Blueprint for perturbative solution of flow and transport in strongly heterogeneous composite media using fractal and variational multiscale decomposition

This paper offers a blueprint for extending perturbative ensemble moment (and potentially other, e.g., Gaussian or perturbative Karhunen-Loéve) solutions of stochastic flow and transport problems in composite, randomly heterogeneous porous media to cases in which medium heterogeneity is arbitrarily large. The proposed approach is based on a novel idea of combined fractal and variational multiscale decomposition. It considers log hydraulic conductivity to be a composite random field with piecewise statistically homogeneous increments characterized by a truncated power variogram, i.e., a truncated fractal. This consideration is supported either exactly or approximately by a growing amount of observational data. It allows decomposing the field into two or more mutually uncorrelated components having relatively small variances associated with a hierarchy of spatial correlation scales. Combining this fractal decomposition with a variational multiscale decomposition formalism, motivated by the multiscale nature of turbulence, allows decomposing flow and transport problems into two or more coupled sets of variational problems each of which is amenable to perturbative solution regardless of how heterogeneous the original field was. The problems are defined on a hierarchy of computational grids having discretization intervals proportional to the correlation scales. The proposed approach does not require introducing effective parameters on any scale; instead, it allows resolving flow and transport ensemble moments fully on multiple scales of spatial resolution. The approach nevertheless allows subgrid closure (another idea inspired by the literature on turbulence) through subgrid refinement or the use of effective subgrid parameters. It additionally offers the critically important capability of conditioning flow and transport analyses on multiscale measurements. Details are presented in the context of steady state single-phase flow in a composite medium.