An Upscaling Method for One - Phase Flow

In this paper we study the problem of determining the eeective permeability on a coarse scale level of problems with strongly varying and discontinuous coeecients deened on a ne scale. The upscaled permeability is deened as the solution of an optimization problem, where the diierence between the ne scale and the coarse scale velocity eld is minimized. We show that it is not necessary to solve the ne scale pressure equation in order to minimize the associated cost-functional. Furthermore, we derive a simple technique for computing the derivatives of the cost-functional needed in the x-point iteration used to compute the optimal permeability on the coarse mesh. Finally, the method is illustrated by several analytical examples and numerical experiments.