Large-scale porous media and wavelet transformations

heterogeneities of such porous media, whose linear sizes are typically on the order of at least a few kilometers, manifest themselves at four widely disparate length scales: • microscopic scale—at the level of pores and grains (typical sizes are up to tens of microns); • mesoscopic scale—at the level of laboratory core plugs (typical length scale on the order of a few centimeters); • macroscopic scale—characteristic length scale of up to tens of meters for large blocks of a porous medium; and • megascopic scale—the entire porous medium. For two important reasons, LSPM are of immense interest. One reason is economical: oil and gas extracted from underground reservoirs are the world's most important energy source. Much of our drinking water comes from groundwater aquifers. Landfills, and their effect on the surrounding soil, are an unavoidable part of modern times. The second reason is political: much of the industrialized world does not produce enough oil and gas domestically and, therefore, relies on oil-exporting countries, most of which are politically unstable. Many wars over the past 100 years, including the latest one in Iraq, have had at least partly to do with oil reservoirs and political control over them. In addition, the problem of natural resource contamination—specifically, groundwater aquifers—has become so severe that no politician who hopes to get elected can ignore it. Regardless of whether we're interested in extracting oil or gas from an underground reservoir, or wish to study how a contaminant may spread in an aquifer, we must be able to model flow and transport phenomena in LSPM. However, such models can be predictive and, thus, useful only if we can characterize and model the LSPM's morphology. If it contains no fractures, we can model a porous medium at the laboratory (or mesoscopic) length scale via a three-dimensional (3D) network of interconnected pore throats—the narrow channels that control a fluid's passage through the medium—and pore bodies—the large chambers where most of the medium's porosity resides and where the pore throats meet. With the advent of sophisticated instruments and techniques over the past two decades, we can now get accurate and detailed images of a porous medium using 3D X-ray computed tomography. 4 Furthermore, we can compute the porous medium's flow and transport properties based on the 3D image with few (if any) approximations. For example, one of a porous medium's most important flow properties, which is its effective permeability …