Page 118 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

106 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

It is noted that the total solutions expressed in Eqs. (5.53) and (5.54) must satisfy
the governing equations that are expressed in Eqs. (5.45) and (5.46). With consid-
eration of Eq. (5.58), the first-order perturbation equations of this system can be
expressed as

2
∂ ˆ p 2
ˆ
2
C = 0, − m ˆ p + m p 0x = 0 (in the downstream region), (5.59)
∂ξ 2
2 ˆ
ˆ
∂ C ∂C 2 ˆ ∂ ˆ p

2

+ p fx − m C − m p exp(−p ξ) − p exp(−p ξ) = 0,
fx
fx
fx
fx
∂ξ 2 ∂ξ ∂ξ
2
∂ ˆ p 2
2
− m ˆ p + m p = 0 (in the upstream region). (5.60)
fx
∂ξ 2
The corresponding boundary conditions of the first-order perturbation problem
are:
∂ ˆ p
ˆ
C = 0, lim = 0 (downstream boundary), (5.61)
x→∞ ∂ξ
∂ ˆ p
ˆ
lim C = 0, lim = 0 (upstream boundary). (5.62)
x→−∞ x→−∞ ∂ξ
Similarly, the interface conditions for this first-order perturbation problem can be
expressed as follows:
ˆ
C = 0, lim ˆ p = lim ˆ p, (5.63)
S→0 − S→0 +
ˆ
∂C ∂ ˆ p ψ(φ 0 ) ∂ ˆ p
lim = ω(φ f − φ 0 ), lim = lim . (5.64)
S→0 ∂n S→0 ∂n ψ(φ f ) S→0 ∂n
−
+
−
Solving Eqs. (5.59) and (5.60) with the boundary and interface conditions (i.e.
Eqs. (5.61) and (5.62)) yields the following analytical results:
1 − β
ˆ
C = 0, ˆ p(ξ) = p 0x 1 − exp(− |m| ξ) (in the downstream region),
1 + β
(5.65)
2 1 − β

ˆ

C(ξ) =−p exp(−p ξ) − exp(σξ) + exp[(|m| − p )ξ] ,
fx fx fx
1 + β 1 + β

1 − β
ˆ p(ξ) = p 1 + exp(|m| ξ) (in the upstream region),
fx
1 + β
(5.66)