Multidomain Modeling for the Simulation of Flow and Transport in a Neighborhood of an Underground Nuclear Waste Repository

In this presentation we are concerned with the problem of developing a numerical simulator for modeling flow and contaminant transport in the vicinity of an underground nuclear waste repository for use in performance assessment. Clearly reasonably accurate and reliable simulators are needed, but as performance assessment requires many simulations, it is also important that the simulators be very efficient. The simulations under consideration necessarily involve large scale calculations both in space and in time, and yet must take into account important heterogeneities in the domain. This suggests the use of multidomain modeling: the domain is divided into more nearly homogeneous subdomains and efficient numerical solvers are used for each subdomain. The calculations on the various subdomains are related through transmission or interface conditions, and parallel computing and high level programming languages can then be used for numerical implementation. Around the waste depository the porous medium is made up of several regions of different permeability, the permeability varying over several orders of magnitude from one region to the next. In the method developed here the domain is partitioned into homogeneous subdomains and nonoverlapping domain decomposition is used for both the flow calculations and to solve the transport equation. The single phase flow equation is an elliptic equation coupling a conservation equation with Darcy's law. As the Darcy velocity is the unknown of interest, a mixed finite element method is used to solve the equation within the subdomains. In order to be able to use general hexahedral grids, Kuznetsov-Repin macroelements are used. Continuity of the pressure and continuity of the flux are enforced at the interfaces between the subdomains using a balancing domain decomposition method. This method has the advantage of being very robust with respect to large contrasts in permeability. The transport equation is of advection-diffusion type. To approximate the solution to this equation an operator splitting technique is used: an advection equation is solved alternately with a diffusion-dispersion-decay equation. The parabolic diffusion-dispersion-decay equation is solved implicitly using mixed finite elements while the advection equation is solved explicitly using cell centered finite volumes with upstream weighting. In this way we are able to use a smaller time step for the advection equation when the advection coefficient is large. The domain decomposition methods used for the pressure equation extend also to these time dependent problems. In subdomains of high permeability, the natural time scale is much more rapid than in neighboring subdomains with lower permeability, and a much smaller time step size must be used for the advection step is these subdomains. However if only one time step size is used for all of the advection steps, it is the more rapid time scale that is imposed and the much smaller time step size must be used throughout the domain even if the size of the region in which it is required is relatively small. This suggests the use of different time step sizes for different subdomains. One of the many advantages of using domain decomposition techniques is that they lend themselves readily to parallelization. However, implementing the subdomain coupling requires complex programming. This is done efficiently using the programming environment OCamlP3l. This is a recently developed part of the strongly typed functional programming language OCaml which can be used for parallel computing. Realistic three dimensional numerical experiments from the Andra project will be shown.Presenters: name: Sboui, Amel affiliation: INRIA ROCQUENCOUR