Derivation of the Conventional and a Generalized Form of Darcy’s Law from the Langevin Equation

The conventional and a generalized form of Darcy’s law for the absolute permeability of porous media at low flow rates has been derived using the Langevin equation. It can account for the conventional viscous resistance and additionally a term referred to as thermal resistance. The latter is hypothesized to stem from the relationship between the mean and molecular random velocity along the mean flow direction. The key to note when using the Langevin equation as the starting point for the derivation is that the frictional resistance occurring are the same whether the fluid or the object is moving relative to each other. Hence, a spherical particle moving through a stagnant fluid will experience the same friction as a fluid flowing past a stagnant spherical particle. Experimental data for absolute permeability at different temperatures indicate a temperature dependency, which can be accounted for by the thermal resistance term, which so far has been neglected. A procedure for matching and predicting pressure variation and absolute permeability vs. temperature is described. The generalized form of Darcy’s law presented should be of interest in all sciences where fluid transport in porous media occurs at varying temperatures, e.g., in soil science, hydrology, geothermal, chemical and petroleum engineering. It has also been discussed in relation to the Kozeny–Carman equation.