Page 157 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

146 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions

a specific value due to numerical round off and cutoff errors. For this reason, the
numerical simulated mineral precipitation patterns here are approximate represen-
tations of the three fundamental types of mineral precipitation patterns.
The computational model (shown in Fig. 6.1) and related parameters for the ver-
ification example in the previous section (i.e. Sect. 6.3) are used to illustrate three
different types of mineral precipitation in a focusing and mixing system of two reac-
tive fluids. Two different fluid pressure gradients are considered to control the flow
rate (i.e. vertical fluid velocity) within the surrounding rock during the numerical
simulation. To simulate the fast flow rate associated with type 2 (as stated in the
previous section), the fluid pressure gradient is assumed to be a lithostatic pressure
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gradient, which results in a vertical flow rate of 1.7 × 10 m/s within the surround-
ing rock. This gradient is used because it is considered to be near the highest reason-
able fluid pressure gradient likely to be encountered in nature. On the other hand,
in order to simulate the slow flow rate associated with types 1 and 3 (see the pre-
vious section), the excess fluid pressure gradient is assumed to be one percent of
a lithostatic pressure gradient minus a hydrostatic one, which results in a vertical
flow rate of 1.7 × 10 –11 m/s within the surrounding rock. It is known that for fluid
pressure gradient dominated flow, the flow pattern around a permeable fault zone is
dependent only on the contrast in permeability between the fault and the surround-
ing rocks, the geometry of the fault zone and the inflow direction relative to the axis
of the fault (Phillips 1991, Zhao et al. 1999d) and is independent of the pressure
gradient along the fault. This means that although the fluid pressure gradient is dif-
ferent in the three simulations, the streamline pattern within and around the fault
zone must be identical for all three cases, as demonstrated by the related numerical
result shown in Fig. 6.2. In all cases, fluid flow converges into the fault zone at the
lower end and diverges out of the fault zone at the upper end (see also Phillips 1991,
Zhao et al. 1999d).

6.4.4.1 The First Type of Chemical Reaction Pattern
The first type of chemical reaction pattern results from an approximate representa-
tion of Type 1 in Sect. 6.4.3; the background flow velocity within the surrounding
rock is not exactly equal to zero in the numerical simulation but instead is one per-
cent of the lithostatic pressure gradient minus the hydrostatic gradient. Here two
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controlling chemical reaction rates, namely 10 (1/s) and 10 –11 (1/s) are considered
to investigate the effect on chemical reaction patterns. Due to flow focusing, the
maximum vertical flow velocity within the fault zone is about 3.16 × 10 –10 m/s.
Figure 6.9 shows the concentration distributions of the two chemical reactants
and the corresponding chemical product at two time instants of t = 500,000 and
t = 800,000 years. The distribution of the concentration of the chemical product
comprises a lenticular shape within the fault zone. This coincides with what is
expected from the previous theoretical analysis. For a fault zone of width 250 m,
the optimal chemical reaction rate calculated from Eq. (6.33) is 4.8 × 10 –15 (1/s) so
that chemical equilibrium can be attained within the whole width of the fault zone.
Since the optimal chemical reaction rate is much smaller than the two chemical