A topological and manifold framework for porous media Modeling: A review of engineering applications and mathematics

Porous media exhibit intricate networks of pores, grains, and channels that influence flow and transport processes. While imaging and pore-scale simulations have advanced structural characterization, current approaches often overlook the topological complexity of pore geometry. This study presents a comprehensive topological and manifold framework for modeling porous media. By employing tools from topology and geometry, including Hausdorff space, Euler characteristics, Betti numbers, Minkowski functionals, Hadwiger's theorem, and manifold analysis, we provide a unified method to capture pore connectivity, void distribution, and flow potential. The framework highlights how topological invariants can explain percolation thresholds, ganglion dynamics, and phase connectivity in multiphase systems. This perspective bridges theory and application, offering new pathways to improve permeability, wettability, and multiphase flow in natural and engineered porous systems calculations. This paper integrates algebraic topology with digital rock physics to address connectivity problems that traditional geometric methods cannot. The proposed approach enhances the mathematical foundations of porous media analysis and establishes a platform for applications in groundwater flow, oil recovery, CO 2 storage, and filtration technologies. This framework can be implemented to quantitatively model and optimize flow transport and connectivity in complex porous networks, providing a practical bridge between topological theory and engineering applications.