Page 88 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

4 Algorithm for Simulating Fluid-Rock Interaction Problems 75
(3 − β)(1 + β) k(φ 0 )
Zh critical = , β = , (4.7)
2(1 − β) k(φ f )

where Zh critical is the critical Zhao number of the fluid-rock interaction system;
k(φ 0 ) is the initial permeability corresponding to the initial porosity of the porous
rock; k(φ f ) is the final permeability corresponding to the final porosity, φ f ,ofthe
porous rock.
Using the concepts of both the Zhao number and the corresponding critical one,
the instability of a chemical dissolution front in a fluid-rock interaction system can
be determined. The focus of this chapter is to deal with fluid-rock interaction sys-
tems of subcritical Zhao numbers, while the focus of the next chapter is to deal with
fluid-rock interaction systems of critical and supercritical Zhao numbers.
In terms of numerical modelling of coupled reactive species transport phenomena
in pore-fluid saturated porous rock masses, we have divided the reactive transport
problems into the following three categories (Zhao et al. 1998a). In the first category
of reactive species transport problem, the time scale of the convective/advective flow
is much smaller than that of the relevant chemical reaction in porous rock masses so
that the rate of the chemical reaction can be essentially taken to be zero in the numer-
ical analysis. For this reason, the first category of species transport problem is often
called the non-reactive mass transport problem. In contrast, for the second category
of reactive species transport problem, the time scale of the convective/advective
flow is much larger than that of the relevant chemical reaction in pore-fluid satu-
rated porous rock masses so that the rate of the chemical reaction can be essen-
tially taken to be infinite, at least from the mathematical point of view. This means
that the equilibrium state of the chemical reaction involved is always attained in
this category of reactive species transport problem. As a result, the second category
of reactive species transport problem is called the quasi-instantaneous equilibrium
reaction transport problem. The intermediate case between the first and the sec-
ond category of reactive species transport problem belongs to the third category
of reactive species transport problem, in which the rate of the relevant chemical
reaction is a positive real number of finite value. Another significant characteris-
tic of the third category of reactive species transport problem is that the detailed
chemical kinetics of chemical reactions must be taken into account. It is the chem-
ical kinetics of a chemical reaction that describes the reaction term in a reactive
species transport equation. Due to different regimes in which a chemical reaction
proceeds, there are two fundamental reactions, namely homogeneous and heteroge-
neous reactions, in geochemistry. For homogeneous reactions, the chemical reaction
takes place solely between reactive aqueous species. However, for heterogeneous
reactions, the chemical reaction takes place at the surfaces between reactive aque-
ous species and solid minerals. This implies that both solid and fluid phases need
to be considered in the numerical modelling of reactive species transport problems
with heterogeneous reactions. Although significant achievements have been made
for the numerical modelling of non-reactive species and quasi-instantaneous equi-
librium reaction transport problems, research on the numerical modelling of the
third category of reactive species transport problem is rather limited (Steefel and