Particle‐based transport model with Markovian velocity processes for tracer dispersion in highly heterogeneous porous media

Abstract

Monte Carlo (MC) studies of flow in heterogeneous porous formations, in which the log‐conductivity field is multi‐Gaussian, have shown that as the log‐conductivity variance σY2 increases beyond about 0.5, the one‐point velocity probability density functions (PDFs) deviate significantly from Gaussian behavior. The velocity statistics become more complex due to the formation of preferential flow paths, or channels, as σY2 increases. Methods that employ low‐order approximations (e.g., truncated perturbation expansions) are limited to small σY2 and are unable to represent the complex velocity statistics associated with σY2 > 1. Here a stochastic transport model for highly heterogeneous domains (i.e., σY2 > 1) is proposed. In the model, the Lagrangian velocity components of tracer particles are represented using continuous Markovian stochastic processes in time. The Markovian velocity process (MVP) model is described using a set of first‐order ordinary and stochastic differential equations, which are easy to solve using a particle‐based method. Once the MVP model is calibrated based on velocity statistics from MC simulations of the flow (hydraulic head and velocity), the MVP‐based model can be used to describe the evolution of the tracer concentration field accurately and efficiently. Specifically, the MVP model is validated using MC simulations of longitudinal and transverse tracer spreading due to a point‐like injection in the domain. We demonstrate that for σY2 > 1, the ensemble‐averaged tracer cloud remains markedly non‐Gaussian for relatively large travel distances from the point source. The MVP transport model captures this behavior and reproduces the velocity statistics quite accurately.