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, Ytrue(boldxtrue)=ln⁡Ktrue(boldxtrue) (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, ΔY, 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 ΔY 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 ΔY in a unified and consistent manner has been recently proposed by us. This new “generalized sub‐Gaussian (GSG)” model has the form Ytrue(boldxtrue)=Utrue(boldxtrue)Gtrue(boldxtrue) where Gtrue(boldxtrue) is (generally, but not necessarily) a multiscale Gaussian random field and Utrue(boldxtrue) 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 K=eY 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 Kt=eYt to second order in σYt (the standard deviation of Yt) yields results which approach those we obtain upon perturbing K=eY to second order in σY even as the corresponding series diverges. Our analysis is rendered mathematically tractable by considering mean‐uniform steady state flow in an unbounded, two‐dimensional 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 k = true〈U2true〉/〈U〉2 ( true〈true〉 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,...

Recent advances in scalable non-Gaussian geostatistics: The generalized sub-Gaussian model

“…The GSG model is comprehensive and unique in that it captures simultaneously observed scaling behaviors of (a) the probability density distribution and (b) (statistical) moments of a random function and its increments. It does so by incorporating the concept of a truncated fractal introduced into the literature by Di Federico and Neuman (1997) and Di Federico et al (1999) Neuman (1990Neuman ( , 1991Neuman ( , 1995 to help explain the widely observed dispersivity scale effect (for a recent discussion of this effect and its explanation see Neuman, 2017). Di Federico and Neuman (1997) and Di Federico et al (1999) , it can immediately be down and/or upscaled to any other choices of these scales.…”

“…Potential applications, such as those concerning environmental risk assessment and management, are virtually endless. An early step in this direction was taken by Riva et al (2017) 1 2…”

Solute transport in bounded porous media characterized by generalized sub-Gaussian log-conductivity distributions

“…These concepts have already been employed in preliminary analytical and numerical studies of flow and transport in porous media whose log-conductivity is characterized through a GSG model. Riva et al (2017) present lead-order analytical flow and transport solutions in unbounded GSG log-conductivity fields under mean-uniform flow. Libera et al (2017) rely on a numerical Monte Carlo framework to analyze the joint effects of a GSG heterogeneous log-conductivity field and a temporally variable pumping rate on solute breakthrough curves (BTCs) detected at a pumping well operating in a twodimensional domain.…”

“…Libera et al (2017) rely on a numerical Monte Carlo framework to analyze the joint effects of a GSG heterogeneous log-conductivity field and a temporally variable pumping rate on solute breakthrough curves (BTCs) detected at a pumping well operating in a twodimensional domain. Besides spatial dimensionality, three major limitations can be identified for the analytical study by Riva et al (2017): ( ) macrodispersion is the only transport metric analyzed, ( ) the joint impact of local dispersivity combined with the heterogeneous advection driven by is not evaluated, and ( ) output uncertainty due to finite size of the medium is not considered.…”

Grid convergence for numerical solutions of stochastic moment equations of groundwater flow

“…Moment differential equations of groundwater flow have been recently applied to field settings (Riva et al, 2009;Bianchi Janetti et al, 2010;Panzeri et al, 2015), to non-Gaussian fields (e.g., Hristopulos, 2006;Riva et al, 2017) and have been embedded in geostatistical inverse modeling approaches (Hernandez et al 2003), stochastic pumping test interpretation (Neuman et al, 2004(Neuman et al, , 2007, or reactive solute transport (e.g., Hu et al, 2004). Most recent developments have allowed embedding stochastic MEs of transient groundwater flow in data assimilation/integration and parameter estimation approaches, e.g., via ensemble Kalman filter (Li and Tchelepi, 2006;Panzeri et al, 2013Panzeri et al, , 2015.…”