A joint velocity‐concentration PDF method for tracer flow in heterogeneous porous media

Abstract

The probability density function (PDF) of the local concentration of a contaminant, or tracer, is an important component of risk assessment in applications that involve flow in heterogeneous subsurface formations. In this paper, a novel joint velocity‐concentration PDF method for tracer flow in highly heterogeneous porous media is introduced. The PDF formalism accounts for advective transport, pore‐scale dispersion (PSD), and molecular diffusion. Low‐order approximations (LOAs), which are usually obtained using a perturbation expansion, typically lead to Gaussian one‐point velocity PDFs. Moreover, LOAs provide reasonable approximations for small log conductivity variances (i.e., σY2 < 1). For large σY2, however, the one‐point velocity PDFs deviate significantly from the Gaussian distribution as demonstrated convincingly by several Monte Carlo (MC) simulation studies. Furthermore, the Lagrangian velocity statistics exhibit complex correlations that span a wide range of scales, including long‐range correlations due to the formation of preferential flow paths. Both non‐Gaussian PDFs and complex long‐range correlations are accurately represented using Markovian velocity processes (MVPs) in the proposed joint PDF method. LOA methods can be generalized to some extent by presuming a certain shape for the concentration PDF (e.g., a β PDF fully characterized by the concentration mean and variance). The joint velocity‐concentration PDF method proposed here does not require any closure assumptions on the shape of the marginal concentration PDF. The Eulerian joint PDF transport equation is solved numerically using a computationally efficient particle‐based approach. The PDF method is validated with high‐resolution MC reference data from Caroni and Fiorotto (2005) for saturated transport in velocity fields, which are stationary in space and time, for domains with σY2 = 0.05, 1, and 2 and Péclet numbers ranging from 100 to 10,000. PSD is modeled using constant anisotropic dispersion coefficients in both the reference MC computations and our PDF method.