Page 93 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

80 4 Algorithm for Simulating Fluid-Rock Interaction Problems

In other words, the use of a constant surface area cannot simulate the dependent
nature of the chemical reaction rate on a change in the amount of the dissolving
mineral during the heterogeneous chemical reaction process. In addition, the use of
a constant surface area cannot automatically terminate the dissolution reaction, even
though the dissolving mineral is absolutely exhausted in the rock mass. Since the
surface area of a dissolving mineral is strongly dependent on its generalized con-
centration, we can establish the following relation between the surface area and the
generalized concentration of the dissolving mineral:

A j q
= α j C , (4.14)
Gj
V f
where α j and q are positive real numbers.
Like the surface area of the dissolving minerals, the values of both α j and q are
dependent on the constituents, packing form, grain size and so forth of the minerals.
For simple packing of loose uniform particles of the minerals, these values can be
determined analytically. For example, in the case of packing uniform circles in a
two dimensional domain, q = 1/2, whereas in the case of packing uniform spheres
in a three dimensional domain, q = 2/3. However, for a real rock mass in pore-fluid
saturated hydrothermal/sedimentary basins, the values of both α j and q need to be
determined by field measurements and laboratory tests.
Inserting Eq. (4.14) into Eq. (4.13) yields the following equation:

q Q j
r j = α j C k j 1 − . (4.15)
Gj
K j
Clearly, this equation states that once the dissolving mineral is gradually con-
sumed in the rock mass, the generalized concentration of this mineral continuously
evolves to zero and therefore, its dissolution rate is automatically set to be zero
when it is completely depleted in the numerical analysis. This is the first advantage
of introducing the new concept of the generalized concentration for solid minerals.
Another advantage of introducing the concept of the generalized concentration
for solid minerals is that both the first type and second type of transport equations
(i.e. Eqs. (4.8) and (4.9)) can be solved simultaneously. Since several heterogeneous
chemical reactions take place simultaneously in dissolution problems involving
multiple minerals, it is important to solve simultaneously all the transport equations
with heterogeneous reaction terms in fluid-rock interaction problems, if we want to
simulate the chemical kinetics of these heterogeneous chemical reactions correctly.
If the transient process of a heterogeneous chemical reaction is of interest,
then the dissolution and precipitation of minerals can result in the variation of
porosity with time. This indicates that the linear average velocity of pore-fluid,
which is involved in the advection/convection term of a reactive aqueous species
transport equation, varies with time during the transient analysis. Even for a hori-
zontal aquifer, in which the horizontal Darcy velocity may be constant, the related
average linear velocity still varies with time, because it is inversely proportional