Appendix A Nomenclature
A.1 Roman Letters
AL2 area; A˛ L2T 2 Helmholtz free energy of ˛ phase; BL aquifer thickness; bL hydraulic aperture; 3 1 bk bk .ML / Freundlich sorption coefficient of species k; bk 1 Freundlich sorption exponent of species k; p bk rate constants of species k, .p D 0;1;:::;N/; CL1=2T 1 Chezy roughness coefficient; CL 1 moisture capacity; C 1 L inverse moisture capacity; 3 Ck ML mass concentration of species k; 3 Cks ML maximum mass concentration of species k; 3 Ckw ML prescribed concentration of species k at well point xw; Cr 1 Courant number; cL2T 2% 1 specific heat capacity; cF 1 Forchheimer form-drag constant; D spatial dimension; DL diameter or thickness; D 2nd-order differential operator or dispersion coeffi- cient; 2 1 Dk L T tensor of hydrodynamic dispersion of species k; ? 2 1 Dk L T nonlinear (extended) tensor of hydrodynamic disper- sion of species k; 3 1 DN k L T D B Dk, depth-integrated tensor of hydrodynamic dispersion of species k;
H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 809 © Springer-Verlag Berlin Heidelberg 2014 810 A Nomenclature
2 1 Dk;0 L T tensor of diffusion of species k; 2 1 Dmech L T tensor of mechanical dispersion; 3 1 DN mech L T D B Dmech, depth-integrated tensor of mechanical dispersion; 2 1 Dk L T coefficient of molecular diffusion of species k in porous medium; 2 1 DM k L T coefficient of molecular diffusion of species k in open fluid body; 2 1 s D L T D Œ" C .1 "/ =." c/, thermal diffusivity; 1 1 T d T D 2 Œrv C .rv/ , rate of deformation (strain) tensor; dL characteristic length, thickness or diameter; da L2 microscopic differential area; dS L2 projected planar area of averaging volume; dV L3 averaging volume; dv L3 microscopic differential volume; E ML2T 2 internal (thermal) energy; EL2T 2 specific internal (thermal) energy density; E ML 1T 2 Young’s (or elastic) modulus of the solid phase; E# L2T 2 activation energy; e 1 D g=kgk, gravitational unit vector; e. ; ; / 1 e vector with respect to local coordinates, defined by (K.2); e 1 strain tensor; e, e error and error vector, respectively; ei 1 ith component of base vector; eNi 1 ith component of tangent coordinate vector; F extensive quantity; F surface or interface function; f function or intensive quantity; f B L bottom bounding surface of aquifer; fD 1 Darcy-Weisbach roughness coefficient; f 1 viscosity relation function; fN 1 Newton-Taylor roughness coefficient; 2 f LT interfacial drag term of fluid momentum exchange; f T L top bounding surface of aquifer; 2 2 f ML T deviatoric drag term of fluid momentum exchange; G ML 1T 2 shear modulus of the solid phase; 1 GL first derivative of relative permeability kr with respect to pressure head ; G number of grout zones of BHE; A.1 Roman Letters 811
G˛ L2T 2 Gibbs free energy of ˛ phase; g LT 2 gravity vector; gLT 2 Dkgk, gravitational acceleration; gi 1 ith component of base vector of transformed coordi- nates; gNi 1 ith component of tangent vector of transformed coor- dinates; HL height; H ˛ L2T 3 supply (source/sink) of energy of ˛ phase; 2 3 H˛ L T bulk source/sink term of energy of ˛ phase; 1 3 He ML T overall source/sink term of internal energy; 3 HNe MT D BHe, depth-integrated source/sink term of inter- nal energy; H ? L2T 3 modified heat source/sink term without well func- tion; hL hydraulic head; he L characteristic element length or height; I 1 ionic strength; I functional; J Jacobian matrix; 1 =F M LT Forchheimer coefficient; 1 2 =H M L T non-Fickian HC dispersion coefficient; 1 =N H M LT D B =H , depth-integrated non-Fickian HC disper- sion coefficient; 2 1 jk ML T mass flux of species k; 1 1 jNk ML T D B jk, depth-integrated mass flux of species k; 3 1 jS MT % entropy flux; 3 jT MT heat flux; j arbitrary flux vector; jQ smoothed flux; 1 K LT D k 0g= 0, tensor of hydraulic conductivity; K ML2T 2 kinetic energy; Keq 1 equilibrium constant; d 1 3 s Kk M L D k= , distribution coefficient of species k; 3 Km ML Michaelis-Menten’s half-saturation constant; k L2 porous-medium permeability tensor (phase- independent); kf L2 intrinsic porous-medium permeability tensor of f phase; kk 1 Langmuir numerator sorption coefficient of species k; 812 A Nomenclature
1 3 kk M L Langmuir denominator sorption coefficient of species k; kr 1 relative permeability; L L 1 symmetric gradient operator; LL length, macroscopic scale length; L differential operator, PDE (system) operator; Lek 1 Lewis number of species k; ` mesh level or refinement level; M number of phases present in the system; ML1=3T 1 Manning roughness coefficient; M f ML 1T 1% 1 Soret coefficient of phase f ; M M mass; m 1 specific unit vector; m 1 VG-curve fitting parameter; m total number of Gauss points; mk M molecularP mass of species k; ˛ N D ˛ N , total number of chemical species; e NI 1 shape function of element e at node I ; N o number of reactants; N ˛ number of chemical species in the ˛ phase; NBHE number of nodes of single BHE; NBN number of nodes per element; ND number of Dirichlet nodes; NDOF number of degrees of freedom; NE number of finite elements;
NEp number of finite elements agglomerated into parti- tion p; NEQ number of equations; NP number of points (nodes); NPA number of partitions of agglomerated finite elements; NPAD number of disjoint partitions of agglomerated finite elements; NS number of slices; 3 Nk MLT Dufour coefficient of species k; Nr number of chemical reactions; N˙ number of patch elements; Nu 1 NusseltP number; ˛ N D ˛.N 1/, essential number of chemical species; NW number of wells; n 1 outward-directed unit normal vector; A.1 Roman Letters 813
nk 1 concentration exponent of species k; n 1 pore size distribution index; n number of Gauss points in each direction; P LT 1 vector of accretion on a phreatic surface; PLT 1 groundwater recharge (infiltration rate) on a phreatic surface; Pet 1 thermal Darcy-Peclet´ number; Pg 1grid(mesh)Peclet´ number; Pr 1 Prandtl number; p ML 1T 2 (thermodynamic) pressure; 1 2 pc ML T capillary pressure; 1 2 pmech ML T mechanical pressure; Q˛ T 1 supply (source/sink) of mass of ˛ phase; 1 Q˛ T bulk source/sink term of mass of ˛ phase; 1 QT D Qh C Qhw, bulk source/sink term of flow; QLTN 1 depth-integrated bulk source/sink term of mass; 1 QEOB T correction sink/source term of extended Oberbeck- Boussinesq approximation; Qn arbitrary integral boundary balance flux; 3 1 Qnh L T integral boundary balance flux of liquid (positive inward-directed); 1 Qh T supply term of flow; 1 Qhw T specific liquid sink/source function of wells; 1 QN h LT depth-integrated supply term of flow; 1 QN hw LT depth-integrated specific liquid sink/source function of wells; 3 1 Qk ML T zero-order kth-species mass sink/source function; 2 1 QN k ML T depth-integrated zero-order kth-species mass sink/source function; 3 1 Qkw ML T specific kth-species mass sink/source function of wells; 2 1 QN kw ML T depth-integrated kth-species mass sink/source func- tion of wells; 3 1 Qr L T total refrigerant flow discharge of BHE; 1 3 QT ML T He QT w, overall heat source/sink term without well function; 3 QN T MT HNe QN T w, depth-integrated heat source/sink term without well function; 1 3 QT w ML T specific heat sink/source function of wells; 3 QN T w MT depth-integrated heat sink/source function of wells; 3 1 Qw L T discharge of single well w (pumping rate); f 1 f s q LT D "f .v v /, Darcy velocity; 814 A Nomenclature q LT 1 D qf , Darcy velocity of fluid; qN L2T 1 D B q, depth-integrated Darcy velocity; qn arbitrary boundary flux (positive outward-directed); 1 qnh LT D q n, normal boundary flux of liquid (positive outward-directed); 2 1 qNnh L T DNq n, depth-integrated normal boundary flux of liquid; 2 1 qnkC ML T normal boundary mass flux of species k (positive outward-directed); 1 1 qNnkC ML T depth-integrated normal boundary mass flux of species k; 3 qnT MT normal boundary heat flux (positive outward- directed); 3 qNnT MLT depth-integrated normal boundary heat flux; 1 qh LT prescribed Neumann boundary liquid flux; 2 1 qNh L T prescribed depth-integrated Neumann boundary liq- uid flux; 2 1 qkC ML T prescribed Neumann boundary mass flux of species k of the dispersive part; 2 1 qkC ML T prescribed Neumann boundary mass flux of species k of the total (convective plus dispersive) part; 1 1 qNkC ML T prescribed depth-integrated Neumann boundary mass flux of species k of the dispersive part; 1 1 qNkC ML T prescribed depth-integrated Neumann boundary mass flux of species k of the total (convective plus dispersive) part; 3 qT MT prescribed Neumann boundary heat flux of the dispersive part; 3 qT MT prescribed Neumann boundary heat flux of the total (convective plus dispersive) part; 3 qNT MLT prescribed depth-integrated Neumann boundary heat flux of the dispersive part; 3 qNT MLT prescribed depth-integrated Neumann boundary heat flux of the total (convective plus dispersive) part; RL2T 2% 1 8.314 J/ıK mole, molar gas constant; RL radius of pipe or wellbore; 1 2 3 RN, RNn M L T % thermal resistance and thermal resistance of material n, respectively; 1 1 3 R, Rn M L T % specific thermal resistance and specific thermal resis- tance of material n, respectively; 1 1 3 Ra, Rb M L T % (specific) internal borehole thermal resistance and (specific) borehole thermal resistance, respectively; A.1 Roman Letters 815 P 3 1 ˛ ˛ Rk ML T D ˛ "˛.rk C Rk /, bulk rate of reaction of species k in all phases;P Q 3 1 ˛ O Rk ML T D Rk C ˛ "˛#kCk D Rk C Qkw C Qk, deduced bulk reaction rate; 3 1 ROk ML T RQk Qkw Qk, modified bulk reaction rate; 3 1 RNk ML T D BRk , depth-integrated reaction bulk rate of species k; N 3 1 RQk ML T D BRQk, depth-integrated reaction bulk rate; ˛ 3 1 Rk ML T heterogeneous reaction rate of species k of ˛ phase; Ra 1 Rayleigh number; Rat 1 thermal Rayleigh number; Rak 1 solutal Rayleigh number of species k; Re; Rep 1 Reynolds number and pore Reynolds number, respectively; s s S 1 D 1CŒ.1 "/ c =." c/, thermal storage coefficient (thermal retardation factor); s 1 saturation of fluid in the void space "; sL arc length of curve; se 1 effective saturation; sp 1 pseudo-saturation; sr 1 residual (irreducible) saturation; ss 1 maximum saturation; 2 1 T L T D kB 0g= 0, tensor of transmissivity; T% temperature; Ti % inlet temperature; To % outlet temperature; Tw % prescribed temperature at well point xw; T 1 tortuosity; TBE./ L3 total balance error; Tuk 1 Turner (or buoyancy) number of species k; t 1 unit tangent vector; ts ML 1T 2 elasticity matrix of the solid phase; tT time; t0 T initial time; tend T final time; t1=2k T reaction half-life of species k; 3 UC ML mass-mechanical coefficient; 1 2 UT ML T thermo-mechanical coefficient; u L displacement; u LT 1 vector of refrigerant fluid velocity; u LT 1 Dkuk, magnitude of refrigerant fluid velocity; V MLT 1 (linear) momentum; v LT 1 velocity, pore velocity; v˛ LT 1 velocity of the ˛ phase; v˛s LT 1 D v˛ vs , relative velocity of the ˛ phase to the s phase; vLT 1 Dkvk, magnitude of velocity; 3 1 vm ML T Michaelis-Menten’s maximum growth rate; W ˛ L2T 3% 1 supply (source/sink) of entropy of ˛ phase; W LT 1 velocity of macroscopic interface; W./ 1 well function; w LT 1 velocity of microscopic interface; w 1 spatial weighting function; w L width; X L Lagrangian material coordinates; A.2 Greek Letters 817 x L Eulerian spatial coordinates; x L macroscopic position vector; xw L position of well w; x1;x2;x3 L Cartesian coordinates; x;y;z L Cartesian coordinates; Z L 1 D 1 m 1, unit-canceling coefficient; z L axial or vertical coordinate; z L nodal vector of vertical coordinates; zk 1 charge on the kth species; A.2 Greek Letters ˛L 1 curve fitting parameter, sorptive number; ˛ 1 upwind parameter, .0 ˛ 1/; ˛ 1 specific solutal expansion coefficient of a single- species solute, density ratio; ˛k 1 specific solutal expansion coefficient of species k, density ratio; 2 1 ˛L MT % longitudinal thermodispersivity; 1 1 1 ˛NL ML T % specific longitudinal thermodispersivity; 2 1 ˛T MT % transverse thermodispersivity; 1 1 1 ˛NT ML T % specific transverse thermodispersivity; ˇ% 1 thermal expansion coefficient; 1 3 ˇck M L D ˛k=.Cks Ck0/, solutal expansion coefficient of species k; ˇL L longitudinal dispersivity; ˇT L transverse dispersivity; ˇTH L horizontal transverse dispersivity; ˇTV L vertical transverse dispersivity; L2 areal boundary; ˛ 1 ˛ phase distribution function; f M 1LT 2 compressibility of f phase; k 1 activity coefficient of species k; Δ 1 Boolean8 matrix; < C1; r D 0 ı.r/L D D , Dirac delta function of spatial : 0; r ¤ 0 dimension of r, D D 1; 2; 3; ı 1 error tolerance; ıL microscopic scale length; ı 1 curve fitting exponent; 818 A Nomenclature ı 1 constant of friction slope relationship; δ unit or identity matrix; ıij Kronecker symbol; 1 strain pseudovector; ε Levi-Civita tensor; " 1 volume fraction, porosity (void space); 1 error tolerance; 2 1 (dimensionless) residual error tolerance; 1 (dimensionless) AMR-specific error tolerance; Î 1 (dimensionless) particle tracking error tolerance; 3 1 2 L T (dimensional) residual error tolerance; "e 1 D ".1 sr /, specific yield (storativity of phreatic aquifer); "ij k permutation symbol; η 1 transformed (local) coordinate vector; 1 local coordinate; Eulerian angle; 1 weighting coefficient, .0 1/; 1 D s", moisture content; 1 #k T decay rate of species k; k 1 Henry sorptivity coefficient of species k; .A/ 1 condition number of matrix A; Λ MLT 3% 1 tensor of hydrodynamic thermodispersion; ΛN ML2T 3% 1 D B Λ, depth-integrated tensor of hydrodynamic thermodispersion; 3 1 Λ0 MLT % tensor of thermal conductivity; 3 1 Λmech MLT % tensor of mechanical thermodispersion; MLT 3% 1 coefficient of thermal conductivity of liquid; ˛ MLT 3% 1 coefficient of thermal conductivity of ˛ phase; r MLT 3% 1 coefficient of thermal conductivity of refrigerant fluid; s MLT 3% 1 coefficient of thermal conductivity of solid; f ML 1T 1 dilatational viscosity of the fluid phase; s ML 1T 2 LameK constant of the solid phase; f ML 1T 1 dynamic viscosity of the fluid phase; l 1 1 N 0 ML T specific dynamic reference viscosity of the liquid phase; 2 2 k L T chemical potential of kth-species; s ML 1T 2 LameK constant of the solid phase; 1 Poisson’s ratio; k; kr 1 stoichiometric coefficient of species k (and reac- tion r); A.2 Greek Letters 819 ML 3 mass density; 3 k ML mass density of species k; ML 3% 4 Stefan-Boltzmann constant; σ ML 1T 2 stress tensor; σB ML 1T 2 bottom surface momentum exchange vector; σT ML 1T 2 top surface momentum exchange vector; τ ML 1T 2 deviatoric stress tensor; generalized friction factor; L2T 3% 1 net production of entropy; M 1LT 2 coefficient of skeleton compressibility; ˚L2T 1 potential function; 1 ˚h T liquid transfer (colmation/leakage) coefficient; 1 ˚Nh LT depth-integrated liquid transfer coefficient; ˚k 1 D Lek=S , ratio of diffusivities of heat and species k; 1 ˚kC LT mass transfer coefficient of species k of the dispersive part; 1 ˚kC LT mass transfer coefficient of species k of the total (convective plus dispersive) part; 2 1 ˚NkC L T depth-integrated mass transfer coefficient of species k of the dispersive part; N 2 1 ˚kC L T depth-integrated mass transfer coefficient of species k of the total (convective plus dispersive) part; 3 1 ˚T MT % heat transfer coefficient of the dispersive part; 3 1 ˚T MT % heat transfer coefficient of the total (convective plus dispersive) part; 3 1 ˚NT MLT % depth-integrated heat transfer coefficient of the dispersive part; N 3 1 ˚T MLT % depth-integrated heat transfer coefficient of the total (convective plus dispersive) part; arbitrary function, azimuthal coordinate or Eulerian angle; 'k 1 adsorption function of species k; 1 D tnC1= tn, maximum rate of time step change; aniso 1 ratio of anisotropy of hydraulic conductivities; aniso 1 anisotropy factor of solid-phase thermal conductivi- ties; 1 D . 0/= 0, buoyancy coefficient; L2T 1,L3T 1 streamfunction for 2D and axisymmetric problems, respectively; scalar variable or Eulerian angle; L pressure head; 820 A Nomenclature a L air-entry pressure head; c L capillary fringe pressure head; ˝L3 domain or volume; ω T 1 vorticity; ! 1 2 .0; 1/, coating factor; !k 1 mass fraction or specific density of species k; 1 local coordinate; 1 local coordinate; r L 1 Nabla (vector) operator; A.3 Subscripts 0 reference or initial; ˛ phase index taking a value of l, g,ands; subscript designates bulk quantities; ˛k phase which contains the species k; ˇ phase index taking a value of l, g,ands; subscript designates bulk quantities; C Cauchy-type boundary; D Dirichlet-type boundary; reversal; eq equilibrium; ex external; F discrete feature; f fluid phase taking a value of l,andg; subscript designates bulk quantities; g gaseous phase; subscript designates bulk quantities; g grout; phase index taking a value of l,andg; subscript designates bulk quantities; H coarse mesh level; h fine mesh level; het heterogeneous; hom homogeneous; int intermediate; i;j;l;d spacial coordinate or global node indices; i pipe-in or internal; I;J local node indices; k; m; n species indicator; k BHE component index; A.4 Superscripts 821 n time plane; n layer, material; L longitudinal; l liquid phase; subscript designates bulk quantities; m Michaelis-Menten; mech mechanical; N Neumann-type boundary; o pipe-out or outer; s solid phase; subscript designates bulk quantities; P porous medium; R Robin-type boundary; r reaction or refrigerant; T transverse; w well; A.4 Superscripts ae aerobic; anae anaerobic; ˛ phase index taking a value of l, g,ands; superscript designates intrinsic quantities; B bottom; ˇ phase index taking a value of l, g,ands; superscript designates intrinsic quantities; crit critical; d drying; diff diffusive; e element counter; eff effective; f fluid phase taking a value of l,andg; superscript designates intrinsic quantities; f w freshwater; phase index taking a value of l,andg; superscript designates intrinsic quantities; g gaseous phase; superscript designates intrinsic quantities; g grout; g global; H coarse mesh level; h fine mesh level; I interface; i internal; 822 A Nomenclature in inward; l liquid phase; superscript designates intrinsic quantities; K iteration number to restart; k iteration counter; m Michaelis-Menten; o outer; opt optimal; out outward; pipe (BHE) system; p predictor; r refrigerant; s solid phase; superscript designates intrinsic quantities; s slice or soil; sw saltwater; T transpose; T top; TB top and bottom; w wetting; iteration counter; 4 Delta configuration; r gradient type; Y configuration; C forward reaction; backward reaction; A.5 Special Symbols ./ ./ scalar (dot) product; ./ ./ vector (cross) product or Cartesian product of sets; ./ ˝ ./ tensor (dyadic) product; ./P T 1 D @=@t, differentiation with respect to time t; ./R T 2 D @2=@t2, 2nd derivative with respect to t; ./« T 3 D @3=@t3, 3rd derivative with respect to t; ./0 L 1 D @=@x, differentiation with respect to the 1D space coordinate x; ./00 L 2 D @2=@x2, 2nd derivative with respect to x; ./000 L 3 D @3=@x3, 3rd derivative with respect to x; ./.n/ L n D @n=@xn, .n D 4;5;:::/, nth derivative with respect to x; ./O normalized or approximate quantity; A.5 Special Symbols 823 .N/˛ intrinsic mass average of ˛ phase; hi˛ intrinsic volume average of ˛ phase; hi˛ volume average of ˛ phase; r L 1 gradient vector; rO 1 dimensionless gradient vector; 1 ri L D @=@xi .i D 1; 2; 3/, gradient operator in xi coordinate direction; 1 rz L D @=@z, gradient operator in z coordinate direction; r. ; ; / 1 gradient vector with respect to local coordinates, defined by (K.1); @=@t T 1 time derivative; D˛=Dt./ T 1 D @./=@t C v˛ r./, Eulerian material derivative of P ˛ phase; Q summation notation; product notation; jj absolute value of scalar or determinant of matrix; kk vector norm; kkRMS RMS error norm; kkL1 maximum error norm; Œ molarity; dc diagonal matrix (tensor); fg activity; Appendix B Coleman and Noll Method The functional dependence for the N.2 C D/ C M.2 C 2D C D2/ constitutive variables listed in (3.105) is chosen as (3.106). This functional form is restricted in that the assumed dependence (3.106)of(3.105) may not violate the entropy inequality (3.69) for any process. This is the object of the Coleman and Noll method [94]. Using (3.103) the Clausius-Duhem inequality (3.69) can be written as h N ˛ X 1 Ds A˛ DsT ˛ X Ds !˛ D " ˛ C S ˛ . ˛ k / C ˛ T ˛ Dt Dt k Dt ˛ k XN ˛ ˛s ˛ ˛ ˛ ˛ ˛ v rA C S rT . k r!k / C k ˛ XN ˛ 2 j f ˛ v˛s C A˛ 1 v˛s ˛!˛ .Q˛ C Q˛ / C T rT ˛ C 2 k k ex T ˛ k N ˛ N ˛ N ˛ i X X rT ˛ X σ˛Wd˛ C ˛.r˛ C R˛/ C j˛ r ˛ j˛ ˛ C k k k k k T ˛ k k k k k ˛ ˛ Wex 0 (B.1) where the interface relation (3.42) with Table 3.4 have been applied to introduce the relative velocity v˛s. The differentials in (B.1) will be developed for the chosen independent variables (3.106). For instance, the use of the chain rule to Ds A˛=Dt yields: DsA˛ X @A˛ Ds" X @A˛ Ds ˇ X @A˛ Ds vfs D f C C C ˇ fs Dt @"f Dt @ Dt @v Dt f ˇ f H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 825 © Springer-Verlag Berlin Heidelberg 2014 826 B Coleman and Noll Method ˛ X @A˛ Dsdf @A˛ Dss X XN @A˛ Ds !ˇ W C C k C @df Dt @s Dt ˇ Dt f ˇ k @!k ˛ X XN @A˛ Dsr!ˇ X @A˛ Ds T ˇ X @A˛ Ds rT ˇ k C C .ˇ D s; f / ˇ Dt @T ˇ Dt @rT ˇ Dt ˇ k @r!k ˇ ˇ (B.2) The same expansion takes place for rA˛: X @A˛ X @A˛ X @A˛ rA˛ D r" C r ˇ C r.vfs/ C f ˇ fs @"f @ @v f ˇ f ˛ X @A˛ @A˛ X XN @A˛ Wr.df / C r.s / C r!ˇ C @df @s ˇ k f ˇ k @!k ˛ X XN @A˛ X @A˛ X @A˛ r.r!ˇ/ C rT ˇ C r.rT ˇ/.ˇD s; f / ˇ k @T ˇ @rT ˇ ˇ k @r!k ˇ ˇ (B.3) ˛ and similar to r k . Furthermore, the conservation equation (3.48) for mass of the ˛ phase .˛ D f; s/ can be written with (3.103)as Ds ˛ Ds" " D v˛s r." ˛/ " ˛.δWd˛/ C " ˛.Q˛ C Q˛ / ˛ ˛ (B.4) ˛ Dt ˛ ˛ ˛ ex Dt Equation (B.4) is substituted into (B.2). Then substitution of (B.2)and(B.3)into (B.1) yields: X X " ˛ @A˛ f " f X @A D vfs . ˛ r" / f . r" /C ˛ f f f T @ "f T @" f ˛ N f X @Af X @Af X . rC ˛/ C . rT ˛/ C S f rT f . f r!f / C f f @C ˛ k @T ˛ k k ˛ k ˛ k ˛ X ˛ XN X ˇ ˇ " 2 " @A ˛ A˛ 1 v˛s ˛!˛ C ˇ ˛ Q˛ C Q˛ T ˛ 2 k k T ˇ @ ˛ ex ˛ k ˇ n N ˇ N ˛ o X " X " X @ ˛ rT ˇ ˇ jˇ jˇ ˇ C ˇ j˛ k T ˇ2 T k k T ˇ k @T ˇ ˇ k k B Coleman and Noll Method 827 n X " σf X " ˛ @A˛ X " @A df W f . ˛ f δ/ C . v s ˝ /C T f T ˛ @ f T @vfs f ˛ N ˛ o X " X @ ˛ . ˛ .j˛ ˝ k // T ˛ k @vfs ˛ k X X X " @ ˛ r!ˇ ˛ j˛ k k T ˛ k ˇ ˇ k ˛ @!k Xn XN ˛ o ˛ ˛ ˛ ˛ ˛ "˛ k .rk C Rk / C "˛ Wex ˛ k h " σs X " ˛ @A˛ X " ˛ @A˛ dsW s . ˛ sδ/ C ˛ T s T ˛ @ s T ˛ @s ˛ ˛ N ˛ i X " f X @Af X " X X @ ˛ C . f vfs ˝ / C . ˛ .j˛ ˝ k // T f @v s T ˛ k @v s f ˛ k h i X Ds " X " ˛ @A˛ f @A˛ s @A˛ C f ˛ C ˛ f s Dt T @"f "f @ "s @ f ˛ h N ˛ i X X " X @ ˛ C r" ˛ j˛ k f ˛ k T @"f f ˛ k h X X " @A " s @As C r f .vfs v s/ C s vfs T @ f T s @ f f N ˛ i h i X " X @ ˛ X Dsvfs X " ˛ @A˛ ˛ j˛ k C ˛ T ˛ k @ f Dt T ˛ @vfs ˛ k f ˛ h N ˛ i X X " ˛ @A˛ X " X @ ˛ C r.vfs/W ˛ v˛s ˝ ˛ j˛ ˝ k T ˛ @vfs T ˛ k @vfs f ˛ ˛ k h i X Dsdf X " ˛ @A˛ C W ˛ Dt T ˛ @df f ˛ h N ˛ i X X " ˛ @A˛ X " X @ ˛ C r.df /W ˛ v˛s ˝ ˛ j˛ ˝ k T ˛ @df T ˛ k @df f ˛ ˛ k h N ˛ i X X " ˛ @A˛ X " X @ ˛ C r.s /W ˛ v˛s ˝ ˛ j˛ ˝ k T ˛ @s T ˛ k @s f ˛ ˛ k 828 B Coleman and Noll Method N ˇ h i X X Ds !ˇ X " ˛ @A˛ " ˇ C k ˛ C ˇ ˇ Dt T ˛ ˇ T ˇ k ˇ k ˛ @!k h i X DsT ˇ X " ˛ @A˛ " ˇ C ˛ ˇ S ˇ Dt T ˛ @T ˇ T ˇ ˇ ˛ N ˇ h i h i X X Dsr!ˇ X " ˛ @A˛ X DsrT ˇ X " ˛ @A˛ C k ˛ C ˛ Dt T ˛ ˇ Dt T ˛ @rT ˇ ˇ k ˛ @r!k ˇ ˛ N ˇ h N l i X X X " f @Af X " X @ ˛ C r.r!ˇ/W f vfs ˝ ˛ j˛ ˝ l k T f ˇ T ˛ l ˇ ˇ k f @r!k ˛ l @r!k h N ˛ i X X " f @Af X " X @ ˛ C r.rT ˇ/W f vfs ˝ ˛ j˛ ˝ k T f @rT ˇ T ˛ k @rT ˇ ˇ f ˛ k 0 for .˛ D s; f / .ˇ D ˛/ . D f/ .f D 1;:::;M 1/ .k.¤ l/ D 1;:::;N˛/ (B.5) The terms in the square brackets of (B.5) represent coefficients of quantities which the constitutive functions do not depend on by assumption (3.106). Therefore, the necessary and sufficient condition for to be non-negative for all independent thermodynamic states is that these coefficients have to vanish. Hence, all bracketed terms in (B.5) must be equal to zero. In doing so, the following restrictions result: ˛ fs f ˇ ˇ • A must be independent of v ; d ; r!k ; rT • Af is independent of s and ¤f ˛ fs f s ˇ ˇ • k must be independent of v ; d ; ; r!k ; rT Furthermore, we find X " ˛ @A˛ " ˇ ˛ D ˇ ˇ T ˛ ˇ T ˇ k ˛ @!k X ˛ ˛ ˇ "˛ @A "ˇ D S ˇ .˛; ˇ D s; f / (B.6) T ˛ @T ˇ T ˇ ˛ X " ˛ @A˛ " σs X " ˛ @A˛ ˛ D s C . ˛ sδ/ T ˛ @s T s T ˛ @ s ˛ ˛ Appendix C Thermally Variable Fluid Density Expansion ˇf .T f / The temperature-dependent fluid density f ranging between 0 and 100 ıC can be fit by a 6th order polynomial with a high accuracy as 2 3 4 5 6 f .T f / D a C bT f C cT f C dTf C eT f C fTf C gT f in [g/l] (C.1) with the coefficients for water a D 9:998396 102 b D 6:764771 10 2 c D 8:993699 10 3 d D 9:143518 10 5 (C.2) e D 8:907391 10 7 f D 5:291959 10 9 g D 1:359813 10 11 where the temperature T f is in ıCanda represents the fluid density at T f D 0. Taking also into consideration of mass fraction and pressure dependencies we can f argue the measured curve (C.1) is related to a reference mass fraction !k0 and f a reference pressure p0 . Indeed, the coefficients (C.2) have been derived for a f f freshwater condition !k0 D 0.k D 1;:::;N 1/. Introducing a reference f temperature T0 we can then obtain directly from (C.1) an expression for the f reference fluid density 0 ,viz., ˇ ˇ ˇ ˇ f f f f f ˇ ˇ f ˇ f 2 ˇ f 3 0 D .p0 ;!k0;T0 / D a f f C b f f T0 C c f f T0 C d f f T0 p0 ;!k0 p0 ;!k0 p0 ;!k0 p0 ;!k0 ˇ ˇ ˇ ˇ f 4 ˇ f 5 ˇ f 6 Ce f f T0 C f f f T0 C g f f T0 (C.3) p0 ;!k0 p0 ;!k0 p0 ;!k0 H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 829 © Springer-Verlag Berlin Heidelberg 2014 830 C Thermally Variable Fluid Density Expansion ˇf .T f / To find the thermally variable fluid density expansion ˇf .T f / of the fluid density function (3.199) NXf 1 f f f f f f f f f f f f D 0 1 C .p p0 / C ˛k .!k !k0/ ˇ .T /.T T0 / kD1 (C.4) f f f f a Taylor series expansion for around T0 , !k0 and p0 is used: ˇ @ f ˇ f .T f ;!f ;pf / D f .T f ;!f ;pf / C ˇ .T f T f / C k „ 0 ƒ‚k0 0… f f f f 0 @T T0 ;!k0;p0 f 0 ˇ ˇ @2 f ˇ @3 f ˇ 1 ˇ .T f T f /2 C 1 ˇ .T f T f /3 C 2 f 2 f f f 0 6 f 3 f f f 0 @T T0 ;!k0;p0 @T T0 ;!k0;p0 ˇ ˇ @4 f ˇ @5 f ˇ 1 ˇ .T f T f /4 C 1 ˇ .T f T f /5 C 24 f 4 f f f 0 120 f 5 f f f 0 @T T0 ;!k0;p0 @T T0 ;!k0;p0 ˇ @6 f ˇ 1 ˇ .T f T f /6 C 720 f 6 f f f 0 @T T0 ;!k0;p0 f ˇ ˇ NX 1 @ f ˇ @ f ˇ ˇ .!f !f / C ˇ .pf pf / f f f f k k0 f f f f 0 (C.5) @! T0 ;!k0;p0 @p T0 ;!k0;p0 kD1 „ k ƒ‚ … „ ƒ‚ … f f f f 0 0 ˛k providing a 6th-order accuracy for the temperature T f while only a linear 1st-order f f approximation is deemed to be sufficient for !k and p dependencies. By utilizing n f f n f f f (C.1) we can evaluate the above @ =@T at T0 , !k0 and p0 according to f ˇ @ ˇ 2 3 4 5 ˇ D b C 2cT f C 3dT f C 4eT f C 5f T f C 6gT f f f f f 0 0 0 0 0 @T T0 ;!k0;p0 2 f ˇ @ ˇ 2 3 4 ˇ D 2c C 6dT f C 12eT f C 20f T f C 30gT f f 2 f f f 0 0 0 0 @T T0 ;!k0;p0 3 f ˇ @ ˇ 2 3 ˇ D 6d C 24eT f C 60f T f C 120gT f f 3 f f f 0 0 0 @T T0 ;!k0;p0 4 f ˇ (C.6) @ ˇ 2 ˇ f f 4 f f f D 24e C 120f T0 C 360gT0 @T f T ;! ;p ˇ 0 k0 0 @5 f ˇ ˇ f 5 f f f D 120f C 720gT0 @T f T ;! ;p ˇ 0 k0 0 @6 f ˇ ˇ D 720g f 6 f f f @T T0 ;!k0;p0 C Thermally Variable Fluid Density Expansion ˇf .T f / 831 As the result the fluid density f (C.5) yields NXf 1 f f f f f f f f f f f .T ;!k ;p / D 0 1 C ˛k .!k !k0/ C .p p0 / kD1 f f 2 f 3 f 4 f 5 f f C.b C 2cT0 C 3dT0 C 4eT0 C 5f T 0 C 6gT0 /.T T0 / f f 2 f 3 f 4 f f 2 C.c C 3dT0 C 6eT0 C 10f T0 C 15gT0 /.T T0 / f f 2 f 3 f f 3 C.d C 4eT0 C 10f T0 C 20gT0 /.T T0 / f f 2 f f 4 C.e C 5f T 0 C 15gT0 /.T T0 / f f f 5 C.f C 6gT0 /.T T0 / f f 6 Cg.T T0 / (C.7) From (C.7)to(C.4) we finally obtain the expression for computing the nonlinear thermally variable density expansion to be used in (C.4) h f f 1 f f 2 f 3 f 4 f 5 ˇ .T / D f .b C 2cT0 C 3dT0 C 4eT0 C 5f T 0 C 6gT0 / 0 f f 2 f 3 f 4 f f C.c C 3dT0 C 6eT0 C 10f T0 C 15gT0 /.T T0 / f f 2 f 3 f f 2 C.d C 4eT0 C 10f T0 C 20gT0 /.T T0 / f f 2 f f 3 C.e C 5f T 0 C 15gT0 /.T T0 / f f f 4 C.f C 6gT0 /.T T0 / i f f 5 Cg.T T0 / (C.8) f with 0 computed from (C.3) and the coefficients .a;b;c;d;e;f;g/ as given by (C.2). As the consequence of the nonlinear thermally variable fluid density expansion the term .1= f /@ f =@T f in (3.197) is no longer a constant ˇf . Instead, we have f f f f f @ˇ .T / f f 1 @ ˇ .T / C @T f .T T0 / D P f f @T f f f f N 1 f f f f f f f 1 C .p p0 / C kD1 ˛k .!k !k0/ ˇ .T /.T T0 / (C.9) 832 C Thermally Variable Fluid Density Expansion ˇf .T f / where ˇf .T f / is taken from (C.8) and the derivation of ˇf .T f / in (C.9) becomes f f h @ˇ .T / 1 2 3 4 D .c C 3dT f C 6eT f C 10f T f C 15gT f / @T f f 0 0 0 0 0 f f 2 f 3 f f C2.d C 4eT0 C 10f T0 C 20gT0 /.T T0 / f f 2 f f 2 C3.e C 5f T 0 C 15gT0 /.T T0 / C4.f C 6gT f /.T f T f /3 0 0 i f f 4 C5g.T T0 / (C.10) Appendix D Parametric Models for Variably Saturated Porous Media D.1 Definitions For the sake of simplicity the fluid/liquid phase indices f; l will be omitted for the symbols used here. We introduce the capillary pressure head of the liquid phase defined as p D c (D.1) 0g where pc is the capillary pressure (3.221)or(3.257), 0 is the reference density of the liquid phase and g is the gravitational acceleration. In an unsaturated water-air system is usually negative, so is sometimes termed as suction.Furthermore, we define the effective saturation of the liquid phase as s sr se D (D.2) ss sr where sr 0 is the residual (or irreducible) saturation of liquid and ss is the maximum saturation of liquid. We note that ss is usually unity. D.2 Analytic Saturation s. / and Inverse Capillary Pressure .s/ Relations D.2.1 Van Genuchten (VG) Relationship Van Genuchten [539] proposed the analytic function 8 < 1 <0 n m for se D : .1 Cj˛ j / (D.3) 1 for 0 H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 833 © Springer-Verlag Berlin Heidelberg 2014 834 D Parametric Models for Variably Saturated Porous Media which has gained wide acceptance in practice, where ˛, m and n are positive curve VG-fitting parameters. The parameter n is known as the pore size distribution index. Equation (D.3) is used to express the relation s. / n m s D sr C .ss sr / 1 Cj˛ j (D.4) and its inverse .s/ 1 1 1 D s m 1 n (D.5) ˛ e Their first derivatives give1 @s mn˛j˛ jn 1 C D D .s s / @ .1 Cj˛ jn/mC1 s r (D.6) 1 1 @ 1 1 1 1 m n .1C / C D D se 1 .se/ m @s mn˛.ss sr / where C and C 1 are known as the moisture capacity and the inverse moisture capacity, respectively. D.2.2 Brooks-Corey Relationship Brooks and Corey [58]andCorey[100] introduced the relationship 8 < 1 < 1 n for ˛ se D j˛ j (D.7) : 1 1 for ˛ 1 where ˛ and n represent positive curve fitting coefficients. The limit . ˛ / in (D.7) 1 can be identified as an air-entry pressure head a D ˛ .Since n s D sr C .ss sr /j˛ j (D.8) 1 The second derivative of s. / with respect to reads for the VG relation @2s .m C 1/mn2˛2.˛ /2n 2 m.1 n/n˛2.˛ /n 2 D C .s s /: @ 2 Œ1 C .˛ /n mC2 Œ1 C .˛ /n mC1 s r As seen s is continuously differentiable at D 0 if n 1.However,if1 1 1 D s n (D.9) ˛ e their first derivatives are @s n˛ C D D .s s / @ j˛ jnC1 s r 1 (D.10) C 1 @ 1 .1 n / C D D se @s n˛.ss sr / D.2.3 Haverkamp Relationship Haverkamp [528, 529] proposed the empirical equation 8 ˛ < for <0 ˛ CjZ jˇ se D : (D.11) 1 for 0 where ˛ and ˇ are positive curve fitting coefficients and Z 1 m 1 is a unit- canceling coefficient. With ˇ 1 s D sr C .ss sr /˛.˛ CjZ j / (D.12) and its inverse 1 Z 1 ˇ D ˛.se 1/ (D.13) their first derivatives yield @s ˛ˇjZ jˇ 1 C D D .s s / @ .˛ CjZ jˇ/2 s r (D.14) 1 @ ˛ 1 C 1 D D ˛.s 1 1/ ˇ 2 e @s ˇse .ss sr / 836 D Parametric Models for Variably Saturated Porous Media D.2.4 Exponential Relationship Gardner [185] and Rijtema [443] proposed an exponential relation in the form ˛. a/ e for < a se D (D.15) 1 for a which is applicable to analytic (exact) solutions of unsaturated flow problems, e.g., [490], where ˛ is the only positive fitting coefficient, sometimes termed as sorptive number,and a 0 is the air-entry pressure head to be prescribed. Since ˛. a/ s D sr C .ss sr /e (D.16) and its inverse 1 s sr D ln C a (D.17) ˛ ss sr their first derivatives are @s C D D ˛.s s /e˛. a/ @ s r (D.18) @ 1 C 1 D D @s ˛.s sr / D.2.5 Linear Relationship A simple linear relationship can be given in the form 8 ˆ <ˆ c for < < c a s D c a (D.19) e ˆ 1 :ˆ for a 0 for c to approximate the capillary pressure in a capillary fringe of thickness a c at a phreatic surface, where c < a <0is the capillary fringe pressure head and a 0 is the given air-entry pressure head. With c s D sr C .ss sr / (D.20) c a D.3 Analytic Relative Permeability kr .s/ and kr . / Relations 837 and its inverse D c . c a/se (D.21) their first derivatives are @s s s C D D s r @ c a (D.22) @ C 1 D D c a @s ss sr D.2.6 Time-Centered Analytic Moisture Capacity Evaluation Mass balance accuracy in the numerical modeling of unsaturated flow is affected to a large extent by the actual treatment of the moisture capacity C D C. /. Commonly, the evaluation of C (and accordingly C 1) in time t is done at the current time CnC1 D C. .tnC1//,wheren C 1 corresponds to the new time plane. Alternatively, for stability reasons it can be useful to evaluate analytically C between the previous and current time stages as [297] C D .1 !/Cn C !CnC1 (D.23) where C corresponds now a time-weighted capacity, subscripts n and n C 1 denote previous and current time plane, respectively, and ! is a time-weighting coefficient varying between 0 and 1. The time-centered analytic moisture capacity results for 1 using ! D 2 in (D.23). D.3 Analytic Relative Permeability kr .s/ and kr . / Relations D.3.1 Van Genuchten-Mualem (VGM) Relationship In a statistical approach the porous medium is conceptualized as a collection of interconnected cylindrical pores, where for each pore a laminar Poiseuille-like flow is considered having a parabolic flow profile in a tube. Based on this approach, Mualem [379] derived for the relative permeability f.s / 2 k D s e with r e f.1/ Z (D.24) x 1 f.x/ D d 0 . / 838 D Parametric Models for Variably Saturated Porous Media where is a pore connectivity parameter. Applying the VG relation (D.5)–(D.24) Z Z 1 ! 1 x 1 x m n f.x/ D d D ˛ d (D.25) . / 1 0 0 1 m we can derive a closed solution for kr .s/ after some manipulations as [536] h i 1 2 m m kr .s/ D se 1 .1 se / (D.26) if the so-called Mualem assumption holds 1 m D 1 n (D.27) 1 The exponent in (D.26) usually set to D 2 accounts in Mualem’s interpretation for tortuosity and connectivity so that in a physical sense >0.However, is sometimes considered as a free fitting parameter and even negative can frequently be determined (e.g., [416]). Equivalently, by using the VG relation (D.3) we can express (D.26)asthe function kr . / in the form 5m m m 2 kr . / D .1 C '/ 2 .1 C '/ ' with the auxiliary variable (D.28) ' D .˛ /n The first derivative of kr . / (D.28) with respect to yields 5m @kr G D D .n 1/˛.˛ /n 1.1 C '/ 2 .1 C '/m 'm @ h i 1 5' 2 1 .1 C '/m 1 (D.29) .˛ / 2.1 C '/ 2 D.3.2 Brooks-Corey Relationship Brooks and Corey [58]andCorey[100] suggested the following function ı kr .s/ D se (D.30) D.3 Analytic Relative Permeability kr .s/ and kr . / Relations 839 where se isgivenby(D.7)andı is a prescribed exponent. Brooks and Corey expre- 2 2 ssed the exponent ı by the pore size distribution index n in the form : ı n C 3. Using (D.7) we can write (D.30) as the function kr . / according to nı kr . / Dj˛ j (D.31) where its first derivative with respect to is @k G D r D nı˛j˛ j .nıC1/ (D.32) @ D.3.3 Modified van Genuchten Relationship In contrast to the VGM parametric model consisting of the kr .s/ relation (D.26) derived from the VG .s/ function (D.5) under the Mualem assumption (D.27), it is often advantageous to formulate the relative permeability kr .s/ in the same form as (D.30) ı kr .s/ D se (D.33) but now in combination with the VG-retention curve (D.3), where the fitting exponent ı can be prescribed independently. A typical value for the exponent is ı 3 as suggested by Irmay [286]. A simple linear relationship is obtained be setting ı D 1. Inserting the VG-relation (D.3)into(D.32) we obtain the modified VG kr . / relation in the form n mı kr . / D .1 Cj˛ j / (D.34) 2Using the Mualem relation (D.24) for the Brooks-Corey function (D.9)itis Z Z x 1 x 1 f.x/D d D ˛ n d 0 . / 0 and we can derive 2 C C2 n kr .s/ D se 1 It is obvious, with D 2 we find ı kr .s/ D se : where 2 ı D C 5 n 2 2 which is slightly different to the Brooks-Corey exponent n C 3. 840 D Parametric Models for Variably Saturated Porous Media with its first derivative @k j˛ jn 1 G D r D mnı˛ (D.35) @ .1 Cj˛ jn/mıC1 In this alternative formulation there is no need for the Mualem assumption (D.27) and the fitting parameters n and m can be used independently. D.3.4 Haverkamp Relationship Haverkamp [528, 529] proposed a relationship for the relative permeability kr . /, which has a characteristic quite similar to the corresponding retention curve (D.11), viz., A k . / D (D.36) r A CjZ jB where A and B are positive curve fitting parameters and Z 1 m 1 is a unit- canceling coefficient. The first derivative with respect to gives @k ABjZ jB 1 G D r D (D.37) @ .A CjZ jB /2 D.3.5 Exponential Relationship For the relative permeability Gardner [185] and Rijtema [443] proposed the relation kr .s/ D se (D.38) where the exponential expression (D.15) is used for the effective saturation se. Inserting (D.15)into(D.38) it yields ˛. a/ kr . / D e (D.39) with its first derivative @kr G D D ˛e˛. a/ (D.40) @ D.4 Spline Approximation of Retention and Relative Permeability Curves 841 Table D.1 Analytic parametric models for retention and relative permeability curves Parametric model se kr References h i 1 p 1 2 m m (1) van Genuchten-Mualem n m se 1 .1 se / [379, 536, 539] (VGM) .1 Cj˛ j / 1 (2) Modified van sı [100, 539] n m e Genuchten .1 Cj˛ j / 1 (3) Brooks-Corey sı [58, 100] j˛ jn e ˛ A (4) Haverkamp [62, 528, 529] ˛ CjZ jˇ A CjZ jB ˛. a / (5) Exponential e se [185, 443] c (6) Linear se c a D.3.6 Linear Relationship For the relative permeability a linear relationship is given by kr .s/ D se (D.41) if the linear expression (D.19) is used to express the effective saturation se.Using (D.19)in(D.41) we obtain c kr . / D (D.42) c a with its first derivative @k 1 G D r D (D.43) @ c a The analytic parametric models described above are summarized in Table D.1. D.4 Spline Approximation of Retention and Relative Permeability Curves3 Analytic relations for the retention (capillary pressure) and relative permeability curves are not always suitable, because they may not describe experimental data suf- ficiently well in certain cases. This is exemplified for a capillary-pressure-saturation 3This section is contributed by V. Mirnyy (DHI-WASY). 842 D Parametric Models for Variably Saturated Porous Media Fig. D.1 Cubic spline curves 1.0 of s. / vs. analytic VG fitting function based on real 0.9 experimental data 0.8 spline knots spline curve 0.7 monotonic spline curve VG fitting 0.6 0.5 0.4 saturation s [1] 0.3 0.2 0.1 0 −2.0 −1.5 −1.0 −0.5 0 pressure head ψ [m] curve as shown Fig. D.1, where a cubic spline graph fits the experimental values very well in contrast to an analytic VG curve. The idea to automate the way from experimental data to analytic curves leads to the application of splines that can be derived directly from the experimental sample points .xi ;yi /, i D 0;:::;P of the curves, where xi stands for pressure head values i and saturation values si of the retention and relative permeability curves, respectively, and yi stands for saturation values si and relative permeability values kri of the retention and relative permeability curves, respectively. Application of cubic splines to the physical characteristics of soils appeared in a couple of works during past few years. Kastanek and Nielson [303] applied classical interpolating cubic splines to the saturation-pressure relation, while allowing the introduction of ‘virtual data points’ to achieve necessary curve properties, such as monotonicity. Classical cubic interpolating splines that go exactly through the measured values are appropriate if the number of experimental data points is small but each measurement is of sufficiently low uncertainty (i.e., measurement errors are insignificant). In contrast to the classic cubic splines, monotonic spline approximation methods are available to enforce curve monotonicity and even positivity in the first derivatives, which has a particular relevance for the capillary pressure curve. For the following definition and analysis of different kinds of splines it is referred to [115]. D.4 Spline Approximation of Retention and Relative Permeability Curves 843 Table D.2 Conditions to compute cubic interpolating spline coefficients (Note that 0 denotes differentiation with respect to x) Number of Condition Range constraints Si .xi / D yi i D 0;:::;P 1P Si .xiC1/ D yiC1 i D 0;:::;P 1P 0 0 Si .xiC1/ D SiC1.xiC1/iD 0;:::;P 2P 1 00 00 Si .xiC1/ D SiC1.xiC1/iD 0;:::;P 2P 1 D.4.1 Definition of Cubic Spline Spline S is a piecewise polynomial continuous function defined on the interval Œx0;xP and represents a very flexible approach of data fitting. Let the interval be covered by P disjoint subintervals Œxi ;xiC1 with a D x0 x1 ::: xP 1 xP D b; i D 0;:::;P 1 (D.44) The given points x0;:::;xP are called spline knots. Interpolating spline assumes that the spline curves go exactly through the given values y0;:::;yP at knots x0;:::;xP . These are the initial data for an interpolating spline: x x0 x1 ... xP y y0 y1 ... yP The polynomial in the interval Œxi ;xiC1 is defined as Si .x/. Then the spline function reads 8 ˆ S0.x/ x0 x x1 <ˆ S1.x/ x1 x x2 S.x/ D : : (D.45) ˆ : : :ˆ : : SP 1.x/ xP 1 x xP Cubic polynomials are chosen in the following form 2 3 Si .x/ D ai C bi .x x0/ C ci .x x0/ C di .x x0/ (D.46) to achieve a continuous spline curve up to second derivative. The set of 4P polynomial coefficients ai ;bi ;ci ;di .i D 0;:::;P 1/ must be determined from several conditions. Most of them are listed in Table D.2. The first and the second row of the table describe that the polynomials Si and SiC1 are connected in the point .xiC1;yiC1/. The third and the forth row define continuity of the first and the second derivatives, respectively. Totally, the table contains 4P 2 conditions. Two conditions are still required to determine 4P 844 D Parametric Models for Variably Saturated Porous Media polynomial coefficients uniquely. Since there are no continuity constraints in the end points, it is natural to gain the missing two conditions in the first and the last knot of the spline. There are different ways for treating BC’s. Some of them are: 0 0 • Fixed slope at the end points: S .x0/ D C0;S.xP / D CP 00 00 • Natural spline: S .x0/ D S .xP / D 0 000 • Not-a-knot condition: S is continuous in x1 and xP 1 Table D.2 and selected BC’s result in a linear algebraic system of 4P equations with 4P unknowns, which has a unique solution. D.4.2 s. / Dependency Specific BC’s are employed to approximate the saturation-pressure head curve via 00 cubic interpolating splines. We apply a natural spline condition S .x0/ D 0 in the 0 first knot and a fixed slope condition S .xP / D 0 at the last point. D.4.2.1 Continuation to Minus Infinity Since spline function is defined on an interval Œa; b , we have to advance the spline approximation to the interval Œ 1; C1 . If we claim that the last given spline knot xP D 0 and S.xP / D 1, then a simple constant continuation to plus infinity is SC1.x/ D 1 (D.47) 0 The fixed slope condition S .xP / D 0 ensures a smooth transition from SP to SC1. In opposite direction, when x ! 1, the saturation-pressure head curve should tend to 0. It can be realized using exponential function of the form bx S 1.x/ D ae (D.48) where parameters a and b can be calculated from two smoothness conditions in the 0 0 first spline knot: S 1.x0/ D S0.x0/ and S 1.x0/ D S0.x0/. Defining S0.x0/ as y0 0 and S0.x0/ as m0 we obtain m0x0 m m0 y 0 y .x x0/ a D y0e 0 ;bD and S 1.x/ D y0e 0 (D.49) y0 D.4.2.2 Monotonic Spline Curve Interpolating cubic splines described above are though continuous up to the second derivative, but may become non-monotonic even when spline knots are monotonic D.4 Spline Approximation of Retention and Relative Permeability Curves 845 Fig. D.2 Cubic spline curves 2.5 of first derivatives @s. /=@ vs. analytic VG fitting spline derivative function 2.0 monotonic spline VG fitting ] −1 1.5 s/ [m 1.0 0.5 derivative 0.0 −0.5 −2.0 −1.5 −1.0 −0.5 0 pressure head [m] (y0 y1 ::: yP 1 yP /. An example of such a behavior is shown in Fig. D.2, where the first derivative of the cubic spline becomes negative. For some parametric curves such fluctuations might be acceptable, while the monotonicity of the saturation-pressure head dependency is important due to its physical meaning. Monotone interpolation can be accomplished by using cubic Hermite spline with the tangents mi modified to ensure the monotonicity of the resulting spline. Cubic Hermit spline is generally continuous up to the first derivative only. Thus, monotonicity is achieved loosing continuity of the second derivative comparing to the cubic interpolating spline. The tangents mi will be computed by using the Fritsch-Carlson method [176] as follows: 1. Compute the slopes of the secant lines between successive points: yiC1 yi i D ;iD 0;:::;P 1 xiC1 xi 2. Initialize the tangents at every data point as the average of the secants C m D i 1 i ;iD 1;:::;P 1 i 2 For the end points one-sided differences are used: m0 D 0 and mP D P 1. 3. If two successive points have equal values .yi D yiC1/,thensetmi D miC1 D 0, as the spline connecting these points must be flat to preserve monotonicity. Next steps 4 and 5 are omitted. 846 D Parametric Models for Variably Saturated Porous Media 4. Let ˛i D mi = i and ˇi D miC1= i .If˛i D 0 or ˇi D 0, then the input data points are not monotone. In such cases, piecewise monotonic curves can still be generated by choosing mi D miC1 D 0, although global monotonicity is not possible. 5. To prevent overshoot and ensure monotonicity, the function .2˛ C ˇ 3/2 .˛;ˇ/ D ˛ 3.˛ C ˇ 2/ must have a value greater than zero. One simple way to satisfy this constraint is 2 to restrict the magnitude of vector .˛i ;ˇi / to a circle of radius 3. Thatq is, if ˛i C 2 2 2 ˇi >9,thensetmi D i ˛i i and miC1 D i ˇi i ,where i D 3= ˛i C ˇi . For this algorithm the only one path is required. To evaluate the cubic monotonic spline at an arbitrary point x,wefindtheinterval Œxk ;xkC1 , such that xk x xkC1 by using a binary search algorithm [315]. We define, ı D xkC1 xk and t D .x xk /=h. Then the interpolant is S.x/Q D ykh00.t/ C ımkh10.t/ C ykC1h01.t/ C ımkC1h11.t/ (D.50) where Pii are the basis functions for the cubic Hermite spline: 3 2 2 h00.t/ D 2t 3t C 1 D .1 C 2t/.1 t/ 3 2 2 h10.t/ D t 2t C t D t.1 t/ 3 2 2 (D.51) h01.t/ D 2t C 3t D t .3 2t/ 3 2 2 h11.t/ D t t D t .t 1/ Figure D.1 shows the classical VG fitting function for the saturation-pressure head dependency that poorly approximates experimental data given as spline knots. On the other hand, it has a well-shaped first derivative behavior (Fig. D.2). Differences between cubic interpolating and cubic monotonic spline curves are hardly visible in Fig. D.1, however, the first derivative plots (Fig. D.2) reveal clearly the positivity for the monotonic spline and the lost of the smoothness. Appendix E Heat Transfer and Thermal Resistance for Wall Configurations E.1 Conduction Heat Transfer E.1.1 Single and Composite Plane Wall First, let us consider the thermal conduction through a single plane wall of thickness d made of solid material with a thermal conductivity s as shown in Fig. E.1a. The temperatures at the two inner and outer solid surfaces of the wall are fixed at T1 and TC with T1 >TC . For steady conditions without heat supply and constant thermal conductivity s, the heat transport equation, e.g., Table 3.5 (assuming " D 0 for thermal conduction in solids), is for the 1D solid problem 2 s @ T @ 2 D 0 jT D 0 (E.1) @x1 @x1 where jT is the Fourier heat flux in 1D, cf. (3.176). We assume that jT is aligned with the boundary heat flux qnT normal to the wall surfaces, i.e., s @T qnT D jT D (E.2) @x1 Using the BC’s T.x1 D 0/ D T1 and T.x1 D d/ D TC (E.3) integration of (E.1) gives the following linear temperature distribution x T D T C .T T / 1 (E.4) 1 C 1 d H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 847 © Springer-Verlag Berlin Heidelberg 2014 848 E Heat Transfer and Thermal Resistance for Wall Configurations Fig. E.1 One-dimensional a b heat conduction through (a) single and (b) composite s s s plane wall of solid 1 2 T1 T1 T T 2 C TC q q nT nT d d1 d2 Applying (E.4), we obtain from (E.2) the heat flux relation s q D .T T / (E.5) nT d C 1 which corresponds to a Cauchy-type boundary flux condition, e.g., (6.40), where the s heat transfer coefficient yields ˚T D =d with T D T1. Second, the same procedure is applicable to a composite wall as shown in Fig. E.1b. Due to energy conservation @qnT =@x1 D 0 the heat flux qnT is constant and we find s s 1 2 qnT D .T2 T1/ D .TC T2/ (E.6) d1 d2 and eliminate T2 to obtain 1 qn D .TC T1/ (E.7) T d1 d2 s C s 1 2 This can be further generalized to a composite wall consisting of n material layers: 1 qn D .TC T1/ (E.8) T d1 d2 d3 dn s C s C s C :::C s 1 2 3 n Once qnT has been determined, the temperatures at the inner material interfaces can be computed as E.1 Conduction Heat Transfer 849 ab L r2 r s 1 r3 s s 2 1 r 1 r 2 T1 T1 T2 q qn TC nT TC T Fig. E.2 Radial heat conduction through (a) single and (b) composite circular pipe wall of solid d1 T2 D T1 qnT s 1 d1 d2 T3 D T1 qnT s C s (E.9) 1 2 d1 d2 dn 1 Tn D T1 qnT s C s C :::C s 1 2 n 1 E.1.2 Single and Composite Circular Pipe Wall First, we consider a single circular pipe wall of inside radius r1, outside radius r2, length L and thermal conductivity s asshowninFig.E.2a. At the inside surface and at the outside surface of the circular pipe wall constant temperatures T1 and TC , respectively, are prescribed with T1 >TC . For steady-state condition without heat supply and constant thermal conductivity s, the heat transport equation reads for the axisymmetric problem (cf. Sect. 2.1.6) @ @T @ s r D 0 q D 0 (E.10) @r @r @r nT where r corresponds to the radial coordinate, supplemented by the BC’s T.r D r1/ D T1 and T.r D r2/ D TC (E.11) In the same analytical procedure as done for the single plane wall we obtain the temperature distribution for the single circular pipe wall as 850 E Heat Transfer and Thermal Resistance for Wall Configurations TC T1 r T D T1 C ln (E.12) ln.r2=r1/ r1 which leads to the radial heat flux s qnT D .TC T1/ (E.13) r ln.r2=r1/ as a function of the radial coordinate qnT D qnT .r/. In analogy to the composite plane wall we find for the composite circular pipe wall as shown in Fig. E.2b the following heat flux relation: 1 qn D .TC T1/ (E.14) T ln.r2=r1/ ln.r3=r2/ r s C s 1 2 In generalization, the heat flux for a n layered circular pipe wall system leads to 1 qnT D .TC T1/ (E.15) ln.r2=r1/ ln.r3=r2/ ln.r4=r3/ ln.rnC1=rn/ r s C s C s C :::C s 1 2 3 n E.2 Heat Transfer Coefficient, Thermal Resistance and Specific Thermal Resistance From above the heat flux qnT can be expressed qnT D ˚T .TC T/ (E.16) which represents a Cauchy-type boundary heat flux condition, cf. (6.40), where TC is the known (external or ambient) temperature on the boundary and T.D T1/ is the internal temperature. It can be recognized as a specific form of the Newton’s law of cooling [43, 447]. In (E.16) ˚T is the heat transfer coefficient which is given for the configurations: 8 ˆ s ˆ d single plane wall ˆ s ˆ singlepipewall <ˆ r ln.r2=r1/ ˚T D 1 composite plane wall (E.17) ˆ d1 d2 dn ˆ s C s C:::C s ˆ n ˆ 1 2 ˆ 1 composite pipe wall : ln.r2=r1/ ln.r3=r2/ ln.rnC1=rn/ r s C s C:::C s 1 2 n This heat transfer relation has a direct analogy to electric resistance. According to the Ohm’s law the electric resistance is defined as the ratio of the voltage difference E.2 Heat Transfer Coefficient, Thermal Resistance and Specific Thermal Resistance 851 to the current flow. A thermal resistance RN can accordingly be defined as the ratio of the temperature difference to the associated rate of heat transfer. Thus .T T / 1 RN D R C D (E.18) qnT d ˚T A R where qnT d represents the integral of the heat flux over the surface (boundary) , which is approximated by Z qnT d AqnT (E.19) where A corresponds to the exchange area. On the other hand, the specific thermal resistance R defines the thermal resistance per unit length and represents a material property. It is related to the heat transfer coefficient as 1 R D D RLN (E.20) ˚T S A where S D L is the specific surface and L is a length. With these definitions, the thermal resistances and specific thermal resistances of material i for a plane wall and a circular pipe wall, respectively, are 8 ˆ di < s plane wall N A i Ri D ˆ (E.21) :ˆ ln.riC1=ri / s circular pipe wall 2 L i 8 ˆ di < s plane wall S i Ri D ˆ (E.22) :ˆ ln.riC1=ri / s circular pipe wall 2 i where for the circular pipe wall the specific exchange area is S D 2 r.The advantage of using specific thermal resistances is in particular for analyzing heat conduction through composite materials, where we can write for a n layered structure 1 qnT D .TC T/ SR (E.23) P n R D R1 C R2 C :::C Rn D iD1 Ri forming a serial thermal resistance R composed of their partial resistance values Ri .i D 1;:::;n/according to (E.22). 852 E Heat Transfer and Thermal Resistance for Wall Configurations a b R2 R4 R2 R1 R3 R1 R4 R3 R4 s s s s s s s s s s 2 1 3 4 4 2 1 1 3 4 T1 T1 T2 T2 T3 T3 q q TC nT TC nT R2 R1 R2 T2 R1 R4 T1 TC T1 T2 T3 TC T3 R3 R4 R3 Fig. E.3 Serial and parallel heat transfer through a composite circular pipe wall with material zones showing its thermal circuit It can also be possible that the composite wall is additionally structured in zones having different thermal properties as exemplified in Fig. E.3 for a circular pipe wall. It generates a combined serial and parallel heat transfer. It is usual to indicate the thermal resistance by its thermal network in direct analogy to electrical resistances in electric circuits (Fig. E.3). The serial-parallel heat transfer results for the first case (Fig. E.3a) as 1 1 1 qnT D C .TC T/ (E.24) 2S R1 C R2 R3 C R4 and for the second case (Fig. E.3b) as 1 qnT D .TC T/ (E.25) 1 S R1 C 1 C R4 1 C 1 2 R2 R3 where the partial thermal resistances Ri are given from (E.22). In such a way thermal resistances for more complex configurations can be developed, see e.g., [43, 237, 447]. E.3 Thermal Circuits of Prototypical Configurations in Boreholes 853 E.3 Thermal Circuits of Prototypical Configurations in Boreholes Borehole heat exchanger (BHE) represents a typical application of thermally interacting components of circular pipe geometries (cf. Sect. 13.5 and Appendix M) for which the effective thermal resistances can be derived by using the above technique. There are two prototypical configurations consisting of three thermal resistors: Delta configuration and Y configuration. E.3.1 Delta Configuration Considering a borehole cross-section containing two inner pipes denoted by sub- script i (pipe-in) and subscript o (pipe-out) as shown in Fig. E.4. The borehole wall is in direct contact with the surrounding soil denoted by subscript s. In the Delta configuration (Fig. E.4a), in electrical engineering referred to as the Pi configuration, there is a series-parallel combination of the three thermal resistors between pipe-in and pipe-out, pipe-in and soil as well as pipe-out and soil, for which the equivalent specific thermal resistance can be expressed by 1 1 1 ▵ D C (E.26) R Rio Ris C Ros or R .R C R / R▵ D io is os (E.27) Rio C Ris C Ros where Rio, Ris and Ros correspond to the thermal resistances of pipe-in – pipe- out, pipe-in – soil and pipe-out – soil, respectively. In the BHE context [237]the ▵ resistance R is usually termed as internal borehole thermal resistance Ra D R and with Rb the borehole thermal resistance is defined as 1 1 1 RisRos D C or Rb D (E.28) Rb Ris Ros Ris C Ros which measures only the thermal resistance in a parallel heat transfer between the two pipes with the surrounding soil. The Delta configuration has been preferred in BHE modeling by Eskilson and Claesson [159], see Sect. 13.5 and Appendix M for more. E.3.2 Y Configuration The Y configuration, in electrical engineering referred to as the Wye or T config- uration, describes a series combination of three thermal resistors. In the borehole 854 E Heat Transfer and Thermal Resistance for Wall Configurations ab Rio i o i o Rig Rog io g Ris Ros g Rgs s s s Fig. E.4 (a) Delta and (b) Y configuration for three thermal resistors in a borehole containing two inner pipes the resistances with the backfill, the grout component denoted by subscript g,is additionally considered. As shown in Fig. E.4b for this circuit there is a serial connection of resistor Rgs occurring between grout and soil to a parallel connection of the resistors Rig and Rog occurring between pip-in and grout as well as pipe-out and grout, respectively, which can be expressed by R D Rgs C Rio 1 (E.29) D Rgs C 1 C 1 Rig Rog in which Rio represents the equivalent thermal resistance of the parallel circuit between Rig and Rog. The resulting thermal resistance R of (E.29) can be considered as an internal borehole thermal resistance Ra D R for the Y configuration. In contrast to the Delta configuration, the Y configuration entails that no direct thermal interaction between pipe-in and pipe-out exists. In BHE modeling such a Y configuration has been preferred by Al-Khoury et al. [8] and Al-Khoury and Bonnier [7]. The prototypical Delta and Y configurations represent basic circuits for BHE’s, which can be enriched and extended by further inner components such as wall materials, grout zones and different pipe arrangements, so as thoroughly described in Appendix M. Appendix F Optimality of the Galerkin Method The optimality of the Galerkin method indicates that the Galerkin criterion (choos- ing weighting function equal to the basis function) is the best approximation possible compared to any other approximation according to an appropriate measure of error. This is particularly true for 2nd-order PDE’s of elliptic type and is expressed by the inequality kekE; G kekE; O (F.1) where kekE; G and kekE; O are the energy (Hilbert space) norm errors (8.22)pro- duced by the Galerkin method and by any other approximation method, respectively. This error estimate is central in FEM and formulated by C«ea’s lemma, see e.g., [84, 193, 555], and tells that, at least valid for elliptic PDE’s, there is never a better approximation than the Galerkin approximation. To prove (F.1) we follow Lewis et al. [345] and consider a steady-state diffusion equation of elliptic type (cf. Sect. 8.3) written in the form: r .D r / D H on ˝ rw .D r /d˝ D wH d˝ wqN d (F.3) ˝ ˝ N subject to the Neumann-type BC: qN D .D r / n on N ,wherew D w.x/ is a weighting function, which is at least once-differentiable, and D .x/ is the exact solution. The weak form (F.3) must be valid for every w. Let us the define the error in the Galerkin approximation by e G D O G (F.4) H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 855 © Springer-Verlag Berlin Heidelberg 2014 856 F Optimality of the Galerkin Method where O G is the Galerkin-based approximate solution, and the error in an approxi- mation e O D O O (F.5) when any other approximate solution O O is used. Applying the Galerkin approxi- mation O G to the weak form (F.3), it must satisfy Z Z Z rw .D r O G/d˝ D wH d˝ wqN d (F.6) ˝ ˝ N for every w. Subtracting (F.6) from (F.3) it yields Z rw .D re G/d˝ D 0 (F.7) ˝ for every w. Let us define the energy (Hilbert space) norm error of a function f according to (8.22) in the following form Z 2 kf kE D rf .D rf/d˝ (F.8) ˝ we can expand (F.5)to e O D O O D . O G/ C . O G O O/ D e G C . O G O O/ (F.9) and find 2 2 O O O O 2 kekE; O DkekE; G C 2k. G O/e GkE Ck G OkE (F.10) Using (F.8)and(F.10) leads to Z Z 2 O O kekE; O D re G .D re G/d˝C 2 r. G O/ .D re G/d˝C ˝ Z ˝ r. O G O O/ D r. O G O O/ d˝ ˝ (F.11) Since (F.7) is valid for every weighting function w, the second term on the RHS of (F.11)mustvanish Z r. O G O O/ .D re G/d˝ D 0 (F.12) ˝ F Optimality of the Galerkin Method 857 and it follows from (F.11) 2 2 O O 2 kekE; O DkekE; G Ck G OkE (F.13) O O 2 From the definition of the error measure (F.8) it results that k G OkE 0 is strictly non-negative and the optimality condition (F.1) becomes now apparent. Appendix G Isoparametric Finite Element Shape Functions and Their Derivatives Standard isoparametric elements in 1D, 2D and 3D can be found in finite element textbooks, e.g., [590]. Pyramidal 3D elements are described by Zgainski et al. [586]. G.1 One Dimension e Table G.1 One-dimensional isoparametric shape functions NI and their derivatives at node I of element ˝N e <1 of type: (a) line and (b) parabola e e @NI Type INI @ Element in global and local coordinates (a) 1 1 .1 / 1 linear 2 2 12 2 1 .1 C / 1 1 2 2 2 (-1) (+1) x 1 1 1 quadratic (b) 1 2 . 1/ 2 1 1 2 3 2 1 2 2 2 (-1) (0) (+1) 1 1 3 . C 1/ C x 2 2 1 H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 859 © Springer-Verlag Berlin Heidelberg 2014 860 G Isoparametric Finite Element Shape Functions and Their Derivatives (1,0) (1,1) (1,-1) 4 3 5 (1,-1) (1,1) 3 2 (1,0) 2 (0,1) (0,-1) 6 2 7 8 1 (0,1) 4 1 1 3 (0,0) (-1,-1) (-1,1) (-1,1) (-1,-1) (-1,0) 1 1 1 x x x 4 2 2 3 5 3 6 2 3 linear quadratic linear 7 4 8 1 1 1 2 2 2 x x x Element in global and local coordinates of type: (a) triangle, (b) quadrilateral and 2 < e N ˝ 2 / 2 / 2 / 2 / C 2 2 C 2 2 C / C 2 / / 2 2 / / / 2 of element / 2 2 / C C C C I C C C .1 .1 .1 e I 1 4 1 4 1 2 .1 .1 1 .1 .1 . . . .1 . @ 1 4 1 4 1 4 1 4 1 4 1 2 1 4 @N 0 1 / / 2 2 / / 2 2 C C 2 2 2 2 C / C 2 / 2 2 / / / / 2 2 2 / / C C C and their derivatives at node C C C C e .1 .1 .1 I e I 1 4 1 4 1 2 .1 .1 1 .1 .1 . . .1 . . N @ 1 4 1 4 1 4 1 4 1 2 1 4 1 4 @N 1 0 1/ 1/ 1/ 1/ C C / / 2 / 2 / / /. / / /. /. / /. C C C C C /.1 /.1 2 2 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 C C C C C e I .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 1 4 1 4 1 4 1 4 1 4 1 2 1 4 1 2 1 4 1 2 1 4 1 2 1 1 2 3 1 2 3 4 1 2 3 4 5 6 7 8 IN Two-dimensional isoparametric shape functions (c) curved quadrilateral G.2 Two Dimensions Table G.2 Type (a) (b) (c) G.3 Three Dimensions 861 (1,1,-1) (1,1,1) (1,1,0) 3 7 3 (1,-1,-1) 2 (1,0,1) (1,-1,0) (1,0,-1) (1,-1,1) 3 6 (0,0,1) 5 2 2 (0,1,0) 5 (0,1,-1) (-1,1,-1) 4 8 (-1,1,1) 3 (0,0,1) 4 6 (0,0,0) (0,1,1) 4 (-1,1,0) 1 1 1 5 1 4 (1,0,0) 2 of type: (a) tetrahedron, (-1,-1,1) (0,0,1) (-1,-1,-1) 3 (0,0,-1) (-1,-1,0) 3 3 7 6 3 3 2 2 5 6 2 2 5 4 4 8 4 < 1 1 1 1 x x x x 1 5 1 4 1 1 e 2 2 2 2 x x x x N ˝ 3 3 3 3 x x x x Element in global and local coordinates / / / / / / / / C 2 2 of element C 2 2 / / / / / C C / I .1 .1 /.1 /.1 .1 .1 /.1 /.1 /.1 /.1 /.1 /.1 C C C C C C .1 .1 .1 .1 .1 Œ1 Œ1 Œ1 Œ1 e I 1 2 1 2 1 2 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 4 1 .1 .1 .1 .1 .1 @ 1 2 1 2 1 2 1 8 1 8 1 8 1 8 @N 1 0 0 1 / / / / 1 1 / / / / C 1 C 1 C C C C / / / /.1 /.1 / /.1 /.1 / / / /.1 /.1 C / /.1 /.1 C C C C C C C .1 .1 .1 .1 .1 .1 Œ.1 Œ.1 e I 1 2 1 2 1 8 1 8 1 8 1 8 1 4 1 4 1 .1 .1 .1 .1 .1 .1 Œ.1 Œ.1 and their derivatives at node @ 1 2 1 2 1 8 1 8 1 8 1 8 1 4 1 4 @N 0 0 0 0 1 0 e I N / / / / 1 1 / / / / C 1 C 1 C C C C / / / /.1 /.1 / /.1 /.1 / / / /.1 /.1 C / /.1 /.1 C C C C C C C .1 .1 .1 .1 .1 .1 Œ.1 Œ.1 e I 1 2 1 2 1 8 1 8 1 8 1 8 1 4 1 4 1 .1 .1 .1 .1 .1 .1 Œ.1 Œ.1 @ 1 2 1 2 1 8 1 8 1 8 1 8 1 4 1 4 @N 0 0 0 1 0 0 1 1 1 1 C / / / / C / / / / C C C C / / C / / / / /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 C C C C C C /.1 /.1 / / / / /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 C C C C C C C C e I .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 .1 Œ.1 Œ.1 Œ.1 Œ.1 1 2 1 2 1 2 1 2 1 2 1 2 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 8 1 4 1 4 1 4 1 4 1 Three-dimensional linear isoparametric shape functions IN 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 7 8 1 2 3 4 5 (b) triangular prism (pentahedron), (c) quadrilateral prism (hexahedron) and (d) square pyramid G.3 Three Dimensions Table G.3 Type (a) (b) (c) (d) 862 G Isoparametric Finite Element Shape Functions and Their Derivatives 11 17 5 5 17 11 16 4 4 3 15 16 6 10 10 18 3 6 2 11 (1,1,0) 12 (-1,1,0) 13 (-1,-1,-1) 14 (0,-1,-1) 15 (1,-1,-1) 16 (1,0,-1) 17 (1,1,-1) 18 (0,1,-1) 19 (-1,1,-1) 20 (-1,0,-1) 15 18 12 7 14 19 1 8 20 x 7 2 12 19 14 9 13 (0,1,1) 1 8 2 20 x 1 (-1,-1,1) 2 (0,-1,1) 3 (1,-1,1) 4 (1,0,1) 5 (1,1,1) 6 7 (-1,1,1) 8 (-1,0,1) 9 (-1,-1,0) 10 (1,-1,0) 9 1 13 3 x Element in global andcoordinates local of curved hexahedral element 3 2/ 2/ 2/ 2/ 2/ 2/ 2/ 2/ C / C C / / C / < / / / / C C e C C C C N C C ˝ / /.1 /.1 / /.1 /.1 / / / 2 / / 2 / / /.1 /.1 / 2 /.1 /.1 C C / /. /. 2 / /. /. /. /. C /. /. C C C C C C C C C /.1 /.1 /.1 /.1 /.1 /.1 2 2 2 /.1 2 /.1 2 /.1 2 /.1 /.1 /.1 /.1 /.1 2 2 /.1 /.1 /.1 /.1 2 /.1 2 /.1 /.1 2 /.1 2 /.1 /.1 /.1 /.1 /.1 C C C C C C /.1 C of element 1 1 1 1 C C C C C C C C C .1 .1 .1 .1 .1 .1 .1 .1 I e I 1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 Œ.1 .1 Œ.1 Œ.1 Œ.1 .1 Œ.1 .1 Œ. Œ. Œ. Œ. @ 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 8 1 8 1 8 @N 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 C / C C / / C / / / / / C C C C C C C C / /.1 /.1 /.1 /.1 / / / / 2 / / 2 / /.1 /.1 / /.1 /.1 / C C 2 / /. /. 2 C / /. /. /. /. C /. /. C C C C C C C C /.1 /.1 /.1 /.1 /.1 /.1 2 2 2 /.1 2 /.1 2 /.1 2 /.1 /.1 /.1 /.1 /.1 and their derivatives at node 2 2 /.1 /.1 /.1 /.1 2 /.1 2 /.1 2 /.1 2 /.1 C C /.1 /.1 /.1 /.1 /.1 C C C C /.1 e C I 1 1 1 1 C C C C C C C C C N .1 .1 .1 .1 .1 .1 .1 .1 e I 1 4 1 2 1 2 1 4 1 4 1 4 1 2 1 2 Œ. Œ. Œ.1 .1 Œ.1 .1 .1 Œ. Œ. Œ.1 .1 Œ.1 @ 1 8 1 8 1 8 1 4 1 8 1 4 1 4 1 8 1 8 1 8 1 4 1 8 @N 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 2/ .1 C / C C / / C / / / / / C C C C C C C C / /.1 /.1 / /.1 /.1 / / / 2 / / 2 / / /.1 /.1 / /.1 /.1 / C C 2 /. /. 2 / /. /. /. C /. /. /. C C C C C C C C C /.1 /.1 /.1 /.1 2 2 /.1 /.1 /.1 2 /.1 2 /.1 2 /.1 2 /.1 /.1 /.1 /.1 2 2 /.1 /.1 /.1 /.1 2 /.1 2 /.1 2 /.1 2 /.1 /.1 /.1 /.1 C C /.1 /.1 C C C C /.1 C 1 1 1 1 C C C C C C C C C .1 .1 .1 .1 .1 .1 .1 .1 e I 1 2 1 2 1 4 1 4 1 4 1 2 1 2 1 4 Œ. Œ.1 Œ.1 Œ.1 Œ. .1 .1 Œ. Œ.1 .1 Œ.1 Œ. @ 1 8 1 8 1 4 1 8 1 8 1 4 1 4 1 8 1 8 1 4 1 8 1 8 @N / / / / 2 / / / 2 2 / / / / / / / 2 / / / / / / C C C C C C C C 2/ .1 2/ .1 2/ .1 2/ .1 /.1 /.1 2/ .1 /.1 /.1 2/ .1 2/ .1 2/ .1 2 2 /.1 /.1 /.1 /.1 2 2 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 C C C C C C C C C C C C /.1 /.1 /.1 /.1 2 2 2 2 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 /.1 C C /.1 /.1 /.1 /.1 C C C C C C Three-dimensional quadratic isoparametric shape functions C C C C . . . . . . . . e I Œ.1 .1 Œ.1 .1 Œ.1 .1 Œ.1 .1 .1 .1 .1 .1 Œ.1 .1 Œ.1 .1 Œ.1 .1 Œ.1 .1 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 4 1 4 1 4 1 4 1 8 1 4 1 8 1 4 1 8 1 4 1 8 1 4 6 12 17 5 18 9 10 11 13 15 19 type 3 7 4 8 Table G.4 IN 1 20 16 2 14 Appendix H Analytical Evaluation of Element Matrices and Vectors The element matrices and vectors appearing in (8.104) for the discretized ADE will be analytically evaluated for certain types of elements, where we assume constant coefficients within each element. A necessary and sufficient condition for the analytical evaluation is that the Jacobian is constant.Thisisalwaysgivenfor the linear 1D element, the linear 2D triangle and the linear 3D tetrahedron, for which the Jacobians are independent of the element shapes. However, to evaluate analytically quadrilateral, hexahedral, pentahedral or pyramidal elements, geometric simplifications are necessary to attain constant Jacobians for undisturbed elements. These can be shown for a quadrilateral element if simplifying to a rectangle or parallelogram, for a hexahedral element if simplifying to a brick or parallelepiped, for a pentahedral element if simplifying to a triangular prism with parallel top and bottom surfaces and for a pyramidal element if simplifying to a pyramid with parallelogram or rectangular base and oblique shape. H.1 Linear 1D Element We consider the linear 2-node element e as shown in Fig. H.1 with the shape functions at the nodes 1 and 2 (cf. Appendix G,TableG.1a) N e D 1 .1 / 1 2 (H.1) e 1 N2 D 2 .1 C / and their derivatives e @N1 1 @ D 2 e (H.2) @N2 1 @ D 2 H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 863 © Springer-Verlag Berlin Heidelberg 2014 864 H Analytical Evaluation of Element Matrices and Vectors Fig. H.1 Basis functions e 1 e N I NI .I D 1; 2/ of the linear 1D element N e N e 2 1 1 e 2 xxe xxe x = 1 = 2 = -1 e = +1 ∆x for . 1 1/.Furthermore,wehavefortheelement,cf.(8.71) e e e e x D N1 x1 C N2 x2 (H.3) and with (8.117)and(8.120) we obtain the Jacobian (and its determinant) @x @N e @N e xe J e DjJ ejD D 1 xe C 2 xe D (H.4) 11 @ @ 1 @ 2 2 which is constant, and the inverse Jacobian according to (8.119) 1 2 .J e / 1 D D (H.5) jJ ej xe where xe is the element length (Fig. H.1). Then, the divergence term (8.118) becomes with (H.2) 0 1 ! @N e e 1 1 rN1 e 1 @ @ A xe e D .J / e D (H.6) rN @N2 1 2 @ xe Using (8.122)itis e e xe d˝ D dx DjJ jd D 2 d (H.7) Thus, the matrices and vectors of the ADE convective form (8.104)to(8.105) become for element e: R e e e e e RK e N1 N1 N1 N2 e O D ˝e e e e e d˝ N2 N1 N2 N2 R 2 2 e .1 / 1 e e e 21 D RK 1 x d D RK x 4 1 1 2 .1 C /2 2 6 12 H.1 Linear 1D Element 865 R e e e e e e e N1 .q rN1 /N1 .q rN2 / e A D ˝e e e e e e e d˝ N2 .q rN1 /N2 .q rN2 / R e .1 / 1 e e 11 D q 1 x d D q 2 xe 1 .1 C / 1 C 2 2 11 R e e e e e e e rN1 .D rN1 / rN1 .D rN2 / e C D ˝e e e e e e e d˝ rN2 .D rN1 / rN2 .D rN2 / R e 1 1 e e 1 1 D D 1 x d D D . xe /2 1 11 2 xe 11 R e e e e e e e N1 N1 N1 N2 e R D ˝e .# C Q / e e e e d˝ N2 N1 N2 N2 R 2 2 e e .1 / 1 e e e e 21 D # CQ 1 x d D .# CQ / x 4 1 1 2 .1 C /2 2 6 12 R e e e e R e e e e e e N1 N1 N1 N2 e N1 .D rN1 / n N1 .D rN2 / n e B D e ˚ d e d C e e e e N e e e e N2 N1 N2 N2 O N2 .D rN1 / n N2 .D rN2 / n D D D D 2 2 1 1 1 1 e .1 / 1 e .1 / 1 D ˚ D 4 1 2 .1 C /2 2 xe .1 C / 1 C D 2D1 D 2D1 10 e 11 D ˚ e D (H.8) xe 01 11 e NO R e R e e e e N1 e e N1 e H D e ˚ d e q d C C e N N e N2 N2 D 1D 1 D 1D 1 e e 1 e 1 1 1 D ˚ C qN D ˚ e e qe 2 1 C 2 1 C C 1 N 1 D 2D1 D 2D1 R e P e e N1 e Q D ˝e H e d˝ w w.xw/Qw.t/ N2 R P P e 1 1 e e e 1 D H x d .x /Q .t/ D H x .x /Q .t/ 2 1 1 C 2 w w w w 2 1 w w w w (H.9) where for convenience we assume constant parameters (storage coefficient RK e,flux qe, dispersion De, decay rate #e, flow supply Qe, transfer coefficient ˚ e and source/sink H e) within the element. Finally, the spatially discretized ADE in the convective form (8.99) can be summarized as 866 H Analytical Evaluation of Element Matrices and Vectors Fig. H.2 Linear 3-node triangle in the global and 3 (0,1) local coordinate system y 3 2 1 1 2 (0,0) (1,0) x ! e X d 1 RK e xe 21 dt qe 11 De 1 1 .#e CQe/ xe 21 6 e C 2 C xe C 6 12 d 2 11 11 12 e dt e e 10 De 11 1 e e 1 e 1 H e xe 1 C˚ xe e ˚ C C qN 2 01 11 2 1 1 1 P NO C w w.xw/Qw.t/ D 0 (H.10) Similar expressions can be obtained for the divergence form of ADE (8.98). H.2 Linear 2D Triangle We consider the linear 3-node triangular element e as shown in Fig. H.2 (cf. Appendix G,TableG.2a) having the following shape functions e N1 D 1 e N2 D (H.11) e N3 D and local derivatives e e @N1 @N1 @ D 1; @ D 1 e e @N2 @N2 @ D 1; @ D 0 (H.12) e e @N3 @N3 @ D 0; @ D 1 for .0 ; 1/. Furthermore, we have for the triangular element, cf. (8.71) x D N exe C N exe C N exe 1 1 2 2 3 3 (H.13) e e e e e e y D N1 y1 C N2 y2 C N3 y3 H.2 Linear 2D Triangle 867 e e where xI ;yI .I D 1; 2; 3/ correspond to the Cartesian coordinates of the vertices (nodes) of the triangle. Using (H.12)and(H.13)in(8.116), (8.119), and (8.120)we obtain the Jacobian e e e e e x2 x1 y2 y1 J D e e e e ; (H.14) x3 x1 y3 y1 the inverse Jacobian 1 ye ye ye ye .J e / 1 D 3 1 1 2 e e e e e (H.15) jJ j x1 x3 x2 x1 and its determinant e e e e e e e e e e e jJ jDx1 .y2 y3 / C x2 .y3 y1 / C x3 .y1 y2 / D 2A (H.16) which is twice the area Ae of the triangle. We recognize that the triangle has the advantage of having a constant Jacobian (and its inverse) independent of the element shape in the Cartesian coordinates (we shall see further below this is not the case for the quadrilateral). The global derivatives (8.118) result with (H.12)in 0 1 @N e e e e e 1 e e e e 1 y3 y1 y1 y2 @ @ A 1 y2 y3 1 S11 rN1 D 2Ae e e e e e D 2Ae e e D 2Ae e x x x x @N1 x x S 1 3 2 1 @ 3 2 „ƒ‚21 … e 0 1 S e1 @N e e e e e 2 e e e e 1 y3 y1 y1 y2 @ @ A 1 y3 y1 1 S12 rN2 D 2Ae e e e e e D 2Ae e e D 2Ae e x x x x @N2 x x S 1 3 2 1 @ 1 3 „ƒ‚22 … e 0 1 S e2 @N e e e e e 3 e e e e 1 y3 y1 y1 y2 @ @ A 1 y1 y2 1 S13 rN3 D 2Ae e e e e e D 2Ae e e D 2Ae e x x x x @N3 x x S 1 3 2 1 @ 2 1 „ƒ‚23 … e S e3 (H.17) introducing the coordinate matrix e e e e e e e e e e S11 S12 S13 y2 y3 y3 y1 y1 y2 S D e e e D e e e e e e (H.18) S21 S22 S23 x3 x2 x1 x3 x2 x1 where eI are nodal base vectors defined as 0 1 0 1 0 1 1 0 0 @ A @ A @ A e1 D 0 e2 D 1 e3 D 0 (H.19) 0 0 1 868 H Analytical Evaluation of Element Matrices and Vectors With (8.122)and(H.16)aswellas(8.123)and(H.15)itis d˝e DjJ e jd d D 2Aed d (H.20) and 8 e ̂ < l12d on edge 12; D 0 d e D e ̂ : l13d on edge 13; D 0 (H.21) e ̂ l23d on edge 23; D 1 where ! @x e e p @ x x le D D 2 1 D .xe xe /2 C .ye ye/2 12 @y ye ye 2 1 2 1 @ 2 1 ! @x e e p @ x x le D D 3 1 D .xe xe /2 C .ye ye/2 13 @y ye ye 3 1 3 1 @ 3 1 ! ! @x @x e e p @ @ x x le D D 3 2 D .xe xe /2 C .ye ye/2 23 @y @y ye ye 3 2 3 2 @ @ 3 2 (H.22) Thus, the matrices and vectors of the ADE convective form (8.104)and(8.105) become for element e: 0 1 e e e e e e R N1 N1 N1 N2 N1 N3 e RK e @ e e e e e eA e O D ˝e N2 N1 N2 N2 N2 N3 d˝ e e e e e e N3 N1 N3 N2 N3 N3 0 1 0 1 R R .1 /2 .1 / .1 / 211 1 1 K e e RK e @ 2 A e R A @ A D 0 0 .1 / 2A d d D 12 121 .1 / 2 112 0 1 e e e e e e e e e R N1 .q rN1 /N1 .q rN2 /N1 .q rN3 / e @ e e e e e e e e e A e A D ˝e N2 .q rN1 /N2 .q rN2 /N2 .q rN3 / d˝ e e e e e e e e e N3 .q rN1 /N3 .q rN2 /N3 .q rN3 / 0 1 e e e e e e e R R .1 /.S11 C S21/.1 /.S12 C S22/.1 /.S13 C S23/ q 1 1 @ e e e e e e A e D 2Ae 0 0 .S11 C S21/ .S12 C S22/ .S13 C S23/ 2A d d e e e e e e .S11 C S21/ .S12 C S22/ .S13 C S23/ 0 1 e e e e e e .S11 C S21/.S12 C S22/.S13 C S23/ qe @ e e e e e e A D 6 .S11 C S21/.S12 C S22/.S13 C S23/ e e e e e e .S11 C S21/.S12 C S22/.S13 C S23/ 0 1 e e e e e e e e e R rN1 .D rN1 / rN1 .D rN2 / rN1 .D rN3 / e @ e e e e e e e e e A e C D ˝e rN2 .D rN1 / rN2 .D rN2 / rN2 .D rN3 / d˝ e e e e e e e e e rN3 .D rN1 / rN3 .D rN2 / rN3 .D rN3 / 0 1 R R ..S e /2 C .S e /2/.Se S e C S e S e /.Se S e C S e S e / e 11 21 11 12 21 22 11 13 21 23 D 1 1 @ e e e e e 2 e 2 e e e e A e D .2Ae /2 0 0 .S12S11 C S22S21/ ..S12/ C .S22/ /.S12S13 C S22S23/ 2A d d e e e e e e e e e 2 e 2 .S13S11 C S23S21/.S13S12 C S23S22/ ..S13/ C .S23/ / H.2 Linear 2D Triangle 869 0 1 e 2 e 2 e e e e e e e e ..S11/ C .S21/ /.S11S12 C S21S22/.S11S13 C S21S23/ De @ e e e e e 2 e 2 e e e e A D 4Ae .S11S12 C S21S22/ ..S12/ C .S22/ /.S12S13 C S22S23/ e e e e e e e e e 2 e 2 .S11S13 C S21S23/.S12S13 C S22S23/ ..S13/ C .S23/ / 0 1 e e e e e e R N1 N1 N1 N2 N1 N3 e e e @ e e e e e eA e R D ˝e .# C Q / N2 N1 N2 N2 N2 N3 d˝ e e e e e e N3 N1 N3 N2 N3 N3 0 1 R R .1 /2 .1 / .1 / e e 1 1 @ 2 A e D .# C Q / 0 0 .1 / 2A d d .1 / 2 0 1 .#e C Qe/Ae 211 D @121A (H.23) 12 112 0 1 e e e e e e R N1 N1 N1 N2 N1 N3 e e @ e e e e e eA e B D e ˚ N N N N N N d C 2 1 2 2 2 3 e e e e e e N3 N1 N3 N2 N3 N3 0 1 e e e e e e R N1 .D rN1 / n N1 .D rN2 / n N1 .D rN3 / n @ e e e e e e A e e N .D rN / n N .D rN / n N .D rN / n d N 2 1 2 2 2 3 O e e e e e e N3 .D rN1 / n N3 .D rN2 / n N3 .D rN3 / n Z 0 1 1 .1 /2 .1 / .1 / p D ˚ e @.1 / 2 A :::.d ;d / 0 .1 / 2 Z 0 1 1 .1 /.S e n C S e n /.1 /.S e n C S e n /.1 /.S e n C S e n / e 11 1 21 2 12 1 22 2 13 1 23 2 D @ e e e e e e A 2Ae .S11n1 C S21n2/ .S12n1 C S22n2/ .S13n1 C S23n2/ e e e e e e 0 .S11n1 C S21n2/ .S12n1 C S22n2/ .S13n1 C S23n2/ p :::.d ;d / 8 0 1 ˆ 210 ˆ e ˆ ˚ @120A le ˆ 6 12 ˆ 0000 1 ˆ e e e e e e ˆ .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ ˆ e ˆ D @.S e n C S e n /.Se n C S e n /.Se n C S e n /A le C on edge 12̂ ˆ 4Ae 11 1 21 2 12 1 22 2 13 1 23 2 12 ˆ ˆ 000 ˆ 0 1 NO ˆ 201 ˆ e ˆ ˚ @000A le <ˆ 6 13 0102 1 D e e e e e e ˆ .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ ˆ De @ A e ̂ ˆ 4Ae 000l13C on edge 13 ˆ e e e e e e ˆ .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ ˆ 0 1 NO ˆ ˆ 000 ˆ ˚ e @ A e ˆ 021 l23 ˆ 6 ˆ 0012 1 ˆ ˆ 000 ˆ De @ e e e e e e A e ̂ ˆ 4Ae .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ l23 on edge 23 : e e e e e e .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ NO 870 H Analytical Evaluation of Element Matrices and Vectors 0 1 0 1 e e R N1 R N1 e e e @ eA e e @ eA e H D e ˚ N d e q N d C C 2 N N 2 e e N3 N3 0 1 0 1 R 1 R 1 e e 1@ Ap e 1@ Ap D ˚ C 0 :::.d ;d / qN 0 :::.d ;d / 8 0 1 0 1 ˆ 1 1 ˆ ˚ e e qe ˆ C @1A N @1A le C on edge 12̂ ˆ 2 2 12 ˆ 0 0 ˆ 0 1 0 1 < 1 1 e e e D ˚ C @0A qN @0A le C on edge 13̂ (H.24) ˆ 2 2 13 ˆ 1 1 ˆ 0 1 0 1 ˆ 0 0 ˆ e e e ˆ ˚ C @ A qN @ A e ̂ : 2 1 2 1 l23 on edge 23 1 1 0 1 e R N1 P e e @ eA e Q D ˝e H N2 d˝ w w.xw/Qw.t/ N e 30 1 R R 1 P e 1 1 @ A e D H 0 0 2A d d w w.xw/Qw.t/ (H.25) 0 1 1 P H eAe @ A D 3 1 w w.xw/Qw.t/ 1 where for convenience we assume constant parameters (storage coefficient RK e,flux qe, dispersion De, decay rate #e, flow supply Qe, transfer coefficient ˚ e and source/sink H e) within the element. In (H.24) the components of the unit normal vector n in the Cartesian x and y coordinate direction on the outflow boundary NO are indicated by n1 and n2, respectively. Finally, the spatially discretized ADE in the convective form (8.99) can be summarized as 0 1 0 1 e 0 1 ( d 1 " e e e e e e X 211 dt .S11 C S21/.S12 C S22/.S13 C S23/ RK e Ae @ A B d e C qe @ A 121 @ 2 A C .S e C S e /.Se C S e /.Se C S e / 12 dt 6 11 21 12 22 13 23 e e e e e e e e 112 d 3 .S C S /.S C S /.S C S / dt 11 21 12 22 13 23 0 1 e 2 e 2 e e e e e e e e ..S11/ C .S21/ /.S11S12 C S21S22/.S11S13 C S21S23/ De @ e e e e e 2 e 2 e e e e A C 4Ae .S11S12 C S21S22/ ..S12/ C .S22/ /.S12S13 C S22S23/ e e e e e e e e e 2 e 2 .S11S13 C S21S23/.S12S13 C S22S23/ ..S13/ C .S23/ / 0 1 0 1 0 1 0 1 ! 211 e 210 201 000 e e e ˚ C .# CQ /A @121A C @120A le C @000A le C @021A le 12 6 12 13 23 112 000 102 012 H.3 Linear 3D Tetrahedron 871 0 1 .S e n C S e n /.Se n C S e n /.Se n C S e n / De 11 1 21 2 12 1 22 2 13 1 23 2 @.S e n C S e n /.Se n C S e n /.Se n C S e n /A le 4Ae 11 1 21 2 12 1 22 2 13 1 23 2 12 000 0 1 e e e e e e .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ @ A e C 000l13 e e e e e e .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ 0 1 ! 000 @ e e e e e e A e C .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ l23 e e e e e e .S11n1 C S21n2/.S12n1 C S22n2/.S13n1 C S23n2/ NO 0 1 0 1 0 1 0 1 0 1 # e " # ) 1 e e e 1 1 0 e e 1 @ eA ˚ C qN @ A e @ A e @ A e H A @ A 2 1 l12 C 0 l13 C 1 l23 1 e 2 3 3 0 1 1 1 X C w.xw/Qw.t/ D 0 (H.26) w Similar expressions can be obtained for the divergence form of ADE (8.98). H.3 Linear 3D Tetrahedron We consider the linear 4-node tetrahedral element e as shown in Fig. H.3 (cf. Appendix G,TableG.3a) having the following shape functions e N1 D 1 e N2 D e (H.27) N3 D e N4 D and local derivatives e e e @N1 @N1 @N1 @ D 1; @ D 1 @ D 1 e e e @N2 @N2 @N2 @ D 1; @ D 0 @ D 0 e e e (H.28) @N3 @N3 @N3 @ D 0; @ D 1 @ D 0 e e e @N4 @N4 @N4 @ D 0; @ D 0 @ D 1 872 H Analytical Evaluation of Element Matrices and Vectors Fig. H.3 Linear 4-node 4 tetrahedron in the global and (0,0,1) local coordinate system 4 z 3 (0,0,0) y 1 3 2 1 (0,1,0) (1,0,0) 2 x for .0 ; ; 1/. It is for the tetrahedral element, cf. (8.71) e e e e e e e e x D N1 x1 C N2 x2 C N3 x3 C N4 x4 e e e e e e e e y D N1 y1 C N2 y2 C N3 y3 C N4 y4 (H.29) e e e e e e e e z D N1 z1 C N2 z2 C N3 z3 C N4 z4 e e e where xI ;yI ; zI .I D 1; 2; 3; 4/ correspond to the Cartesian coordinates of the vertices (nodes) of the tetrahedron.Using (H.28)and(H.29)in(8.115)and(8.119) we obtain the Jacobian 0 1 e e e e e e x2 x1 y2 y1 z2 z1 e @ e e e e e eA J D x3 x1 y3 y1 z3 z1 ; (H.30) e e e e e e x4 x1 y4 y1 z4 z1 the inverse Jacobian 0 1 Ae Ae Ae 1 11 12 13 .J e / 1 D @ e e e A e A21 A22 A23 (H.31) jJ j e e e A31 A32 A33 where e e e e e e e e e e A11 D z1.y4 y3 / C z3.y1 y4 / C z4.y3 y1 / e e e e e e e e e e A12 D z1.y2 y4 / C z2.y4 y1 / C z4.y1 y2 / e e e e e e e e e e A13 D z1.y3 y2 / C z2.y1 y3 / C z3.y2 y1 / e e e e e e e e e e A21 D z1.x3 x4 / C z3.x4 x1 / C z4.x1 x3 / e e e e e e e e e e A22 D z1.x4 x2 / C z2.x1 x4 / C z4.x2 x1 / (H.32) e e e e e e e e e e A23 D z1.x2 x3 / C z2.x3 x1 / C z3.x1 x2 / e e e e e e e e e e A31 D y1 .x4 x3 / C y3 .x1 x4 / C y4 .x3 x1 / e e e e e e e e e e A32 D y1 .x2 x4 / C y2 .x4 x1 / C y4 .x1 x2 / e e e e e e e e e e A33 D y1 .x3 x2 / C y2 .x1 x3 / C y3 .x2 x1 / and its determinant e e e e e e e e e e e jJ jD.x2 x1 /A11 C .x3 x1 /A12 C .x4 x1 /A13 D 6V (H.33) H.3 Linear 3D Tetrahedron 873 which is six times the volume V e of the tetrahedron. We see that the tetrahedron has the important advantage of having a constant Jacobian (and its inverse) independent of the element shape in the Cartesian coordinates (we shall see further below this is not the case for the other 3D elements). The global derivatives (8.118) result with (H.28)in 0 1 @N e 0 1 0 1 1 0 1 e e e e @ e e e S A11 A12 A13 B C .A11 C A12 C A13/ 11 1 B @N e C 1 1 e @Ae Ae Ae A 1 @ .Ae C Ae C Ae /A @ e A rN1 D 6V e 21 22 23 @ @ A D 6V e 21 22 23 D 6V e S21 e e e e e e e e A31 A32 A33 @N1 .A31 C A32 C A33/ S @ „ ƒ‚31 … e 0 1 S e1 e 0 1 @N1 0 1 0 1 e e e e e A A A B @ C A S12 11 12 13 B @N e C 11 e 1 @Ae Ae Ae A 1 1 @Ae A 1 @ e A rN2 D 6V e 21 22 23 @ @ A D 6V e 21 D 6V e S22 e e e e e e A31 A32 A33 @N1 A31 S @ „ ƒ‚32 … e 0 1 S e2 @N e 0 1 0 1 1 0 1 e e e e @ e S A11 A12 A13 B e C A12 13 e 1 @ e e e A B @N1 C 1 @ e A 1 @ e A rN3 D 6V e A21 A22 A23 @ @ A D 6V e A22 D 6V e S23 e e e e e e A31 A32 A33 @N1 A32 S @ „ ƒ‚33 … e 0 1 S e3 @N e 0 1 0 1 1 0 1 e e e e @ e S A11 A12 A13 B e C A13 14 e 1 @ e e e A B @N1 C 1 @ e A 1 @ e A rN4 D 6V e A21 A22 A23 @ @ A D 6V e A23 D 6V e S24 e e e e e e A31 A32 A33 @N1 A33 S @ „ ƒ‚34 … e S e4 (H.34) introducing the coordinate matrix 0 1 0 1 e e e e e e e e S11 S12 S13 S14 A10 A11 A12 A13 e @ e e e e A @ e e e e A S D S21 S22 S23 S24 D A20 A21 A22 A23 (H.35) e e e e e e e e S31 S32 S33 S34 A30 A31 A32 A33 with e e e e e e e e e e A10 D z2.y3 y4 / C z3.y4 y2 / C z4.y2 y3 / e e e e e e e e e e A20 D z2.x4 x3 / C z3.x2 x4 / C z4.x3 x2 / (H.36) e e e e e e e e e e A30 D y2 .x3 x4 / C y3 .x4 x2 / C y4 .x2 x3 / where eI are nodal base vectors defined as 0 1 0 1 0 1 0 1 1 0 0 0 B0C B1C B0C B0C e D B C e D B C e D B C e D B C (H.37) 1 @0A 2 @0A 3 @1A 4 @0A 0 0 0 1 874 H Analytical Evaluation of Element Matrices and Vectors With (8.122)and(H.33)aswellas(8.123)and(H.31)itis d˝e DjJ e jd d d D 6V ed d d (H.38) and 8 ˆ 2Ae d d on area 123;̂ D 1 ; D 0 ˆ 123̂ < e ̂ e 2 d d on area 124; D 1 ; D 0 d D 124 ̂ (H.39) ˆ 2Ae d d on area 134; D 1 ; D 0 :ˆ 134̂ 2Ae d d on area 234;̂ D 1 234̂ where 0 1 0 1 @x @x 0 1 @ @ e B C B C A13 p e B @y C B @y C @ e A e 2 e 2 e 2 2 D @ A @ A D A D .A / C .A / C .A / 123 @ @ 23 13 23 33 @z @z e A33 0 @ 1 0 @ 1 @x @x 0 1 @ @ e B C B C A12 p e B @y C B @y C @ e A e 2 e 2 e 2 2 D @ A @ A D A D .A / C .A / C .A / 124 @ @ 22 12 22 32 @z @z e A32 0 @ 1 0 @ 1 @x @x 0 1 @ @ e B C B C A11 p e B @y C B @y C @ e A e 2 e 2 e 2 2 D @ A @ A D A D .A / C .A / C .A / 134 @ @ 21 11 21 31 @z @z e A31 0 @ 1@ 0 1 @x @x @x @x 0 1 @ @ @ @ e B C B C A10 p e B @y @y C B @y @y C @ e A e 2 e 2 e 2 2 D @ A @ A D A D .A / C .A / C .A / 234 @ @ @ @ 20 10 20 30 @z @z @z @z e A30 @ @ @ @ (H.40) Thus, the matrices and vectors of the ADE convective form (8.104)and(8.105) become for element e: 0 1 e e e e e e e e N1 N1 N1 N2 N1 N3 N1 N4 R B e e e e e e e eC e RK e BN2 N1 N2 N2 N2 N3 N2 N4 C e O D ˝e @ e e e e e e e eA d˝ N3 N1 N3 N2 N3 N3 N3 N4 e e e e e e e e N4 N1 N4 N2 N4 N3 N4 N4 0 1 .1 /2 .1 / .1 / .1 / R R R B 2 C D RK e 1 1 1 B.1 / C 0 0 0 @ .1 / 2 A .1 / 2 6V ed d d H.3 Linear 3D Tetrahedron 875 0 1 2111 B C RK e e B1211C D V @ A 20 1121 1112 0 1 e e e e e e e e e e e e N1 .q rN1 /N1 .q rN2 /N1 .q rN3 /N1 .q rN4 / R B e e e e e e e e e e e e C e BN2 .q rN1 /N2 .q rN2 /N2 .q rN3 /N2 .q rN4 /C e A D ˝e @ e e e e e e e e e e e e A d˝ N3 .q rN1 /N3 .q rN2 /N3 .q rN3 /N3 .q rN4 / e e e e e e e e e e e e N4 .q rN1 /N4 .q rN2 /N4 .q rN3 /N4 .q rN4 / e R R R q 1 1 1 D 6V e 0 0 0 0 1 e e e e e e e e e e e e e e e e N1 .S11 C S21 C S31/N1 .S12 C S22 C S32/N1 .S13 C S23 C S33/N1 .S14 C S24 C S34/ B e e e e e e e e e e e e C B .S11 C S21 C S31/ .S12 C S22 C S32/ .S13 C S23 C S33/ .S14 C S24 C S34/ C @ e e e e e e e e e e e e A .S11 C S21 C S31/ .S12 C S22 C S32/ .S13 C S23 C S33/ .S14 C S24 C S34/ e e e e e e e e e e e e .S11 C S21 C S31/ .S12 C S22 C S32/ .S13 C S23 C S33/ .S14 C S24 C S34/ 6V ed d d 0 1 e e e e e e e e e e e e .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ e B e e e e e e e e e e e e C q B.S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/C D 24 @ e e e e e e e e e e e e A .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ e e e e e e e e e e e e .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ 0 1 e e e e e e e e e e e e rN1 .D rN1 / rN1 .D rN2 / rN1 .D rN3 / rN1 .D rN4 / R B e e e e e e e e e e e e C e BrN2 .D rN1 / rN2 .D rN2 / rN2 .D rN3 / rN2 .D rN4 /C C D ˝e @ e e e e e e e e e e e e A rN3 .D rN1 / rN3 .D rN2 / rN3 .D rN3 / rN3 .D rN4 / e e e e e e e e e e e e rN4 .D rN1 / rN4 .D rN2 / rN4 .D rN3 / rN4 .D rN4 / d˝e R R R De 1 1 1 D .6V e /2 0 0 0 0 e 2 e 2 e 2 e e e e e e e e e e e e e e 1 ..S11/ C .S21/ C .S31/ /.S11S12 C S21S22 C S31S32/.S11S13 C S21S23 .S11S14 C S21S24 B e e e e C B CS31S33/ CS31S34/ C B e e e e e e e 2 e 2 e 2 e e e e e e e e C B.S12S11 C S22S21 C S32S31/ ..S12/ C .S22/ C .S32/ /.S12S13 C S22S23 .S12S14 C S22S24C B e e e e C B CS32S33/ CS32S34/ C B e e e e e e e e e e e e e 2 e 2 e e e e C B.S13S11 C S23S21 C S33S31/.S13S12 C S23S22 C S33S32/ ..S13/ C .S23/ .S13S14 C S23S24C B e 2 e e C B C.S33/ / CS33S34/ C @ e e e e e e e e e e e e e e e e e 2 e 2 A .S14S11 C S24S21 C S34S31/.S14S12 C S24S22 C S34S32/.S14S13 C S24S23 ..S14/ C .S24/ e e e 2 CS34S33/ C.S34/ / 6V ed d d De D 36V e 0 e 2 e 2 e 2 e e e e e e e e e e e e e e 1 ..S11/ C .S21/ C .S31/ /.S11S12 C S21S22 C S31S32/.S11S13 C S21S23 .S11S14 C S21S24 B e e e e C B CS31S33/ CS31S34/ C B e e e e e e e 2 e 2 e 2 e e e e e e e e C B.S11S12 C S21S22 C S31S32/ ..S12/ C .S22/ C .S32/ /.S12S13 C S22S23 .S12S14 C S22S24C B e e e e C B CS32S33/ CS32S34/ C B e e e e e e e e e e e e e 2 e 2 e e e e C B.S11S13 C S21S23 C S31S33/.S12S13 C S22S23 C S32S33/ ..S13/ C .S23/ .S13S14 C S23S24C B e 2 e e C B C.S33/ / CS33S34/ C @ e e e e e e e e e e e e e e e e e 2 e 2 A .S11S14 C S21S24 C S31S34/.S12S14 C S22S24 C S32S34/.S13S14 C S23S24 ..S14/ C .S24/ e e e 2 CS33S34/ C.S34/ / 876 H Analytical Evaluation of Element Matrices and Vectors 0 1 N eN e N eN e N eN e N eN e B 1 1 1 2 1 3 1 4 C R BN eN e N eN e N eN e N eN e C e .# e C Qe / B 2 1 2 2 2 3 2 4 C d˝e R D ˝e @ e e e e e e e e A N3 N1 N3 N2 N3 N3 N3 N4 e e e e e e e e N4 N1 N4 N2 N4 N3 N4 N4 R R R e e 1 1 1 D .# C Q / 0 0 0 0 1 .1 /2 .1 / .1 / .1 / B C B.1 / 2 C B C 6V e d d d @ .1 / 2 A .1 / 2 0 1 2111 B C .#e CQe/V e B1211C D @ A (H.41) 20 1121 1112 0 1 N eN e N eN e N eN e N eN e B 1 1 1 2 1 3 1 4 C R B e e e e e e e eC e e N2 N1 N2 N2 N2 N3 N2 N4 e B D e ˚ B C d C @ e e e e e e e eA N3 N1 N3 N2 N3 N3 N3 N4 e e e e e e e e N4 N1 N4 N2 N4 N3 N4 N4 0 1 e e e e e e e e N1 .D rN1 / n N1 .D rN2 / n N1 .D rN3 / n N1 .D rN4 / n R B e e e e e e e e C BN2 .D rN1 / n N2 .D rN2 / n N2 .D rN3 / n N2 .D rN4 / nC e B C N @ e e e e e e e e A O N3 .D rN1 / n N3 .D rN2 / n N3 .D rN3 / n N3 .D rN4 / n e e e e e e e e N4 .D rN1 / n N4 .D rN2 / n N4 .D rN3 / n N4 .D rN4 / n d e R R e 1 1 . ; / D ˚ 0 0 0 1 .1 /2 .1 / .1 / .1 / B C B.1 / 2 Cp B C ::: .d d ;d d ;d d / @ .1 / 2 A .1 / 2 R R De 1 1 . ; / 6V e 0 0 0 P P P P 1 e e e e .1 / S ni .1 / S ni .1 / S ni .1 / S ni B P i i1 P i i2 P i i3 P i i4 C B S e n S e n S e n S e n C B Pi i1 i Pi i2 i Pi i3 i Pi i4 i C @ e e e e A Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni e e e e i Si1ni i Si2ni i Si3ni i Si4ni p ::: .d d ;d d ;d d / H.3 Linear 3D Tetrahedron 877 8 0 1 ˆ 2110 ˆ B C ˆ ˚ e B1210C e ˆ @ A  ˆ 12 1120 123 ˆ ˆ 00000P P P P 1 ˆ e e e e ˆ Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni ˆ B e e e e C ˆ De BPi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni C e ̂ ˆ e @ A  C on area 123 ˆ 18V S e n S e n S e n S e n 123 ˆ i i1 i i i2 i i i3 i i i4 i ˆ 0000 ˆ N ˆ 0 1 O ˆ 2101 ˆ B C ˆ ˚ e B1201C e ˆ @ A  ˆ 12 0000 124 ˆ ˆ 01102P P P P 1 ˆ e e e e ˆ Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni ˆ B e e e e C ˆ De B i Si1ni i Si2ni i Si3ni i Si4ni C e ̂ ˆ e @ A  C on area 124 ˆ 18V P 0000P P P 124 < e e e e i Si1ni i Si2ni i Si3ni i Si4ni D 0 1 NO ˆ 2011 ˆ B C ˆ ˚ e B0000C e ˆ @ A  ˆ 12 1021 134 ˆ ˆ 01012P P P P 1 ˆ e e e e ˆ i Si1ni i Si2ni i Si3ni i Si4ni ˆ B C ˆ De BP 0000P P P C e ̂ ˆ e @ A  C on edge 134 ˆ 18V S e n S e n S e n S e n 134 ˆ Pi i1 i Pi i2 i Pi i3 i Pi i4 i ˆ S e n S e n S e n S e n ˆ i i1 i i i2 i i i3 i i i4 i ˆ 0 1 NO ˆ 0000 ˆ B C ˆ ˚ e B0211C e ˆ @ A  ˆ 12 0121 234 ˆ ˆ 0112 ˆ 0 1 ˆ P 0000P P P ˆ B e e e e C ˆ e B S ni S ni S ni S ni C ˆ D Pi i1 Pi i2 Pi i3 Pi i4 Ae on edge 234̂ ˆ 18V e @ e e e e A 234̂ ˆ Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni : e e e e i Si1ni i Si2ni i Si3ni i Si4ni NO (H.42) P e e e e with i Si1ni D S11n1 C S21n2 C S31n3 and so forth, 0 1 0 1 e e N1 N1 R B eC R B eC e e e BN2 C e e BN2 C e H D e ˚ d e q d C C @ eA N N @ eA N3 N3 e e N4 N4 0 1 R R 1 1 1 . ; /B Cp D ˚ e e B C ::: .d d ;d d ;d d / C 0 0 @ A 0 1 R R 1 1 1 . ; /B Cp qe B C ::: .d d ;d d ;d d / N 0 0 @ A 878 H Analytical Evaluation of Element Matrices and Vectors 8 0 1 0 1 ˆ 1 1 ˆ B C B C ˆ ˚ e e B1C qe B1C ˆ C N Ae C on area 123̂ ˆ 3 @ A 3 @ A 123̂ ˆ 1 1 ˆ 0 0 ˆ 0 1 0 1 ˆ 1 1 ˆ B C B C ˆ e e e ˆ ˚ C B1C qN B1C e ̂ ˆ 3 @ A 3 @ A  C on area 124 <ˆ 0 0 124 1 1 D 0 1 0 1 (H.43) ˆ 1 1 ˆ B C B C ˆ ˚ e e B0C qe B0C ˆ C N Ae C on area 134̂ ˆ 3 @ A 3 @ A 134̂ ˆ 1 1 ˆ ˆ 011 011 ˆ 0 0 ˆ ˆ e e B C e B C ˆ ˚ C B1C qN B1C e ̂ ˆ @ A @ A  on area 234 :ˆ 3 1 3 1 234 1 1 0 1 e N1 R B eC P e e BN2 C e Q D ˝e H @ eA d˝ w w.xw/Qw.t/ N3 e N4 0 1 1 R R R P 1 1 1 B C D H e B C 6V ed d d .x /Q .t/ 0 0 0 @ A w w w w (H.44) 0 1 1 B C P H eV e B1C D @ A w.xw/Qw.t/ 4 1 w 1 where for convenience we assume constant parameters (storage coefficient RK e, flux qe, dispersion De, decay rate #e, flow supply Qe, transfer coefficient ˚ e and source/sink H e) within the element. In (H.42) the components of the unit normal vector n in the Cartesian x , y and z coordinate direction on the outflow boundary NO are indicated by n1, n2 and n3, respectively. Finally, the spatially discretized ADE in the convective form (8.99) can be summarized as 0 1 d e 0 1 1 ( B dt C 2111 B e C X B C d 2 RK e V e B1211C B dt C @ A B e C 20 1121 B d 3 C e @ dt A 1112 e d 4 dt 0 1 " e e e e e e e e e e e e .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ B C e B.S e C S e C S e /.Se C S e C S e /.Se C S e C S e /.Se C S e C S e /C q B 11 21 31 12 22 32 13 23 33 14 24 34 C C 24 @ e e e e e e e e e e e e A .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ e e e e e e e e e e e e .S11 C S21 C S31/.S12 C S22 C S32/.S13 C S23 C S33/.S14 C S24 C S34/ H.3 Linear 3D Tetrahedron 879 De C 36V e 0 1 ..Se /2 C .Se /2 C .Se /2/.Se Se C Se Se C Se Se /.Se Se C Se Se C Se Se /.Se Se C Se Se C Se Se / B 11 21 31 11 12 21 22 31 32 11 13 21 23 31 33 11 14 21 24 31 34 C B C B.Se Se C Se Se C Se Se / ..Se /2 C .Se /2 C .Se /2/.Se Se C Se Se C Se Se /.Se Se C Se Se C Se Se /C B 11 12 21 22 31 32 12 22 32 12 13 22 23 32 33 12 14 22 24 32 34 C B e e e e e e e e e e e e e 2 e 2 e 2 e e e e e e C @.S11S13 C S21S23 C S31S33/.S12S13 C S22S23 C S32S33/ ..S13/ C .S23/ C .S33/ /.S13S14 C S23S24 C S33S34/A e e e e e e e e e e e e e e e e e e e 2 e 2 e 2 .S11S14 C S21S24 C S31S34/.S12S14 C S22S24 C S32S34/.S13S14 C S23S24 C S33S34/ ..S14/ C .S24/ C .S34/ / 0 1 0 1 0 1 0 1 2111 2110 2101 2011 B C e B C B C B C .#e CQe/V e B1211C ˚ B1210C e B1201C e B0000C e C @ A C @ AA ̂ C @ AA ̂ C @ AA ̂ 20 1121 12 1120 123 0000 124 1021 134 1112 0000 1102 1012 0 1 0P P P P 1 ! e e e e 0000 S ni S ni S ni S ni e Pi i1 Pi i2 Pi i3 Pi i4 B0211C D B S e n S e n S e n S e n C C B CAe BPi i1 i Pi i2 i Pi i3 i Pi i4 i CAe @ A 234̂ e @ e e e e A 123̂ 0121 18V i Si1ni i Si2ni i Si3ni i Si4ni 0112 0000 0P P P P 1 e e e e Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni B e e e e C B i Si1ni i Si2ni i Si3ni i Si4ni C e C @ AA ̂ P 0000P P P 124 e e e e i Si1ni i Si2ni i Si3ni i Si4ni 0P P P P 1 e e e e i S ni i S ni i S ni i S ni B i1 i2 i3 i4 C BP 0000P P P C e C @ e e e e AA134̂ Pi Si1ni Pi Si2ni Pi Si3ni Pi Si4ni e e e e i Si1ni i Si2ni i Si3ni i Si4ni 0 1 0 1 e 0000 ! # 1 P P P P B C B S e n S e n S e n S e n C e C BPi i1 i Pi i2 i Pi i3 i Pi i4 i CAe B 2 C @ S e n S e n S e n S e n A 234̂ @ eA Pi i1 i Pi i2 i Pi i3 i Pi i4 i 3 e e e e NO i Si1ni i Si2ni i Si3ni i Si4ni e 4 0 1 0 1 0 1 0 1 " 1 1 1 0 # e e e B C B C B C B C ˚ C qN B1C e B1C e B0C e B1C e @ AA ̂ C @ AA ̂ C @ AA ̂ C @ AA ̂ 3 1 123 0 124 1 134 1 234 0 1 1 1 0 1 1 ) e e B C X H V B1C @ A C w.xw/Qw.t/ D 0 (H.45) 4 1 w 1 Similar expressions can be obtained for the divergence form of ADE (8.98). 880 H Analytical Evaluation of Element Matrices and Vectors H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians H.4.1 Linear Quadrilateral Element as Parallelogram or Rectangle The linear 4-node quadrilateral element as described in Appendix G (Table G.2b) has the following shape functions e 1 N1 D 4 .1 /.1 / N e D 1 .1 C /.1 / 2 4 (H.46) e 1 N3 D 4 .1 C /.1 C / e 1 N4 D 4 .1 /.1 C / and the local derivatives e e @N1 1 @N1 1 @ D 4 .1 /; @ D 4 .1 / e e @N2 1 @N2 1 @ D 4 .1 /; @ D 4 .1 C / e e (H.47) @N3 1 @N3 1 @ D 4 .1 C /; @ D 4 .1 C / e e @N4 1 @N4 1 @ D 4 .1 C /; @ D 4 .1 / for . 1 ; 1/. It is for the quadrilateral element, cf. (8.71) e e e e e e e e x D N1 x1 C N2 x2 C N3 x3 C N4 x4 e e e e e e e e (H.48) y D N1 y1 C N2 y2 C N3 y3 C N4 y4 e e where xI ;yI .I D 1; 2; 3; 4/ correspond to the Cartesian coordinates of the vertices (nodes) of the quadrilateral. Using (H.47)and(H.48)in(8.116) we obtain the Jacobian ! e e e e e e e e e e e e e e e e e 1 . x1 C x2 C x3 x4 C .x1 x2 C x3 x4 // . y1 C y2 C y3 y4 C .y1 y2 C y3 y4 // J D 4 e e e e e e e e e e e e e e e e . x1 x2 C x3 C x4 C .x1 x2 C x3 x4 // . y1 y2 C y3 C y4 C .y1 y2 C y3 y4 // (H.49) We see from (H.49) that the Jacobian J e of the quadrilateral is a function of the local coordinates ; , which makes an analytical evaluation impossible. To eliminate the e e e e e e e ; dependency from J we have to enforce x1 x2 C x3 x4 D 0 and y1 y2 C e e y3 y4 D 0, which become valid for the parallelogram and the rectangle as shown in Fig. H.4. The parallelogram is characterized by the geometric conditions: e e e e e e e e e e ıx D x4 x1 D x3 x2 Lx D x2 x1 D x3 x4 e e e e e e e e e e (H.50) ıy D y2 y1 D y3 y4 Ly D y4 y1 D y3 y2 H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians 881 Fig. H.4 Linear 4-node y parallelogram in the global e 3 (-1,1) (1,1) x and local coordinate system 4 3 4 e 2 Ly e 1 y 1 2 e (-1,-1) (1,-1) Lx x Then, the Jacobain of the parallelogram ! e e e 1 Lx ıy J D 2 e e (H.51) ıx Ly becomes constant. It yields the inverse Jacobian ! 1 Le ıe .J e / 1 D y y e e e (H.52) 2jJ j ıx Lx and the determinant e 1 e e e e 1 e jJ jD 4 .Lx Ly ıxıy/ D 4 A (H.53) e e which corresponds to the quarter of the parallelogram area. We note that ıx D ıy D 0 for a rectangular element aligned to the global coordinate axes. Now, it is possible to evaluate the element matrices and vectors for the parallelogram and the rectangle in a similar way as done in Sect. H.2 for the triangle. H.4.2 Linear Hexahedral Element as Parallelepiped or Brick The linear 8-node hexahedral element as described in Appendix G (Table G.3c) has the following shape functions e 1 N1 D 8 .1 /.1 /.1 C / e 1 N2 D 8 .1 C /.1 /.1 C / e 1 N3 D 8 .1 C /.1 C /.1 C / e 1 N4 D 8 .1 /.1 C /.1 C / 882 H Analytical Evaluation of Element Matrices and Vectors e 1 N5 D 8 .1 /.1 /.1 / e 1 N6 D 8 .1 C /.1 /.1 / e 1 N7 D 8 .1 C /.1 C /.1 / e 1 N8 D 8 .1 /.1 C /.1 / (H.54) and the local derivatives e e e @N1 1 @N1 1 @N1 1 @ D 8 .1 /.1 C /; @ D 8 .1 /.1 C /; @ D 8 .1 /.1 / e e e @N2 1 @N2 1 @N2 1 @ D 8 .1 /.1 C /; @ D 8 .1 C /.1 C /; @ D 8 .1 C /.1 / e e e @N3 1 @N3 1 @N3 1 @ D 8 .1 C /.1 C /; @ D 8 .1 C /.1 C /; @ D 8 .1 C /.1 C / e e e @N4 1 @N4 1 @N4 1 @ D 8 .1 C /.1 C /; @ D 8 .1 /.1 C /; @ D 8 .1 /.1 C / e e e @N5 1 @N5 1 @N5 1 @ D 8 .1 /.1 /; @ D 8 .1 /.1 /; @ D 8 .1 /.1 / e e e @N6 1 @N6 1 @N6 1 @ D 8 .1 /.1 /; @ D 8 .1 C /.1 /; @ D 8 .1 C /.1 / e e e @N7 1 @N7 1 @N7 1 @ D 8 .1 C /.1 /; @ D 8 .1 C /.1 /; @ D 8 .1 C /.1 C / e e e @N8 1 @N8 1 @N8 1 @ D 8 .1 C /.1 /; @ D 8 .1 /.1 /; @ D 8 .1 /.1 C / (H.55) for . 1 ; ; 1/. It is for the hexahedral element, cf. (8.71) X8 X8 X8 e e e e e e x D NI xI ;yD NI yI ; z D NI zI (H.56) I D1 I D1 I D1 e e e where xI ;yI ; zI .I D 1;:::;8/ correspond to the Cartesian coordinates of the vertices (nodes) of the hexahedron. Using (H.55)and(H.56)in(8.115) we obtain the Jacobian 0 1 .a x C bx C cx C dx /.a y C by C cy C dy /.a z C bz C cz C dz/ e 1 @ A J D 8 .a x C bx C ex C dx /.a y C by C ey C dy /.a z C bz C ez C dz/ .a x C cx C ex C bx /.a y C cy C ey C by /.a z C cz C ez C bz/ (H.57) where 9 a D xe C xe C xe xe xe C xe C xe xe > x 1 2 3 4 5 6 7 8 > a D ye C ye C ye ye ye C ye C ye ye > y 1 2 3 4 5 6 7 8 > a D ze C ze C ze ze ze C ze C ze ze > z 1 2 3 4 5 6 7 8 > e e e e e e e e => a x D x1 x2 C x3 C x4 x5 x6 C x7 C x8 D e e C e C e e e C e C e a y y1 y2 y3 y4 y5 y6 y7 y8 > (H.58) a D ze ze C ze C ze ze ze C ze C ze > z 1 2 3 4 5 6 7 8 > e e e e e e e e > a x D x1 C x2 C x3 C x4 x5 x6 x7 x8 > e e e e e e e e > a y D y1 C y2 C y3 C y4 y5 y6 y7 y8 ;> e e e e e e e e a z D z1 C z2 C z3 C z4 z5 z6 z7 z8 H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians 883 z e Ly (-1,1,1) 4 e 4 (1,1,1) 1 y zx (1,-1,1) 3 3 (-1,-1,1) 1 2 e 8 e Lx e e 8 yx 2 zy Lz (-1,1,-1) 7 5 5 (1,1,-1) e 7 (-1,-1,-1) yz 6 e (1,-1,-1) 6 xy e xz x Fig. H.5 Linear 8-node parallelepiped in the global and local coordinate system 9 e e e e e e e e bx D x1 x2 C x3 x4 C x5 x6 C x7 x8 > e e e e e e e e > by D y1 y2 C y3 y4 C y5 y6 C y7 y8 > e e e e e e e e > bz D z1 z2 C z3 z4 C z5 z6 C z7 z8 > > c D xe C xe C xe xe C xe xe xe C xe > x 1 2 3 4 5 6 7 8 > c D ye C ye C ye ye C ye ye ye C ye > y 1 2 3 4 5 6 7 8 => c D ze C ze C ze ze C ze ze ze C ze z 1 2 3 4 5 6 7 8 (H.59) e e e e e e e e > dx D x1 x2 C x3 x4 x5 C x6 x7 C x8 > D e e C e e e C e e C e > dy y1 y2 y3 y4 y5 y6 y7 y8 > D e e C e e e C e e C e > dz z1 z2 z3 z4 z5 z6 z7 z8 > e e e e e e e e > ex D x1 x2 C x3 C x4 C x5 C x6 x7 x8 > e e e e e e e e > ey D y1 y2 C y3 C y4 C y5 C y6 y7 y8 ;> e e e e e e e e ez D z1 z2 C z3 C z4 C z5 C z6 z7 z8 It is obvious from (H.57) that the Jacobian J e of the hexahedron is still a function of the local coordinates ; ; . Any analytical evaluation needs that the ; ; dependencies in J e vanish, i.e., all coefficients b; c; d and e of (H.59)must be zero. This is valid for the parallelepiped and the brick as shown in Fig. H.5,where the following geometric conditions hold: 9 ıe D xe xe D xe xe D xe xe D xe xe > xz 4 1 3 2 8 5 7 6 > ıe D xe xe D xe xe D xe xe D xe xe > xy 4 8 1 5 3 7 2 6 > ıe D ye ye D ye ye D ye ye D ye ye > yx 1 5 2 6 4 8 3 7 > e e e e e e e e e > ıyz D y2 y1 D y3 y4 D y6 y5 D y7 y8 = ıe D ze ze D ze ze D ze ze D ze ze zx 2 1 6 5 3 4 7 8 > (H.60) ıe D ze ze D ze ze D ze ze D ze ze > zy 8 5 4 1 3 2 7 6 > e e e e e e e e e > Lx D x2 x1 D x3 x4 D x6 x5 D x7 x8 > e e e e e e e e e > Ly D y4 y1 D y3 y2 D y8 y5 D y7 y6 ;> e e e e e e e e e Lz D z4 z8 D z1 z5 D z3 z7 D z2 z6 884 H Analytical Evaluation of Element Matrices and Vectors Then, the Jacobain of the parallelepiped 0 1 Le ıe ıe B x yz zxC e 1 @ e e e A J D 2 ıxz Ly ızy (H.61) e e e ıxy ıyx Lz becomes constant. It yields the inverse Jacobian 0 1 .Le Le ıe ıe /.ıe ıe ıe Le/.ıe ıe ıe Le / 1 B y z yx zy zx yx yz z yz zy zx y C .J e / 1 D @ e e e e e e e e e e e e A e .ıxyızy ıxzLz /.LxLz ızxıxy/.ıxzızx ızy Lx/ 4jJ j e e e e e e e e e e e e .ıxzıyx ıxyLy /.ıyzıxy ıyxLx /.Lx Ly ıyzıxz/ (H.62) and the determinant e 1 e e e e e e e e e e e e e e e 1 e jJ jD 8 ŒLx .Ly Lz ıyxızy/ C ıxz.ıyxızx ıyzLz / C ıxy.ıyzızy ızxLy / D 8 V (H.63) e which corresponds to one-eighth of the parallelepiped volume. We note that ıxz D e e e e e ıxy D ıyx D ıyz D ızx D ızy D 0 for a brick element aligned to the global coordinate axes. The element matrices and vectors can now be analytically evaluated for the parallelepiped and the brick element in a similar way as done in Sect. H.3 for the tetrahedron. H.4.3 Linear Pentahedral Element as Triangular Prism with Parallel Top and Bottom Surfaces The linear 6-node pentahedral element as described in Appendix G (Table G.3b) has the following shape functions e 1 N1 D 2 .1 /.1 C / e 1 N2 D 2 .1 C / N e D 1 .1 C / 3 2 (H.64) e 1 N4 D 2 .1 /.1 / e 1 N5 D 2 .1 / e 1 N6 D 2 .1 / and the local derivatives e e e @N1 1 @N1 1 @N1 1 @ D 2 .1 C /; @ D 2 .1 C /; @ D 2 .1 / e e e @N2 1 @N2 @N2 1 @ D 2 .1 C /; @ D 0; @ D 2 H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians 885 e e e @N3 @N3 1 @N3 1 @ D 0; @ D 2 .1 C /; @ D 2 e e e @N4 1 @N4 1 @N4 1 @ D 2 .1 /; @ D 2 .1 /; @ D 2 .1 / e e e @N5 1 @N5 @N5 1 @ D 2 .1 /; @ D 0; @ D 2 e e e @N6 @N6 1 @N6 1 @ D 0; @ D 2 .1 /; @ D 2 (H.65) for .0 ; 1/ and . 1 1/. It is for the pentahedral element, cf. (8.71) X6 X6 X6 e e e e e e x D NI xI ;yD NI yI ; z D NI zI (H.66) I D1 I D1 I D1 e e e where xI ;yI ; zI .I D 1;:::;6/ correspond to the Cartesian coordinates of the vertices (nodes) of the pentahedron. Using (H.65)and(H.66)in(8.115) we obtain the Jacobian 0 1 .a x C bx /.a y C by /.a z C bz/ e 1 @ A J D 2 .a x C cx /.a y C cy /.a z C cz/ (H.67) .a x C bx C cx/.a y C by C cy /.a z C bz C cz/ where 9 a D xe C xe xe C xe > x 1 2 4 5 > e e e e > a y D y C y y C y > 1 2 4 5 > a D ze C ze ze C ze > z 1 2 4 5 > e e e e > a x D x1 C x3 x4 C x6 = e e e e a y D y1 C y3 y4 C y6 > (H.68) e e e e > a z D z1 C z3 z4 C z6 > > a D xe xe > x 1 4 > e e > a y D y1 y4 ;> e e a z D z1 z4 9 b D xe C xe C xe xe > x 1 2 4 5 > e e e e > by D y1 C y2 C y4 y5 > e e e e = bz D z C z C z z 1 2 4 5 (H.69) c D xe C xe C xe xe > x 1 3 4 6 > e e e e > cy D y1 C y3 C y4 y6 ;> e e e e cz D z1 C z3 C z4 z6 It is obvious from (H.67) that the Jacobian J e of the hexahedron is still a function of the local coordinates ; ; . Any analytical evaluation needs that the ; ; dependencies in J e vanish, i.e., all coefficients b and c of (H.69)mustbe 886 H Analytical Evaluation of Element Matrices and Vectors z e e xy (0,1,1) Ly y 3 1 (0,0,1) e e 3 1 L e zx (1,0,1) z Lx 2 e 2 6 (0,1,-1) yx 4 e 4 zy (0,0,-1) 5 6 e 5 e (1,0,-1) xz yz x Fig. H.6 Linear 6-node triangular prism with parallel top and bottom surfaces in the global and local coordinate system zero. This is valid for the triangular prism with parallel top and bottom surfaces as shown in Fig. H.6, where the following geometric conditions hold: e e e e e ıxy D x3 x1 D x6 x4 e e e ıxz D x1 x4 e e e e e ıyx D y2 y1 D y5 y4 e e e ıyz D y1 y4 e e e e e ızx D z2 z1 D z5 z4 (H.70) e e e e e ızy D z3 z1 D z6 z4 e e e e e Lx D x2 x1 D x5 x4 e e e e e Ly D y3 y1 D y6 y4 e e e Lz D z1 z4 Then, the Jacobain of the triangular prism with parallel top and bottom surfaces 0 1 e e e Lx ıyx ızx e B e e e C J D @ıxy Ly ızy A (H.71) e e e ıxz ıyz Lz becomes constant. The inverse Jacobian results 0 1 .Le Le ıe ıe /.ıe ıe ıe Le/.ıe ıe ıe Le / 1 B y z yz zy zx yz yx z yx zy zx y C .J e / 1 D @ e e e e e e e e e e e e A e .ıxzızy ıxyLz /.LxLz ızxıxz/.ıxyızx ızyLx/ (H.72) jJ j e e e e e e e e e e e e .ıxyıyz ıxzLy /.ıyxıxz ıyzLx/.LxLy ıyxıxy/ H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians 887 with the determinant e e e e e e e e e e e e e e e e e jJ jDLx.Ly Lz ıyzızy/ C ıxy.ızxıyz ıyxLz / C ıxz.ıyxızy ızxLy / D 2V (H.73) which is twice the volume of the triangular prism. The special case represents a triangular prism which is vertical and has horizontal top and bottom surfaces, so e e e e that ıxz D ıyz D ızx D ızy D 0. Then, it simplifies 0 1 Le ıe 0 B x yx C e @ e e A J D ıxy Ly 0 e 00Lz 0 1 Le Le ıe Le 0 1 B y z yx z C .J e/ 1 D @ e e e e A e ıxyLz LxLz 0 (H.74) jJ j e e e e 00LxLy ıxyıyx e e e e e e e e e jJ jD.LxLy ıxyıyx/Lz D 2A Lz D 2V e 1 e e e e 1 e e e e e e e e e where A D 2 .LxLy ıxyıyx/ D 2 Œx1 .y2 y3 / C x2 .y3 y1 / C x3 .y1 y2 / is the base area of the triangular prism. In using these relations the element matrices and vectors for the triangular prismatic element can be analytically evaluated in a similar way as done in Sect. H.3 for the tetrahedron. H.4.4 Linear Pyramidal Element with Parallelogram or Rectangular Base and Oblique Shape The linear 5-node pyramidal element as described in Appendix G (Table G.3d) has the following shape functions e 1 N1 D 4 Œ.1 /.1 / C 1 e 1 N2 D 4 Œ.1 C /.1 / 1 e 1 (H.75) N3 D 4 Œ.1 C /.1 C / C 1 e 1 N4 D 4 Œ.1 /.1 C / 1 e N5 D and the local derivatives e e e @N1 1 @N1 1 @N1 1 @ D 4 Œ.1 / 1 ; @ D 4 Œ.1 / 1 ; @ D 4 Œ1 .1 /2 e e e @N2 1 @N2 1 @N2 1 @ D 4 Œ.1 / 1 ; @ D 4 Œ.1 C / C 1 ; @ D 4 Œ1 C .1 /2 888 H Analytical Evaluation of Element Matrices and Vectors e e e @N3 1 @N3 1 @N3 1 @ D 4 Œ.1 C / C 1 ; @ D 4 Œ.1 C / C 1 ; @ D 4 Œ1 .1 /2 e e e @N4 1 @N4 1 @N4 1 @ D 4 Œ.1 C / C 1 ; @ D 4 Œ.1 / 1 ; @ D 4 Œ1 C .1 /2 e e e @N5 @N5 @N5 @ D 0; @ D 0; @ D 1 (H.76) for . 1 ; 1/ and .0 1/. It is for the pyramidal element, cf. (8.71) X5 X5 X5 e e e e e e x D NI xI ;yD NI yI ; z D NI zI (H.77) I D1 I D1 I D1 e e e where xI ;yI ; zI .I D 1;:::;5/ correspond to the Cartesian coordinates of the vertices (nodes) of the pyramid. Using (H.76)and(H.77)in(8.115) we obtain the Jacobian 0 1 a x C bx a y C by a z C bz B 1 1 1 C e 1 B C J D @ a x C bx a y C by a z C bz A (H.78) 4 1 1 1 a x C .1 /2 bx a y C .1 /2 by a z C .1 /2 bz where 9 e e e e a x D x C x C x x > 1 2 3 4 > e e e e > a y D y1 C y2 C y3 y4 > e e e e > a z D z C z C z z > 1 2 3 4 > e e e e > a x D x1 x2 C x3 C x4 = e e e e a y D y1 y2 C y3 C y4 > (H.79) e e e e > a z D z z C z C z > 1 2 3 4 > e e e e e > a x D x1 x2 x3 x4 C 4x5 > > a D ye ye ye ye C 4ye > y 1 2 3 4 5 ;> e e e e e a z D z1 z2 z3 z4 C 4z5 9 e e e e > bx D x1 x2 C x3 x4 = b D ye ye C ye ye (H.80) y 1 2 3 4 ;> e e e e bz D z1 z2 C z3 z4 It is obvious from (H.78) that the Jacobian J e of the pyramid is still a func- tion of the local coordinates ; ; . Any analytical evaluation needs that the e ; ; dependencies in J vanish, i.e., all coefficients bx, by and bz of (H.80)must be zero. This is valid for the pyramid with a parallelogram base and oblique shape as shown in Fig. H.7, where the following geometric conditions hold: H.4 Simplified Element Shapes in 2D and 3D to Attain Constant Jacobians 889 e xz z 5 y 5 (0,0,1) e e 4 yz zy 4 e 3 e Lz 3 (-1,1,0) (1,1,0) zx 1 2 1 2 e (-1,-1,0) Ly (1,-1,0) e e e xy Lx yx x Fig. H.7 Linear 5-node (oblique) pyramid with a parallelogram base in the global and local coordinate system e e e e e ıxy D x4 x1 D x3 x2 e e 1 e e e e ıxz D x5 4 .x1 C x2 C x3 C x4 / e e e e e ıyx D y2 y1 D y3 y4 e e 1 e e e e ıyz D y5 4 .y1 C y2 C y3 C y4 / e e e e e ızx D z2 z1 D z3 z4 (H.81) e e e e e ızy D z4 z1 D z3 z2 e e e e e Lx D x2 x1 D x3 x4 e e e e e Ly D y4 y1 D y3 y2 e e 1 e e e e Lz D z5 4 .z1 C z2 C z3 C z4/ Then, the Jacobain of the pyramid with a parallelogram base 0 1 1 Le 1 ıe 1 ıe B 2 x 2 yx 2 zxC e B 1 e 1 e 1 e C J D @ 2 ıxy 2 Ly 2 ızyA (H.82) e e e ıxz ıyz Lz becomes constant. The inverse Jacobian results 0 1 2.Le Le ıe ıe /2.ıe ıe ıe Le/.ıe ıe ıe Le / B y z yz zy zx yz yx z yx zy zx y C e 1 1 B C .J / D @2.ıe ıe ıe Le /2.Le Le ıe ıe /.ıe ıe ıe Le /A 4jJ ej xz zy xy z x z zx xz xy zx zy x e e e e e e e e e e e e 2.ıxyıyz ıxzLy /2.ıyxıxz ıyzLx/.LxLy ıyxıxy/ (H.83) 890 H Analytical Evaluation of Element Matrices and Vectors with the determinant e 1 e e e e e e e e e e e e e e e 3 e jJ jD 4 Lx.Ly Lz ıyzızy/ C ıxy.ızxıyz ıyxLz / C ıxz.ıyxızy ızxLy / D 4 V (H.84) e e e which is three quarters of the pyramid volume. We note that ıxz D ıyz D ızx D e ızy D 0 for a right pyramid, where the apex is aligned directly above the center of the base. The element matrices and vectors can now be analytically evaluated for the pyramid with a parallelogram or rectangular base and possibly oblique shape in a similar way as done in Sect. H.3 for the tetrahedron. Appendix I Parameters in Relation to Selected Problem Class, Medium Type and Dimension The following tables summarize the essential parameters (material/constitutive relationships and BC-related parameters) required for solving the governing flow, mass and heat transport equations as listed in Tables 3.7 and 3.9–3.11 for porous- media problems as well as in Tables 4.5–4.7 for fractured media (discrete feature) problems. They depend on both the problem class (flow, mass transport, heat transport, thermohaline problem), the type of medium (porous medium, fractured medium), the free-surface or variably saturated media formulation and the dimen- sion (3D, 2D (vertical, axisymmetric, horizontal)) of the chosen problem. Generally, the parameters can be functions of space and time, e.g., K11 D K11.x;t/ K22 D K22.x;t/ K33 D K33.x;t/ : (I.1) : out out ˚h D ˚h .x;t/ : : H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 891 © Springer-Verlag Berlin Heidelberg 2014 892 I Parameters in Relation to Selected Problem Class, Medium Type and Dimension I.1 Flow Table I.1 Parameters for 3D flow in porous media (with and without mass transport) Item Symbol Unit Default Reference(s) Axis-parallel anisotropy: 4 1 Conductivity [Kxx] K11 10 ms 1 Sect. 7.4.1 4 1 Conductivity [Kyy] K22 10 ms 1 4 1 Conductivity [Kzz] K33 10 ms 1 General and shaped-derived anisotropy (optional): m 4 1 Conductivity [K1m] K1 10 ms 1 Sects. 7.3.1 and 7.3.2 m 4 1 Conductivity [K2m] K2 10 ms 1 m 4 1 Conductivity [K3m] K3 10 ms 1 Eulerian angles only for general anisotropy: ı 0 Sect. 7.3.1 ı 0 ı 0 In(C)/out( )flow on top/bottom P10 4 md 1 0(9.4) 4 Density ratio ˛k 10 0(3.199)and(3.275) Specific yield "e 1 0:2 (3.296)and(9.4) 1 4 Specific storage coefficient So m 10 (4.25)and(9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 Table I.2 Parameters for 3D flow in porous media (at heat transport) Item Symbol Unit Default Reference(s) Axis-parallel anisotropy: 4 1 Conductivity [Kxx] K11 10 ms 1 Sect. 7.4.1 4 1 Conductivity [Kyy] K22 10 ms 1 4 1 Conductivity [Kzz] K33 10 ms 1 General and shaped-derived anisotropy (optional): m 4 1 Conductivity [K1m] K1 10 ms 1 Sects. 7.3.1 and 7.3.2 m 4 1 Conductivity [K2m] K2 10 ms 1 m 4 1 Conductivity [K3m] K3 10 ms 1 Eulerian angles only for general anisotropy: ı 0 Sect. 7.3.1 ı 0 ı 0 In(C)/out( )flow on top/bottom P10 4 md 1 0(9.4) (continued) I.1 Flow 893 Table I.2 (continued) Item Symbol Unit Default Reference(s) Constant thermal expansion coefficient: Expansion coefficient (constant) ˇ10 4 K 1 0(3.199) Variable thermal expansion (optional): Expansion coefficient (variable) ˇ.T / 10 4 K 1 0(C.4)and(C.8) Specific yield "e 1 0:2 (3.296)and(9.4) 1 4 Specific storage coefficient So m 10 (4.25)and(9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 Table I.3 Parameters for 3D flow in porous media (at thermohaline transport) Item Symbol Unit Default Reference(s) Axis-parallel anisotropy: 4 1 Conductivity [Kxx] K11 10 ms 1 Sect. 7.4.1 4 1 Conductivity [Kyy] K22 10 ms 1 4 1 Conductivity [Kzz] K33 10 ms 1 General and shaped-derived anisotropy (optional): m 4 1 Conductivity [K1m] K1 10 ms 1 Sects. 7.3.1 and 7.3.2 m 4 1 Conductivity [K2m] K2 10 ms 1 m 4 1 Conductivity [K3m] K3 10 ms 1 Eulerian angles only for general anisotropy: ı 0 Sect. 7.3.1 ı 0 ı 0 In(+)/out(-)flow on top/bottom P10 4 md 1 0(9.4) 4 Density ratio ˛k 10 0(3.199)and(3.275) Constant thermal expansion coefficient: Expansion coefficient (constant) ˇ10 4 K 1 0(3.199) Variable thermal expansion (optional): Expansion coefficient (variable) ˇ.T / 10 4 K 1 0(C.4)and(C.8) Specific yield "e 1 0:2 (3.296)and(9.4) 1 4 Specific storage coefficient So m 10 (4.25)and(9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 894 I Parameters in Relation to Selected Problem Class, Medium Type and Dimension Table I.4 Parameters for vertical or axisymmetric 2D flow in porous media (with and without mass transport) Item Symbol Unit Default Reference(s) 4 1 Conductivity [Kmax] Kmax 10 ms 1 Sect. 7.2 Anisotropy factor [Kmin/Kmax] aniso 11(7.9) Angle from C x-axis to Kmax ı 0Fig.7.2 4 Density ratio ˛k 10 0(3.199)and(3.275) 1 4 Specific storage coefficient So m 10 (4.25), (9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 Table I.5 Parameters for vertical or axisymmetric 2D flow in porous media (at heat transport) Item Symbol Unit Default Reference(s) 4 1 Conductivity [Kmax] Kmax 10 ms 1 Sect. 7.2 Anisotropy factor [Kmin/Kmax] aniso 11(7.9) Angle from C x-axis to Kmax ı 0Fig.7.2 Constant thermal expansion coefficient: Expansion coefficient (constant) ˇ10 4 K 1 0(3.199) Variable thermal expansion (optional): Expansion coefficient (variable) ˇ.T / 10 4 K 1 0(C.4)and(C.8) 1 4 Specific storage coefficient So m 10 (4.25)and(9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 Table I.6 Parameters for vertical or axisymmetric 2D flow in porous media (at thermohaline transport) Item Symbol Unit Default Reference(s) 4 1 Conductivity [Kmax] Kmax 10 ms 1 Sect. 7.2 Anisotropy factor [Kmin/Kmax] aniso 11(7.9) Angle from C x-axis to Kmax ı 0Fig.7.2 4 Density ratio ˛k 10 0(3.199)and(3.275) Constant thermal expansion coefficient: Expansion coefficient (constant) ˇ10 4 K 1 0(3.199) Variable thermal expansion (optional): Expansion coefficient (variable) ˇ.T / 10 4 K 1 0(C.4)and(C.8) 1 4 Specific storage coefficient So m 10 (4.25)and(9.2) 4 1 Source(C)/sink( ) Qh 10 d 0(9.2) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Unsaturated properties as listed in Table I.10 I.1 Flow 895 Table I.7 Parameters for horizontal 2D flow in porous media (confined aquifer) Item Symbol Unit Default Reference(s) 4 2 1 Transmissivity [Tmax] Tmax 10 m s 1 Sect. 7.2,(3.302) Anisotropy factor [Tmin/Tmax] aniso 11(7.9) Angle from C x-axis to Tmax ı 0Fig.7.2 4 Specific storage coefficient SNo 1 10 (3.299)and(9.11) 4 1 Source(C)/sink( ) QN h 10 md 0(9.12) N in 4 1 In-transfer coefficient ˚h 10 md 0(6.7)and(6.9) N out 4 1 Out-transfer coefficient ˚h 10 md 0(6.7)and(6.9) Table I.8 Parameters for horizontal 2D flow in porous media (unconfined aquifer) Item Symbol Unit Default Reference(s) 4 1 Conductivity [Kmax] Kmax 10 ms 1 Sect. 7.2 Anisotropy factor [Kmin/Kmax] aniso 11(7.9) Angle from C x-axis to Kmax ı 0Fig.7.2 Aquifer bottom elevation f B m 0 (3.283)and(9.7) Aquifer top elevation f T m 103 (3.283)and(9.31) Specific yield "e 1 0:2 (3.296)and(9.8) 1 4 Specific storage coefficient So m 10 (4.25)and(9.8) 4 1 Source(C)/sink( ) QN h 10 md 0(9.8) a in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) b out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) a N in 4 1 For integral third kind BC (Sect. 6.5.4) depth-integrated ˚h Œ10 md (6.9)isused b N out 4 1 For integral third kind BC (Sect. 6.5.4) depth-integrated ˚h Œ10 md (6.9)isused Table I.9 Parameters for flow in fractured media (discrete features) Item Symbol Unit Default Reference(s) 1D fracture type: Cross-sectional area A m2 1Table4.8 2D fracture type: Thickness B m 10 3 Table 4.8 Darcy flux law: Conductivity K10 4 ms 1 1(4.38), Table 4.5 Hagen-Poiseuille flux law: Hydraulic aperture b m 10 3 Table 4.9 Manning-Strickler flux law: Roughness coefficient M m1=3 s 1 30 Tables 4.9 and 4.4 Specific yield "e 1 1 (4.29), Table 4.5 1 4 Specific storage coefficient So m 10 (4.25), Table 4.5 Source(C)/sink( ) Q10 4 d 1 0Table4.5 4 Density ratio ˛k 10 0(4.63)and(3.265) (continued) 896 I Parameters in Relation to Selected Problem Class, Medium Type and Dimension Table I.9 (continued) Item Symbol Unit Default Reference(s) Constant thermal expansion coefficient: Expansion coefficient (constant) ˇ10 4 K 1 0(4.63)and(3.265) Variable thermal expansion (optional): Expansion coefficient (variable) ˇ.T / 10 4 K 1 0(4.63)and(C.4) in 4 1 In-transfer coefficient ˚h 10 d 0(6.7)and(6.8) out 4 1 Out-transfer coefficient ˚h 10 d 0(6.7)and(6.8) Table I.10 Parameters for unsaturated porous media (analytical parametric models summarized in Table D.1) (Spline approximations alternatively exist to input experimental sample points for the parametric curves of saturation s and relative permeability kr , see Sect. D.4 of Appendix D) Item Symbol Unit Default Reference(s) Porosity " 1 0:3 (3.219) Residual saturation sr 1 0:0025 (D.1) Maximum saturation ss 1 1:0 (D.1) van Genuchten-Mualem parametric model: Fitting coefficient ˛ m 1 4:1 (D.3) Fitting exponent n 1 1:964 (D.3) Modified van Genuchten parametric model: Fitting coefficient ˛ m 1 4:1 (D.3) Fitting exponent n 1 1:964 (D.3) Fitting exponent m 1 0:491 (D.3) Fitting exponent ı 1 3:4 (D.32) Brooks-Corey parametric model: Fitting coefficient ˛ m 1 4:1 (D.7) Fitting exponent n 1 1:964 (D.7) Fitting exponent ı 1 3:4 (D.29) Haverkamp parametric model: Fitting coefficient ˛ 1 4:1 (D.11) Fitting exponent ˇ 1 1:964 (D.11) Fitting coefficient A 1 3:4 (D.35) Fitting exponent B 1 0:491 (D.35) Exponential parametric model: Sorptive number ˛ m 1 3:4 (D.15)and(D.38) Air-entry pressure head a m0 (D.15)and(D.38) Linear parametric model: Fringe pressure head c m 4:1 (D.19)and(D.41) Air-entry pressure head a m0 (D.19)and(D.41) Hysteresisa a Parametric models under hysteretic conditions require a double dataset for drying and wetting curves I.2 Mass Transport 897 I.2 Mass Transport Table I.11 Parameters for 3D, vertical or axisymmetric mass transport of species k at liquid phase l in porous media Item Symbol Unit Default Reference(s) Porosity " 1 0:3 (3.219) Henry sorption: Sorption coefficient k 10(5.65), Table 3.8 Freundlich sorption: 1 1 bk Sorption coefficient bk (mg l ) 0(5.65), Table 3.8 Sorption exponent bk 10(5.65), Table 3.8 Langmuir sorption: Numerator sorption coefficient kk 10(5.65), Table 3.8 1 Denominator sorption coefficient kk l(mg) 0(5.65), Table 3.8 9 2 1 Molecular diffusion Dk 10 m s 1(3.184), Table 3.7 Longitudinal dispersivity ˇL m 5 (3.184), Table 3.7 Transverse dispersivity ˇT m 0:5 (3.184), Table 3.7 Nonlinear (non-Fickian) dispersion: 2 1 HC dispersion coefficient =H m dg 0 (3.272), Table 3.7 Chemical reactions for single-species solute transport: First-order decay: 4 1 Decay rate #k 10 s 0 (5.73), Table 3.7 Michaelis-Menten: 4 1 1 Maximum growth rate vm 10 mg l s 0 (5.90) 1 Half-saturation constant Km mg l 0 (5.90) Chemical reactions for multispecies mass transport: 4 1 Rate constant kk 10 s 0 Sect. 5.5 ˙ ) Reaction kinetics editor 3 1 Source(C)/sink( ) Qk gm d 0 a in 1 In-transfer coefficient ˚kC md 0(6.22)and(6.24) b out 1 Out-transfer coefficient ˚kC md 0(6.22)and(6.24) a in For the divergence form ˚kC is input b out For the divergence form ˚kC is input 898 I Parameters in Relation to Selected Problem Class, Medium Type and Dimension Table I.12 Parameters for 2D horizontal mass transport of species k at liquid phase l in porous media (unconfined and confined aquifer) Item Symbol Unit Default Reference(s) Confined aquifer: Aquifer thickness B m 1:0 (3.283) Porosity " 1 0:3 (3.219) Henry sorption: Sorption coefficient k 10(5.65), Table 3.8 Freundlich sorption: 1 1 bk Sorption coefficient bk (mg l ) 0(5.65), Table 3.8 Sorption exponent bk 10(5.65), Table 3.8 Langmuir sorption: Numerator sorption coefficient kk 10(5.65), Table 3.8 1 Denominator sorption coefficient kk l(mg) 0(5.65), Table 3.8 9 2 1 Molecular diffusion Dk 10 m s 1(3.184), Table 3.10 Longitudinal dispersivity ˇL m 5 (3.184), Table 3.10 Transverse dispersivity ˇT m 0:5 (3.184), Table 3.10 Nonlinear (non-Fickian) dispersion: 1 HC dispersion coefficient =N H mdg 0 (3.272), Table 3.10 Chemical reactions for single-species solute transport: First-order decay: 4 1 Decay rate #k 10 s 0 (5.73), Table 3.10 Michaelis-Menten: 4 1 1 Maximum growth rate vm 10 mg l s 0 (5.90) 1 Half-saturation constant Km mg l 0 (5.90) Chemical reactions for multispecies mass transport: 4 1 Rate constant kk 10 s 0 Sect. 5.5 ˙ ) Reaction kinetics editor 2 1 Source(C)/sink( ) QN k gm d 0 a in 1 In-transfer coefficient ˚kC md 0(6.22)and(6.24) b out 1 Out-transfer coefficient ˚kC md 0(6.22)and(6.24) a in For the divergence form ˚kC is input. For confined conditions or integral third kind BC N in 2 1 (Sect. 6.5.4) ˚kC Œm d (6.23)and(6.25)isused b out For the divergence form ˚kC is input. For confined conditions or integral third kind BC N out 2 1 (Sect. 6.5.4) ˚kC Œm d (6.23)and(6.25)isused Table I.13 Parameters for 3D, vertical or axisymmetric mass transport of species k at solid phase s in porous media Item Symbol Unit Default Reference(s) Solid volume fraction "s 1 0:7 (3.219) First-order decay: 4 1 Decay rate #k 10 s 0 (5.73), Table 3.7 (continued) I.2 Mass Transport 899 Table I.13 (continued) Item Symbol Unit Default Reference(s) Chemical reactions for multispecies mass transport: 4 1 Rate constant kk 10 s 0 Sect. 5.5 ˙ ) Reaction kinetics editor 3 1 Source(C)/sink( ) Qk gm d 0 Table I.14 Parameters for 2D horizontal mass transport of species k at solid phase s in porous media (unconfined and confined aquifer) Item Symbol Unit Default Reference(s) Confined aquifer: Aquifer thickness B m 1:0 (3.283) Solid volume fraction "s 1 0:7 (3.219) First-order decay: 4 1 Decay rate #k 10 s 0 (5.73), Table 3.7 Chemical reactions for multispecies mass transport: 4 1 Rate constant kk 10 s 0 Sect. 5.5 ˙ ) Reaction kinetics editor 3 1 Source(C)/sink( ) Qk gm d 0 Table I.15 Parameters for mass transport of species k at liquid phase l in fractured media (discrete features) Item Symbol Unit Default Reference(s) 1D fracture type: Cross-sectional area A m2 1Table4.8 2D fracture type: Thickness B m 10 3 Table 4.8 Darcy flux law only: Porosity " 1 1:0 (4.7) Henry sorption coefficient k 10(5.65), Table 3.8 9 2 1 Molecular diffusion Dk 10 m s 1(4.67), Table 4.6 Longitudinal dispersivity ˇL m 5 (4.68), Table 4.6 Transverse dispersivity ˇT m 0:5 (4.68), Table 4.6 Chemical reactions for single-species solute transport: 4 1 Decay rate #k 10 s 0 (4.66), Table 4.6 Chemical reactions for multispecies mass transport: 4 1 Rate constant kk 10 s 0 Sect. 5.5 ˙ ) Reaction kinetics editor 3 1 Source(C)/sink( ) Qk gm d 0 a in 1 In-transfer coefficient ˚kC md 0(6.22)and(6.24) b out 1 Out-transfer coefficient ˚kC md 0(6.22)and(6.24) a in For the divergence form ˚kC is input b out For the divergence form ˚kC is input 900 I Parameters in Relation to Selected Problem Class, Medium Type and Dimension I.3 Heat Transport Table I.16 Parameters for 3D, vertical or axisymmetric heat transport in porous media Item Symbol Unit Default Reference(s) Porosity " 1 0:3 (3.219) Volumetric heat capacity of fluid c 106 Jm 3 K 1 4:2a (3.208), Table 3.7 Volumetric heat capacity of solid s cs 106 Jm 3 K 1 2:52b (3.208), Table 3.7 Heat conductivity of fluid Jm 1 s 1 K 1 0:65 (3.172), Table 3.7 Heat conductivity of solid s Jm 1 s 1 K 1 3 (3.172), Table 3.7 3D problems only: s s Anisotropy factor Œ zz= xx;yy aniso 1 1 (7.26) Longitudinal dispersivity ˇL m 5 (3.238), Table 3.7 Transverse dispersivity ˇT m 0:5 (3.238), Table 3.7 Source(C)/sink( )offluid H ? Jm 3 d 1 0(13.3), Table 3.7 s ? 3 1 Source(C)/sink( ) of solid Hs Jm d 0(13.3), Table 3.7 c in 2 1 1 In-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) d out 2 1 1 Out-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) a D 1;000 kg m 3, c D 4;200 Jkg 1 K 1 b s D 2;650 kg m 3, cs D 950 Jkg 1 K 1 c in For the divergence form ˚T is input d out For the divergence form ˚T is input Table I.17 Parameters for 2D horizontal heat transport in porous media (unconfined and confined aquifer) Item Symbol Unit Default Reference(s) Confined aquifer: Aquifer thickness B m 1:0 (3.283) Porosity " 1 0:3 (3.219) Volumetric heat capacity of fluid c 106 Jm 3 K 1 4:2a (3.208), Table 3.10 Volumetric heat capacity of solid scs 106 Jm 3 K 1 2:52b (3.208), Table 3.10 Heat conductivity of fluid Jm 1 s 1 K 1 0:65 (3.172), Table 3.10 Heat conductivity of solid s Jm 1 s 1 K 1 3 (3.172), Table 3.10 Longitudinal dispersivity ˇL m 5 (3.238), Table 3.10 Transverse dispersivity ˇT m 0:5 (3.238), Table 3.10 Source(C)/sink( )offluid B H ? Jm 2 d 1 0Table3.10 s ? 2 1 Source(C)/sink( ) of solid B Hs Jm d 0Table3.10 c in 2 1 1 In-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) d out 2 1 1 Out-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) a D 1;000 kg m 3, c D 4;200 Jkg 1 K 1 b s D 2;650 kg m 3, cs D 950 Jkg 1 K 1 c in For the divergence form ˚T is input. For confined conditions or integral third kind BC N in 1 1 1 (Sect. 6.5.4) depth-integrated ˚T [J m d K ](6.41)and(6.43)isused d out For the divergence form ˚T is input. For confined conditions or integral third kind BC N out 1 1 1 (Sect. 6.5.4) depth-integrated ˚T [J m d K ](6.41)and(6.43)isused I.3 Heat Transport 901 Table I.18 Parameters for heat transport in fractured media (discrete features) Item Symbol Unit Default Reference(s) 1D fracture type: Cross-sectional area A m2 1Table4.8 2D fracture type: Thickness B m 10 3 Table 4.8 Darcy flux law only: Porosity " 1 1:0 (4.7) Volumetric heat capacity of fluid c 106 Jm 3 K 1 4:2a (4.75), Table 4.7 Darcy flux law only: Volumetric heat capacity of solid scs 106 Jm 3 K 1 2:52b (4.75), Table 4.7 Heat conductivity of fluid Jm 1 s 1 K 1 0:65 (4.76), Table 4.7 Darcy flux law only: Heat conductivity of solid s Jm 1 s 1 K 1 3 (4.76), Table 4.7 Longitudinal dispersivity ˇL m 5 (4.68), Table 4.7 Transverse dispersivity ˇT m 0:5 (4.68), Table 4.7 Source(C)/sink( )offluid H Jm 3 d 1 0(4.74), Table 4.7 Darcy flux law only: s 3 1 Source(C)/sink( ) of solid Hs Jm d 0(4.74), Table 4.7 c in 2 1 1 In-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) d out 2 1 1 Out-transfer coefficient ˚T Jm d K 0(6.40)and(6.42) a D 1;000 kg m 3, c D 4;200 Jkg 1 K 1 b s D 2;650 kg m 3, cs D 950 Jkg 1 K 1 c in For the divergence form ˚T is input d out For the divergence form ˚T is input Appendix J Elements of PVST for Solving the Mixed ψ s Based Form of Richards’ Equation J.1 Jacobian J for the Pressure Head ψ as Primary Variable The derivative of the residual (10.64) with respect to the pressure head ψnC1 at the new time plane n C 1 and the current iterate yields the following expression .i;j;l D 1;:::;NP/: @R i;nC1.ψnC1; snC1/ Jij .ψnC1; snC1/ D @ j;nC1 1 2 3 4 5 D Jij C Jij C Jij C Jij Jij Oij.snC1/ D C Dij.ψnC1/C tn B @s il l;nC1 C t @ (J.1) n j;nC 1 @O .ψ / l;n il nC1 l;nC1 1 P 1 l;n C @ j;nC1 tn @Dil.snC1/ l;nC1 @ j;nC1 @Fi .snC1/ @ j;nC1 where the matrices O, B, D and the RHS-vector F are defined in (10.61). The partial Jacobians in (J.1) are obtained as follows 1 Oij.snC1/ Jij D C Dij.ψnC1/ (J.2) tn H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 903 © Springer-Verlag Berlin Heidelberg 2014 904 J Elements of PVST for Solving the Mixed ψ s Based Form of Richards’ . . . X Z 2 e 1 e Jij D Ni " Cj;nC1ıij d˝ e t e ˝ n (J.3) no summation over i and j Z X 3 e j;nC1 j;n 1 P e Jij D Ni So Cj;nC1ıij 1 j;n d˝ e t e ˝ n (J.4) no summation over i and j Z X 4 e e e Jij D rNi K f Nj Gj;nC1 rNl l;nC1 d˝ e e ˝ (J.5) no summation over j Z X 5 e e e e Jij D rNi K f Nj .1 C /Gj;nC1 e d˝ C e e Z ˝ re e (J.6) Ni Nj Gj;nC1qh d re N no summation over j with @s. / j;nC1 Cj;nC1 D (J.7) @ j;nC1 and @k . / r j;nC1 Gj;nC1 D (J.8) @ j;nC1 The derivatives Cj;nC1 and Gj;nC1 are given functions which can be evaluated either analytically from the parametric models summarized in Appendix D or numerically from chord slope approximations given in Sect. J.3 for the known variables s and at the iterate , the global node j and the time plane n C 1. Here, CnC1 is the moisture capacity function known from the standard unsaturated flow modeling. J.2 Jacobian J s for the Saturation s as Primary Variable The derivative of the residual (10.64) with respect to the saturation snC1 at the new time plane n C 1 and the current iterate yields the following expression .i;j;l D 1;:::;NP/: J.2 Jacobian J s for the Saturation s as Primary Variable 905 s @R i;nC1.ψnC1; snC1/ Jij .ψnC1; snC1/ D @sj;nC1 s1 s2 s3 s4 s5 D Jij C Jij C Jij C Jij Jij @ Oil.snC1/ l;nC1 D C Dil.ψnC1/ C tn @sj;nC1 B ij C t (J.9) n @O .ψ / il nC1 l;nC1 l;n 1 P 1 l;n C @sj;nC1 tn @Dil.snC1/ l;nC1 @sj;nC1 @Fi .snC1/ @sj;nC1 where the matrices O, B, D and the RHS-vector F are defined in (10.61). The partial Jacobians in (J.9) are obtained as follows s1 Oij.snC1/ 1 Jij D C Dij.ψnC1/ Cj;nC1 tn (J.10) no summation over j s2 Bij Jij D (J.11) tn Z X s3 e j;nC1 j;n 1 P e Jij D Ni So ıij 1 j;n d˝ e t e ˝ n (J.12) no summation over i and j Z X s4 e e 1 e Jij D rNi K f Nj Gj;nC1 Cj;nC1 rNl l;nC1 d˝ e e ˝ (J.13) no summation over j Z X s5 e e e 1 e Jij D rNi K f Nj .1 C /Gj;nC1 Cj;nC1 e d˝ C e e Z ˝ 1 re e (J.14) Ni Nj Gj;nC1 Cj;nC1qh d re N no summation over j 906 J Elements of PVST for Solving the Mixed ψ s Based Form of Richards’ . . . with the inverse moisture capacity @ .s / 1 j;nC1 1 Cj;nC1 D D (J.15) @sj;nC1 Cj;nC1 which can be evaluated either analytically from the parametric models summarized in Appendix D or numerically by using chord slope approximations given in Sect. J.3. Notice, it is necessary to use the pressure head instead of the hydraulic 1 head h to evaluate the moisture capacity functions Cj;nC1 and Cj;nC1. Actually, Cj;nC1 can also be expressed by h since @s=@ D @s=@h, but the inverse moisture 1 capacity Cj;nC1 is not simply invertible for h because @ =@s D @h=@s @z=@s. It is important to note that in the derivation of the residual R with respect to the saturation s no assumptions are implied for spatial derivations of inherent parameters. Thus, (J.9) is also valid for inhomogeneous porous media. This is an essential difference to transformations for developing the common s form of the Richards’ equation as discussed in Sect. 10.3 or the Kirchhoff transformation as introduced in Sect. 10.4. J.3 Chord Slope Approximations of Saturation Derivatives In contrast to analytical derivatives in form of the moisture capacity CnC1 (J.7) 1 and its inverse CnC1 (J.15) chord slope approximations can be useful and effective. Within the GLS predictor-corrector one-step Newton scheme the derivative terms p p are evaluated by using the predicted solutions ψnC1 and snC1 for the current time plane n C 1. For instance, a simple first-order accurate finite difference approximation of CnC1 would lead to si;nC1 si;n Ci;nC1 D (J.16) i;nC1 i;n For the GLS predictor-corrector one-step Newton technique only one iteration per time step is employed. Thus, the iterates indicated by the superscript can be replaced by the predictors denoted by the superscript P . This yields p p si;nC1 si;n Ci;nC1 D p (J.17) i;nC1 i;n It can be easily seen that this derivative is nothing more than the quotient of the acceleration vectors for the saturation and the pressure head p sPi;n Ci;nC1 D (J.18) Pi;n J.3 Chord Slope Approximations of Saturation Derivatives 907 which represents a chord slope approximation of the saturation derivative applied to the first-order accurate BE scheme. A corresponding second-order accurate chord slope approximation suited for the TR scheme can be similarly derived [141]: 2 p 2 t .s si;n/ C t .si;n si;n 1/ Cp D n 1 i;nC1 n (J.19) i;nC1 2 p 2 tn 1. i;nC1 i;n/ C tn . i;n i;n 1/ 1p The chord slope approximations for the inverse moisture capacity Ci;nC1 yield equivalent expressions. Note here that limitations exist for the chord slope approximations if the denominator of (J.18)and(J.19) tends to zero. Practically, below an absolute minimum difference tolerance (typically we use 10 18 for the pressure head and 10 8 for the saturation) the evaluation of the derivative becomes an analytical (exact) procedure. Appendix K Integral Functions of the Frolkovic-Knabnerˇ Algorithm (FKA) K.1 Transformations in Local Coordinates The FKA computations are performed on generalized (local) coordinates η (8.68). The mapping from the local coordinates η to the global ones x is given by (8.114) and requires the transformation Jacobian J e,(8.115)–(8.117), for an element e. Using this transformation we have 0 1 @ B @ C @ @ A e e 1 r. ; ; / D @ D J r; rD.J / r. ; ; / (K.1) @ @ for the derivatives and 0 1 e @ A e e 1 e. ; ; / D e D J e; e D .J / e. ; ; / (K.2) e for the gravitational unit vector (3.261). Using these relationships the equivalent formulation of the Darcy velocity q D kr Kf .rh C e/ in local coordinates is given by e 1 e q D kr Kf .J / .r. ; ; /h C J e/ (K.3) or e 1 q D kr Kf .J / .r. ; ; /h C e. ; ; // (K.4) H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 909 © Springer-Verlag Berlin Heidelberg 2014 910 K Integral Functions of the Frolkovic-Knabnerˇ Algorithm (FKA) K.2 FKA Formulation Introducing the following integral functions [177, 312] for each element e Z e e H D H . ; ; / D . ; ; /e . ; ; /d Z0 e e H D H . ; ; / D . ; ; /e . ; ; /d (K.5) Z0 e e H D H . ; ; / D . ; ; /e . ; ; /d 0 and since 0 1 e @H B @ C B e C B @H C D e (K.6) @ @ A . ; ; / e @H @ we can write the Darcy velocity (K.4) in an equivalent form on element level 0 1 @ e @ .h C H / B C e 1 @ @ e A q D kr Kf .J / @ .h C H / (K.7) @ e @ .h C H / e e e The integral functions H , H , H allow us to obtain the same spatial variability for both the h term and the term. The consistency for (K.7) in the definition of (11.60) can be proved. Assuming the gravity acts in the z direction, i.e., .x; y; z/ D .x0;y0; z/, we can write Z Z z. ; ; / e H D .x0;y0; z. ; ; //e d D ez .x0;y0; /d (K.8) 0 z0 e e where .x0;y0; z0/ D .x.0; 0; 0/; y.0; 0; 0/; z.0;0;0//and similarly for H and H . e e e In the FEM the functions h, H , H , H are interpolated by their nodal basis functions: XNBN e e h D NJ . ; ; /hJ J D1 XNBN e e e H D NJ . ; ; /H J J D1 e e e K.3 The Nodal Quantities H J ;H J ;H J of the Integral Functions 911 XNBN e e e H D NJ . ; ; /H J J D1 XNBN e e e H D NJ . ; ; /H J (K.9) J D1 and we obtain the velocity (K.7) in the discretized formulation 0 1 e e @ e N .hJ C H J / NJ . ; ; / XBN B @ C e 1 @ e e @ e A q D kr Kf .J / .hJ C H J / @ NJ . ; ; / (K.10) J e e @ e .hJ C H J / @ NJ . ; ; / which represents a fully consistent approximation of the Darcy velocities. We solve e e e (K.10) for given hydraulic heads h and the values of the H ;H ;H functions e e e at the nodes J . The nodal quantities H J ;H J ;H J are dependent on the finite element types and will be evaluated next for linear elements in two and three dimensions. In doing this, the buoyancy coefficient in the gravity term is interpolated according to XNBN e e D NJ . ; ; / J (K.11) J D1 e where J are the buoyancy coefficient values at node J .Using from (11.2)the finite element expansion of the buoyancy coefficient reads X X X ˛k e e e e e D . N C Ck0/ ˇ.T /. N T T0/ (K.12) Cks Ck0 J kJ J J k J J or X e ˛k e e e D .C Ck0/ ˇ.T /.T T0/ (K.13) J Cks Ck0 kJ J k in relation to the expansion (K.11) e e e K.3 The Nodal Quantities H J ;H J ;H J of the Integral Functions K.3.1 2D Linear Triangular Element The linear triangle is described in Sect. G.2 of Appendix G, where its shape functions and its local derivatives are given in Table G.2a. Its Jacobian J e (H.14) appears independent of the local coordinates . ; / and the gravitational unit vector e. ; / in the local coordinates (K.2) is also constant. Accordingly, we can write 912 K Integral Functions of the Frolkovic-Knabnerˇ Algorithm (FKA) Z Z e H D . ; /e . ; /d D e . ; /d 0 Z 0 X3 D e N e. ; / e d D J J (K.14) Z0 J D1 e e e D e .1 / 1 C 2 C 3 d 0 2 e 2 e e D e . 2 / 1 C 2 2 C 3 e e e and similarly for H . From the integrals we find the nodal values for H and H as e e H .0; 0/ D H 1 D 0 e e 1 e e H .1; 0/ D H 2 D 2 e . 1 C 2/ (K.15) e e H .0; 1/ D H 3 D 0 e e H .0; 0/ D H 1 D 0 e e H .1; 0/ D H 2 D 0 (K.16) e e 1 e e H .0; 1/ D H 3 D 2 e . 1 C 3/ Now we can express the gravity term (K.6) in local coordinates as 0 1 e @H @ @ A e. ; / D e @H @ e e e ! @N1 e @N2 e @N3 e @ H 1 C @ H 2 C @ H 3 D e e e (K.17) @N1 e @N2 e @N3 e H 1 C H 2 C H 3 @ @ ! @ e . e C e/ D 1 1 2 2 e e e . 1 C 3/ representing a consistent approximation in which the density is appropriately averaged in the gravitational direction. K.3.2 2D Linear Quadrilateral Element The linear quadrilateral is described in Sect. G.2 of Appendix G, where its shape functions and its local derivatives are given in Table G.2b. While for this element the Jacobian J e (H.49) is in general space-dependent, the gravitational unit vector e. ; / in local coordinates (K.2) takes the special form e e . / D (K.18) e e . / e e e K.3 The Nodal Quantities H J ;H J ;H J of the Integral Functions 913 Similarly to the above triangular element, we can compute the integral functions e e H , H at the corner nodes J for the linear quadrilateral element as e e 1 e e H . 1; 1/ D H 1 D 4 e . 1/.3 1 C 2/ e e 1 e e H .1; 1/ D H 2 D 4 e . 1/. 1 C 3 2/ e e 1 e e (K.19) H .1; 1/ D H 3 D 4 e .1/.3 3 C 4/ e e 1 e e H . 1; 1/ D H 4 D 4 e .1/. 3 C 3 4/ e e 1 e e H . 1; 1/ D H 1 D 4 e . 1/.3 1 C 4/ e e 1 e e H .1; 1/ D H 2 D 4 e .1/.3 2 C 3/ e e 1 e e (K.20) H .1; 1/ D H 3 D 4 e .1/. 2 C 3 3/ e e 1 e e H . 1; 1/ D H 4 D 4 e . 1/. 1 C 3 4/ The gravity term (K.6) written in local coordinates yields 0 1 e @H @ @ A e. ; / D e @H @ e e e e ! @N1 H e @N2 H e @N3 H e @N4 H e @ 1 C @ 2 C @ 3 C @ 4 (K.21) D e e e e @N1 e @N2 e @N3 e @N4 e H 1 C H 2 C H 3 C H 4 @ @ @ @ e e e e 1 e . 1/.1 /. 1 C 2/ C e .1/.1 C /. 3 C 4/ D 4 e e e e e . 1/.1 /. 1 C 4/ C e .1/.1 C /. 2 C 3/ For the linear quadrilateral element the consistent approximation (K.21) can be recognized as the consistent formulation previously introduced by Voss [550], where the gravity term is averaged in a directional manner, so for instance e e 1 e .1/. 3 C 4/ e. D1; D1/ D 2 e e (K.22) e .1/. 2 C 3/ K.3.3 3D Linear Pentahedral (Triangular Prismatic) Element The linear pentahedral (triangular prismatic) element is described in Sect. G.3 of Appendix G, where its shape functions and its local derivatives are given in Table G.3b. Since the Jacobian J e (H.67) is space-dependent, the gravitational unit vector e. ; ; / in local coordinates (K.2)is 0 1 0 1 e e . / @ A @ A e D e . / (K.23) e e . ; / 914 K Integral Functions of the Frolkovic-Knabnerˇ Algorithm (FKA) e e e The integral functions H , H , H at the corner nodes J for the linear pentahedral element are then e e H .0;0;1/D H 1 D 0 e e 1 e e H .1;0;1/D H 2 D 2 e .1/. 1 C 2/ e e H .0;1;1/D H 3 D 0 e e (K.24) H .0; 0; 1/ D H 4 D 0 e e 1 e e H .1; 0; 1/ D H 5 D 2 e . 1/. 5 C 6/ e e H .0; 1; 1/ D H 6 D 0 e e H .0;0;1/D H 1 D 0 e e H .1;0;1/D H 2 D 0 e e 1 e e H .0;1;1/D H 3 D 2 e .1/. 1 C 3/ e e (K.25) H .0; 0; 1/ D H 4 D 0 e e H .1; 0; 1/ D H 5 D 0 e e 1 e e H .0; 1; 1/ D H 6 D 2 e . 1/. 4 C 6/ e e 1 e e H .0;0;1/D H 1 D 4 e .0; 0/.3 1 C 4/ e e 1 e e H .1;0;1/D H 2 D 4 e .1; 0/.3 2 C 5/ e e 1 e e H .0;1;1/D H 3 D 4 e .0; 1/.3 3 C 6/ e e 1 e e (K.26) H .0; 0; 1/ D H 4 D 4 e .0; 0/. 1 C 3 4/ e e 1 e e H .1; 0; 1/ D H 5 D 4 e .1; 0/. 2 C 3 5/ e e 1 e e H .0; 1; 1/ D H 6 D 4 e .0; 1/. 3 C 3 6/ The gravity term (K.6) written in local coordinates gives 0 e 1 @H B @ C B e C e D B @H C . ; / @ @ A e @H @ 0 e e e e e e 1 @N1 e @N2 e @N3 e @N4 e @N5 e @N6 e @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5 C @ H 6 B e e e e e e C B @N1 e @N2 e @N3 e @N4 e @N5 e @N6 e C D @ @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5 C @ H 6A e e e e e e @N1 e @N2 e @N3 e @N4 e @N5 e @N6 e @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5 C @ H 6 0 1 e e e e e .1/.1 C /. 1 C 2/ C e . 1/.1 /. 5 C 6/ 1 @ e e e e A D 4 e .1/.1 C /. 1 C 3/ C e . 1/.1 /. 4 C 6/ e e e e e e 2Œe .0; 0/.1 /. 1 C 4/ C e .1; 0/ . 2 C 5/ C e .0; 1/ . 3 C 6/ (K.27) e e e K.3 The Nodal Quantities H J ;H J ;H J of the Integral Functions 915 K.3.4 3D Linear Hexahedral (Brick) Element The linear hexahedral (brick) element is described in Sect. G.3 of Appendix G, where its shape functions and its local derivatives are given in Table G.3c. The e Jacobian J (H.57) is space-dependent and the gravitational unit vector e. ; ; / in local coordinates (K.2) reads 0 1 0 1 e e . ; / @ A @ A e D e . ; / (K.28) e e . ; / e e e The integral functions H , H , H at the corner nodes J for the linear hexahedral element can be derived as e e 1 e e H . 1; 1; 1/ D H 1 D 4 e . 1; 1/.3 1 C 2/ e e 1 e e H .1; 1; 1/ D H 2 D 4 e . 1; 1/. 1 C 3 2/ e e 1 e e H .1;1;1/D H 3 D 4 e .1; 1/.3 3 C 4/ e e 1 e e H . 1; 1; 1/ D H 4 D 4 e .1; 1/. 3 C 3 4/ e e 1 e e (K.29) H . 1; 1; 1/ D H 5 D 4 e . 1; 1/.3 5 C 6/ e e 1 e e H .1; 1; 1/ D H 6 D 4 e . 1; 1/. 5 C 3 6/ e e 1 e e H .1; 1; 1/ D H 7 D 4 e .1; 1/.3 7 C 8/ e e 1 e e H . 1; 1; 1/ D H 8 D 4 e .1; 1/. 7 C 3 8/ e e 1 e e H . 1; 1; 1/ D H 1 D 4 e . 1; 1/.3 1 C 4/ e e 1 e e H .1; 1; 1/ D H 2 D 4 e .1; 1/.3 2 C 3/ e e 1 e e H .1;1;1/D H 3 D 4 e .1; 1/. 2 C 3 3/ e e 1 e e H . 1; 1; 1/ D H 4 D 4 e . 1; 1/. 1 C 3 4/ e e 1 e e (K.30) H . 1; 1; 1/ D H 5 D 4 e . 1; 1/.3 5 C 8/ e e 1 e e H .1; 1; 1/ D H 6 D 4 e .1; 1/.3 6 C 7/ e e 1 e e H .1; 1; 1/ D H 7 D 4 e .1; 1/. 6 C 3 7/ e e 1 e e H . 1; 1; 1/ D H 8 D 4 e . 1; 1/. 5 C 3 8/ e e 1 e e H . 1; 1; 1/ D H 1 D 4 e . 1; 1/.3 1 C 5/ e e 1 e e H .1; 1; 1/ D H 2 D 4 e .1; 1/.3 2 C 6/ e e 1 e e H .1;1;1/D H 3 D 4 e .1; 1/.3 3 C 7/ e e 1 e e H . 1; 1; 1/ D H 4 D 4 e . 1; 1/.3 4 C 8/ e e 1 e e (K.31) H . 1; 1; 1/ D H 5 D 4 e . 1; 1/. 1 C 3 5/ e e 1 e e H .1; 1; 1/ D H 6 D 4 e .1; 1/. 2 C 3 6/ e e 1 e e H .1; 1; 1/ D H 7 D 4 e .1; 1/. 3 C 3 7/ e e 1 e e H . 1; 1; 1/ D H 8 D 4 e . 1; 1/. 4 C 3 8/ 916 K Integral Functions of the Frolkovic-Knabnerˇ Algorithm (FKA) For the hexahedral (brick) element the consistent formulation of the gravity term in form of the integral functions, (K.29)–(K.31), is equivalent to the formulation given by Leijnse [336]. This should be exemplified for the e component of the gravity term: P e 8 @NJ e 1 e e e D J D1 @ H J D 8 e . 1; 1/.1 /.1 C /. 1 C 2/C e .1; 1/.1 C /.1 C /. e C e/C 3 4 (K.32) e e e . 1; 1/.1 /.1 /. 5 C 6 /C e e e .1; 1/.1 C /.1 /. 7 C 8/ K.3.5 3D Linear Pyramidal Element The linear pyramidal element is described in Sect. G.3 of Appendix G,where its shape functions and its local derivatives are given in Table G.3d. Since the e Jacobian J (H.78) is space-dependent, the gravitational unit vector e. ; ; / in local coordinates (K.2)is 0 1 0 1 e e . ; / @ A @ A e D e . ; / (K.33) e e . ; ; / e e e The integral functions H , H , H at the corner nodes J for the linear pyramidal element are then e e 1 e e H . 1; 1; 0/ D H 1 D 4 e . 1; 0/.3 1 C 2/ e e 1 e e H .1; 1; 0/ D H 2 D 4 e . 1; 0/. 1 C 3 2/ e e 1 e e H .1;1;0/D H 3 D 4 e .1; 0/.3 3 C 4/ (K.34) e e 1 e e H . 1; 1; 0/ D H 4 D 4 e .1; 0/. 3 C 3 4/ e e H .0;0;1/D H 5 D 0 e e 1 e e H . 1; 1; 0/ D H 1 D 4 e . 1; 0/.3 1 C 4/ e e 1 e e H .1; 1; 0/ D H 2 D 4 e .1; 0/.3 2 C 3/ e e 1 e e H .1;1;0/D H 3 D 4 e .1; 0/. 2 C 3 3/ (K.35) e e 1 e e H . 1; 1; 0/ D H 4 D 4 e . 1; 0/. 1 C 3 4/ e e H .0;0;1/D H 5 D 0 e e 1 e e e e H . 1; 1; 0/ D H 1 D 4 e . 1; 1; 0/. 1 2 C 3 4/ e e 1 e e e e H .1; 1; 0/ D H 2 D 4 e .1; 1; 0/. 1 C 2 3 C 4/ e e 1 e e e e H .1;1;0/D H 3 D 4 e .1; 1; 0/. 1 2 C 3 4/ (K.36) e e 1 e e e e H . 1; 1; 0/ D H 4 D 4 e . 1; 1; 0/. 1 C 2 3 C 4/ e e 1 e e e e e H .0;0;1/D H 5 D 8 e .0; 0; 1/. 1 C 2 C 3 C 4 C 4 5/ e e e K.3 The Nodal Quantities H J ;H J ;H J of the Integral Functions 917 Then, the gravity term (K.6) written in local coordinates reads 0 e 1 @H B @ C B e C e D B @H C . ; / @ @ A e @H @ 0 e e e e e 1 @N1 e @N2 e @N3 e @N4 e @N5 e @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5 B e e e e e C B @N1 e @N2 e @N3 e @N4 e @N5 e C D @ @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5A e e e e e @N1 e @N2 e @N3 e @N4 e @N5 e @ H 1 C @ H 2 C @ H 3 C @ H 4 C @ H 5 0 1 e e e e 4Œe . 1; 0/.1 1 /. 1 C 2/ C e .1; 0/.1 C C 1 /. 3 C 4/ B e e e e C B4Œe . 1; 0/.1 /. 1 C 4/ C e .1; 0/.1 C C /. 2 C 3/ C B n 1 1 C 1 B e e e e C D B Œe . 1; 1; 0/ C e .1;1;0/ 1 .1 /2 . 1 C 2 3 C 4/C C 16 B C @ Œe .1; 1; 0/ C e . 1; 1; 0/ 1 C . e e C e e /C A .1 /2 o 1 2 3 4 e e e e e 2e .0; 0; 1/. 1 C 2 C 3 C 4 C 4 5/ (K.37) Appendix L Formulation of Hydraulic Head BC’s for Variable-Density Problems L.1 Problem Description In formulating flow BC’s the prescription of a hydraulic head (first kind Dirichlet- type or third kind Cauchy-type) BC at a given boundary portion is a common task (Sect. 6.3.1). However, in modeling variable-density problems such as saltwater intrusion or geothermal processes these hydraulic head BC’s have to consider the specific definition of the hydraulic head h (3.260), viz., p h D C z (L.1) 0g which must be appropriately related to a reference fluid density 0 (cf. Sect. 3.8.6.1). A typical example is the saltwater intrusion from a sea into a coastal aquifer as schematized in Fig. L.1, where at the sea side the boundary is imposed by a given hydraulic head distribution h.z/. On the other hand, at the sea the hydraulic head can be measured in form of a piezometric head hs which is related to the actual fluid density of saltwater s , while the reference fluid density 0 typically refers to the freshwater. L.2 Reference Potential from Measured Heads A measurement of a piezometric head is normally related to the actual fluid density. It can be expressed for saltwater by p hs D C z (L.2) sg where s is the fluid density at known salinity Cs and temperature Ts: s D .Cs;Ts/. (Note that the salinity Cs concerns a single-species concentration, where for the sake of simplicity we can drop the species index k.) It is obvious that H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 919 © Springer-Verlag Berlin Heidelberg 2014 920 L Formulation of Hydraulic Head BC’s for Variable-Density Problems z A sea freshwater 0 saltwater s B p gz = – s hfz= s Fig. L.1 Saltwater intrusion in a coastal aquifer with related BC’s the piezometric head hs cannot be directly used as a BC. Instead, it has to be transformed to the hydraulic head h (L.1), which is related to the reference fluid density 0. This can be simply done under considering the following relationships: Expanding (L.2)by 0 p 0 hs D C z (L.3) 0g s we obtain if introducing (L.1) 0 0 hs D h C 1 z (L.4) s s and finally s s 0 h D hs z (L.5) 0 0 For the fluid density relation we find from (3.265), (3.278)or(11.2) s 0 D ˛ ˇ.Ts/.Ts T0/ (L.6) 0 where ˛ is the specific solutal expansion coefficient (density ratio) defined by (3.275)andˇ.Ts/ is the thermal expansion coefficient defined in (11.2). Inserting (L.6)into(L.5) it results h D .1 C ˛/hs ˛z C ˇ.Ts/.Ts T0/.z hs / (L.7) The relation (L.7) is to be used to calculate the (freshwater) hydraulic heads h from (saltwater) piezometric heads hs measured at a known saltwater density s (at known salinity Cs and temperature Ts). Under isothermal conditions (L.7) reduces to h D .1 C ˛/hs ˛z (L.8) L.3 Hydrostatic Condition 921 L.3 Hydrostatic Condition Let us consider the pressure distribution in the vertical z direction of gravity g under hydrostatic conditions. We assume that the density D .z/ is varying linearly in the depth as shown in Fig. L.2: z D 1 C . 1 2/ H ; H z 0 (L.9) The fluid is hydrostatic for the vertical problem if dp D g dz Z z (L.10) p.z/ D g . /d z1 which yields with (L.9) 1 2 2 p D p1 g 1z C 2H z (L.11) The hydraulic head h (L.1) related to the reference density 0 is then 1 0 1 1 2 2 h D h1 z z (L.12) 0 2H 0 At boundaries where hydrostatic conditions can be imposed two cases are com- monly of interest as illustrated in Fig. L.3: 1. A constant saltwater density in the depth and 2. A linear increase of density as typical in a transition zone. From (L.12) we obtain with 1 D 2 D s for the case of a constant saltwater density (case 1), written for the sake of simplicity under isothermal conditions h D hs ˛z (L.13) For the case 2 with 1 D 0, 2 D s, we get from (L.12) for a linear saltwater density (case 2) ˛ 2 h D h0 C 2H z (L.14) The hydraulic head at the depth z D H is then h D hs C ˛H for the constant ˛ density and h D h0 C 2 H for the linear density relationship. Corresponding expressions can be derived for nonisothermal conditions if using (L.6), viz., h D hs C Œ˛ ˇ.Ts/.Ts T0/ H (L.15) 922 L Formulation of Hydraulic Head BC’s for Variable-Density Problems Fig. L.2 Hydrostatic z condition in a depth of H p 1 1 under a linear density g z gradient D 1 C. 1 2/ H H density pressure p p 2 2 Fig. L.3 Two interesting case 1 case 2 density profiles for a z (constant) (linear) hydrostatic BC s 0 g H density density s s for the constant density relationship and 1 h D h0 C 2 Œ˛ ˇ.Ts/.Ts T0/ H (L.16) for the linear density relationship. Based on these relations we are able to specify head conditions along boundaries with a given fluid density profile. Let us consider the example as shown in Fig. L.4 where a transition zone at a vertical boundary should be modeled for a saltwater intrusion process. The fluid density D .z/ varies linearly through the transition zone with a thickness of ı. At such a boundary a hydraulic head condition h D h.z/ L.3 Hydrostatic Condition 923 g 0 z h 0 transition zone h 1 H s h 2 density hydraulic head Fig. L.4 Boundary with a predefined saltwater-freshwater transition zone has to be imposed. From (L.14)to(L.13) we obtain the following sample values for the hydraulic head profile under isothermal conditions as illustrated in Fig. L.4: ˛ h1 D h0 C 2 ı ı (L.17) h2 D h1 C ˛.H ı/ D h0 C ˛ H 2 where h0 represents the hydraulic head at the boundary which is related to the freshwater density 0. Appendix M BHE Modeling: Numerical and Analytical Approaches M.1 Types of BHE In this section, different types of BHE with their individual pipe and grout components are described. They form highly slender cylindrical boreholes. The BHE systems are represented by 1D schematizations, where the pipe and grout components have a reduced spatial dimension. They imply that the variation of the temperature is along the vertical axis. The heat fluxes normal to the contact surfaces for the 1D pipe and grout components are modeled by heat transfer relations. M.1.1 Double U-Shape Pipe (2U) The double U-shape pipe (2U) exchanger is a cylindrical borehole consisting of two inner pipes forming a U-shape and filled with a grout material, cf. Fig. 13.1. Basically, the grout can be considered as a homogeneous impervious material and could be schematized by only one component so as proposed in [6–8]. However, to improve the approximation of the inner pipe-to-grout heat transfer we introduce a larger number of grout components in the extended TRCM, which correlates with the number of pipes of BHE [29]. This has some advantages: (1) A better accuracy results in modeling the transient behavior of U-shape pipe exchangers. (2) It allows a much higher flexibility in configuration of U-shape pipe systems, particularly, the U-shape pipes can be arranged crosswise or side by side. (3) Furthermore, the flow through double U-shape pipe configurations can be parallel or serial. In total, we schematize a 2U exchanger by eight components (Fig. M.1): • Two pipes-in (denoted as i1 and i2) • Two pipes-out (denoted as o1 and o2) • Grout material, which is subdivided into four zones (denoted as g1, g2, g3, g4) H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 925 © Springer-Verlag Berlin Heidelberg 2014 926 M BHE Modeling: Numerical and Analytical Approaches T T i1 o1 T T T s g1 g3 ro o i r R2U R2U 1 o1 T T fig fog g1 g4 T R2U i gg1 1 T R2U Tg T i o1 i gs 1 g4 r i r 2U 1 o1 2U C2U Cg Rgs g R2U s gg2 2U 2U Ts R Rgg o 2U gg1 1 ro o Rgs R2U 2 r 2U gg2 2U i2 Cg Cg T R2U T T T i2 gs g3 g2 o2 2U T Rgg i i T g2 1 r r g3 2U R2U o2 i2 Rfog fig Tg T 4 w g2 Ts T T o2 i2 D Fig. M.1 Inner pipe-grout heat flux resistance relationships of a 2U borehole consisting of four pipe components and four grout zones The four pipe components i1, i2, o1 and o2 transfer heat across their cross-sectional areas and exchange fluxes across their surface areas. The radial heat transfer from the pipes is directed to the grout zones gi .i D 1;:::;4/. The grout zones gi .i D 1;:::;4/exchange heat directly to the surrounding soil (the porous matrix with the filled fluid in the void space) denoted as s and to other contacted grout zones too. It can be seen that, as physically occurring, the heat coupling only occurs via the grout zones gi .i D 1;:::;4/, which work as intermediate media that transfer heat from one pipe to another and vice versa. Only the grout zones exchange heat with the surrounding soil s because there is no direct thermal contact between the pipes i1, i2, o1 and o2 with the soil s. The 2U system involves several material and geometric parameters, which are either given by the manufacturer of the heating system or determined experimen- tally. These relations are used to express the overall thermal resistance between the 2U borehole and the soil. The usual practice is to lump the effects of the 2U components into effective heat transfer coefficients representing the reciprocal of the sum of the thermal resistances between the different components (cf. Appendix E). The inner pipe-grout heat flux resistance relationships are shown in Fig. M.1. M.1.2 Single U-Shape Pipe (1U) The single U-shape pipe (1U) exchanger can be easily degenerated from a 2U configuration when dropping the second U-tube. A 1U configuration only consists of four components (Fig. M.2): • One pipe-in (denoted as i1) • One pipe-out (denoted as o1) • Grout material, which is subdivided into two zones (denoted as g1, g2) M.1 Types of BHE 927 Ts Ts T T g1 g2 1U 1U Rgs Rgs o o r r C1U 1U i1 o1 g Cg T T T T g1 g2 i1 o1 R1U 1U gg 1U Rfig Rfog ri ri i1 w o1 T T i1 o1 Ts D Fig. M.2 Inner pipe-grout heat flux resistance relationships of a 1U borehole consisting of two pipe components and two grout zones Similar to the 2U exchanger the U-tube of the 1U configuration transfers heat in radial direction to the grout zones gi .i D 1;:::;2/, while the grout material zones exchange heat directly to the surrounding soil s and to the adjacent grout zone. The corresponding inner pipe-grout heat flux resistance relationships are shown in Fig. M.2. M.1.3 Coaxial Pipe with Annular (CXA) and Centered (CXC) Inlet These types of BHE consist only of three components (Figs. M.3 and M.4): • One pipe-in (denoted as i1) • One pipe-out (denoted as o1) • Grout material considered in one zone (denoted as g1) Such coaxial BHE systems represent pipe-in-pipe installations, where two principal cases occur. In the case of the CXA exchanger the pipe-out is configured inside the pipe-in as shown in Fig. M.3 forming an annular inlet and a centered outlet. Accordingly, the heat exchange to the grout material g1, which is in contact to the surrounding soil s, is only performed via the pipe-in i1. On the other hand, the pipe- in i1 exchanges heat with the pipe-out o1 component. The coaxial pipes can also be installed with interchanged inlet and outlet. This represents the CXC exchanger, where the pipe-in is configured inside the pipe-out as shown in Fig. M.4 forming a centered inlet and an annular outlet. Here, the heat exchange to the grout material g1 is only performed via the pipe-out o1. 928 M BHE Modeling: Numerical and Analytical Approaches Fig. M.3 Inner pipe-grout heat flux resistance T Tg s 1 ro relationships of a CXA i1 Ts borehole with annular inlet CXA Rgs i r CXA i1 Cg To o T 1 ro g1 i 1 r CXA o1 Rfig CXA T Rff i1 T T o1 i1 T g1 Ts D Fig. M.4 Inner pipe-grout Ts heat flux resistance Tg 1 ro relationships of a CXC o1 Ts borehole with centered inlet CXC Rgs i r CXC o1 Cg T i1 Tg ro 1 i i1 r CXC i1 Rfog T RCXC o1 ff T T i1 o1 T g1 Ts D M.2 Thermal Resistances Thermal resistance is a measure of material’s ability to resist heat transfer through its surface and contact zone (cf. Appendix E). Thermal resistances are determined from the physical, material and geometric engineering parameters of the different BHE configurations as shown in Fig. M.1 for the 2U exchanger, in Fig. M.2 for the 1U exchanger, in Fig. M.3 for the CXA exchanger and in Fig. M.4 for the CXC exchanger. As indicated the interaction between the different components of the pipe exists between the pipe-in and grout zone(s), the pipe-out and grout zone(s) as well as the pipe-in and pipe-out. The thermal resistances due to heat conduction in the grout material are derived by adding correction terms gained from numerical simulations to well-known 2D heat conduction shape factors. These resistances then are divided in such a manner, that the grout points are suitably located to obtain accurate transient computation results, see [29]. The following specific thermal resistances can be derived. M.2 Thermal Resistances 929 M.2.1 2U Exchanger The thermal resistance between the pipes and grout zones is caused by the advection of the pipe flow and thermal conductivity of the pipe wall material specified separately for pipe-in and pipe-out 2U 2U 2U 2U R D R C R a C R b .k D i1 \ i2/ (M.1) fig advk conk con 2U 2U 2U 2U R D R C R a C R b .k D o1 \ o2/ (M.2) fog advk conk con M.2.1.1 Thermal Resistance Due to the Advective Flow of Refrigerant in the Pipes 1 R2U D .k D i1; o1; i2; o2/ (M.3) advk r Nuk where r is the thermal conductivity of the refrigerant. In (M.3) the Nusselt numbers, Nuk.k D i1; o1; i2; o2/, differ between laminar and turbulent flow [544], viz., 8 ˆ 4:364 ˆ ˆ for laminar flow if Re < 2; 300 ˆ k ˆ h i ˆ 2=3 ˆ . =8/Re Pr d i ˆ pk k k < 2=3 1 C 1C12:7 k=8.Pr 1/ L Nuk D (M.4) ˆ for turbulent flow if Re 104 ˆ k ˆ n h io ˆ 2=3 ˆ .0:0308=8/104Pr d i ˆ p k ˆ .1 k/4:364 C k 2=3 1 C ˆ 1C12:7 0:0308=8.Pr 1/ L :ˆ 4 for flow in transition range if 2,300 Rek <10 in which Pr represents the Prandtl number and Rek are the Reynolds numbers defined as r r 2U i c jukj d Pr D ; Re D k .k D i1; o1; i2; o2/ (M.5) r k . r = r / i i i where dk are the inner diameters of the pipes dk D 2rk.k D i1; o1; i2; o2/. Furthermore, L corresponds to the length of the pipe and 9 2 = k D 1:8 log10 Rek 1:5 (M.6) Rek 2;300 ; k D 104 2;300 .0 k 1/ 930 M BHE Modeling: Numerical and Analytical Approaches It is 8 ˆ Qr ˆ for parallel discharge < 2 .ri /2 2U k jukj D .k D i1; o1; i2; o2/ (M.7) ˆ Q :ˆ r for serial discharge i 2 .rk/ where Qr is the total refrigerant flow discharge of the 2U exchanger. M.2.1.2 Thermal Resistances Due to the Pipes Wall Material and Grout Transition o i 2U ln.rk =rk/ R a D .k D i1; o1; i2; o2/ (M.8) conk 2 k where i1; o1 ; i2; o2 correspond to the thermal conductivities of the pipe wall material. 2U 2U 2U Rconb D x Rg (M.9) with p D2C4d2 ln p o x2U D 2 2do (M.10) ln. D / 2do and D2Cd 2 s2 arcosh. o / s s2 R2U D 2Ddo 3:098 4:432 C 2:364 (M.11) g 2 g D D2 P where D denotes the borehole diameter, d D 1 d o is the average outer diameter o 4 k k p o o of the pipes dk D 2rk .k D i1; o1; i2; o2/ and s D w 2 corresponds to diagonal distances of pipes (see Fig. M.1). M.2.1.3 Thermal Resistance Due to Inter-grout Exchange 2R2U.R2U 2x2UR2U/ R2U D gs ar1 g (M.12) gg1 2U 2U 2U 2U 2Rgs Rar1 C 2x Rg 2R2U.R2U 2x2UR2U/ R2U D gs ar2 g (M.13) gg2 2U 2U 2U 2U 2Rgs Rar2 C 2x Rg M.2 Thermal Resistances 931 with 2 2 s do arcosh. d 2 / R2U D o (M.14) ar1 2 g 2 2 2s do arcosh. d 2 / R2U D o (M.15) ar2 2 g M.2.1.4 Thermal Resistance Due to Grout-Soil Exchange 2U 2U 2U Rgs D .1 x /Rg (M.16) M.2.2 1U Exchanger It is 1U 1U 1U 1U R D R C R a C R b .k D i1/ (M.17) fig advk conk con 1U 1U 1U 1U R D R C R a C R b .k D o1/ (M.18) fog advk conk con M.2.2.1 Thermal Resistance Due to the Advective Flow of Refrigerant in the Pipes 1 R1U D .k D i1;o1/ (M.19) advk r Nuk where Nuk is given by the expressions (M.4)–(M.6) in which the refrigerant fluid velocity for 1D pipe is Q ju j1U D r .k D i1;o1/ (M.20) k i 2 2 .rk/ M.2.2.2 Thermal Resistance Due to the Pipes Wall Material and Grout Transition o i 1U ln.rk =rk/ R a D .k D i1;o1/ (M.21) conk 2 k 1U 1U 1U Rconb D x Rg (M.22) 932 M BHE Modeling: Numerical and Analytical Approaches with p 2 2 D C2do ln 2d x1U D o (M.23) ln. pD / 2do and D2Cd 2 w2 arcosh. o / w R1U D 2Ddo 1:601 0:888 (M.24) g 2 g D where w corresponds to distances between the pipes (see Fig. M.2). M.2.2.3 Thermal Resistance Due to Inter-grout Exchange 2R1U.R1U 2x1UR1U/ R1U D gs ar g (M.25) gg 1U 1U 1U 1U 2Rgs Rar C 2x Rg with 2 2 2w do arcosh. d 2 / R1U D o (M.26) ar 2 g M.2.2.4 Thermal Resistance Due to Grout-Soil Exchange 1U 1U 1U Rgs D .1 x /Rg (M.27) M.2.3 CXA Exchanger It is CXA CXA CXA CXA R D R C R a C R (M.28) ff advo1 advi1 cono1 RCXA RCXA RCXA RCXA fig D b C coni1 C b (M.29) advi1 con M.2 Thermal Resistances 933 M.2.3.1 Thermal Resistance Due to the Advective Flow of Refrigerant in the Pipes 1 RCXA D (M.30) advo1 r Nuo1 CXA 1 dh R a D (M.31) advi1 r o Nui1 do1 1 d RCXA D h (M.32) advb r i i1 Nui1 di1 8 ˆ 4:364 ˆ ˆ for laminar flow if Reo1 < 2; 300 ˆ ˆ h i ˆ 2=3 ˆ . =8/Re Pr d i < po1 o1 o1 2=3 1 C 1C12:7 o1 =8.Pr 1/ L Nuo1 D ˆ 4 ˆ for turbulent flow if Reo1 10 ˆ n h io ˆ 2=3 ˆ .0:0308=8/104Pr d i ˆ p o1 ˆ .1 o1 /4:364 C o1 2=3 1 C L :ˆ 1C12:7 0:0308=8.Pr 1/ 4 for flow in transition range if 2,300 Reo1 <10 (M.33) 8 h i o 0:04 ˆ 0:102 do1 ˆ 3:66 C 4 do i ˆ o1 C0:02 di1 ˆ di ˆ i1 ˆ for laminar flow if Rei1 < 2; 300 ˆ h i ˆ ( o 0:84 o 0:6 ) ˆ h i do1 do1 ˆ 2/3 0:86 C 1 0:14 ˆ . =8/Re Pr di di ˆ pi1 i1 dh i1 i1 ˆ 2=3 1 C do ˆ 1C12:7 i1=8.Pr 1/ L o1 ˆ 1C i ˆ di1 ˆ 4 <ˆ for turbulent flow if Rei1 10 ( ) Nui1 D o 0:04 ˆ 0:102 do1 ˆ .1 i1/ 3:66 C 4 do i C ˆ o1 di1 ˆ i C0:02 ˆ ( di1 ˆ ˆ .0:0308=8/104Pr ˆ p ˆ i1 2=3 ˆ 1C12:7 0:0308=8.Pr 1/ ˆ h i ˆ o 0:84 o 0:6 ) ˆ h i do1 do1 ˆ 2=3 0:86 i C 1 0:14 i ˆ dh di1 di1 ˆ 1 C o ˆ L do1 ˆ 1C i ˆ di1 : 4 for flow in transition range if 2,300 Rei1 <10 (M.34) where r r ju jCXA i CXA c o1 do1 ju11j dh (M.35) Pr D r ; Reo1 D . r = r / ; Rei1 D . r = r / 934 M BHE Modeling: Numerical and Analytical Approaches and 9 d D d i d o > h i1 o1 > > 2 => k D 1:8 log10 Rek 1:5 > (M.36) Rek 2; 300 > k D .0 k 1/ > 104 2; 300 ;> .k D i1;o1/ 9 Qr > ju jCXA D => o1 i 2 2 .ro1/ Q > (M.37) ju jCXA D r ;> i1 i 2 o 2 2 Œ.ri1/ .ro1 / M.2.3.2 Thermal Resistance Due to the Pipes Wall Material and Grout Transition ln.ro=ri / RCXA D k k .k D i1;o1/ (M.38) conk 2 k CXA CXA CXA Rconb D x Rg (M.39) with p D2C.d o /2 p i1 ln o CXA 2di1 x D D (M.40) ln. o / di1 and ln.D=d o / RCXA D i1 (M.41) g 2 g M.2.3.3 Thermal Resistance Due to Grout-Soil Exchange CXA CXA CXA Rgs D .1 x /Rg (M.42) M.2.4 CXC Exchanger CXC CXC CXC CXC R D R C R a C R (M.43) ff advi1 advo1 coni1 RCXC D RCXC C RCXC C RCXC (M.44) fog b cono1 b advo1 con M.2 Thermal Resistances 935 M.2.4.1 Thermal Resistance Due to the Advective Flow of Refrigerant in the Pipes 1 RCXC D (M.45) advi1 r Nui1 CXC 1 dh R a D (M.46) advo1 r o Nuo1 di1 1 d RCXC D h (M.47) advb r i o1 Nuo1 do1 with 8 ˆ 4:364 ˆ ˆ for laminar flow if Rei1 < 2; 300 ˆ ˆ h i ˆ i 2=3 <ˆ . i1=8/Rei1Pr di1 p 2=3 1 C 1C12:7 i1=8.Pr 1/ L Nui1 D ˆ for turbulent flow if Re 104 ˆ i1 ˆ n h io ˆ i 2=3 ˆ .0:0308=8/104Pr d ˆ p i1 ˆ .1 i1/4:364 C i1 2=3 1 C :ˆ 1C12:7 0:0308=8.Pr 1/ L 4 for flow in transition range if 2,300 Rei1 <10 (M.48) 8 h i o 0:04 ˆ 0:102 di1 ˆ 3:66 C 4 do i ˆ i1 do1 ˆ i C0:02 ˆ do1 ˆ ˆ for laminar flow if Reo1 < 2; 300 ˆ ˆ h i ˆ ( o 0:84 o 0:6 ) ˆ h i di1 di1 ˆ 2/3 0:86 C 1 0:14 ˆ . =8/Re Pr di di ˆ po1 o1 dh o1 o1 ˆ 2=3 1 C L do ˆ 1C12:7 o1 =8.Pr 1/ 1C i1 ˆ di ˆ o1 ˆ for turbulent flow if Re 104 <ˆ o1 ( ) Nuo1 D ˆ ˆ o 0:04 ˆ 0:102 di1 ˆ .1 o1 / 3:66 C 4 do i C ˆ i1 do1 ˆ i C0:02 ˆ ( do1 ˆ ˆ .0:0308=8/104Pr ˆ p ˆ o1 2=3 ˆ 1C12:7 0:0308=8.Pr 1/ ˆ h i ˆ o 0:84 o 0:6 ) ˆ h i di1 di1 ˆ 2=3 0:86 i C 1 0:14 i ˆ dh do1 do1 ˆ 1 C o ˆ L di1 ˆ 1C i :ˆ do1 4 for flow in transition range if 2,300 Reo1 <10 (M.49) 936 M BHE Modeling: Numerical and Analytical Approaches where r r CXC i CXC c jui1j di1 juo1 j dh (M.50) Pr D r Rei1 D . r = r / Reo1 D . r = r / and 9 d D d i d o > h o1 i1 > > 2 => k D 1:8 log10 Rek 1:5 > (M.51) Rek 2; 300 > k D .0 k 1/ > 104 2; 300 ;> .k D i1;o1/ 9 CXC Qr > jui1j D = 2 .ri /2 i1 (M.52) Q > ju jCXC D r ;> o1 i 2 o 2 2 Œ.ro1 / .ri1/ M.2.4.2 Thermal Resistance Due to the Pipes Wall Material and Grout Transition ln.ro=ri / RCXC D k k .k D i1;o1/ (M.53) conk 2 k CXC CXC CXC Rconb D x Rg (M.54) with p D2C.d o /2 p o1 ln o CXC 2do1 x D D (M.55) ln. o / do1 and ln.D=d o / RCXC D o1 (M.56) g 2 g M.2.4.3 Thermal Resistance Due to Grout-Soil Exchange CXC CXC CXC Rgs D .1 x /Rg (M.57) M.2 Thermal Resistances 937 M.2.5 Notes to Negative Thermal Resistances of Grout for 2U and 1U Exchangers In dependence on geometric measures for 2U and 1U exchangers negative thermal 2U 2U 1U resistances for grout Rgg1;Rgg2;Rgg may occur. This is caused by the applied TRCM conception of grout zones and can be accepted in both numerical and analytical BHE models. However, the following constraints have to be satisfied: 1 1 1 C >0 R2U 2R2U gg1 gs (M.58) 1 1 1 C >0 2U 2U Rgg2 2Rgs for 2U exchangers and 1 1 1 C >0 (M.59) 1U 1U Rgg 2Rgs for 1U exchangers. In cases where (M.58)or(M.59) are violated the values of x2U and x1U, respectively, have to be reduced until the constraints (M.58)and(M.59) are met. The following correction procedure is applied: 2U;1U 2 2U;1U 1. If (M.58)or(M.59) are violated reduce xnew D 3 xold and check (M.58)or (M.59). 2U;1U 1 2U;1U 2. If (M.58)or(M.59) are still violated reduce xnew D 3 xold and check (M.58) or (M.59). 2U;1U 3. If (M.58)or(M.59) are again violated set xnew D 0. M.2.6 Direct Use of Borehole Thermal Resistances Ra and Rb Obtained from Thermal Response Test (TRT) From practical point of view it could be useful to specify directly thermal resistances which have been measured in the field. Such field-related thermal resistances result for instance from Thermal Response Tests (TRT’s), e.g., [25]. In such cases the borehole thermal resistance Rb and the internal borehole thermal resistance Ra are determined according to the definition introduced by Hellstrom¨ [237]given for a Delta configuration of thermal circuit such as described in Sect. E.3.1 of Appendix E. With known Rb and Ra the complete set of thermal BHE resistances can be determined in dependence on the chosen analytical or numerical solution strategy as follows. 938 M BHE Modeling: Numerical and Analytical Approaches M.2.6.1 Analytical BHE Solution 2U Exchanger: Replace (M.1), (M.2)and(M.16)by 2U Rfig D 2Rb 2U Rfog D 2Rb (M.60) 2U Rgs D 2Rb respectively, as well as (M.12)and(M.13)by 2U 8Rb.Ra 2Rb/ Rgg1 D 4Rb Ra (M.61) 2U 2U Rgg2 D Rgg1 respectively. 1U Exchanger: Replace (M.17), (M.18), and (M.27)by 1U Rfig D Rb 1U Rfog D Rb (M.62) 1U Rgs D Rb respectively, as well as (M.25)by 1U 2Rb.Ra 2Rb/ Rgg D (M.63) 4Rb Ra CXA Exchanger: Replace (M.28), (M.29), and (M.42)by CXA Rff D Ra CXA Rb Rfig D 2 (M.64) CXA Rb Rgs D 2 respectively. M.2 Thermal Resistances 939 CXC Exchanger: Replace (M.43), (M.44), and (M.57)by CXC Rff D Ra CXC Rb Rfog D 2 (M.65) CXC Rb Rgs D 2 respectively. M.2.6.2 Numerical BHE Solution 2U Exchanger: Defining R2U D 1 .R2U C R2U C R2U C R2U / adv 4 advi1 advi2 advo1 advo2 (M.66) 2U 1 2U 2U 2U 2U R D .R a C R a C R a C R a / con 4 coni1 coni2 cono1 cono2 we replace (M.11), (M.14), and (M.15)by 2U 2U 2U Rg D 4Rb Radv Rcon p 2U 2U 2U .2 C 2/R .Ra R R / R2U D g adv con (M.67) ar1 2U 2U 2U Rg C Ra Radv Rcon p 2U 2U Rar2 D 2Rar1 respectively. 1U Exchanger: Defining R1U D 1 .R1U C R1U / adv 2 advi1 advo1 (M.68) 1U 1 1U 1U R D .R a C R a / con 2 coni1 cono1 we replace (M.24)and(M.26)by 1U 1U 1U Rg D 2Rb Radv Rcon (M.69) 1U 1U 1U Rar D Ra 2.Radv C Rcon/ respectively. 940 M BHE Modeling: Numerical and Analytical Approaches CXA Exchanger: Replace (M.28)and(M.41)by CXA Rff D Ra (M.70) RCXA R RCXA RCXA g D b b coni1 advi1 respectively. CXC Exchanger: Replace (M.43)and(M.56)by CXC Rff D Ra (M.71) RCXC R RCXC RCXC g D b b cono1 advo1 respectively. M.3 Heat Transfer Coefficients A heat transfer coefficient ˚T represents the reciprocal of the effective specific thermal resistance R with its specific exchange area S (cf. Sect. E.2 of Appendix E), viz., 1 ˚ D (M.72) T RS Accordingly, the specific thermal resistances Rfig, Rfog, Rgg1, Rgg2, Rff , Rgg and Rgs given by the relationships in the preceding Sect. M.2 for the 2U, 1U, CXA and CXC configurations can be used to express the corresponding heat transfer coefficients ˚fig, ˚fog, ˚gg1, ˚gg2, ˚ff , ˚gg and ˚gs for the BHE configurations as follows: 2U Exchanger: 9 2U 1 1 > ˚fig D 2U S > Rfig i > > 2U 1 1 > ˚fog D 2U S > Rfog o => 2U 1 1 ˚gg1 D 2U S (M.73) Rgg1 g1 > > ˚ 2U D 1 1 > gg2 R2U Sg2 > gg2 > 2U 1 1 ;> ˚gs D 2U Rgs Sgs M.3 Heat Transfer Coefficients 941 Table M.1 Specific S 2U 1U CXA CXC exchange surfaces S i i i for the BHE configurations Si di1;i2 di1 di1 – i i i So do1;o2 do1 – do1 i i Sio –– do1 di1 1 Sg1 2 D D– – Sg2 D––– 1 1 Sgs 4 D 4 D D D 1U Exchanger: 9 1U 1 1 > ˚fig D 1U S > Rfig i > > ˚ 1U D 1 1 => fog R1U So fog (M.74) 1U 1 1 > ˚gg D 1U > Rgg Sg1 > > 1U 1 1 ; ˚gs D 1U Rgs Sgs CXA Exchanger: 9 CXA 1 1 > ˚fig D CXA S > Rfig i => CXA 1 1 ˚ D CXA (M.75) ff R Sio > ff > CXA 1 1 ;> ˚gs D CXA Rgs Sgs CXC Exchanger: 9 CXC 1 1 > ˚fog D CXC S > Rfog o => CXC 1 1 ˚ D CXC (M.76) ff R Sio > ff > CXC 1 1 ;> ˚gs D CXC Rgs Sgs where the related specific exchange areas of the BHE configurations are listed in Table M.1. 942 M BHE Modeling: Numerical and Analytical Approaches M.4 Analytical Solution of the Local Problem In this section, the BHE equations are solved by Eskilson and Claesson’s analytical method [159]. This method is applied to the different BHE configurations by using the extended TRCM [29]. Explicit relations result for temperatures of the pipes and the grout zones at vertical position and time. They are coupled to the soil temperature due to the incorporated heat transfer between BHE and the soil system. The analytical method has shown highly efficient, precise and robust, however, not applicable to short-term processes (a temporal scale in order of seconds, minutes or few hours). M.4.1 Local Steady-State Condition with Given Temperature at Borehole Wall The present analytical solution is only valid for local steady-state heat transport and given temperature Ts D Ts.z;t/ at borehole wall. It was firstly derived by Eskilson and Claesson [159] for heat transfer between two pipes and the borehole wall assuming a Delta configuration of the corresponding three thermal resistors ▵ ▵ ▵ with their specific thermal resistances R1 , R2 and R12 as illustrated in Fig. M.5,see also Sect. E.3.1 of Appendix E. We extend this analytical method to 2U, 1U, CXA and CXC configurations of BHE. In accordance with Fig. M.5, the local steady-state heat balance equations for fluid in pipe-in and pipe-out read @T T T T T Ai r cr juj i1 D q q D s i1 i1 o1 @z 1 12 R▵ R▵ 1 12 (M.77) i r r @To1 Ts To1 Ti1 To1 A c juj D q2 C q12 D ▵ C ▵ @z R2 R12 which have to be solved for the pipe(s)-in temperature Ti1.z/ and pipe(s)-out temperature To1 .z/,wherethez coordinate is directed downward, superscript r refers to the refrigerant fluid, u is the refrigerant fluid velocity directed positive downward in z direction (i.e., positive in pipe-in i1 and negative in pipe-out o1). They represent 1D heat transport equations for each pipe in the vertical z direction, where the vertical heat conduction within the pipes is neglected. It is further assumed that the inner cross-sectional area of pipe-in and pipe-out is equal, i.e., i i i A D Ai D Ao. The local steady-state condition limits the application of (M.77)to a time scale larger than [159] 5 " c C .1 "/ scs t>tsteady D D2 (M.78) limit 4 " C .1 "/ s M.4 Analytical Solution of the Local Problem 943 Fig. M.5 Borehole cross Ts D section, heat flow R q q 12 components and thermal 1 2 circuit of Delta configuration i1 o1 (Modified from [159]) Ti To D 1 1 RD R 1 2 q 12 s i The time for the refrigerant to circulate through the borehole is 2A L=Qr . Accord- ingly, Eq. (M.77) can only describe transient input variations of inlet temperature and pumping rate on a time scale larger than [159] steady i 2L t>tlimit C A (M.79) Qr The specific thermal flux .z;t/exchanging heat of the borehole with the adjacent soil s is given from (M.77) according to Ts Ti1 Ts To1 .z;t/D ▵ C ▵ (M.80) R1 R2 M.4.2 Eskilson and Claesson’s Analytical BHE Solution The coupled equations (M.77) can be solved by using Laplace transforms [159]. It yields Z z Ti1.z;t/ D Ti1.0; t/f1.z/ C To1 .0; t/f2.z/ C Ts. ; t/f4.z /d 0 Z .0 z L/ z To1 .z;t/ D Ti1.0; t/f2.z/ C To1 .0; t/f3.z/ Ts. ; t/f5.z /d 0 (M.81) The functions f1;f2;:::;f5 are given by the expressions ˇz f1.z/ D e Œcosh. z/ ı sinh. z/ ˇ f .z/ D eˇz 12 sinh. z/ 2 ˇz f3.z/ D e Œcosh. z/ C ı sinh. z/ (M.82) 944 M BHE Modeling: Numerical and Analytical Approaches ˇ ˇ f .z/ D eˇz ˇ cosh. z/ .ıˇ C 2 12 / sinh. z/ 4 1 1 ˇ ˇ f .z/ D eˇz ˇ cosh. z/ C .ıˇ C 1 12 / sinh. z/ 5 2 2 where 1 1 1 ˇ ˇ ˇ D ˇ D ˇ D ˇ D 2 1 1 R▵Ai r cr u 2 R▵Ai r cr u 12 R▵ Ai r cr u 2 r 1 2 12 .ˇ C ˇ /2 1 ˇ C ˇ D 1 2 C ˇ .ˇ C ˇ /ıD ˇ C 1 2 (M.83) 4 12 1 2 12 2 The following BC’s are applied Ti1.0; t/ D Ti .t/ Ti1.L; t/ D To1 .L; t/ (M.84) where Ti .t/ represents the inlet temperature. Using (M.84)in(M.82)and(M.83) the outlet temperature To.t/ is given as To.t/ D To1 .0; t/ (M.85) M.4.3 Solutions for 1U and 2U Configurations It is assumed that the pipes are arranged symmetrically within the borehole, i.e., ▵ ▵ R2 D R1 (M.86) so that 9 1 > ˇ D ˇ D > 2 1 ▵ i r r > R1 A c u > > 1 > ˇ D => 12 ▵ i r r R12A c u > (M.87) ˇ D q0 > > 2 > D ˇ C 2ˇ12ˇ1 > 1 > > 1 ; ı D .ˇ12 C ˇ1/ M.4 Analytical Solution of the Local Problem 945 Hence, (M.82) simplifies f1.z/ D cosh. z/ ı sinh. z/ ˇ f .z/ D 12 sinh. z/ 2 f3.z/ D cosh. z/ C ı sinh. z/ (M.88) ˇ ˇ f .z/ D ˇ cosh. z/ .ıˇ C 2 12 / sinh. z/ 4 1 1 ˇ ˇ f .z/ D ˇ cosh. z/ C .ıˇ C 1 12 / sinh. z/ 5 2 2 In using (M.84)theEq.(M.81) can be equalized at z D L and solved for the outlet temperature To.t/,viz., Z L f1.L/ C f2.L/ Ts. ; t/Œf4.L / C f5.L / To.t/ D Ti .t/ C d f3.L/ f2.L/ 0 f3.L/ f2.L/ (M.89) With known inlet temperature Ti .t/ from the BC (M.84) and outlet temperature To.t/ from (M.89) the temperature distributions Ti1 and To1 as a function of z and t are obtained after evaluating the integrals in (M.81). It yields1 Z z Ti1.z;t/ D Ti .t/f1.z/ C To.t/f2.z/ C Ts. ; t/f4.z /d 0Z z (M.90) To1 .z;t/ D Ti .t/f2.z/ C To.t/f3.z/ Ts. ; t/f5.z /d 0 Note that for the 2U configuration we assume Ti2 D Ti1 and To2 D To1 .The integrals in (M.90) are performed elementwise, where the solid temperature Ts at the borehole wall is numerically approximated as a linear function from the nodal finite element solution at time t. For example Z z X e e e e Ts .z1;t/C Ts .z2;t/ e e Ts. ; t/f4.z /d F4.z; z2; z1/ (M.91) 0 e e 2 e2.z1;z2/ 1 The integrals f4.z / and f5.z / result for 1U and 2U configurations, respectively, R b ˇ1 b ıˇ1 ˇ2ˇ12 b F4.z;a;b/ D a f4.z /d D sinh. .z //ja C C 2 cosh. .z //ja R b ˇ2 b ıˇ2 ˇ1ˇ12 b F5.z;a;b/ D a f5.z /d D sinh. .z //ja C 2 cosh. .z //ja 946 M BHE Modeling: Numerical and Analytical Approaches e e where z1; z2 represent the vertical coordinates of the lower and upper nodes, respectively, of element e. The temperature distributions for the grout zones are derived from horizontal steady-state heat flow balances at the grout points, where the total heat exchange between the grout zones, pipe(s) and soil is assumed in equilibrium, such that T T T T T T g1 s C g1 i1 C g1 g2 D 0 1U 1U 1U Rgs Rfig Rgg (M.92) T T T T T T g2 s C g2 o1 C g2 g1 D 0 1U 1U 1U Rgs Rfog Rgg given for the 1U exchanger (Fig. M.2)and T T T T T T T T T T g1 s g1 i1 g1 g2 g1 g3 g1 g4 0 2U C 2U C 2U C 2U C 2U D Rgs Rfig Rgg2 Rgg1 Rgg1 T T T T T T T T T T g2 s C g2 i2 C g2 g1 C g2 g3 C g2 g4 D 0 2U 2U 2U 2U 2U Rgs Rfig Rgg2 Rgg1 Rgg1 (M.93) T T T T T T T T T T g3 s C g3 o1 C g3 g4 C g3 g1 C g3 g2 D 0 2U 2U 2U 2U 2U Rgs Rfog Rgg2 Rgg1 Rgg1 T T T T T T T T T T g4 s C g4 o2 C g4 g3 C g4 g1 C g4 g2 D 0 2U 2U 2U 2U 2U Rgs Rfog Rgg2 Rgg1 Rgg1 given for the 2U exchanger (Fig. M.1). Accordingly, the temperature distribution for the two grout zones Tg1.z;t/and Tg2.z;t/of the 1U configuration can be derived as h i T .z;t/ T .z;t/ T .z;t/ T .z;t/ s C o1 C s C i1 u R1U R1U 1U 1U 1U 1U 1 gg gg Rgs R Rgs R T .z;t/ D fog fig g1 1U 2 2 .Rgg / u1 1 T .z;t/ T .z;t/ T .z;t/ 1 T .z;t/ D g1 C o1 C s (M.94) g2 1U 1U 1U Rgg Rfog Rgs u1 with 1 1 1 u D C C (M.95) 1 1U 1U 1U Rfig Rgs Rgg and the temperature distribution for the four grout zones Tg1.z;t/, Tg2.z;t/, Tg3.z;t/ and Tg4.z;t/of the 2U configuration results in M.4 Analytical Solution of the Local Problem 947 h 2T .z;t/ 2T .z;t/ 2T .z;t/ T .z;t/D T .z;t/D s C o1 C s C g1 g2 2U 2U 2U Rgs Rfog Rgs i 2T .z;t/ v i1 u v 2U 2 2 2 Rfig v u2 1 T .z;t/ 2T .z;t/ 2T .z;t/ 1 T .z;t/D T .z;t/D g1 C o1 C s (M.96) g3 g4 2U 2U v Rfog Rgs u2 with 2 2 1 u D C C 2 2U 2U Rfig Rgs v 2U 2U Rgg1Rgg2 v D 2U 2U (M.97) 2.Rgg1 C Rgg2/ 1U 1U 2U 2U ▵ ▵ assuming Rfig D Rfog and Rfig D Rfog. The thermal resistances R1 and R12 are given by 9 R▵ D R1U C R1U > 1 fig gs => 1U 1U 2 1U 2 for 1U configuration (M.98) .u1R R / .R / > R▵ D fig gg fig ;> 12 1U Rgg and 9 R2U C R2U > R▵ D fig gs => 1 2 > for 2U configuration (M.99) .R2U/2 1 > R▵ D fig .u2v / ; 12 4 2 v M.4.4 Solution for CXA Configuration For coaxial BHE pipes with annular inlet there is ▵ R2 D1 (M.100) 948 M BHE Modeling: Numerical and Analytical Approaches so that 9 1 > ˇ1 D > R▵Ai r cr u > 1 > > ˇ2 D 0 > > 1 > ˇ12 D ▵ => R Ai r cr u 12 (M.101) ˇ > ˇ D 1 > 2 > q > 2 > ˇ1 > D C ˇ12ˇ1 > 4 > ;> 1 ˇ1 ı D ˇ12 C 2 Hence, (M.82) simplifies ˇz f1.z/ D e Œcosh. z/ ı sinh. z/ ˇ f .z/ D eˇz 12 sinh. z/ 2 ˇz f3.z/ D e Œcosh. z/ C ı sinh. z/ (M.102) ˇz f4.z/ D e ˇ1 cosh. z/ ıˇ1 sinh. z/ ˇ ˇ f .z/ D eˇz 1 12 sinh. z/ 5 The outlet temperature To.t/ is determined by Z L f1.L/ C f2.L/ Ts. ; t/Œf4.L / C f5.L / To.t/ D Ti .t/ C d f3.L/ f2.L/ 0 f3.L/ f2.L/ (M.103) and the temperature distributions Ti1 and To1 are obtained from the integral expressions2 Z z T .z;t/ D T .t/f .z/ C T .t/f .z/ C T . ; t/f .z /d i1 i 1 o 2 Z s 4 0 z (M.104) To1 .z;t/ D Ti .t/f2.z/ C To.t/f3.z/ Ts. ; t/f5.z /d 0 2 The integrals f4.z / and f5.z / result for the CXA configuration R F .z;a;b/ D b f .z /d 4 a 4 h i ˇ1 b b b D 2 ˇ2 exp.ˇ.z //ja . ı C ˇ/cosh. .z //ja . C ıˇ/ sinh. .z //ja R h i b ˇ1ˇ12 b ˇ b b F5.z;a;b/ D a f5.z /d D 2 ˇ2 exp.ˇ.z //ja sinh. .z //ja cosh. .z //ja M.4 Analytical Solution of the Local Problem 949 With the equilibrium condition (Fig. M.3) T T T T g1 s C g1 i1 D 0 (M.105) CXA CXA Rgs Rfig the temperature distribution for the grout zone Tg1.z;t/yields RCXA T .z;t/D fig T .z;t/ T .z;t/ C T .z;t/ (M.106) g1 CXA CXA s i1 i1 Rfig C Rgs ▵ ▵ The thermal resistances R1 and R12 are given by ▵ CXA CXA R1 D Rfig C Rgs ▵ CXA (M.107) R12 D Rff M.4.5 Solution for CXC Configuration For coaxial BHE pipes with centered inlet there is ▵ R1 D1 (M.108) so that 9 ˇ D 0 > 1 > 1 > ˇ D > 2 R▵Ai r cr u > 2 > > 1 > ˇ12 D ▵ => R Ai r cr u 12 (M.109) ˇ > ˇ D 2 > 2 > q > 2 > ˇ1 > D C ˇ12ˇ2 > 4 > ;> 1 ˇ2 ı D ˇ12 C 2 Hence, (M.82) simplifies ˇz f1.z/ D e Œcosh. z/ ı sinh. z/ ˇ f .z/ D eˇz 12 sinh. z/ 2 ˇz f3.z/ D e Œcosh. z/ C ı sinh. z/ (M.110) 950 M BHE Modeling: Numerical and Analytical Approaches ˇ ˇ f .z/ D eˇz 2 12 sinh. z/ 4 ˇz f5.z/ D e ˇ2 cosh. z/ C ıˇ2 sinh. z/ The outlet temperature To.t/ is determined by Z L f1.L/ C f2.L/ Ts. ; t/Œf4.L / C f5.L / To.t/ D Ti .t/ C d f3.L/ f2.L/ 0 f3.L/ f2.L/ (M.111) and the temperature distributions Ti1 and To1 are obtained from the integral expressions3 Z z Ti1.z;t/ D Ti .t/f1.z/ C To.t/f2.z/ C Ts. ; t/f4.z /d 0Z z (M.112) To1 .z;t/ D Ti .t/f2.z/ C To.t/f3.z/ Ts. ; t/f5.z /d 0 With the equilibrium condition (Fig. M.4) T T T T g1 s C g1 o1 D 0 (M.113) CXC CXC Rgs Rfog the temperature distribution for the grout zone Tg1.z;t/yields RCXC T .z;t/D fog T .z;t/ T .z;t/ C T .z;t/ (M.114) g1 CXC CXC s o1 o1 Rfog C Rgs ▵ ▵ The thermal resistances R1 and R12 are given by ▵ CXC CXC R2 D Rfog C Rgs (M.115) ▵ CXC R12 D Rff 3 The integrals f4.z / and f5.z / result for the CXC configuration R h i b ˇ2ˇ12 b ˇ b b F4.z;a;b/ D a f4.z /d D ˇ2 2 exp.ˇ.z //ja sinh. .z //ja cosh. .z //ja R b F5.z;a;b/ D f5.z /d a h i ˇ2 b b b D ˇ2 2 exp.ˇ.z //ja .ˇ ı/cosh. .z //ja C .ıˇ /sinh. .z //ja M.4 Analytical Solution of the Local Problem 951 M.4.6 Resulting Exchange Terms For solving the coupled matrix system (13.37) for the BHE temperature T and soil temperature T s the following exchange terms have to be known: RO s.T s/, RO s .T / and R . From the preceding sections we find: 8 0 1 ˆ Ti1.zi ;t/ ˆ B C ˆ BTi2.zi ;t/C ˆ B C ˆ BT .z ;t/C ˆ B o1 i C ˆ BT .z ;t/C ˆ B o2 i C ˆ B C 2U configuration ˆ BTg1.zi ;t/C ˆ B C ˆ BTg2.zi ;t/C ˆ @ A < Tg3.zi ;t/ O s s R .T / D ˆ 0Tg4.zi ;t/1 .1 i NBHE/ (M.116) ˆ ˆ Ti1.zi ;t/ ˆ B C ˆ BTo1 .zi ;t/C ˆ @ A 1U configuration ˆ T .z ;t/ ˆ g1 i ˆ T .z ;t/ ˆ 0 g2 i 1 ˆ ˆ Ti1.zi ;t/ ˆ @ A :ˆ To1 .zi ;t/ CXA and CXC configurations Tg1.zi ;t/ where zi corresponds to the z coordinate at the BHE nodal point i, the pipe and grout temperatures are given by the analytical expressions (M.90), (M.94), (M.96), (M.104), (M.106), (M.112), and (M.114), which are dependent on the unknown soil temperature Ts.z;t/.Furthermore,itis Z T T O s i1 o1 R .T / D ▵ C ▵ dz (M.117) z R1 R2 and Z 1 1 R D δ ▵ C ▵ dz (M.118) z R1 R2 with 0 1 0 1 Ti1.z1;t/ To1 .z1;t/ B C B C B Ti1.z2;t/ C B To1 .z2;t/ C Ti1 D B : C ; To1 D B : C (M.119) @ : A @ : A Ti1.zNBHE ;t/ To1 .zNBHE ;t/ 952 M BHE Modeling: Numerical and Analytical Approaches ▵ ▵ where R1 and R2 are given by (M.86), (M.98), (M.99), (M.100), (M.107), (M.108), and (M.115) for the corresponding BHE configurations. M.5 Numerical Solution of the Local Problem In this section, the BHE equations are solved by the numerical method which is based on the ideas given by Al-Khoury et al. [8], Al-Khoury and Bonnier [7], and Al-Khoury [6]. This method is applied to the different BHE configurations by using the extended TRCM [29]. The governing BHE heat transport equations are discretized by 1D finite element discrete feature elements (DFE’s), cf. Chap. 14, which are coupled to the global heat transport equation for the soil (porous medium) via heat transfer relations. In comparison to the analytical method (Sect. M.4)the numerical method is more general and applicable both to short-term and long- term analyses, however, can be less efficient and stable in particular for long-term simulations. M.5.1 Basic BHE Equations of Heat Transport in Pipes and Grout Zones The BHE represents a closed pipe system, where a refrigerant fluid is circulating with a given velocity u. The 2U configuration consists of 8 borehole components (2 pipes-in, 2 pipes-out and 4 grout zones, Fig. M.1), the 1U configuration consists of 4 borehole components (1 pipe-in, 1 pipe-out and 2 grout zones, Fig. M.2)and each of the CXA and CXC configurations consists of 3 borehole components (1 pipe-in, 1 pipe-out and 1 grout zone, Figs. M.3 and M.4), which are schematized by the corresponding number of 1D DFE’s as shown in Fig. M.6. Their 1D heat transport equations are given as follows for the pipe(s) @T r cr k C r cr u rT r .Λr rT / D H in ˝ @t k k k k r (M.120) with Cauchy-type BC: .Λ rTk/ n D qnTk on k for k D i1; o1; .i2; o2/ and for the grout zone(s) @T gcg k r . grT / D H in ˝ @t k k k g (M.121) with Cauchy-type BC: . rTk/ n D qnTk on k for k D g1;.g2;.g3;g4// M.5 Numerical Solution of the Local Problem 953 2U 1U CXA and CXC 7 3 2 1 1 6 1 4 2 5 2 3 4 8 3 g3 i1 o1 o1 i1 g1 g2 i1 g1 o1 o2 g4 i2 g1 g2 11 9 15 10 6 4 13 14 5 8 5 6 12 16 7 pipe-in pipe-out grout Fig. M.6 BHE pipe(s)-in, pipe(s)-out and grout zone components for 2U, 1U, CXA and CXC configurations discretized by 1D DFE’s. Local node numbering is used for the vertical 1D representations of the components which are considered impervious, where r r r r Λ D . C c ˇLkuk/δ (M.122) is the hydrodynamic thermodispersion for the refrigerant, rD@=@z is defined here for the vertical 1D line direction z along the BHE ‘well’ axis and the boundary heat fluxes qnTk are summarized in Table M.2 for the related BHE components k.Note CXA CXC that ˚ff ¤ ˚ff due to the different pipe radii for pipe-in and pipe-out in a coaxial pipe installation. M.5.2 Finite Element Discretization of the BHE Equations The BHE Eqs. (M.120)and(M.121) are discretized by finite elements. Introducing the spatial weighting function w the following weak statements result for the pipe(s) components Z h i r r @Tk r w c C u rTk Crw .Λ rTk/ d˝ D ˝ @t Z k Z w qnTk d C wHk d˝ for k D i1; o1; .i2; o2/ (M.123) k ˝k and for the grout zone(s) components of the BHE Z h i g g @Tk g w c Crw . rTk/ d˝ D ˝ @t Z Z k w qnTk d C wHk d˝ for k D g1;.g2;.g3;g4// (M.124) k ˝k 954 M BHE Modeling: Numerical and Analytical Approaches / / / / g1 o1 / o1 T i1 T g1 T T T i1 o1 o1 g1 s .T .T .T .T .T CXC CXC ff fog CXC CXC CXC ff fog gs ˚ ˚ ˚ ˚ ˚ / / / / i1 g1 / i1 T T o1 g1 T T T o1 i1 g1 i1 s .T .T .T .T .T CXA CXA ff fig CXA CXA CXA fig ff gs ˚ ˚ ˚ ˚ ˚ –– / / / / / g1 / g2 g2 g1 / / T T o1 i1 T T g1 g2 T T T T i1 g2 o1 g1 g1 g2 s s .T .T .T .T .T .T .T .T 1U 1U 1U 1U fig gg fog gg 1U 1U 1U 1U fig fog gs gs ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ k ––– ––– ––– ––– / / / / g1 g2 g4 g3 g1 g1 g3 g3 T T T T .T .T .T .T 1 1 / / / 1 1 / 2U 2U 2U 2U i1 gg gg i2 o1 gg gg o2 g2 g3 g4 g1 for the BHE components ˚ ˚ T T T ˚ ˚ T .T .T .T .T k 2U 2U 2U 2U fig fig fog fog nT / / / / q ˚ ˚ ˚ ˚ g4 g2 g2 g4 / / g1 g3 g4 / g2 / T T T .T .T .T T .T i2 o1 o2 / i1 / / / 1 1 1 1 T T T T g1 g2 g3 g4 2U 2U 2U 2U gg gg gg gg T T T T ˚ ˚ ˚ ˚ g2 g1 g4 g3 g1 g2 g3 g4 s s s s .T .T .T .T 2 2 2 2 / / / / .T .T .T .T .T .T .T .T 2U 2U 2U 2U gg gg gg gg D g1 g2 g3 g4 2U 2U 2U 2U 2U 2U 2U 2U fig fig fog fog gs gs gs gs ˚ ˚ ˚ T T T T ˚ k ˚ ˚ ˚ ˚ ˚ ˚ ˚ ˚ nT 2U 1U CXA CXC q Boundary heat fluxes o2 i2 g2 g3 o1 g1 g4 Table M.2 k i1 M.5 Numerical Solution of the Local Problem 955 where the boundary heat exchange qnTk is specified in Table M.2.UsingGFEM (M.123)and(M.124) lead to the following matrix system P TP C L T D W R s T s (M.125) with 8 0 1 ˆ Pi1 0000000 ˆ B C ˆ B 0 P 000000C ˆ B i2 C ˆ B C ˆ B 00Po1 00000C ˆ B C ˆ B 000Po2 0000C ˆ B C 2U ˆ B 0000P 000C ˆ B g1 C ˆ B C ˆ B 00000Pg2 00C ˆ @ A < 000000Pg3 0 000000 0P P D ˆ g4 (M.126) ˆ 0 1 ˆ Pi1 000 ˆ B C ˆ B 0 Po1 00C ˆ @ A 1U ˆ 00P 0 ˆ g1 ˆ 000Pg2 ˆ 0 1 ˆ P 00 ˆ B i1 C ˆ @ A :ˆ 0 Po1 0 CXA or CXC 00Pg1 8 0 1 ˆ Ki1 0 Rio 0 Ri1 000 ˆ B C ˆ B 0 K 000 R 00C ˆ B i2 i2 C ˆ B C ˆ B Rio 0 Ko1 000 Ro1 0 C ˆ B C ˆ B 000Ko2 000 Ro2 C ˆ B C 2U ˆ B R 000K R R R C ˆ B i1 ig g2 g1 g1C ˆ B C ˆ B 0 Ri2 00Rg2 Kig Rg1 Rg1C ˆ @ A ˆ 00 Ro1 0 Rg1 Rg1 Kog Rg2 ˆ ˆ 000 Ro2 Rg1 Rg1 Rg2 Kog <ˆ 0 1 Ki1 Rio Ri1 0 B C L D ˆ B Rio Ko1 0 Ro1 C (M.127) ˆ @ A 1U ˆ Ri1 0 Kig Rg1 ˆ 0 R R K ˆ o1 g1 og ˆ 0 1 ˆ K R R ˆ B i1 io i1C ˆ @ A ˆ Rio Ko1 0 CXA ˆ ˆ Ri1 0 Kig ˆ ˆ 0 1 ˆ K R 0 ˆ B i1 io C ˆ @ A :ˆ Rio Ko1 Ro1 CXC 0 Ro1 Kog 956 M BHE Modeling: Numerical and Analytical Approaches 8 0 1 8 0 1 8 0 1 ˆ 0 ˆ W ˆ T ˆ B C ˆ B i1 C ˆ B i1 C ˆ B 0 C ˆ BW C ˆ BT C ˆ B C ˆ B i2C ˆ B i2C ˆ B 0 C ˆ BW C ˆ BT C ˆ B C ˆ B o1 C ˆ B o1 C ˆ B 0 C ˆ B C ˆ B C ˆ B C ˆ BWo2 C ˆ BTo2 C ˆ B C ˆ B C ˆ B C 2U ˆ B RsC ˆ BWg1C ˆ BTg1C ˆ B C ˆ B C ˆ B C ˆ B RsC ˆ BWg2C ˆ BTg2C <ˆ @ A <ˆ @ A <ˆ @ A Rs Wg3 Tg3 s R D Rs ; W D W ; T D T ˆ 0 1 ˆ 0 g41 ˆ 0 g41 ˆ 0 ˆ Wi1 ˆ Ti1 ˆ B C ˆ B C ˆ B C ˆ B 0 C ˆ BWo1 C ˆ BTo1 C ˆ @ A ˆ @ A ˆ @ A 1U ˆ Rs ˆ Wg1 ˆ Tg1 ˆ ˆ ˆ ˆ Rs ˆ 0Wg21 ˆ 0Tg21 ˆ 0 1 ˆ ˆ ˆ 0 ˆ Wi1 ˆ Ti1 ˆ @ A ˆ @ A ˆ @ A :ˆ 0 :ˆ Wo1 :ˆ To1 CXA or CXC Rs Wg1 Tg1 (M.128) and similarly for TP ,where 8 X Z ˆ r r e <ˆ c Ni Nj d˝ for k D i1; o1; .i2; o2/ e e Z˝k Pk D X (M.129) ˆ g g e : c Ni Nj d˝ for k D g1;.g2;.g3;g4// e e ˝k Ki1 D Ci1 C Ri1 C Rio Ki2 D Ci2 C Ri2 Ko1 D Co1 C Ro1 C Rio Ko2 D Co2 C Ro2 (M.130) Kig D Gig C Ri1 C 2Rg1 C Rg2 C Rs Kog D Gog C Ro1 C 2Rg1 C Rg2 C Rs Z X r r e Ck D Ni c u rNj rNi .Λ rNj / d˝ for k D i1;o1;.i2;o2/ e Xe Z˝k g e Gig D Gog D rNi . rNj /d˝ for k D .g1; g2; g3; g4/ ˝e Xe Z k e Wk D Ni Hk d˝ for 8k e e ˝k (M.131) and the boundary heat exchange matrices as listed in Table M.3,inwhich.i; j D 1;:::;NBHE/ and e runs over the associated numbers of 1D DFE’s. The symbols e e e e ˝i1;i2; i1;i2 denote the domain and surface of pipe(s)-in, ˝o1;o2 ; o1;o2 for M.5 Numerical Solution of the Local Problem 957 e e e pipe(s)-out and ˝gi; gi .i D 1;:::;G/ for the grout zones ( g1 g4 for all grout zones) of finite element e (G is the number of grout zones: 4 for 2U, 2 for 1U and 1 for CXA and CXC configurations). Furthermore, the following heat transfer matrices are defined: 8 < 0000 Rs Rs Rs Rs 2U T Rs D R s D 00 R R 1U (M.132) : s s 00 Rs CXA or CXC and R D GRs (M.133) which appear in the coupled matrix system (13.44). Analytical integration of matrices (M.129)–(M.131) and those of Table M.3 is performed for linear 1D elements in accordance with Sect. H.1 of Appendix H.In doing so, the volume and surface elements are A ze d˝e D Adz D AjJ ejd D d 2 S ze d e D Sdz D SjJ ejd D d (M.134) 2 where jJ ejD ze=2 is the Jacobian (H.4), ze is the length of element e, is the local coordinate, S is the specific exchange surface given in Table M.1 and A is a cross-sectional area given in Table M.4 for the inner pipes and grout zones. The used linear 1D element e consists of two nodes with each DOF temperature variables for the BHE components i1;i2;o1;o2;g1;g2;g3;g4 and with only one temperature variable for the soil s as depicted in Fig. M.7. In using these relationships we find the matrices for element e as follows: i r r e 21 P e D Ai c z i1;i2 6 12 i r r e 21 P e D Ao c z o1;o2 6 12 (M.135) Ai gcg ze 21 P e D g g1;g2 6 12 Ao gcg ze 21 P e D g g3;g4 6 12 958 M BHE Modeling: Numerical and Analytical Approaches e e e d d d j j j N N N i i i N N N CXC CXC CXC gs fog ff ˚ ˚ ˚ e e e o1 i1 g1 Z Z Z e e e 0 X X X e e e d d d j j j N N N i i i N N N CXA CXA gs CXA ff fig ˚ ˚ ˚ e e e i1 o1 g1 Z Z Z e e e X 0 00 X X e e e e d d j j d d N N j j i i N N N N i i N N 1U 1U gg gs 1U ˚ ˚ 1U fog fig ˚ ˚ e e e e i1 o1 g1;g2 g1;g2 Z Z Z Z e e e e X X X 000 X e e e e e d d d j j d j d j j N N N i i i N N i i N N N 1 2 N N 2U 2U 2U gg gg gs 2U fog 2U fig ˚ ˚ ˚ ˚ ˚ g4 g4 g4 e e e e e i1;i2 o1;o2 g1 g1 g1 Z Z Z Z Z e e e e e 2UX X 00 X X 1UX CXA CXC Boundary heat exchange matrices i1 i2 o1 o2 io g1 g2 s R R R R Table M.3 R R R R M.5 Numerical Solution of the Local Problem 959 Table M.4 Cross-sectional areas A for the BHE configurations according to Figs. M.1–M.4 A 2Ua 1U CXA CXC i i 2 i 2 i 2 o 2 i 2 Ai .ri / .ri / Œ.ri / .ro / .ri / Ai .ri /2 .ri /2 .ri /2 Œ.ri /2 .r o/2 o o o o o i 2 2 2 Ai 1 D .r o/2 1 D .r o/2 D .r o/2 – g 4 4 i 2 4 i 4 i o 1 D2 o 2 1 D2 o 2 D2 o 2 Ag 4 4 .ro / 2 4 .ro / – 4 .ro / a In case of 2U exchangers it is assumed that the radii for the two pipes-in and two pipes-out are i i i o o o i i i o o o equal, i.e., it is defined: ri D ri1 D ri2, ri D ri1 D ri2, ro D ro1 D ro2 and ro D ro1 D ro2 2U, DOF = 8 1U, DOF = 4 CXA/CXC, DOF = 3 soil, DOF = 1 1 TTT T T TTT 1 Ti TTog T g 1 T TT 1 T i12i oogg 1 2 1234gg1 11121 i111og1 s 1 e e e e 2 TTTi T T TTT 2 T TT T 2 T TT 2 T 12i oogg 1 2 1234gg2 i1112og g2 i111og2 s 2 Fig. M.7 1D element e used for BHE components of 2U, 1U, CXA or CXC configuration and for soil s i r r 11 i kΛr k 1 1 Ce D Ai c u C Ai i1;i2 2 11 ze 11 (M.136) i r r 11 i r 1 1 Ce D Ao c . u/ C AokΛ k o1;o2 2 11 ze 11 Ai g 1 1 Ge D g ig ze 11 (M.137) Ao g 1 1 Ge D g og ze 11 i e 1 W e D Ai Hi1;i2 z i1;i2 2 1 i e 1 W e D AoHo1;o2 z o1;o2 2 1 (M.138) Ai H ze 1 W e D g g1;g2 g1;g2 2 1 Ao H ze 1 W e D g g3;g4 g3;g4 2 1 960 M BHE Modeling: Numerical and Analytical Approaches e 21 e 21 Re D ˚fig Si z D z i1;i2 6 6Rfig 12 12 e 21 e 21 Re D ˚fog So z D z o1;o2 6 6Rfog 12 12 e 21 e 21 Re D ˚ff Sio z D z io 6 12 6Rff 12 (M.139) e 21 e 21 Re D ˚gg1 Sg1 z D z g1 6 6Rgg1 12 12 e 21 e 21 Re D ˚gg2 Sg2 z D z g2 6 6Rgg2 12 12 e 21 e 21 Re D ˚gs Sgs z D z s 6 12 6Rgs 12 If the advective part in the heat transport equations of the BHE pipes becomes dominant, wiggles in the temperature solutions can occur and the spatial discretiza- tion with the standard GFEM becomes insufficient. 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Methods Eng. 11(1), 131– 143 (1977) Index Absorption, 55 Aquifuge, 46 Accuracy, 7 Aquitard, 46, 420, 509, 747 Activity, 53, 174 Arrhenius kinetics, 188–189 coefficient, 54 Assembly, 262, 279–283 Adams-Bashforth method, 299, 302 parallelization, 282–283 Adaptive Average operator mesh refinement, 776–786 Boltzmann, 62, 65 time marching process, 304 intrinsic mass, 62, 65 Adiabatic boundary condition, 205 intrinsic volume, 62, 65 Adsorption, 55 and quantities, 61–63 function, 118, 180 Averaging theorem, 63, 65 isotherms, 177 Axisymmetric problems, 40 Advancing front technique, 764–765 Advection, 54 Advection-dispersion equation, 244 Balance equations Algebraic multigrid method, 373–374 convective form, 71 Amplification divergence form, 70 factor, 309 general, 69 matrix, 308 macroscopic, 66–76 Anisotropy, 227–235, 435 vertically averaged, 76–79 axis-parallel, 233 Balance error, 379, 472 randomly distributed, 438 Bandwidth, 357 Springer’s method, 231–233 Barycentric variables, 68 thermal factor, 234 BASD technique, 416–421, 531, 534 Aperture, 47, 152 Base vectors, 24, 718, 867, 873 corrected, 163 Basis functions, 248 hydraulic, 159 finite element, 266–274 Aquiclude, 46 Benchmark, 7, 583 Aquifer, 46 Bifurcation, 553 averaging, 64–65 Borehole, 222 balance equations, 76–79 abandoned, 224, 746 confined, 46, 408, 429 heat exchanger, 673, 683–692, 925–960 perched, 47 analytical solution, 697, 942–952 phreatic, 47 coaxial pipe, 927 system, 46 double U-shape pipe, 925 unconfined, 47, 406, 413, 429 heat transfer coefficients, 940–941 H.-J.G. Diersch, FEFLOW, DOI 10.1007/978-3-642-38739-5, 987 © Springer-Verlag Berlin Heidelberg 2014 988 Index numerical solution, 952–960 Capillary barrier, 514–522 single U-shape pipe, 926 Capillary diffusivity, 451 thermal resistances, 928–940 Capillary pressure, 109–111 Boundary conditions, 48, 193–226 assumption, 122 1st kind (see Dirichlet-type) head, 833 2nd kind (see Neumann-type) Celia et al.’s approximation method, 458, 471 3rd kind (see Cauchy-type) Cellular convection. See Rayleigh-Benard´ adiabatic (see natural (no-heat flux)) convection Cauchy-type, 49, 196, 200, 205 Channel flow, 154, 722 Dirichlet-type, 48, 195, 199, 204 Chemical implementation, 349–350 component (see species) essential, 48 constituent (see species) free exit (see outflow) equilibrium, 175 free surface, 216–217 kinetics, 172 gradient-type, 220–221 potential, 75, 106 integral, 219–220 reaction, 167–191, 625 mixed (see Robin-type) species, 52 multilayer well, 221–224 essential number, 119 natural Chezy law, 155 no-flux, 49, 195 Cholesky method, 356 no-heat flux, 205 Chord slope approximation, 906–907 no-mass flux, 199 Clausius-Duhem Neumann-type, 48, 195, 199, 204 entropy production, 778 outflow, 224–226 inequality, 76, 825–828 Robin-type, 49, 201, 206 Clogging. See Colmation seepage face, 217–218 Coating factor, 171 surface ponding, 218 Coleman and Noll method, 84, 95, 825–828 Boundary element method, 240 Collocation Boundedness, 6, 317 point, 255 Boussinesq approximation. See Oberbeck- subdomain, 256 Boussinesq approximation Colmation, 196 Bowyer-Watson algorithm, 767 Compressibility Brackish water, 45, 541 fluid, 103 Brine, 45, 100, 541 skeleton, 117 Brinkman term, 119 Concentration, 53 Brooks-Corey relationship, 447, 834–835, mass, 53, 125 838–839 maximum, 127 Budget analysis, 391–398 molar, 53, 177 discrete feature, 722–724 Condition number, 360 flow, 427–429, 485–487, 573–575 Conduction, 55 heat, 573–575, 681–683 hydraulic, 55 mass, 573–575, 640–642 thermal, 55, 847–850 Buoyancy coefficient, 124, 128, 546 Conductivity Buoyancy-driven flow, 537–624 hydraulic, 124 Buoyancy number. See Turner number isotropic, 234 Burnett and Find’s dispersion tensor, 626 relative, 111 thermal, 97, 116, 234 Confined aquifer, 46, 408, 429 Cea’s´ lemma, 256 Conformity, 757 CAD, 12, 758 Conjugate gradient method, 362–363 Capacity Conservation moisture, 449 energy, 74 inverse, 451 mass, 71–72 specific heat, 105 momentum, 73 Index 989 principles, 66 Cubic law, 152 species mass, 72–73 Curl, 39 Conservativity, 6, 399 local, 399 Consistency, 6 Dam seepage, 522–524 Consistent boundary flux method, 391–398 Danckwerts condition. See Boundary Consistent mass, 293 conditions, natural (no-mass flux) Consistent velocity approximation, 568–573 Darcy Constitutive theory, 80–111 equation, 121, 122, 151 Constraints, 208–216 flow, 94 Content velocity, 116, 139, 148, 405, 427, 485, 909 moisture, 497 depth-integrated, 407 soil water (see Moisture) Darcy-Brinkman-Forchheimer equation, 121 Continuum, 3, 57 Darcy-Forchheimer equation, 121 mechanics, 9 Darcy-Peclet´ number, 551 Convection, 54, 548–557 Darcy-Weisbach law, 155 cellular (see Rayleigh-Benard)´ Davies relationship, 54 double-diffusive, 138, 554–557, 611–624 Decay, 118, 167, 181–182, 627, 642, 654 finger, 556, 611 rate, 181 forced, 54 Deformation tensor, 81 free, 54, 552–554 rate of fluid phase, 83 mixed, 54 rate of solid phase, 81 monotonic, 555 Degradation kinetics, 187–188, 652–654, oscillatory, 554, 555, 557 663–671 Rayleigh-Benard,´ 548 Degrees of freedom, 255, 266, 402, 404 regimes, 552–557 Delaunay-Vorono¨ı method, 766–770 stable, 555 Density, 53 stationary, 553 bulk mass, 54 thermohaline, 138, 554–557 entropy, 53 Convective form fluid, 102–103 advection-dispersion equation, 259–260 internal energy, 53 balance equation, 71 mass, 53 Convergence, 6, 360 momentum, 53 criterion, 376, 472–473 ratio, 128 quadratic, 377 specific, 53, 54 radius, 377, 379 thermally variable, 829–832 super-, 385 Density-driven flow, 537–624 Coordinates Derived quantities, 382–398 Cartesian, 28 De Saint-Venant equation, 154 cylindrical, 33 Desorption, 55 Eulerian, 29 rate-limited, 654–659 Lagrangian, 29 Diagonal tensor, 26 material, 29 Diffusion, 54 Coordinate system, 28 coefficient, 100 orthogonal, 31 molecular, 99 rotated, 35 tensor, 99 Coordinate transformation, 31, 267, 274, 283, thermo-, 99 291, 577, 760, 761 Diffusion-type flow equation, 155, 711 Courant-Friedrichs-Lewy condition, 316 Diffusivity, 86 Courant number, 314 capillary, 451 Crank-Nicolson scheme, 298, 310, 312, 347 thermal, 549 Cross effects, 94, 98, 99 Directional cosines, 35, 228–230, 716 Cross product. See Vector product DFE, 716–720 Crout method, 353–357 Direct solution methods, 352–358 990 Index Discrete feature, 141–165 inequality, 87–89 elements, 425, 711 Envelope, 357 modeling, 711–750 Equations of state, 89, 102–108 Disjoint domain partitioning, 282 Equilibrium Dispersion chemical, 175 Burnett and Find’s tensor, 626 condition, 547, 949, 950 hydrodynamic, 99 constant, 175 mechanical, 70, 99 hydrostatic, 540 numerical, 346, 562–563 pre-, 176 Scheidegger-Bear model, 97, 99 restrictions, 85–86 thermo- thermodynamic, 85, 116 hydrodynamic, 97 two-site, 659 mechanical, 97 Error norm, 250 Dispersivity Errors, 5, 250–252, 344–349 longitudinal, 97, 99 Essential boundary condition, 48 numerical, 328 Eulerian angles, 36, 228 transverse, 97, 99 Euler method Displacement, 30, 81, 96 backward, 298, 299 Dissipation, 74 forward, 298, 299 anisotropic balancing, 328 Excavation, 483 Distribution coefficient, 131, 179 Expansion coefficient, 103 Divergence, 39 solutal, 103, 128 Divergence form thermal, 103 advection-dispersion equation, 257–259 variable thermal, 103 balance equation, 70 Exponential relationship, 836, 840 Divergence theorem. See Gauss’s integral Extensive quantity, 52, 66 theorem Divergenceless. See Solenoidal Domain, 48 Fault, 142, 711 partitioning, 282 FEFLOW, ix Dot product. See Scalar product development, x Double-diffusive convection, 554–557, Fick’s law, 99, 156 611–624 Fill-in, 357 fingering, 556, 611 reduction, 357–358 Double dot product, 25 Filtration velocity. See Darcy velocity Drainage, 526–533 Fingering regime, 556 Drawdown, 429 Finite difference method, 240, 256 Dufour effect, 98 Finite element mesh, 262, 757–786 Dupuit assumption, 64 Finite element method, 239–404 Dupuit-Forchheimer relationship, 116 Bubnov-Galerkin (see Galerkin) Dyadic product, 24 extended, 242 Galerkin, 256 least squares, 256, 332–339 Einstein’s summation convention, 23 Petrov-Galerkin, 256, 321–327 Elasticity matrix, 96 Finite volume method, 240, 256 Elder problem, 593–601 Fixed point iteration, 376 long-heater, 782 Flooding, 533–536 Energy, 52 Fluctuation, 63 internal, 53, 103–106 Forced convection, 54 kinetic, 53 Forchheimer total, 66 coefficient, 94 Entropy, 52 flow, 94 balance, 74–75 term, 121 Index 991 Fourier heat flux, 98, 158, 847 Fourier flux, 98, 158, 847 Fracture, 47, 60, 141, 142, 711, 724–729, radiative, 676 736–740 specific capacity, 105 inclined, 731–735 transfer, 847–854 network, 741–744 coefficient, 205, 850, 940–941 Fractured porous rock, 47 parallel, 852 Free convection, 54, 552–554 serial, 851 Free surface, 51, 131, 145, 216 transport, 673–709 computation, 412–424 Heaviside function, 415 Freshwater, 45, 103, 544 Hele-Shaw cell, 538, 583, 593, 601, 611 Freundlich adsorption, 131, 179, 626 Henry adsorption, 131, 178–179, 626 Friction, 74, 119, 121 Henry problem, 585–589 slope, 154 Heterovalence, 172 Frolkovic-Knabnerˇ algorithm, 572–573, Hook’s law, 95 909–917 Hoopes and Harlemann’s two-well problem, Functional spaces, 250–252 340 Horton-Rogers-Lapwood problem, 548–550 Hydraulic Galerkin method, 256 aperture, 159 Gardner’s problem, 488–489 conduction, 55 Gaussian elimination, 352–353 conductivity, 124 Gauss-Legendre quadrature, 288 head, 123 Gauss points, 289, 291, 386, 389, 487 radius, 145, 159 Gauss’s integral theorem, 40, 258, 259, 265, Hydrostatic condition, 546–548, 569–570, 584, 395 921–923 Geothermal Hysteresis, 110, 479–483 energy extraction, 2, 538, 683 energy storage system, 2, 538 heat pumps, 673 Identity matrix. See Kronecker symbol Ghyben-Herzberg relation, 540, 545 Incidence matrix, 279 Gibbs free energy, 85 Index notation, 23 Gibbs’ notation. See Symbolic notation Infiltration, 47, 134, 196 GIS, 12, 759 Initial conditions, 48, 194 GMRES method, 364–365 Inner product, 254 Gradient operator. See Nabla operator, 63, 65, Intensive quantity, 52, 66 77, 82, 144, 713 Interface, 51 Gravity, 73, 88, 123, 131, 537, 546, 570 boundary, 51 projected, 575–580 graphical, 13 GRIDBUILDER, 765 sharp, 544 Groundwater, 1, 45 Interpolation, 802–808 divide, 47 transfinite, 762 flow, 405–444 Ion exchange, 177 perched, 47 Irrotational, 42 recharge, 47, 134 Isoparametric elements, 268, 859–863 whirls, 435 Isotropy, 234 Iterative solution methods, 358–374 Hagen-Poiseuille flow, 151, 159, 222 Half-life, 181 Jacobian, 30, 33, 37, 284 Hanging nodes, 757, 775 inverse, 285 Haverkamp relationship, 835, 840 matrix, 32, 378 Heat partial, 378, 415 flux, 96–98 Juxtaposition, 11 992 Index Kantorovich method, 248 fixed, 422 Kinematics, 80–82 generation, 760–770 Kinetics moving, 416 Arrhenius type, 188–189 nonconformal, 757 chemical, 172 optimal, 691 degradation type, 187–188 prismatic, 772–774 editor, 190, 652, 658, 662, 670 refinement, 774–776 Monod type, 189–190 smoothing, 771–772 Kirchhoff integral transform, 453 structured, 758 Kriging, 13 superelement, 759 Kronecker symbol, 24 unstructured, 758 Meshless method, 240 Method of characteristics, 240 Lanczos Method of weighted residuals, 252–256 bi-conjugate gradient stabilized method, Michaelis-Menten mechanism, 168, 183–186 367 Mixed conjugate gradient square method, 366 boundary conditions (see Boundary Langmuir adsorption, 131, 178, 626 conditions Robin-type) Laplacian convection, 54, 579 equation, 43 finite element formulation, 402 operation, 39 Richards’ equation s form, 449 smoothing, 772 Model Law of mass action, 175 conceptual, 5 LBB condition, 404 development, 9 Leaching, 202 final equations, 121–137 Leakage, 50, 196 mathematical, 4 Leipniz’s integral rule, 41 numerical, 9 Levi-Civita tensor, 24 simulation, 9 Lewis number, 551, 555, 612 standard equations, 138–139 Locking, 282 MODFLOW, 412 Lumping, 295–296 Moisture capacity, 449 inverse, 451 Macroscopization, 69 content, 497 Manning-Strickler law, 155, 159 Molar concentration, 53, 177 Mapping, 32 Molarity, 53 Mass Monod kinetics, 168, 189–190 concentration, 53, 125 Monovalence, 172 maximum, 127 Moving density, 53 interface, 145 fraction, 54 mesh, 416–421 matrix surface, 131, 133 consistent, 293 Mualem assumption, 838 lumping, 295–296 Multigrid methods, 367–374 species flux, 98–100 algebraic, 373–374 transfer coefficients, 201 geometric, 370–373 transport, 625–671 Multilayer well Material derivative, 31 boundary condition, 221–224 Matrix solution methods, 351–374 incorporation, 425–426 Mesh Multilevel nested dissection method, 358 adaptive refinement, 776–786 alignment, 757 conformal, 757 Nabla operator, 31, 34 derefinement, 774–776 Navier-Stokes equation, 92, 150, 151, 154, 402 Index 993 Newton iteration method, 377–379, 415, 465, Petrov-Galerkin method, 256, 321–327 467, 803 least square, 332–339 modified, 380 Phase, 51 one-step, 382, 465, 474, 564 fluid, 58, 122 quasi-, 380 gaseous, 58 Newton-Raphson method. See Newton liquid, 58 iteration method nonwetting, 109 Newton’s law of cooling, 50, 850 solid, 58 Newton’s viscosity law, 92, 150 stagnant, 122 Newton-Taylor law, 155 wetting, 109 Nodal reordering, 357–358 Phenomenological equations, 86, 89–100 Non-Fickian law, 100, 581–583, 606 Phreatic aquifer, 47 Norm Phreatic surface. See Free surface energy, 251 Picard iteration method, 375–377, 415, 421, error, 250 458 maximum, 252 one-step, 382, 564 root mean square, 252 Piezometric head. See Hydraulic head vector, 25 Pinching, 774 Normalized vector, 25 Poisson’s ratio, 95 Numerical dispersion, 346, 562–563 Pollock’s tracking method, 789–792 Nusselt number, 551, 929 Pore diameter, 150 Reynolds number, 119 Oberbeck-Boussinesq approximation, 125–127 scale, 479 extended, 126 size distribution index, 483, 834, 839 Optimality, 6 velocity, 549, 788 Galerkin method, 855–857 Porosity, 58 Onate˜ and Bugeda’s criterion, 781 drainable (see Specific yield) Zienkiewicz and Zhu’s criterion, 780 fillable (see Specific yield) ORTHOMIN method, 363–364 time-varying, 483–485 Outflow boundary condition, 224–226 Porous medium, 57–139 Overland flow, 154, 722 Potential chemical, 75 flow, 42 Peclet´ number, 314 function, 42 Partial differential equations pseudo-, 43 elliptic, 246 reference, 919 hyperbolic, 246 Prandtl number, 929 parabolic, 246 Preconditioning, 360–362 Particle tracking, 786–796 Predictor-corrector methods, 299–308 Pollock’s method, 789–792 Pressure Runge-Kutta method, 792–796 aquifer, 46 Partitions, 282 capillary, 109 Pathline, 44, 786 head, 123, 833 Perched mechanical, 91 aquifer, 47 thermodynamic, 84, 91 water table, 509–512 Primary variable switching technique, 454, Permeability, 94, 111 465–477, 903–907 relative, 111, 837–841 Principle saturated, 111 admissibility, 83 tensor, 94, 111 equipresence, 83 Permutation symbol, 24 objectivity, 83 Perturbation, 552, 553, 596, 605, 611, 623 Profiling, 350 initial, 620 Projection, 27, 331, 807 994 Index Pseudopotential, 43 Reverse Cuthill-McKee method, 358 Pseudo-unsaturated flow, 422 Reynolds number, 119, 929 Pumping wells, 198 Reynolds’ transport theorem, 41 Richards’ equation, 139, 447 s-form, 450 Rational thermodynamics, 3 mixed s form, 449 Raviart-Thomas element, 404 standard h-form, 449 Rayleigh-Benard´ convection, 548 Richardson iteration, 359 Rayleigh number Robustness, 8 critical, 552, 553 Roughness coefficients, 155, 163 effective, 563 corrected Manning, 164 solutal, 550 Runge-Kutta tracking method, 792–796 thermal, 550 Rayleigh-Ritz method, 252 Reaction, 100–101 Saline water, 45 bulk rate, 101 Salt dome problem, 589–593 chemical, 167–191 Salt lake problem, 601–605 consecutive, 168, 182–183 Saltpool problem, 607–610 decay, 181–182 Saltwater, 45, 107, 541, 544 equilibrium, 101 encroachment (see Intrusion) heterogeneous, 100, 167 intrusion, 45, 538 homogeneous, 100, 167 upconing, 538, 591 irreversible, 101, 167 Saturated zone, 47 kinetic, 101, 167 Saturation, 109 kinetics editor, 190, 652, 658, 662, 670 Scalar product, 23 reversible, 101, 167 Scheidegger-Bear dispersion model, 97, 99, serial-parallel, 649–652 156, 224, 626 stoichiometry, 172 Schur complement, 689 surface, 168 Schwarz’s inequality, 254 Red-green triangulation, 775 Seepage Reference temperature, 84, 103, 106, 829 face, 217, 523 Refrigerant, 684 velocity (see Darcy velocity) velocity, 931 Self-adjoint differential operator, 253 Relative permeability, 111, 837–841 Sequential iterative approach, 566, 635 Representative elementary volume, 58–60 Shape functions, 266, 859–863 aquifer, 64 global, 268–274 Residual, 253, 254, 300, 377, 379, 467, 469, local, 268–274 472 Sharp interface approximation, 544–546 control, 474 Shear error, 255, 472 modulus, 95 vector, 361 strain, 82 Resistance, 50 stress, 92, 154 specific, 50 Sherwood number, 551, 597 thermal, 207, 851 Shock capturing, 329–332 thermal, 851 Simulation model, 9 borehole, 853 Smoothing internal borehole, 853, 854 global, 384–385, 571 Retardation, 180–181 Heaviside function, 415 derivative, 119, 131 local, 385–388, 571 factor, 119, 131 mesh, 771–772 thermal, 549 Soil water content. See Moisture, content Retention curve, 111, 833–846 Solenoidal, 42 drying, 479 Solute, 102 wetting, 479 single-species, 102, 628 Index 995 Solvent, 102 Superconvergent patch recovery, 388–390 Soret effect, 99 Supercriticality, 556 Sorption, 55 Superelement mesh, 759 Freundlich, 131 Surface, 51 Henry, 131 condition, 145 isotherm, 118 material, 145 kinetic, 659–662 ponding, 218 Langmuir, 131 reaction, 168 Sorptive parameter, 453 runoff, 47 Species. See Chemical, species specific exchange, 941 Specific stress, 147 density, 53 velocity, 134 heat capacity, 105 water, 45 resistance, 50 Symbolic notation, 23 yield, 135 Symmetric tensor, 26 Spectral element method, 240 Symmetrization, 332 Spline approximation, 111, 447, 841–846 Springer’s method, 231–233 Temperature, 75, 106 Stability, 6, 317, 344–349 reference, 84, 103, 106, 829 time integration, 308–317 system, 116 Static condensation, 688 Tensor product. See Dyadic product Steady state, 55, 138 Test functions. See Weighting functions approximation, 175 Theis’ problem, 429 Stoichiometric coefficient, 172 Thermal Stoichiometry, 172 conduction, 55 Stokes’ assumption, 92, 150 conductivity, 234 Stokes’ theorem, 41 resistance, 851 Storage coefficient borehole, 853 specific, 128 internal borehole, 853, 854 thermal, 549 resistance and capacity model, 685 Storativity. See Storage coefficient specific resistance, 207, 851 Strain, 96 Thermodispersion normal components, 82 hydrodynamic, 97 pseudovector, 81 mechanical, 97 shear components, 82 Thermodynamics, 3, 9 tensor, 73, 81, 147 first law of, 52, 66, 74 rate, 81 rational, 3 Streamfunction, 43, 798 second law of, 52, 69, 75–76, 79 Streamline, 43, 787 Thermohaline convection, 554–557 computation, 796–802 Time integration, 293–317 stabilization, 333 explicit scheme, 298 upwind, 327–329 implicit scheme, 298 Stress predictor-corrector methods, 299–308 deviatoric fluid tensor, 86, 91–93 stability, 308–317 deviatoric solid tensor, 96 trapezoid rule, 298, 299 solid tensor, 94–96 Time step, 294 surface, 147 adaptive control, 299, 304–308 wind, 154 critical, 311 Subsurface target-based control, 477 hydrology, 122, 405 Tortuosity, 100, 838 water, 1, 45 Total balance error, 472 Suction, 833 Total dissolved solids, 107, 540 Superconvergence, 385 Trajectory, 44, 488 996 Index Transfer independent, 83 coefficients, 196 switching, 454, 465–477, 903–907 heat, 205 Variably saturated porous media, 445–536 mass, 201 Variational functional, 253 condition, 50 Vector product, 23 Transfinite interpolation, 762 Velocity, 31, 38, 723 Transformation barycentric, 68 coordinate, 31, 267, 274, 283, 291, 577, consistent, 568–573 760, 761 continuous, 383, 788 methods, 453–455 Darcy, 116, 139, 148, 383, 405, 427, 485, Transmissivity, 136, 227, 230 909 Transport depth-integrated, 407 mapping, 760 divergenceless (see Solenoidal) theorem, 63, 65 interface, 51, 63, 78, 146 Transpose, 25 normal, 49 Trapezoid rule, 298, 299, 302 particle, 67 Trial functions. See Basis functions pore, 549, 788 TRIANGLE, 770 refrigerant, 931 Turner number, 551, 591, 612 relative, 82, 83, 825 solid phase, 81 surface, 134 Unconfined aquifer, 47, 406, 413, 429 total, 723 Unit matrix. See Kronecker symbol Verification, 7 Unit vector, 25 Virtual radius, 690 Unsaturated Viscosity flow, 445–536 dilatational, 91 porous media, 139 dynamic, 91, 106–108 zone, 1, 47, 134 reference fluid, 107 Upstream weighting, 487–488 relation function, 108, 124 Upwind, 317–339, 562–563 Void space, 58, 109 full, 325, 329 Voigt notation, 81 least squares, 332–339 Vorticity, 39, 799 methods, 317–339 function, 799 parameter, 321 optimal, 325 shock capturing, 329–332 Water table, 47, 133, 429 streamline, 327–329 Weak forms, 257–262 Upwinding. See Upwind, methods Weighting functions, 254, 255, 321, 330 Well bore, 222, 425 Well doublet system, 692–696 Vadose zone. See Unsaturated, zone Well-type singular point conditions, 198, 203, Validation, 5, 7 207 van Genuchten-Mualem relationship, 837–838 Wells, 198 van Genuchten relationship, 447, 833–834 Whirls, 435 modified, 839–840 Wiggles, 6, 318, 320, 960 Variable-density flow, 537–624 Variables dependent, 83 Young’s modulus, 95