Page 119 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.1 Mathematical Background of Chemical Dissolution Front Instability Problems 107

where
ψ(φ 0 ) k(φ 0 )
β = = , (5.67)
ψ(φ f ) k(φ f )

%
2
2
(p ) + 4m − p

fx fx
σ = . (5.68)
2
Substituting Eq. (5.66) into Eq. (5.64) yields the following equation for the
growth rate of the small perturbation:
−p %
fx 2 2

ω(m) = [−p − (p ) + 4m + (1 − β) |m|). (5.69)
fx
fx
(1 + β)(φ f − φ 0 )
Equation (5.69) clearly indicates that the planar dissolution front of the reactive
transport system, which is described by the above-mentioned second special prob-
lem, is stable to short wavelength (i.e. large wavenumber m) perturbations but it is
unstable to long wavelength (i.e. small wavenumber m) perturbations.
Letting ω(m) = 0 yields the following critical condition, under which the reac-
tive transport system can become unstable.
(3 − β)(1 + β)
&
p & =− , (5.70)
fx critical
2(1 − β)
&
where p & is the critical value of the generalized dimensionless pressure gra-
fx critical
dient in the far upstream direction as x approaching negative infinite (Zhao et al.
&
2008e). Since p & is usually of a negative value, the following critical Zhao
fx critical
number is defined to judge the instability of the reactive transport system:
(3 − β)(1 + β)
&
Zh critical =−p & = . (5.71)
fx critical
2(1 − β)
Thus, the Zhao number of the reactive transport system can be defined as follows:

∗
p L ∗ k(φ f )L p

fx fx v flow V p
Zh =−p =− =− . (5.72)
fx =
p ∗ φ f μD(φ f ) φ f D(φ f ) k chemical A p C eq
Using Eqs. (5.71) and (5.72), a criterion can be established to judge the instability
of a chemical dissolution front associated with the particular chemical system in this
investigation. If Zh > Zh critical , then the chemical dissolution front of the reactive
transport system becomes unstable, while if Zh < Zh critical , then the chemical dis-
solution front of the reactive transport system is stable. The case of Zh = Zh critical
represents a situation where the chemical dissolution front of the reactive transport