Page 151 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

140 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions
D
optimal
k . (6.29)
R = 2
l chemical
diffusion
Both l chemical and the optimal chemical reaction-rate also have clear physical
diffusion
meanings: The physical meaning of l chemical is that for parallel flow with a given
diffusion
solute diffusion/dispersion coefficient, a chemical reaction with a given reaction rate
can only reach equilibrium within the distance diffused by the solute in the direction
perpendicular to the fluid flow within the time scale of chemical equilibrium. Since
l chemical is directly proportional to the square root of the solute diffusion/dispersion
diffusion
coefficient, the greater the solute diffusion/dispersion coefficient, the larger l chemical
diffusion
in the direction perpendicular to the fluid flow. This implies that a large solute
diffusion/dispersion coefficient can result in a chemical reaction, with a given
reaction rate, reaching equilibrium over a large distance normal to the direction of
fluid flow. By contrast, the physical meaning of the optimal chemical reaction rate
is that for a given l chemical , a chemical reaction with a given reaction rate can reach
diffusion
equilibrium if the chemical reaction proceeds at this chemical reaction rate within
the time scale of chemical equilibrium. Since the optimal chemical reaction rate is
inversely proportional to the square of l chemical , the larger l chemical in the direction
diffusion diffusion
perpendicular to the fluid flow, the smaller the optimal chemical reaction rate. This
implies that a large chemical equilibrium length due to solute diffusion/dispersion
in the direction perpendicular to the parallel fluid flow requires a relatively slow
optimal chemical reaction rate, so that for a given solute diffusion/dispersion
coefficient, the chemical reaction can reach equilibrium within this long chemical
equilibrium length in the direction perpendicular to the parallel fluid flow.
The combined use of these two numbers, Da and Z, can express the interaction
between solute advection, diffusion/dispersion and chemical kinetics. Note that for
a given fault zone involving pore-fluid flow focusing and mixing, it is possible to
define three different types of mineral precipitation patterns in Z-Da number space.
For this purpose, the thickness of the fault zone is chosen as the characteristic length
of the Z Number, while the length of the fault zone is chosen as the characteristic
length of the Da Number. The relationships between Da, Z and mineral precipitation
types are shown in Fig. 6.7.

6.4.3 Chemical Reaction Patterns due to Mixing and Focusing
of Two Reactive Fluids in Permeable Fault Zones

The theoretical understanding of the interaction between solute advection, solute
diffusion/dispersion and chemical kinetics presented in the previous section is, in
principle, useful for investigating chemical reaction patterns resulting from chem-
ical equilibrium associated with fluid flow in all kinds of porous rocks. Chemical
reaction patterns arising from flow focusing and mixing of two reactive fluids within
permeable vertical fault zones are the subject of this section.