Percolation and flow in geological formations: upscaling from microscopic to megascopic scales

Modelling transport processes in a disordered system with broadly distributed heterogeneities, such as a field-scale porous medium, is often hampered by the fact that a very large set of equations (of the order of several millions) has to be solved repeatedly if detailed information about the system's morphology is available. Solving such a large set is often not practical. We discuss a novel and very efficient method for scale up of such systems such that no important information about their characteristics is lost at any length scale, and at the same time the number of transport equations to be solved is reduced drastically to a manageable level. The method, which is applicable to any type of heterogeneous medium, including one with percolation disorder, uses a wavelet transformation to coarsen the original fine-scale description of the system, such that finer resolution is maintained in regions of high transport properties, whereas coarser property description is applied to the rest of the system. The computational cost of the method is several orders of magnitude less than those of the most efficient methods currently available. The performance of the method is demonstrated by its application to calculation of the effective flow properties of several models of field-scale porous media. The method is equally applicable to other disordered media with broadly distributed heterogeneities, such as amorphous semiconductors.