A Stochastic Non-Gaussian Velocity Model for Tracer Dispersion in Heterogeneous Porous Media

To model tracer transport in porous media, computationally expensive Monte Carlo (MC) techniques or low-order approximation methods (LOAM) are applicable [1]. The latter are inexpensive but limited to relatively homogeneous media with low conductivity or transmissivity variations, and approximately Gaussian one-point velocity statistics. MC studies have shown that heterogeneous media lead to distinctly skewed non-Gaussian velocity distributions [2]. In addition to MC and LOAM, continuous time random walk (CTRW) or Lévy motion (LM) approaches were proposed for the modeling of dispersion in highly heterogeneous media, e.g, fractured rock [3,4]. Both models involve discontinuous stochastic processes for the displacement of tracer particles. The parameters that determine these processes, however, are not always easy to identify. In this work, a new particle-based model for the simulation of tracer dispersion in homogeneous and heterogeneous porous media is presented. Other than in CTRW or LM models, a continuous stochastic process for the Lagrangian velocity of a tracer particle is formulated. The suggested formulation encompasses Gaussian and skewed velocity statistics, and the model parameters can be related more easily to medium characteristics. Numerical simulations of the tracer plume evolution in the Borden tracer experiment and of breakthrough curves in homogeneous and uniformly heterogeneous sand packs are successfully validated with experimental data [5,6]. Non-Fickian dispersion behavior resulting from the scale effect (plume-size dependent dispersivities) and skewed velocity statistics is demonstrated and analyzed. [1] Zhang, Y. K. and D. Zhang (2004). "Forum: The state of stochastic hydrology." Stochastic Environmental Research and Risk Assessment 18(4): 265-265. [2] Salandin, P. and V. Fiorotto (1998). "Solute transport in highly heterogeneous aquifers." Water Resources Research 34(5): 949-961. [3] Benson, D. A., R. Schumer, et al. (2001). "Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests." Transport in Porous Media 42(1): 211-240. [4] Berkowitz, B., A. Cortis, et al. (2006). "Modeling non-Fickian transport in geological formations as a continuous time random walk." Reviews of Geophysics 44(2). [5] Freyberg, D. L. (1986). "A Natural Gradient Experiment on Solute Transport in a Sand Aquifer. 2. Spatial Moments and the Advection and Dispersion of Nonreactive Tracers." Water Resources Research 22(13): 2031-2046. [6] Silliman, S. E. and E. S. Simpson (1987). "Laboratory Evidence of the Scale Effect in Dispersion of Solutes in Porous-Media." Water Resources Research 23(8): 1667-1673.