Theoretical analysis of non-Gaussian heterogeneity effects on subsurface flow and transport

Abstract

Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi-Gaussian log hydraulic conductivity (or transmissivity) fields, Ydx_5ln Kdx_ (x being a position vector), characterized by one or (less frequently) a multiplicity of spatial correlation scales. Yet Y and many other variables and their (spatial or temporal) increments, DY, are known to be generally non-Gaussian. One common manifestation of non-Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments DY are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale-dependent distributions of Y and DY in a unified and consistent manner has been recently proposed by us. This new `` generalized sub-Gaussian (GSG)'' model has the form Ydx_5Udx_Gdx_ where Gdx_ is (generally, but not necessarily) a multiscale Gaussian random field and Udx_ is a nonnegative subordinator independent of G. The purpose of this paper is to explore analytically, in an elementary manner, lead-order effects that non-Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Recognizing that perturbation expansion of hydraulic conductivity K5eY diverges when Y is sub-Gaussian, we render the expansion convergent by truncating Y's domain of definition. We then demonstrate theoretically and illustrate by way of numerical examples that, as the domain of truncation expands, (a) the variance of truncated Y (denoted by Yt) approaches that of Y and (b) the pdf (and thereby moments) of Yt increments approach those of Y increments and, as a consequence, the variogram of Yt approaches that of Y. This in turn guarantees that perturbing Kt5eYt to second order in rYt (the standard deviation of Yt) yields results which approach those we obtain upon perturbing K5eY to second order in rY even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean-uniform steady state flow in an unbounded, twodimensional domain of mildly heterogeneous Y with a single-scale function G having an isotropic exponential covariance. Results consist of expressions for (a) lead-order autocovariance and cross-covariance functions of hydraulic head, velocity, and advective particle displacement and (b) analogues of preasymptotic as well as asymptotic Fickian dispersion coefficients. We compare these theoretically and graphically with corresponding expressions developed in the literature for Gaussian Y. We find the former to differ from the latter by a factor k5hU2 i= hUi 2 (h i denoting ensemble expectation) and the GSG covariance of longitudinal velocity to contain an additional nugget term depending on this same factor. In the limit as Y becomes Gaussian, k reduces to one and the nugget term drops out. Plain Language Summary Much of the stochastic groundwater literature is devoted to the analysis of flow and transport in Gaussian or multi- Gaussian log hydraulic conductivity fields, Y(x), (x being a position vector). Yet Y, as well as many other variables and their increments DY, are known to be generally non- Gaussian. One common manifestation of non- Gaussianity is that whereas frequency distributions of Y often exhibit mild peaks and light tails, those of increments are generally symmetric with peaks that grow sharper, and tails that become heavier, as separation scale or lag between pairs of Y values decreases. A statistical model that captures these disparate, scale- dependent distributions of Y and DY in a unified and consistent manner has been recently proposed by us. This new generalized sub- Gaussian (GSG) model has the form Y(x) 5U(x) G(x) where G(x) is (generally, but not necessarily) a multi- scale Gaussian random field and U(x) is a non- negative subordinator independent of G. The purpose of this paper is to explore analytically lead-order effects that non-Gaussian heterogeneity described by the GSG model have on the stochastic description of flow and transport. Our analysis is rendered mathematically tractable by considering mean uniform steady state flow in an unbounded, two-dimensional domain of mildly heterogeneous Y.