Variational geometric approach to the thermodynamics of porous media

Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper we derive the equations of motion for the dynamics of a deformable porous media which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics. This theory extends the recently developed variational derivation of the mechanics of deformable porous media to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid-matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics. The resulting equations give us the general version of possible friction forces incorporating thermodynamics, Darcy-like forces and friction forces similar to those used in the Navier-Stokes equations. The equations of motion are valid for arbitrary dependence of the kinetic and potential energies on the state variables. The results of our work are relevant for geophysical applications, industrial applications involving high pressures and temperatures, food processing industry, and other situations when both thermodynamics and mechanical considerations are important.