A Area m2

Ar Flow area m2

A0 Initial area m2

a Vectorial acceleration (eq. 2.79a) m/s2 [ad] Dimensionless parameter ad AMP Annual temperature amplitude in spring-and groundwater K B Skempton coefficient (chapter 2) ad b Biot-Willis coefficient (chapter 2) ad ba Aquifer thickness m bk Unitary basis for interpolation ad bs Thickness of semiconfining layer m b0 Constant parameter associated with the rock pore size ad C Biot poroelastic coefficient (chapter 2) Pa C(x) Molecular concentration of the substance mol/m3

CB Matrix of the poroelastic coefficients Pa CB Drained bulk compressibility 1/Pa CD Consolidation diffusion coefficient m2/s Cf Isothermal compressibility of the fluid 1/Pa Cm Mass fraction of NaCl kg/kg = ad Cp Heat capacity at constant pressure J/K Cs Non-jacketed volumetric compressibility 1/Pa Csol Solute concentration in the source fluid (volume referred:

mol/m3 or kg/m3) mol/kg or

kg/kg = ad CT Bulk isothermal compressibility 1/Pa CU Undrained bulk compressibility 1/Pa CV Volumetric heat capacity at constant volume J/m3/K Cw Isothermal compressibility of water 1/Pa C0 Experimental constant ad C Unjacketed compressibility of the pore volume 1/Pa

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ck Compaction coefficient 1/m ci Specific heat capacity at constant pressure (i = f , l, r, s, v, w) J/kg/K cm Salinity concentration (mass fraction) % or ad cp Isobaric specific heat capacity J/kg/K cV Specific heat capacity at constant volume J/kg/K Coi Courant’s magnitude (i = x, y) ad D Tensor of molecular diffusion (section 4.8.1) or dispersion tensor

(section 4.8.2 and 4.8.3) m2/s

D/Dt Advective derivative 1/s D∗ Molecular diffusivity m2/s Df Fractal dimension – DI Eigenvalues of the diffusion tensor D (I = X ,Y ,Z) m2/s Dm Mechanical or hydrodynamic dispersion coefficient m2/s Dv Symmetrical tensor of spatial changes of fluid velocity 1/s dg Representative grain diameter for the porous medium m dni Nodal distance m d0 Initial diameter m dpw/dt Pressure drawdown in the well Pa/s dS Differential surface or area m2

dT/dz Temperature gradient K/m dVi Differential volume (i = f ,B, s) m3 dV Differential porous volume m3

E Young’s modulus of elasticity (chapter 2) Pa E2rr Error of quadratic order in the approximation –Ei Energy flow (i = f , l, r, v) J/m2/s E1 Vector space of linear functions on the meshM1 – E1(u) Exponential integral – ei Volumetric internal energy of (i = f , l, v, r, s) J/m3 F Force N FE1 Total energy flow, single-phase fluid J/m2/sFE2 Total energy flow, two-phase fluid J/m2/s Fe Electrical resistivity factor adFi Vectorial flow of mass (i = l, v, r) kg/m2/sFM1 Momentum of the fluid, per unit volume of porous rock kg/m2/sFM2 Momentum of the two-phase fluid, per unit volume of rock kg/m2/s Fni Average flux at the interface Sni kg/m2/s Fs Helmholtz free internal energy J Fx Force in the x direction N ff Volumetric Helmholtz potential or free energy (i = f , s,w) J/m3 fv Volumetric fraction of incondensible gas in the gaseous phase ad fϕ Friction factor ad G Shear or rigidity modulus (chapter 2) Pa GP Electric geothermal power (electric Watt) We GS Gibbs free internal enthalpy J GU Rigidity undrained modulus Pa GF Depth of groundwater table below earth’s surface m g Acceleration of gravity m/s2

gf Gibbs specific free enthalpy of the fluid J/m3

gS Volumetric Gibbs potential of the skeleton J/m3

H Inverse of the poroelastic expansion coefficient Pa H (x) Heaviside distribution –

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HS Skeleton enthalpy J h Hydraulic head, piezometric or hydraulic height (water level

in the well in hydraulics) m

hi Specific enthalpy (i = f , l,w, r, s, S, v) J/kg hT Heat transfer coefficient J/s/m2/K hwell Water level in the well m I Unit matrix – ih Hydraulic gradient m/m = ad J Joule-Thomson coefficient K/Pa J (t) Determinant of the Jacobian matrix – K Tensor of absolute permeability at any point (x, y, z)

of the porous medium m2

KB Bulk modulus of the porous rock Pa Kd Distribution coefficient kg/kg = ad KH Hydraulic conductivity tensor m/s KHL Vertical conductivity of the semiconfining layer m/s KI Principal value of the hydraulic conductivity tensor KH

(matrix coefficient) (I = X ,Y ,Z) m/s

Ki Bulk modulus of the fluid (i = w, f ) Pa Ks Elastic modulus of the solid phase (chapter 2) Pa KU Elastic undrained bulk modulus Pa k Intrinsic permeability m2

ke Effective permeability m2

ki Absolute permeability (1 Darcy = 9.86923 × 10−13 m2) m2 kni Average permeability of the medium at the interface Sni m2

krg0 Gas relative permeability for imbibition ad kri Relative permeability (i = l,w, o, g, s, v) ad krj Relative permeability: ( j = nw = non-wetting phase,

w = wetting phase ad

krw0 Water relative permeability for drainage ad kTi Thermal conductivity tensor for anisotropic media

(i = l,w, o, g, s, v, r) W/m/K

kTi Thermal conductivity (i = l,w, o, g, s, v, r) W/m/K kTni Thermal conductivity at the interface Sni W/m/K kz Kozeny constant (≈ 0.5 m2) (eq. 2.2b) m2 L Tensorial differential operator – L Dripping or leakage factor 1/s Lkj Linear Lagrange polynomials of j-rank in the interval Ik – Lv Latent heat of vaporization J/mol or J/kg M Biot coefficient characterizing the fluid elastic properties Pa Mi Mass of (i = f , l, v, r,w) kg M1 Mesh of the reservoir – Mϕ Multiple-porosity reservoir – mf Fluid mass content per unit reference volume kg/m3

m0 Fluid mass content (reference or initial) kg/m3

[mD] Millidarcy (1mD = 10−3 D = 9.86923 ×10−16 m2 ≈ 10−15 m2) – N Biot Tangent modulus Pa N ∗(x) Concentration of the diffusive substance number of

NM Number of molecules of the substance number of molecules

NV Number of finite volumes –

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n Unit normal vector – Nk Multiple permeability reservoir m2

P Parameter space (chapter 7) – PA Arithmetic average – PB1 Budiansky’s average 2D – PB2 Budiansky’s average 3D – PC Continuity average – PG Geometric average – PgA General arithmetic average – PgG General geometric average – PHS Hashin-Shtrikman average 2D – PHS2 Hashin-Shtrikman average 3D – PL Linear average – PL1 Linear Lagrange average – Pp Parallel average – PS Serial average – Pw Weighted average – PwG Weighted geometric average – pa Atmospheric pressure Pa pB Fluid pressure in the Brinkman region Pa pC Capillary pressure in a porous medium Pa pCe Entry capillary pressure Pa pCmax Maximum capillary pressure Pa pD Fluid pressure in the Darcy region Pa pd Differential pressure Pa pE Equilibrium pressure Pa pe Effective pressure Pa pf Pore pressure of the fluid Pa pFmax Maximum fracture capillary pressure Pa pi Pressure (i = S, l,w, o, g, s, v) Pa pk Confining lithostatic pressure Pa pNS Pressure in the Navier-Stokes region Pa pnw Non-wetting pressure Pa psv Vapor saturation pressure for a flat interface Pa pU Non-isothermal undrained fluid pressure Pa p Average reservoir pressure Pa Q Heat exchanged between the system and its surroundings J Qf Vectorial flow of convective energy related to the fluid movement W/m2 QE Thermal energy J QH Volumetric heat generation W/m3

Qn Flow rate averaged in each Vn kg/s Qni Averaged flow rate at the interface kg/s QVT Total volumetric pumping rate m3/s QV Volumetric fluid extraction or injection rate m3/s q∗ Solute flux number of

molecules/m2/s qD Vector of convective-dispersive flux of the dissolved solute kg/m2/s qD Convective-dispersive flux of the solutes dissolved

in the fluid of the porous rock kg/m2/s

qDi Convective-dispersive flux of the solutes dissolved in the fluid of the porous rock in i direction (i = x, y, z)


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the 3 qH Power of a heat source W qij Interaction between medium Vi and Vj kg/m3/s qN Rate of recharge/extraction in the reservoir m/s qn Normal component of the heat flow J/m2/s qT Vectorial heat flow W/m2 qV Volumetric flow rate per unit volume of the sources

or sinks of water 1/s

R Inverse of the unconstrained storage coefficient Pa Rd Retardation factor ad Re Reynolds number ad Re Error or residual of the approximation ue – Rg Ideal gas constant (8.314472 J/K/mol) J/K/mol RI Resistivity index ad Rj Rate of solute production in the reaction number j

of a total of NR different reactions kg/m3

r Radial distance m r Position vector of a fluid particle in V m rC Radius of the well casing m rM Average radius of curvature of the interface m rw Radius of the well m S Entropy (thermodynamics) J/K S Storativity of an aquifer (or storage coefficient) ad Se Effective liquid saturation ad Sgc Critical gas saturation ad Si Saturation (i = w: wetting phase, nw: non-wetting phase) ad Sj Saturation ( j = o,w, g, l) ad Sni Boundary at the interface m2

Snw General normalized wetting-phase saturation ad Snwi Initial saturation of the non-wetting phase ad Snwimb Normalized water saturation by imbibition ad Sp Storage for pressure and neglecting gravity (eq. 4.56b) m · s2/kg Srg Water saturation at irreducible gas saturation ad Srl Irreducible water saturation ad Srs Residual steam saturation by imbibition ad Srw Residual water saturation ad Ssp Specific storage of an aquifer (or specific storage coefficient) 1/m SV Steam saturation ad Sw Wetting-phase saturation ad Sdrainajew Drainage water saturation ad Simbibitionw Imbibition water saturation ad Swc Critical water saturation ad Swcg Critical water saturation for gas ad Swi Irreducible water saturation ad Swr Residual saturation of Sw ad si Volumetric entropy (i = S, s, f , w) J/K/m3 TA Characteristic average reservoir temperature ◦C or K Ti Temperature (i = f , l, r, s, v, w) ◦C or K Tj Average temperature ( j = f , l, v, r, w) ◦C or K Tsat Saturation temperature ◦C or K Tv Transmissivity tensor of the aquifer m2/s

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T0 Reference temperature (initial temperature) ◦C or K t Time s t Unit tangential vector – tE Commercial exploitation time of the reservoir s t0.5 Value of half-life for the radioactive substance s U Internal energy J Us Solid internal energy J UT Total thermal energy J u Vector displacement m u(x) Continuous function u of the space E1 (chapter 5) – uf Vector displacement of the fluid particles m us Vector displacement of the solid particles m ux Pore fluid velocity m/s Vi Arbitrary volume of phase i m3

Vj Volume (i = B, F , f , l, r, s, v, w) m3 VP Volume of the pores m3

V Effective pore volume m3

V0 Initial volume m3

v Poisson’s coefficient (chapter 2) ad vA Average pore velocity of the fluid through the porous material m/s vB Brinkman fluid velocity m/s vC Velocity of the contaminant m/s vD Darcy fluid velocity m/s vf Darcy flux, or volumetric discharge per unit area, or Darcy

velocity, or seepage velocity m/s

vi Vectorial velocity (i = f , s) m/s vi Speed component in the direction of the turbulence m/s vNS Navier-Stokes fluid velocity m/s vr Radial flow velocity m/s vs Speed of sound m/s vx Darcy velocity (eq. 4.111) m/s W Work done by the system (W < 0) or on the system (W > 0) J W (u) Well function ad w(x, y) Weight function – Xl Liquid quality ad Xv Steam quality ad x Position vector m z0 Reference depth m∇T Vector thermal gradient K/m ∇xp Pressure gradient vector in one dimension Pa/m

Greek symbols Meaning Unit (SI)

α Experimental correction coefficient (eq. 2.155) ad αA Areal thermal expansion coefficient at constant pressure 1/K αL Longitudinal dispersivity m αij Dimensionless constant ad αT Transverse dispersivity m αx Linear thermal expansion 1/K

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βF Volumetric fraction of fractures ad βf Experimental factor (eq. 4.81) ad βij Thermoelastic tensor Pa/K Boundary of domain (∂) – γa Coefficient of molecular diffusion m2/s γC Composed thermal expansivity 1/K γi Thermal expansivity (i = B, w, f ) 1/K γm Global pore-fluid thermal expansion coefficient 1/K γU Undrained thermal expansivity 1/K γϕ Termal expansivity of the pores 1/K ef Change in internal energy J GS Useful energy able to be totally transformed into work J Hs Skeleton-stored enthalpy J h Drawdown or change in hydraulic head M hf Change in fluid enthalpy J h/ z Hydraulic gradient m/m = ad mf Change in fluid mass content kg/m3

p/ x Pressure gradient in one dimension Pa/m qz/ z Heat flow vertical gradient W/m Us Matrix-stored energy J VB Representative elementary volume (REV) m3

ρf Change of fluid density kg/m3

ϕ Change in effective porosity ad ϕ Phase difference between the maximum and minimum

Earth’s surface and corresponding groundwater temperatures (chapter 8)

K or ◦C

δ(x) Dirac distribution – δH Hydraulic diffusivity m2/s δij Unit tensor – δQ Heat exchanged per unit volume J/m3

δT Thermal diffusivity m2/s δW Volumetric work J/m3

δw Diffusivity of water m2/s δzn Small normal compression of the fracture relative to z m ∂ = Boundary of the domain – ε Overall volumetric strain ad εi Volumetric strain of (i = B, s, f ) ad εj Evaporation rate ( j = v) and condensation rate ( j = l) kg/m2/s εl Condensation rate ad εn Volumetric strain in direction n ad εs Volumetric deformation of the solid phase ad εT Total vectorial strain ad εt Volumetric strain in direction t ad εV Volumetric deformation ad εx Axial strain ad εz Vertical strain ad νf Fluid kinematic effective viscosity m2/s ζ Biot variation of fluid content ad η Mobility m · s/kg ηG Geothermal-electricity conversion factor ad (or as %) ηH Hydraulic diffusion coefficient m2/s θ Angular component –

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ni n Matrix of flow coefficients in the finite volume method λ Drained Lamé coefficient (chapter 2) Pa λf Coefficient of fluid viscosity of dilatation or compression Pa · s λijkl Thermoelastic tensor Pa λR Radioactive decay constant 1/s λU Undrained Lamé modulus Pa λϕ Pore size distribution coefficient ad μe Effective viscosity Pa · s μf Fluid dynamic viscosity Pa · s μij Fluid viscosity when crossing Sij Pa · s ν Phase velocity field (vector) m/s νni Phase velocity between two finite volumes Vn and Vi m/s νU Undrained Poisson’s coefficient ad ρ Global density kg/m3

ρb Bulk density of the rock kg/m3

ρi Density (i = f , g, l, r, s, v, w) kg/m3 ρij Fluid density when crossing Sij kg/m3

ρT Average density kg/m3

ρ0 Fluid density in a reference state kg/m3

σ Stress Pa σ Stress tensor Pa σB Bulk stress tensor Pa σdry Stress in dry rock Pa σf Fluid stress tensor Pa σH Hydrostatic stress Pa σlg Liquid-gas stress Pa σM Average of the stress tensor σ components Pa σsg Solid-gas stress Pa σsl Solid-liquid stress Pa σT Symmetric two-order tensor in four dimensions Pa σT Total poroelastic stress vector Pa σw Surface tension of water Pa σX Axial stress Pa σY Lateral confining stress Pa σZ Vertical stress Pa σ sZ Vertical solid stress Pa σ0 Superficial tension Pa τ Tangential shear stress Pa τf Tortuosity factor ad τdry Tangential stress in dry rock Pa τij Terzaghi effective stresses Pa τw Tangential stress Pa υ Specific volume (i = f , g, l, r, s, v, w) m3/kg υM Liquid phase molar volume m3/mol Differential model of model function ϕ (chapter 7) – ϕ = ϕeff Effective porosity ad (or as %) ϕ(t) Model function; effects of causes (chapter 7) – ϕF Fracture porosity ad ϕk Unitary basis function {k = 0, n} – ϕM Matrix porosity ad ϕs Solid volume fraction ad

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as ϕ1 Surface reference porosity ad (or as %) χ Earth’s physical parameter (density, electrical conductivity,

elasticity, etc.) (chapter 7) –

S Energy dissipation function Pa/s 1M Total volumetric flow rate (single-phase) exchanged between

Vn and its surroundings kg/m3/s

2M Total volumetric flow rate (two-phase) exchanged between Vn and its surroundings


Spatial domain occupied by the porous rock – e Finite element volume – ω Boltzmann’s transformation ad

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“references” — 2010/6/2 — 19:55 — page 439 — #1

“references” — 2010/6/2 — 19:55 — page 440 — #2

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“references” — 2010/6/2 — 19:55 — page 456 — #18

“book-series-page” — 2010/6/3 — 13:08 — page 479 — #1

Series Editors: Jochen Bundschuh & Mario César Suárez Arriaga


Publisher: CRC/Balkema, Taylor & Francis Group

1. Numerical Modeling of Coupled Phenomena in Science and Engineering Editors: M.C. Suárez Arriaga, J. Bundschuh & F.J. Domínguez-Mota 2009 ISBN: 978-0-415-47628-7

2. Introduction to the Numerical Modeling of Groundwater and Geothermal Systems Editors: J. Bundschuh & M.C. Suárez Arriaga 2010 ISBN: 978-0-415-40167-8