Humidity transfer in unsaturated heterogeneous porous media by homogenization

In the paper the one-equation model of humidity transfer in unsaturated macroscopically heterogeneous porous media is presented. The homogenization method by two-scale asymptotic expansions is used to derive the upscaled form of the Richard equation, which is commonly used when the medium is considered as macroscopically homogeneous. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics of the porous medium. The domain of validity of the model is explicitly given, namely: the length-scales separation, the characteristic time scale condition and the ratio of the hydrodynamic characteristics being of the same orders of magnitude. The effective capillary capacity and the effective hydraulic conductivity for an equivalent continuum are defined in terms of geometry and local hydrodynamic characteristics of the porous medium. A procedure of determination of the effective suction curve and the effective hydraulic conductivity curve as functions of the average water content for any type of the macroscopic heteregeneity for which the method can be applied, is provided. Since the problem is non-linear this procedure involves the resolution of a local boundary value problem formulated over a period for each value of suction. In two or three-dimensional cases, this problem can be solved using the numerical methods for any geometry of the medium. In a one-dimensional case it was shown that the analytical solution gives the well-known results of harmonic and arithmetic mean.