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Nomenclature Appendix A Nomenclature A.1 Roman Letters AL2 area; A˛ L2T 2 Helmholtz free energy of ˛phase; BL aquifer thickness; bL hydraulic aperture; 3 1bk 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; CL1 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 ML1T 2 Young’s (or elastic) modulus of the solid phase; E# L2T 2 activation energy; e 1 Dg=kgk, gravitational unit vector; e.;;/ 1 evector 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 ML1T 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; gLT2 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 k0g=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 L1 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 ML1T 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; PLT1 groundwater recharge (infiltration rate) on a phreatic surface; Pet 1 thermal Darcy-Peclet´ number; Pg 1grid(mesh)Peclet´ number; Pr 1 Prandtl number; p ML1T 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
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