Pore Interaction Network Modeling

• Pore Interaction Network is a formalism for representing and simulating flow, transport, or reaction phenomena in porous media by modeling discrete pores and their interconnections.
• It employs calibrated parameters and Dirichlet-to-Neumann maps derived from finite-element simulations to bridge microscopic details with macroscopic behavior.
• The approach rigorously enforces mass conservation and enables efficient solution of global systems via sparse Schur complement reduction and advanced Krylov methods.

A pore interaction network is a formalism for representing and simulating flow, transport, or reaction phenomena in porous media by explicitly modeling the discrete spatial structure and local physical couplings between pores. The concept underlies contemporary pore-network modeling (PNM), including both classical edge-node approaches and advanced multi-physics or calibrated domain decomposition models. Nodes represent individual pores (or clusters of void space), while edges encode direct interactions, typically via throats or shared interfaces. Each interaction is parameterized by conductance or analogous couplings calibrated from geometry, physical theory, or data-driven methods. The pore interaction network efficiently bridges the microscopic (voxel or continuum description) and the macroscopic (effective-medium) scales by leveraging the sparse, physically motivated connectivity of complex pore architectures.

1. Mathematical Foundation and Network Construction

A pore interaction network consists of $N$ pore body subdomains $\Omega_i$ whose union defines the resolved pore space, typically segmented from micro-CT or other imaging data. Pore bodies are coupled through internal interfaces $\Gamma_{ij}$, where fluid or other transport phenomena occur. The network is constructed by defining:

• Nodes: Each pore subdomain $\Omega_i$ corresponds to a node.
• Edges: Each shared boundary $\Gamma_{ij}$ (or physical throat) between pores $i$ and $j$ becomes an edge.

For classical PNM, the connectivity is encoded in the node–edge incidence matrix $C \in \{-1,0,1\}^{N \times |\mathcal E|}$, where each edge corresponds to a physical throat, and graph edges represent allowed local transport routes. In advanced frameworks such as the domain-decomposition pore-network method (DD-PNM), the network nodes represent finite-element subdomains and edges encapsulate calibrated, high-fidelity interface response operators (Wang et al., 15 Oct 2025).

2. Physical Coupling: Dirichlet-to-Neumann Maps and Conductance Calibration

The local interaction between neighboring pores is rigorously captured through local Dirichlet-to-Neumann (DtN) operators $G_i$ for each pore subdomain. $G_i$ maps prescribed interface (“traction”) conditions on $\Gamma_{ij}$ to normal fluxes, as computed from finite-element solutions to the Stokes equations on body-fitted meshes:

$$G_i = [g_{i,sr}]_{s,r=1}^{m_i}, \hspace{1em} g_{i,sr} = -[f_i^{(s)}]^T \widetilde f_i^{(r)} \text{ with } \widetilde f_i^{(r)} \text{ from local FE solves}.$$

This operator is symmetric and negative semidefinite, strictly enforcing local mass conservation.

To recover a classical 1-D pore-network model, half-throat hydraulic conductivities $k_{i,r}$ are fitted by minimizing the Frobenius-norm misfit between off-diagonal elements of the FE DtN map and those of the standard 1-D network DtN analogue $G_i^*(D_i)$, enabling direct parameterization of the network with physical or numerically derived conductances (Wang et al., 15 Oct 2025).

$$G_i^*(D_i) = D_i - \frac{D_i 1 1^T D_i}{1^T D_i 1}, \qquad D_i=\mathrm{diag}(k_{i,1},\ldots,k_{i,m_i}).$$

The physical throat conductance between two neighboring pores is given by the harmonic average

$$k_\omega = \left( \frac{1}{k_{i,\omega}} + \frac{1}{k_{j,\omega}} \right)^{-1}$$

ensuring consistent mass transfer and satisfying pairwise symmetry.

3. Global System and Sparse Schur Complement Reduction

The global interaction among all pores is summarized in the Schur-complement system for the unknown interface tractions $p \in \mathbb R^m$ (collection of all internal faces):

$$S p = F, \hspace{1em} S = -\sum_{i=1}^N (R_i^u)^T G_i^{uu} R_i^u, \hspace{1em} F = -\sum_{i=1}^N (R_i^u)^T G_i^{uk} R_i^k p^k$$

where $R_i^u$/$R_i^k$ select unknown/known (boundary) faces. $S$ is symmetric positive-definite under the DD-PNM formalism and enforces discrete local/global mass conservation, with the property that the assembled system is solvable for arbitrary Dirichlet (inlet/outlet) conditions and admits efficient solution via Krylov methods, e.g., conjugate gradient with algebraic multigrid preconditioning.

Upon solution of the global Schur complement, full velocity and pressure fields in each pore can be reconstructed, preserving local solenoidal structure and exact flux continuity across internal interfaces (Wang et al., 15 Oct 2025). The resulting pore-body pressures and throat fluxes define the reduced PNM equations

$$q_\ell = -k_\ell (\bar p_i - \bar p_j), \hspace{1em} C q = 0,$$

$C\,\mathrm{diag}(k)\,C^T\,\bar p=0,$

where $k_\ell$ are the calibrated throat conductances.

4. Algorithmic Workflow and Performance

The DD-PNM computational pipeline is:

• For each pore, assemble FE matrices and right-hand-sides, solve the saddle-point Stokes problem for all interface faces, and form the discrete DtN matrices.
• Extract interface conductance matrices and assemble the global Schur system.
• Solve the sparse symmetric positive-definite system for the interface tractions.
• Optionally reconstruct local FE solution fields using precomputed shape function expansions.
• Calibrate and output the equivalent classical pore-network by fitting conductances.
• Solve PNM equations for macroscopic effective properties or simulate pore-scale responses.

A canonical 2D case (rectangular domain with 40 randomly placed inclusions) with body-fitted FE meshes of $\approx 10^5$ DOFs was reduced via DD-PNM to a $150 \times 150$ Schur system and solved in $O(100)$ iterations, yielding $L^2$ pressure and $H^1$ velocity errors of $10^{-3}$ relative to full FE reference, and mass conservation to machine precision locally and globally. The extracted network predicts pore pressures to within 5% of the full solution (Wang et al., 15 Oct 2025).

5. Interpretive Significance and Advantages

The pore interaction network formalism:

• Directly links high-fidelity, mesh-based simulations to low-dimensional network models by explicit operator-based parameter calibration.
• Rigorously enforces conservation principles at all scales.
• Admits incorporation of arbitrary physical boundary/interface conditions, and can accommodate geometric disorder, throat variability, and complex topology.
• Enables systematic analysis of the validity of linear-network approximations and quantifies network reduction error.
• Provides a principled route to upscaling fundamental transport properties from image-resolved geometries or synthetic microstructures.

The approach generalizes to multiphysics settings or coupled fields (e.g., coupling Stokes flow and electrokinetic transport or reactive flows by extending local subdomain solves and their DtN closures), and is extensible to generalized block-diagonal operator splitting, as encountered in multi-phase and non-linear problems.

6. Context Within Broader Pore-Network Modeling

The DD-PNM pore interaction network sits at the intersection of classical 1-D PNM, direct numerical simulation, and multi-physics network theory. In contrast to empirically assigned conductances, the DtN calibration procedure provides a systematic, data-consistent basis for parameter assignment. This contrasts with circle-packing or maximal-ball approaches that extract only approximate geometric proxies for network construction.

Other models—such as those for catalysis deactivation, colloidal filtration, or electrokinetics—instantiate the same graph-theoretic and physical coupling principles, but with context-specific conductance definitions and local update rules. The formal mapping from resolved subdomain (FE or voxel-based) solutions to interaction operators is the distinguishing feature of the DD-PNM approach (Wang et al., 15 Oct 2025). Its accuracy and conservation properties provide a quantitative bridge between high-resolution simulation and efficient pore-scale network simulation.