Page 87 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

74 4 Algorithm for Simulating Fluid-Rock Interaction Problems

generalized dimensionless pore-fluid pressure gradient (Zhao et al. 2008e), the cor-
responding dimensionless Zhao number of a single mineral dissolution system can
be defined as follows:

ν flow V p
, (4.1)
Zh =
φ f D(φ f ) k chemical A p C eq
! !
where ν flow is the Darcy velocity of the pore-fluid flow; φ f and D φ f are the
final porosity of the porous medium and the corresponding diffusivity of chemical
species after the completion of soluble mineral dissolution; C eq is the equilibrium
concentration of the chemical species; V p is the average volume of the soluble grain;
A p is the averaged surface area of the soluble grain; k chemical is the rate constant of
the chemical reaction.
To understand the physical meanings of each term in the Zhao number, Eq. (4.1)
can be rewritten in the following form:

Zh = F Advection F Diffusion F Chemical F Shape , (4.2)

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 chem-
ical 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 = ν flow , (4.3)

1
, (4.4)
F Diffusion =
φ f D(φ f )

1
F Chemical = , (4.5)
k chemical C eq

V p
F Shape = . (4.6)
A p

Equations (4.2), (4.3), (4.4), (4.5) and (4.6) clearly indicate that the Zhao number
is a dimensionless number that can be used to represent the geometrical, hydrody-
namic, thermodynamic and chemical kinetic characteristics of a fluid-rock system in
a comprehensive manner. The condition under which a chemical dissolution front in
a two-dimensional fluid-saturated porous rock becomes unstable can be expressed
by the critical value of this dimensionless number as follows: