Modeling Permeability Hysteresis in Two- and Three-Phase Flow via Relaxation

Wednesday, January 9, 2002 - 2:00pm - 3:00pm
Keller 3-180
Bradley Plohr (University at Albany (SUNY))
Two-phase flow in a porous medium can be modeled, using Darcy's law, in terms of the relative permeability functions of the two fluids (say, oil and water). The relative permeabilities generally depend not only on the fluid saturations but also on the direction in which the saturations are changing. During water injection, for example, the relative oil permeability kro falls gradually until a threshold is reached, at which stage the kro begins to decrease sharply. This stage is termed imbibition. If oil is subsequently injected, then kro does not recover along the imbibition path, but rather increases only gradually until another threshold is reached, whereupon it rises sharply. This second stage is called drainage, and the type of flow that occurs between the imbibition and drainage stages is called scanning flow. Changes in permeability during scanning flow are approximately reversible, whereas changes during drainage and imbibition are irreversible.

In our lecture, we describe a model of permeability hysteresis based on relaxation. The distinctive features of our model are that it (a) allows the scanning flow to extend beyond the drainage and imbibition curves and (b) treats these two curves as attractors of states outside the scanning region. Through a rigorous study of traveling waves, we determine the shock waves that have diffusive profiles, and by means of a formal Chapman-Enskog expansion, we make a connection between our model and a standard one in the limit of vanishing relaxation time. Numerical experiments confirm our analysis.