Probability density functions of hydraulic head and velocity in three‐dimensional heterogeneous porous media

Abstract

In this study, we assess probability density functions of hydraulic heads and specific discharges in three‐dimensional bounded heterogeneous porous media by Monte Carlo (MC) simulation. We discuss their empirical shapes and demonstrate that the intuitive use of obvious information on boundedness leads to parametric distribution functions, which fit surprisingly well. On the basis of statistical moments of hydraulic heads and velocities up to fourth order, we discuss the spatial dependence of the empirical distributions and their dependence on the variance of log conductivity. Comparison of the first and second central moment to the results from classical numerical first‐order second‐moment (FOSM) analysis reveals that FOSM predicts these moments surprisingly close for hydraulic heads. On the basis of this fact, we demonstrate that fitting the chosen parametric distributions for hydraulic heads to FOSM moments is promising for the sake of estimating exceedance probabilities. Our MC scenarios vary in variance of log conductivity (0.125 to 5.0), in the type of multivariate dependence, in correlation scale and types of boundary conditions. Our study illustrates that in contrast to the common assumption, FOSM is a reasonable choice for evaluating multivariate and univariate moments for heads, if used in conjunction with additional information on distribution shapes. In the absence of utilizable additional information, we demonstrate that second‐moment methods are mostly inadequate for assessing distributions accurately. Significant deviations from Gaussian distributions occurred for discharge components even at a variance of log conductivity as low as 0.125, and we found that the distributions of transverse discharge components are extremely fat‐tailed. The observed non‐Gaussianity questions the results of approximate approaches in solute flux and dispersion studies where velocity fields are assumed to be multi‐Gaussian and then directly represented by or generated from their covariances. The main implication is to apply more accurate schemes such as exact non‐local methods, extensive MC or higher‐order stochastic Galerkin approaches, and to include higher‐order moments, at least if no additional assumptions on the shape of distributions are available or justifiable.