Page 116 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

104 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

equations for the dimensionless variables of the problem in both the downstream
region and the upstream region can be expressed below:

2
C = 1, ∇ p = 0, φ = φ 0 (in the downstream region), (5.45)

2
∇• (∇C + C∇ p) = 0, ∇ p = 0, φ = φ f (in the upstream region).
(5.46)
If the chemical dissolution front is denoted by S(x,τ) = 0, then the dimension-
less pressure, chemical species concentration and mass fluxes of both the chemical
species and the pore-fluid should be continuous on S(x,τ) = 0. This leads to the
following interface conditions for this moving-front problem:

lim C = lim C, lim p = lim p, (5.47)
S→0 − S→0 + S→0 − S→0 +

∂C ∂ p ψ(φ 0 ) ∂ p
lim = ν front (φ f − φ 0 ), lim = lim , (5.48)
S→0 ∂n S→0 ∂n ψ(φ f ) S→0 ∂n
−
−
+
where n is the unit normal vector of the moving dissolution front.
When the planar dissolution front is under stable conditions, the base solutions
for this special problem can be derived from Eqs. (5.45) and (5.46) with the related
boundary and interface conditions (i.e. Eqs. (5.43), (5.44), (5.47) and (5.48)). The
resulting base solutions are expressed as follows:

C(ξ) = 1, p(ξ) = p ξ + p , (in the downstream region), (5.49)
0x C1 φ = φ 0

C(ξ) = exp(−p ξ), p(ξ) = p ξ + p , φ = φ f (in the upstream region),

C2
fx
fx
(5.50)
where p C1 and p C2 are two constants to be determined. For example, p C1 can be
determined by setting the dimensionless pressure p(ξ) to be a constant at a pre-
scribed location of the downstream region, while p C2 can be determined using the
pressure continuity condition at the interface between the upstream and downstream
regions. Other parameters are defined below:
ψ(φ f ) p fx

ξ = x − ν front τ, p = p , ν front =− . (5.51)
0x fx
ψ(φ 0 ) φ f − φ 0
Therefore, if the finite element method is used to solve the second special prob-
lem, the accuracy of the finite element simulation can be conveniently evaluated
by comparing the numerical solutions with a complete set of analytical solutions
including porosity, the location of the chemical dissolution front, the dimensionless
chemical-species concentration and the dimensionless pressure.