An implicit-explicit scheme and adaptive multiscale approximation of generalized Forchheimer flow in fractured porous media

In this work, we consider the generalized Forchheimer flow model for fractured porous media. We construct a semi-discrete system based on the finite-volume method with an embedded fracture model. For the time approximation of the nonlinear semi-discrete problem, we consider two schemes: (i) linearized Implicit scheme (LI) and (ii) Explicit-Implicit-Null scheme (EIN). In the LI scheme, we use a linearization from the previous time level to produce a linear system at each time level. This scheme requires an update of the problem operator at each time level. In the EIN scheme, we use an additive representation of the problem operator to separate the linear part of the operator and apply explicit approximation for the nonlinear part to produce a linear system for each time layer. We write the two schemes in general (canonical) form and present the stability estimate. We show that the EIN scheme for the nonlinear flow model, with a careful choice of the additive representation of the operator, satisfies the necessary properties for stability. In the next step, we reduce the size of the linear discrete system by constructing an accurate multiscale approximation on a coarse grid, where the EIN scheme gives us an efficient choice of operator for the multiscale basis construction. The multiscale space is constructed based on solving the local linear spectral problem to accurately encapsulate the high-contrast permeability coefficient adaptively by introducing several macroscale variables. We show that the adaptive approach based on the leading local flow modes accurately approximates flow processes in fractured porous media. To reduce the error associated with a nonlinear flow pattern, we construct an adaptive local multiscale enrichment of the multiscale space based on the local residual. We show that a few adaptive iterations of the online enrichment can significantly reduce error and produce a highly accurate solution for the generalized Forchheimer flow model.