Upscaling Using Haar Wavelets

In the context of flow in porous media, up-scaling is the coarsening of a geological model and it is at the core of water resources research and reservoir simulation. An ideal up-scaling procedure preserves heterogeneities at different length-scales but reduces the computational costs of dynamic simulations. A number of up-scaling procedures have been proposed. We present a block renormalization algorithm using Haar wavelets which provide a representation of data based on averages and fluctuations. The single-phase flow of an incompressible fluid dominated by viscous forces is described by Darcy's law. Combining this with flux conservation leads to the pressure equation which can be solved to obtain pressure in terms of the hydraulic conductivity. A Haar wavelet transform is applied to the discretized Laplace like equation in a finite-differences scheme. By transforming the terms in the flow equation and assuming that the change in scale does not imply a change in governing physical principles, a new equation is obtained, identical in form to the original. Having decoupled pressure into averages and differences, we can perform a mean-field approximation obtaining a solution to the coarse scale problem which is a good approximation to the averaged fine scale pressure profile. Since the expression for the coarse conductivity corresponds to a simple averaging scheme, no matrix operations are required in up-scaling and a great advantage is obtained when solving for pressure on the coarse rather than on the fine scale. This technique is efficiently implemented within a block renormalization framework, where the fine grid is progressively coarsened to the desired level. Consequently, it is ensured that features of the system are preserved at all scales larger than the grid resolution and that the method does not break down in the presence of strong heterogeneity. These results are a first step to-wards examining the effect of the fluctuation terms which have been so far discarded. It is envisaged that an analysis beyond mean-field would allow us to obtain higher order and more useful coarse conductivity values, perhaps at the expense of a longer computational time. An attempt to apply the method to two-phase flow equations is made and ideas to extend the technique to irregular grids are examined.