  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.
