A model for hysteretic constitutive relations governing multiphase flow: 2. Permeability‐saturation relations

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Abstract
A theoretical model is described for the prediction of relative permeability‐saturation (k‐S) relations in two‐phase (air‐water) and three‐phase (air‐oil‐water) porous media systems subject to arbitrary saturation paths. Integral expressions for air, water, and oil realtive permeabilities are presented which extend the nonhysteretic relative permeability model of Parker et al. (1987) to accomodate effects of pore blockage by air trapped in water and oil phases and oil trapped in the water phase. The parametric model for saturation‐pressure (S‐P) relations and fluid entrapment of paper 1 (Parker and Lenhard, this issue) is employed in the integral equations to enable derivation of closed‐form expressions for air, water, and oil relative permeabilities as functions of current fluid saturations and saturation history. Three‐phase k‐S relations are calculated for main drainage and imbibition paths for a hypothetical soil to illustrate usage of the model and to evaluate the magnitude of fluid entrapment effects on relative permeabilities. Water permeability‐saturation relations are predicted to exhibit mild hysteretic effects except at high saturations, while hysteresis in air permeability‐saturation relations is much more pronounced. Predicted hysteresis in oil permeability is low at low water saturations but becomes quite marked as water saturation increases. Predictions of k‐S‐P relations for a hypothetical NAPL contamination scenario are presented using model parameters determined for a sandy soil by two methods in paper 1 (Parker and Lenhard, this issue). The results indicate that hysteresis and nonwetting fluid entrapment effects on k‐S‐P relations may be quite substantial. Sensitivity to calibration method is found to be rather small.