A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

Abstract

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.

Appendix

The goal of this appendix is to demonstrate that the rotation of the w- and o-regions is negligible on the macroscopic scale and, thus, to eliminate rotation from the macroscopic picture of the motion.

1.1 Statement

Rotation of the w- and o-regions may, on average, be disregarded. The value of the corresponding velocity $W_{Z}$ is equal to zero outside the regions of negligible volume.

In order to demonstrate the validity of this statement, we apply the curl operator to the Brinkman Eq. (13) for the w-phase. The gradient of pressure cancels out:

$$ - \frac{{\mu_{{w}} }}{K}\nabla \times {\mathbf{u}}_{{w}} + \mu_{\text{e},w} \Delta (\nabla \times {\mathbf{u}}_{{w}} ) = 0 $$

(A1)

That is, the curl $\nabla \times {\mathbf{u}}_{{w}}$ obeys the Helmholtz equation with a negative coefficient. Its solution is zero, provided that the boundary conditions do not contain vortices, which is true for most of the typical problem statements. In rare cases where $\nabla \times {\mathbf{u}}_{{w}}$ is nonzero on the boundary of a region of interest, the solution of Eq. (A1) is damped exponentially with the distance from the boundary, and the characteristic damping distances are of the order of $\sqrt K$ (Tikhonov and Samarskii 1963). For typical values of permeabilities, these distances are comparable to the pore sizes, and much smaller than the size of a region of interest. Hence, for the most part of the region it may be asserted that

$$ \nabla \times {\mathbf{u}}_{{w}} = 0 $$

(A2)

It should also be remarked that the last equality would be valid automatically if the microscale Brinkman equation could be reduced to the Darcy law.

By the Stokes circulation theorem, it follows from Eq. (A2) that circulation of vector ${\mathbf{u}}_{{w}}$ over contour $B_{{wo}}$ (Fig. 1d) is equal to zero:

$$ \oint\limits_{{B_{{wo}} }} {w_{Z} {\text{d}}s} = 0 $$

(A3)

The value of $W_{Z}$ is the average of $w_{Z}$ over the surface $A_{{wo}}$ and, hence, is superposition of integrals (A3). Therefore, it is also equal to zero. The statement is proven.