Calculus with combinatorial differential forms for fluid flow analysis in porous and fractured media

TL;DR

The paper addresses the challenge of predicting transport in porous and fractured media where fabric heterogeneity undermines continuum assumptions. It introduces a discrete calculus based on Forman's combinatorial differential forms on a Forman-subdivided, quasi-cubical cell complex to represent fluxes and conservation laws directly in matrix form, while mapping XCT-derived fabric into 3D voids, 2D void faces, and 1D void edges. Key contributions include (i) a flexible mapping of voluminous and expansive voids to higher- and lower-dimensional cells, (ii) a complete discrete transport framework with cell-local conductivities and a material-dependent Laplacian, and (iii) validation across four rocks showing permeability predictions in good agreement with CFD and experiments, with substantial efficiency advantages. This approach enables fabric-aware, scalable predictions of macroscopic transport and offers a pathway to coupled multi-physics and evolving-fabric problems in porous media.

Abstract

The fabric of porous and fractured media contains solid regions (grains) and voids. The space conducting fluids is a system of connected voids with variable geometries. Relative to the grain sizes, the voids can be voluminous with three comparable large extensions, narrow expansive with two comparable large extensions and one smaller extension, and thin long with one comparable large extension and two smaller extensions. The widely used representation of void spaces by systems of spheres connected by cylinders (pore network models) is an acceptable approximation for some special cases, but not for most porous and fractured media. We propose a flexible method for modelling such media by mapping their measured fabric's characteristics - void and grain volume distributions and shapes - onto polyhedral tessellations of space. The map assigns voluminous voids and grains to polyhedrons (3D), narrow expansive voids to some polyhedral faces (2D), and thin long voids to some polyhedral edges (1D), as dictated by experimental data. The analysis of transport through such discrete structures with components of different dimensions is performed by a novel mathematical method, which uses combinatorial differential forms to represent physical properties and their fluxes, as well as structure-preserving operators on such forms to formulate the conservation laws exactly and directly in matrix form, ready for computation. The method allows for individual material properties, such as conductivity, to be assigned to voids of all dimensions, so that the three types of voids are suitably represented. Publicly available XCT images of four different rocks are used to test the method.

Figures (23)

• Figure 1: Pathway for development of calculus on complex materials' systems
• Figure 2: Triangular mesh $\mathcal{M}$ (a) and its Forman subdivision $\mathcal{K}$ (b)
• Figure 3: The cochain complex $(C^\bullet \mathcal{K}, \delta)$ and the metric-dependent operations on the cochain spaces.
• Figure 4: Binary CT images and segmented images, where blue areas are pore spaces and black areas are grain spaces. (a) and (b) Bentheimer sandstone; (c) and (d) Doddington sandstone; (e) and (f) Estaillades carbonate; (g) and (h) Ketton carbonate.
• Figure 5: Reconstructed 3D material domains and pore systems based on CT images: (a) Bentheimer sandstone; (b) Doddington sandstone; (c) Estaillades carbonate; (d) Ketton carbonate.