Special Session 108: New Developments in porous media

Multiscale Numerical Methods For Solving Nonlinear Forchheimer Equation in Highly Heterogeneous Porous Media

Manal Alotibi
Texas A&M University
Co-Author(s):    Manal Alotaibi, Yalchin Efendiev, Eric Chung.
Abstract. In this talk, I will present a local multiscale model reduction for nonlinear flows in heterogeneous porous media. I will consider generalized Forchheimer equations. The generalized Forchheimer equation describes flows at Darcy scales and arises when the pore-scale velocity is large. We consider the two term law form of Forchheimer equation and write the resulting system in terms of a degenerate nonlinear flow equation for the pressure. Our multiscale model reduction can be considered a generalization of recently introduced upscaling and numerical homogenization techniques, where the authors consider problems with scale separation. In the proposed approach, we construct local reduced-order model by constructing appropriate snapshot spaces and local spectral problems within the framework of Generalized Multiscale Finite Element Method (GMsFEM). To save the computational time, we use empirical interpolation techniques in estimating the nonlinear terms. I will discuss the use of adaptive procedures both in offline and online stages of the computation. We present numerical and theoretical results for the proposed method.