Page 23 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.1 Governing Equations of the Problem 9

element analysis, the asymptotic approach concept needs to be combined with the
finite element method in a different fashion (Zhao et al. 1997a).

2.1 Governing Equations of the Problem

For a two-dimensional fluid-saturated porous medium, if Darcy’s law is used to
describe pore-fluid flow and the Oberbeck-Boussinesq approximation is employed
to describe a change in pore-fluid density due to a change in pore-fluid temperature,
the governing equations of a natural convection problem, known as the steady-state
Horton-Rogers-Lapwood problem (Nield and Bejan 1992, Zhao et al. 1997a), for
incompressible pore-fluid can be expressed as

∂u ∂ν
+ = 0, (2.1)
∂x ∂y

K x ∂ P
u = − + ρ f g x , (2.2)
μ ∂x

K y ∂ P
v = − + ρ f g y , (2.3)
μ ∂y
2 2
∂T ∂T ∂ T ∂ T
ρ f 0 c p u + ν = λ ex + λ ey , (2.4)
∂x ∂y ∂x 2 ∂y 2
ρ f = ρ f 0 [1 − β T (T − T 0 )], (2.5)

λ ex = φλ fx + (1 − φ)λ sx , λ ey = φλ fy + (1 − φ)λ sy , (2.6)

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 temperature
of the porous material; K x and K y are the permeabilities of the porous material 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; ρ f 0 and T 0 are the reference density and temperature;
λ fx and λ sx are the thermal conductivities of the pore-fluid and rock mass in the x
direction; λ fy and λ sy are the thermal conductivities of the pore-fluid and rock mass
in the y direction; c p is the specific heat of the pore-fluid; g x and g y are the gravity
acceleration components in the x and y directions; φ and β T are the porosity of the
porous material and the thermal volume expansion coefficient of the pore-fluid.
It is noted that Eqs. (2.1), (2.2), (2.3) and (2.4) are derived under the assumption
that the porous medium considered is orthotropic, in which the y axis is upward in
the vertical direction and coincides with the principal direction of medium perme-
ability as well as that of medium conductivity.