Wavelet Transforms for the Simulation of Flow Processes in Porous Geologic Media

Wavelets are localized small waves that exhibit the characteristic oscillatory behavior of waves with an amplitude that declines rapidly to zero. Their properties include orthogonality or bi-orthogonality, a natural multiresolution and, often, compact support. These properties can be used to repeatedly rescale a signal or a function, decomposing it to a desirable level, and obtaining and preserving trend and detail data at all scales that allow re-composition of the original signal. The overall goal of this work is to create a set of wavelet-based (WB) numerical methods using different wavelet bases for application to the solution of the PDEs of interest to petroleum engineering, namely the solution of the PDEs of fluid flow through porous and fractured media. A particular emphasis of the study is in processes associated with ultra-low permeability media such as shale oil and shale gas reservoirs. To address the problem, we developed WTFS (wavelet transform flow simulator), a new flow simulator written in MATLAB that couples wavelet transform with a standard finite-difference scheme. In the current state of development, the WB numerical solution is verified against analytical solutions of 1D problems for liquid flow through porous media and is validated through comparisons to numerical solutions for problems of 2D and 3D flow through porous media obtained from a conventional numerical simulator.