A theoretical model of hysteresis and dynamic effects in the capillary relation for two-phase flow in porous media

It is well known that the relationship between capillary pressure and saturation, in two-phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit representation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P-c-S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P-c-S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P-c-S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.