Elyes Ahmed

Mats Kirkesæther Brun, Elyes Ahmed, Inga Berre et al.

This paper concerns splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE's, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare three approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from mechanics; these approaches have in common a simultaneous application of the fixed-stress splitting scheme on both the non-linearity and the coupling structure of the problem. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and validate our results through several numerical examples.

Elyes Ahmed, Alessio Fumagalli, Ana Budiša

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches.