Page 136 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

6.1 Problems between Fluids Mixing, Heat Transfer and Redox Chemical Reactions 125

and decoupling procedure can be extended to deal with the high order redox chem-
ical reactions. For this reason, a redox chemical reaction of the second order is
considered as follows:

k f
A + B ⇔ AB, (6.9)
k b
where A and B are two chemical reactants; AB is the chemical product due to this
redox chemical reaction; k f and k b are the forward and backward reaction rates of
this redox chemical reaction. It needs to be pointed out that Eq. (6.9) represents a
k f k f
−
+
+
−
class of redox chemical reactions such as H +OH ⇔ H 2 O, Na +Cl ⇔ NaCl,
k b k b
k f
Ca 2+ + CO 2− ⇔ CaCO 3 and so forth in geochemical systems. It is clear that in the
3
k b
−
+
first chemical reaction example, chemical reactants A and B are H and OH , while
chemical product AB is H 2 O. In the second chemical reaction example, chemical
reactants A and B are Na and Cl , while chemical product AB is NaCl. Similarly,
−
+
in the third chemical reaction example, chemical reactants A and B are Ca 2+ and
2−
CO , while chemical product AB is CaCO 3 .
3
From the chemical reaction point of view, the general chemical reaction
source/sink terms due to the redox chemical reaction expressed by Eq. (6.9) can
be written as follows:
R A =−k f r n f −1 C A C B + k b r n b −1 C AB , (6.10)
R B =−k f r n f −1 C A C B + k b r n b −1 C AB , (6.11)
R AB = k f r n f −1 C A C B − k b r n b −1 C AB , (6.12)

where C A , C B and C AB are the concentrations of chemical species A, B and AB; R A ,
R B and R AB are the chemical reaction source/sink terms associated with chemical
species A, B and AB; n f and n b are the orders of the forward and backward reac-
tions respectively; r is a quantity of unity value to balance the unit of the reaction
source/sink terms due to different orders of chemical reactions so that it has a recip-
rocal unit of the chemical species concentration. For the redox reaction expressed by
Eq. (6.9), the forward reaction is of the second order, while the backward reaction
is of the first order. Since a redox system allows chemical reactions to be proceeded
toward both the product and the reactant directions, the orders of the forward reac-
tion (i.e. the chemical reaction proceeds toward the product direction) and backward
reaction (i.e. the chemical reaction proceeds toward the reactant direction) can be
determined from the related chemical kinetics.
It is noted that the accumulation or diffusion of chemical species in the rock
matrix may lead to some change in porosity, which in turn affects permeabil-
ity and fluid flow in the rock matrix (Zhao et al. 2001d, Xu et al. 2004). This