A Fokker–Planck approach for probability distributions of species concentrations transported in heterogeneous media

Abstract

a b s t r a c tWe identify sufficient conditions under which evolution equations for probability density functions (PDF) of random concentrations are equivalent to Fokker-Planck equations. The novelty of our approach is that it allows consistent PDF approximations by densities of computational particles governed by Itô processes in concentration-position spaces. Accurate numerical solutions are obtained with a global random walk (GRW) algorithm, stable, free of numerical diffusion, and insensitive to the increase of the total number of computational particles. The system of Itô equations is specified by drift and diffusion coefficients describing the PDF transport in the physical space, provided by up-scaling procedures, as well as by drift and mixing coefficients describing the PDF transport in concentration spaces. Mixing models can be obtained similarly to classical PDF approaches or, alternatively, from measured or simulated concentration time series. We compare their performance for a GRW-PDF numerical solution to a problem of contaminant transport in heterogeneous groundwater systems.

Consistency issues in PDF methods

“…The same result has been obtained by equating the stochastic average of the operator from the left hand side of (1) applied to a test function with that corresponding to the right hand side of (1) and by integrating by parts [1,2,5,7].…”

“…It is also readily to check, by using (16), that the FokkerPlanck equation (30) takes the form of the PDF equation (2) only if Θ is a constant (see e.g. [7]), which implies a uniform position PDF p x (x, t) = Θ (x, t).…”

“…Brought to you by | MIT Libraries Authenticated Download Date | 5/9/18 7:06 PM that is, as an arithmetic average of position PDFs p (α) x = Ωc α p cα,x (c α , x, t)dc α associated to each chemical species, where p cα,x (c α , x, t) are marginal PDFs of p(c, x, t) [7]. The natural conjecture is that the position PDF p x (x, t) corresponds to the expectation of the arithmetic mean of the species concentrations,…”

Solute transport in aquifers with evolving scale heterogeneity

“…The memory coefficient of the center of mass is readily obtained, similarly to (28), by a change of the integration limits in the expression (31) of the center of mass dispersion coefficients:…”

“…For instance, the superposition of 2 fields has the variance 0.2 and the integral scale λ 2 = 3 m and that of seven fields (the largest one investigated here) has the variance 0.7 and λ 7 = 28 m. The GRW algorithm has been developed by Vamoş et al [30]. Implementation details and applications for transport in saturated aquifers can be found in [24,19,27,23,28]. In our GRW numerical simulations we estimate dispersion coefficients for processes starting from instantaneous point injections by the mean half-slope of the variances defined in (3).…”