Technical Notes: The Anisotropy of Relative Permeability
Authors: Prats Michael , Lake Larry W. 
Published: February 2015

Technical Notes
Relative permeabilities are used routinely in reservoir engineering, especially in numerical simulation, to describe multiphase flow through permeable media under the influence of 3D pressure fields and 1D gravitational body forces. Because gravitational body forces are directionally anisotropic (they point toward the center of the Earth) and pressure fields and body forces act differently on the fluids, relative permeabilities must depend on the direction of flow (i.e., be directionally anisotropic) even in homogeneous and isotropic reservoirs.
In practice, the same relative permeabilities, usually measured in horizontal flows, are used to determine both horizontal and vertical flows. The one exception is when gravity drainage is considered or known to be important, such as in downward gas displacements, in which cases special care is taken to determine how vertical downward flows respond to the combined effects of both pressure gradients and gravitational body forces. The reason for the exceptional treatment is that relatively small oil saturations are often attained under gravity drainage; sometimes less than 0.1 pore volume, especially if film flow occurs.
These conditions and results also may occur in high-permeability reservoirs undergoing gas displacement, such as thermal floods. And, of course, differences in relative permeability under vertical and horizontal flow at the same potential gradient and oil saturation mean that the oil relative permeability curves are directionally anisotropic. Directional anisotropy in relative permeability is not a mere matter of principle; it is a known and important phenomenon.
Even though the anisotropic behavior described above when drainage is important is well known, and its implications regarding the directional anisotropy of relative permeabilities are apparent, we have been slow to embrace the concept that relative permeabilities are directionally anisotropic. Simply stated, relative permeabilities are not measured, used, and discussed as being directionally anisotropic.
Let us emphasize that this discussion is not about the effective permeability anisotropy resulting from the anisotropies in the absolute permeability of a formation associated with small-scale heterogeneities attributable to its depositional and sedimentological environment. These effects are well known and accounted for by the industry, which routinely considers the ratio of vertical to horizontal absolute permeabilities to be generally less than one. We are arguing here for a more basic effect, that directional anisotropy in relative permeabilities is present even in a homogeneous and isotropic medium.
That the directional anisotropy exists in principle does not mean it is universally important. But it is clearly important where gravity drainage is a dominant mechanism.
Both industry and academia should inquire further into the full implications of the concept and of the pore-level phenomena associated with it. Specifically, it would seem important to have the option to use directionally anisotropic relative permeabilities in simulation models. To our knowledge, such capability is not generally available.
Michael Prats, SPE Honorary Member Larry W. Lake, SPE Honorary Member