Upscaling conductivity and porosity in three-dimensional heterogeneous porous media

Abstract

An empirical formula is presented to upscale the conductivity of 3-dimensional heterogeneous porous media, in which the distribution of local-scale conductivity is non-Gaussian with a high variance. The upscaled conductivity is determined as a function of the volumetric proportion, the spatial connectivity and the statistical geometric length of high-permeable inclusions, and the arithmetic mean of conductivities of all hydrofacies. A systematic comparison to other traditional upscaling methods indicates that this empirical formula provides a better estimation of the equivalent conductivity. In the second part of this study, numerical experiments of solute migration reveal that porosity also needs to be upscaled to capture the transport of contaminants in a heterogeneous medium using an effective or upscaled homogeneous medium. This is due to the tendency of contaminants to be preferrentially transported by 3-dimensional pathways composed of high-permeable materials in heterogeneous aquifer systems. The apparent difference between the actual transport velocity of contaminants and the upscaled velocity, based on the equivalent conductivity, forces upscaling of porosity. Further systematic analyses demonstrate that the upscaled porosity follows a non-linear trend as the content of high-permeable sediments decreases. Resultant upscaled porosity, with values varying between 0.004 and 1.5, is beyond the definition of the traditional porosity on the representative elementary volume (REV) scale. When the content of high-permeable materials is less than 30%, the upscaling of porosity is critical in the simulation of the contaminant transport in a heterogeneous medium using an upscaled, homogeneous counterpart.