Page 90 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

4.2 Development of the Term Splitting Algorithm 77

the dissolution rate of this particular mineral must be identically equal to zero. This
indicates that the variation in the amount of the dissolving mineral should be con-
sidered in the dissolution rates of the minerals. Otherwise, the numerical modelling
may violate the real mechanism of the related chemical kinetics and produce incor-
rect numerical results.
Thus, in order to effectively and efficiently use the finite element method
for solving fluid-rock interaction problems in pore-fluid saturated hydrother-
mal/sedimentary basins, new concepts and numerical algorithms need to be devel-
oped for dealing with the following fundamental issues: (1) Since the fluid-rock
interaction problem involves heterogeneous chemical reactions between reactive
aqueous chemical species in the pore-fluid and solid minerals in the rock masses,
it is necessary to develop a new concept involving the generalized concentration of
a mineral, so that two types of reactive mass transport equations, namely the con-
ventional mass transport equation for the aqueous chemical species in the pore-fluid
and the degenerated mass transport equation for the solid minerals in the rock mass,
can be solved simultaneously in computation. (2) Since the reaction area between
the pore-fluid and mineral surfaces is basically a function of the generalized concen-
tration of the solid mineral, there is a need to consider the appropriate dependence
of the dissolution rate of a dissolving mineral on its generalized concentration in the
numerical analysis. (3) Considering the direct consequence of the porosity evolution
with time in the transient analysis of fluid-rock interaction problems, the problem of
variable mesh Peclet number and Courant number needs to be converted into a prob-
lem involving constant mesh Peclet and Courant numbers, so that the conventional
finite element method can be directly used to solve fluid-rock interaction problems.
Taking the above-mentioned factors into account, we focus this study on the
numerical modelling of mixed solid and aqueous species transport equations with
consideration of reaction terms from heterogeneous, isothermal chemical reactions.
This means that we will concentrate on the development of new concepts and algo-
rithms so as to solve the fluid-rock interaction problems effectively and efficiently,
using the finite element method. For this purpose, a fluid-rock interaction problem,
in which K-feldspar (KAlSi 3 O 8 ) and/or muscovite (KAl 3 Si 3 O 10 (OH) 2 ) are dissolved
and muscovite and/or pyrophyllite (Al 2 Si 4 O 10 (OH) 2 ) are precipitated, is considered
as a representative example in this chapter.

4.2 Development of the Term Splitting Algorithm

For fluid-rock interaction problems in pore-fluid saturated hydrothermal/
sedimentary basins, heterogeneous chemical reactions take place at the interface
between the reactive aqueous species in the pore-fluid and solid minerals in the
rock mass. This means that we need to deal with two types of transport equa-
tions in the numerical modelling of fluid-rock interaction problems. The first is
the conventional transport equation with the advection/convection term and the
diffusion/dispersion term for reactive aqueous species in the pore-fluid. The second
is the degenerated transport equation, in which the advection/convection term and