Combinatorial differential forms for multi-dimensional fluid flow in porous media: A unified framework for volumetric pores, fractures, and channels

Abstract

We present a novel mathematical framework for modelling fluid flow in porous media that naturally accommodates the mixed-dimensional nature of real pore spaces. Unlike traditional pore network models that reduce complex geometries to one-dimensional flow between idealised pores, or computationally intensive direct numerical simulations, our approach uses cell complexes with combinatorial differential forms to represent flow through volumetric pores (3D), sheet-like voids and fractures (2D), and narrow channels (1D) simultaneously. The method maps experimentally measured pore space characteristics onto polyhedral tessellations where different void types are assigned to cells of appropriate dimensions. Flow equations are formulated using calculus with combinatorial differential forms, yielding exact conservation laws directly in matrix form. We validate the approach using X-ray computed tomography images of four different rocks: Bentheimer sandstone, Doddington sandstone, Estaillades carbonate, and Ketton carbonate. For each rock, we generate 30 statistically equivalent realisations to investigate fabric-property relationships. The method achieves substantial computational efficiency compared to direct numerical simulations while maintaining accuracy comparable to pore-scale CFD and lattice-Boltzmann methods. Beyond efficiency, the framework provides scientific insight by explicitly linking pore-space topology to macroscopic permeability, enabling systematic exploration of how connectivity and dimensional transitions in the pore network control flow. The framework’s structure-preserving formulation and ability to assign different material properties to features of different dimensions make it particularly suitable for studying evolving pore structures, multiphase flow, and coupled processes in heterogeneous porous media relevant to groundwater systems and subsurface hydrology.