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6.4 Applications of the Proposed Decoupling Procedure 143

Here the factor 2 arises because of the symmetry of geometry; l chemical is mea-
diffusion
sured from the centre of the fault.
As indicated earlier in this chapter, there are three possible cases in which chemi-
cal equilibrium can be attained in the conceptual model shown in Fig. 6.8. The three
types of interest are: Type 1: chemical equilibrium is attained just at the lower tip of
the fault; Type 2: chemical equilibrium is attained at the upper tip of the fault; and
Type 3: chemical equilibrium is attained somewhere between the lower tip and upper
tip of the fault. As mentioned above, analytical solutions to Eqs. (6.1), (6.2), (6.3),
(6.4) and (6.5) are difficult but we can gain some insight into the types of chem-
ical reaction patterns that are possible by considering some limiting cases below.
In the forthcoming section, we numerically examine more realistic and applicable
situations.

6.4.3.1 Type 1: Chemical Equilibrium is Attained at the Lower Tip
of the Fault
In this limiting case, the fluid velocity is zero in the whole system. Substituting a
zero velocity into Eq. (6.30) yields a zero value of the starting position of mineral
precipitation within the fault zone. In this situation, it is possible for chemical equi-
librium to be attained throughout the whole length of the fault, depending on the
actual residence time of the two chemical reactants within the system. To enable
chemical equilibrium to be reached within the whole fault length, L fault , the resi-
dence time for which the two chemical reactants should exist in the system is equal
to the time interval, t diffuse , for the solute to diffuse from the lower tip to the upper
tip of the fault:

L 2 fault
t diffuse = . (6.32)
D
Consider a specific example: Eq. (6.32) indicates that for a solute diffu-
2
sion/dispersion coefficient of 10 –10 m /s and a vertical fault 1 km long, the two
8
chemical reactants need to be present for 10 16 s (i.e. about 3 × 10 years) at the
bottom of the conceptual model in order for chemical equilibrium to be reached
within the whole length of the fault. Since the required duration for the existence of
these two chemical reactants is quadratically proportional to the length of a fault, it
9
is increased to about 25 × 10 16 s (i.e. about 7.5 × 10 years) to enable the chemi-
cal reaction to reach equilibrium within the whole length of a 5 km long fault. In
this example, where the fluid flow is negligible, the time required to reach chemical
equilibrium within the whole length of a fault 5 km long is greater than the age of
the Earth.
As indicated in Eq. (6.31), the thickness of mineral precipitation within the
fault zone is dependent on both the chemical reaction rate and the solute diffu-
sion/dispersion coefficient. For a given solute diffusion/dispersion coefficient, the
optimal
optimal chemical reaction rate k , which will enable chemical reactions to attain
R
equilibrium across the whole width of the fault zone, is as follows: