  In this work we propose upscaling method for nonlinear Forchheimer flow in
highly heterogeneous porous media. The generalized Forchheimer law is
considered for incompressible and slightly-compressible single-phase flows. We
use recently developed analytical results Aulisa et al. (2009) and write the
resulting system in terms of a degenerate nonlinear flow equation for the
pressure with the nonlinearity that depends on the pressure gradient. The
coarse scale parameters for the steady state problem are determined so that the
volumetric average of velocity of the flow in the domain on fine scale and on
coarse scale are close enough. A flow-based coarsening approach is used, where
the equivalent permeability tensor is first evaluated following the streamline
of the existing linear cases, and successively modified in order to take into
account the nonlinear effects. Compared to previous works Durlofsky and
Karimi-Fard (2009) and Peszynska et al. (2009), our approach relies on recent
analytical results of Aulisa et al. (2009) and combines it with rigorous
mathematical upscaling theory for monotone operators. The developed upscaling
algorithm for nonlinear steady state problems is effectively used for variety
of heterogeneities in the domain of computation. Direct numerical computations
for average velocity and productivity index justify the usage of the coarse
scale parameters obtained for the special steady state case in the fully
transient problem. Analytical upscaling formulas in stratified domain are
obtained for the nonlinear case. They correlate with high accuracy with
numerical results.
