Multiscale Asynchronous Algorithms Based on the Superelements Method for Multiphase Flow

The typical scales of pore pressure and fluid saturations variations are different both for space and temporal variables. The multiscale method is based on using the coarse grid for pore pressure equation and fine grid for saturation equations. The time step for these equations may be different too. The essential feature of the method is the basic function construction by solving the one phase stationary equations. These basis functions have the same peculiarities determined by the fine grid structures. The solution of pore pressure equations may be constructed as linear span of these basic functions for Buckley-Leverett equations. In general case it is possible to construct the tensor total permeability coefficients and to up-scale the relative permeability. The method of calculating these variables is the dissipative energy integral approximation. As a result we have the non-linear parabolic equation for pore pressure with tensor coefficients. The implicit schemes on the fine grid are considered for saturation equation. The general method is high resolution method. It is essential this method is a high resolution one. The results of modeling some problems are represented.