Page 156 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

6.4 Applications of the Proposed Decoupling Procedure 145

of α mp is greater than zero but less than one. A value of α mp equal to zero represents
the first limiting case (Type 1), while a value of α mp equal to one represents the
second limiting case (Type 2), as discussed above.
Similarly, the width of chemical equilibrium within the fault can be expressed as
follows:
W mp = β mp W fault , (6.36)

where W mp is the width of chemical equilibrium within the fault zone; W fault is
the width of the fault; β mp is a coefficient expressing the ratio between the width
of the fault and the width of chemical equilibrium within the fault zone; that is,
"
β mp = W mp W fault . In this case, the value of β mp is greater than zero but less than
one. A value of β mp equal to zero represents the first limiting case (Type 2), whilst
a value of β mp equal to one represents another limiting case (i.e. the limiting case
of Type 3), in which case the width of chemical equilibrium within the fault zone is
just equal to that of the fault.
If both the solute diffusion/dispersion coefficient and the width of chemical equi-
librium within the fault zone are known, the corresponding optimal chemical reac-
tion rate in this case can be estimated from the following formula:

4D 4D
optimal
k = = . (6.37)
R 2 2 2
W β W
mp mp fault
Thus, the corresponding optimal flow rate in this particular case is:
optimal 4φDα mp L fault
V optimal = φk L mp = . (6.38)
R 2 2
β W fault
mp
Equation (6.38) indicates that for a particular pattern of mineral precipitation
within the fault zone, there exists an optimal flow rate such that chemical equilib-
rium is attained within the region associated with this particular mineral precipita-
tion pattern.

6.4.4 Numerical Illustration of Three Types of Chemical Reaction
Patterns Associated with Permeable Fault Zones

The theoretical analysis carried out in the previous section predicts that there exist
three fundamental types of chemical reaction patterns associated with permeable
vertical fault zones due to two fluids mixing and focusing. In order to illustrate
these different types of chemical reaction patterns, the finite element method is used
to solve the coupled problem numerically involving fluid mixing, heat transfer and
chemical reactions expressed by Eqs. (6.1), (6.2), (6.3), (6.4) and (6.5). In theory,
it is possible to predict exactly the three fundamental types of chemical reaction
patterns. However, in numerical practice, it is difficult to control the flow rate of