Numerical Analysis of the Advection-Diffusion of a Solute in Porous Media with Uncertainty

We consider the problem of numerically approximating the solution of the coupling of the flow equation in a porous medium, with the advection-diffusion equation in the presence of uncertainty on the permeability of the medium. Random coefficients are classically used in the flow equation to modelize uncertainty. More precisely, we propose the numerical analysis of a method developed to compute the mean value of the spread of a solute introduced at the initial time, and the mean value of the macro-dispersion, defined as the temporal derivative of the spread. We consider a Monte-Carlo method to deal with the uncertainty, i.e. with the randomness of the permeability field. The flow equation is solved using a finite element method. The advection-diffusion equation is seen as a Fokker-Planck equation, and its solution is hence approximated thanks to a probabilistic particular method. The spread is indeed the expected value of a function of the solution of the corresponding stochastic differential equation, and is computed using an Euler scheme for the stochastic differential equation and a Monte-Carlo method. Error estimates on quantities generalizing the mean spread and the mean macro-dispersion are established, under some assumptions including the case of random fields of lognormal type (i.e. neither uniformly bounded from above nor below with respect to the random parameter) with low regularity, which is pertinent on an application point of view and rises several mathematical diffculties.