Page 274 - Geothermal Energy Systems Exploration, Development, and Utilization

250 5 Geothermal Reservoir Simulation
5.2.2.2 Liquid Flow in Deformable Porous Media
Incompressible fluid flow in deformable porous media is described by the following
fluid mass balance equation,

∂p k l
∂u
S s −∇ · ∇p + ρ g∇z +∇ · = q f (5.3)
∂t µ ∂t
For describing nonlinear flow behavior we use the Forchheimer equation (Forch-
heimer, 1914)
p = a 1 Q + a 2 Q 2 (5.4)
which states that the pressure drop can be described by a quadratic function of the
flow rate. If the quadratic term dominates as the case for large flow rates, nonlinear
flow behavior can be expressed as a pressure-dependent permeability.
−1/2
l
µ ρ D h p
k(p) = √ (5.5)
ρ l λ L
5.2.2.3 Thermoporoelastic Deformation
Thermoporoelastic deformation is described by the momentum balance equation
in the terms of stress tensor as
∇· (σ − α b pI − β T EI T) + ρg = 0 (5.6)
The density of the porous medium composed of two phases, liquid and solids is
l
ρ = nρ + (1 − n)ρ s
Displacement is the primary variable to be solved by substituting the constitutive
law for stress–strain behavior
σ = C ε (5.7)
1 T
ε = (∇u + (∇u) ) (5.8)
2
The fourth-order elasticity tensor C is

C := λδ ij δ kl + 2Gδ ik δ jl (5.9)
Allsymbols arelistedatthe endof the chapter.
Different numerical methods are applied to solve the above system of par-
tial differential equations. FEFLOW (TH) (Kolditz and Diersch, 1993; Diersch,
2002), FRACTure (THM) (Kohl, 1992), and GeoSys/RockFlow (THMC) (Wang and
Kolditz, 2007) use the FEM. SHEMAT (THC) is based on the finite difference
method (Clauser, 2003).

5.3
Reservoir Characterization

A characterization of geothermal reservoirs is complicated due to several
facts: