Multiscale Modeling with Online Correction for Darcy–Forchheimer Flow

In this research, we present an algorithm of the Online generalized multiscale finite element method (Online GMsFEM) for Darcy–Forchheimer model in fractured media. The mathematical model describes a nonlinear Darcy flow with a high inertial effect and flow speed. The fine grid approximation is carried out by the finite element method (FEM) and to approximate lower dimensional fractures we use a discrete fracture model (DFM). We use a model reduction approach called Online GMsFEM, that is based on local residuals of coarse neighborhoods. The online multiscale basis functions are constructed in each local domain based on residuals. Such online multiscale basis functions make it possible take into account the effects of nonlinear coefficients in the Darcy–Forchheimer equation. We present numerical results in two-dimensional heterogeneous fractured domain. We investigate the accuracy of the method, depending on the influence of nonlinearity. The numerical experiments show that the Online GMsFEM has high accuracy for such nonlinear problems. The accuracy weakly depends on the magnitude of the nonlinear part of the equation.