Our work with x-ray micro-CT images of complex porous materials has required the development of topologically valid and efficient algorithms for studying and quantifying their intricate structure. As an example, simulations of two-phase fluid displacements in a porous rock depend on network models that accurately reflect the connectivity and geometry of the pore space. These network models are usually derived from curve skeletons and watershed basins. Existing algorithms compute these separately and may give inconsistent results. We have recently shown that Forman's discrete Morse theory provides a unifying framework for producing both topologically faithful skeletons and compatible pore space partitions. Although this work is grounded in image analysis, the theory and algorithms may be extended to simplicial and more general complexes, and our formulation of discrete unstable and stable sets may also be relevant to general combinatorial vector fields of interest in dynamical systems. This talk is based on joint work with Olaf Delgado-Friedrichs and Adrian Sheppard.