Abstract
In this paper, the solution of the Darcy-Forchheimer model in high contrast heterogeneous media is studied. This problem is solved by a mixed finite element method (MFEM) on a fine grid (the reference solution), where the pressure is approximated by piecewise constant elements; meanwhile, the velocity is discretized by the lowest order Raviart-Thomas elements. The solution on a coarse grid is performed by using the mixed generalized multiscale finite element method (mixed GMsFEM). The nonlinear equation can be solved by the well known Picard iteration. Several numerical experiments are presented in a two-dimensional heterogeneous domain to show the good applicability of the proposed multiscale method.
Highlights
The Darcy-Forchheimer equation is commonly used for describing the high velocity flow near oil and gas wellbores and fractures, which is a correction formula of the well known Darcy’s law by supplementing a nonlinear velocity quantity as follows: μk−1 u + βρ |u| u + ∇ p = 0, (1)where μ, k, ρ and β represent the viscosity, the permeability, the density, and the dynamic viscosity coefficient of the fluid, respectively. β is mentioned as the Forchheimer coefficient, whose values stand for the nonlinear intensity
In [12,13,14], we developed a mixed generalized multiscale finite element method (GMsFEM), where we enriched a multiscale space by new degrees of freedom, which was obtained by solving local spectral problems on the snapshot space
We conducted a numerical study of the solution of the Darcy-Forchheimer model in high contrast heterogeneous media
Summary
Girault et al in [1] proved the existence and uniqueness of the solution of the Darcy-Forchheimer model They considered mixed finite element methods by piecewise constant and nonconforming Crouzeix-Raviart elements to approximate the velocity and the pressure, respectively. In [8], the authors introduced the mixed multiscale finite element methods and presented the main convergence results for the solution of second order elliptic equations with heterogeneous coefficients, which oscillate rapidly. In [12,13,14], we developed a mixed generalized multiscale finite element method (GMsFEM), where we enriched a multiscale space by new degrees of freedom, which was obtained by solving local spectral problems on the snapshot space. We construct an efficient algorithm of the mixed generalized multiscale finite element method to make an approximation of the problem on the coarse grid.
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