Predicting flow and transport processes in the subsurface is of major interest in the geosciences. Quantitative predictions may be derived by simulating pore networks, i.e., graph-based representations of porous media. The most crucial part of this concept is to represent the uncertainty of the pore structure of the subsurface, that is, to find pore networks, which adequately represent the topological and metric characteristics of a porous medium. Thinning is commonly used to obtain pore networks from digital images of porous media. However, it typically yields ambiguous partitions of the porous medium into pores and grains. This proposal develops a dual approach for subdividing (an image of) a pore space into pores and defining a corresponding pore network. The pore network and the grain matrix will be derived from digital images of porous media and be represented by a dual pair of cellular complexes, which uniquely determine the pores, the grains, and the formation of pores and pore throats by grains. The networks will be used to analyze the uncertainty in pore structure of subsurface systems and to simulate multiphase flow.
Groundwater is abused with pollution sources ranging from leaking sewers, landfills, small workshops and garages, to large industrial plants. Evaluating the contamination that occurs from such sources, assessing the risk to human and ecological health, and planning cleanup strategies can all be facilitated by the use of mathematical models. These models describe how fluids flow and contaminants are transported and react in such systems. A key problem in modeling subsurface systems is their very complex pore geometry. This research will develop a novel and mathematically rigorous approach for representing these complex pore geometries by pore networks (in the simplest case: spheres connected by tubes) that adequately and uniquely represent the connectivity and geometry of the original soils. This new approach relates the topology (connectivity) of the pore space to the topology of the solid phase by using advanced concepts from mathematical graph theory. The resulting networks will answer many questions that are concerned with the fate and transport of hazardous chemicals and liquids in the subsurface.
