Modelling of block-scale macrodispersion as a random function

Abstract

Numerical modelling of solute dispersion in natural heterogeneous porous media is facing several challenges. Amongst these we highlight the challenge of accounting for high-frequency variability that is filtered out by homogenization at the subgrid scale and the uncertainty in the dispersive flux for transport under non-ergodic conditions. These two effects when combined lead to inaccurate representation of the dispersive fluxes. We propose to compensate for this deficiency by defining a block-scale dispersion tensor and modelling it as a random space function ℳij. The derived dispersion tensor is a function of several length scales and time. Grid blocks will be assigned dispersion coefficients generated from the ℳij distribution. We will show the dependence of ℳij on the spatial variability of the conductivity field, on the contaminant source size, on the travel time and on the grid-block scale. For an ergodic source, a statistically uniform conductivity field and very large grid blocks, ℳij is equal to the macrodispersion coefficients proposed by Dagan (J. Fluid Mech., vol. 145, 1984, p. 151) with zero variance. For an ergodic source and non-uniform conductivity field with a finite-size grid block, ℳij approaches the model proposed by Rubin et al. (J. Fluid Mech., vol. 395, 1999, p. 161). In both cases, ℳij is defined by its mean value with zero variance. ℳij is subject to uncertainty when the source is non-ergodic and when the grid block is defined by a finite scale. When the grid-block scale approaches zero, which means that the spatial variability is captured completely on the numerical grid, ℳij approaches zero with zero variance. In addition, we provide a complete statistical characterization of ℳij by invoking the concept of minimum relative entropy, thus providing upper bounds on the uncertainty associated with ℳij.