ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Unconfined aquifers
figures from Kruseman and de Ridder (1991)
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Variations from “normal” drawdown hydrographs Unconfined aquifers
Early time: when pumping starts, drawdown has not reached
the water table; water comes from elastic storage only (Ss)
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Hydrology Program, Prof. J. Wilson 1 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Unconfined aquifers
Intermediate time: now, a moving water table is being drained (Sy); there is a strong 3-D component to flow The vertical flow component means that more energy is required
to get water out of the aquifer—flowpaths are longer 3
Variations from “normal” drawdown hydrographs Unconfined aquifers
Late time: now, most water is coming from far away from the well; large annular area means a small drawdown will yield a lot of water Flow is now close to 1-D Jacob-type flow (or Theis flow)— but storage is still from S y 4
Hydrology Program, Prof. J. Wilson 2 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Unconfined aquifers
Slopes are determined by T; intercepts are determined by S
t t o early o late S = 10-4 S = 10-1
base figure from Kruseman and de Ridder (1991)
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Variations from “normal” drawdown hydrographs Leakage
Flow through aquitards is common
An aquifer receiving leakage is a “semi-confined aquifer” or [“leaky aquifer”]
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Hydrology Program, Prof. J. Wilson 3 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Leakage
figures from Kruseman and de Ridder (1991)
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Variations from “normal” drawdown hydrographs Partial penetration
(Dawson and Istok, 1991) 8
Hydrology Program, Prof. J. Wilson 4 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Partial penetration
(Kruseman and de Ridder, 1991)
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Variations from “normal” drawdown hydrographs Large diameter wells
The specific yield (even in a confined aquifer, a pumping well must be open to the atmosphere) of a well bore is essentially 1; this high storage term results in ‘well bore storage’ effects—the water produced in early time is coming mainly from stored water in the well bore, not inflow to the well bore from the aquifer
If T is low, S is very low, or Q is very high, even a “small” well bore can be “large”
At low T, well bores above ~2 cm can show storage effects; at high T, a well might have to be over 0.5 m in diameter before storage effects are observed
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Hydrology Program, Prof. J. Wilson 5 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs Large diameter wells
(Kruseman and de Ridder, 1991)
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Variations from “normal” drawdown hydrographs Boundary effects
(Kruseman and de Ridder, 1991)
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Hydrology Program, Prof. J. Wilson 6 ERTH403/HYD503, NM Tech Fall 2006
Variations from “normal” drawdown hydrographs In-well pumping tests
It’s best to avoid using data collected in the pumping well •can be difficult to get measuring instruments in •data can be bad •could damage pump or measuring equipment •permeability adjacent to the well bore can be very different than in the bulk of the aquifer •bentonite clay “skin” of low permeability •gravel pack •well screen
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Variations from “normal” drawdown hydrographs Skin effects
skin or poor screen
gravel pack or very well developed well
base figure from Kruseman and de Ridder (1991)
Can get T (slope is fine), but estimates of S will be off—you should never try to estimate S from a pumping well 14
Hydrology Program, Prof. J. Wilson 7 ERTH403/HYD503, NM Tech Fall 2006
Superposition
• Today
– Superposition
– Image Well Theory
– Well Test Analysis
Charles Harvey, MIT
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Motivation:
How will the aquifer How can we use the answers respond … to these questions to interpret well tests for model diagnostics and parameter – to pumping at a different rate estimation? from the same well? For example: – to injection at a constant rate through the same well? – Variable pumping rates • especially “Recovery Tests” – if the pumping rate varies in time? – Effects of aquifer boundaries • Especially lateral stream and barrier boundaries – If there are two or more wells pumping or injecting at different locations?
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Hydrology Program, Prof. J. Wilson 8 ERTH403/HYD503, NM Tech Fall 2006
Linearity • The answers to these and other questions relies on the concept of linearity – We “superpose” the effects of each event or well separately, and then – sum their effects. • Many problems of well hydraulics, and indeed of groundwater hydrology in general, can be considered essentially linear and solved via superposition. • It’s a very useful concept – But should be tested to see if the system is actually behaving approximately linear. – Testing linearity is not a subject of this class.
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Reminder: Linear Models Recall the concept of an operator L( )
Given '( ) T ' '( ) eg, L( ) S &r # = ( $ ! – constants a & b 't r 'r % 'r "
– states (dependent variables) s1 & s2 then an operator L( ) is linear iff
L(a s1 ) = a L(s1 )
L(s1 + s2 ) = L(s1 ) + L(s2 )
L(a s1 + b s2 ) = a L(s1 ) + b L(s2 )
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Hydrology Program, Prof. J. Wilson 9 ERTH403/HYD503, NM Tech Fall 2006
Example application of Linear Model Rules
Let 3 – s1 & s2 be the drawdowns due to unit (e.g. 1 m /d) pumping at each of two wells, – Let constants a & b, respectively, represent the multiplier to find the actual pumping rate at each well.
Then the total drawdown … r1
due to pumping rate a at well 1 is s = a s1 r2 Pumping well 1 due to unit pumping at the two wells is s = s1 + s2 Pumping well 2 Point of interest due to arbitrary pumping at the two wells is s = a s1 + b s2 19
Model Define si as the drawdown caused Due to linearity: by a single well, well index i, – say due to a “unit” pumping rate 3 Drawdown due to arbitrary (eg, Q=Qu=1m /d) – is sometimes called the influence pumping at two wells is or response function. s = a s1 + b s2 Let’s illustrate how we use Q1 Q2 linearity to answer our = W (u1 ) + W (u2 ) questionsfor well hydraulics 4 T 4 T aQ bQ – where the response of an aquifer = u W (u ) + u W (u ) to pumping by a well at a constant 4 T 1 4 T 2 rate is described by the Theis model, or its Cooper-Jacob logarithmic approximation.
r1 Most of what follows can be generalized to other aquifer/well where : 2 2 models; you can find this in your r1 S r2 S u1 = ; u2 = r text and various references. 4Tt 4Tt 2 Pumping well 1 r1, r2 = distance to Pumping well 2 wells 1 & 2, respectively Point of interest
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Hydrology Program, Prof. J. Wilson 10 ERTH403/HYD503, NM Tech Fall 2006
Answers to questions: How will the aquifer respond …
– to pumping at a different aQ rate from the same well? s = u W (u) 4 T We’ll take a look at a simple version of this – to injection at a constant Q problem, step changes in rate through the same s = u W (u) pumping, not involving a 4 T convolution integral. well?
r 2 S W ( ) – if the pumping rate varies 1 t 4T (t ) s = Q( ) d in time? 4 T 0 t – If there are two or more aQu bQu wells pumping or injecting s = W (u1 ) + W (u2 ) at different locations? 4 T 4 T
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Varying pumping rate Can be handled by superposing pumping and injection in our well so as to recreate the desired pumping history. Q In practice assume pumping history is Q2 piecewise constant, i.e. step changes in pumping rate. Q1 0 Suppose, e.g.: 0 t1 t Q = 0, t<0, 1 1 0 Q = Q , 0 Then superpose drawdowns. This is most 0 t1 clearly seen in the simple case of a t recovery test (next slide). 22 Hydrology Program, Prof. J. Wilson 11 ERTH403/HYD503, NM Tech Fall 2006 Well Test Analysis: What happens when we shut off a How can we use the well that has been pumping? answers to these Does drawdown recover instantly? questions to interpret well tests for model Q diagnostics and parameter estimation? 0 For example: t – Variable pumping rates 0 especially “Recovery Tests” ? – Effects of aquifer boundaries s especially lateral stream and barrier boundaries t 23 For a realistic recoveryR model,ec weo useve superposition.ry Test Pretend the well keeps pumping at the same constant rate +Q, but, at the time it shuts off add an imaginary well injecting water into the aquifer at constant rate –Q at the same location; use superposition to add the results. Because Q + -Q = 0 for t’>0, we are modeling the situation where then Q=0. 24 (Freeze and Cherry, 1979) Hydrology Program, Prof. J. Wilson 12 ERTH403/HYD503, NM Tech Fall 2006 Recovery Test s t’ -s Harvey, MIT 25 Recovery Test Recovery tests can be important, as there is no problem maintaining a constant pumping rate. Using semilog model, the forward model is: Q 2.25 T t Time 0 < t s = ln 2 4 T r S Time 0 < t’ Superposing solutions: Q 2.25 T t 2.25 T t' s s s ln ln total = pumping + "injection" = 2 2 4 T r S r S Q t stotal = ln 4 T t' t: time since pumping started t’: time since pumping stopped 26 Hydrology Program, Prof. J. Wilson 13 ERTH403/HYD503, NM Tech Fall 2006 Recovery Test Inversion: Q t Q t stotal = ln T = ln 4 T t' 4 s t' To be used with semilog plot: late time 0 2.3 Q t T = log 4 s t' s early time 2.3 Q $ 10t' 2.3 Q log (t/t’) T = log& ) = 4 " #slc % t' ( 4 " #slc When the pump is shut off, t/t’ = While at very large time, t/t’ 1 The curved line at early time is due to the fact that the injection well has u > 0.01. ! 27 Multiple Wells If there are two or more wells, say with well number index ( )i and each pumping at (time varying) rate Qi then s(t) = s1(t)+ s2(t) +s3(t)+…= !si’s which works if each of the Qi’s are constant or if they are time varying Qi(t) with different time histories. If the pumping rates are constant wrt time then the r drawdown is given by 1 Q1 Q2 Q3 s = W (u ) + W (u ) + W (u ) + ... r 4"T 1 4"T 2 4"T 3 2 Pumping well 1 2 Pumping well 2 ri S ui = ; toi = start time for well i Point of interest 4T (t ! toi ) 28 Hydrology Program, Prof. J. Wilson 14 ERTH403/HYD503, NM Tech Fall 2006 Image Well Theory With multiple wells, not all of the wells need be real. -Q(t) Q(t) We can use the concept of virtual or “image” wells to superpose drawdowns in order to mimic Dirichlet and d d Neumann boundary image well real well conditions. constant head boundary 29 How would represent a straight line constant head Dirichlet boundary located a distance d from the pumping well? If the well is pumping at rate Often called a recharge boundary Q(t)=Q1(t), causing drawdown s(t)= s1(t), then place a mirrior image injection well, injecting at the same rate -Q(t) Q(t) Q2(t) = - Q1(t), at a distance d on the other side of the boundary, causing drawdown s2(t) = -s1(t) [ie, with drawup s1(t)]. d d Then s(t)=s1(t)+s2(t) image well real well Along the boundary, where r1=r2, the drawdown from well 1 is balanced by the drawup from well 2, such that s(t)=0, preserving the constant head BC. constant head boundary Q , & 2.25 T t # & 2.25 T t #) Q & r # s = s + s . ln$ ! - ln$ ! = ln$ 2 ! real image 4 T * $ r 2 S ! $ r 2 S !' 2 T $ r ! / + % 1 " % 2 "( / % 1 " 30 Hydrology Program, Prof. J. Wilson 15 ERTH403/HYD503, NM Tech Fall 2006 How would represent a straight line constant head Dirichlet boundary located a distance d from the pumping well? -Q(t) Q(t) d d image well real well constant head boundary (Freeze and Cherry, 1979) 31 How would represent a straight line no-flux Dirichlet boundary located a distance d from the pumping well? If the real well is pumping at rate Q(t)=Qr(t), Often called a barrier boundary where r = real, causing drawdown s(t)= sr(t), then place a mirrior image pumping well, pumping at the same rate Q(t) Q(t) Qi(t)= Q(t) = +Qr(t), where i = image, at a distance d on the other side of the boundary, causing drawdown si(t) Then s(t)=s (t)+s (t) r i d d Along the boundary, where rr=ri, image well real well the gradient from real well is balanced by the gradient from image well, such that !s/ !x=0, preserving the no flux BC. Using the semilog model: no flow Q & , 2.25 T t ) , 2.25 T t )# boundary s = s + s - ln* ' + ln* ' real image 4 T $ * r 2 S ' * r 2 S '! y . % + r ( + i (" x 32 Hydrology Program, Prof. J. Wilson 16 ERTH403/HYD503, NM Tech Fall 2006 How would represent a straight line no-flux Dirichlet boundary located a distance d from the pumping well? Q(t) Q(t) d d image well real well (Schwartz and Zhang, 2003) no flow boundary y (Freeze and Cherry, 1979) x 33 How would represent a straight line no-flux Dirichlet boundary located a distance d from the pumping well? image well Q(t) no flow d boundary d Q(t) real well (De Wiest, 1965) 34 Hydrology Program, Prof. J. Wilson 17 ERTH403/HYD503, NM Tech Fall 2006 What about other geometries? You can use image wells in other geometric patterns to mimic more complex geometric domains (see e.g., Bear, 1972) Well near a “corner”: Image pumping well Real pumping well Image pumping well Image pumping well 35 Well Test Analysis with Boundaries • How can we use image- • Previous classes: the well-test equations made the assumption well theory to interpret that the aquifer is of infinite well tests? horizontal extent. • We use image well theory to • For example: handle nearby boundaries. • The image well results can be – Variable pumping rates – represented in log-log, semilog, • especially “Recovery and derivative plots and Tests” – used to diagnose the presence of – Effects of aquifer different types of BCs, and boundaries – to estimate their parameters, such as distance d above, as well as T • especially lateral stream and S. and barrier boundaries 36 Hydrology Program, Prof. J. Wilson 18 ERTH403/HYD503, NM Tech Fall 2006 Well Test Analysis with Boundaries 0 Influence of recharge boundary Constant head or s recharge boundary log t Q & r # s s s ln 2 = real + image ' $ ! 2 ( T % r1 " Influence of 0 Barrier or barrier boundary no-flow s Slope = m boundary Slope = 2m log t Q & , 2.25 T t ) , 2.25 T t )# s = s + s - ln* ' + ln* ' real image 4 T $ * r 2 S ' * r 2 S '! . % + r ( + i (" 37 Well Test Analysis with Boundaries 0 How far to the boundary? s We pick so that sR = sI Q 2.25TtR sR = ln 2 4"T rR S Q 2.25Tt I log t sI = ln 2 4"T rI S ! 0 2.25TtR 2.25TtI tR tI " ln 2 = 2 " 2 = 2 s ! rR S rI S rR rI 1 " % 2 tI ! rI = rR $ ' ! # tR & log t 38 ! Hydrology Program, Prof. J. Wilson 19 ERTH403/HYD503, NM Tech Fall 2006 Well Test Analysis with Boundaries Where is the boundary located? observation well Q real pumping well observation well 39 Hydrology Program, Prof. J. Wilson 20