Section: New Results

Models and simulations for flow and transport in porous media

Flow and transport in highly heterogeneous porous medium

Participants : Jocelyne Erhel, Grégoire Lecourt, Géraldine Pichot.

This work is done in the context of the H2MNO4 project (see 8.1.1 ), the H2OGUILDE project (see 8.1.4 ), the HEMERA project (see 8.1.2 ). Computations are partly done with GENCI supercomputers (see 8.1.6 ), using the platform H2OLab (see 5.1 ) and the software GWNUM, GWUTIL, PARADIS (see 5.3 , 5.2 , 5.5 ).

This work was done in collaboration with A. Beaudoin, from University of Poitiers (Pprime) and J.-R. de Dreuzy, from Geosciences Rennes (who is on leave until 2013 at UPC, Barcelona, Spain, see 8.2.1 ). It is also done in collaboration with A. Debussche, from ENS-Cachan-Rennes/Ipso Inria team. It was also done in the context of Grégoire Lecourt's internship (M2, INSA Rennes).

It has been presented at a conference (plenary talk) [26] and a paper is submitted to a journal.

Models of hydrogeology must deal with both heterogeneity and lack of data. We consider in this paper 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 and in turn 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.

Solving flow equations in highly heterogeneous porous medium

Participant : Thomas Dufaud.

This work was done in collaboration with L. Berenguer and D. Tromeur-Dervout, from University of Lyon (ICJ).

It is published in a journal [12] .

This paper is devoted to the acceleration by Aitken’s technique of the convergence of the Schwarz domain decomposition method applied to large scale 3D problems with non separable linear operators. These operators come from the discretization of groundwater flow problems modeled by the linear Darcy equation, where the permeability field is highly heterogeneous and randomly generated. To be computationally efficient, a low-rank approximation of the Aitken’s formula is computed from the singular value decomposition of successive iterated solutions on subdomains interfaces. Numerical results explore the efficiency of the solver with respect to the random distribution parameters, and specific implementations of the acceleration are compared for large scale 3D problems. These results confirm the numerical behavior of the methodology obtained on 2D Darcy problems [49] .

Transport in discontinuous porous medium

Participants : Lionel Lenôtre, Géraldine Pichot.

This work was done in collaboration with A. Lejay, from Inria Nancy, in the context of the H2MNO4 project ( 8.1.1 ).

It is published in a journal [16] .

We propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian Motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.

Adaptive stochastic collocation method for an elliptic problem with random data

Participants : Jocelyne Erhel, Mestapha Oumouni.

This work is done in collaboration with Z. Mghazli, from the university of Kenitra, Morocco, in the context of the joint PhD supervision and the HYDRINV project (see 8.3.8 , 8.3.4 ).

This work has been presented at two conferences [43] [42] .

Stochastic collocation methods are frequently used for elliptic equations with random coefficients. However, sparse grid methods are quite expensive and adaptive approaches are designed to save computations.

Reactive transport

Participants : Édouard Canot, Jocelyne Erhel, Souhila Sabit.

This work is done in the context of the MOMAS GNR ( 8.1.7 ), the contract with Andra ( 7.1 ) and the C2S@EXA project (see 8.1.3 ). Computations use the software GRT3D (see 5.6 ).

It has been presented at a conference and a workshop [35] [36] .

Modeling reactive transport of contaminants in porous media is a complex time-dependent problem, due to combining the difficulties of modeling transport and chemistry, especially the coupling between them. In this work, we are interested to solve this type of coupling. Several methods have been developed for the resolution for solving this type problem. We choose to solve this problem by a global approach, which considers all the equations as a whole system of differential algebraic equations (DAE), which come from the spatial-only discretization of the equations (method of lines). This approach uses implicit schemes, which imply solving many large linear systems with the Jacobian matrix. The differential algebraic system (DAE) is solved by the solver IDA Sundials. Our new technique is implemented in the GRT3D software; we have observed that the CPU time increases very fast with the size of the system. Our aim is thus to reduce this computation time. Profiling tools have shown that an important part of this computation is due to the linear solving related to the Jacobian matrix. We focus our effort on improving this part, by exploiting the 3x3 block-structure of the Jacobian matrix, via a Gaussian block elimination technique. Our simulations are performed on academic test cases, which involve few chemical components (4 to 5) for both 1D and 2D geometries, giving a number of unknowns up to 72000. First results have shown that our technique is very promising, because the CPU time is reduced by approximately 40 After this part, we eliminated the tracer in our test cases. In GRT3D-SL software, we calculated the concentrations directly without using the Logarithms and with this software, we have reduced the CPU time to 50 %.