Page 53 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

3.1 Statement of the Coupled Problem and Solution Method 39

problem describing the steady-state pore-fluid flow, heat transfer and mass trans-
port/chemical reactions in a porous medium, the corresponding governing equations
are expressed as follows:

∂u ∂v
+ = 0, (3.1)
∂x ∂y

K x ∂P
u = − , (3.2)
μ ∂x

K y ∂P
v = − + ρ f g , (3.3)
μ ∂y
2 2
∂T ∂T ∂ T ∂ T
ρ f 0 c u + v , (3.4)
p = λ ex 2 + λ ey 2
∂x ∂y ∂x ∂y

2 2
∂C i ∂C i ∂ C i ∂ C i
ρ f 0 u + v = ρ f 0 D ex + D ey + φR i (i = 1, 2, ..., N),
∂x ∂y ∂x 2 ∂y 2
(3.5)
N

ρ f = ρ f 0 1 − β T (T − T 0 ) − β Ci (C i − C 0 ) , (3.6)
i=1
λ ex = φλ fx + (1 − φ)λ sx , λ ey = φλ fy + (1 − φ)λ sy , (3.7)

D ex = φD fx , D ey = φD fy , (3.8)

where u and v are the horizontal and vertical velocity components of the pore-fluid
in the x and y directions respectively; P is the pore-fluid pressure; T is the temper-
ature of the porous medium; C i is the normalized concentration (in a mass fraction
form relative to the pore-fluid density) of chemical species i; N is the total num-
ber of the active chemical species considered in the pore-fluid; K x and K y are the
permeabilities of the porous medium in the x and y directions respectively; μ is
the dynamic viscosity of the pore-fluid; ρ f is the density of the pore-fluid and g is
the acceleration due to gravity; ρ f 0 , T 0 and C 0 are the reference density, reference
temperature and reference normalized concentration of the chemical species used in
the analysis; λ fx and λ sx are the thermal conductivities of the pore-fluid and solid
matrix in the x direction; λ fy and λ sy are the thermal conductivities of the pore-fluid
and solid matrix in the y direction; c p is the specific heat of the pore-fluid; D fx and
D fx are the diffusivities of the chemical species in the x and y directions respec-
tively; φ is the porosity of the porous medium; β T and β Ci are the thermal volume
expansion coefficient of the pore-fluid and the volumetric expansion coefficient due
to chemical species i; R i is the chemical reaction term for the transport equation of
chemical species i.