Page 120 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

108 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

system is neutrally unstable, implying that the introduced small perturbation can be
maintained but it does not grow in the corresponding reactive transport system.
Clearly, Eq. (5.72) indicates that for the reactive chemical-species transport con-
sidered in this investigation, the dissolution-enhanced permeability destabilizes the
instability of the chemical dissolution front, while the dissolution-enhanced diffu-
sivity stabilizes the instability of the chemical dissolution front. If the shape factor
"
of soluble grains is represented by θ = V p A p , then an increase in the shape fac-
tor of soluble grains can destabilize the instability of the chemical dissolution front,
indicating that the instability likelihood of a porous medium comprised of irregu-
lar grains, is higher than that of a porous medium comprised of regular spherical
grains. Similarly, an increase in either the equilibrium concentration of the chemi-
cal species or the chemical reaction constant of the dissolution reaction can cause
the stabilization of the chemical dissolution front, for the reactive chemical-species
transport considered in this investigation.
To understand the physical meanings of each term in the Zhao number, Equation
(5.72) can be rewritten in the following form:

Zh = F Advection F Diffusion F chemical F Shape , (5.73)

where F Advection is a term to represent the solute advection; F Diffusion is a term to
represent the solute diffusion/dispersion; F chemical is a term to represent the chemi-
cal kinetics of the dissolution reaction; F Shape is a term to represent the shape fac-
tor of the soluble mineral in the fluid-rock interaction system. These terms can be
expressed as follows:

F Advection = v flow , (5.74)

1
, (5.75)
F Diffusion =
φ f D(φ f )

1
F chemical = , (5.76)
k chemical C eq

V p
F Shape = . (5.77)
A p

Equations (5.73), (5.74), (5.75), (5.75) and (5.77) clearly indicate that the Zhao
number is a dimensionless number that can be used to represent the geometrical,
hydrodynamic, thermodynamic and chemical kinetic characteristics of a fluid-rock
system in a comprehensive manner. This dimensionless number reveals the intimate
interaction between solute advection, solution diffusion/dispersion, chemical kinet-
ics and mineral geometry in a reactive transport system.