Modification of the fundamental theorem for transport phenomena in porous media

To examine convection in porous media, numerous analyses and numerical simulations have been conducted based on the macroscopic governing equations. In deriving the macroscopic governing equations, the phenomena intrinsic to porous media, such as Darcy's flow resistance, Forchheimer's flow resistance, and dispersion, are modeled using the theorem of the local volume average of a gradient (or a divergence). The theorem has been widely accepted as fundamental in the theory of convection in porous media; however, certain questions relating to the correctness of the theorem (the continuous derivative of the volume average and the pressure correction for Darcy's law) have been raised. In this study, we modify the conventional theorem for the local volume average of a gradient (or a divergence) to solve the aforementioned questions. First, we introduce the concept of a point mass to describe the reference point for the movement of the fluid phase, and we derive the Reynolds transport theorem in the macroscopic field that corresponds to the continuum of porous media. Then, we examine the definition of divergence at a point to obtain the relation between the microscopic description that employs the velocity vector u and quantity B of the fluid particle and the macroscopic description that employs the reference velocity vector u0 and the reference quantity B0 of the point-mass particle, and we derive the modified theorem for the local volume average of a gradient (or a divergence). Furthermore, we derive the governing equations for porous media with the aid of the modified theorem.