Wavelet Transforms for the Simulation of Flow Processes in Porous Geologic Media
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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 biorthogonality, 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 research 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. To address the problem, a new flow simulator WTFGS (Wavelet Transform Flow and Geomechanics Simulator) has been developed, which is written in MATLAB, where this code 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.