Page 149 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

138 6 Fluid Mixing, Heat Transfer and Non-Equilibrium Redox Chemical Reactions

product. A conceptual model is presented here to investigate this interaction and the
corresponding influence on chemical reaction patterns.

6.4.2 Theoretical Analysis of Mineral Precipitation Patterns
in a Focusing and Mixing System Involving
Two Reactive Fluids

Although it is difficult, if not impossible, to obtain analytical solutions for the cou-
pled problem expressed by Eqs. (6.1), (6.2), (6.3), (6.4) and (6.5) in general cases, it
is possible to gain theoretical understanding through analytical solutions for the cou-
pled problem in some limiting cases (Zhao et al. 2007a). This understanding can be
achieved by converting the reactive transport equation (i.e. Eq. (6.5)) into a dimen-
sionless one so that the controlling processes associated with the reactive chemical
species transport can be identified. This means that it is possible to investigate the
relationship between the controlling processes associated with reactive chemical
species transport so that the overall structure of their solutions can be understood.
Since there are three major controlling processes, namely solute advection, solute
diffusion/dispersion and chemical kinetics, that may play dominant roles in deter-
mining chemical reaction patterns, we need to consider the relationships between
the time scales for these three controlling processes. For this purpose, we first con-
sider a dimensionless parameter known as the Damk¨ ohler number, Da, (Steefel and
Lasaga 1990, Ormond and Ortoleva 2000) to express the relative time scales of
solute advection and reaction kinetics:

φk R l
Da = = Time Scale for Solute Advection/
V (6.24)
Time Scale for Chemical Reaction,

where V is the characteristic Darcy velocity of the system; l is the characteris-
tic length of the controlling process in the system; k R is the controlling chemical
-1
reaction rate with units of [s ]; φ is the porosity of the porous medium. When the
time scale for solute advection is equal to the time scale for chemical kinetics, the
Damk¨ ohler number is equal to one. In this case, the chemical equilibrium length
scale of the system can be expressed as follows:

V
chemical
l = , (6.25)
advection
φk R
where l chemical is the chemical equilibrium length due to solute advection for a given
advection
chemical reaction. Below we refer to l chemical as the advection chemical equilibrium
advection
length. It is clear that if a chemical reaction rate is given, there exists an optimal
flow rate such that the chemical reaction can reach equilibrium beyond the advection
chemical equilibrium length determined from Eq. (6.25). Thus, for a given l chemical ,
advection
the corresponding optimal flow rate, V optimal , for which the chemical reaction can
reach equilibrium beyond the given l chemical , is as follows:
advection