Page 114 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

102 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

C eq ψ(φ 0 )p + v front φ 0 (C eq − ρ s ) + v front φ f ρ s = 0, (5.28)
ox

ψ(φ 0 )p + v front φ 0 − ψ(φ f )p − v front φ f = 0. (5.29)
ox fx
Solving Eqs. (5.28) and (5.29) simultaneously results in the following analytical
solutions:

−ψ(φ 0 )p C eq u 0x C eq
0x
v front = = , (5.30)
φ 0 C eq + (φ f − φ 0 )ρ s φ 0 C eq + (φ f − φ 0 )ρ s
ψ(φ f )[φ 0 C eq + (φ f − φ 0 )ρ s ]

p 0x = p , (5.31)
fx
ψ(φ 0 )[φ 0 C eq + (φ f − φ 0 )(ρ s + C eq )]
where u 0x is the Darcy velocity in the far downstream of the flow as x approaches
positive infinite. Using Darcy’s law, u 0x can be expressed as

φ 0 C eq + (φ f − φ 0 )ρ s
u 0x = u fx , (5.32)
φ 0 C eq + (φ f − φ 0 )(ρ s + C eq )
where u fx is the Darcy velocity in the far upstream of the flow as x approaches
negative infinite.
If the finite element method is used to solve this special problem, the accuracy
of the finite element simulation can be conveniently evaluated by comparing the
numerical solutions with the analytical ones for both the propagation speed of the
planar dissolution front (i.e. v front ) and the Darcy velocity in the far downstream of
the flow as x approaches positive infinite (i.e. u 0x ).

5.1.2.2 The Second Special Case (Base Solutions for a Stable State)
Since the solid molar density greatly exceeds the equilibrium concentration of the
chemical species, a small parameter can be defined as follows:

C eq
ε = << 1. (5.33)
ρ s
To facilitate the theoretical analysis in the limit case of ε approaching zero, the
following dimensionless parameters and variables can be defined.

x y z
x = , y = , z = , (5.34)
L ∗ L ∗ L ∗

C p
u
C = , p = ,
u = , (5.35)
C eq p ∗ u ∗