Stochastic flow and transport through multifractal porous media

Abstract

Stochastic theories of flow and transport in aquifers have relied on the linear perturbation approach that is accurate for flow fields with log-conductivity variance cr2 less than unity. Several studies have found that the linear perturbation ignores terms that have significant effects on the spectra of the hydraulic gradient VH and specific discharge q when (Y2 exceeds unity. In this thesis we study flow and transport when the hydraulic conductivity K is an isotropic lognormal multifractal field. Unlike the perturbation approach, results obtained are nonlinear even though several simplifying assumptions are made. The spectral density of F = in (K) for this type of field is SF (k) o kl-D where D is the space dimension. It is found that under this condition, the hydraulic gradient VH and specific discharge q are also multifractal; whose renormalization properties under space contraction involve random scaling and random rotation of the fields. Analytical expressions that are functions of D and the codimension parameter of F, CK 'are obtained for the renormalization properties and marginal distributions of VH and q . Because of the boundary conditions, the fields VH and q are anisotropic at large scales but become isotropic at very small scales. The mean specific flow decreases as the scaling range of F increases, at a rate that is dependent on D and CK. Flow simulations on a plane validate the analytical results. The multifractal properties of VH and q are used to derive their spectral density tensors, the macrodispersivities, and the effective conductivity of the medium. The spectra obtained account for the random rotation of the VH and q at smaller scales. Spectra for VH and q are anisotropic at large scales but become isotropic at small scales.

(cont.) The scale of isotropy depends on D and CK. The linear perturbation approach does not capture this important feature and further gives incorrect amplitudes and power decays of the spectral density tensors. Using the spectra of q the macrodispersivities are computed and compared with results from the linear perturbation approach. Reflecting the properties of the spectral density of q, the macrodispersivities for the nonlinear theory are isotropic at small travel distances and are anisotropic at large travel distances. In the ergodic case when the spatial averages of all fields of interest are close to their ensemble averages, it is found that our expression for effective conductivity Keff corresponds to a formula conjectured by Matheron [1967]. Using the scaling properties of the inverse of the velocity field (also known as slowness), we derive expressions for the first passage time distribution FPTD and mean plume concentration for transport in a multifractal K field. The theoretical results of FPTD for the nonlinear theory are fitted by regression methods to data from field experiments and from numerical simulations and compared with results from the continuous time random walk CTRW and two-phase transport model. Results of the nonlinear theory are found to be more suitable for predicting non-Fickian transport. The CTRW model is more suited for transport in statistically inhomogeneous media. Both the CTRW and two-phase models are suitable for modeling Fickian transport ...