A multiscale flux basis for mortar mixed discretizations of reduced Darcy-Forchheimer fracture models

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

Non-Local Multi-Continuum method (NLMC) for Darcy-Forchheimer flow in fractured media

This work presents the application of the non-local multicontinuum method (NLMC) for the Darcy-Forchheimer model in fractured media. The mathematical model describes a nonlinear flow in fractured porous media with a high inertial effect and flow speed. The space approximation is constructed on the sufficiently fine grid using a finite volume method (FVM) with an embedded fracture model (EFM) to approximate lower dimensional fractures. A non-local model reduction approach is presented based on localization and constraint energy minimization. The multiscale basis functions are constructed in oversampled local domains to consider the flow effects from neighboring local domains. Numerical results are presented for a two-dimensional formulation with two test cases of heterogeneity. The influence of model nonlinearity on the multiscale method accuracy is investigated. The numerical results show that the non-local multicontinuum method provides highly accurate results for Darcy-Forchheimer flow in fractured media