Transport in Porous Media

, Volume 122, Issue 3, pp 713–744 | Cite as

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

  • Alexander A. Shapiro


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.


3D two-phase flow Porous medium Interface Hydrodynamic modeling Numerical solution 

List of symbols





Contour (boundary of a surface)


Specific liquid–liquid interface


Average velocity of the separating surface


Local velocity of the separating surface

\( {\mathbf{D}},D \)

Auxiliary vector in the generalized Darcy laws (or its component)

\( {\mathbf{E}} \)

Vector of a basis of the curvilinear system of coordinates on the microscale, associated with the interface


Fractional flow


Friction function



H, h

Coefficients for numerical solution


Relative permeability


Absolute permeability

L, l

Coefficients for numerical solution


Arbitrary microscale property


Macroscale average of property m



\( {\mathbf{n}},n \)

Vector of normal direction (or its component)


Mass source or sink (in the macroscale mass balance equations)


Representative elementary volume (r.e.v.)


Saturation (volume fraction) of the w-phase




Characteristic timescale

\( {\mathbf{U}},U \)

Volume average flow rate of a phase on the macroscale (or its component)

\( {\mathbf{u}},u \)

Microscale phase velocity (or its component)



\( {\mathbf{W}},W \)

Vector of average phase flow rate near the interface (or its component)


Local phase velocity near the separating surface


Local coordinate near the interface, in direction of expansion/contraction of the w-phase


Local coordinate near the interface, in direction of its rotation


Cartesian (fixed) coordinate directed along the saturation gradient

y, z

Cartesian (fixed) coordinates in the space orthogonal to x


\( \gamma \)

Auxiliary parameter




Characteristic distance




Characteristic relaxation time



Interface (in the definitions of the unit normal vectors)




Driving force


Direction of the saturation gradient












Surface phase




Interface between w- and o-phase

x, y, z, Y, Z

Components of a vector along the corresponding coordinate


Infinitesimal element



This work was initiated several years ago in the framework of the ADORE project, supervised by Professor Erling H. Stenby (DTU). Hadise Baghooee has performed a number of useful checks and computations in the framework of her Master thesis work. Professor Pavel G. Bedrikovetsky (University of Adelaide, Australia) and Dr. Sidsel Marie Nielsen (Technical University of Denmark) are kindly acknowledged for a number of useful discussions. Dr. Hanne Pernille Andersen is kindly acknowledged for professional language editing.


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Authors and Affiliations

  1. 1.CERE – Center for Energy Resources Engineering, Department of Chemical and Biochemical EngineeringTechnical University of Denmark – DTUKgs. LyngbyDenmark

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