On the multiscale characterization of effective hydraulic conductivity in random heterogeneous media: A historical survey and some new perspectives

•Multiscale characterization of the effective conductivity distributions.•Improved analytical results using the energy dissipation formulation.•Efficient numerical estimator valid at all scales using that formulation.•Compare scale dependence of full pdf, mean and variance with two other estimators. In large scale heterogeneous aquifer simulations, determining the appropriate coarsening scale λ to define an effective hydraulic conductivity Keff is a challenging task, that involves a trade-off between accuracy and cost. Efficiently adjusting the scale λ is then key, in particular for uncertainty quantification. In this paper, we obtain improved analytical results for the variance of Keff, valid at any scale, in the context of energy dissipation formulation. Using this formulation, we then derive an efficient Keff numerical estimator, and compare it with those of the potential-flow average and permeameter formulations in 2D, for lognormal and binary media, over a wide range of λ and of heterogeneity. We analyze the probability density function (pdf), mean, and variance, of these estimators, comparing them with the analytical results. In the lognormal case, the pdf’s are rather similar for the three estimators, and remain lognormal at all scales. In the binary case, slow convergence to an asymptotic regime is observed close to the percolation threshold. [Display omitted]