Numerical simulation of non‐Fickian transport in geological formations with multiple‐scale heterogeneities

Abstract

[1] We develop a numerical method to model contaminant transport in heterogeneous geological formations. The method is based on a unified framework that takes into account the different levels of uncertainty often associated with characterizing heterogeneities at different spatial scales. It treats the unresolved, small-scale heterogeneities (residues) probabilistically using a continuous time random walk (CTRW) formalism and the largescale heterogeneity variations (trends) deterministically. This formulation leads to a Fokker-Planck equation with a memory term (FPME) and a generalized concentration flux term. The former term captures the non-Fickian behavior arising from the residues, and the latter term accounts for the trends, which are included with explicit treatment at the heterogeneity interfaces. The memory term allows a transition between non-Fickian and Fickian transport, while the coupling of these dynamics with the trends quantifies the unique nature of the transport over a broad range of temporal and spatial scales. The advection-dispersion equation (ADE) is shown to be a special case of our unified framework. The solution of the ADE is used as a reference for the FPME solutions for the same formation structure. Numerical treatment of the equations involves solution for the Laplace transformed concentration by means of classical finite element methods, and subsequent inversion in the time domain. We use the numerical method to quantify transport in a two-dimensional domain for different expressions for the memory term. The parameters defining these expressions are measurable quantities. The calculations demonstrate long tailing arising (principally) from the memory term and the effects on arrival times that are controlled largely by the generalized concentration flux term.

Model of dispersive transport across sharp interfaces between porous materials

“…This is however inconsistent with the experimental evidence [1], where the solute transported in the F→C direction is observed to arrive faster with respect to the one transported in the C→F direction. Similarly, the adoption of a Fokker-Planck (FP) constitutive relationship j in [14] predicts faster solute arrival times are for the C→F flow direction and thus does not explain the results in [1].…”

Transport in Heterogeneous Porous Media

“…The average normalised flux concentration of chloride, j at the outlet of the apparatus (bold points) vs. time/min. It also shows the best ADE (dotted line) and CTRW fit (see section 3.2) (bold line) for these results by Cortis et al (2004). They show clearly that using the ADE gave poor predictions of the early and late time periods that is characteristic of anomalous transport.…”

“…The CTRW formulation has not only reproduced field scale numerical models as discussed earlier in the report (c.f. section 3.2 paragraph 2) but has been successful in simulating tracer migration in several field/laboratory scale experiments (Berkowitz and Scher , 1998;Berkowitz et al, 2000;Kosakowski et al, 2001;Levy and Berkowitz , 2003;Cortis et al, 2004), as described in section 3.1. In summary, each study demonstrated non-Gaussian dispersion that is typical in porous systems, showed it could not be modelled using traditional average approaches and then applied CTRW theory coupled with an appropriate ψ(t) to reproduce the given results.…”

“…Metre-scale laboratory based experiments Silliman and Simpson, 1987;Levy and Berkowitz , 2003), work on pore-scale samples (Hoffman et al, 1996;Cortis et al, 2004) and field tests of heterogeneous (Sudicky, 1986;Boggs et al, 1992) and fractured Sidle et al, 1998) porous media are also characterised by unusual arrival distributions. These results are termed anomalous as they do not conform to the Gaussian solutions obtained from formulations with average reservoir coefficients.…”

An integrated approach to shallow aquifer characterization: combining geophysics and geostatistics

“…These features are commonly referred to as unresolved heterogeneities and are typically accounted for in modeling through the use of dispersion parameters. Also, these features have been suggested as possible contributors to anomalous transport behavior and preferential flow path formation [5,9,10,25,45]; however, the precise influence of these heterogeneities is largely unknown. Many of the software packages currently available for heterogeneity modeling [e.g., Geostatistical Software Library (GSLIB), FLUVSIM, and transition probability geostatistics (TPROGS)] do not account for process-based geologic tendencies and rely on oversimplifications of subsurface geometry [4].…”