Page 121 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.2 Segregated Algorithm for Simulating the Morphological Evolution 109
5.2 Proposed Segregated Algorithm for Simulating
the Morphological Evolution of a Chemical Dissolution Front

Although analytical solutions can be obtained for the above-mentioned special
cases, it is very difficult, if not impossible, to predict analytically the compli-
cated morphological evolution process of a planar dissolution front in the case of
the chemical dissolution system becoming supercritical. As an alternative, numer-
ical methods are suitable to overcome this difficulty. Since numerical methods are
approximate solution methods, they must be verified before they are used to solve
any new type of scientific and engineering problem. For this reason, the main pur-
pose of this section is to propose a numerical procedure for simulating how a planar
dissolution front evolves into a complicated morphological front. To verify the accu-
racy of the numerical solution, a benchmark problem is constructed from the theo-
retical analysis in the previous section. As a result, the numerical solution obtained
from the benchmark problem can be compared with the corresponding analytical
solution. After the proposed numerical procedure is verified, it will be used to sim-
ulate the complicated morphological evolution process of a planar dissolution front
in the case of the chemical dissolution system becoming supercritical.

5.2.1 Formulation of the Segregated Algorithm for Simulating
the Evolution of Chemical Dissolution Fronts

In this section, Eqs. (5.40), (5.41) and (5.42) are solved using the proposed numer-
ical procedure, which is a combination of both the finite element method and the
finite difference method. The finite element method is used to discretize the geo-
metrical shape of the problem domain, while the finite difference method is used to
discretize the dimensionless time. Since the system described by these equations is
highly nonlinear, the segregated algorithm, in which Eqs. (5.40), (5.41) and (5.42)
are solved separately in a sequential manner, is used to derive the formulation of the
proposed numerical procedure.
For a given dimensionless time-step, τ + Δτ, the porosity can be denoted by
φ τ+Δτ = φ τ + Δφ τ+Δτ , where φ τ is the porosity at the previous time-step and
Δφ τ+Δτ is the porosity increment at the current time-step. Using the backward dif-
ference scheme, Eq. (5.42) can be written as follows:

ε
' (
+ (1 − C τ+Δτ ) Δφ τ+Δτ = (φ f − φ τ )(1 − C τ+Δτ ), (5.78)
Δτ

where C τ+Δτ is the dimensionless concentration at the current time-step; Δτ is the
dimensionless time increment at the current time-step.
Mathematically, there exist the following relationships in the finite difference
sense: