Section: New Results

Models and simulations for flow and transport in porous media

Flow and transport in highly heterogeneous porous medium

Participants : Jean-Raynald de Dreuzy, Jocelyne Erhel, Géraldine Pichot.

Grants: H2MN04 8.2.1 , H2OGuilde 8.2.4 , HEMERA 8.2.2

Software: PARADIS, H2OLab

Publications: [13]

Abstract: Models of hydrogeology must deal with both heterogeneity and lack of data. We consider a flow and transport model for an inert solute. The conductivity is a random field following a stationary log normal distribution with an exponential or Gaussian covariance function, with a very small correlation length. The quantities of interest studied here are the expectation of the spatial mean velocity, the equivalent permeability and the macro spreading. In particular, the asymptotic behavior of the plume is characterized, leading to large simulation times, consequently to large physical domains. Uncertainty is dealt with a classical Monte Carlo method, which turns out to be very efficient, thanks to the ergodicity of the conductivity field and to the very large domain. These large scale simulations are achieved by means of high performance computing algorithms and tools.

Diffusion processes in porous media

Participants : Lionel Lenôtre, Géraldine Pichot.

Grants: H2MN04 8.2.1

Software: SBM 5.1.7 , PALMTREE 5.2.1

Publications: [21]

Conferences: [41] , [43] , [42]

Abstract: We present some recent results about Monte Carlo simulations in media with interfaces. By nature, porous media are extremely heterogeneous. We consider a one-dimensional advection-diffusion equation with piecewise constant coefficients. Without drift term, the Skew Brownian Motion permits to develop several exact algorithms with constant time step. We aim at adding the drift term and dealing with higher dimensional problems.

Adaptive stochastic model for flow and transport with random data

Participants : Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.7 , joint Ph-D 8.4.8

Publications: [27] .

Conferences: [46] .

Thesis:[11] .

Abstract: This work presents a development and an analysis of an effective approach for partial differential equation with random coefficients and data. We are interesting in the steady flow equation with stochastic input data.

A projection method in the one-dimensional case is presented to compute efficiently the average of the solution.

An anisotropic sparse grid collocation method is also used to solve the flow problem. First, we introduce an indicator of the error satisfying an upper bound of the error, it allows us to compute the anisotropy weights of the method. We demonstrate an improvement of the error estimation of the method which confirms the efficiency of the method compared with Monte Carlo and will be used to accelerate this method by the Richardson extrapolation technique.

We also present a numerical analysis of a probabilistic method to quantify the migration of a contaminant in random media. We consider the previous flow problem coupled with the advection-diffusion equation, where we are interested in the computation of the mean extension and the mean dispersion of the solute. The flow model is discretized by a mixed finite elements method and the concentration of the solute is the density of the solution of a stochastic differential equation, which is discretized by an Euler scheme. We present an explicit formula of the dispersion and optimal a priori error estimates.

Reactive transport

Participants : Édouard Canot, Jocelyne Erhel, Souhila Sabit.

Grants: H2MN04 8.2.1 , ANDRA 7.1 , MOMAS 8.2.7 , C2SEXA 8.2.3

Software: GRT3D.

Publications: [52] ,[30] .

Conferences: [20] ,[40] ,[50] , [32] .

Abstract: Numerical simulations are essential for studying the fate of contaminants in aquifers, for risk assessment and resources management. In this study, we deal with reactive transport models and show how a Newton method can be used efficiently. Numerical experiments illustrate the efficiency of a substitution technique. Moreover, it appears that using logarithms in the chemistry equations lead to ill conditioned matrices and increase the computational cost.