Page 122 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

110 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks
! ! !
∂ φC Δ φ τ+Δτ C τ+Δτ Δφ τ+Δτ Δ C τ+Δτ
ε = ε = εC τ+Δτ + εφ τ+Δτ ,
∂τ Δτ Δτ Δτ
(5.79)
∂φ Δ (φ τ+Δτ ) ! !
ε = ε = 1 − C τ+Δτ φ f − φ τ+Δτ , (5.80)
∂τ Δτ

) * ) *
∗
∗
∇• D (φ)∇C =∇ • D (φ τ+Δτ )∇C τ+Δτ , (5.81)
) * ) * ) *
∇• Cψ (φ)∇ p = C∇• ψ (φ)∇ p +∇ p • ψ (φ)∇C
∗
∗
∗
) ∗ *
= C τ+Δτ ∇• ψ (φ τ+Δτ )∇ p τ+Δτ (5.82)
) *
+∇ p τ+Δτ • ψ (φ τ+Δτ )∇C τ+Δτ
∗
Substituting Eqs. (5.79), (5.80), (5.81) and (5.82) into Eq. (5.41) yields the fol-
lowing finite difference equation:
ε 1 ) ∗ *

φ τ+Δτ + (φ f − φ τ+Δτ ) C τ+Δτ −∇ • D (φ τ+Δτ )∇C τ+Δτ
Δτ ε
) *
−∇ p τ+Δτ • ψ (φ τ+Δτ )∇C τ+Δτ (5.83)
∗
ε 1 !
= φ τ+Δτ C τ + φ f − φ τ+Δτ
Δτ ε
Similarly, Eq. (5.40) can be rewritten in the following discretized form:
) * ) *
∇• ψ (φ)∇ p =∇ • ψ (φ τ+Δτ )∇ p = (1 − C τ+Δτ )(φ f − φ τ+Δτ ). (5.84)
∗
∗
τ+Δτ
Using the proposed segregated scheme and finite element method, Eqs. (5.78),
(5.83) and (5.84) are solved separately and sequentially for the porosity, dimension-
less concentration and dimensionless pressure at the current time-step. Note that
when Eq. (5.78) is solved using the finite element method, the dimensionless con-
centration at the current time-step is not known. Similarly, when Eq. (5.83) is solved
using the finite element method, the dimensionless pressure at the current time-step
remains unknown. This indicates that these three equations are fully coupled so that
an iteration scheme needs to be used to solve them sequentially. At the first iter-
ation step, the dimensionless concentration at the previous time-step is used as a
reasonable guess of the dimensionless concentration at the current time-step when
Eq. (5.78) is solved for the porosity. In a similar way, the dimensionless pressure at
the previous time-step is used as a reasonable guess for the current time-step when
Eq. (5.83) is solved for the dimensionless concentration. The resulting approximate
porosity and dimensionless concentration can be used when Eq. (5.84) is solved for
the dimensionless pressure. At the second iteration step, the same procedure as used
in the first iteration step is followed, so that the following convergence criterion can
be established after the second iteration step.