Page 108 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

96 5 Simulating Chemical Dissolution Front Instabilities in Fluid-Saturated Porous Rocks

the propagation problem of the dissolved mineral front in fluid-saturated porous
medium. In the petroleum industry, the secondary recovery of oil by acidifying the
oil field to uniformly increase porosity and hence the yield of oil is associated with
the propagation of the acid-dissolved material front in porous rocks. More impor-
tantly, due to the ever-increasing demand for mineral resources and the likelihood
of the exhaust of the existing ore deposits, it is imperative to develop advanced tech-
niques to explore for new ore deposits. Towards this goal, there is a definite need
to understand the important physical and chemical processes that control ore body
formation and mineralization in the deep Earth (Raffensperger and Garven 1995,
Zhao et al. 1997a, 1998a, 1999b, d, 2000b, 2001b, d, Gow et al. 2002, Ord et al.
2002, Schaubs and Zhao 2002, Zhao et al. 2002c, 2003a). According to modern
mineralization theory, ore body formation and mineralization is mainly controlled
by pore-fluid flow focusing and the equilibrium concentration gradient of the con-
cerned minerals (Phillips 1991, Zhao et al. 1998a). Since the chemical dissolution
front can create porosity and therefore can locally enhance the pore-fluid flow, it
becomes a potentially powerful mechanism to control ore body formation and min-
eralization in the deep Earth.
Although analytical solutions can be obtained for some reactive transport prob-
lems with simple geometry, it is very difficult, if not impossible, to predict analyti-
cally the complicated morphological evolution of a chemical dissolution front in the
case of the chemical dissolution system becoming supercritical. As an alternative,
numerical methods are suitable to overcome this difficulty. Since numerical meth-
ods are approximate solution methods, they must be verified before they are used
to solve any new type of scientific and engineering problem. For this reason, it is
necessary to derive the analytical solution for the propagation of a planar dissolution
front within a benchmark problem, the geometry of which can be accurately simu-
lated using numerical methods such as the finite element method (Zienkiewicz 1977,
Lewis and Schrefler 1998) and the finite difference method. This makes it possible
to compare the numerical solution obtained from the benchmark problem with the
derived analytical solution so that the proposed numerical procedure can be verified
for simulating chemical-dissolution-front propagation problem in the fluid-saturated
porous medium.

5.1 Mathematical Background of Chemical Dissolution Front
Instability Problems in Fluid-Saturated Porous Rocks

5.1.1 A General Case of Reactive Multi-Chemical-Species
Transport with Consideration of Porosity/Permeability
Feedback

For a pore-fluid-saturated porous medium, Darcy’s law can be used to describe
pore-fluid flow and Fick’s law can be used to describe mass transport phenom-
ena respectively. If both the porosity change of the porous medium is caused by