Page 112 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

100 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

planar dissolution front propagates in the full space. Since the dissolution front is a
plane, the problem described in Eqs. (5.11), (5.12) and (5.13) degenerates into a one-
dimensional problem. For this particular case, analytical solutions can be obtained
for both the propagation speed of the dissolution front and the downstream pressure
gradient of the pore-fluid. The second special case to be considered is an asymptotic
problem, in which the solid molar density greatly exceeds the equilibrium concen-
tration of the chemical species, implying that the region of a considerable porosity
change propagates very slowly within the fluid-saturated porous medium. In this
particular case, it is possible to derive a complete set of analytical solutions for the
pore-fluid pressure, chemical species concentration and porosity within the fluid-
saturated porous medium. In addition, it is also possible to investigate the reactive
infiltration instability associated with the dissolution front propagation in this par-
ticular case (Chadam et al. 1986).

5.1.2.1 The First Special Case

In this special case, the planar dissolution front is assumed to propagate in the pos-
itive x direction, so that all quantities are independent of the transverse coordinates
y and z. For this reason, Eqs. (5.11), (5.12) and (5.13) can be rewritten as follows:

∂φ ∂ ∂p
− ψ(φ) = 0, (5.15)
∂t ∂x ∂x
∂ ∂ ∂C ∂p A p
(φC) − φD(φ) + Cψ(φ) + ρ s k chemical (φ f − φ)(C − C eq ) = 0,
∂t ∂x ∂x ∂x V p
(5.16)
∂φ A p
+ k chemical (φ f − φ)(C − C eq ) = 0. (5.17)
∂t V p

If the chemical species is initially in an equilibrium state and fresh pore-fluid is
injected at the location of x approaching negative infinite, then the boundary condi-
tions of this special problem are expressed as
∂p
lim C = 0, lim φ = φ f , lim = p fx (Upstream boundary),
x→−∞ x→−∞ x→−∞ ∂x
(5.18)

∂p
lim C = C eq , lim φ = φ 0 , lim = p 0x (Downstream boundary),

x→∞ x→∞ x→∞ ∂x
(5.19)

where φ 0 is the initial porosity of the porous medium; p is the pore-fluid pressure
fx
gradient as x approaching negative infinite in the upstream of the pore-fluid flow; p
0x
is the unknown pore-fluid pressure gradient as x approaching positive infinite in the

downstream of the pore-fluid flow. Since p drives the pore-fluid flow continuously
fx