Examination of Darcy's law for flow in porous media with variable porosity

Henry Darcy's experimental studies in 1856 of saturated water flow through a homogeneous porous medium contained in a vertical column have provided the basis for the quantitative description of fluid flow in a wide variety of both natural and engineered porous medium environmental systems. Extrapolation of Darcy's original observations and conclusions has led to several commonly applied equations used to model flow in porous media. This work examines this original experimental study, summarizes the appropriate mathematical expressions that ensue directly from the data, and indicates expressions in common use that are suggested, but not actually supported, by the data. The paradoxes that exist in the common approaches for the case of a porous medium with a spatially variable porosity are illustrated. A modified form of Darcy's law, and also of the Hubbert potential, is derived based upon fundamental notions of averaging. The modified form of Darcy's law derived here reduces to the conventional form for a homogeneous porous medium.