Page 109 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

5.1 Mathematical Background of Chemical Dissolution Front Instability Problems 97

chemical dissolution of soluble solid minerals within the porous medium and the
feedback effect of such a change on the variation of permeability and diffusivity
are taken into account, the governing equations of the coupled nonlinear problem
between porosity, pore-fluid flow and reactive multi-chemical-species transport in
the pore-fluid-saturated porous medium can be expressed as follows:
∂
(ρ f φ) +∇ • (ρ f φ
u linear ) = 0, (5.1)
∂t
k(φ)
u

u = φ
linear =− ∇ p, (5.2)
μ
∂
(φC i ) +∇ • (φC i
u linear ) =∇ • [φD i (φ)∇C i ] + R i (i = 1, 2,... N),
∂t
(5.3)
where
linear is the averaged linear velocity vector within the pore space of the
u
u
porous medium;
is the Darcy velocity vector within the porous medium; p and C i
are pressure and the concentration (moles/pore-fluid volume) of chemical species i;
μ is the dynamic viscosity of the pore-fluid; φ is the porosity of the porous medium;
D i (φ) is the diffusivity of chemical species i; ρ f is the density of the pore-fluid; N
is the total number of all the chemical species to be considered in the system; R i
is the source/sink term of chemical species i due to the dissolution/precipitation of
solid minerals within the system; k(φ) is the permeability of the porous medium.
It is noted that in Eqs. (5.1), (5.2) and (5.3), the chemical species concentration,
the fluid density and averaged linear velocity of the pore-fluid are defined in the
pore space, while the source/sink term and the Darcy velocity of the pore fluid are
defined in the whole medium space (Phillips 1991, Nield and Bejan 1992, Zhao
et al. 1994c).
Since the diffusivity of each chemical species is considered as a function of
porosity, a common phenomenological relation can be used for describing this func-
tion (Bear 1972, Chadam et al. 1986).

3 5
q
D i (φ) = D 0i φ ≤ q ≤ , (5.4)
2 2
where D 0i is the diffusivity of chemical species i in pure water.
To consider the permeability change caused by a change in porosity, an equa-
tion is needed to express the relationship between permeability and porosity. In
this regard, Detournay and Cheng (1993) state that “The intrinsic permeability k
is generally a function of the pore geometry. In particular, it is strongly dependent
on porosity φ. According to the Carman-Kozeny law (Scheidegger 1974) which
is based on the conceptual model of packing of spheres, a power law relation of
2
k ∝ φ 3 " (1 − φ) exists. Other models based on different pore geometry give sim-
ilar power laws. Actual measurements on rocks, however, often yield power law
relations with exponents for φ significantly larger than 3.” Also, Nield and Bejan