Using Wavelets as a Computational and Theoretical Tool for Homogenization

• Main Author: Watkins, Laura Lee
• Format: Others
• Published: DigitalCommons@USU 2005
• Subjects: wavelet; computational; theoretical; tool; homogenization; Mathematics
• Online Access: https://digitalcommons.usu.edu/etd/7151; https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=8251&context=etd

Since the cost of petroleum fluctuates widely, it is advisable to optimize extraction of oil and other hydrocarbon products form existing oil reserves. Because of the costs involved in recovering oil from a reservoir, predicting reservoir performance can be a useful tool for determining whether continued extraction might be profitable. This can be done using computer simulations of the physical processes involved such as pressure/head, fluid velocities, and so forth. Fluid flow within a reservoir occurs at a very small scale relative to the size of the reservoir. This size difference makes performing simulations at the physically appropriate scale unfeasible. Homogenization is a technique used in reservoir simulation to upscale small scale dependent behavior, such as a permeability tensor, to make simulation feasible. To calculate a homogenized permeability tensor, the solution to a system of uncoupled elliptic partial differential equations must be found repeatedly throughout the reservoir. Generally, the solution to the system of differential equations is approximated numerically using finite element or finite difference methods. We explore using wavelets as a means of characterizing homogenization in reservoir simulations in the search for fast algorithms for computing equivalent tensors. In addition to the analogy developed between homogenization and wavelets, proofs of convergence results from homogenization within the wavelet characterization are considered.