Details

The following coupled partial and ordinary differential equations describe the mechanism governing the displacement of oil from a porous medium:

,

,

with the following initial and boundary conditions:

,

,

, and

where:

diffusion coefficient

interstitial solvent velocity

fraction of stagnant pore space

mass transfer coefficient

concentration of solvent in the flowing pore space

concentration of solvent in the stagnant pore space

length

time

inlet solvent concentration

initial solvent concentration in the stagnant pore space

initial solvent concentration in the stagnant pore space.

Solutions of these equations exhibit the following characteristics:

1. The effluent solvent concentration profile exhibits the more realistic asymmetry observed in laboratory experiments compared to the symmetric results obtained by solutions of the advection convection equation; the asymmetric tail is the result of stagnant oil being recovered by molecular diffusion.

2. Increasing the solvent velocity increases the rate of solvent and thus the rate of oil removal.

3. For sufficiently high mass transfer coefficient or low fraction of stagnant pore space, the modified equations reduce to the advection convection equation.

Reference

[1] M. Bai and D. Elsworth "On the Modeling of Miscible Flow in Multi-component Porous Media," Transport in Porous Media, 21(1), 1995 pp. 19–46. doi:10.1007/BF00615333.