## Theory of porous media

In the mid-19^{th} century, the theory of porous media were unknown, it was a long vacuum since the development of volume fractions by Woltman in 1794. However, the fundamental discoveries were found by Delesse, Fick and Darcy. Delesse develop the concept of surface fraction in 1848, Fick studied the problem of diffusion through membrane in 1855, and then Darcy in 1856 investigated the permeability of water running through sand as a basic study in multiphase continuum [1].

The Darcy equation actually was derived from the Navier-Stokes equation by assuming stationary, creeping, incompressible flow through porous medium. Since the Darcy’s equation becomes invalid for liquid for high velocity, or gases for very low or very high velocities, the corrections was build by Brinkmann. Brinkman’s filtration equation is usually used to describe the low-Reynolds-number flow in porous media in situations where velocity gradients are non-negligible [2].

References:

- R. de Boer, Theory of porous media: Highlights in the historical development and current state, Springer-Verlag, Berlin-Heidelberg, 2000.
- F. J. Valdes-Parada, J. A. Ochoa-Tapia, and J. Alavarez-Ramirez, On the effective viscosity for the Darcy-Brinkmann equation,
*Physica A*, 2007,**385**, 69-79.