Page 155 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

144 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions
4D
optimal
k = . (6.33)
R 2
W
fault
6.4.3.2 Type 2: Chemical Equilibrium is Attained at the Upper Tip
of the Fault

This is another limiting case, in which the fluid rate (i.e. velocity) has reached a
critical value in the fault zone. The fundamental characteristic of this limiting case
is that if the fluid rate is equal to or greater than this critical value, the starting point
of chemical equilibrium is at or beyond the upper tip of the fault, indicating that the
chemical equilibrium of the chemical product and therefore mineral precipitation
cannot be attained within the fault zone. For a given chemical reaction rate, this
critical flow rate within the fault zone, V critical , can be expressed as follows:
V critical = φk R L fault . (6.34)

Equation (6.34) is useful for estimating the critical flow rate for a vertical fault.
For example, if the porosity of a vertical fault is 0.3 and the chemical reaction
rate is 10 –11 (1/s), then the corresponding critical flow rate is 3 × 10 –9 m/s for a
vertical fault 1 km long. If both the fault porosity and the chemical reaction rate
remain unchanged, the corresponding critical flow rate of the fault is increased to
–9
about 15 × 10 m/s for a vertical fault 5 km long. Suppose the flow-focusing factor
(Phillips, 1991) of the vertical fault is of the order of 15, the corresponding critical
flow rate within the surrounding rock for a vertical fault is 10 –9 m/s. This critical
flow rate can be easily exceeded when the fluid within the surrounding rock of the
fault zone is under a lithostatic pressure gradient. If the permeability of the sur-
2
2
rounding rock is 10 –16 m and the dynamic viscosity of the fluid is 10 –3 Ns/m ,
then the flow rate induced by the lithostatic pressure gradient within the surround-
ing rock is 1.7 × 10 –9 m/s. This implies that the flow rate induced by a lithostatic
pressure gradient within the surrounding rock is too high to enable minerals to be
precipitated within the vertical fault zone.

6.4.3.3 Type 3: Chemical Equilibrium is Attained Somewhere
between the Lower Tip and Upper Tip of the Fault

In this case, the equilibrium of the reaction product is attained within the fault zone.
The starting position of the chemical equilibrium is measured from the lower tip of
the fault and can be expressed by the following equation:

L mp = α mp L fault , (6.35)

where L mp is the starting position of chemical equilibrium within the fault zone in
the flow direction; α mp is a coefficient to express the relative relationship between
the length of the fault and the starting position of chemical equilibrium within the
"
fault zone; that is α mp = L mp L fault . It is clear that in this particular case, the value