Page 123 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.2 Segregated Algorithm for Simulating the Morphological Evolution 111
⎛ - -
N φ N C 2
. .
! 2 . k k−1
k−1
k
.
E = Max ⎝ / φ − φ , / C − C ,
i,τ+Δτ i,τ+Δτ i,τ+Δτ i,τ+Δτ
i=1 i=1
⎞
-
N p
.
.
p i,τ+Δτ − p i,τ+Δτ ⎠ < E,
/ k k−1 ! 2 ⎟
i=1
(5.85)
where E and E are the maximum error at the k-th iteration step and the allowable
error limit; N φ , N and N p are the total numbers of the degrees-of-freedom for the
C
porosity, dimensionless concentration and dimensionless pressure respectively; k is
the index number at the current iteration step and k − 1 is the index number at the
previous iteration step; φ k , C k and p k are the porosity, dimensionless
i,τ+Δτ i,τ+Δτ i,τ+Δτ
concentration and dimensionless pressure of node i at both the current time-step and
the current iteration step; φ k−1 , C k−1 and p k−1 are the porosity, dimension-
i,τ+Δτ i,τ+Δτ i,τ+Δτ
less concentration and dimensionless pressure of node i at the current time-step but
at the previous iteration step. It is noted that k ≥ 2inEq. (5.85).
The convergence criterion is checked after the second iteration step. If the con-
vergence criterion is not met, then the iteration is repeated at the current time-step.
Otherwise, the convergence solution is obtained at the current time step and the
solution procedure goes to the next time-step until the final time-step is reached.
5.2.2 Verification of the Segregated Algorithm for Simulating
the Evolution of Chemical Dissolution Fronts

The main and ultimate purpose of a numerical simulation is to provide numeri-
cal solutions for practical problems in a real world. These practical problems are
impossible and impractical to solve analytically. Since numerical methods are the
basic foundation of a numerical simulation, only an approximate solution can be
obtained from a computational model, which is the discretized description of a con-
tinuum mathematical model. Due to inevitable round-off errors in computation and
discretized errors in temporal and spatial variables, it is necessary to verify the pro-
posed numerical procedure so that meaningful numerical results can be obtained
from a discretized computational model. For this reason, a benchmark problem, for
which the analytical solutions are available, is considered in this section.
Figure 5.1 shows the geometry and boundary conditions of the coupled problem
between porosity, pore-fluid pressure and reactive chemical-species transport within
a fluid-saturated porous medium. For this benchmark problem, the dimensionless-
pressure gradient (i.e. p =−1) is applied on the left boundary, implying that there

fx
is a horizontal throughflow from the left to the right of the computational model. In
this case, the Zhao number of the reactive transport system is unity. The dimension-
less height and width of the computational model are 5 and 10 respectively. Except
for the left boundary, the initial porosity of the porous medium is 0.1, while the