Network models of dissolution of porous media

We investigate the chemical dissolution of porous media using a network model in which the system is represented as a series of interconnected pipes with the diameter of each segment increasing in proportion to the local reactant consumption. Moreover, the topology of the network is allowed to change dynamically during the simulation: as the diameters of the eroding pores become comparable with the interpore distances, the pores are joined together thus changing the interconnections within the network. With this model, we investigate different growth regimes in an evolving porous medium, identifying the mechanisms responsible for the emergence of specific patterns. We consider both the random and regular network and study the effect of the network geometry on the patterns. Finally, we consider practically important problem of finding an optimum flow rate that gives a maximum increase in permeability for a given amount of reactant.

💡 Research Summary

The paper presents a novel network‑based framework for simulating chemical dissolution in porous media, extending earlier fixed‑topology models by allowing the connectivity of the pore network to evolve as the pores enlarge. The porous medium is represented as a lattice of cylindrical pipes (or “throats”). Each pipe’s diameter d_i grows in proportion to the local reactant consumption according to the update rule d_i(t+Δt)=d_i(t)+α·R_i·Δt, where α is a material‑transfer coefficient and R_i is the reaction rate determined by the local concentration and kinetic law. When a pipe’s diameter reaches a prescribed fraction (≈0.8) of the distance to a neighboring pipe, the two throats are merged, creating a new node and a new conduit. This dynamic rewiring mimics the physical process whereby adjacent pores coalesce once their walls have been sufficiently eroded.

Fluid flow through the network is governed by the Laplace equation for pressure, ∇·(k∇p)=0, with conduit conductance k_i∝d_i^4 (Hagen–Poiseuille law). The pressure drop across each pipe determines the volumetric flow rate, which together with the concentration field sets the local Damköhler number Da_i=R_i·L_i/U_i, while the Péclet number Pe=U_i·L_i/D characterises the relative importance of advection versus diffusion. By varying Pe and Da the authors explore distinct dissolution regimes.

Two initial topologies are examined: (1) a fully random graph, where connections are assigned without spatial order, and (2) a regular square lattice, which preserves geometric symmetry. Simulations are performed under a fixed inlet‑outlet pressure difference, with a finite total amount of reactant C_total introduced into the system. The authors monitor the evolution of permeability (or bulk hydraulic conductivity) K(t) and quantify the total permeability gain ΔK after the reactant is exhausted.

The results reveal three characteristic patterns. In the low‑Pe, high‑Da regime, reaction dominates over transport, leading to highly localized erosion. A few throats expand dramatically, merge, and form “finger‑like” dominant channels that carry the majority of the flow. This channelization yields a rapid increase in K but also creates a highly heterogeneous structure. In the high‑Pe, low‑Da regime, advection spreads reactant uniformly, producing relatively homogeneous widening of all throats; permeability rises more slowly but the medium remains isotropic. An intermediate regime exhibits a balance between the two extremes, generating a network of moderately enlarged pathways that still retain some degree of connectivity diversity.

A central contribution is the identification of an optimal flow rate Q* that maximizes ΔK for a given C_total. By numerically integrating the instantaneous permeability growth dK/dt over the dissolution time t_f(Q) for each imposed flow, the authors construct ΔK(Q) curves that are typically concave, with a clear maximum at intermediate Q. Physically, too low a flow limits reactant delivery to the interior pores, confining dissolution to the inlet region; too high a flow dilutes the reactant, reducing the effective reaction rate. The optimal Q corresponds to a Péclet number where advection and reaction are balanced, providing the most efficient use of the chemical budget.

The dynamic rewiring mechanism also influences pattern selection. In random networks, stochastic variations in local connectivity cause the first few mergers to dictate the eventual channel layout, often resulting in a single dominant conduit that shortcuts the system. In regular lattices, symmetry constraints delay the emergence of a single dominant path; instead, several comparable channels develop, preserving a more regular spacing. Nonetheless, even in the regular case, once a throat reaches the merging threshold it can break the symmetry and steer growth preferentially along a diagonal or a particular axis, depending on boundary conditions.

The authors discuss model limitations and future extensions. The current representation treats pores as one‑dimensional tubes and assumes reactions occur only at the tube walls; real rocks exhibit three‑dimensional grain structures, variable surface chemistry, and mechanical deformation as dissolution proceeds. Incorporating grain‑scale mechanics, precipitation, and multi‑species transport would improve realism. Moreover, the merging rule is based on a fixed geometric criterion; a more physics‑based criterion (e.g., stress‑controlled collapse) could capture pore collapse phenomena observed in carbonate dissolution.

In summary, this work advances porous‑media dissolution modeling by coupling diameter growth with topological evolution, thereby reproducing experimentally observed channelization, quantifying the dependence of pattern formation on Pe and Da, and providing a practical method for determining the flow rate that yields the greatest permeability enhancement per unit of reactant. The framework offers a valuable tool for engineers and geoscientists seeking to design efficient acidizing treatments, enhance oil recovery, or predict the long‑term evolution of subsurface flow pathways.