In this paper, we show how very relevant mathematical works of P.I. Plotnikov, publicized by L.C. Evans and M. Portilheiro, can be used to model the effects of cyclic hysteresis phenomena for flows in unsaturated porous media. We draw particular attention to the example of a model for water–air flows, simplified for purposes of illustration. First, an unstable “spinodal” interval is artificially introduced. Then S.L. Sobolev’s method of dynamic regularization allows associating with the continuity equations additional information in the form of entropy type inequalities. The asymptotic limits of viscous approximate solutions generate effects of irreversibility and the expected clockwise hysteresis loop.
Seismoelectric coupling in unsaturated porous media: theory, petrophysics, and saturation front localization using an electroacoustic approach
