A non-equilibrium model of a porous medium saturated with immiscible fluids

A model of a thermoelastic porous medium, saturated with two immiscible fluids, is considered. It is assumed that there are no phase transitions, the contribution of pulsations to the stress and kinetic energy is small, and that the components of the medium are in thermodynamic equilibrium. The non-equilibrium of the state, related to the finite time of redistribution of the fluids among the pores of the channels due to the presence of surface forces, is taken into account. A general form of the governing relations, necessary and sufficient to satisfy the principles of thermodynamic compatibility and independence of the choice of system of coordinates, is obtained. It is shown that the establishment of equilibrium is accompanied by dissipation due to capillary forces, which does not lead to seepage dissipation or thermal dissipation. For the case when the deformation of the skeleton and the deviation of the mean porous pressure and the temperature from the initial values are small, while the saturation and the non-equilibrium parameter undergo finite changes, an approximation of the potential of the skeleton is proposed in the form of a quadratic expansion in small parameters. A feature of the expansion is the presence of an initial value of the potential, which depends on the saturation and non-equilibrium. The relationship between the thermodynamic potential and the non-equilibrium kinetics, related to the requirement that the dissipation by the capillary forces should be non-negative, is determined. A generalized Darcy's law is formulated, which takes cross terms into account. It is shown that the proposed approximations enable key effects, which accompany the motion of immiscible fluids in a porous medium, to be described. (C) 2009 Elsevier Ltd. All rights reserved.