Derivation of the Mechanical Energy Equation in the Homogeneous and Heterogeneous regions of Porous Media

Modeling single- or multiphase flow in porous media is usually focused on the governing equations for mass and momentum transport, which yield the velocity and pressure at the pore- or Darcy-scales. However, in many applications, it is important to determine the work (or power) needed to induce flow in porous media, and this can be achieved when the mechanical energy equation is taken into account.. At the macroscopic scale, this equation is usually postulated to be the result of the inner product of Darcy's law and the seepage velocity. In this work we derive the macroscopic mechanical energy equation using the method of volume averaging. Our analysis shows that the result of averaging the pore-scale version of the mechanical energy equation to the Darcy scale is not, in general, the expected product of Darcy's law and the seepage velocity. As a matter of fact this result is only applicable in the bulk homogeneous region of the porous medium and, in the derivation of this result, the symmetry properties of the permeability tensor are determinant. Furthermore, near the porous medium boundaries, a more novel version of the mechanical energy equation is obtained, which incorporates additional terms that take into account the rapid variations of structural properties taking place in this particular portion of the system. This analysis can be further readily applied to multiphase and compressible flows in porous media and in many other multiscale systems. These applications are currently being studied by our workgroup.