大象传媒

Advisors: Eirik Keilegavlen, Florin Radu and Jan Martin Nordbotten

Short description of project:

The key to reliable simulations of flow in fractured porous media is the聽proper design, analysis and implementation of numerical methods. These聽methods should take into account the specific properties of the聽underlying model, while at the same time be flexible enough to handle聽variability of the model's components. The particular features of聽fractured porous media we concern ourselves with are the complex聽geometry of the fracture network and the disparity in scales in the聽model parameters. The model we study is based on interpretation of聽fractures and the porous rock as a mixed-dimensional geometry, and the聽resulting system of partial differential equations is highly coupled and聽parameter-dependent. In this thesis, we build upon the common approaches聽to discretization of the flow problem and deliver a numerical solution聽by constructing efficient numerical solvers and preconditioners. The two聽main topics of our research are the design of preconditioners to finite聽element discretization of the linear flow model and the development of聽linearization methods to the non-linear model.

In the first part, we consider the fact that our flow problem reveals聽the saddle-point structure. This motivates to see how some established聽approaches to preconditioning saddle-point problems work under the聽mixed-dimensional complexity. We construct the preconditioners to the聽classical solving approaches, such as Krylov subspace methods, based on聽the well-posedness of our saddle-point system. As the goal of any聽preconditioner is to approximate the inverse of the coefficient operator聽of the system, the principal idea of our approach is to find that聽inverse mapping that is equivalent in terms of norms on the given聽function spaces. In this way, we can ensure that the preconditioned聽numerical solvers will converge more rapidly, independently of values of聽the discretization and physical parameters. In our case, we are able to聽derive two such preconditioners by identifying two different topologies聽on the given discrete finite-element spaces. In fact, one of the聽approaches leads to a general framework to preconditioning聽mixed-dimensional elliptic problems that can be applied to other聽problems with a similar hierarchical structure as the model of flow in聽fractured porous media.

Finally, we study a choice of non-linear and time-dependent flow models聽that appear in cases of enhanced conductivity of the fractures and聽compressibility of the fluid. The development of the iterative solution聽methods for our problem considers the natural domain decomposition聽setting imposed by the fracture network and the standard linearization聽methods are adapted to the mixed-dimensional setting. By using聽non-matching grids, we can employ a multiscale method to the interface聽problem to handle the dominating computational cost in each iteration of聽the non-linear solver, namely the repeated solving process on the rock聽matrix subdomains. The flexibility of the method is showcased by聽successfully applying it to several non-linear flow models.

Link to thesis at BORA-UiB:聽