Upscaling conductivity and porosity in three-dimensional heterogeneous porous media

An empirical formula is presented to upscale the conductivity of 3-dimensional heterogeneous porous me- dia, in which the distribution of local-scale conductivity is non-Gaussian with a high variance. The upscaled conductiv- ity 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 con- ductivities of all hydrofacies. A systematic comparison to other traditional upscaling methods indicates that this em- pirical formula provides a better estimation of the equivalent conductivity. In the second part of this study, numerical ex- periments 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 homo- geneous 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 po- rosity. Further systematic analyses demonstrate that the up- scaled 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 be- yond 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.