Page 139 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

128 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions

chemical reaction are finite. For this reason, Equation (6.18) can be rewritten into
the following form:

2 2
∂C AB ∂C AB ∂C AB ∂ C AB ∂ C AB
φ + u + ν − D ex + D ey
∂t ∂x ∂y ∂x 2 ∂y 2 (6.22)
3 2 4
− φk b K e rC − [K e r(C I + C II ) + 1] C AB + K e rC I C II = 0.
AB
2
Since Eq. (6.22) is strongly nonlinear with the nonlinear term, φk b K e rC ,the
AB
Newton-Raphson algorithm is suitable for solving this equation.
Using the proposed decoupling procedure, the coupled problem between fluids
mixing, heat transfer and redox chemical reactions in fluid-saturated porous rocks
can be solved in the following five main steps: (1) For a given time step, the coupled
problem described by Eqs. (6.1), (6.2), (6.3), (6.4) and (6.5) with the related bound-
ary and initial conditions are solved using the conventional finite element method;
(2) After the pore-fluid velocities are obtained from the first step, mass transport
equations of the chemical reaction rate invariants (i.e. Eqs. (6.16) and (6.17)) with
the related boundary and initial conditions are then solved using the existing finite
element method; (3) The chemical reaction source/sink terms involved in Eq. (6.22)
is determined from the related redox chemical reaction so that Eq. (6.22) can be
solved using the Newton-Raphson algorithm; (4) According to the definitions of the
chemical reaction rate invariants, C I = C A +C AB and C II = C B +C AB , the chemical
reactant concentrations (i.e. C A and C B ) can be determined from simple algebraic
operations; (5) Steps (1–4) are repeated until the desired time step is reached. These
solution steps have been programmed into our research code.

6.3 Verification of the Decoupling Procedure

The main and ultimate purpose of a numerical simulation is to provide numerical
solutions for practical problems in a real world. Since numerical methods are the
basic foundation of a numerical simulation, only can an approximate solution 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, at least
from the qualitative point of view, the proposed numerical procedure so that mean-
ingful numerical results can be obtained from a discretized computational model.
For this reason, a testing coupled problem, for some aspects of which the analytical
solutions are available, is considered in this section.
Figure 6.1 shows the geometry of the coupled problem between pore-fluids mix-
ing, heat transfer and redox chemical reactions around a vertical geological fault
within the crust of the Earth. For this problem, the pore-fluid pressure is assumed
to be lithostatic, implying that there is an upward throughflow at the bottom of the
computational model. The height and width of the computational model are 10 km