Homogenization of Relative Permeabilities Curves for Two-phase Flow in Porous Media Using an Optimization Method

Grid coarsening remains essential in practical reservoir studies in order to get acceptable simulation time. This implies being able to upscale two-phase flow in particular the relative permeability. Upscaling can be divided in two stages: homogenization and mesh changing. Optimization gets involved here in the homogenization part. We proceed by identification between fine grid simulation on both representative heterogeneous regions and homogeneous equivalent region. We start with a mesh containing heterogeneous rock type. Each rock type has its own relative permeability curve and these curves are homogenized throughout the mesh with a unique relative permeability curve. In order to do this, the exit oil flow rates from the heterogeneous rock type case are considered as a reference solution. We then simulate the same flow except for the unique effective relative permeability curve. The exit oil flow rates from the two simulations are extracted to build a least squares objective function. The effective relative permeability curve (kr) is the main parameter of the optimization problem : the end points of a Brooks-Corey relative permeabilities model are used to look for a minimum objective function value.