Page 111 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.1 Mathematical Background of Chemical Dissolution Front Instability Problems 99

If the molar density (i.e. moles per volume) of the soluble grains is denoted by ρ s ,
then the source/sink term of chemical species i due to the dissolution/precipitation
of solid minerals within the system can be expressed as follows:

N

$ χ i
R i =−χ i ρ s k chemical D p A p C − K eq
i
i=1
(5.10)
N

A p $ χ i
= −χ i ρ s k chemical (φ f − φ) C i − K eq .
V p
i=1
5.1.2 A Particular Case of Reactive Single-Chemical-Species
Transport with Consideration of Porosity/Permeability
Feedack

If the pore-fluid is incompressible, the governing equations of a reactive single-
chemical-species transport problem in a fluid-saturated porous medium can be writ-
ten as follows:

∂φ
−∇ • [ψ(φ)∇ p] = 0, (5.11)
∂t

∂ A p
(φC) −∇ • [φD(φ)∇C + Cψ(φ)∇ p] + ρ s k chemical (φ f − φ)(C − C eq ) = 0,
∂t V p
(5.12)

∂φ A p
+ k chemical (φ f − φ)(C − C eq ) = 0, (5.13)
∂t V p

k(φ)
ψ(φ) = , (5.14)
μ
where C and C eq are the concentration and equilibrium concentration of the single
chemical species. Other quantities in Eqs. (5.11), (5.12), (5.13) and (5.14) are of the
same meanings as those defined in Eqs. (5.1), (5.2) (5.3), and (5.9).
Note that Eqs. (5.11) and (5.12) can be derived by substituting the linear average
velocity into Eqs. (5.1) and (5.3) with consideration of a single-chemical species.
It needs to be pointed out that for this single-chemical-species system, it is
very difficult, even if not impossible, to obtain a complete set of analytical solu-
tions for the pore-fluid pressure, chemical species concentration and porosity within
the fluid-saturated porous medium. However, in some special cases, it is possible
to obtain analytical solutions for some variables involved in this single-chemical-
species system. The first special case to be considered is a problem, in which a