xn+1 − xn|
k1,n = ∆xn · f (xn, yn) (5.96) ∆x k k = ∆x · f x + n , y + 1 2,n n n 2 n 2 ∆x k k = ∆x · f x + n , y + 2 3,n n n 2 n 2
k4,n = ∆xn · f (xn + ∆xn, yn + k3)
Example for application of the Runge Kutta method: An approximation solution for the differential equation y0 = xy1/3 with y (1) = 1 is looked for. With x0 = 1 and h = 0, 1 we get a 4th order Runge Kutta formula (see equation 5.96):
k1 = 0, 1 · f (1, 1) = 0, 1
k2 = 0, 1 · f (1, 05; 1, 05) = 0, 10672
k3 = 0, 1 · f (1, 05; 1, 05336) = 0, 10684
k4 = 0, 1 · f (1, 1; 1, 10684) = 0, 11378
177 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
We calculate: 1 y = 1 + (0, 1 + 0, 21344 + 0, 21368 + 0, 11378) = 1, 10682 1 6 The analytical value is y = 1, 10326. The corresponding value with the Euler mehtod is y = 1, 01, i.e. the Runge Kutta method yields better results. However the increment must be likewise smaller selected if a higher accuracy is demanded.
5.4.2.3 Predictor Corrector Method
The Predictor Corrector method is a two-step procedure. In the first step an auxiliary ∗ value yb is computed and then yb. Thus an increased numerical expenditure develops, but the accuracy rises substantially compared to one-step method. Besides Runge Kutta method Predictor Corrector method represents the most substantial integration procedure. The predictor step in the simplest form, like in the Euler method, a rectangle formula for the computation of the integral is used:
b Z ∗ yb = ya + f(x, y)dx ≈ ya + f(a, ya) · (b − a) (5.97) a We can also write:
b Z ∗ dy 0 yb = ya + f(x, y)dx ≈ ya + · (b − a) = ya + ya(b − a) (5.98) dx x=a a
The difference is that in the predictor step the wanted value yb will not be computed, ∗ but as a first approximation of the value yb is regarded. As the second step, the corrector step, the integral will be calculated by the trapezoid formula, while the value ∗ yb is used as upper value in the trapezoid formula: (b − a) y = y + · (f(a, y ) + f(b, y∗)) (5.99) b a 2 a b Similar to the predictor step here the basis of output differential equation can be formulated: (b − a) 0 0 y = y + · y + y∗ (5.100) b a 2 a b
178 5.4. Methods for Numerical Integration
Also this procedure can be expanded to n subintervals of the range a to b and computed iteratively. Then the predictor and the corrector step for the n + 1th interval is:
∗ 0 yn+1 = yn + ∆x · (f(xn, yn)) = yn + ∆x · yn
∆x ∆x 0 0 y = y + · f(x , y ) + f(x , y∗ ) = y + · y + y∗ (5.101) n+1 n 2 n n n+1 n+1 n 2 n n+1
with ∆x = |xn+1 − xn|
A series of procedures were developed. The above procedure possesses the disadvan- tage that there is a relative large residue, i.e. a residual value error, which grows proportionally with ∆x2 (O(∆x2)). The advantage lies in a relatively simple incre- ∗ ment control, only the value f(xn+1; yn+1) or the derivative of yn+1 is to be computed. A very widespread Predictor Corrector method is the Adam Bashforth Moulton scheme. This method is very stable. In contrast to the simple Predictor Corrector method several supporting places of integration steps are needed here. Thus the ap- proximation area will not be made a rectangle any longer, but poly-line is used for boundary. The frequently used 3rd order Adam Bashforth:
For the predictor step:
0 ∆x y∗ = y + (5 f(x , y ) − 16f(x , y ) + 23 f(x , y )) (5.102) n+1 n 12 n−2 n−2 n−1 n−1 n n
∆x 0 0 0 = y + 5y − 16y + 23y n 12 n−2 n−1 n
Then the corrector step is:
∆x 0 y = y + −f(x , y ) + 8f(x , y ) + 5f(x , y∗ ) (5.103) n+1 n 12 n−1 n−1 n n n+1 n+1
∆x 0 0 0 = y + −y + 8 y + 5y∗ n 12 n−1 n n+1 This method yields a residue, grows proportionally with 4th power of ∆x (∼ ∆x4), i.e. O(∆x4). The disadvantage is that the intervals n−1 and n−2 must be calculated again if changing increment for the interval n. So it is necessary to calculate the intervals n, n − 1 and n − 2 with the same increment ∆x. This can lead to an increased numerical expenditure when strong gradient oscillation.
179 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
Example for application of the Predictor Corrector method: An approxima- 1 tion solution for the differential equation y0 = xy 3 with y(1) = 1 is looked for. The accuracy ε ≤ 10−5. For each forward step the simple Euler formula is used as a predictor. It presupposes a first estimation of yn+1. Here x0 = 1 and h = 0.05.
y (1, 05) ≈ 1 + 0, 05 · 1 = 1, 05 Then the differential equation is:
1 y0 (1, 05) = 1, 05 · 1, 05 3 = 1, 0661
The Euler formula will be modified for Corrector (according to trapezoidal rule): 1 y = y + h y0 + y0 n+1 n 2 n n+1 It yields: y (1, 05) ≈ 1 + 0, 025 (1 + 1, 0661) = 1, 05165 With this new value of the differential equation y0(1.05) will be corrected to 1.0678; afterwards the corrector is used again and yields the result:
y (1, 05) ≈ 1 + 0, 025 (1 + 1, 0678) = 1, 0517
Further calculations confirm these four decimal places, so that the desired accuracy is reached. It is noticed that the same accuracy can be achieved with increment h = 0.01 in simple Euler formula.
Generally we iterate until it converges if it exists. Afterwards we can continue with the next interval in order to start again with a simple predictor formula.
180 5.4. Methods for Numerical Integration
5.4.2.4 Exercises to numerical solutions of ODE
Exercises to 5.4.2:
1. Solve the following differential equation with numerical methods by using few intervals to x = 1, i.e. 0, 5; 0, 2 and 0, 1.
y0 = −xy2 mit y(0) = 2
a) Use the simple Euler method. Converge the results to the analytical value y(1) = 1? b) Use the Runge Kutta method of 4th order and a predictor corrector method and compare the results
2. Solve the differential equation dx + t2x = 0 mit x(0) = 3 dt at the time t = 3 with the Euler method. Use the increments ∆t = 0, 1; 0, 05 and 0, 01.
3. Solve the differential equation dx π t − x = t2 cos t mit x( ) = π dt 2 for the time t = 2π by the Euler method. Use the increments ∆t = 0, 1π and ∆t = 0, 05π.
4. The following differential equation applies to the concentration C [mg] in sorption of pollutants at the soil matrix:
˙ T1C + C = K
−1 T1 is a time constant and K a constant. T1 = 1d , K = 100 The concentration at t = 0 is C(0) = 0. a) Solve the ODE with the Euler method (h = 0, 1d) and calculate the con- centration for the time t = 1d. b) Outline the time process of concentration change.
181 CHAPTER 5. ORDINARY DIFFERENTIAL EQUATIONS
5. The padding to the remainder holes of the former brown coal open pit caused by the rise of groundwater under natural conditions will last too long time. Therefore
external supply is introduced to the filling procedure (ht=0 = 0) for acceleration (see figure 5.20). Set up the differential equation for the filling up procedure h(t), without consid- eration of the aquifer and contingent groundwater formation rate. Describe the solution by means of numerical methods.
Figure 5.20: filling procedure of a remainder hole
6. Following ODE is given: dh + k · h = g dt −1 −1 with ht=0 = 0, g = 0, 015m · s und k = 0, 01s Solve the ODE with numerical methods.
182 Part II
Partial differential equations of underground processes
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management
Chapter 6
Overview
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 6. OVERVIEW
There are no generally valid solutions for partial differential equations (PDE), which are characterised by consideration of functional dependency on several arguments. Be- cause of this reason the following selected PDE’s, which plays an important role in hydrogeology, will be discussed. The groundwater flow equation and the convection dispersion equation predominantly stand in the foreground. Different mathematical methods, such as analytical or numerical solutions will be introduced according to the complexity of the equation, the number of independent parameters and the con- sideration of inhomogeneity, anisotropy, as well as nonlinearity.
The physical processes are divided into so called quantity flow and material transporta- tion. The application of energy and mass conservation law lead to following coupled partial differential equation, for material process only transportation is displayed:
• the dynamic basic equation of flow processes:
~v = −k grad h (6.1)
• the balance equation of flow processes: δh div ~v = S − w (6.2) 0 δt • the boundary conditions of flow processes:
initial and boundary conditions of 1., 2. and 3. type
This equation system must be set up in the modelling of material and energy trans- portation for each substance contained in water or in immiscible material processes for each group of materials and for each phase in the multi-phase system (liquid (water, oils), solid (stone matrix), gaseous (air, gases)). Balance equations must be defined for each subsystem, which consist of the following parts:
• the dynamic basic equation for transport processes:
~ transport by dispersion: ~g1 = D grad P (6.3)
transport by convection: ~g2 = ~vP (6.4)
• the balance equations for the transport processes: δP div ~g = (n + α) − w (6.5) 0 δt g
186 • the boundary conditions for the transport processes:
initial and boundary conditions of 1., 2. and 3. type
The connection of the equations within each subsystem is given by the exchange terms. In the subsystems it is made via internal reaction terms. The following balance equation applies to a balance area, which is also designated as representative elementary volume (REV):
transport = internal reaction + storage + exchange + external sources
The chemical reaction equations (material change processes) and biological growth processes can be added to these basic equations. The mathematical model thereby consists of a system of ordinary or partial differential equations and algebraic equations, whose coefficients are usually a function of place, time and potential. Thus the system is nonlinear and local and time variant. The processes in the soil and groundwater range are characterized by a high complexity, a bad condition, a large range of time constants and a high uncertainty of initial parameters.
The basic equations can be summarized in each case for the flow and the material process, and yield two nonlinear partial differential equations of second order:
• the conduction equation for the flow process (parabolic PDE):
δh div (k grad h) = S − w (6.6) (x,y,z) 0 δt
• the convection diffusion equation for the transport process (hyperbolic PDE): δP div D~ grad P − ~vP = (n + α) − w (6.7) 0 δt g Depending upon the relationsship of the dispersion D~ grad P to the convection (~vP ) of the transport process the property of these PDE’s varies among predominantly parabolic, hyperbolic or first order PDE. If the convection converges to zero (~vP −→ 0), the PDE is of parabolic type, for D~ grad P −→ 0 we get a first order PDE.
The connection of the volume and material flow is characterised by the critical values of hte water property (temperatur T , material concentration C, kinematical viscosity
187 ν and density ρ) and by the critical values of the underground flow processes (filter δp velocity ~v, storage contents change C · as well as internal flow sources and sinks w). δt This complex form of the system description is often approximated by simplified forms, in which one or more processes are neglected or the dependence on one or other argu- ments is ignored. A fundamental simplification results from the decoupled approach of flow and transportation processes and chemical kinetics. Substantial simplification can be also achieved by reduction of the multidimensional area to a local coordinate or time variable.
This procedure will be demonstrated exemplary by often used and significant engineer- ing examples in the following chapters.
6.1 One dimensional flow equation
Under the prerequisite of simplified flow conditions, the view in the cylindrical coor- dinate space as well as the integration over the height of z by a transformation, e.g. the so called Girinskij potential Φ, we get the following equations for the rotational symmetric flow field:
d2Φ 1 dΦ w steady flow: + + = 0 (6.8) dr2 r dr k d2Z 1 dZ Z “leakier” flow (Leakyaquifer): + − = 0 (6.9) dr2 r dr B2 ∂2Z 1 ∂Z Z ∂Z non steady flow: + − = a (6.10) ∂r2 r ∂r B2 ∂t
These equations and other analytic solutions (see section 8.1 Theis’ well equation, page 206) were found by Theis. The importance of these equation is that they supply useful results with a local character (approx. 200 m expansion, e.g. foundation pit) for many engineering investigations, which fulfils the hydraulic geological conditions. In addition they form the basic procedures for the indirect parameter investigation, e.g. for the so called pumping test evaluations (see section 14.1 pumping test evaluation, page 386).
Similar conditions occur at parallel ditch flow, and the PDE has the following form:
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber ∂2Z w ∂Z parallel ditch flow: − = a (6.11) ∂x2 k ∂t
The one dimensional processes are also meaningful for the investigation of transporta- tion procedures in stream tube connected with pollution.
6.2 Horizontal plane groundwater flow equation
The horizontal plane groundwater flow equation represents a fundamental principle for the flow processes apart from the well equation (see equations 6.8 to 6.10). A simplified aquifer characterized by means of the Dupuit assumption (see section 7.1 Dupuit assumption and balance equation, page 194), and an integral transform for the description of the profile permeability, the transmissibility T :
∂z div (T grad z ) = S R − w (6.12) (x,y) R ∂t N
This equation can be built separately for each groundwater story and the coupling between the aquifer can be achieved by hydraulic windows. This equation forms the basics of most hydraulic geological region models, also for the mining districts of the Central Germany and the Lusatia area.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 6.3 One dimensional material transfer
For the material transfer the modeling of the one dimensional processes also plays important role since on the one hand it is partly analytically solvable and on the other hand it is basics for the model measuring (e.g. the so called column test). Also it is backwards indirect Parameter estimation (e.g. tracer test). As example equations can be derived:
• heat transport due to precipitation in the unsaturated soil zone:
∂ ∂ pw dnw water (index w): (kw ( + z)) = − ww (6.13) ∂z ∂z pw dt
∂ ∂ pL dnL air (index L): (kL ( + z)) = − wL (6.14) ∂z ∂z pL dt • one dimensional transport: ∂C ∂C convection: ε = −q (6.15) ∂t ∂x ∂2C ∂C dispersion: = a (6.16) ∂z2 ∂t ∂2C ∂C ∂C dispersion and convection: MD − q = ε + λC − w (6.17) 1 ∂x2 ∂x ∂t The three cases are differentiated, with whether λ and/or ω are equal or unequal to zero.
6.4 Multiphase flow
Simultaneous effects of several phases in the porous medium, soil or aquifer are con- sidered in the modeling of the multiphase flow. In the literature the relations of three phase system are illustrated. According to the equation 6.7 on page 187 with neglecting the dispersion portion:
∂(Φp S ) div (p ~v ) + α α = p (6.18) α α ∂t α
In this case α represents a general fluid phase. Within the three phase system water (α = ω), NAPL (n) and air (a) will be considered. NAPL is the abbreviation for petroleum products (Non Aqueous Phase Liquid - NAPL).
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 6.4. Multiphase flow
The Darcy law can be extended to the multiphase system by neglecting the theorem of momentum between the fluid phases:
k · krα ~vα = − (grad pα + pα · ~g) (6.19) µα
The reciprocal effects between the individual phases will be described by additional equations, secondary conditions:
sw + sn + sa = 1 (the pore area is filled out by the sum of the three phases)
pn · pw = PCnw(Sw,Sa) (Capillary pressure saturation relationship)
pa · pn = PCan(Sw,Sa)
krα = krα(Sw,Sa) (relative permeability saturation relationship)
In many practical applications the nonlinear equation system is limited by the assump- tion that air is infinitely mobile in each case a movement phase for the water phase and for the NAPL phase.
The analytical solution from Buckley and LeverettT will be used for the one dimensional displacement procedure of oil through water, which describes the transient procedures. To describe the relative permeability saturation curve the Corey function can be set:
∗4 krw = S (6.20)
∗ 2 ∗2 krn = (1 − S ) · (1 − S ) (6.21)
∗ (Sw − Swr) with : S = ; Swr = Snr = 0, 2 (1 − Swr − Snr)
For the dimensional case and the investigation of three phase system air/NAPL/water the following beginnings are examined: For the capillary pressure saturation re- lationship the Parker formula is used:
191 CHAPTER 6. OVERVIEW
1 n 1 vG n PCnw = pn − pw = (Se − 1) vG (6.22) αvG · −βnw (1 − nvG)
1 S + S − S nvG 1 n w wr (1−n ) n PCan = pa − pn = (( ) vG − 1) vG (6.23) αvG · −βan 1 − Swr
Sw − Swr σaw σaw with : Se = ; βnw = ; βan = 1 − Swr σnw σan
For the relative permeability saturation relationship for the non aqueous phase liquid(NAPL) a model by Stone will be used :
Sn(1 − Swr)krnw · kran krn = (6.24) (1 − Sw)(Sn + Sw − Swr)
Hereby krnw and kran stand for the relative permeability saturation relationship in NAPL phase of a two phase system (water/NAPL) and (air/NAPL). The parame- ters Snr and krncw were occupied with the values 0 and 1 in the original form of the textscStone model. For the water phase the relationship the Parker formula is used:
2 (nvG−1) n p nvG vG krw = Se 1 − 1 − Se (6.25) (nvG − 1)
Sw − Swr with: Se = (1 − Swr)
192 Chapter 7
Horizontal plane Groundwater flow equation
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 7.1 Dupuit assumption and ballance equation
The description of the rotationally symmetric groundwater flow field is based on hori- zontal plane flow processes, in which the vertical flow vector is neglected. The transfer of the three dimensional flow regime into a two dimensional mathematical description takes place with consideration of the
Dupuit assumption:
• The potential lines h = const. run parallelly to the z axis. This means that the
vertical component of the groundwater flow (vz → 0) is equal to zero. This can be realized by an infinitely large vertical flow resistance (specific permeability
coefficient in z direction (kz → ∞) or by a no gradient gauge level: ∂h v = = 0 z ∂z
• The horizontal speed is constant during the entire through flow height of the aquifer. It means the vertical gradients of the horizontal flow components are equal to zero. ∂v ∂v x = y = 0 (7.1) ∂z ∂z • The horizontal speed is proportional to the decline gradient of free the surface according to the Darcy law:
∂h v = −k · (7.2) y y ∂y ∂h v = −k · (7.3) x x ∂x
The force equilibrium law is set up under the condition that only pressure force, gravity force, capillary force and internal friction are effective. Inertia forces, adhesive force, turbulent friction forces and others are small enough to be negligible. Since the groundwater movement is regarded as saturated filter flow, we know following law from Darcy:
~v = −k · grad h (7.4)
The Darcy law is only as long valid as its by its derivation existing preconditions fulfil. Thus it loses its validity if the above neglected forces increase. For the practical
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 7.1. Dupuit assumption and ballance equation groundwater flow procedures the validity of the Darcy law can be assumed with sufficient accuracy. Only directly in the proximity of well with large filter velocity a breach of this law can occur. With the Dupuit assumption the balance equation for horizontal plane groundwater flow is built up. The specific flow rate~q , refer to flow field width of 1m, can be calculated to:
D Z ~q = ~vdz (7.5)
z=a
D through flow thickness M aquifer thickness in confined D = aquifer zR positon of the free groundwater surface in unconfined
Then the balance equation is:
D Z ∂h div ~q = n + S dz · − w (7.6) 0 0 ∂t z=a
w sources/sinks
n0 storage coefficient at the free groundwater surface due to gravimetric effects
S0 elastic storage coefficient, which works within the aquifer
The summary expression for the storage capability is designated with S as general storage coefficient:
D Z S = n0 + S0 dz (7.7) z=a zR n + R S dz unconfined 0 0 S = z=0 aquifer (7.8) M R S0 dz confined z=0
195 CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
If the gravimetric storage coefficient is substantially larger than the sum of all elastic effects in vertical direction: z Z R S0 dz (7.9) z=a
The storage coefficient S can take the following values: n0 ≈ na ≈ ne unconfined S = M aquifer (7.10) R S0 dz confined z=0 It results that the storage coefficient is only dependent on the gravimetric coefficient in the case of a free groundwater surface and a small through flow thickness aquifer (D << 100m). For the effect water height h: h confined h = aquifer zR unconfined
Thus the balance equation, also as continuity equation, is written in the form: ∂h div ~q = S · − w (7.11) ∂t
With the equations 7.5 and 7.6 we get the horizontal plane groundwater flow equation in the following form:
D Z ∂h div − k dz grad h = S · − w (7.12) ∂t z=a
According to definition of h, grad h is independent on z thus it can be pulled out of integral. For further writing simplification the integral of permeability coefficient, the term transmissibility T , is introduced:
D Z T = k dz (7.13)
z=a This integral of transmissibility will be analysed numerically poorly as the permeability coefficient is only expressed as step function and not a continuous function.
196 Thus the horizontal plane groundwater flow equation in the representation of the water height is: ∂h div (T grad h) = S · − w confined ∂t aquifer (7.14) ∂z div (T grad z ) = S · R − w unconfined R ∂t
7.2 Potential illustration
An integral transform for solving partial differential equation of underground flow pro- cesses in the former chapter was used, which yields the value of transmissibility. Now in this section another integral transform is applied, the so called Girinskij potential Φ also a relatively simple solution, thus the horizontal plane groundwater flow equation is illustrated in potential expression.
The Girinskij potential Φ is defined as:
D Z Φ(x, y) = g(z) · (h(x, y, z) − z) dz (7.15)
z=a
In this equation the function g(z) characterizes the dependence of permeability coeffi- cient k on height z: k(x, y, z) = k(x, y) · g(z) (7.16)
For the following considered unstratified auqifer follows:
g(z) = 1
∂h Here with the validity of the Dupuit assumption ( = 0; h 6= f(z)) and the as- ∂z sumption of lower bound of the aquifer a equal to zero (a = 0), the integral yields two solutions: D Z z2 D Φ(x, y) = (h − z) dz = h · z − (7.17) 2 0 z=0 2 M M · h − confined = 2 aquifer (7.18) z2 R unconfined 2
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
With consideration of the Darcy law the specific volume flow is:
D Z ~q = ~vdz (7.19)
z=0 and with ~v = −k grad h follows:
D Z ~q = (−k · grad h)dz (7.20)
z=0
Since k and h by definition are not functions of z, we can write k outside the integral and exchange the order of the deviation(the gradient) and the integration. We get:
D Z ~q = −k · grad h dz (7.21)
z=0
~q = −k · grad Φ (7.22)
The horizontal plane groundwater flow equation in potential form:
∂h div (k grad Φ) = S − w confined ∂t aquifer (7.23) ∂z div(k grad Φ) = S R − w unconfined ∂t
This PDE can be transferred into a uniform potential expression, if the definition for the Girinskij potential is used separately, according to confined and unconfined conditions:
∂h ∂h ∂Φ S = S · · (7.24) ∂t ∂Φ ∂t 1 ∂Φ S ∂Φ M 2 S · · = with Φ = Mh − confined M ∂t M ∂t 2 = 2 aquifer 1 ∂Φ S ∂Φ zR S · √ · = with Φ = ununconfined 2Φ ∂t zR ∂t 2 (7.25)
198 7.2. Potential illustration
We get:
Φ M h = + (7.26) M 2 ∂h 1 = (7.27) ∂Φ M or:
√ zR = 2Φ (7.28) ∂z 1 R = √ (7.29) ∂Φ 2Φ
Assuming a homogeneous, isotropic aquifer, i.e. k = const., then k can be calcu- lated from the divergence and the division of the right side. With introduction of the transmissibility and the geohydraulic time constant we get:
∂Φ w div (grad Φ) = a − ∂t k with:
S a = (7.30) T D Z k · M confined T = k dz = GWL (7.31) z=0 k · zR unconfined
Now we have found a universally valid PDE, which is linear and analytically solvable.
However it must be noted that the linearity is not exact under free groundwater surface conditions, i.e. with unconfined aquifer, since the geohydraulic time constant a is a function of zR. In this case a temporal average value will be taken for T and also for a. The following approximation has been well proved for a:
a − 2a a˜ ≈ t=0 t (7.32) 3
For extreme drawdown ratios over 10% of groundwater level this equation is only approximately valid. Often the water level change of a standpipe , i.e. the drawdown, is of interest instead of the Girinskij potential.
199 CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
Therefore the potential difference between the output potential Φ0 and the current potential Φ is used. In unstratified aquifer, i.e. with k(z) = const. and thus g(z) = 0 follows:
D D Z Z Z = Φt=0 − Φ = (ht=0 − z) dz − (ht − z) dz (7.33) z=0 z=0 M(ht=0 − h) confined = aquifer (z2 − z2 ) Rt=0 R unconfined 2
The subscript 0 stands for conditions at time point t = 0, i.e. Φ0 = Φt=0, h0 = ht=0, zR0 = zRt=0. In some ciations the subscript n (Φn, hn, zRn) is used for it.
Inserting this into the PDE we get: ∂Z w0 div (grad Z) = a + (7.34) ∂t k
By definition w0 is the supply quantity caused by the change of potential Z, is by definition, while w represents the supply quantity, which affects from the outside of aquifer, e.g. the natural groundwater replenishment:
w0 w w = − (7.35) k k k t=0
Practically the following two cases are interested:
• w0 = 0, i.e. supply conditions won’t change when the regarded groundwater level varies, and w0 Z • = , i.e. the difference of potential Z causes an additional proportional k B2 supply (see section 8.1.3, page 231 supply from neighbouring layers)
If we add a supply factor B in all cases, which approaches to infinite for the first case (B ⇒ ∞), we get the general form, the standard form of the horizontal planes groundwater flow equation:
∂Z Z div (grad Z) = a + (7.36) ∂t B2
200 To solve this PDE it is necessary to transfer the general vectorial differential expression into a coordinate related expression (see section 2.2 arithmetic rules of the vector algebra, page 57).
By using cartesian coordinates we get a PDE to analyse the groundwater flow processes in connection with the ditch flow (see to section 6.1 one dimensional flow equation, page 188). Cylindrical coordinates lead to a representation, which is very useful for rotationally symmetrical problems (see section 8.1 Theis well equation).
7.3 Boundary conditions
Each flow process takes place in a locally and temporally defined area, i.e. it represents a closed system, which is connected with its environment under certain conditions. Information, energy and material can be exchanged through such couple conditions. They are called boundary conditions. While the system is described by the PDE and generally valid for all conditions, a unique solution will be achieved by boundary conditions. The effect of the boundary conditions is identical to determination of the integration constant by solving a differential equation. Boundary conditions are impressed to the regarded system from the outside and influence independently of state variables.
It is differentiated between boundary conditions (conditions at certain local points) and initial conditions (conditions of reference to a time point). Besides force equilib- rium and mass conservation, the initial- and boundary conditions serve explicit math- ematical descriptions of the original process, the flow process. They are regarded as a part of the mathematical model.
7.3.1 Initial conditions
In dynamic systems relative time will be discussed. The absolute time point, from which the system behaviour is changing from static into movement, is regarded as starting point with relative time t = 0. The initial conditions serve to define the dynamic system state at this point. Since the state variable of horizontal planes groundwater flow is the piezometric head h or the situation of free surface, the initial condition is a matter of potentials within the systems , i.e. the groundwater height or the pertinent transform potential. In the model this will be a function of h, zR or Φ dependent on
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION location. Other conditions are also necessary for the maintenance of steady state at the border. These are from the same type like the boundary conditions. The difference is they are valid for t < 0.
7.3.2 Randbedingungen
Three different kinds of boundary conditions differ in physical action modes in the groundwater flow (see figure 7.1):
1. 1st type (Dirichlet condition) 2. 2nd type (Neumann condition) 3. 3rd type (Cauchy condition)
Boundary conditions are general functions of place and time. We differentiate boundary conditions between influence inside of flow field (e.g. well, lakes, rivers, precipitation, evaporation), and outside effect at the edge (e.g. delimitation of flow field by rivers or barriers). It is characteristic for boundary conditions that its effect is independent on the flow conditions (e.g. groundwater level) of the investigation area. Generally it is nearly impossible to find a complete analytical expression for geohydraulic boundary conditions.
• 1st type boundary condition (Dirichlet condition) work,
if the hydraulic potential (e.g. h, zR, Z, Φ) on the boundary is known as a function of the time t and independent on the potential, i.e. the system variables of the investigation area. This appears e.g. in rivers, lakes or drainage:
ϕ = ϕ(x, y, t) (7.37)
• 2nd type boundary conditions (Neumann condition) work, if the source intensity distribution and thus the hydraulic potential gradient on the bound are known as a function of time t. This may be aroused for example by wells with constant flow rate, supply due to groundwater regeneration, sealing of sheet pile wall or underground structures:
grad ϕ = grad ϕ(x, y, t) (7.38)
• 3rd type boundary condition (Cauchy condition) work, if in general a temporally constant flow resistance exists between a surface with known potential distribution and the boundary of flow field. Such boundary
202 7.3. Boundary conditions
conditions work in rivers with colmation bottom layer as well as flow resistance of lift wells:
ϕ + A grad ϕ = B(A and B are definite constants) (7.39)
In the figure 7.1 the effect of boundary conditions on an aquifer is illustrated. We recognize that the flow rates of 1st and 3rd type boundary conditions dependent on the difference between the effect potential (water level h) in the aquifer and the boundary conditions. Therefore the flow rate can vary in amount and direction. In 2nd type boundary condition the potential of the boundary condition (possibly re- garded as negative or positive pressure) can vary accordingly with the potential of the aquifer.
With the numerical models (see to section 9.1.1 numerical methods, e.g. finite differ- ences methods, page 246) additional boundary conditions arise in the course of the definition of the model borders. In contrast to the original procedure, which possesses an infinite spatial expansion, the numerical models are spatially limited due to the fi- nite computing capacity (memory space, computing speed). Thus a considerable error arises, which must be reduced or eliminated by suitable measures (see section 9.1.1 finite differences method, page 246).
Another problem related to boundary conditions arises in the interaction investigation of surface and aquifer systems . A volume flow appears between the surface and the aquifer or the unsaturated soil zone. Depending upon potential conditions an ex- or infiltration of surface water can come out or into the aquifer. If this filtration stream is substantially smaller than the volume flow within water, or the filtration amount is substantially smaller than the entire storage volume of surface water, the surface water has an effect of boundary condition on the groundwater. In the other case, if the potential of surface water varies due to the filtration phenomenon, the surface water may be not considered as boundary condition, but components of the system and are coupled to the aquifer model according to suitable mathematical relations.
203 CHAPTER 7. HORIZONTAL PLANE GROUNDWATER FLOW EQUATION
Figure 7.1: effect of boundary conditions on an aquifer
204 Chapter 8
Analytical Solutions
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 8.1 Theis well equation (rotationally symmetrical flow)
The computation of rotationally symmetric flow field, i.e. the solution of partial dif- ferential equation, represents a primary task of geohydraulics.
Such procedures can be described by means of horizontal plane groundwater flow equa- tion in standard form, which is deduced in the section 7.2 potential representation, page 197: ∂Z Z div(grad Z) = a + ∂t B2 Changes of groundwater flow conditions with employment of vertical filter wells were first calculated by Theis in the year 1935 and replenished by several other authors (e.g. Thiem, Jacob, Cooper, Neumann, Hantush and others). Due to the importance of this task numerous publications and text books refer to this topic (for instance Busch/Luckner/Tiemer, Geohydraulik, and others).
8.1.1 General solution
On the basis of the partial differential equation we are looking for a solution for the standard form of the horizontal plane groundwater flow equation, i.e. the drawdown of the groundwater level as a function of place and time, if a change of boundary condition takes place at relative time point t = 0.
Figure 8.1: Coordinate System for a rotationally symmetrical well
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 8.1. Theis well equation (rotationally symmetrical flow)
According to the transfer of the vectorial to the coordinate based differential operators, here in cylindrical coordinates (see figure 8.1, see section 2.2 arithmetic rules of vector algebra, page 57), we know that the flow field is rotationally symmetric (Z(α) = 0) and not dependent on the local coordinate z (Z(z) = 0):
∂Z Z div(grad Z) = a + (8.1) ∂t B2 ∂2Z 1 ∂Z ∂Z Z + = a + (8.2) ∂r2 r ∂r ∂t B2
Thus the solution of this partial differential equation, the Z, depends only on the time t and the radius r (distance between well and calculation point). Under the above prerequisite of the parameters of the aquifer, the permeability coefficient k and the storage coefficient S change neither with the height of z nor with the angle α in the regarded area.
The simplest solution can be achieved if following initial and boundary conditions are considered and the aquifer is regarded as an infinitely expanded, homogeneous and isotropic flow field (see figure 8.2).
initial condition: Z(r,t=0) = 0 (8.3)
external boundary condition: lim Z(r,t) = 0 (8.4) r→∞ ∂Z internal boundary condition: lim2πrk = −V˙ = const (8.5) r→0 ∂r
The internal boundary condition can be technically realized, if a vertical filter well is arranged at the origin of the cylindrical coordinates (r = 0) , which conveys a constant flow rate V˙ starting from time point t = 0 (see figure 8.1).
The solution of the corresponding simplified partial differential equation ∂2Z 1 ∂Z ∂Z + = a (8.6) ∂r2 r ∂r ∂t was found by Theis under the aforementioned conditions:
V˙ ar2 S Z(r, t) = W (σ), mit σ = und a = (8.7) 4πk 4t T
207 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.2: Infinitively expanded aquifer
Both groundwater generation rates and supply from neighbouring layers are here taken into account. Remarks in addition are in the section 8.1.3 supply from neighbouring layers, page 231.
The so called well function W (σ) is specified as the integral of an exponential function, which in the analysis is known as the exponential integral Ei(x) and defined as follows:
∞ Z e−xt Ei(x) = dt (8.8) t 1 ∞ X xn Ei(x) = γ + ln x + (8.9) n · n! n=1
γ is the Euler constant and is given by:
∞ ! X 1 γ = lim − ln (n) (8.10) n→∞ n n=1
γ ≈ 0, 5772156649
With x = −σ, W (σ) is: ∞ Z eσ W (σ) = − Ei(−σ) = dσ (8.11) σ 1
208 8.1. Theis well equation (rotationally symmetrical flow)
This integral is not elementarily solvable, but expressed as an infinite series.
σ2 σ3 σ4 σn W (σ) = −γ − ln(σ) + σ − + − + ··· (−1)n+1 (8.12) 2 · 2! 3 · 3! 4 · 4! n · n! σ2 σ3 σ4 σn W (σ) = − ln(Cσ) + σ − + − + ··· (−1)n+1 2 · 2! 3 · 3! 4 · 4! n · n! ∞ X σn W (σ) = − ln(Cσ) + (−1)n+1 n · n! n=1
with C = eγ
γ ≈ 0, 5772156649
C ≈ 1, 7810724
To solve equation 8.1.1 series development and substitution method can be applied and similar methodology such as Bessel function (see section 5.2.2.3 differential equation of type c, 142).
Table 8.1 contains the values of well function W (σ) for the range 1 · 10−12 ≤ σ ≤ 9.
209 CHAPTER 8. ANALYTICAL SOLUTIONS
Table 8.1: Well function W (σ) for the range 1 · 10−12 ≤ σ ≤ 9 5 − 10 · 0411 3437 6462 9488 2514 5540 8566 9 2602 9187 1423 4368 7386 , , , , , , , , , , , , 24 0 1 4 6 8 , 11 13 15 17 20 22 24 1 5 − 10 · 1589 4614 7640 0666 3692 6718 9744 8 3106 0269 2591 5545 8563 , , , , , , , , , , , , 77 0 2 4 6 8 , 11 13 15 18 20 22 24 3 4 − 10 · 2924 5950 8976 2001 5027 8053 1079 7 3738 1508 3916 6879 9899 , , , , , , , , , , , , 16 0 2 4 6 8 , 11 13 15 18 20 22 25 1 4 − 10 · 4465 7491 0517 3543 6569 9595 2620 6 4544 2953 5448 8420 1440 , , , , , , , , , , , , 60 0 2 4 6 9 , 11 13 16 18 20 22 25 3 6289 9314 2340 5366 8392 1418 4444 5 5598 4679 7261 0242 3263 , , , , , , , 00115 , , , , , , 0 2 4 7 9 0 11 13 16 18 20 23 25 8520 1546 4572 7598 0623 3649 6675 4 7024 6813 9482 2472 5495 , , , , , , , 00378 , , , , , , 0 2 4 7 9 0 11 14 16 18 21 23 25 1397 4423 7449 0474 3500 6526 9552 3 0143 9057 9591 2349 5348 8371 , , , , , , , , , , , , , 0 0 2 5 7 9 12 14 16 19 21 23 25 2426 5451 8477 1503 4529 7555 0581 3607 2 0496 2227 3547 6394 9402 , , , , , , , , , , , , , 0 1 3 5 7 10 12 14 17 19 21 24 26 9357 2383 5409 8435 1460 4486 7512 0538 1 2194 8229 0379 3315 6332 , , , , , , , , , , , , , 0 1 4 6 8 10 13 15 17 20 22 24 27 00 01 02 03 04 05 06 07 08 09 10 11 12 + − − − − − − − − − − − − σ 10 10 10 10 10 10 10 10 10 10 10 10 10 · · · · · · · · · · · · · mantis. 1 1 1 1 1 1 1 1 1 1 1 1 1 exponent
210 8.1. Theis well equation (rotationally symmetrical flow)
From the definition of the drawdown or Girinskij potential the inverse transform to the physical dimensions water level h or zR and the drawdown s can be accomplished:
V˙ Z = Φ − Φ = W (σ) (8.13) n 4πk 2 D M Z Mh − confined Φ = (h − z)dz = 2 auqifer (8.14) 2 zR z=a unconfined 2 and hn − h confined s = aquifer (8.15) zRn − zR unconfined
For confined flow conditions:
Z = Φn − Φ M 2 M 2 = Mh − − Mh − n 2 2
= M (hn − h) Z V˙ s = = W (σ) with T = k · M (8.16) gesp. M 4πT or Z V˙ h = h − = h − W (σ) (8.17) n M n 4πT
For unconfined flow conditions:
Z = Φn − Φ z2 z2 = Rn − R 2 2 s q V˙ s = z − z2 − 2Z = z − z2 − W (σ) (8.18) ungesp. Rn Rn Rn Rn 2πk or s q V˙ z = z2 − 2Z = z2 − W (σ) (8.19) R Rn Rn 2πk
211 CHAPTER 8. ANALYTICAL SOLUTIONS
This sharp separation between confined and unconfined groundwater conditions and modelling by means of the Theis solution are not consistently implemented in all literature. This can also be applied to graphic methods for pumping test evaluation. In the case of very thick aquifer, for instance in North Germany the drawdown only amounts to a few percentage of thickness, it can be possibly calculated with the formula for confined aquifer.
From above derived formulas of the drawdown for different aquifers we know that under unconfined groundwater conditions the position change of the free surface represents a strongly nonlinear process.
Note that the well formula is not valid in the proximity of a well with r → r0. The reason is that it does not fulfil the prerequisites, which were the derivation of a rota- tionally symmetrical flow equation. So the vertical flow component which should be vz = 0 can not be applied in the proximity of the well. Also the effective well radius ∗ r0 is mostly not exactly confirmed. The storage effects and the flow resistances in the well area are hardly predictable (see section 14.2 pumping test simulator, page 393). In spite of some restrictions the well formula is a fundamental calculation formula in geohydraulics for computation of groundwater level height drawdown according to a volume stream.
The potential series W (σ) (see equation 8.12, page 209) already strongly converge for values σ < 0.03, so that terms of higher order of σ are small enough to be negligible. Thus W (σ) can be computed in the case of an error ε < 1 only by the logarithmic function. This approximation was advanced by Cooper & Jacob in 1946:
W (σ) ≈ − ln (C · σ) (8.20)
2, 246 T · t W (σ) ≈ ln (8.21) S · r2
This simplified formula has great importance for many practical applications. In par- ticular all graphic procedures of pumping test evaluation (see section 14.1 pumping test evaluation, page 386) are based on this formula (also see table 8.2).
The relative error, which results from the approximation, can be computed as follows: ∞ n n P (−1) σ
W (σ)C&J − W (σ) n=1 n · n! ε = = (8.22) W (σ) ∞ (−1)n+1σn − ln(Cσ) + P n=1 n · n!
212 8.1. Theis well equation (rotationally symmetrical flow)
Since the series for σ < 1 converge very fast, it is only necessary to include the first term of the series to the error estimation. All further terms will be substantially smaller than the linear term and thereby can be neglected. The second term only contributes a portion, which is squarely smaller than the first one.
σ ε = − ln(Cσ) + σ
In table 8.2 the error for the approximation by Cooper & Jacob depending on σ is listed.
Table 8.2: Error of COOPER and JACOB formula as a function of σ
σ σ ε = Minimale Zeit t (σ − ln(C · σ)) C = 1, 7811 0, 25000 23, 61% 1, 00 · r2 · a 0, 20000 16, 23% 1, 25 · r2 · a 0, 15000 10, 20% 1, 67 · r2 · a 0, 10000 5, 48% 2, 50 · r2 · a 0, 07500 3, 59% 3, 33 · r2 · a 0, 05000 2, 03% 5, 00 · r2 · a 0, 03000 1, 01% 8, 33 · r2 · a 0, 02500 0, 80% 10, 00 · r2 · a 0, 01000 0, 25% 25, 00 · r2 · a 0, 00750 0, 17% 33, 33 · r2 · a 0, 00500 0, 11% 50, 00 · r2 · a 0, 00250 0, 05% 100, 00 · r2 · a 0, 00100 0, 02% 250, 00 · r2 · a 0, 00075 0, 01% 333, 33 · r2 · a
Simultaneously we get an estimation for the relation of computation place (r), compu- tation time (t) and approximation errors (ε). It is also recognized that the geohydraulic S time constant (a = T ) representatively determines the stop accuracy. Thus the more computation time point approaches to steady state, the more exact the computation is. For the unsteady transition region the approximation of Cooper & Jacob is not well applicable and large approximation errors yield.
The computational evaluation of well function W (σ) can be substantially simplified by
213 CHAPTER 8. ANALYTICAL SOLUTIONS a recursive expression of sum formula.
a ≤ε Xn W (σ) = − ln(Cσ) + an (8.23) n=1 (−1)σ(n − 1) with a = a n n−1 n2
and a1 = σ
Thus it yields:
• Only as many terms must be calculated as for a given accuracy are necessary (e.g. the change between two sum terms). • The computation of each sum term requires only a multiplication.
8.1.2 Consideration of special effects
The general solution of well equation according to Theis only applies to a very ideal aquifer. So it is assumed e.g. as homogeneous and isotropic. Further more an infinite expansion is supposed. Only one well is considered in the solution, which conveys from the time t0 with constant flow and is arranged as singularity with a radius of rBr = 0m at the coordinate origin. These idealizations can not be found in practice with real aquifers. However for some practice relevant conditions results can be obtained based on Theis solution, if appropriate auxiliary computations and substitutions are accom- plished. Such are for example the consideration of technical well radii, imperfection of the wells and boundary conditions, as well as the laminated aquifers and supply from neighbouring layers. These special effects are regarded as additional potential in well equation, which are established and dismantled.
8.1.2.1 Imperfect well
Imperfect boundary conditions, particular wells, appear, if the boundary conditions or wells do not act on the entire thickness of the aquifer. In the case of wells it happens if the working filter pipe length is smaller than the thickness of the aquifer through flow. With the imperfect wells it is assumed a potential loss results from through flow thickness near the imperfect well smaller than the actual aquifer. Besides we can suppose that the average flow path via redirecting is longer than the geometrical distance r. Figure 8.3 shows conditions with different filter installation.
214 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.3: imperfect well with filter in the a) upper, b) lower, c) middle part of the aquifer
Thereby are: C length of the full pipe
D flow through thickness
L Length of the filter pipe within the flow through thickness
M thickness of the aquifer
zR position of the free groundwater surface
ϕV filter losses
215 CHAPTER 8. ANALYTICAL SOLUTIONS
We can describe the potential loss as follows:
V˙ Z(r, t) = (W (σ) + ϕ ) (8.24) 4πk V r D αL αD L ϕV = 2 ln − ln − 1 − (8.25) L r0 r0 D
α = 0, 735 1 + (H − 1)4 (8.26)
2C H = (8.27) D − L We recognize that a stronger sink occurs due to the imperfection, comparing with the case of a perfect well.
8.1.2.2 Multi-well plants
In practical multi-well plants are meaningful. In seldom cases e.g. foundation pit drainage or a ground water works only one well is operated. The computation for such multi-well plants is possible based on principle of superposition. The solutions, i.e. the partial drawdown potentials, which apply to the individual wells, will be superposed, i.e. overlaid together (see figures 8.4 and 8.5). The principle of superposition can be only applied in linear systems. Relating to the Theis solution this means that the superposition can be only used in the potential expression. When the superposed potential is formed, the inverse transform of physical dimension drawdown or water level can be accomplished. Since the connection between potential and drawdown for the confined aquifer is linear, the superposition in this case could be also exceptionally applied to the drawdown. In principle:
n X ZGes.x,y,t = Z (ri, t) i=1 n ˙ ! X Vi = W (σ ) 4πk i i i=1 n 1 X r2a = V˙ · W i 4πk i i 4t i=1
Thereby are:
• ri distances between the individual well and the computation point Px,y • a geohydraulic time constant for the entire area a = const.
216 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.4: multi-well plant (Br = well)
˙ First we calculate the individual drawdown portions of Z(ri, t) due to the well effects Vi and then sum them up. Afterwards the conversion in the total drawdown takes place according to groundwater conditions (confined or unconfined) and equations 8.16 and 8.18 (see page 211).
8.1.2.3 Variable conveying curve of a well
The Theis solution assumes for the internal boundary condition, that the flow rate affects on the aquifer from time point t = 0. In practice it often happens that, this condition is not fulfilled. Because of technical/technological criteria a variation of pump capacity is often required. This problem play an important role, if the groundwater level after switching off the pump is in the so called rising phase.
Also here a solution based on Theis formula can be obtained by means of superposition principle. The basic idea consists of the fact that time-dependent conveyor capacity is set as summation of temporally transfer step functions. Figuratively we can imagine n fictitious pumps, which are put on the same well successively according to the conveying stages and switched on (see figure 8.6 and 8.7).
Subsequently we check the individual fictitious partial conveying capacities in the total
217 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.5: superposition of the drawdown potentials
drawdown potential and add these accordingly (see figure 8.8):
m X ZGes,r,t = Z(r, ti) i=1 m ˙ ! X Vi = W (σ ) 4πk i i i=1 m 1 X r2a = V˙ · W (8.28) 4πk i i 4t i=1 i
If we introduce the real time t and the starting times τi of conveying capacity, we get:
m X ZGes,r,t = Z(r, t − τi) i=1 m 1 X · r2a = V · W (8.29) 4πk i i 4(t − τ ) i=1 i
˙ We can also calculate the partial conveying capacities Vi from the real conveying ca- ˙ pacity at time t of Vreal,i,t by subtracting those time stages from this:
218 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.6: Virtual conveying flows with time-dependent conveying curve
Figure 8.7: composite conveying curve
· · ·
V i,t = V real,i,t − V real,i−1,t−τi (8.30) m 1 X · · r2a Z = V − V W (8.31) Ges.r,t 4πk real,i,t real,i−1,t−τi i 4(t − τ ) i=1 i
Observing this formula it is recognized that the rising phase can be computed. In this case the last partial conveying amount is negative (see figure 8.9). The sign reversal means that this component current is not exfiltration but as treated as infiltration and thus not leads to a drawdown, but an increase of the groundwater level compared with the foregoing time.
219 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.8: drawdown potential
The described method for computation of drawdown potentials with temporally con- veying curves can be also used, if the conveying curves are not step functions, but as continuous, concave or convex functions. In this case the function will be approxi- mated by a step function (see figure 8.10), whereby step height and width may be not constant. We do not have to assume an equidistant quantization (see section 11.3.5 approximation of signals, page 312). The decision between necessary accuracy and expenditure here is of importance for editors.
The methods for computation of multi-well plants and variable conveying curves can be also summarized, so we get a solution for the superposition of both effects:
n m ! 1 X X · r2a Z = V W i (8.32) Ges.r,t 4πk j,i j,i 4 (t − τ ) i=1 j=1 j,i
The drawdown potentials are to be superposed first over all conveying stages of one well and afterwards over all wells.
220 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.9: groundwater rising
Figure 8.10: approximation of a continuous conveying curve
221 CHAPTER 8. ANALYTICAL SOLUTIONS
8.1.2.4 Limitations
• 1st and 2nd type boundary conditions
The solution of the well equation according to Theis is derived for the infinitely ex- panded aquifer. For special limitations of the aquifer the boundary conditions of Theis well equation can be modified into a geometrically simple form by means of superpo- sition principle in order to find out a solution. The basic idea is to arrange a virtual well with drawdown potential in the overlay in such a way that the same hydraulic effect as real a well can be exactly obtained like 1st and 2nd type boundary condition, i.e. a constant change of potential or a constant influx at the flow limitation. These are special cases, which occur very often in practice. This method of arranging virtual sources or sinks in a potential field for model building of special boundary conditions is designated as reflection method in general potential theory, which applies to many different potential fields (e.g. thermal conduction, elec- trostatic and magnetic fields). The realization of different kinds of boundary conditions is under consideration of different volume flow directions (ex- or infiltration).
With a limitation in 1st type boundary conditions, we try to keep the drawdown potential value at zero by means of a virtual infiltration well with same vertical distance l between the real well and the boundary condition (ZRand = 0) (see figures 8.11 and 8.12).
With 2nd type boundary conditions the flow rate at the boundary will be by defi- nition remained constant, here at value zero (dZRand/dr = 0). We can model this with a virtual conveying well, which is axially symmetric and is pressurized with the same conveying capacity (see figures 8.13 and 8.14).
222 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.11: 1st type boundary condition modelling by a virtual well
Figure 8.12: consideration of one-sided 1st type boundary condition
223 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.13: 2nd type boundary condition modelling by a virtual well
Figure 8.14: consideration of one-sided 2nd type boundary condition
224 8.1. Theis well equation (rotationally symmetrical flow)
One-sided, linear aquifer limitation and a conveying well at the point Br(x, y) yield the drawdown potential Z at the computation point P (x, y):
V˙ Z(r, t) = (W (r, t) + (−1)mW (ρ, t)) (8.33) 4πk real virt st 1 1 boundary condition m = nd 2 2 boundary condition and the distance from well to computation point:
2 2 2 r = (xBr − xP ) + (yBr − yP ) (8.34) or the distance from the virtual well to the computation point:
2 2 2 ρ = (xBrvirt − xP ) + (yBrvirt − yP ) (8.35)
For multi-well arrangements or variable conveying curves the superposition must be carried on accordingly . The transformation of the drawdown potentials into real drawdown is accomplished according to the arithmetic rules for confined or uncon- fined aquifers and equations 8.16 and 8.18 (see page 211) (see section 7.2 potential illustration, page 197).
Under 1st type boundary conditions a method for computation of the final steady state can be derived from above equation with consideration of Cooper & Jacob approximation:
V˙ Z = (W (r, t) − W (ρ, t)) (r,t) 4πk real virtuell V˙ Z (r, t → ∞) = (− ln (Cσ ) + ln (Cσ )) stat 4πk r ρ ˙ V σρ Zstat (r) = ln 4πk σr S · r2 S · ρ2 σ = bzw. σ = (8.36) 4 · T · t 4 · T · t V˙ ρ Z (r) = ln (8.37) stat 2πk r
225 CHAPTER 8. ANALYTICAL SOLUTIONS
We recognize that the final steady state of the drawdown potential of the aquifer with one side limited 1st type boundary condition proportionally depends on the ratio of conveying capacity to permeability and proportionally on the logarithm of distance.
Taking the same considerations on the steady case of a aquifer with one side limited 2nd type boundary condition, then we get that the drawdown potential approaches infinitely. This is technically impossible. In the practice this means, it takes an infinite time to drain the aquifer circumscribed by sheet pile wall completely.
• 3rd type boundary condition
Real boundary conditions are characterized by the fact that due to their effects the definitions and conditions for the derivation of the mathematical model are not fulfilled. These are imperfection of 1st type boundary condition and additional flow resistances between 1st type boundary condition and the aquifer. Such flow resistances are e.g. colmation layers of water surface. Both effects cause an additional potential decrease between the boundary condition and the drawdown potential point P . These effects correspond to 3rd type boundary condition (see section 7.3.2 boundary conditions, page 202). The additional potential decrease depends on the flow quantity, which flows between 1st type boundary condition and the aquifer.
In connection with the analytical solution of the well equation such 3rd type boundary conditions can be solved in such a way, that we find out the equivalent flow resistance of a part of the aquifer, which causes the same potential decrease with ideal 1st type boundary conditions. In the model we shift the real boundary condition by a virtual auxiliary length ∆L away from the well (see figure 8.15). Thus the influence of the boundary condition on the drawdown potential is reduced.
We differentiate two kinds of extra lengths by the origin, imperfection or by colmation bottom layer.
In the first case of imperfection, i.e. the boundary condition does not extend over the entire flow through thickness, the extra length is determined by the following diagramm (see figure 8.16).
In lakes with 3rd type boundary conditions it can be always assumed that the extra length amounts to:
∆L1 = 0, 43 · D (8.38)
In the colmation bottom layers the length of equivalent aquifer, which causes the same
226 8.1. Theis well equation (rotationally symmetrical flow)
Figure 8.15: consideration of one-sided 3rd type boundary condition
decrease potential like the colmation layer, can be computed in such way that hydraulic resistances are equated:
1 ∆L R = · 2 (8.39) hydrGW L k D · b
1 M 0 RhydrKOL = 0 · (8.40) k ∆L2 · b
According to the equation of these two hydraulic resistances the extra length is:
r k · D · M 0 ∆L = (8.41) 2 k0
227 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.16: dependence of the auxiliary length on the standardized river width
D flow through thickness of the aquifer
k0 permeability coefficient of the colmation layer Thereby are : k permeability of the aquifer
M 0 thickness of the colmation layer
8.1.2.5 Multilateral boundary
Besides the linear one-sided boundaries, described in the preceding sections, which are going through investigated area to infinity, which fulfil the correspondent conditions of the Theis well equation solution, many practical cases are characterized by the fact that the boundary conditions are not linear and/or multilateral boundary conditions rise at the same time. In these cases it is a matter of multilateral limited aquifer systems. These nonlinear boundary conditions, e.g. the confluence of several receiving streams, will be approximated piecewise by linear boundary conditions. It has to be noticed that the linear boundary conditions are to be considered with an infinite length.
On the basis of superposition the effect of boundary condition can be computed as additive overlays of the different linear processes. The reflecting method can be applied
228 8.1. Theis well equation (rotationally symmetrical flow) here again (see figure 8.17). However it must be considered that the virtual wells (reflecting well) at each linear boundary are also to be reflected and lead to further virtual wells. The combination of different boundary conditions (1st and 2nd type) is possible.
In principle arbitrary angle arrangements between the multilateral boundaries can be considered mathematically based on analytical geometry. Practically this method has limits and only with comfortable computer programs realizeable (e.g. CAE Groundwa- ter/Theis). The restriction of perpendicular or parallel standing bounds is for simpler applications. With the perpendicular or parallel bounds the number of repeated reflec- tion will be estimated, i.e. how large is the influence of nth reflection well. Since the drawdown potential is proportional to the W (σ) function and W (σ) decreases strongly for large σ nascence, the distance between the nth reflection well and the computation point plays a dominative role. It is pointed out that σ grows quadratically with the distance r.
229 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.17: multilateral boundary conditions with corresponding virtual wells
230 8.1. Theis well equation (rotationally symmetrical flow)
8.1.3 Supply from neighbouring layers
In former sections a homogeneous aquifer was presupposed. This is in the rarest cases justifiable. The question, whether a vertical soil profile is to be regarded as homo- geneous, laminated or as impervious layer, depends on the variation of the aquifer parameters, the permeability coefficient k and the storage coefficient S (n0 or S0 · D). In practice the following distinctions are generally accepted. If we consider two abut- ting soil consditions with the permeability coefficients k1 and k2, then the following classification can be carried out according to real precision demand: ≤ 20 homogeneous k1 = ≥ 50 and ≤ 100 laminated aquifer k2 ≥ 150 verticaly limited
The first case, the homogeneous aquifer, leads to the Theis well equation solution, the third, the vertically limited, to groundwater storey. In the second case, the laminated aquifer, a supply of the better permeable comes from more badly conducting layer due to larger capillary forces in the layer with larger permeability coefficient (blotting paper effect). This supply was considered by the supply factor B in the general well equation. The laminated aquifer is also called leakage aquifer and the supply factor B the leakage factor. All three cases are transferable into one another and represent only simplified computation possibilities. Furthermore the borders between the computation possibilities are not rigid, but cross over into each other.
This supply factor describes the portion of groundwater regeneration, which results in a potential change in the aquifer and originates from semipermeable layer. These supply rates are thereby calculated under the quasi-stable potential conditions in the semipermeable layer.
Thereby three cases of spatial arrangements of the good and semipermeable layers can arise: the supply from above (from hanging), the supply from below (from lying) and the combination of both. The aquifer lies between two semipermeable layers. The groundwater level is understood as piezometer head h for the semipermeable layer in all three cases.
The supply factor is computed for the three cases as follows: the aquifer is expressed by thickness M and the permeability coefficient k, the semipermeable layer by thickness
Mn and the permeability kn:
231 CHAPTER 8. ANALYTICAL SOLUTIONS
r k M supply from the layer above Ko r k (h + h) B = n supply from the layer below (8.42) Ku 2 r k M supply form the layer above and below (Ko + Ku)
kn with: Kn = Mn
The general well equation in polar coordinates is:
∂2Z 1 ∂Z ∂Z Z + · = a + (8.43) ∂r2 r ∂r ∂t B2
Hantush has found the solution of the drawdown potential:
V˙ ar2 Z (r, t) = W σ, r σ = (8.44) 4πk B 4t
It is designated also as well function of a semipermeable aquifer, leaky aquifer. Also here the inverse transform from the potential plane into the physical dimension water level or drawdown still must be carried out.
The function W (σ, r/B) is again a shorter notation for the exponential integral Ei, here with an extended argument. The derivation of this solution is topic of section 8.1.1 general solution of the well equations, page 206.
∞ Z “ 2 ” r −σ− r 1 W σ, = e 4B2σ dσ (8.45) B σ σ
This function exists as diagram (see figure 8.18). There are simplifications for different ranges of the parameters σ or r/B, so that in practical tasks this complicated formula does not have to be applied:
232 8.1. Theis well equation (rotationally symmetrical flow)
V˙ W(σ, r ) generally valid 4πk B ˙ V 2r W(σ) σ > Z(r, t) = 4πk B (8.46) ˙ V K r long time o( B ) 2πk V˙ ln 1, 12 B long time and r < 0, 03B 2πk r
In the application of this solution for the leaky aquifer it must be noted that the supply is set as constant value and thus steady groundwater flow conditions in less permeable aquifer, semipermeable aquifer are presupposed. The storage capacities of these layers are neglected.
The error is not too large for the lying, since in such cases confined groundwater conditions are predominant there, which possess smaller storage effects. In hanging also only a small error occurs under confined conditions. Larger errors can come up if a free groundwater surface exists in the lying. Particularly in the evaluation of pumping tests this simplified assumption emerges as not acceptable. In this case the supply factor must be increased empirically.
Figure 8.18: function r/B dependent on σ
233 8.2 Exercises to analytical solutions
1. Compute the drawdown s for the groundwater observation well (GWOW) with distance r and at the times t, which results from water conveying V˙ in the well for consecutively infinitely expanded aquifer (see figure 8.19) and state the result −3 m S s graphically. k = 1 · 10 s ; M = 10m; S = 0, 001; a = T = 0, 1 m2 · m3 r0 = 0, 25m; V = 0, 015 s ; hn = 16m r = 5m; 10m; 20m; 50m t = 1min; 2min; 5min;10min; 20min; 30min; 45min; 60min; 90min; 120min
Figure 8.19: infinitely expanded aquifer with well (Br) and GWOW (GWBR)
2. Compute the drawdown in the GWOW (r = 10m) for the aquifer from exercise 1 (see figure 8.19) every 10 minutes until maximally 100 minutes, if that flow rate of conveying well is subject to following stagger time. Plot the solution.
h m3 i volume flow s 0, 005 0, 010 0, 015 0, 020 0, 025 0, 030 0
extraction start [min] 0 10 20 30 40 50 60
3. A foundation pit should be lowered in an aquifer near a river. The centre of the foundation pit is 100m far away from the river, the drainage well is 80m away. Three wells are arranged parallel to the river, which have 25m distant from each
other. The diameter of the wells is r0 = 0.3m and the conveying capacity is V˙ = 0.015m3/s. The width of the river is B = 20m and a colmation layer 0 −6 m 0 k = 3 · 10 s , M = 1m. (see figure 8.20) The properties of the aquifer are:
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 8.2. Exercises to analytical solutions
−4 m k = 5 · 10 s ; n0 = 0, 20; hn = 15m; M = 20m. Will a drawdown of 2.5m be achieved in 10 days in the centre of the foundation pit?
Figure 8.20: aquifer with imperfect river, well and foundation pit
4. Please apply the analytical solution of the well flow to check whether the centre of ˙ m3 the foundation pit is drained after 7 days with conveying capacity of V = 0, 01 s , r0 = 0, 30m and a security of 0.5m (see figure 8.21).
Figure 8.21: aquifer with well and foundation pit
5. In a pumping test in an infinitely expanded aquifer the following water levels are measured as a function of the distance to the well after pumping 120min (see figure 8.22). Compute the water deficit (volume) of the drawdown funnel, if the
235 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.22: groundwater level dependent on radius
aquifer has the following characteristic values: m −1 hn = 16m, M = 10m, k = 0, 001 s , S0 = 0, 0001m ,n0 = 0, 20.
˙ l 6. A constant flow rate of V = 25 s is conveyed from a well (Br(100m,500m)), which connects an ideal river (x = 0, −∞ < y < +∞). The well has a radius of r = 0, 35m. The aquifer is characterized by the following parameters: 0 m h = 15m, M = 17m,k = 1 · 10−3 , S = 0, 0002m−1, n = 0, 25. n s 0 0 a. Calculate the final steady state (the portion of temporal functionality should be smaller than 0.001) for the point P (200m, 600m) and b. the time point , from when to calculate with this Tips: Work as long as possible with general symbols.
7. A simulation system is to be developed for a induced recharge water works (see figure 8.23) with parallel flow regime. The river should be considered as idealized boundary condition. Compute the final steady state under these hydraulic conditions based on the analytical well equation solution. · −3 m l l k = 1 · 10 s ; hF l = 15m; zR0 = 15m; S = 0, 25;V = 50 s ; q = 0, 001 s·m2 ; −5 m b = 100m; kKolm = 5 · 10 s ; MKolm = 1m; Establish the solution a. with a idealized river b. with consideration of a real river (colmation and imperfection).
236 8.2. Exercises to analytical solutions
Figure 8.23: aquifer with river, well and influx
8. In geohydraulics pumping tests are used for the determination of the aquifer parameters. Under certain conditions the drawdown can be determined according to the Theiss/Jakob/Cooper formula.
· V 2, 25 · T · t s = ln 4 · πT r2 · S
By using this formula deduct an equation to determine the k value for a local point P , which is with a distance r away from well. The determination of the k
value is based on the use of the drawdown value s1 at the time t1 and s2 at the time t2. The relation of measurement period t1 : t2 amounts to 1 : 10.
9. Compute the drawdown in the GWOT (see figure 8.24) for the time point t = 10h, m3 if a flow rate of 0, 2 s is conveyed for 5h in the well and afterwards the pumps were switched off. m −1 ht=0 = 10m, M = 15m,k = 0, 0001 s , S0 = 0, 0001m , n0 = 0, 25 10. Compute the drawdown at point P after one year based on induced recharge for a groundwater recovery plant. The conveying rate of each well is: 25l · s−1 −3 −1 k = 2 · 10 m ··· ; hn = 15m; S = 0, 25 −5 −1 k = 1 · 10 m · s ; M = 1m; B = 25m; r1 = 250m; r2 = 500m; rP = 375m
11. From two wells to a river (without colmation and perfect) a constant flow 25l·s−1 is conveyed. Compute the drawdown at the point P after one month and the final steady state. Coordinates:
237 CHAPTER 8. ANALYTICAL SOLUTIONS
Figure 8.24: Infinitely expanded aquifer with a conveying well
well 1: x = 750m y = 100m
well 2: x = 700m y = 400m
point P: x = 1000m y = 500m −3 −1 hn = 15m, n0 = 0, 25, k = 10 m · s 12. Calculate the change of the groundwater level after 10 days for the following schematic groundwater plant by means of Theis well equation (see figure 8.25). ˙ 3 −1 −1 −1 Given: V = 0, 001m · s , S0 = 0, 001m , n0 = 0, 25, k = 0, 001m · s
Figure 8.25: real river with pumping well (artificial groundwater recharge plant)
238 8.2. Exercises to analytical solutions
13. Calculate the change of groundwater condition after 10 days for the following schematic groundwater plant by means of Theis well equation (see figure 8.26). ˙ 3 −1 −1 −1 Given: V = 0, 001m · s , S0 = 0, 001m , n0 = 0, 25, k = 0, 001m · s
Figure 8.26: groundwater plant with well and sheet pile wall
14. Calculate the drawdown at the gauge for time point t = 15h, if a flow rate of V˙ = 0, 1m3s−1 is conveyed for 10h in the well and afterwards the pumps were switched off (see figure 8.27). −1 −1 Given: ht=0 = 40m, M = 50m, k = 0, 0001ms , S0 = 0, 0001m , n0 = 0, 25
Figure 8.27: induced recharge plant with well and river
239 CHAPTER 8. ANALYTICAL SOLUTIONS
240 Chapter 9
Numerical methods
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 9. NUMERICAL METHODS
The horizontal plane groundwater flow equation in general form of a nonlinear partial differential equation is not completely solvable. By local and the time coordinates quantization a numerically solvable discontinuous model can be set up. In the liter- ature the term discretisation is also often used instead of quantization and the discontinuous model is called discrete model. This misuse of the words in the literature is caused by careless separation of indepen- dent and dependent variables. The quantization of an independent variable leads to the discontinuous, while the quantization of a dependent variable leads to the discrete model.
The discontinuous models can be simply adapted to the structures of data models, which are generally existent in the form of samples.
The quantization of local variables should first be regarded independent of the time variable. This is justifiable in any case of steady processes, but in unsteady procedures these classes of the variables can be also regarded as independent of each other. Only in special interpolation scheme some connections of these two quantization procedures come up and must be treated separately.
We assume the continuous function of the piezometer head or the position of free groundwater surface in the original (zR(x,y,t)), then we get a discontinuous function
(zR(xi,yi,ti)) by the discontinuous simulation. Subsequently, the problem editor will try again to approximate a continuous function from it. Following demands result from setting up tasks for the execution of quantization:
The demands are contradictory to some extent. Above all the demands of the theo- retical informa- tion side contradict practical application. Thus a search of an optimal organization of tactile points will be also carried out in quantization.
• No information loss appears in the quantization of function, since otherwise the continuous cannot be retrieved one to one from the discontinuous function. • No redundant data processing is caused by quantization, i.e. the distance of the tactile point is not too small to select.
Further demands concerning quantization result from the used simulator and the ex- isting input data:
• the quantized field is designed in such a way that the hydro geological and techni- cal/ technological conditions of the original can be clearly, physically descriptively and with high accuracy taken into consideration. • quantization must allow a simple simulation.
242 The demands are contradictory to some extent. Above all the demands of the theo- retical information side contradict practical application. Thus a search of an optimal organization of tactile points will be also carried out in quantization.
243 9.1 Methods of local quantization
The quantized values of the x- and y- axis yield tactile points in the x-y plane, which can be distributed arbitrarily. These points are connected with straight lines, and we get a reticular structure. The tactile point function is from nodes. The function values at the tactile points or the nodes are supposed to be known as possible solution of the simulation so that the values can be interpolated along the net lines and within the mesh. If the function value (e.g. water level, temperature, concentration) is represented as z-axis, then we get a three-dimensional plane, which is supported by the given function values at the nodes. With respect to practical simulation we differentiate various network configurations, which can be divided according to the following criteria (see table 9.1 and figure 9.1).
Table 9.1: Einteilung von Netzformen
Coordinate reference coordinaten true coordinate independent Quantisation equidistant arbitrary quantisized Topology regular irregular Geometry orthogonal triangular
The different network configurations possess advantages and disadvantage concerning the fulfilment of initial demands. The regular network configurations allow a relatively simple subsequent treatment by means of simulators. However they poorly fulfil the demand that hydro geological conditions should be considered under minimum simu- lation expenditure. In addition a redundant data processing cannot be avoided due to small increments. The irregular network configurations, particularly the coordinate- independent, can not be well simulated. The network configuration can be however well adapted to the hydro geological and technical/technological conditions by the ar- bitrary distribution of the nodes. The substantial advantage in the arbitrary node density distribution is that, they can lie as at close quarters or as far away as required. Thus a minimized redundancy of data processing is realized with minimum number of nodes. The disadvantage of the irregular network configurations is a complicated execution of the simulation. We can make a compromise if we transfer the arbitrary network configurations into topologically regular, but geometrically irregular triangle nets. the topological organization of the net is crucial for the execution of the simu- lation. There are always six connections from a net point to neighbouring points in the topologically regular triangle net. Finally lots of connections exist to neighbouring nodes in a arbitrary triangle net.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 9.1. Methods of local quantization
Figure 9.1: network configurations
245 CHAPTER 9. NUMERICAL METHODS
Independent of the organization of network configuration a further effect appears in the modelling of hydraulic areas by means of numerical models. The generally infinite flow conditions in an aquifer are artificially limited by the finite expansion of mathematical nd ∂h models. The edges of model belong to 2 type boundary conditions ( ∂n = const). Thus ˙ at the edge of the model a volume flow VRand = 0 is outwards forced. And the barriers of model edges are set to be equal. This is often contrast to the hydraulic conditions of original process. This effect is recognizable from the fact that the equipotential lines, e.g. water level, the so called isolines, which in principle stand perpendicularly at the model edge. We can minimize this error in order to apply natural 1st and 3rd boundary conditions to the model boundary such as rivers and lakes, as well as considering possible colmation effects. With know hanging influx or phreatic divide, in particular with groundwater basin borders, we can input these at the model border as 2nd boundary condition.
There are different ways to describe the geohydraulic behaviour within the mesh. The most known ones are method of finite differences (FDM), the method of finite volumes (FVM) and the method of finite elements (FEM). In FDM it is assumed that the exchange takes place along the mesh borders and in the nodes. In contrast the basic idea of FEM consists in the fact that the network mesh is regarded as continuum and the reciprocal effect with the neighbour elements is achieved by energy, impulse and mass exchange perpendicularly through mesh bound. In the following FDM will be described in detail. The other quantization procedures such as FEM and FVM are only roughly represented.
9.1.1 Finite Differencen Method
As previously mentioned, no time-dependent procedures are considered in local dis- cretisation. Therefore the derivation of local quantizations is independent of whether an unsteady or steady flow field. On this account the following assertions are derived on the basis of the steady flow equation. They can be also applied to the unsteady flow regime under consideration of the mathematical conditions (see section 9.2 time quantization, page 270).
246 9.1. Methods of local quantization
9.1.1.1 Balance equation
The differential equation of the horizontal plane groundwater flow in steady case is:
div (T · grad h) = 0 (9.1) h confined with h = aquifer zR unconfined D k(x, y, z)M confined Z and T = = k(x, y, z)dz k(x, y, z)zR unconfined z=a
The steady flow is characterized by the fact that no storage procedures appear. This partial differential equation can be written in cartesian coordinates as follows: karte- sischen Koordinaten wie folgt schreiben:
∂ ∂h ∂ ∂h T + T = 0 (9.2) ∂x ∂x ∂y ∂y
If the finite difference method is applied to this equation with reference to a coordinate, topologically regular grid, then this is equated with transfer of derivatives in difference quotients:
∆ ∆h ∆ ∆h T + T = 0 (9.3) ∆x ∆x ∆y ∆y
The quantization error, which occurs during this transition, has a quadratic order (O(x, y) ∼ ∆x2, ∆y2). We can also describe this error as the deviation of the secant, which is used in the difference quotient, and the tangent, which is defined by the derivative. This deviation is dependent on the gradients, i.e. the slope of the tangent.
Also the geometrical size of the boundary conditions is changed. In the continuum each boundary condition has an arbitrary finite geometrical expansion. In the dis- continuous field each boundary condition can accept only one expansion, which is an integral multiple of the quantization increment ∆x and ∆y. Generally this means that a substantially larger effect area is arranged in the discontinuous simulated network and the original of this case is the boundary condition. In this relation the effect of
247 CHAPTER 9. NUMERICAL METHODS
Figure 9.2: transition between continuum and resistance network
the finite expanded net is again pointed out. The net edges lead to a limitation of flow ˙ field, since no flow rate flows at the borders (VRand = 0). In original however it does not have to be so. For this reason the edge of model should agree with the position of the hydraulic boundary conditions if possible.
Apart from the mathematical derivative the quantization procedure can be also phys- ically justified. For this purpose the continuum is covered with an appropriate mesh. At the individual nodes the balance equations, here the mass balance equations, are set up. The connections are represented by flow resistances between the net nodes.
The mass balance equation at the point Pn,m results from the sum of water quantities, which flow along the net lines (see figure 9.3). The sum must be equal to zero accord- ing to definition (under steady condition steps no storage effect). Thus the balance equation is:
˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m = 0
248 9.1. Methods of local quantization
Figure 9.3: balance for node n, m
The flow rates can also be represented as quotient of differences of the water level or pressure and hydraulic flow resistances or as product of differences of the water level or pressure and hydraulic conductivity (general Darcy law, also see theory of pipe- and channel hydraulics), and we get:
Gx n−1,m(hn,m − hn−1,m) + Gx n,m(hn,m − hn+1,m)+ (9.4)
Gy n,m−1(hn,m − hn,m−1) + Gy n,m(hn,m − hn,m+1) = 0 or arranged according to water height:
hn−1,m(−Gx n−1,m)+
hn,m−1(−Gy n,m−1)+
hn,m(Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m)+ (9.5)
hn,m+1(−Gy n,m)+
hn+1,m(−Gx n,m) = 0
If we go through the grid net in columns, i.e. the nodes of row 1 to my are processed within the columns 1 to nx, then it yields an equation system with nx · my rows and
249 CHAPTER 9. NUMERICAL METHODS with nx · my unknown quantities. This high dimensional equation system, in practical until several hundred thousand or millions, can be solved according to Gauss total step iteration, Gauss Seidel or by means of iterative methods, in particular the method of preconditioned conjugate gradients (PCG) (see section 1.3 linear equation system, page 23ff).
If we arrange these equations according to the elements hn,m we get an equation system (see table 9.3), which has the characteristic diagonal shape. If we represent this as matrix equation, we get the coefficient matrix G with nx · my columns and just as many as rows. This coefficient matrix is only located in certain places, namely in the diagonals, the two directly bordering second diagonals and two further second diagonals, which is nx distance from main diagonal. All other elements of the matrix are equal to zero. Since this matrix is still symmetrical, the actual significant value space reduces to 3 · nx · my elements (see section 1.2.1 band matrix, page 13). The solution function, i.e. the water levels hn,m at the nodes, is represented by a column vector nx · my. In contrast on the right side a column vector nx · my stands likewise.
An example of the quantization of a two-dimensional aquifer by means of a net with four columns and four rows (see figure 9.4) as well as the equation system will be demonstrated.
Figure 9.4: 4 x 4 grid net
This example yields following 16 equations (see table 9.2):
250 9.1. Methods of local quantization
Table 9.2: equation system for two dimensional aquifer quantization 0 = 0 ) = +1 ) = 0 ) = 0 ) = 0 ))) = 0 = 0 ) = 0 )) = 0 = 0 = 0 ) = 0 2 3 4 2 2 3 4 2 3 4 3 n,m , , , , , , , , , , , 4 4 4 1 2 2 2 3 3 3 1 ) = 0 h h h h h h h h h h h h h − − − − − − − − − − − − − 1 2 3 1 1 2 3 1 2 3 2 3 n,m , , , , , , , , , , , , 4 4 4 1 2 2 2 3 3 3 1 1 h ( h h h h h h h h h h h h ( ( ( ( ( ( ( ( ( ( ( ( 2 3 2 3 1 2 3 1 1 1 2 3 , , , , , , , , , , , , 4 4 4 1 2 2 2 3 3 3 1 1 y n,m y y y y y y y y y y y y G G G G G G G G G G G G G )+ ,m )+ )+ )+ )+ ))+ )+ )+)+ )+ )+ = 0 = 0 = 0 )+ +1 1 1 1 2 3 4 2 3 4 2 3 4 n , , , , , , , , , , , , 3 4 2 2 2 2 3 3 3 4 4 4 h h h h h h h h h h h h h − − − − − − − − − − − − − 1 1 1 2 3 4 2 3 4 2 3 4 n,m , , , , , , , , , , , , 2 2 2 3 3 3 2 3 1 1 1 1 h ( h h h h h h h h h h h h ( ( ( ( ( ( ( ( ( ( ( ( 1 1 1 2 3 4 2 3 4 2 3 4 , , , , , , , , , , , , 2 3 1 2 2 2 3 3 3 1 1 1 x n,m x x x x x x x x x x x x G G G G G G G G G G G G G )+ 1 equation − n,m )+ )+ )+ )+ )+ )+ )+ )+ )+ = 0 )+ )+ )+ h 1 2 3 1 2 3 1 2 3 1 2 3 , , , , , , , , , , , , 2 2 2 3 3 3 4 4 4 1 1 1 − h h h h h h h h h h h h n,m − − − − − − − − − − − − h 2 3 4 2 3 4 2 3 4 2 3 4 ( , , , , , , , , , , , , 1 2 2 2 3 3 3 4 4 4 1 1 1 − h h h h h h h h h h h h ( ( ( ( ( ( ( ( ( ( ( ( 1 2 3 1 2 3 1 2 3 1 2 3 , , , , , , , , , , , , 2 2 2 3 3 3 4 4 4 1 1 1 y n,m y y y y y y y y y y y y G G G G G G G G G G G G G )+ ,m 1 − n )+ )+ )+ )+ )+ )+ )+ )+ )+ )+ )+ )+ h 1 2 3 4 1 2 3 4 1 2 3 4 , , , , , , , , , , , , 1 1 1 1 2 2 2 2 3 3 3 3 − h h h h h h h h h h h h n,m − − − − − − − − − − − − h 1 2 3 4 1 2 3 4 1 2 3 4 ( , , , , , , , , , , , , 2 2 2 2 3 3 3 3 4 4 4 4 ,m h h h h h h h h h h h h 1 ( ( ( ( ( ( ( ( ( ( ( ( 1 2 3 4 1 2 3 4 1 2 3 4 − , , , , , , , , , , , , 1 1 1 1 2 2 2 2 3 3 3 3 x n x x x x x x x x x x x x G G G G G G G G G G G G G 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 Nr. n,m 1 2 3 4 n col
251 CHAPTER 9. NUMERICAL METHODS
Table 9.3: equation system in diagonal shape
row 1
h1,1 h1,2 h1,3 h1,4
column Nr. · · · · Gx 1,1 1,1 −Gy 1,1 +Gy 1,1 Gy 1,1 1 1,2 −Gy 1,1 +Gx 1,2 −Gy 1,2 +Gy 1,2 Gy 1,2 1,3 −Gy 1,2 +Gx 1,3 −Gy 1,3 +Gy 1,3 Gy 1,3 1,4 −Gy 1,3 +Gx 1,4 2,1 −Gx 1,1 2 2,2 −Gx 1,2 2,3 −Gy 1,3 2,4 −Gy 1,4 3,1 3 3,2 3,3 3,4 4,1 4 4,2 4,3 4,4
252 9.1. Methods of local quantization
Table 9.4: continuation 1
row 2
h2,1 h2,2 h2,3 h2,4
column Nr. · · · · 1,1 −Gx 1,1 1 1,2 −Gx 1,2 1,3 −Gx 1,3 1,4 −Gx 1,4 Gx 1,1 2,1 +Gx 2,1 −Gy 2,1 +Gy 2,1 Gx 1,2 +Gy 2,1 2 2,2 −Gy 2,1 −Gy 2,2 +Gx 2,2 +Gy 2,2 Gx 1,3 +Gy 2,2 2,3 −Gy 2,2 −Gy 2,3 +Gx 2,3 +Gy 2,3 Gx 1,4 2,4 −Gy 2,3 +Gy 2,3 +Gx 2,4 3,1 −Gx 2,1 3 3,2 −Gx 2,2 3,3 −Gx 2,3 3,4 −Gx 2,4 4,1 4 4,2 4,3 4,4
253 CHAPTER 9. NUMERICAL METHODS
Table 9.5: cotinuation 2
row 3
h3,1 h3,2 h3,3 h3,4
column Nr. · · · · 1,1 1 1,2 1,3 1,4 2,1 −Gx 2,1 2 2,2 −Gx 2,2 2,3 −Gx 2,3 2,4 −Gx 2,4 Gx 2,1 3,1 +Gx 3,1 −Gy 3,1 +Gy 3,1 Gx 2,2 +Gy 3,1 3 3,2 −Gy 3,1 −Gy 3,2 +Gx 3,2 +Gy 3,2 Gx 2,3 +Gy 3,2 3,3 −Gy 3,2 −Gy 3,3 +Gx 3,3 +Gy 3,3 Gx 2,4 3,4 −Gy 3,3 +Gy 3,3 +Gx 3,4 4,1 −Gx 3,1 4 4,2 −Gx 3,2 4,3 −Gx 3,3 4,4 −Gx 3,4
254 9.1. Methods of local quantization
Table 9.6: continuation 3
row 4 RS
h4,1 h4,2 h4,3 h4,4
column Nr. · · · · 1,1 = 0 1 1,2 = 0 1,3 = 0 1,4 = 0 2,1 = 0 2 2,2 = 0 2,3 = 0 2,4 = 0 3,1 −Gx 3,1 = 0 3 3,2 −Gx 3,2 = 0 3,3 −Gx 3,3 = 0 3,4 −Gx 3,4 = 0 Gx 3,1 4,1 −Gy 4,1 = 0 +Gy 4,1 Gx 3,2 4 4,2 −Gy 4,1 +Gy 4,1 −Gy 4,2 = 0 +Gy 4,2 Gx 3,3 4,3 −Gy 4,2 +Gy 4,2 −Gy 4,3 = 0 +Gy 4,3 Gx 3,4 4,4 −Gy 4,3 = 0 +Gy 4,3
255 CHAPTER 9. NUMERICAL METHODS
The calculation of conductivity can be achieved according to following scheme (see figure 9.5)
Figure 9.5: allocation of hydraulic parameter to compensatory conductivity
The smallest quantization area is the area, which results from around the regarded point Pn,m with the edge lengths ∆x and ∆y. According to the quantization regula- tions this area will be uniformly parameterised, i.e. the parameters of the aquifer such as the k value, the storage coefficient S, the position of the aquiclude a, the through flow thickness D and the current piezometer head or the situation of the free ground- water surface zR are considered as independent of location x, y within this planning element. A change of these values can take place only at the borders of the planning element. There however large jumps may arise. Thus the parameters and state vari- ables of the aquifer are reflected by means of FDM with discontinuous functions in the quantized net. These usually contradict continuous functions in original process. For the flow processes this means this planning element is regarded as homogeneous aquifer with horizontal bed and horizontal groundwater level. According to the quan- tization step an interpolation between the nodes is not allowed. This corresponds to
256 9.1. Methods of local quantization the same statements, which apply to quantized signals (see Graeber: textbooks to the lectures ”Automatisierungstechnik” or ”Grundwassermesstechnik”). If we consider these premises, the individual hydraulic conductivity can be defined. Generally: A G = k · (9.6) l whereby A is the perpendicularly through flow area and l is the parallel through flow length. A results from the flow width b and the through flow thickness D, which is equal to the thickness of the confined aquifer or the free groundwater height zR. The differential conductivity of a streamline is: dz · b dG = k · (9.7) l or the total conductivity of a perpendicularly through flow area results from paral- lel connection of the individual streamline conductivity. The parallel connection is regarded as summation or integration of the differential conductivity:
D Z b G = k dz l z=a D b Z G = kdz l z=a T · b G = l with: D Z T = kdz
z=a M confined D = aquifer zR unconfined This integral expression of the transmissibility will be evaluated numerically poorly, since the permeability coefficient is a step function and can not be represented as continuous function. Thus the transmissibility is always a piecewise linear function of the variable z and thereby can be written as sum formula:
n X T = kl((zl+1 − zl)Γ1 + (zR − zl)Γ2) (9.8) l=0
257 CHAPTER 9. NUMERICAL METHODS with:
0 zR ≤ zl+1 Γ1 = 1 zR > zl+1 0 zR > zl+1 Γ2 = 1 zl ≤ zR ≤ zl+1
zl are the absolute heights of the different soil layers and kl are the pertinent per- meability coefficients. The single conductivity (indices O, W, S, N) is computed as follows:
∆y ∆y G = T = 2 · T x n,m O n,m ∆x n,m ∆x 2
Gx n,mW = Gx n,m O (9.9)
Since it is presupposed that in the planning element n, m all parameters, including though flow thickness, are constant, each pair of conductivities can be assumed equal:
∆x ∆x G = T = 2 · T y n,m S n,m ∆y n,m ∆y 2
Gy n,m N = Gx n,m S (9.10)
The interconnection of two partial conductivities, e.g. the Gxn,mO and the Gxn+1,mW results in the conductivity between two nodes. It is know from the fluid engineering or electro-technology that the series connection of two resistances is their sum:
258 9.1. Methods of local quantization
R = R1 + R2 (9.11) 1 1 G = = (9.12) R R1 + R2 1 = 1 + 1 G1 G2 G · G = 1 2 G1 + G2 with:
Gxn,mO · Gxn+1,mW Gxn,m = (9.13) Gxn,mO + Gxn+1,mW ∆ym ∆ym 2Tn,m · 2Tn+1,m = ∆xn ∆xn+1 ∆ym ∆ym 2Tn,m + 2Tn+1,m ∆xn ∆xn+1 1 ∆ym 2Tn,m · Tn+1,m ∆xn ∆xn+1 = 1 1 Tn,m + Tn+1,m ∆xn ∆xn+1 2T · T ∆y = n,m n+1,m m Tn,m∆xn+1 + Tn+1,m∆xn For equidistant decomposition, i.e.:
∆xn = ∆xn+1 = ∆x bzw.
∆ym = ∆ym+1 = ∆y we get: 2Tn,m · Tn+1,m ∆y Gxn,m = (Tn,m + Tn+1,m) ∆x Generally: ∆ym Gxn,m = T ∆xn it yields: T · T T = 2 n,m n+1,m (9.14) Tn,m + Tn+1,m
This harmonious averaging also corresponds to the hydraulic real condition very well, in which the smaller permeability coefficient or the smaller transmissibility dominantly intersperses. The computation of transmissibility is complicated in the steady case with
259 CHAPTER 9. NUMERICAL METHODS unconfined aquifer, since here it is a matter of a value, which depends on the solution of the equation system. In the example of an unstratified aquifer the transmissibility is:
Tn,m = kn,m · zRn,m (9.15)
In this case it is necessary to compute the equation system iteratively. And we start (1) with an estimated value zR n,m. The better this estimated value to the true solution zR n,m approaches, the fewer iteration steps have to be implemented. The first approx- (1) imation for the transmissibility Tn,m can be computed by means of the estimated value and the coefficient matrix with the appropriate conductivity can be developed. It leads (2) to the solution zR n,m, an improved approximation of the exact solution zR n,m. This is again used for the computation of the improved transmissibility Tn,m11(2). The proce- dure is continued, until the deviation of two approximations is smaller than a certain limit ε. (i) (i+1) z − z R n,m R n,m < ε (i) zR n,m
9.1.1.2 Consideration of boundary conditions
The preceding accomplishment for the quantization of the continuum applies to the case of uninfluenced groundwater flow. For a unique simulation boundary conditions must work as in the original procedure, and no groundwater flow comes about without it. With the discontinuous models on the basis of finite differences method the boundary conditions affect in principle on the nodes. Especialy by consideration of 1st type boundary conditions difficulties occure.
In the numerical simulation at least on one node a 1st type boundary condition must work. Models, which are endued with 2nd or 3rd type boundary conditions, do’t yield unique simulation results. The case of the infinitely expanded aquifer, which can be calculated by means of the Theis analytical solution, does not exist in discontinuous models.
In the following the realization possibilities will be indicated for the different kinds of boundary conditions. The 2nd type boundary condition, which affects on a node (see figure 9.6), can be considered as follows based on the balance equation:
260 9.1. Methods of local quantization
Figure 9.6: consideration of 2nd type boundary condition
˙ ˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + VRB 2. Art n,m = 0
˙ ˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m = −VRB 2. Art n,m (9.16) | {z } | {z } unknown quantities known quatities
Thus the 2nd type boundary condition can be directly written on the right side of the equation. With introduction of the potential differences for the volume flow this equation turns into to the accustomed matrix equation for grid network, whereby for the nodes with 2nd type boundary condition the right side is different from zero.
With 3rd type boundary condition (see figure 9.7) we proceed in the same way. The balance equation is built for the node, on which the boundary condition affects. The boundary condition 3rd type is by definition a potential decrease of a flow resistance, which is nothing else but a variable flow rate. With a 3rd type boundary condition the potential difference between a given potential (e.g. water level of a receiving stream or another surface water) and the water level at the point of aquifer, where the boundary condition affects, will be built:
261 CHAPTER 9. NUMERICAL METHODS
Figure 9.7: consideration of a 3rd type boundary condition
˙ ˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + VRB 3. Art n,m = 0
˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + Ganstrm n,m(hn,m − hF lu) = 0
˙ ˙ ˙ ˙ Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + Ganstrm n,m · hn,m = Ganstrm n,m · hF lu | {z } | {z } unknown quantities known quatities
(9.17)
With introduction of potential difference for all volume flows:
Gx n−1,m(hn,m − hn−1,m)+
Gy n,m−1(hn,m − hn,m−1)+
Gx n,m(hn,m − hn+1,m)+ (9.18)
Ganstrom¨ n,mhn,m+
Gy n,m(hn,m − hn,m+1) = Ganstrom¨ n,mhF luss n,m
262 9.1. Methods of local quantization or arranged according to h:
hn−1,m(−Gx n−1,m)+
hn,m−1(−Gy n,m−1)+
hn,m(Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m + Ganstrom¨ n,m)+ (9.19)
hn,m+1(−Gy n,m)+
hn+1,m(−Gx n,m) = Ganstrom¨ n,mhF luss n,m
Thus we recognize that with existence of a 3rd type boundary condition the main diagonal also has further addends besides the right side different from zero.
1st type boundary conditions will be treated in the discontinuous model as a spe- cial case of 3rd type boundary condition with evanescent flow resistance. Since the balance equations of the nodes orient flow rates, the potential of the boundary condi- tion won’t arise explicitly. Hydraulically this mathematical step is quite meaningfully interpretable, since the 3rd type boundary conditions can be also regarded as combi- nation of a 1st type boundary condition and a flow resistance. If the flow resistance approaches to zero, i.e. the potential loss between groundwater level and surface water disappears, this is equated with the influence of a 1st type boundary condition. Based on this conclusion the derivation can be taken over according to the above statements of 3rd type boundary condition:
hn−1,m(−Gx n−1,m)+
hn,m−1(−Gyn,m−1)+
hn,m(Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m + GRB 1.Art n,m)+ (9.20)
hn,m+1(−Gy n,m)+
hn+1,m(−Gx n,m) = GRB 1.Art n,mhF luss n,m
In this equation we set Ganstrom¨ n,m = GRB 1.Art n,m since it is a matter of mathematical st shaping of a 1 type boundary condition. An evanescent resistance, i.e. Ranstroem → 0, means a conductivity GRB1.Art n,m → ∞. This is not numerically realizable here. Since
GRB1.Art n,m has influent conductivity in additive linkage to the others at the node
263 CHAPTER 9. NUMERICAL METHODS
n, m, it is sufficient that the conductivity GRB1.Art n,m dominates the summation of conductivity. This is given, if the following inequation is fulfilled:
GRB 1.Art n,m >> Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m
This inequation can be regarded as fulfilled if:
GRB 1.Art n,m = 100 · (Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m)
Due to the numerical instabilities within the equation solution GRB1.Art n,m should not be select too large.
In some simulation programs 1st type boundary conditions are also interpreted as infinitely large storage effects. This however only works in the application of unsteady flow regime.
If the boundary conditions are located outside the nodes, which is in the majority of cases, and not all the field element boundary condition properties are arranged, then the boundary condition can be connected with four of resistances of neighbouring nodes (see figure 9.8). The computation of resistances and the associated allocation of boundary condition effect on the neighbouring nodes can be achieved according to geometrical conditions, i.e. according to the distance between boundary condition and nodes and the appropriate effect range. The effect range results from the gravity centre of the representative area between the connecting lines of gravity centres and the adjacent nodes. And the gravity centres should have coordinatesxMn,m; yMn,m.
Figure 9.8: outside nodes lying boundary condition
264 9.1. Methods of local quantization
b G = T with (9.21) n,m,RB n,m l
q 2 2 b = (xMn,m − xMn+1,m) + (yMn,m − yMn+1,m) (9.22)
q 2 2 l = (xn,m − xRBn,m) + (yn,m − yRBn,m) (9.23)
The mathematical formulation of different kinds of boundary conditions can be achieved according to above forms for 1st , 2nd , 3rd type boundary conditions.
265 CHAPTER 9. NUMERICAL METHODS
9.1.2 Finite Element Method
The finite element method (FEM) represents a further kind of continuum transfer into a quantized representation. It differs from the FDM (see section 9.1.1 finite differences method) in the determination of planning elements and parameters. While in the FDM the balance is essentially computed on the basis of the node equation, in the FEM takes the balance over the boundaries of the planning elements.
Arbitrary compounds in the form of polyhedrons can be selected as planning elements. In the two-dimensional level the planning elements become planes, which are formed by arbitrary closed polygons. The simplest form is the formation of triangle elements. Each higher order planning elements can be formed by the summary of more triangles.
In the planning elements potential functions, e.g. water level, piezometer head, con- centration distribution, temperature or others, are presupposed as linear, homogeneous conditions. Thus the distribution can be computed analytically within the planning elements. This computation can be achieved by means of variation calculation or according to the Galerkin method.
According to the principle of the variation calculation an approximation function ∗ P( x, y) of the quantized continuums is looked for the potential distribution P(x, y) (e.g. h, zR, ϕ) in the entire regarded area G. Since by definition linear system status dominates within the planning element, for a triangle element:
∗ P(x,y) = a + b · x + c · y (9.24)
In any cases this equation must fulfil the potential distribution at the supporting place, the triangle points i, j, k:
∗ Pi (x,y) = a + b · xi + c · yi
∗ Pj (x,y) = a + b · xj + c · yj
∗ Pk (x,y) = a + b · xk + c · yk
Then we get three equations with three unknown quantities a, b, and c, which can be solved. As matrix equation:
266 9.1. Methods of local quantization
−1 a 1 x y P ∗ i i i b = 1 x y · P ∗ (9.25) j j j ∗ c 1 xk yk Pk
∗ To insert the solution into the equation about P( x, y) yields the solution of searched function. For the planning element m the potential distribution can be expressed as following: ∗ ∗ ∗ ∗ Pm(x, y) = Wi(x, y) · Pi + Wj(x, y) · Pj + Wk(x, y) · Pk The weight functions W (x, y) are linear in x and y direction in the region G and subject to orthogonal conditions, i.e.: 1 n = m Wn(xm, ym) = 0 n 6= m
m, n ∈ i, j, k
Individually:
1 W (x, y) = [(x · y − x · y ) + (y − y ) · x + (x − x ) · y] i 2∆ j k k j j k k j 1 W (x, y) = [(x · y − x · y ) + (y − y ) · x + (x − x ) · y] j 2∆ k i i k k i i k 1 W (x, y) = [(x · y − x · y ) + (y − y ) · x + (x − x ) · y] k 2∆ i j j i i j j i 1 mit : ∆ = · [x (y − y ) + x (y − y ) + x (y − y )] 2 i j k j k i k i j ∆ is the area of the planning triangle, and:
Wi(x, y) + Wj(x, y) + Wk(x, y) = 1 (9.26)
P ∗ is continuous in the entire range G. If the basic values are known at the nodes, the function is representable in the whole area. In the following it is to be demonstrated ∗ how to determine the basic values Pi of the continuum such that the potential values Pi of the quantized area are adapted in best way. This is achieved according to Galerkin method. The set up of the differential equations apply to the continuum exactly. In
267 CHAPTER 9. NUMERICAL METHODS the case of the horizontal plane groundwater flow equation (see equation 7.14, page 197): ∂ ∂z ∂ ∂z T R + T R = 0 ∂x ∂x ∂y ∂y In quantized system in contrast only following approximation is valid:
∂ ∂z∗ ∂ ∂z∗ T R + T R = r(x, y) 6= 0 ∂x ∂x ∂y ∂y r(x, y) is designated as residue. The better the quantization area to the continuum ∗ ∗ adapts, the smaller the residue is. The approximation solution Pi , here concretely zRi, converges for the continuum solution P or zR or other expressions in case of infinitely small planning elements or infinitely large number of supporting places, nodes and planning elements. And the residue becomes zero.
According to the weighted residues method the approximation solution will be searched in such a way that the residue disappears at the weighted mean. This is accomplished with the following expression for each planning element:
ZZ Wi(x, y) · r(x, y) dG = 0 (9.27) G
Under constant transmissibility condition in individual planning elements, the residue can be written by definition:
ZZ ∂2z∗ ∂2z∗ W (x, y) · T · R + R dx · dy = 0 (9.28) i ∂x2 ∂y2 G It was still considered that the time dependence should be identically zero. This applies to the steady processes. But also the unsteady processes can be treated in such way, since the time dependence is regarded independent of local quantization (see section 9.2 time quantization method, page 270). This area integral must be solved for each supporting point or net point. We get n equations weighting functions with ∗ n unknowns. For the approximation solution zR at the net point a linear substitution was selected, which however has no second derivative. On this account the integral must be transformed with the 1st Green formula . Generally:
ZZ ∂2w ∂2w ZZ ∂w ∂u ∂w ∂u Z ∂w u · + dxdy = − · + · dxdy + u dB ∂x2 ∂y2 ∂x ∂x ∂y ∂y ∂n G G B (9.29)
268 9.1. Methods of local quantization
Thereby B means the boundary of the area and n is unit normal, which stands per- pendicularly on the boundary and is directed towards outside. w(x, y) and u(x, y) are arbitrary scalar potential functions. This equation applies to the residue:
ZZ ∂z∗ ∂W ∂z∗ ∂W Z ∂z∗ − T · R · i + R · i dx dy + T · W R dB = 0 (9.30) ∂x ∂x ∂y ∂y i ∂n G B The line integral of the boundary B describes the potential-independent flow over the boundary and can be regarded as boundary condition for the planning element n: Z ∂z∗ Z T · W R dB = W · q0 dB = V˙ i ∂n i n B B
0 ˙ whereby qn means the specific flow rate per unit length and V is the influx to the node n. In the same way internal boundary conditions e.g. wells can be also considered.
Similar to the finite differences method a high-dimensional equation system also devel- ops here in the FEM. In contrast to FDM the FEM does not produce diagonal band matrices, but an irregular matrix structure develops as a function of the number of affecting planning elements. This leads to a increased numerical expenditure in the solution equation system.
269 9.2 Time quantization methods
On the basis of general form of horizontal planes groundwater flow equation ∂h h confined div (T grad h) = −S mit h = aquifer ∂t zR unconfined the independent of treatment on the left side, local functionality, on the right side, the time dependence of a quantization must be undergone. Quantization of place dependence can be achieved by the described method in the section 9.1. The temporal derivative must be transferred into a difference quotient, since otherwise there are no possible simple numerical treatment. The construction of an appro- priate equation system is only again possible by this transfer. The transfer from the derivative into a difference quotient should be visualised by first backward difference method as implicit procedure.
Therby we use: ∂h h − h ≈ t t−∆t ∂t ∆t
Figure 9.9: Relationship from tangent to secant with a typical drawdown procedure
Due to the introduction of the temporal difference quotient, a time point must be also arranged at the convection part, i.e. the left side of the equation. Different methods
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 9.2. Time quantization methods for time quantization are differentiated for allocating the time to the left side of the differential equation, i.e. the local flow process (see figure 9.9). The most substantial distinctions lie between one- and multi-step procedures as well as between the explicit and implicit procedures. With the explicit one step method the equation system can be solved directly, since the parameters are known from the preceding time step. The explicit method, also designated as forward difference, however has large disadvantage that for mathematical stability reasons only very small time steps (in minute order of magnitude) are realizable and an extremely large number of time steps (total simulation time divide by the time increments) must be worked out for a test run. The implicit one step method is also called backward difference method. It also yields stable solutions for large time steps and thus represents the standard method for numerical simulation systems. A series procedures are developed to decrease the quantization error (to multi-step method, Predictor Corrector method, higher order method, see figure 9.10), and also a special extrapolation procedure by Graber¨ .
Figure 9.10: Argument association in time quantization of a 1-D-field problem
271 CHAPTER 9. NUMERICAL METHODS
9.2.1 Backward difference - Implicit methods
In the implicit one step method, also designated as backward difference or Liemann method, the local flow part of the horizontal planes groundwater flow equation will be considered to time point t: ∂h h confined div (T grad h) = −S mit h = aquifer (9.31) ∂t zR unconfined
After the transfer from the temporal derivative into the difference quotients: h − h div(T grad h) ≈ −S t t−∆t (9.32) |t ∆t Instead of the squiggly equal sign the equals sign is mostly used, which actually is not exact. If we implement quantization again also on the left side and insert the (local-) conductivity according to physical FDM method, then balance equation at node n, m arises (see figure 9.11): The expression
S ∆xn∆ym = Cn,m (9.33) will be designated as hydraulic storage effect or capacity. The so called time conductivity is introduced for the quotient from capacity and time step. S ∆x ∆y C n m = G = n,m (9.34) ∆t z n,m ∆t This represents the flow rate in connection with the temporal potential, which is re- leased or received from aquifer due to storage effect within the time step ∆t.
˙ VZeit n,m = GZeit t,n,m (ht,n,m − ht−∆t,n,m) (9.35)
h ˙ ˙ ˙ ˙ ˙ i Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + VZeit n,m = 0 t h ˙ ˙ ˙ ˙ i Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + GZeit t,n,m (ht,n,m − h∆t,n,m) = 0 (9.36) t
h ˙ ˙ ˙ ˙ i Vn−1,n + Vn+1,n + Vm−1,m + Vm+1,m + GZeit t,n,m · ht,n,m = GZeit t,n,m · h∆t,n,m t | {z } | {z } known quantities unknown quantities
(9.37)
272 9.2. Time quantization methods
Figure 9.11: consideration of time quantization
d hn−1,m(−Gx n−1,m)+ e
| hn,m−1(−Gy n,m−1)+ |
| hn,m(Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m)+ | (9.38)
| hn,m+1(−Gy n,m)+ |
b hn+1,m(−Gx n,m) c |t 1 = −S ∆x ∆y (h − h ) · n m n,m t n,m t−∆t ∆t
If this is inserted into the upper equation system and within a row we arrange the known and unknown variables, then we get the following system, in which only known
273 CHAPTER 9. NUMERICAL METHODS quantities stand on the right side:
d hn−1,m(−Gx n−1,m)+ e
| hn,m−1(−Gy n,m−1)+ |
| hn,m(Gx n−1,m + Gx n,m + Gy n,m−1 + Gy n,m + Gz n,m)+ | (9.39)
| hn,m+1(−Gy n,m)+ |
b hn+1,m(−Gx n,m) c |t
= hn,m t−∆t(Gz n,m)
Thus the pentadiagonal equation system remains. A known quantity from the preced- ing time step was added on the right side. The main diagonal was also extended by the addends Gz n,m. The developed matrix equation is not explicitly solvable due to the potential dependence of conductivity, in particular the transmissibility with unconfined aquifer. Therefore iteration must be carried out. In the first step the conductivity G(1) is computed according to the initial water levels (zR t−∆t). For groundwater drawdown procedures this means that the transmissibility and the conductivity are assumed too (1) large. The matrix can be solved with these values, and we get water levels zRt , which are too low compared with real situation. Improved conductivity G(2) can be computed (1) with this first approximation zRt , which leads to the second approximation of water (2) height zRt . This exceeds the true solution, since the conductivity was assumed too small and too little discharge was realized. The solutions approach to true value in form of a damped oscillation. This iteration process will be continued until the changes between two iterations do not exceed an error bound any longer. Then we get the so- lution for the time step ∆t. Nevertheless the iteration within the time step remains a quantization error, which grows proportionally with ∆t, since the secant between the points t and t − ∆t is calculated instead of tangent at the point t. Since the groundwater flow processes accordingly approach to an abating exponential function asymptotic steady final state, the time quantization error is not only depen- dent on the increment ∆t, but also dependent on the dynamics of process. In the figures 9.12 and 9.13 the results are displayed to defined time for the example of one dimensional ditch flow with different time increments. The results with the increment ∆t 24 are assumed accurate. We clearly recognize the strong dependence of time quanti- zation error on temporal gradients. In figure 9.14 the convergence of the solution as a function of time increment is observed. At the same time we recognize the small quantization errors of extrapolation.
274 9.2. Time quantization methods
Figure 9.12: Time quantized computed drawdown of a ditch flow
Figure 9.13: Time quantized computed groundwater rise of a ditch flow
275 CHAPTER 9. NUMERICAL METHODS
Figure 9.14: Dependence of the time quantization error on the time increment
276 9.2. Time quantization methods
9.2.2 Mixed methods
The backward difference is often applied for simplifying time quantization like already described. That means the parameters will be determined for the time point t. This leads to difficulties with nonlinear parameters, like under unconfined groundwater con- ditions, since they must be adjusted iteratively. There is therefore a series of attempts, by weighted allocation of local partial derivative as well as the parameter to quantized time, in order to achieve an error decrease. Generally we can write:
(1 − γ) div(T grad h)| t−∆t + γ div (T grad h)|t (9.40) (h − h ) = (1 − γ) S + γS t t−∆t |t−∆t | t ∆t Depending upon the selection of the method (see figure 9.10, page 271) we get the following γ-values: 0 explicit method 1 2 Crank-Nicolson scheme γ = 2 Galerkin weighting 3 1 implicit method
Apart from the one step method (also with iteration cycles), in which only the time point t and t − ∆t play a role, multi-step procedures are frequently used for the sim- ulation to decrease the time quantization error. The Predictor Corrector method is ∆t common. In the Douglas Jones method, a two-step method, a half step 2 will be attended according to implicit solution scheme (γ = 1), and all parameters are adjusted to time t − ∆t and ht−∆t/2 (predictor step) in the substitution. The Crank Nicolson scheme (γ = 1/2) realizes a total step δt (corrector step), whereby all pa- rameters are set to time point t − ∆t/2 (see section 5.4.1 numerical integration, page 161). Very high approximation accuracies can be also obtained that not only the time derivatives at local quantization point Pn,m but also at neighboring nodes are taken into consideration. In simplest form according to the Simpson rule (by an example of one dimensional case): dh 1 dh dh dh ≈ + + (9.41) dt 6 dt |n−1 dt |n dt |n+1
277 CHAPTER 9. NUMERICAL METHODS
A special scheme is suggested by Stoyan, with which all partial derivatives are subject to a controlled weighting. Thus a very stable and exact numerical solution is obtained, which changes into analytical solution for the case of net convection. The disadvantage of this method consists of the fact that we try to reduce the time quantization error ef- fects by the manipulation of the differential equation remainder, usually the parameter of local convection term (the right side of the differential equation). Thus the cause of error remains untouched. It generally leads to no satis- factory solution and numerical instabilities possibly.
278 9.2. Time quantization methods
9.2.3 Extrapolation method
A very effective method to decrease time quantization error is indicated by Graeber in extrapolation method. For the balancing processes, in which the horizontal plane groundwater flow arises, the function h = h(t) is always continuous monotonically increasing or decreasing between the time points t − ∆t and t. It yields that the secant of backward difference is always smaller than the tangent at the point t according to amount. Thus the following inequation:
dh ht − ht−∆t ≤ dt ∆t The inequation can be changed into equation by introducing a correction factor: dh h − h = t t−∆t (9.42) dt K · ∆t Thus the time quantization error is reduced and can converge to zero by proper se- lection of K. K is ≥ 1 by definition. Approximately the local point Pn,m can be represented by the following equivalent circuit diagram (see figure 9.15). H stands for equivalent potential (1st type boundary condition) and R is an equivalent resistance, which summarizes the hydraulic characteristics of the aquifer between the neighbour nodes and the regarded nodes. It could be e.g. the entire quantization network or a st 1 type boundary condition. Ct stands for the effective storage effect of the aquifer in this time step. For this equivalent circuit weg get the solution (see sections 5.2.1 first
Figure 9.15: equivalent circuit diagram of local point Pn,m order ordinary differential equations, page 125 and 12.1 transmission behaviour with first order delay, page 344):
∆t − hn,m t = (H − hn,m t−∆t) 1 − e τ (9.43)
279 CHAPTER 9. NUMERICAL METHODS
whereby τ is designated as time constant τ = RC. In place of the capacity Ct an equivalent resistance Rz can be computed now, which engenders the same groundwater flow as the capacity C for the time t:
hn,m t ˙ dhn,m t Rz n,m t = with: V = Ct (9.44) V˙ dt ∆t − τ 1 − e τ
= ∆t − Cn,m t e τ
The difficulty consists of determining the time constant τ. An analytical expression ∆t cannot be found. Therefore the quotient τ is determined from the system behaviour during the offset procedure. We simulate the system according to backward difference in first step, i.e. with the time resistance Rz n,mt = ∆t/Cn,m t. The change between the potentials hn,m t and hn,m t−∆t serves as basis for the time constant computation. For the case of the drawdown (hn,m t ≥ hn,m t−∆t):
∆t hn,m t−∆t Rz n,m t = (9.45) Cn,m t hn,m t
or
Cn,m t hn,m t Gz n,m t = ∆t hn,m t−∆t and for a groundwater rising process:
∆t (hmax − hn,m t−∆t) Rz n,m t = (9.46) Cn,m t (hmax − hn,m t)
or
Cn,m t (hmax − hn,m t) Gz n,m t = ∆t (hmax − hn,m t−∆t)
Here two relatively simple expressions are developed for the corrected time resistance. These solutions are numerically stable and are characterised by a good convergence behaviour and a very small quantization error. Accordingly a 24 times smaller time step is developed (see figure 9.14, page 276) with application of the backward difference. Using larger time steps means a substantial economisation of computing time.
280 9.3 Exercises of numerical calculation
1. By means of one dimensional steady ditch flow calculate the position of free surface as a function of x, the outflow from head water and the inflow to bottom water (see figure 9.16). Use five quantization elements.
Figure 9.16: stratified aquifer with steady flow regime
2. By means of one dimensional steady ditch flow calculate the position of free surface as a function of x and t (0 to 2d), the outflow from head water and the inflow to bottom water (see figure 9.17). Use five quantization elements and five time steps. Select the time step according to the expected gradients.
Figure 9.17: stratified aquifer with steady flow regime
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 9. NUMERICAL METHODS
3. In a aquifer a tunnel (underground) will be built parallel to a river (see figure 9.18). Compute how does the groundwater condition for steady case change caused by this construction. Select a suitable rough quantization scheme.
Figure 9.18: tunnel construction in an aquifer
4. In a plain tract the polder area is to be protected against floods by means of a dyke construction (see figure 9.19) (according to simplified scheme). −4 m −1 Dyke: k = 10 s , n0 = 0, 15,S0 = 0, 002m −5 m −1 Sealing material: k = 10 s ; n0 = 0, 05; S0 = 0, 001m a) develop the simple discrete scheme to estimate the groundwater flow pro- cesses. b) How much water flows into the polder area per meter dyke length?
Figure 9.19: dyke construction and core seal
282 9.3. Exercises of numerical calculation
5. Expected groundwater flow conditions will be simulated for an induced recharge waterworks (see figure 9.20). · 3 −1 −1 −1 given: V = 0, 001m s , S0 = 0, 0001m , n0 = 0, 25, k = 0, 001ms , zR(t=0) = 10m,
hF l = 10m, M = 15m, b = 1m a) develop the simple quantization scheme with three nodes to estimate the groundwater flow processes according to the given geometry. b) Set up the three node equations for the instationary flow calculation
c) calculate the water level zR(t) in the GWBR for time point t = 1d.
Figure 9.20: aquifer with river and well
283 CHAPTER 9. NUMERICAL METHODS
6. A numerical groundwater flow model is built for an induce recharge waterworks (see figure 9.21) with parallel flow regime. The river is to be considered as idealized boundary condition. · −3 m l l k = 10 s ; hF l = 15m; zR0 = 15m; S = 0, 25; V = 50 s ; q = 0, 001 sm2 ; b = 100m; −5 m kKolm = 5 · 10 s ; MKolm = 1m; B = 6, 357m
Figure 9.21: influence of river and hanging inflow on the aquifer
a) select a suitable simple quantization scheme with maximal five elements, in order to calculate the water level at gauge P in steady case most possible. b) Formulate the balance equations at the centre of the elements and demon- strate it in matrix form. c) calculate the hydraulic conductivity for the flow. d) How does the equation system and the result change, if the river is not idealized, but imperfection a colmation are considered? Outline the solution and roughly estimate the result.
284 Chapter 10
Simulation programm system ASM
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 10. SIMULATION PROGRAMM SYSTEM ASM
Exercises
1. Simulate the drawdown for the given points with in distance r and at the time t, which results from a water conveying V˙ in the well for the given aquifer and state the result graphically.
−3 m S s k = 1 · 10 s ; M = 10m; S = 0, 001; a = T = 0, 1 m2 ; r0 = 0, 25m · m3 V = 0, 015 s ; hn = 16m;
r = 5m; 10m; 20m; 50m
t = 1; 2; 5; 10; 20; 30; 45; 60; 90; 120 [min]
2. Simulate the drawdown at the point r = 10m for the above given aquifer every 10 minutes until maximally 100 minutes, if the flow rate of this conveying well is subject to following stagger time. Plot the solution.
m3 flow rate [ s ] 0, 005 0, 01 0, 015 0, 02 0, 025 0, 03 0
start of conveying [min] 0 10 20 30 40 50 60
3. A foundation pit should be lowered in an aquifer near a river. The centre of the foundation pit is 100m far away from the river; the drainage well is 80m away. Three wells are arranged parallel to the river, which are 25m distant from
each other. The diameter of wells r0 = 0.3m and the conveying capacity is ˙ m3 V = 0.015 s The width of the river is B = 20m and it has a colmation layer of 0 −6 m k = 3 · 10 s ; M‘ = 1m. The properties of the aquifer are: −4 m k = 5 · 10 s ; n0 = 0, 20; hn = 15m; M = 20m. Will a drawdown of 2.5m be achieved in 10 days in the centre of the foundation pit? 4. Verify with the simulation programm ASM whether the centre of the foundation ˙ m3 pit is drained after 7 days with conveying capacity of V = 0.01 s , r0 = 0.30m and a security of 0.5m (see figure 10.1).
l 5. A constant flow rate of 25 s is conveyed from a well (Br), which is located at an ideal river (Br(100m, 500m)). The well has a radius of r0 = 0.35m. T he aquifer is characterized by the following parameters: −3 m hn = 15m, M = 17m, k = 1 · 10 s ,S0 = 0, 0002, n0 = 0, 25.
286 Figure 10.1: aquifer with well and foundation pit
Simulate the final steady state (the portion of temporal functionality should be smaller than 0.001) for the point P (200m, 600m). From when is to calculate with the steady state? 6. Simulate the following one dimensional groundwater flow: a) By means of one dimensional steady ditch flow (see figure 10.2) simulate the position of the free surface as a function of x and investigate the outflow from the head water and the inflow to bottom the water. Use five quantization elements.
Figure 10.2: stratified aquifer with steady flow regime
b) By means of one dimensional unsteady ditch flow (see figure 10.3) simulate the position of the free surface as a function of x and time t. 7. In an aquifer a tunnel (underground) will be built parallel to a river (see figure 10.4). Simulate how does the groundwater condition for the steady case change caused by this construction. Select a suitable rough quantization scheme.
287 CHAPTER 10. SIMULATION PROGRAMM SYSTEM ASM
Figure 10.3: stratified aquifer with unsteady flow regime
Figure 10.4: tunnel construction in an aquifer
8. In a plain tract the polder area is to be protected against floods by means of a dyke construction according to the simple scheme in figure 10.5. a) Determine the time, when a steady flow regime appears, if the flood stands 5m over normal for long time. b) How much water flows into the polder area per meter dyke length? −4 m −1 Dyke: k = 10 s , n0 = 0, 15,S0 = 0, 002m ; −5 m −1 Sealing material: k = 10 s ; n0 = 0, 05; S0 = 0, 001m
9. Model the following horizontal aquifer by means of the program system ASM, which is limited on the right and left side by two perfect complete receiving streams with a water height of 50m. The aquifer possesses a thickness of 20m, a m2 transmissibility of T = 0.01 s , a storage coefficient of S = 0.001 and a porosity ˙ m3 of 0.1. A well with a surveying capacity of V = 0.05 s lies in the centre of the model area. a) Simulate the water level distribution (contour line) after one day well sur-
288 Figure 10.5: dyke construction with core seal
veying. b) Graphically place the water level hydrograph curves at the well every 200m (parallel and perpendicular to the receiving stream). c) Compute the water balance for the model area after one-day surveying, as well as the inflow from left receiving stream. d) Check the hydraulic system of the task of c) the influence of the local and time quantization increments and solution methods. First plot the hydro contour line after one day and compare.
289 CHAPTER 10. SIMULATION PROGRAMM SYSTEM ASM
290 Part III
System theory and Modelling
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management
Chapter 11
Fundamentals
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management The system theory makes up the theoretical frame, with which the control and feedback control engineering can be scientifically investigated. The application of this theory in the water management is particularly important for the process analysis, i.e. for the modelling, as well as for the data extraction, transmission and processing. The system theory yields the fundamentals for the terms information and system. The introduction of information concepts for physical or chemical data enables the possibility to apply different methods of computer science, cybernetics, mathematics and electro technology in planning and realization of water management monitoring control and automation systems.
11.1 Model classification
The modelling or the process analysis can be carried out in theoretical or experimental way.
The physicochemical processes are analysed and mathematically formulated under the help of the scientific laws in the theoretical modelling and process analysis (see section 11.2.1, pages 300). In this way the model structures, if possible, the model parameters of the internal influence mechanism are determinable. Analysing the ob- jects takes place from inside to outside. The mathematical models are scientifically justified. The input- and output signals of the objects are measured and evaluated in the exper- imental modelling and process analysis. Natural or artificial test signals are applied. The analysis of objects takes place from the outside.
The disadvantages of theoretical modelling and process analysis are unreliability with insufficient process knowledge and high expenditure with complex processes. The dis- advantages of experimental modelling and process analysis consist of the only selective model validity in contrast to the necessity in the real experimental process and the difficulty of the scientific interpretation.
Usually it is favourable to combine both methods and to a large extent the model structure are theoretically and the model parameters are experimentally determined.
The justified models play a dominant role for the migration processes. The experi- mental modelling above all gains importance lately. The results of the experimental process analysis, the transfer functions, are usually difficult to interpret or physically imagine with the real processes. Therefore this method often comes across baseless scepticism.
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 11.1. Model classification
Like migration processes models the computers, hard- and software, can be also as- sumed as models (see figure 11.1). An important problem, which arises during pro- cessing of migration problems on computers, is the coupling of migration models to models on computers. In this con- nection the computer is regarded as simulators for the migration processes. However the coupling is impossible trouble free, if the models of simulators are not identical. Such differences arise e.g. in the consideration of the arguments (continuous, discontinuous) or the allocation of parameters and variables of state. In these cases an approximation must be accomplished between the two models.
Figure 11.1: coupling of migration- and simulator model
In investigation area the model basis contains different mathematical models of dy- namic procedures, chemical processes and biological phenomena, i.e. the underground flow processes with the coupled material -, energy -, exchange -, and transformation processes as well as the migration processes in soil- and groundwater region. Thereby basis is generally nonlinear, and coupled material circulation. Simplified, aggregate models can be found for special cases.
Krug classifies the mathematical models into the justified and describable models (see figure 11.2) according to the model development background.
The describable models are used e.g. for ecological systems and population problems. For the modelling of technical systems the class of the justified models is meaningful.
295 CHAPTER 11. FUNDAMENTALS
Figure 11.2: model classification by Krug
These are classified in:
• linear - nonlinear • continuous - discontinuous and • dynamic - static
The question of modelling is in connection with the process control extremely impor- tant. The achievable quality of the control- and feedback control task solution depends particularly on whether sufficient qualitative and quantitative knowledge of plant con- trolled system, here the soil and groundwater region, are available. The concept of mod- elling must be considered closely connected with process analysis. Toepfer/Besch suggests following classification principles for model application in the automation tech- nical investigation area.
Model Classification according to:
• Model extraction method Theoretical model extraction/process analysis (laws of nature) Experimental model extraction/process analysis (experiments) • Model purpose of use Construction-, calculation-, behaviour models handling-, function models • Model notation
296 11.1. Model classification
Mathematical models in equation form/parametric models (equations) Mathematical models in graphic form (signal flow chart), non parametric models (curve, pair of variates) Physical models (analogy model, graphic model)
• Model conclusions Static models Dynamic models • Model adaptability Prediction models Adaptive Models Adapted Models • Relationship of variables Deterministic/stochastic models Linear/non-linear models • Model validity Type models (for classes of Objects) Special models (for concrete object)
The term model is usually not used uniformly. We can understand it as a triangle relation between model, object and subject. The model is characterized not only by, about what the model is, but also by for what it is. The object is also denoted by model original. The relationship between model and original is always a kind of image relation.
The quality of a model is measured by
• relative compatibility in view to the subject • relation fidelity of the image and the behaviour • application simplicity
The status of process analysis, model and simulation within the scope of control and regulation is shown in the following figures 11.3 and 11.4. Thereby the model formu- lation of the soil- and groundwater processes should be understood in further process analysis. This can be also called procedure modelling. In contrary the progress of orig- inal process is reproduced in the simulation. The necessary parameters and variables of state are communicated to models and the process cycle starts.
297 CHAPTER 11. FUNDAMENTALS
Figure 11.3: devices and technical programming realization
298 11.1. Model classification
Figure 11.4: process character of modelling and simulation
299 11.2 Methods of process analysis
By means of process analysis methods a mathematical description of the behaviour of transfer elements is to be extracted. This process is also denote by modelling. These models can be attained in two different ways, on one hand by means of theoretical, on the other hand by means of experimental Process analysis. While the theoretical process analysis yields the model structure, the related parameters must be determined or improved by the experimental analysis. In contrast to the theoretical proceeding an investigation of the output signals takes place as system response to input signals in the experimental process analysis. Both methods form a unity and complement each other, because an experimental analysis without theoretical advance information and a theo- retical analysis without experimental supporting are hardly feasible.
11.2.1 Theoretical process analysis
With the theoretical process analysis based on internal structure the transfer el- ements are tempted in order to find the mathematical descriptions (models) between in- and output variables. Transfer functions, which are formed on the basis of the theoretical process analysis, are always justified by natural laws. They always possess physical or chemical bases in water management application. The characteristics of the theoretical process analysis consist of the fact that:
• the model can be already carried out before the practical realization, • expensive experiments for investigating of important properties can be reduced, • the analysis results with same process type are transferable, • a processing of diverse studies is possible by computer models, • a system optimization can be realized by giveb target functions and constraints, • the connections between technological and constructional data remain, • the process determinant variables are identified in the system and • important statements about the model structure are attained.
The difficulties of this method consist of the fact that:
• the expenditure is very high and the models are complicated, • the necessary process parameters can be achieved often very hard and only inac- curately,
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 11.2. Methods of process analysis
• the proceeding is poor with respect to algorithm and • the physicochemical process must be acquainted sufficiently.
For setting up mathematical models within the scope of theoretical process analysis a fact has been proved that, large systems by division into subsystems and then by individual balance equations (mass-, energy and momentum conservation law as well as source- and sink activities) are analysable.
11.2.2 Experimental process analysis
The experimental process analysis in contrast to the theoretical assumes the in- vestigation of system in- and output signals. The systems are thereby regarded as transfer elements. Artificial experiments are in progress at the original system, and special attention must be dedicated to the choice of input signals. If the execution of experiments is not possible, also natural phenomena (e.g. flood waves) can be used as database. We also mention that the experimental process analysis checks the systems from the outside.
The methods of experimental process analysis are also known as black box method in cybernetics.
Both methods form a unity and complement each other, because an experimental analysis without theoretical advance information and a theoretical analysis without experimental supporting are hardly feasible.
In table 11.1 some selected characteristics of the two kinds of process analysis are compared.
301 CHAPTER 11. FUNDAMENTALS
Table 11.1: comparison of theoretical and experimental process analysis ¨ Ein-/Ausgangs- und Signalmodelle vorwiegend ohne Struktur des realen Prozesses Kein Zusammenhang zwischen den Modellparametern und konstruktiven, technologischen und anderen Daten leicht m¨oglich im untersuchten Bereich und f¨urden speziellen Prozess Modelle sehr g¨unstigf¨urdie L¨osungder Diagnose-, Uberwachungs-, Steuerungs- und Vorhersageaufgabe Modell erst mit Existenz/Inbetriebnahme des Prozeses erstellbar H¨aufig”St¨orung”des Prozesses erforderlich (aktives Experiment) Experimentelle Prozessanalyse ¨ Uberwachungs-, Zustandsmodelle mit innerer Struktur des realen Prozesses Zusammenhang mit konstruktiven technologischen Daten schwierig in großen Bereichen und f¨urverschiedene Prozesse Modelle h¨aufigzu aufw¨andigf¨ur L¨osungder Diagnose-, Steuerungs- und Vorhersageaufgabe Modell bereits in derProjektphase Entwurfs-/ erstellbar Keine ”St¨orung”des Prozesses bei der Modellerstellung Theoretische Prozessanalyse Struktur Parameter der Modelle Modellvereinfachung derG¨ultigkeit Modelle Aufwand Zeitpunkt Beeinflussung des Prozesses Modelleigenschaften
302 11.3 Signal representation
Processes can be characterized by means of signals. In case of water management processes it means that these can be described by in- and output variables (e.g. water level, flow, chemical concentrations, temperature).
We call a function carried by physical variables signal, if it has a parameter, the image function is a variable of the physico-technical space. In principle a signal is represented by a four dimen- sional function x = f(x, y, z, t) mathematically. In the mathematical description of signals we consider the double meaning of symbol ”x”. It acts as general signal note and as character of local coordinate. Often it is more favourable to use as abbreviation of each physical variable than sig nal characteristics. Signal Parameters are called information parameters , and physical variables are signal carriers, by which the signal is carried. Examples are specified in table 11.2.
Table 11.2: classification of information parameter and signal carrier
application information parameter signal carrier electricity 220V electric current 1A electric current hydraulics 10m flow resistance 1m3 current of water Thermik 273K W¨armepotential 1kW heat flow
In communications technology such signals are used e.g. in the form of voltage states, current and power changes. According to above definition signal and information concept can be also applied in other technical systems, like here in water technical processes. For example the water level, the flow rate and the temperature also appear as signal carriers analogously to current and voltage. Therefore the water level is expressed as three dimensional signal x = f(x, y, t). In addition, chemical material concentrations are conceivable as signal.
For more simple representation usually only time is mentioned as independent vari- able in the following consideration. This should be without loss of generality. The implementations also apply exactly to the dependence concerning local coordinates.
The description of signals can be done with a diagram and by means of mathematical functions. The signals are usually defined with a start time t = 0. It is a matter of relative time to an event. In the mathematical description the original procedure
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 11. FUNDAMENTALS in the so called time domain is usually distinguished from the transformed procedure in complex variable domain. Common transformations for signals are Fourier and Laplace transformation.
Different representations have proved their worth for the mathematical description of technical signals. The basic signals are of great importance (see section 11.3.1 basic signal forms, page 304), since they are the basis of all arbitrary signal forms. By means of basic signals arbitrary signals can be generated (see section 11.3.3 signal synthesis, page 309). Likewise arbitrary signal forms can be decomposed into these basic signals (see section 11.3.4 signal analysis, page 310).
11.3.1 Basic signal forms
The most common basic signal forms (see figure 11.5) are the identical magnitude, sine function, step function (unit step) and the DIRAC impulse.
Figure 11.5: basic signal forms
The unit step is a normalised signal with step height (step height = 1) and is represented by 1(t). The Dirac impulse only affects at t = 0 and has an infinite step height there. Unit step and Diracimpulse are connected with each other mathematically by the integration or deviation.
304 11.3. Signal representation
11.3.2 Application of selected test signals
Test signals can be effectively used for the execution of the experimental process anal- ysis (see figure 11.6). With these a special experiment must be accomplished to the real object. It is possible under different technical or technological conditions that only special test signals can be used. Since the same system description develops indepen- dently of test signal type and the different descriptive models are transferable, there are no restrictions in the experimental process analysis method.
Apart from the application of these special test signals the system reaction of natural events, i.e. natural signals, like flood waves, precipitation events etc. can be also drawn on for the determination of transient characteristic. This will be realized by the application of faltung operation (see section 12.3 arbitrary transient characteristic, page 363).
There are following special definitions for the test signals according to Topfer¨ , where the auxiliary term ”unit” always contains a normalisation to the value one.
11.3.2.1 Impulse function
The impulse function is defined as: 0 f¨ur t < 0 A xe(t) = f¨ur 0 ≤ t < ∆t , (11.1) ∆t 0 f¨ur t > ∆t whereby ∆t is the impulse width. The area of impulses amount to:
∆t Z A = xe(t)dt (11.2) 0 The area for a impulse with constant height and a definite impulse duration:
A = xe · ∆t (11.3)
The impulse area embodies an appropriate effect in form mass or energy deposit. A finite pulse width will be always available with technical impulses. If the pulse width is
305 CHAPTER 11. FUNDAMENTALS smaller than a tenth of the smallest time constant (∆T < 0, 1τ) (see section 12.2 second order transient characteristic, page 349), then it can be regarded as an approximately ideal impulse.
If we presuppose the fact that the area remains constant, and for ∆t −→ 0 the pulse amplitude must approach to infinite xe −→ ∞ (see also section 12.2.3 Dirac impulse as input signal, page 355),
lim xe · ∆t = A (11.4) ∆t−→0 xe−→∞ 0 f¨ur t < 0 xe(t) = ∞ f¨ur t = 0 , 0 f¨ur t > 0 whereby the impulse area A has a definite value:
+0 Z A = xe(t)dt (11.5) −0 The so called Dirac impulse, also designated as unit impulse, arises when the impulse area is normalised to value one: 0 f¨ur t < 0 xe(t) δ(t) = = ∞ f¨ur t = 0 A 0 f¨ur t > 0
+∞ Z Aδ(t) = δ(t)dt = 1 (11.6) −∞
11.3.2.2 Step function
The step function is defined as: 0 f¨ur t < 0 xe(t) = xe0 f¨ur t ≥ 0
306 11.3. Signal representation
If the step height is normalized, we get the unit step : x (t) 0 f¨ur t < 0 1(t) = e = (11.7) x e0 1 f¨ur t ≥ 0
Please note that the bar signal already takes value xe0 at the time t = 0.
11.3.2.3 Ramp function
The ramp function, also designated as slope function, is defined as: 0 f¨ur t < 0 xe(t) = c · t f¨ur t ≥ 0
A unit function can be also generated here by normalization. The unit ramp func- tion: x (t) 0 f¨ur t < 0 α(t) = e = (11.8) c t f¨ur t ≥ 0 Among the different illustrated test signals, in particular the unit signals, such connec- tion exists that they are transferable with each other by integration or deviation (see table 11.3).
Table 11.3: relationship between different basic signals
unit impulse unit step unit ramp Dirac impuls δ(t) 1(t) α(t) unit impulse d1(t) d1(t) d2α(t) Dirac impuls δ(t) = δ(t) = = dt dt dt2 δ(t) unit step dα(t) 1(t) = R δ(t) dt 1(t) = 1(t) dt unit ramp α(t) = R 1(t) dt α(t) = R 1(t) dt α(t) = RR δ(t) dt
307 CHAPTER 11. FUNDAMENTALS
Figure 11.6: test signals
308 11.3. Signal representation
11.3.3 Signal syntheses
Several input signals xe are additively merged to an output signal xa at a mixing place in the signal synthesis (overlay).
n X xa = xei (11.9) i=1
The output signal can be determined based on mathematical equation of the mixing place by a signed addition or by means of graphic methods (see figure 11.7).
Figure 11.7: signal syntheses
309 CHAPTER 11. FUNDAMENTALS
11.3.4 Signal analysis
The signal analysis of arbitrary signal forms can be achieved via a decomposition in basic signal forms. The most common method is Fourier series decomposition, with which periodic signal sequences are approximated by different frequencies sinusoidal oscillations. A practically well manageable method with for deadbeat signals, i.e. one time expiring signals, is the approximation through temporally staggered signals. In the following the graphic method should be described, since in contrast to mathematical one it is substantially more simply manageable and more de- scriptive.
The first step with signal analysis by means of bar signals consists of the fact that arbitrary time function is approximated by a echelon form signals (see figure 11.8). The time of echelon flank and the step height should be selected in such a way that the smallest mean error occurs. It has to be proved the integrals of the original curve and the step function draw near. It means that the same areas must be represented by both curves (see script for lecture groundwater meas- uring technique, section error calculation).
Arbitrary signals can be also decomposed into a sum of impulses, particularly into an infinite sum of Dirac impulses. This leads to faltung integral method (see section 12.3 arbitrary transient characteristic, page 363).
Here it should be pointed out again that the exemplary signal representation is ap- plicable to all arguments as time function. The special place dependence in xi and yi direction plays an important role in soil and groundwater range processes as well as in contaminated site treatment.
310 11.3. Signal representation
Figure 11.8: approximation arbitrary signals by bar signals
311 CHAPTER 11. FUNDAMENTALS
11.3.5 Quantization
The display format of signals can be classified according to different criteria. With respect to technology we differentiate signal classifications according to information parameter quantization and according to independent variable quantization.
In the classification according to information parameter quantization we get:
• analogy signals, whose information parameter can be assumed any intermediate value in a metric magnitude and • discrete signals , whose information parameter can be assumed only definite (finitely many) values within certain limits.
The two mentioned classification principles can be also combined and we get the display format shown in the figure 11.9.
Figure 11.9: representation forms of signals
The following subdivisions can be made for the class of independent variable quan- tization:
312 11.3. Signal representation
• continuous signals, for those the information parameter to any value, and • discontinuous signals, for those the information parameter can be only indicated for finitely many values of arguments.
Some measuring instruments and methods are specified in table 11.4 as examples of appearances of different signal forms.
It is still pointed out that the use of term cannot be always kept exactly as the Ger- man industry standards specified. These will be after all conditional due to different application of some terms in the foreign language literature. Thus there are no clear separation between the terms ”discrete” and ”discontinuous”. The word ”discretisa- tion” is often used for ”independent variable quantization”. Also the term ”digital” (digital signals) is often used for discrete measured values, if they are indicated by means of number tablets. Digital measured values are rightly measured values from an information parameter quantization, whereby only the quantization stages ”0” and ”1” (or ”0” and ”L”) are allowed.
Table 11.4: Measuring instruments and methods and signal forms
signal form electrical engineering geohydraulics analog continuous X-Y-writer Schreibpegel Plotter analog discontinuous Punktschreiber Tiefenlot Fallb¨ugelschreiber discret continuous Digitalvoltmeter Widerstandsmesskette (Stufenverschl¨usselung) discret-discontinuous Digitalvoltmeter Brunnenpfeife (Zeitverschl¨usselung)
313 11.4 Transmission systems
The following methods serve the determination of dynamic behaviour of undisturbed systems. Since this is only technically idealized possibility, it must be required that the input signals substantially dominate compared to the disturbing signals.
Further important prerequisites for application of the methods are:
• the system is undisturbed (disturb << wanted signal).
• the system behaviour is linear.
• the system behaviour is time invariant.
• the system has only one input and one output signal.
• the system behaviour is describable by concentrated parameters.
In the system theory, particularly in connection with the experimental process analysis, each process can be represented as so called ”black box”, which is only characterized by the relation of in- and output variables. This ”black box” is then designated as system. A system is always identified by the boundary to its environment and coupled information exchange (see figure 11.10). With respect to the system theory we differen- tiate between concrete and mathematical systems. A concrete system is a spatially delimited part of reality, including some selected connections in its internal structure and its environment. The mathematical systems contain variables, equations or operators.
Figure 11.10: system with its connection to environment
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 11.4. Transmission systems
11.4.1 Mathematical description
Different methods emerge in the description of systems. The approach in connection with technical systems as transmission systems is most widely used. Each system is identified by a input quantity, which is a quantity dedicated by output variables (see figure 11.11). This functional connection between the out- and input variables is called transient characteristic. In the following for each case only the relation of a input- or an output variable is regarded. System with several in- and output variables, so called multi signal systems, can be treated with coupled equation system method.
Figure 11.11: transfer element
The description of the systems by the influence of in- and output signals can take place in diverse forms. The most common are the mathematical equations and the time response diagram of the step response function. Three kinds of definition equations are used for the description of the transient characteristic. According to the basic signal relation, which are applied at input, we get the transit, weight and transfer function:
• The transit function h(t) we get if a unit bar signal is applied at input. This is also designated as step response function:
h(t) = xa |xe=1(t) (11.10)
• The weight function g(t) is yielded if a Dirac impulse act on input, and is also designated as impulse response function:
g(t) = xa |xe=δ(t) (11.11)
• The transfer function G(p) is yielded from Laplace transformation description of output signal if the input signal is the Dirac impulse:
G(p) = Xa |Xe=L{δ(t)} (11.12)
315 CHAPTER 11. FUNDAMENTALS
We differentiate between the time (original) domain and the complex variable domain in the description of the transient characteristic (see figure 11.12), with signals, and the transfer function, a transformation is subordinated. The most common integral transformations are the Fourier and the Laplace transformations (see section 5.3.2 Laplace transformation, page 147). The advantage of the application of transfor- mations consists of the fact that complicated arithmetic operations can be usually simplified with transfer functions in the complex variable domain based on four basic arithmetic operations. The disadvantage includes the poor descriptiveness of the com- plex variable domain as well as the expenditure to transform signals and mathematical models into complex variable domain and back again after solving the transfer func- tion (inverse transformation). While prefabricated correspondences usually exist for the forward direction, the inverse transformation often proves more complex.
The designations of signals and transfer functions are lowercase letters in the time domain, in contrast capital letters in complex variable domain.
Figure 11.12: relationship between time and complex variable domain
The Fourier transformation and its inverse transformation are defined as:
∞ Z Fourier transformation X(jω) = F {x(t)} = x(t)e−jωtdt (11.13)
−∞ ∞ 1 Z inverse transformation x(t) = F −1 {X(jω} = X(jω)ejωtdω; (11.14) 2π −∞
316 11.4. Transmission systems
The Laplace transformation and its inverse transformation are defined as:
∞ Z Laplace transformation X(p) = L {x(t)} = x(t)eptdt (11.15)
0 c+j∞ 1 Z inverse transformation x(t) = L−1 {X(p} = X(p)eptdp (11.16) 2πj c−j∞
While the Fourier transformation is favourable for periodic or periodised signals, the Laplace transformation has been proved its worth for application to bar signals (e.g. switching operation) and impulses (Dirac impulse). Particularly the Laplace transformation is also used for the computation of transient characteristic of arbitrary input signals (see section 12.3 faltung operation, page 363).
These three types, transition , weight and transfer function, the mathematical repre- sentation of transient characteristic, are equivalent, since they are only different mathe- matical computation forms for the same physical or chemical technical processes. They are therefore transferable with each other by means of mathematical connection (see table 11.5). According to the definition, an integral or differential connection exists between the unit bar signal and the Dirac impulse (also see table 11.3, page 307), and the arithmetic rules for Laplace transformation (see section 5.3.2 Laplace transfor- mation, page 5.3.2), yield the connections between different description types.
Table 11.5: relationship between different functions of transient characteristic
h(t) g(t) G(p) t 1 h(t) h(t) = R g(τ)dτ h(t) = L−1 G(p) 0 p dh(t) g(t) g(t) = g(t) = L−1 {G(p)} dt
G(p) G(p) = p L {h(t)} G(p) = L{g(t)}
The methods of experimental process analysis introduced in the following can be only applied under definite conditions. Usually these conditions can be met in the water management processes investigation. Advanced methods are subject to special litera- ture or are still the research subject nowadays.
317 CHAPTER 11. FUNDAMENTALS
11.4.2 Basic transient characteristics
The basic forms of transient characteristics of technical system can be described in proportional, integral and differential forms as well as delay and duration behaviour.
The system designations concerning the transient characteristic are done by capitalized initial letters of the characteristic or by the system response pictogram on a bar signal at input, i.e. via step response. Examples of the particular transient characteristic are summarized in table 11.6.
Table 11.6: transient characteristics
transient characteristic denotation example proportional P leverage integral I filling of a container (V = const) differential D water hammer in a tubing filling of a container (V = f(t)) delay characteristic T flow processes of 1st order 1 (tubing, ground water) filling of cascaded containers delay characteristic T transport processes of 2nd order 2 (tubing, ground water)
duration characteristics TL conveyor, emulsion tube
These basic forms will be described exemplary based on some examples. More detailed descriptions for 1st and 2nd delay elements are given in chapter 12 model regulation.
11.4.2.1 Proportional characteristic =⇒ P element
Example: first class lever Law of lever:
la · Fa = −lb · Fb (11.17)
318 11.4. Transmission systems
l length of the fulcrum
F force at the fulcrum If we set the forces as input and the others as output signals, we get proportional transient characteristic:
lb Fa = − · Fb (11.18) la
xa = K · xe (11.19)
The transfer factor K can be determined either in way of theoretical process analysis from geometrical conditions of the lever arms:
l K = − b (11.20) la or by an experiment, the experimental process analysis, by means of known input signals, e.g. the test signal unit step 1(t), and measured output signal. In this case:
x K = a bzw. (11.21) xe
K = xa |xe=1(t) (11.22)
The transient characteristics are determined by inserting corresponding input signals into the equation 11.19 as follows:
• transit function h(t)
According to definition xe = 1(t) will be inserted. Thus:
h(t) = xa |xe=1(t)
= K · 1(t)
h(t) = K (11.23)
319 CHAPTER 11. FUNDAMENTALS
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal. The weight function is expressed as derivative of transit function:
g(t) = xa |xe=δ(t)
g(t) = K · δ(t) (11.24)
• transfer function G(p) The transfer equation of time domain must be transformed in complex variable domain by means of Laplace operation in this transfer function.
G(p) = Xa |Xe=L{δ(t)}
= L{K · δ(t)}
= K · L{δ(t)}
G(p) = K (11.25)
Since only linear systems are regarded according to prerequisite, K = const . Otherwise L{δ(t)} = 1.
11.4.2.2 Integral characteristic =⇒ I element
Example: Filling procedure with constant flow rate · The filling of a container with surface area A by a constant flow rate V leads to a rise of water height h in this container (see figure 5.20).
If we check the dependence of the rising of water height h on the influent flow rate V˙ , then we find out the following relation:
320 11.4. Transmission systems
Figure 11.13: Filling procedure with constant flow rate
V = A · h (11.26)
t Z · V = V dt (11.27)
0
If we equate these two volumes and solve the equation respective to h, then we get:
t 1 Z · h = V dt (11.28) A 0
· For V = const.:
1 · h = V · t (11.29) A
If we consider again the system as transfer element with xe and xa, then
t Z xa = K xe dt (11.30) 0
This equation represents an integral transient characteristic with a proportionality factor K. This can be also decomposed into a series connection (see section 11.4.3 combined transient characteristic) of a pure P element and a pure I part.
321 CHAPTER 11. FUNDAMENTALS
For xe = const:
xa = K · xe · t (11.31)
Also here the transmission constant K can be determined in two ways, by means of theoretical or experimental process analysis. In the first case, equation 11.28, which was derived on the basis of physical laws, is definite:
1 K = (11.32) A
In the second case, the experimental process analysis based on equation 11.30 yields under the condition xe = const.:
xa = K · xe · t (11.33)
xa1 − xa0 K · xe = (11.34) t1 − t0
Particularly if the unit step xe = 1(t) is used as xe, K can be determined by the straight line slope xa = K · t |xe=1(t):
xa1 − xa0 K = |xe=1(t) (11.35) t1 − t0
For t0 = 0 and xa0 = 0:
xa1 K = |xe=1(t) (11.36) t1
The transient characteristics are determined by inserting corresponding input signals into the equation 11.30 as follows:
• transit function h(t)
According to definition xe = 1(t) will be inserted. Thus:
322 11.4. Transmission systems
h(t) = xa |xe=1(t)
t Z = K 1(t) dt
0
h(t) = K · t (11.37)
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal. The weight function is expressed as derivative of transit function:
g(t) = xa |xe=δ(t)
t Z = K δ(t) dt
0
g(t) = K (11.38)
according to definition:
t Z δ(t)dt = 1
0 • transfer function G(p) The transfer equation of time domain must be transformed in complex variable domain by means of Laplace operation in this transfer function.
G(p) = Xa |Xe=L{δ(t)}
t Z = L{K δ(t) dt}
0 t Z = K · L{ δ(t) dt} = K · L{1}
0 1 G(p) = K · (11.39) p
323 CHAPTER 11. FUNDAMENTALS
Since here applies:
t Z δ(t) dt = 1 (11.40)
0 1 L{1} = (11.41) p
11.4.2.3 Differential characteristic =⇒ D element
Beispiel: Transfer elements with differential behaviour come up in electro-technology and serve in control practice to affect processes with a certain mass or energy deposit in a definite objective. In water management practice it appears in connection with oscillation phenomena such as water hammer in pipes. Differential transient characteristic is characterized by equation 11.42.
dx x = K · e (11.42) a dt
• transit function h(t)
According to definition xe = 1(t) will be inserted. Thus:
h(t) = xa |xe=1(t) d1(t) = K dt
h(t) = K · δ(t) (11.43)
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal. The weight function is expressed as derivative of transit function:
324 11.4. Transmission systems
g(t) = xa |xe=δ(t) dδ(t) = K · dt
g(t) = n.d. (11.44)
• transfer function G(p) The transfer equation of time domain must be transformed in complex variable domain by means of Laplace operation in this transfer function.
G(p) = Xa |Xe=L{δ(t)} dδ(t) = L{K } dt dδ(t) = K · L{ } = K · (pL {δ(t)} − δ(0)) dt
G(p) = K · p (11.45)
11.4.2.4 First order delay =⇒ PT1 element
Example: Filling procedure with variable discharge
If a container is connected with a receiving stream through a hydraulic resistance Rhydr (e.g. gate valve, pipe), the container will have the same water level as in the receiving stream after infinitely long time. The container has a surface area A, and the water level in receiving stream and in the container are hF l and h respectively. The time dependence of water level h is wanted, if the water level hFL has the value hF l over the entire period. h should be equal to zero at time t = 0 (see figure 11.14).
Thus the following equations apply:
325 CHAPTER 11. FUNDAMENTALS
Figure 11.14: Filling procedure with variable flow rate
V = A · h (11.46)
t 1 Z · h = V dt bzw. (11.47) A 0 · dh V = A · (11.48) dt · (h − h) V = F l (11.49) Rhydr dh (h − h) A · = F l dt Rhydr dh h h A · + = F l dt Rhydr Rhydr dh A · R + h = h (11.50) hydr dt F l
If we consider again the system as transfer element with xe and xa, then by considering T = A2 · Rhydr the equation is:
dx T · a + x = Kx (11.51) dt a e
This differential equation has the solution (see section 5.2.1 solution of differential equation, page 125) for the case xe = const:
326 11.4. Transmission systems
− t xa = Kxe 1 − e T (11.52)
Methods for the determination of the parameters K and T are described in detail in section 12.1 model regulation, page 344.
The transient characteristics are determined by inserting corresponding input signals into the equation 11.51 as follows:
• transit function h(t)
According to definition xe = 1(t) will be inserted. Thus:
h(t) = xa |xe=1(t)
− t = 1 − e T · K · 1(t)
− t h(t) = K 1 − e T (11.53)
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal. The weight function is expressed as derivative of transit function:
dh(t) g(t) = x | = a xe=δ(t) dt − t d K 1 − e T = dt
K − t g(t) = e T (11.54) T • transfer function G(p) The transfer equation of time domain must be transformed in complex variable domain by means of Laplace operation in this transfer function. Here we as- sume the differential equation of transient characteristic (see equation 11.51):
G(p) = Xa |Xe=L{δ(t)} dx T a + x = Kx (11.55) dt a e
327 CHAPTER 11. FUNDAMENTALS
The Laplace transformed form (see section 5.3.3 solution of differential equation with Laplace transformation, page 153) is:
T · p · Xa − xa0 + Xa = L{K · δ(t)} (11.56)
with L{δ(t)} = 1 xa0 = 0 K G(p) = X | = (11.57) a Xe=L{δ(t)} 1 + T · p
11.4.2.5 Second order delay =⇒ PT2 element
Example: filling procedure of two cascaded containers with variable discharge
Two containers with the surface areas A1 and A2 as well as the water levels h1 and h2 are cascaded connected one after another. The first one is arranged as the preceding 1st order delay behaviour example and connected receiving stream through the hydraulic resistance Rhydr.1. The second is coupled to the first container through hydraulic resis- tance Rhydr.2. The water level in the receiving stream remains constant value hF l (see figure 11.15).
Figure 11.15: coupled storage cascade
The water level in 1st container results from the derivation according to equation 11.50:
dh A · R 1 + h = h (11.58) 1 hydr.1 dt 1 F l or with T1 = A1 · Rhydr.1:
328 11.4. Transmission systems
dh T 1 + h = h (11.59) 1 dt 1 F l
Similarly for the water level in 2nd container we get:
dh A · R 2 + h = h (11.60) 2 hydr.2 dt 2 1 or with T2 = A2 · Rhydr.2:
dh T 2 + h = h (11.61) 2 dt 2 1
Inserting equation 11.61 into equation 11.59, we get:
d T dh2 + h dh T 2 dt 2 + T 2 + h = h (11.62) 1 dt 2 dt 2 F l d2h dH T T 2 + (T + T ) 2 + h = h (11.63) 1 2 dt2 1 2 dt 2 F l
If we consider again the system as transfer element with xe and xa, and then the equation:
d2x dx T T a + (T + T ) a + x = Kx (11.64) 1 2 dt2 1 2 dt a e
This differential equation has solution (see section 5.2.2.2 solution of differential equa- tion, page 137) in the case of xe = const. The determination of the parameter K,T1 and T2 as well as the solution steps are described in detail in the section 12.2 transient characteristic with 2nd order delay.
t t − T − T xa = Kxe 1 − e 1 1 − e 2 (11.65)
The transient characteristics are determined by inserting corresponding input signals into the equation 11.64 as follows:
329 CHAPTER 11. FUNDAMENTALS
• transient function h(t)
According to the definition xe = 1(t) will be inserted. Thus:
h(t) = xa |xe=1(t)
− t − t = K · 1(t) · 1 − e T1 1 − e T2
− t − t h(t) = K 1 − e T1 1 − e T2 (11.66)
Keep in mind the difference between h (heigth) and h(t) (transient function).
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal. The weight function is expressed as derivative of transit function:
dh(t) g(t) = x | = a xe=δ(t) dt − t − t d K 1 − e T1 1 − e T2 = dt − t − t ! e T1 e T2 g(t) = K + (11.67) T1 T2
• transfer function G(p) The transfer equation of time domain must be transformed in complex variable domain by means of the Laplace operation in this transfer function. Here we assume the differential equation of transient characteristic (see equation 11.64):
G(p) = Xa |Xe=L{δ(t)} d2x dx T T a + (T + T ) a + x = K · x (11.68) 1 2 dt2 1 2 dt a e
The Laplace transformed form (see section 5.3.3 solution of differential equation with Laplace transformation, page 153) is:
2 T1T2 · p · Xa − p · xa0 − xa0 + (T1 + T2) · p · Xa − xa0 + Xa = L{K · δ(t)} (11.69)
330 11.4. Transmission systems
with Lδ(t) = 1 and xa0 = 0
2 T1T2 · p · Xa + (T1 + T2) · p · Xa + Xa = K |Xe=L{δ(t)} (11.70) K 2 = Xa |Xe=L{δ(t)} T1T2 · p + (T1 + T2) · p + 1 K G(p) = (11.71) (1 + T1 · p) · (1 + T2 · p)
11.4.2.6 Duration behaviour =⇒ PTL element
Example: The duration behaviour arises in transportation processes, but a change in the balance occurs, i.e. in the case of pure transport without accumulation effect. Thus this process can be also described by a coordinate transformation. This behaviour also plays an important role, if processes should be considered together with different starting points. In these cases different starting points can be convinced different durations.
The equation for this duration behaviour:
xa(t) = K · xe(t − TL) (11.72)
The transient characteristics are determined by inserting corresponding input signals into the equation 11.72 as follows:
• tranfser function h(t)
According to definition xe = 1(t) will be inserted. Thus:
h(t) = xa |xe=1(t)
= K · 1(t − TL)
h(t) = K · 1(t − TL) (11.73)
• weight function g(t)
In this case the Dirac impulse xe = δ(t) will be used as input signal:
331 CHAPTER 11. FUNDAMENTALS
g(t) = xa |xe=δ(t)
= K · δ(t − TL)
g(t) = K · δ(t − TL) (11.74)
according to the definition:
t Z δ(t)dt = 1
0 • transfer function G(p) The transfer equation 11.72 of time domain must be transformed into complex variable domain by means of the Laplace operation in this transfer function:
G(p) = Xa |xe=L{δ(t)}
= L{K · δ(t − TL)}
= K · L{δ(t − TL)}
= K · e−TLpL{δ(t)}
G(p) = K · e−TLp (11.75)
the LAPLACE transformed form (see section 5.3.2 Laplace transformation , page 147):
L{f(t − a)} = e−apL{f(t)} (11.76) and L{δ(t)} = 1 ist.
11.4.2.7 Overview of basic transient characteristic
An overview of different basic form of the transient characteristic step response func- tions is summarized in figure 11.16. The different kinds of the mathematical represen- tation of transient characteristic, transition, weight and transfer function, for the basic transfer elements are displayed in table 11.7.
332 11.4. Transmission systems
Figure 11.16: basic forms of transient characteristic
333 CHAPTER 11. FUNDAMENTALS
Table 11.7: basic transient characteristic with mathematical description } ) t ( δ ) 2 = xe pT { p L | = 1 L ) p T } pT · impuls p − ) p ( )(1 + t e K K K K 1 a ( · K δ 1 + X { pT K L Dirac ) = (1 + p ( transient function G ) t ( ! δ 2 t = ) T 2 L T xe − | T e ) t T ) t ) t ( − − + t impuls ( t e ( K a 1 ( t δ T 1 x δ T K K δ T · − e description ) = K
t Dirac ( weight function K g 2 t T ) t − ( 1 e ) = L − t xe T T | ) 1 − t ) t e ( ) − t · t δ ( t ( K − · a ( 1 t 1 K T x 1 1 K − · e unit jump ) = K K − t ( 1 h transient function K oder order st nd 1 2 ) t ( e x impulse time delay of delay of properties 1 2 L P proportional ID integral differential PT PT PT
334 11.4. Transmission systems
11.4.3 Combined transient characteristic
The interconnection of linear transfer elements can be ascribed to three basic types,
• the series circuit, • the parallel circuit and • the circle circuit (also designated as feedback or back coupling )
The transfer functions are shown in figure 11.17. And the mathematical description is shown in table 11.8.
Figure 11.17: interconnection of linear transfer elements
335 CHAPTER 11. FUNDAMENTALS
Table 11.8: composite transfer elements ) p ( 2 ) p ) G ( p ) · 2 ( p 2 ) ( G p 1 G ( · ± 1 G ) ) G p p ( ( ± 1 1 1 G G ) = ) = ) = transfer function p p p ( ( ( G G G ) ) t t ( ( 2 2 g g ∗ ± ) ) t t ( ( 1 1 g g ) = ) = t t ( ( weight function g g ) t ( e x )] D ( 1 A ) ) D ( D 2 ( ) 2 B t ( A e ) ± x ) D ) ) ( D 1 D D ( ( ( 2 A 2 2 A A B ) ) ) D D D ( ( ( 1 1 1 differential equation B A B [ ) = ) = t t ( ( a a x x circuit type series parallel circle
336 11.4. Transmission systems
According to the formation law of in series connected transfer elements the trans- fer function of a 2nd order delay element can be calculated as two in series connected 1st order delay elements (see figure 11.18):
Figure 11.18: two series connected PT1 elements
K1 G1(p) = 1 + pT11
K2 G2(p) = 1 + pT12
G(p) = G1(p) · G2(p) (11.77) K K = 1 · 2 1 + pT11 1 + pT12 K G(p) = =⇒ PT2 element (11.78) (1 + pT11) (1 + pT12)
The combined transient characteristic of parallel connected transfer elements can be generated from the addition of individual transfer elements mixture in linear transfer elements. Therefore basic transfer elements are occupied with the same input signal and the outputs are added at mixing place, i.e. the elements are parallel connected (see figures 11.19 and 11.20).
For the PI-element
G1(p) = GP (p) = K1 K G (p) = G (p) = 2 2 I p
G(p) = G1(p) + G2(p) (11.79) K G(p) = G (p) = K + 2 (11.80) PI 1 p
337 CHAPTER 11. FUNDAMENTALS
Figure 11.19: realisation of a PI-element
The transfer function for PI-element:
G1(p) = GP (p) = K1 K G (p) = G (p) = 2 2 I p
G3(p) = GD(p) = K3 · p
G(p) = G1(p) + G2(p) + G3(p) K G(p) = G (p) = K + 2 + K · p (11.81) PID 1 p 3
The circle circuit (feedback systems), which appears in closed control process, is to be explained based on filling level control. This regulation process (seeing figure 11.21) is avowed in Graber¨ ”Grundwassermesstechnik”. We recognize that the forward di- rectional transfer element, filling of the container, has integral transient characteristic; the feedback according to technical construction, float with attached lever and gate
338 11.4. Transmission systems valve, has proportional behaviour. So it re- sults in the computation of total transient characteristic according to figure 11.17 (circle circuit,layout b) and table 11.8, page 336:
G1(p) = GP (p) = K1 K G (p) = G (p) = 2 (11.82) 2 I p G (p) G(p) = 2 (11.83) 1 + G1(p) · G2(p) K2 = p K2 1 + p · K2 1 K2 K1 = = 1 p + K1 · K2 p + 1 K1K2 K 1 G(p) = mit T = (11.84) 1 + T p K1K2
This transfer function corresponds to a behaviour of 1st order delay element. How to attain transient characteristic in experimental away is shown in figure 11.21, and it corresponds likewise to delay characteristic.
339 CHAPTER 11. FUNDAMENTALS
Figure 11.20: realisation of a PID-element
340 11.4. Transmission systems
Figure 11.21: feedback system for water level control
341 CHAPTER 11. FUNDAMENTALS
342 Chapter 12
Modell regulation based on parameters
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management Concrete models and the determination of their characteristics and parameters will be described in the following sections. 1st and 2nd order delay elements play a special role for water management systems. We find this behaviour in filling and transportation procedures not only in geohydraulics, in surface waters, as also in pipes.
Descriptive procedures are described in this section, which can be also partly graph- ically solved. Computational procedures, which are based on method i.e. the adjust- ment of measured values in regression functions, are treated in part IV indirect param- eter identification, page 379. With these procedures optimisation will be applied, e.g. least squares (MKQ).
12.1 Transient characteristic with 1st order delay
12.1.1 Mathematical description
The behaviour of water management systems corresponding to a 1st order delay can be found in all filling procedures with storage effect in connection with flow resistance (see figure 12.1). For the determination of the required hydraulic parameters e.g. the so called pumping tests are used as filling attempts, which can be evaluated by means of the following described methods.
Figure 12.1: equivalent circuit diagram of a transfer element with 1st order delay
According to section 5.2.1 solution methods of ordinary differential equations, page 125, the systems, which consist of a flow resistance and a storage capacity (see figure 12.1) can be described by the following differential equation:
dx RC a + x = Kx (t) (12.1) dt a e dx T a + x = Kx (t) mit T = RC dt a e
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 12.1. Transient characteristic with 1st order delay
The solution of this differential equation with the boundary condition, the input signal is a bar signal (see figure 12.2) (xe t=0 = xe0 ·1(t)) (see section 5.2.1 first order ordinary differential equations, page 125):
− t xa = K · xe0 · 1 − e T (12.2)
t − T h(t) = xa |xe=1(t)= K · 1 − e
Considering the impulse response g(t) is equal to the differential of step response:
dh (t) K − t g (t) = = · e T (12.3) dt T The differential equation can be also solved by means of Laplace transformation (see section 5.3.2 Laplace transformation, page 5.3.2):
dx 1 Kx L a + x = L e dt T a T dx 1 Kx L a + L x = L e dt T a T 1 K pL {x } − x + L {x } = L {x } (12.4) a at=0 T a T e
or xat=0 = 0 K L {x } L {x } = · e a T 1 p + T The transfer function G(p) can be determined from this equation, while according to definition we consider L{δ(t)} = 1 as Laplace transformed Dirac impulse input signal. K G (p) = L {x } | = X | = (12.5) a xe=L{δ(t)} a xe=L{δ(t)} (1 + pT )
These results are already contained in table 11.7, page 334 and have been verified. If such a RC circuit (see figure 12.2) shows a 1st order delay behaviour, then we can conclude backwards uniquely that the 1st order delay behaviour of any system can be alternatively reproduced by such RC circuit. The task of the experimental process analysis is to determine these substitute parameters from the transient characteristic. This does not have to be absolutely physically interpretable. These are parameters,
345 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Figure 12.2: transient characteristic and circuit of a PT1 element which show the same behaviour in the equivalent network as the original system. The equivalent network is a model original procedure.
The determination of the transfer factor K and the time constants T is necessary for clear description of this behaviour. The basic approach of experimental process analysis enables it possible based on a step response function, i.e. the clear regulation of these constants is possible by original reaction on a bar signal.
The transfer factor K can be determined from the transfer element behaviour with 1st order delay for infinite time:
− t xa = K · xe0 · 1 − e T (12.6) with t → ∞ we get: x K = a∞ (12.7) xe∞ For step signals:
xe t=0 = xe t=∞ (12.8) Thus the step response can be also written in following form:
− t xa = xa∞ · 1 − e T (12.9)
From the comparison between the input curve and the output curve (see figure 12.2) or also from above equation we recognize that a proportional behaviour exists in infinite.
346 12.1. Transient characteristic with 1st order delay
There are several ways to determine time constant T :
• determination of time, in which an integer multiple of time constant available • determination of slope at zero point.
12.1.2 Time constant from integer multiples
For the case that the time is an integral multiples of time constant: t x = x 1 − e−n with n = = 0; 1; 2; 3; ... (12.10) at=nT a∞ T
For the step signal as input function applies by definition xe0 = xe∞ and the following table:
t n = 0 1 1, 2 2 3 4 T
xa 0 0, 632 0, 699 0, 865 0, 950 0, 982 xa∞
With: x at=nT = 1 − e−n (12.11) xa∞ This table can be evaluated in such a way that we look for the point on the ordinate, where the ratio xa/xa∞ is a certain value. According to the table a definite ratio between time t and time constant T belongs to this point on the curve (see figure 12.3).
Figure 12.3: time constant determination from its multiple
347 12.1.3 Time constant from slope
The time constant can be also calculated from the tangent at any point t. According to equation 12.9:
− t xa = xa∞ 1 − e T
dxa 1 − t = x e T for t = 0 (12.12) dt T a∞ dx x a | = a∞ dt t=0 T
Figure 12.4: tangent intersection and time constant
We can see an intersection with asymptote of step response function xa∞ during setting u p the straight line equation for tangent. This intersection has a distance tSchn = T .
x x = a∞ t ≡ x T ang T Schn a∞
tSchn = T
Since the measuring errors are largest at the beginning of measurement series, i.e. at time zero, we can also set the tangent at any place. Time difference between tangent point and intersection with the asymptote of step response is then equal to the time constant, because the exponential function possesses a constant slope (see 12.4).
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 12.2. Transient characteristic with 2nd order delay
12.2 Transient characteristic with 2nd order delay
12.2.1 Mathematical description
The behaviour of water management systems corresponding to a 2nd order delay can be found in all transportation procedures with storage effect in connection with flow resis- tance (see figure 12.5). The tracer tests are implemented to determine the associated hydraulic transportation parameters, which can be evaluated by means of following described methods. Also here we can assume the solution of an appropriate differential equation.
Figure 12.5: equivalent circuit diagram of a transfer element with 2nd order delay
According to the equivalent circuit diagram we can set up the following differential equation:
dx R C a1 + x = K x (12.13) 1 1 dt a1 1 e1 dx R C a2 + x = K x (12.14) 2 2 dt a2 2 e2 with coupled conditions:
xe2 = xa1
(12.15) xe = xe1
xa = xa2
349 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS and the time constant:
T1 = R1C1 (12.16)
T2 = R2C2 we get:
dx T a1 + x = K x 1 dt a1 1 e dxa T2 dt + xa = xa1 K2 dxa (T2 +xa) d dt T dxa + x K2 2 dt a T1 + = K1xe dt K2 d2x dx dx T T a + T a + T a + x = K K x 1 2 dt2 1 dt 2 dt a 1 2 e d2x dx T T a + a (T + T ) + x = K x (12.17) 1 2 dt2 dt 1 2 a e We get the solution of this differential equation e.g. through the substitution method (see section 5.2.2.2 differential equation of type b, page 137). The characteristic equa- tion of the homogeneous differential equation:
aλ2 + bλ + c = 0 (12.18)
whereby a = T1T2,
b = T1 + T2 und
c = 1 ist.
b c With the new constants d = a and f = a we get:
λ2 + dλ + f = 0 (12.19) r d d2 λ = − ± − f 1,2 2 4
350 12.2. Transient characteristic with 2nd order delay
With the solution of this quadratic equation we differentiate three cases depending upon radian value. For the regarded technical systems here only the positive case, different from zero radian plays a role.
d2 > f, or .b2 > 2 · c · a 4
2 (T1 + T2) > 2 · (T1 · T2)
Thus we get: r d d2 λ = − ± − f 1,2 2 4 s T + T T + T 2 1 = − 1 2 ± 1 2 − 2 · T1 · T2 2 · T1 · T2 T1 · T2
T1 + T2 1 q 2 = − ± (T1 + T2) − 2 · T1 · T2 2 · T1 · T2 2 · T1 · T2 1 q 2 = − (T1 + T2) ± (T1 − T2) 2 · T1 · T2 1 = (−T1 − T2 ± (T1 − T2)) 2 · T1 · T2 1 1 λ1 = − λ2 = − T1 T2
This yields the solution of differential equation:
t t − T − T xa = xe(t) K1e 1 + K2e 2 (12.20)
The constants K1 and K2 can be determined base on concrete initial- or boundary conditions.
12.2.2 Unit step as input signal(transfer function h(t))
For above transfer element the model parameters can be determined as the behaviour excited by a jump, the step response, in the 1st order delay elements (see figure 12.6).
The behaviour can be described according to the derivation and under the consideration of excitement of a bar signal as follows. 2nd order delay elements, e.g. transportation
351 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Figure 12.6: Step response function and equivalent circuit diagram of a PT2TL element processes, are bonded by their convective portion at delay characteristics. Therefore generally another delay TL, i.e. a time lag, should be considered.
Assuming general solution of differential equation (see equation 12.20) under the special condition of a bar signal xe(t) = xe0 · 1(t):
t−TL t−TL − T − T xa = xe(t) K1e 1 + K2e 2 (12.21)
t−TL t−TL − T − T xa = K · xe0 · 1 − e 1 1 − e 2 (12.22)
The proportional transfer factor K, both time constants T1 and T2 as well as the delay TL must be determined on the basis of complicated structure parameter here. Again selected values of the step response function will be used. The transfer function G(p) nd can be assumed following shape for the 2 order delay elements (PT2TL):
Ke− pTL G (p) = f¨ur T1 6= T2 so called model I (12.23) (1 + pT1) (1 + pT2) Ke− pTL G (p) = f¨ur T1 = T2 so called model II (12.24) (1 + pT )2
The distinction, which type of model deals with appropriate measurement series of characterized transfer element, is achieved by Strejc in table 12.1.
352 12.2. Transient characteristic with 2nd order delay
Table 12.1: classification of model type according STREJC x τ model type aW u xa∞ xa∞ I ≤ 0, 264 ≤ 0104 II > 0264 > 0104
The transfer constant K again results from the behaviour in infinite:
−∞ xat→∞ = K · xe0, da e = 0 x K = a∞ xe0 and thus: t−TL t−TL − T − T xa = xa∞ 1 − e 1 1 − e 2 (12.25)
The delay TL can be read directly from the diagram of the step response (see fig- ure 12.7). The occurrence of a delay must be considered as shift of time axis. The appropriate variables (xaW , xa∞ , τu, xa∞ ) can be taken from figure 12.7.
Figure 12.7: Kenngr¨oßender Sprungantwort
353 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
12.2.2.1 model type I
In this type of model, the case of different time constants T1 6= T2, the conditional equations must be found for both two time constants. According to the literature
[Strejc] it is known that the xa 0,7 = 0, 7 xa∞ is nearly independent of the ratio of two time constants, but is strongly dependent on the sum of two time constants. With an error smaller than 1.7% we can apply:
t0,7xa∞ = 1, 2 (T1 + T2)
t0,7xa∞ T1 = (12.26) 1, 2 1 + T2 T1
Figure 12.8: determination of parameters T1 and T2
On the other hand we can assume that the function value xa0,7/4 = xa(t(0,7xa∞)/4) according to figure 12.8 only depends on the ratio T2/T1. The ratio T2/T1 will be determined from table 12.2.
Thus two equations are available for determination of the time constants and the task is uniquely solvable.
It is still to be noted that these transfer elements for large time (t >> TW ) approxi- mately behave as 1st order transfer elements. Particularly with large difference of time constants the later process is dominated by process with time constant T2, since the processes with time constant T1 already faded away.
354 12.2. Transient characteristic with 2nd order delay
Table 12.2: time constant ratio dependent on step response
xa 0,7/4 T2
xa∞ T1 0, 260 0, 00 0, 200 0, 10 0, 174 0, 20 0, 150 0, 33 0, 135 0, 40 0, 131 0, 50 0, 126 0, 60 0, 125 0, 70 0, 124 0, 80 0, 123 0, 90 0, 123 1, 00
12.2.2.2 model type II
The type of model II is shown in table 12.1 (T1 = T2, i.e. only one time constant), so we can determine the necessary parameters by table 12.3. In this case the transfer function can be even extended to arbitrary integer exponents n:
K epTL G (p) = f¨urModelltyp II (T = T ) (12.27) (1 − pT )n 1 2
The determination of K and TL is independent of model type and can be achieved as described in model type I (see page 353).
Based on table 12.3 we have the possibility to determine the time constant T in different ways. By averaging these values we can obtain a value with a smaller error. This is important since measuring errors of experiment also completely shrink in the parameter estimation.
12.2.3 Dirac impuls as input signal (weight function g(t))
Apart from the possibility for the determination of transfer parameter treated in the preceding section, a further way is described here. The parameters of systems, i.e. time constants, transfer factor etc., should be invariant compared to input signal, since linear systems are considered here. This circumstance allows to use different test signals for
355 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Table 12.3: parameter estimation for model type II
τu TW Tu Ta n xa /xa xa∞ W δ T T T 1 0 0 0 0 1 2 0, 104 0, 264 1 0, 282 2, 718 3 0, 218 0, 323 2 0, 805 3, 695 4 0, 319 0, 352 3 1, 425 4, 463 5 0, 410 0, 371 4 2, 100 5, 119 6 0, 493 0, 384 5 2, 811 5, 699 7 0, 570 0, 394 6 3, 549 6, 226 8 0, 642 0, 401 7 4, 307 6, 711 description of the systems, which however always lead to the same transfer function. In technical test practice usually one or other input signals will be more feasible. In Dirac impulse the impressing of a very large impulse (energy or mass deposit) in a very short time interval ∆T << T (proportional to smallest appeared time constant) is observed. The introduced method here yields expedient value up to a pulse width of ∆T ≤ 0, 1 T . Thus the circumstances are displayed in figure 12.9.
nd Figure 12.9: 2 order transfer element (PT2TL)
In contrast to the preceding section here only transfer elements with same time constant are con- sidered (see equation 12.28). This is designated as model type II in the section 12.2.2.2.
Ke−pTL G (p) = (12.28) (1 + pT )n
356 12.2. Transient characteristic with 2nd order delay
Figure 12.10: impulse response function g(t) for a 2nd order delay element
Table 12.4: variables of impulse response function for 2nd order delay x (T ) T (T x (T )) a m n m a m xa(Tm/2) T (AK) 1,213 2 1 0,368 1,471 3 2 0,271 1,785 4 3 0,224 2,165 5 4 0,196 2,623 6 5 0,175 3,185 7 6 0,159
According to the relationship of the two output values xa(Tm)/xa(Tm/2) (see figure 12.10) the parameters n (number of coupled RC elements = exponent of the denomi- nator polynomial) and T (time constant) will be determined based on table 12.4. The time TM is the point which the impulse response function, the weighting function g(t) reaches maximally (see figure 12.10). A possibly appeared delay is also mentioned here.
Tm/2 stands for half time value to the maximum. Tm, and Tm/2 refer to time axis with TL shifted.
The transfer constant K can be determined by the fourth column in table 12.4. Base on variable A, the impulse area (A = xe · ∆t), technically realizable impulses can also be evaluated with real ∆T ≤ 0, 1 T .
357 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
12.2.4 Exercises to experimental process analysis
· m3 1. Compute the drawdown curve for a conveyor capacity V = 0, 005 s by means · m3 of transfer element method with a flow rate of V = 0, 015 s , if a pumping test yields following values (see table). Present the result graphically.
time [s] draw down [m] time [s] draw down [m]
320 0, 63 2185 0, 92
426 0, 69 2850 0, 96
564 0, 73 3715 0, 99
743 0, 77 4839 1, 03
976 0, 81 6302 1, 06
1279 0, 85 8202 1, 10
1673 0, 88 10.000 1, 14
2. Please determine the transfer function including parameters for the following measurement series, which is originated from a supply function:
t [min] 0 1 2 4 8 15
· h m3 i V s 0 0, 1 0, 1 0, 1 0, 1 0, 1
s [m] 0 0, 1 0, 08 0, 13 0, 19 0, 25
· 3. The following dependence between flow rate V and groundwater drawdown s is found for groundwater position in a pumping test:
· h m3 i V s 0 0, 05 0, 05 0, 05 0, 05 0, 05 0, 05 0, 05 0, 05
s [cm] 0 0 3 8 20 30 35 37 38
t [min] −1 0 1 2 4 10 20 40 100
358 12.2. Transient characteristic with 2nd order delay
· m3 Calculate the drawdown process with a flow rate of V = 0.15 s . Apply the method of transfer functions. Plot the measured value and the result. 4. In pumping test two different position P 1 and P 2 are far away from the infil-
tration well with a distance of r1 = 350m and r2 = 1000m respectively, and following concentrations C of positioning tracer are measured. A steady concen- g tration C(0,t) = 10 m3 is added in the infiltration well.
h i t 107s 0, 5 1, 0 1, 5 2, 0 2, 5 3, 0 3, 5 4, 0 4, 5 5, 0 5, 5
h g i Cr1 m3 2, 4 2, 7 3, 05 3, 55 4, 80 6, 50 7, 95 8, 70 9, 10 9, 20 9, 25
h g i Cr2 m3 2, 4 2, 7 3, 04 3, 32 3, 57 3, 81 4, 01 4, 20 4, 37 4, 53 4, 67 Calculate the transfer functions for this system. 5. In a tracer test 50kg concentrated NaCl solution infiltrates 5min long into the soil at the well. Calculate process of a possible pollutant dispersal, if average 1000kg solution had arrived into the soil. Place the measured values and prognosticated values graphically.
t [min] 24 30 35 40 42 50 60 70 80 90 100 120
mg C l 0 2, 0 7, 0 9, 7 9, 8 7, 5 5, 0 3, 5 1, 5 0, 5 0, 3 0 6. The following groundwater positions were measured by a pumping test:
t [min] 0 15 30 45 60 75 90 105 120 135
s [cm] 16, 00 15, 25 15, 12 15, 07 15, 01 14, 96 14, 95 14, 94 14, 94 14, 938
· » – m3 V s 0 0, 015 0, 015 0, 015 0, 015 0, 015 0, 015 0, 015 0, 015 0, 015 The aquifer has the parameters: m hn = 16m, M = 10m, k = 0, 001 s ,S0 = 0, 0001, n0 = 0, 20 For this model give the first 4 equations of the faltung integral until the time t = 135min.
7. Prognosticate the temperature change in a bank filtration well δF a,p by using the faltung integral, if the measured value for the river is δF l and for the well is δF a,g:
ˆ◦ ˜ δF l C 14, 2 16, 0 17, 7 19, 4 17, 2 16, 0 17, 6 18, 6 14, 8 12, 0 13, 7
ˆ◦ ˜ δF a,g C 8, 5 10, 0 11, 4 14, 0 14, 1 14, 7 15, 4 15, 8 15, 6 14, 9 14, 1
ˆ◦ ˜ δF a,p C
Zeit [d] 0 15 30 45 60 75 90 105 120 135 150
359 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
a) Calculate the temperature change at the 7th time with help of the three equations of the faltung integral and prognosticate the 10th and 11th time. b) Calculate the temperature change at the 5th time with help of five equations of the faltung integral and prognosticate the 10th and 11th time. c) Compare the calculated temperatures in the well with the measured
8. In a column flow test the following impulse response function of a pollutant with concentration 30mg/l was measured (see figure 12.11).
Figure 12.11: impulse response of a column flow test
a) determine the weighting function and the transfer function for these mea- sured values. b) prognosticate the concentration after 160min, if the input concentration is of following characteristic:
t [min] 0 20 40 60 80 100 120 140
mg C l 30 50 80 60 100 50 10 0 9. Following concentrations were measured according to tracer test in a groundwater observation tube. 50kg concentrated NaCl solution infiltrated in this tracer test within 5 hours.
360 12.2. Transient characteristic with 2nd order delay
a) calculate the process of total salt transport in the observation well, if the following individual measured values were obtained. b) places the measuring curve and the computed function graphically.
t [d] 0 1 2 4 5 7 9
mg CNaCl l 0 0 1 2 1, 5 1 0 c) calculate expected breakthrough curve by means of transfer function method and plot if an infiltration of 100kg worked within 2.5 hours. 10. The following groundwater levels were measured in a pumping test (see figure 12.12) a) calculate the water deficit (volume) of drawdown funnel, if the aquifer has the following characteristic values: −1 −1 hn = 16m, M = 10m, k = 0, 001m · s , S0 = 0, 0001m , n0 = 0, 20 b) Compute by means of transfer element method and with a) founded value for · the flow rate V the drawdown curve for conveyor capacity of0, 005m3 · s−1. Set up the first four equations of faltung integral for the model until obser- vation time point t = 1d.
Figure 12.12: groundwater level dependent on radius
11. With a column flow test the transport characteristics of the soil for a certain polutant will be determined. The input signal was an impulse (Dirac impulse). Which three types of transition behoviour we get, if we look at the behaviour of the column as a black-box?
361 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
12. The following output concentrations of a column flow test are measured, after an input impulse of a duration of 10d.
t in d 0 30 60 90 120 150 180 320 340 300
C in mg/l 0 15 45 55 45 20 10 5 2 0
Prognosticate by meas of the faltung integral the output concentration, after a time of 600d, if the following intpu concentrations are measured:
t in d 450 480 510 540 570 600
C in mg/l 100 80 70 60 70
362 12.3 Arbitrary transient characteristic and arbitrary input signals
12.3.1 Introductory
Most natural processes take place one time and are not reproducible. In rarest cases it is also possible to impress arbitrary test signals on natural ecological processes hap- pened only once, in order to determine the type and the parameter of the transient characteristic by means of experimental process analysis. Very often the task consists of one-time natural processes, e.g. flood waves, precipitation discharge events, ground- water formation rates or pollutant disposals, by means of mathematical method to derive the relationship between the input and output behaviour according to experi- mental process analysis, i.e. to determine the transient characteristic. For this reason other methods had to be developed. One of them is the application of faltung in- tegral/Duhamel integral. The basic idea of this method is the decomposition of arbitrary input signal into a sum of impulses, which then possess a special transient characteristic individually. The faltung integral is in particular used for single deadbeat events. Afterwards the portions of the each transferred impulses will be again overlaid. Due to superposition law this method can be only applied in linear or in piecewise linearized systems. The application of Fourier series analysis or syntheses is quoted in periodic functions.
The books to be consulted as literature for this section are: Dyck, S: Grundlagen der Hydrologie Luckner, L.; Schestakov, W. A.: Migrationsprozesse Wernstedt, J.: Experimentelle Prozessanalyse
Furthermore all books can be recommended, in which applications of faltung integrals on technical processes are described. The different notations or the different symbols and abbreviations must be paid attention in a comparative literature study. Following abbreviations according to international standard in system technology (see table 12.5) will be used.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Table 12.5: comparison of applied abbreviations
description intern. Dyck Luckner standard
Diracimpulse δ(t) pn(t) impulse response = g(t) h(t) h(t) weight function step response = h(t) S(t) S(t) transit function
input function xe(t) p(t) R(τ) output function xa(t) q(t) P (t0) delay time τ ∆t τ
12.3.2 decompositions of arbitrary input functions (signal analysis)
While in the preceding sections selected input signals (e.g. step function, Dirac im- pulse) are discussed, here arbitrary input signals are considered. This is very necessary for many tasks of water management, hydrology and geohydraulic. Always, if artifi- cial test signals cannot be used, but natural events must be exploited to experimental systems analysis, only the faltung integral method described in the following can be applied. Time should be considered as independent variables. An application of faltung integral on local variables is also conceivable.
The basic idea of the signal analysis consists of the fact that arbitrary time response of a function can be represented as an infinite sum of selected single signals (see section 11.3.4 signal analysis, page 310). In principle the different signals can be used. The sinusoidal signals are of special meaning, which can be found in well known Fourier series analysis application. The bar signals and the impulses lead to Laplace trans- formation. Therefore periodic and periodization functions are analysed by means of Fourier analysis and unique, deadbeat procedure by means of Laplace transforma- tion.
The arbitrary input signal is decomposed into a sum time shifted impulses in the application of the faltung integral (see figure 12.13).
The effect of a signal on a system is usually characterised by energy or mass flow. It is defined by the respective signal variable and the effect duration, i.e. by the function integral of time. With the approximation of input signal by a sum of individual square pulse, the integral is approximately described by a sum of the products of pulse
364 12.3. Arbitrary transient characteristic and arbitrary input signals
Figure 12.13: approximation of a function by impulse
amplitude xei(τi) and width ∆t :
te n= te Z X∆t xe (t) dt ≈ xei (τi) ∆t (12.29) 0 i=1
The individual impulse of time point τi, which affects as input signal of transmission system, produces individual impulse response functions at system output (see figure
12.14), the weighting functions gi(t − τi). These are superposed and yield system response to the input signal xe(t). It should be noted that the superposition can be only applied for linear systems.
For pulse width ∆t ⇒ 0 the technical impulse approaches Dirac impulse and finite sum in integral representation, whereby an infinite number of impulses is considered.
Figure 12.14: impulse response function g(t) for a 2nd order delay element
365 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
12.3.3 Composition of output functions (signal synthesis)
As already mentioned the output signal by the overlay of sum of individual impulse response functions, results in weighting functions gi(t − τi) at time t. For computation we must note that all preceding impulses in time interval 0 to t contribute, since the weighting functions are not yet faded away at time t.
As recognized from figure 12.15, the output signal xa(t) to time t0 is composed from the time shifted weighting functions portions for the time t0:
n= t0 n= t0 X∆τ X∆τ xa(t0) ≈ xai(t0) = (xei(τi) · g(t0 − τi)) (12.30) i=1 i=1
∆τ is the time lag of the Dirac impulse, the so called aperture time. If we arrange the border crossing to infinitesimal aperture time, the sum changes into integral form, which can be also designated as faltung integral or Duhamel integral:
t Z xa(t) = xe(τ) · g(t − τ)dτ 0
= g(t) ∗ xe(t) (12.31)
In this case ∗ stands for the faltung operation.
We can also interpret faltung integral in such a way that, all impulses of input signal xe(t) in time interval 0 ≤ τ ≤ t contribute to the value to output signal at time point t, which are weighted according to aperture time (t − τ) with the factor g (t − τ) dτ in each case.
366 12.3. Arbitrary transient characteristic and arbitrary input signals
Figure 12.15: overlay of individual step response function
367 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Considering the connection between weight- and transfer function we can also carry out following identical transformation:
dh(t − τ) = g(t − τ) dt t Z d x (t) = (h(t − τ))x (τ)dτ a dt e 0 t d Z = h(t − τ)x (τ)dτ dt e 0 d = [h(t) ∗ x (t)] (12.32) dt e
With application of Laplace transformation the faltung operation changes into mul- tiplication (see section 5.3.2 Laplace transformation, page 147):
L {xa(t)} = L {g(t) ∗ xe(t)}
t Z = L g(t − τ)xe(τ)dτ 0
= G(p) · Xe(p) (12.33)
In practice the numerical execution of faltung operation must be accomplished in a time quantization as its derivation. For a process, which begins from time t = 0, can be described in sum form introduced above as follows:
368 12.3. Arbitrary transient characteristic and arbitrary input signals
xa(t1) = ∆τg(τ1)xe(τ1)
xa(t2) = ∆τg(τ2)xe(τ1) + ∆τg(τ1)xe(τ2)
xa(t3) = ∆τg(τ3)xe(τ1) + ∆τg(τ2)xe(τ2) + ∆τg(τ1)xe(τ3) . .
k X xa(tk) = ∆τ g(τi)xe(τk−i+1) (12.34) i=1
This equation system can be transformed in matrix equation:
x (t ) g(τ ) 0 0 ··· 0 x (τ ) a 1 1 e 1 x (t ) g(τ ) g(τ ) 0 ··· 0 x (τ ) a 2 2 1 e 2 x (t ) = ∆τ g(τ ) g(τ ) g(τ ) ··· 0 · x (τ ) (12.35) a 3 3 2 1 e 3 . . . . . . . . . . .. 0 . xa(tk) g(τk) g(τk−1) g(τk−2) ··· g(τ1) xe(τk)
With different notations tk = k and τk = i, and the introduction of aperture time T = ∆τ:
x (1) g(1) 0 0 ··· 0 x (1) a e x (2) g(2) g(1) 0 ··· 0 x (2) a e x (3) = T · g(3) g(2) g(1) ··· 0 · x (3) (12.36) a e . . . . . . . . . . .. 0 . xa(k) g(i) g(i − 1) g(i − 2) ··· g(1) xe(i)
369 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
The matrix equation can be written for short:
Xa = T · G · Xe (12.37)
or
−1 −1 Xe = T · G · Xa (12.38)
Thereby G−1 is the inverse matrix of G.
12.3.4 Determination of weighting function g(t) for general case
The weighting function g(t) will be determined as follows:
g(t) = xa(t) |xe(t)=δ(t) (12.39)
The experimental determination of g(t) were treated in the preceding sections. The determination of the weighting function can be achieved by a test attempt with an im- pulse as input function (see section 12.3.2 signal analysis, page 364). If a step function is used as input function, then weighting function must be obtained by appropriate differentiation (see section 11.4 transmission systems, table 11.5, page 317).
Experiments on real object will not always be accomplished for regulation of weighting function. Only in case the real input- and output signals can be obtained for compu- tation of g(t). The matrix equation for calculation of output signal (see section 12.3.3 signal synthesis, page 366) can be used for regulation of g(t) or matrix G. x (1) g(1) 0 0 ··· 0 x (1) a e x (2) g(2) g(1) 0 ··· 0 x (2) a e x (3) = T · g(3) g(2) g(1) ··· 0 · x (3) (12.40) a e . . . . . . . . . . .. 0 . xa(k) g(i) g(i − 1) g(i − 2) ··· g(1) xe(i)
370 12.3. Arbitrary transient characteristic and arbitrary input signals
If both the input and output function are known for one observation period, the fol- lowing matrix equation can be developed from the above equation system:
x (1) x (1) 0 0 ··· 0 g(1) a e x (2) x (2) x (1) 0 ··· 0 g(2) a e e x (3) = T · x (3) x (2) x (1) ··· 0 · g(3) (12.41) a e e e . . . . . . . . . . .. 0 . xa(k) xe(i) xe(i − 1) xe(i − 2) ··· xe(1) g(i)
The corresponding matrix equation:
Xa = T · Xe · G (12.42)
bzw.
−1 −1 G = T · Xa · Xe (12.43)
Thus the weighting function g(t) can be described by a value sequence.
In metrological practice the regulation usually looks somewhat different. In above derivation it was presupposed that the process for t < 0 does not exist and started with the first input impulse. Real processes run off independently of observations. Therefore we must begin at arbitrary time point with the observation of input and output signals. The determination accuracy of the weighting function g(t) must be specified by the administrator. The questions are the measuring expenditure and of the main dynamic process criteria for specification of the sample width T and the number of scanning values n. This is the same problem as discontinuous measuring signals treatment and error description (see Graber¨ : ”Lehrbrief Automatisierungstechnik”).
If we always specify that m values are considered for the conditional equations, then 2 · m equations are to be set up, in order to determine m supporting places of the weighting function g(t). Hence we must already observe the process before explicit prognostication about the duration of 2 · m sampling intervals T , i.e. a period of 2 · m · T (see figure 12.16).
371 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
Figure 12.16: formation of discontinuous response signals from measured values
According to this scheme following equation system can be set up:
xa(t4) = T (g (τ4) xe (t1) + g (τ3) xe (t2) + g (τ2) xe (t3) + g (τ1) xe (t4)) (12.44)
xa(t5) = T (g (τ4) xe (t2) + g (τ3) xe (t3) + g (τ2) xe (t4) + g (τ1) xe (t5))
xa(t6) = T (g (τ4) xe (t3) + g (τ3) xe (t4) + g (τ2) xe (t5) + g (τ1) xe (t6))
xa(t7) = T (g (τ4) xe (t4) + g (τ3) xe (t5) + g (τ2) xe (t6) + g (τ1) xe (t7))
Thus we have four equations with four unknown weighting function portions, whereby the equation system is uniquely solvable. Since the observed values, generally measured values of the input signals as well as the output signals are erroneous, in practice more equations are built, which leads to an overdetermined equation system. They will be solved by means of special iterative methods, e.g. Householder method. The solution is then the value range of weighting function portions, which fulfils the equation system with the smallest sum of square deviation.
372 12.3. Arbitrary transient characteristic and arbitrary input signals
12.3.5 Prognosemodelle
In water management using models to prognosticate also shows a main field of ap- plication of faltung operation. The procedure of prognosis contains the algorithm as follows. A precondition of application faltung integral to prognosticate is that the pro- cess was already observed in a lead time, i.e. before a prognosis the in- and output signals of regarding system must be collected by means of suitable measurement. We also speak of the model learning curve in this time. These measured values serve for the determination of weighting function g(t), exactly g(i) or G. How long does the learning curve last, depends on the historical data base, the precision requirement and the desired numerical expenditure.
In the following examples the manipulation will be demonstrated (see figure 12.16, page 372). In this case measured input signal values exist for a range of seven sampling intervals before the forecast horizon. The output signal was measured from the fourth interval. Based on these values following four equations with unknown quantities g1 to g4 can be formulated. Therefore only four supporting places of the weighting function will be proceeded in this example. For real practical tasks this is quantized too roughly:
xa4 = T · (g4xe1 + g3xe2 + g2xe3 + g1xe4) (12.45)
xa5 = T · (g4xe2 + g3xe3 + g2xe4 + g1xe5)
xa6 = T · (g4xe3 + g3xe4 + g2xe5 + g1xe6)
xa7 = T · (g4xe4 + g3xe5 + g2xe6 + g1xe7)
By means of suitable methods for the solution of equation system we get the weighting function portions g1 to g4. These are used into a conditional equation for the first prognosis time step (xa8). Thus the prognosis value can be computed explicitly:
xa8P rog = T · (g4xe5 + g3xe6 + g2xe7 + g1xe8) (12.46)
Parallel to prognosis the process should be further supervised metrologically. In this case we get a new value pair xe8P rog and xa8gem at time point 8. This can be used to calculate new weighting function portions. With retention quantity of weighting function portions the input signal xe1 will not be incorporated into calculation any
373 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
longer. The first equation with the input signal xe1 can however remain in the com- putation, and then five equations are available for the determination of four weighting function portions. This overdetermined equation system is then solved iteratively with an appropriate method, e.g. Householder method. That founded values represent the transient characteristic of the regarding system is possibly better than those with the definite system. This comparison between prognosis and real process is also called constant learning system.
374 12.3. Arbitrary transient characteristic and arbitrary input signals
12.3.6 Exercises and applications for the faltung integral
1. The following groundwater levels were measured in a pumping test (see figure 12.17) a) Calculate water deficit (volume) of drawdown funnel, if the aquifer has the following characteristic values: −1 −1 hn = 16m, M = 10m,k = 0, 001m · s , S0 = 0, 0001m , n0 = 0, 20 b) Compute by means of transfer element method and with a) founded value for the flow rate V˙ the drawdown curve for conveyor capacity of0, 005m3 · s−1. Set up the first four equations of faltung integral for the model until obser- vation time point t = 1d.
Figure 12.17: groundwater level dependent on radius
2. Following groundwater level are measured in a pumping test:
˙ h dm3 i V s 0 15 15 15 15 15 15 15 15 15
s [cm] 16, 00 15, 25 15, 12 15, 07 15, 01 14, 96 14, 95 14, 94 14, 94 14, 938
t [min] 0 15 30 45 60 75 90 105 120 135
m The aquifer has following parameters: hn = 16m, M = 10m, k = 0, 001 s , S0 = 0, 0001, n0 = 0, 20 Set up the first four equations of faltung integral for the model until observation time point t = 135min. 3. With a column flow test the transport parameters of the soil for a certain polutant were determined. The input signal was an impulse (Dirac-Impuls). Wich of the
375 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
three types for describing the transient behaviour we get at the output of the column, if the behaviour is seen as a block box? 4. in an other column flow test the following output concentrations were measured, after the column got an impulse of the duration 10d as input impulse
t in d 0 30 60 90 120 150 180 320 340 300
C in mg/l 0 15 45 55 45 20 10 5 2 0 By using the faltung integral prognosticate the output concentration for the time 600d if the following input concentrations were measured:
t in d 450 480 510 540 570 600
C in mg/l 100 80 70 60 70 50
5. Prognosticate the temperature change in the bank filtration well δF a,p by using the
faltung integral, if for the river δF l and for the well δF a,g the following measured values are known:
◦ δF l [ C] 14, 2 16, 0 17, 7 19, 4 17, 2 16, 0 17, 6 18, 6 14, 8 12, 0 13, 7 ◦ δF a,g [ C] 8, 5 10, 0 11, 4 14, 0 14, 1 14, 7 15, 4 15, 8 15, 6 14, 9 14, 1 ◦ δF a,p [ C] Zeit[d] 0 15 30 45 60 75 90 105 120 135 150
a) Calculate the temperature change from the 7th time by using the three equations of the faltung integral and prognosticate the 10th and 11th time. b) Calculate the temperature change from the 5th time by using the five equa- tions of the faltung integral and prognosticate the 11th time. c) Compare the calculated temperatures in the well with the measured once.
376 Part IV
Indirect parameter identification
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management
12.3. Arbitrary transient characteristic and arbitrary input signals
The indirect parameter identification is treated here as method for parameter estima- tion. It stands in contrast to physical and chemical methods for the direct Parameter estimation which is described in Graber¨ ”Grundwassermesstechnik”. The methods of indirect parameter identification are mathematical, which according to experimen- tal process analysis determines parameter for a transmission system. The transient characteristic can be found by means of experimental or theoretical process analysis. Accordingly the identified parameters are more or less physically/chemically inter- pretable. On all accounts parameters can be found, which well reflect the system behaviour within validity scope of experiment.
379 CHAPTER 12. MODELL REGULATION BASED ON PARAMETERS
380 Chapter 13
Estimation procedures
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 13. ESTIMATION PROCEDURES
In water management practice the experimental process analysis is primarily used for parameter estimation. The model structure is specified by a theoretical process anal- ysis. We try to transfer this model structure into simple mathematical representation. The parameters can be determined by solving conditional equations or by solving pa- rameter approximation problems. Thus there is task, on the basis of structure cognition or assumption, to determine such models, that
• reflects the characteristics of the system so exactly as required and • eliminates the overlaid influences of noise and errors.
In order to satisfy these demands the comparison of output values of original function as input function or an independent variable (time or place) with the model is accom- plished. In the result a change of model parameters is to be made or the model changes until the deviation reaches minimum. The changes can be achieved according to a cer- tain strategy (search algorithms, optimisation programs), statistically (random number generator) or empirically. The visual comparison between two diagrams (original and model output signal) is also possible.
This task is also designated as parameter estimation. In particular the following intro- duced approaches will be classified as iterative estimation method.
In the algorithm or iterative model adjustment (see figure 13.1) we try to let the same input vector, the manipulated vector y affect on the process and model. With 1 a first parameter substitution, the initial parameter, the model output vector xM can be calculated as first approximation. The deviation of these process output vectors is designated as quality of the model adaptation. In water management applications the quadratic evaluation will be carried out. The goal of changing parameters is to minimize the Q value.
382 Figure 13.1: iterative model adjustment
383 CHAPTER 13. ESTIMATION PROCEDURES
384 Chapter 14
Flow parameters
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management 14.1 Pumping test evaluation
14.1.1 Fundamentals
The evaluation of a pumping test, with which e.g. water is conveyed from a well and the drawdown is recorded as a discontinuous function of place and time, is a very significant and representative method to determine geohydraulic parameters. Compared to laboratory procedures it has advantages that:
• is accomplished in undisturbed aquifers and • advances normally integral statements of the aquifer in the regarded flow field section.
The parameter estimation of aquifers occur in the direct laboratory experiment of soil samples, which can be achieved by means of engraver cylinder or cuttings. The disadvantages consist the fact that only a punctiform parameter estimation can be obtained in very inhomogeneous aquifer with this method. Besides the granular structure of soil is destroyed by the sampling and thus another is evaluated in lab. A third difference is that in lab the entire water content is determined, while in nature and with pumping test only the drainable pore volume affects. The representation by means of definite parameter method increases due to the integral character of the pumping tests (see table 14.1).
Table 14.1: difference between pumping test evaluation and laboratory method
charakteristics laboratory method pump test evaluation spatial extend selective integral relevance low high particle grid pertubed regular storage capacity total volume of water drainable volume effort relativly low very high
On the other hand the pumping tests are substantially more complex and expensive than laboratory tests. Therefore the experimental design, execution and special worthy analysis must be attached. Also usually only one time test execution is possible.
For the evaluation of such pumping tests in practice particularly two methods are used:
• the graphic method; in water management practice designated as straight line
Systems Analysis in Water Management Peter-Wolfgang Gr¨aber 14.1. Pumping test evaluation
method and typical curve method and • the search method or optimisation method.
The well incident flow is used as model in the pumping tests. As in the section 8.1 Theis well equation, page 206, deduced, the solution of the partial differential equation according to Theis is suitable for the computation of drawdown processes due to water extraction from well. This model of course can be only assumed approximately for practical flow conditions. The application of model is prohibited to better reflect the original due to too large number of free parameters to adapt. The most substantial restriction of analytical models is the consideration of only one aquifer. The constant of parameter transmissibility T and storage coefficient S as well as the horizontal bed situation can be presupposed in the little spatial expansion of pumping tests as given. Of course it must be also noted that the transmissibility change during drawdown procedure remains negligibly small (linearization around operating point).
The quality function GF in the pumping test evaluation is defined as sum of the square deviation of the measured values at the original process and model results on different local and time points (see figure 14.1):
Figure 14.1: iterative model adjustment in a pumping test
387 CHAPTER 14. FLOW PARAMETERS
n m X X 2 GF = Wi, j · (si, j − sMi, j) (14.1) i=1 j=1 with:
Wi, j weighting factor
s original drawdown value
sM model drawdown value
m maximal number of time steps
n maximal number of local observation points
14.1.2 Practical realisation
The method of least squares (MKQ) is used for parameter adjustment to the measured values. The goal is to minimize the quality function GF . General correspond effective methods were shown in section 4.3 least square method, page 108. It shows that the search strategy on that basis of the nonlinear regression with the utilization of gradients is best suitable for pumping test evaluation. In contrast to Rosenbrock search algorithm the number of search steps will be drastically (factor 10) reduced by Jones (Dammert) spiral method in pumping test evaluation. This pro cedure presupposes that the quality function is constant and differentiable. Both are given in the analytical solution of well function according to Theis.
The program system PSU (Pumping test evaluation) was developed by Beims and Graber¨ for practical realization. This program system (see figure 14.2) has a mod- ular structure, which allow arbitrary model creations of quality function and search algorithm coupled with appropriate main programs.
388 14.1. Pumping test evaluation
Figure 14.2: programme system for pumping test evaluation according to Beims/Graber¨
All the most substantial practical pumping tests can be evaluated with the following specified versions PSU2, PSU5 and PSU8 (see table 14.2).
Table 14.2: Realisierte Programmversionen mit geohydraulischem Schema
programm geohydraulic scheme results unbounded aquifer PSU2 T,S without supply unbounded aquifer PSU5 T,S,B with supply oneside bounded aquifer PSU8 T, S, λ∗ without supply
The parameters in table 14.2 are:
m2 T [ ] transmissibility, profile permeability s S [−] storage coefficient
B [m] supply factor
λ∗ [m] effective boundary condition distance
389 CHAPTER 14. FLOW PARAMETERS
The programs PSUX are written in the form of main program and realise data in- and output as well as the search algorithm control. The fitted values of each search step or only the parameters could serve as output by inserting appropriate control variables, which yield the adjustment according to given error bound. Furthermore a graphic comparison between the measured values of pumping test and the adapted drawdown curve is possible. The search program was realized according to Jones spiral method. It acquires the minimum of the quality function on the basis of nonlinear regression with employment of differential (see figure 14.3 and 14.4). The quality function is to be selected according to the geohydraulic conditions. It is defined as the sum of square deviations between the measured value and the theoretical drawdown curve. In the module the auxiliary programmes are combined with important subprograms to solve well flow equation, e.g. the well functions W (σ) according to Theis and W (σ, B) according to Hantusch as well as Bessel function K0(x) and I0(x).
Figure 14.3: quality mountain in a pumping test evaluation with search procedure of different start points
A special problem exists in the search of parameters B (supply factor) and λ∗ (effective boundary condition distance), since they are not independent of T and S. The search algorithm is however then applied for more parameters if they are independent of each other. In these cases it is a trick to exclude the region in drawdown curve, which depend on different parameters dominantly. So physically it can be justified that, the drawdown strongly depends on the transmissibility T and the storage coefficient S of well vicinity area in the initial phase of a pumping test. In the quasi steady phase the
390 14.1. Pumping test evaluation supply factor B and/or the boundary condition (effective boundary condition distance λ∗) work as further influence variables. A so called stage search is accomplished based on this drawdown curve classification in different phases. The parameters T and S will be searched in phase 1. The measured values of the phase 2 serve to the estimation of supply factor and/or the effective boundary condition distance. The parameters T and S will be applied as known quantity (determined from phase 1) during this phase and thus are not included in search process. The measured values from arising process are combined in a phase 3. With them an improvement of adapted values can be obtained. In this phase the parameters of phase 2 will be as known, i.e. as not adjustable, regarded and again only one search for the two values T and S is carried out.
The pumping test evaluation with the programs PSUX can be of course only as good as measured values; the drawdown values from pumping test are as good as the well incident flow equation model reflects natural processes. For complex geohydraulic conditions we must resort to others, e.g. the pumping test simulator.
The expressiveness of pumping tests or experimental process analysis method generally also de- pends on the used test signal. In classical pumping test this is a step function · with the step height V , the conveyed water quantity. The best results can be achieved by using a Dirac impulse (theoretical impulse with a infinite height and a length of time, which approaches to zero). This is technically not realizable. As compromise the impulse function, the step function carry periodic signals and stochastic signal sequences. The step function is favourable for the determination of final steady state, the so called static behaviour. A combination of different test signals in the Variants
• step function - impulse function • step function - periodic signals or • step function - stochastic signal sequences leads to effective determinations of dynamic transition- and static final state.
391 CHAPTER 14. FLOW PARAMETERS
Figure 14.4: Iteration performance in the pumping test evaluation
392 14.2 Pumping test simulator
The program packet PSUX for the evaluation of pumping tests with corresponding analytical solutions on the basis of Theis well function or Hantusch demonstrated above shows considerable restrictions. So not all characteristics of the aquifer such as anisotropy, stratification, heterogeneity or capillary space and not all kinds of charac- teristics of well design such as imperfectness, well diameter, well bottom inflow, filter losses can be considered with this system. Either with application of natural signals for parameter identification, or with artificially produced test signals like in the pumping tests, it has to be assumed that normally it is a matter of one time procedure, which is not reproducible or changeable due to cost and other reasons. Therefore it is necessary to consider such field tests intensively with the experimental design.
A numerical model was described by Mucha/Paulikova, with which and pumping test can be simulated and interactively evaluated. This model is based on a vertical plane rotation symmetrically quantized flow model, which does not consider simplified assumption. This was converted into a program system WELL, the so called pumping test simulator by Beims/Graber¨ . With it the effect of a pumping test can be demon- strated and optimised on the basis of hypothetical assumption of regarding area. With this model besides the inhomogeneity the existence of multiple aquifers and also key elements well vicinity area can be considered. The transmissibility can be considered in horizontal and vertical inhomogeneity. Furthermore the specific elastic as well as the gravimetric storage coefficient can be processed. The model takes free groundwater flow conditions as a compressible system and the free surface as a mobile bor- der. The transmissibility is computed according to the concrete position of free surface. On the last radius point rn the system is regarded as impermeable, i.e. the discretisation must be chosen in such a way that practically no drawdown appears there (see figure 14.5, page 394).
On the basis a graphic display the effect of different input signals can be demonstrated and at the same time the optimal measuring time points dependent on the distance and the drawdown gradients for real pumping test can be determined. The local situation of the level observation tubes is usually default due to technological conditions.
The pumping test simulator WELL will be applied for following fields:
• determination of flow- and speed relationship in the proximity of well.
• calculation of typical curves for special well- and aquifer conditions.
• interactive pumping test evaluation.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 14. FLOW PARAMETERS
Figure 14.5: structure scheme of pumping test simulator (Beims/Mucha/Graber¨ )
The work with the pumping test simulator WELL represents a substantial complement of pumping test evaluation and interpretation. The exertion as model in the indirect parameter identification can be only realised by means of specially large expenditure.
The pumping test simulator is a vertical finite difference model, whose quantization can be obtained in vertical direction according to geological stratification and well geometry. 0,25 In horizontal direction quantization will be carried out logarithmically (ri = ri+1·10 ). The junctions lie in the centre of gravity of the reticule, i.e. they are in contrast to the geometrical centre around ∆ri outwards shifted:
2 (ri+1 − ri) ∆ri = (14.2) 6(ri+1 + ri) The permeability values are defined as hydraulic conductance between the nodes. The conductance in horizontal direction, for example between the nodes 5/4 and 6/4, under the Dupuit Thiem equation assumption for groundwater flow to a well is:
2πkh,4b4 TF5/4 →= (14.3) ln r6 r5 with
394 14.2. Pumping test simulator
th kh,4 horizontal permeability coefficient of 4 discrete layer
b4 thickness of 4th layer
r5, r6 radii of nodes 5 and 6
The conductance in vertical direction, for example between the nodes 6/2 and 6/3 is:
2kv,2 2kv,3 TF6/2 ↑= π(r72 − r62 ) + (14.4) b2 b3
2π(r72 − r62 )(kv,2b3 + kv,3b2) TF6/2 ↑= (14.5) b2b3 with:
nd rd kv,2, kv,3 vertical permeability coefficient of 2 or 3 layer
nd rd b2, b3 thickness of 2 or 3 layer
r6, r7 radii of the element boundaries 6 and 7
The conductance then yields flowing water quantity, for example between the nodes 5/4 and 6/4: ˙ V5,4,6,4, = TF654 → (H5/4 − H6/4) (14.6) whereby H5/4 and H6/4 are the piezometer head in nodes 5/4 and 6/4. The storage coefficient S designates the relationship of the volume in unit water, which becomes 1m empty during gauge level change, to the total volume of this unit. So the storage factor in the node 6/5 is e. g.:
2 2 SF6/5 = Ss,5b5π(r7 − r6) (14.7)
th Ss,5 specific elastic storage coefficient of 5 layer
th b5 thickness of the 5 layer
r6, r7 radii of the corresponding element
The storage coefficient for the free waster surface e.g. at node 5/1 is:
2 2 SF5,1 = Syπ(r6 − r5) (14.8)
395 CHAPTER 14. FLOW PARAMETERS
Sy gravitation storage coefficient
The released water volume for node 5/1 is:
V5,1 = SF5,1 (H5,1,t − H5,1,t−∆t) (14.9)
H5,1,t and H5,1,t−∆t are potentials at node 5/1 to time point t and t − ∆t, whereby ∆t is the time interval. The storage factor for well is expressed by node 1/1:
2 SF1,1 = πr1, j
th r1, j effective well radius of j layer
The linear well loss is contained in coefficient ϕ:
· V well loss = ϕ 4πT The flow from 4th layer in the well is expressed by the factor:
2πkh/4 · b4c/a TF1/4 = ln r2 + ϕ r1 2 whereby c/a is the relative position of junction 1/4 between water level in well and water level in the node 2/1. The flow in the well can take place through filter or well bottom.
The time discretisation begins with a small increment ∆t and will automatically in- crease according to rules. 0,1 ∆ti+1 = ∆ti · 10 (14.10)
If the discharge flow is not constant, the input of time steps and flow rate are achieved on the basis of each calculation period.
The resulting band matrix with the five diagonal elements will be solve according to a direct method (Gauss method).
396 Chapter 15
Suction power distribution
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 15. SUCTION POWER DISTRIBUTION
Another application field of the parameter identification methods in soil system exists in laboratory scale determination of suction power saturation distribution (SSV) within the unsaturated region. The measuring method for this soil behaviour is specified in Graeber ”Grundwassermesstechnik” and the mathematical model is deducted in the section 6 differential equations of groundwater processes, page 185.
Typical suction power saturation behaviour is connection similar to hysteresis between the suction power pk and the saturation nb. As mathematical model these SSV curves were indicated by an equation from Luckner/Schestakov, whereby A, B, C, and D are four constants, which must be separately determined from the experimentally technically gained upper (drainage) and lower (irrigation) limiting curves of hysteresis branches: Θ − A 1 (1−1/D) Θ = = D (15.1) n − A − B 1 + Cpc The determination of these parameters comes into the indirect parameter identification task range, whereby in this case it is a matter of static characteristic curve approxima- tion. On this account no conclusions about the test signal type have to be made. The dynamic behaviour corresponding to the partial differential equation of this process is not yet evaluated at present.
The estimation of the four parameters thereby can be achieved according to the em- pirical graphic method (method of typical curves) or the mathematical search algo- rithm. The first approach is used in order to obtain the initial value for the search strategy in the experimental design phase and the other is to reach optimal test con- ditions (measuring point selection). The mathematical search algorithm is then used to approximate the mathematical model at founded measured values pairs (pk, nb) as accurately as possible. Also the adjustment is made here by the square quality factor, which represents the deviations between n0 and nm in all adjusted operating points (pressure ranges). Fibonacci method is applied for the minimum search. This has the advantage that it is relatively ”robust”, but stagnates with a very rough approxi- mation. This behaviour also depends on the relative flatness of the quality mountain. It is therefore suggested applying Powell method. It shows good convergence be- haviour. Since the quality mountain, due to its abstract formulation, is constructed in such a way that, minima exist in the range, which do not allow physically mean- ingful interpretation (e.g. negative saturation), and special weighting function will be introduced. As soon as physically conditional limits for the saturation are reached (air or water restsaturation), the quality function acquires appropriate maximal value. The gradient method of the nonlinear regression, as favourably used in pumping test evaluation, conks out here, since the quality mountain runs very flatly and shows kinks at physically conditional edges, which contradicts the required differentiability.
398 BIBLIOGRAPHY
[and82] and, H. F.; Anderson P. M. Wang: Introduction to Groundwater Modeling, Finite Difference and Finite Elements Methods. W. H. Freemann and Company, San Francisco, 1982. [and86] and, L.; Schestakow M. W. Luckner: Migrationsprozesse im Boden- und Grundwasserbereich. Deutscher Verlag f¨urGrundstoffindustrie, Leipzig, 1986. [Bau63] Baule, B.: Die Mathematik Des Naturforschers und Ingenieurs, I - IV. S. Hirzel Verlag, Leipzig, 1963. [Bea79] Bear, J.: Hydraulics of Groundwater. McGraw-Hill, 1979. [Bey73] Beyer, O. U.A.: Mathematik F¨ur Ingenieure, Naturwissenschaftler, Okonomen¨ und Landwirte. BSG B. G. Teubner Verlagsgesellschaft, Leipzig, 1973. [Bus72] Busch, K.-F.; Luckner L.: Geohydraulik. Deutscher Verlag f¨urGrund- stoffindustrie, Leipzig, 1972. [Bus93] Busch, K.-F.; Luckner L.; Tiemer K.: Geohydraulik. Lehrbuch der Hy- drogeologie, Bd. Gebr¨uderBorntr¨ager,Berlin/Stuttgart, 1993. [dNAK73] N. A. Krusemann, G. P.; Ridder de: Untersuchungen und Anwendun- gen von Pumpversuchsdaten. Verlagsgesellsch. R. M¨uller,K¨oln-Braunsfeld,1973. [Dom90] Domenico, P. A.; Schwartz F. W.: Physical and Chemical Hydrogeology. John Wiley & Sons, 1990. [Dyc89] Dyck, S.; Peschke G.: Grundlagen der Hydrologie. Verlag f¨urBauwesen, Berlin, 1989. [Fet80] Fetter, C. W.: Applied Hydrology. Prentice Hall, Englewood Cliffs, NJ 07632, 1980. [Fre79] Freeze, R. A.; Cherry J. A.: Groundwater. Prentice Hall, Englewood Cliffs, NJ 07632, 1979. [Fri88] Fritzsch, W.; Klose J.: Mathematische Modellierung, Simulation und Op- timierung. Taschenbuch Maschinenbau, Bd. 4, Abschn. Verlag Technik, Berlin, 1988. [Gel68] Gellert, W. U.A.: Kleine Enzyklop¨adieMathematik. Bibliographisches In- stitut, Leipzig, 1968. [H¨af92] Hafner,¨ F.; Sames D.; Voigt H.-D.: W¨arme-Stofftransport. Springer Verlag, Berlin/Heidelberg/New York, 1992.
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management CHAPTER 15. SUCTION POWER DISTRIBUTION
[Hea88] Heath, R. C.: Einf¨uhrungin Die Grundwasserhydrologie. R. Oldenburg Ver- lag, M¨unchen, Wien, 1988. [H¨ol92] Holting,¨ B.: Hydrogeologie. Ferdinand Enke Verlag, 1992. [Kin92] Kinzelbach, W.: Numerische Methoden Zur Modellierung Des Transportes Von Schadstoffen im Grundwasser. Oldenbourg, 1992. [KR95] Kinzelbach, W. R. Rausch: Grundwassermodellierung. Gebr¨uder Borntr¨ager,Berlin/Stuttgart, 1995. [Kru89] Krug, W.: Simulation f¨urIngenieure in CAD/CAM-Systemen. Verlag Tech- nik, Berlin, 1989. [Mar86] Marsily de, G.: Quantitative Hydrogeology. Academic Press, Orlando, 1986. [Mat93] Mathess, G.: Lehrbuch der Hydrogeologie, 1 - 3. Gebr¨uderBorntr¨ager, Berlin/Stuttgart, 1993. [NZ94] Ne-Zheng, Sun: Inverse Problems in Groundwater Modeling. Kluwer Aca- demic Publisher, London, 1994. [Phi76] Philippow, E.: Taschenbuch Elektrotechnik, 1. Verlag Technik, Berlin, 1976. [Pin77] Pinder, G. F.; Gray W. G.: Finite Element Simulation in Surface and Subsurface Hydrology. Academic Press, New York, 1977. [Ra71] Remson, I.; Hornberger M. G.; Molz J. F. and: Numericals Methods in Subsurfaces Hydrology. WIley-Interscience, New York, 1971. [T¨op87] Topfer,¨ H.; Besch P.: Grundlagen der Automatisierungstechnik. Verlag Technik, 1987. [Ver70] Verujit, A.: Theory of Groundwater Flow. Macmillian, London, 1970. [Voi04] Voigt, Langguth;: Hydrogeologische Methoden. Springer Verlag, Berlin/Heidelberg, 2004.
400 INDEX
Sarrus’ rule, 20 DARCY-Gesetz, 194 determinant, 19 analogy signals, 312 determinat Anfangsbedingungen, 201 Cramer’s, 28 Anfangswertaufgabe, 171 VANDERMOND, 76 associative law, 16, 55 differences back coupling, 335 divided, 82 balance equation, 186, 248 differentiation basic equation of vectors, 57 dynamic, 186 digital signals, 313 basic signals DIRAC impilse, 306 step function (unit step) DIRICHLET condition, 202 DIRAC impuls, 304 discontinuous signals, 313 blance equation, 195 discrete signals, 312 boundary condition discretisation, 242 2nd type, 260 distributive law, 16, 55 rd 3 type, 261 divergence, 59 CAUCHY, 202 DUHAMEL DIRICHLET, 202 integral, 366 NEUMANN, 202 Dupuit assumption, 194 surface groundwater models, 203 boundary condition elementary volume rd 3 type, 226 representative, 187 boundary conditions, 201 equation st 1 type, 222, 263 conduction, 187 nd 2 type, 222 convection diffusion, 187 capillary pressure saturation relationship, equation system 191 over determined, 22 CAUCHY condition, 202 solvable, 22 characteristic equation, 138 undetermined, 22 circle circuit, 335 estimation, 108 commutative law, 16, 55 EULER constant, 208 conduction equation, 187 Example continuous signals, 313 matrix convection diffusion equation, 187 addition, 14 cross product, 56 example curl, 60 interpolation with
Peter-Wolfgang Gr¨aber Systems Analysis in Water Management INDEX
analytical power function method, Exercises 77 algebraic expressions, 8 calculation leading principal minors, exercises 38 analytical solutions of the ground- CHOLESKY decomposition, 44 water flow computation of equation systems with THEIS well equation, 234 Cramer’s rule, 28 experimental process analysis, 358 Gauss elemination, 25 faltung integral, 375 inverse matrices, 31 interpolation divergence, 60 LAGRANGE, 105 EULER method, 174 LAPLACE transform, 158 gradient, 58 numerical interpolation, 72 calculation of LAGRANGE interpolation, 79 goundwater processes LAPLACE transform, 147 finite difference method, 281 LU decomposition, 35 numerical integration, 169, 181 matrix setting up ODE’s, 121 inverse, 17 solution of ODE multiplication, 15 first order, 133 NEWTON interpolation, 87 solutions of ODE of higher order, Predictor Corrector method, 180 143 rectangle rule, 162 solving of equation systemes with RUNGE KUTTA method, 177 Cramer’s rule, 50 separation of variables, 126 Gauss’ algorithm, 50 setting up DE’s, 117 exponential expressions, 6 SIMPSON rule, 166 faltung solution integral, 366 third order, 141 operation, 366 solution of an ODE system feedback, 335 LAPLACE transform, 156 flow solution of ODE rotation symmetrical, 188 inhomogeneous, 131 flow field second order, 140 parallel spline interpolation, 97 ditch flow, 188 transpose matrix, 10 force equilibrium law, 194 vector calculus, 62 examples GALERKIN method, 266 solution of ODE GAUSS law, 59 LAPLACE transform, 153 Girinskij potential, 197
402 INDEX gradient, 58 subtraction, 14 GREEN formula, 268 symmetric, 10 groundwater flow equation transpose, 9 horizontal plane, 197 unitmatrix, 11 method identical magnitude, 304 Ansatz(ODE), 127 impulse separation of variables, 126 DIRAC impulse, 304 substitution, 136 information parameters, 303 variation of constants, 129 integral transform mixing place, 309 continuous, 145 model discrete, 146 discontinuous, 242 interpolation discrete, 242 spline, 91 modelling spline functions theoretical, 294 cubic, 93 interpolation polynomial NABLA operator, 57 LAGRANGE, 78 NEUMANN condition, 202
LAPLACE operator, 58 operator law LAPLACE, 58 anticommutative, 57 NABLA, 57 associative, 55 commutative, 55 parallel circuit, 335 distributive, 55, 57 partial fractions, 155 leakage factor, 231 polynomial length LAGRANGE, 78 vector, 55 process analysis logarithmic expressions, 6 theoretical, 294, 300 product magnitude of vectors, 55 vector, 55 integration, 130 matrix scalar, 55 addition, 14 vector, 56 band matrix, 12 product rule, 130 determinant, 19 division, 14 quantization, 242 exponentiation, 16 reflection method, 222 inverse, 14, 16 rotation, 60 multiplication, 14 square, 14 scalar product, 55
403 INDEX series ciurcuit, 335 signal, 303 signal carriers, 303 signals, 312 sine function, 304 spline funktions cubic, 93 step function (unit step), 304 storage coefficiant, 195 transferfunction, 315 transit function, 315 transmissibility, 196 unit impulse, 306 ramp function, 307 step, 307 vector product, 56 weight function, 315 well function THEIS, 208
404