## Section: New Results

### Models and simulations for flow and transport in porous media

#### Simulating Diffusion Processes in Discontinuous Media: Benchmark Tests

Participant : Géraldine Pichot.

Grants: H2MN04 8.2.1

Software: SBM 5.2.2

Publications: [33]

Abstract: We present several benchmark tests for Monte Carlo methods for simulating diffusion in one-dimensional discontinuous media, such as the ones arising the geophysics and many other domains. These benchmarks tests are developed according to their physical, statistical, analytic and numerical relevance. We then perform a systematic study on four numerical methods.

#### Uncertainty Quantification and High Performance Computing for flow and transport in porous media

Participants : Jean-Raynald de Dreuzy, Jocelyne Erhel.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Publications: [18]

Abstract: Stochastic models use random fields to represent heterogeneous porous media. Quantities of interest such as macro dispersion are then analyzed from a statistical point of view. In order to get asymptotic values, large scale simulations must be performed, using High Performance Computing. Non-intrusive methods are well-suited for two-level parallelism. Indeed, for each simulation, parallelism is based on domain decomposition for generating the random input and the flow matrix, parallel linear solvers and parallel particle tracker. Also, several simulations, corresponding to randomly drawn samples, can run in parallel. The balance between these two levels depends on the resources available. The software PARADIS implements flow and transport with random data and computation of macro dispersion. Simulations run on supercomputers with large 3D domains.

#### Computation of macro spreading in 3D porous media with uncertain data

Participants : Jean-Raynald de Dreuzy, Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Publications: [15]

Abstract: We consider an heterogeneous porous media where the conductivity is described by probability laws. Thus the velocity, which is solution of the flow equation, is also a random field, taken as input in the transport equation of a solute. The objective is to get statistics about the spreading and the macro dispersion of the solute. We use a mixed finite element method to compute the velocity and a lagrangian method to compute the spreading. Uncertainty is dealt with a classical Monte-Carlo method, which is well-suited for high heterogeneities and small correlation lengths. We give an explicit formulation of the macro dispersion and a priori error estimates. Numerical experiments with large 3D domains are done with the software PARADIS of the platform H2OLab.

#### A combined collocation and Monte-Carlo method for advection-diffusion equation of a solute in random porous media

Participants : Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Publications: [14]

Abstract: In this work, we present a numerical analysis of a method which combines a deterministic and a probabilistic approaches to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow equation in a random porous medium coupled with the advection-diffusion equation. Quantities of interest are the mean spread and the mean dispersion of the solute. The means are approximated by a quadrature rule, based on a sparse grid defined by a truncated Karhunen-Loève expansion and a stochastic collocation method. For each grid point, the flow model is solved with a mixed finite element method in the physical space and the advection-diffusion equation is solved with a probabilistic Lagrangian method. The spread and the dispersion are expressed as functions of a stochastic process. A priori error estimates are established on the mean of the spread and the dispersion.

#### An adaptive sparse grid method for elliptic PDEs with stochastic coefficients

Participants : Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Publications: [31] .

Abstract: The stochastic collocation method based on the anisotropic sparse grid has become a significant tool to solve partial differential equations with stochastic inputs. The aim is to seek a vector of weights and a convenient level of interpolation for the method. The classical approach uses an a posteriori approach on the solution, which causes an additional prohibitive cost.

In this work, we discuss an adaptive approach of this method to calculate the statistics of the solution. It is based on an adaptive approximation of the *inverse* diffusion parameter. We construct an efficient error indicator which is an upper bound of the error on the solution. In the case of unbounded variables, we use an appropriate error estimation to compute suitable weights for the method. Numerical examples are presented to confirm the efficiency of the approach, and to show that the cost is considerably reduced without loss of accuracy.

#### Numerical analysis of stochastic advection-diffusion equation via Karhunen-Loève expansion

Participants : Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Abstract: In this work, we present a convergence analysis of a probabilistic approach to quantify the migration of a contaminant, under the presence of uncertainty on the permeability of the porous medium. More precisely, we consider the flow problem in a random porous medium coupled with the advection-diffusion equation and we are interested in the approximation of the mean spread and the dispersion of the solute. The conductivity field is represented by a Karhunen-Loève (K-L) decomposition of its logarithm. The flow model is solved using a mixed finite element method in the physical space. The advection-diffusion equation is computed thanks to a probabilistic Lagrangian method, where the concentration of the solute is the density function of a stochastic process. This process is solution of a stochastic differential equation (SDE), which is discretized using an Euler scheme. Then, the mean of the spread and dispersion are expressed as functions of the approximate stochastic process. A priori error estimates are established on the mean of the spread and of the dispersion. Numerical examples show the effectiveness of this approach.

#### About a generation of a log-normal correlated field

Participants : Jocelyne Erhel, Mestapha Oumouni.

Grants: HYDRINV 8.4.5 , H2MN04 8.2.1

Software: GENFIELD 5.2.3

Publications: in preparation

Abstract: Uncertainty quantification often requires the generation of large realizations of stationary Gaussian random field over a regular grid.

This paper compares and analyzes the commonalities between the methods and approaches for simulating stationary Gaussian random field. The continuous spectral method is the classical approach which discretizes its spectral density to construct an approximation of the field. When the spectral density and the covariance functions decrease rapidly to zero at infinity, we prove that the spectral method is computationally attractive.

We compare also the classical methods used to simulate the field defined by its covariance function, namely the Discrete Spectral method, the Circulant Embedding approach, and the Discrete Karhunen-Loève approximation. We have found that under some assumptions on the covariance all these latter methods give the same simulations of a stationary Gaussian field on a regular grid, which are very efficient with the Fast Fourier Transform algorithm.

#### A global model for reactive transport

Participants : Édouard Canot, Jocelyne Erhel.

Grants: H2MN04 8.2.1 , MOMAS 8.2.5 , C2SEXA 8.2.3

Software: GRT3D 5.2.1

Thesis: [11]

Abstract: In some scientific applications, such as groundwater studies, several processes are represented by coupled models. For example, numerical simulations are essential for studying the fate of contaminants in aquifers, for risk assessment and resources management. Chemical reactions must be coupled with advection and dispersion when modeling the contamination of aquifers. This coupled model combines partial differential equations with algebraic equations, in a so-called PDAE system, which is nonlinear. A classical approach is to follow a method of lines, where space is first discretized, leading to a semi-discrete differential algebraic system (DAE). Several methods have been designed for solving this system of PDAE.

In this study, we propose a global method which uses a DAE solver, where time is discretized by an implicit scheme. Then, each time step involves a nonlinear system of equations, solved by a modified Newton method. Thanks to the DAE solver, the time step is adaptively chosen in order to ensure accuracy and convergence. Moreover, the Jacobian in the nonlinear iterations is freezed as long as Newton converges fast enough, saving a lot of CPU time.

However, the size of the nonlinear system is quite large, because it involves both the differential and the algebraic variables. We show how to eliminate the differential variables, in order to reduce the size. This is equivalent to a so-called Direct Substitution Approach, but it keeps the nice features of DAE solvers.

Classicaly, the concentrations of chemical species are defined with their logarithms, assuming that they are strictly positive. This simplifies the computation of the mass action laws in the chemistry model and the computation of their derivatives. However, when a species does not exist, its concentration is replaced by a very small value and this may lead to an ill-conditioned Jacobian. We propose to use directly the concentrations, without logarithms, so that the Jacobian is then well-conditioned. Therefore, Newton method converges much faster without logarithms, allowing larger time steps and saving many computations.

We illustrate our method with two test cases, provided by the french agency for nuclear waste (ANDRA) and by the group MOMAS. We can compare our results with either analytical or other numerical solutions and show that our method is quite accurate. We also show that reducing the number of unknowns is very efficient and that dealing without logarithms reduces drastically the CPU time.

#### A chemistry model with precipitation-dissolution

Participant : Jocelyne Erhel.

Grants: H2MN04 8.2.1 , MOMAS 8.2.5

Internship: Tangi Migot (Master M2, INSA and University, Rouen)

Publications: [36]

Abstract: In this study, we focus on precipitation and dissolution chemical reactions, because they induce numerical difficulties.

We consider a set of solute species and minerals, with precipitation occuring when a saturation threshold is reached. A challenge is to detect which minerals are dissolved and which minerals are precipitated. This depends on the total quantities of chemical species. We propose an analytical approach to build a phase diagram, which provides the interfaces between the different possible cases. We illustrate our method with three examples arising from brine media and acid mine drainage.

#### Coupled models for salted aquifers

Participants : Édouard Canot, Jocelyne Erhel.

Grants: H2MN04 8.2.1 , MOMAS 8.2.5 , HYDRINV 8.4.5

Software: GEODENS and SELSAUM (from Tunis)

Internship: Marwen ben Refifa (Ph-D, ENIT, Tunis)

Publications: in preparation

Abstract: We study gravity driven problems in salted aquifers, when many species are present together with high concentrations. In this framework, we couple flow, transport and chemistry by using a fixed point approach. We interfaced two codes developed in Tunis: GEODENS for density driven flow and transport, and SELSAUM for geochemistry. This latter provides also the density of salted water.