Page 115 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.1 Mathematical Background of Chemical Dissolution Front Instability Problems 103
t
τ = ε, (5.36)
t ∗

where τ is a slow dimensionless time to describe the slowness of the chemical
dissolution that takes place in the system. Other characteristic parameters used in
Eqs. (5.34), (5.35) and (5.36) can be expressed as follows:

V p
∗ ∗
∗
t = , L = φ f D(φ f )t , (5.37)
k chemical A p C eq
φ f D(φ f ) φ f D(φ f )
∗ ∗
p = , u = , (5.38)
ψ(φ f ) L ∗
φD(φ) ψ(φ)
∗ ∗
D (φ) = , ψ (φ) = , (5.39)
φ f D(φ f ) ψ(φ f )
Substituting Eqs. (5.34), (5.35), (5.36), (5.37), (5.38) and (5.39) into Eqs. (5.11),
(5.12) and (5.13) yields the following dimensionless equations:
∂φ
∗
ε −∇ • [ψ (φ)∇ p ] = 0, (5.40)
∂τ
∂ ∂φ
∗
∗
ε (φC) −∇ • [D (φ)∇C + Cψ (φ)∇ p ] − = 0, (5.41)
∂τ ∂τ
∂φ
ε + (φ f − φ)(C − 1) = 0. (5.42)
∂τ
Similarly, the boundary conditions for this special case can be expressed in a
dimensionless form as follows:

∂ p
lim C = 1, lim φ = φ 0 , lim = p 0x (downstream boundary),
x→∞ x→∞ x→∞ ∂x
(5.43)
∂ p
lim C = 0, lim φ = φ f , lim = p fx (upstream boundary).
x→−∞ x→−∞ x→−∞ ∂x
(5.44)

It is noted that the propagation front due to chemical dissolution divides the prob-
lem domain into two regions, an upstream region and a downstream region, relative
to the propagation front. Across this propagation front, the porosity undergoes a
jump from its initial value into its final value. Thus, this dissolution-front propa-
gation problem can be considered as a Stefan moving boundary problem (Chadam
et al. 1986). In the limit case of ε approaching zero, the corresponding governing