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arxiv: 2604.09637 · v1 · submitted 2026-03-20 · 🧮 math.AP · cs.NA· math.NA

A mathematical model for colloids deposition in porous media combined with a moving boundary at the microscale: Solvability and numerical simulation

Christos Nikolopoulos , Michael Eden , Adrian Muntean This is my paper

Pith reviewed 2026-05-15 07:35 UTC · model grok-4.3

classification 🧮 math.AP cs.NAmath.NA
keywords colloidal depositionporous mediamoving boundarymultiscale modelweak solvabilityfinite element methodreaction-diffusion
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The pith

Colloid deposition model in porous media admits weak solutions when pores do not clog

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a two-scale reaction-diffusion system that tracks colloidal particles diffusing, aggregating, fragmenting, and depositing inside a porous medium whose internal geometry evolves as solid cores grow or shrink. It proves existence of weak solutions to the resulting strongly nonlinear parabolic system provided the cores stay separated, so the pore space remains connected without topology changes. The macroscopic transport equation draws its coefficients from cell problems solved on the current microscale geometry, allowing the model to capture how local deposition alters global flow and storage. Numerical experiments with a two-scale finite element scheme then illustrate the resulting shifts in the effective dispersion tensor.

Core claim

The central claim is that the coupled macro-micro evolution problem, consisting of an effective macroscopic transport equation whose coefficients come from microscopic cell problems on moving solid-core boundaries, possesses weak solutions in the non-clogging regime; a two-scale finite-element scheme approximates these solutions and shows that progressive local deposition changes the dispersion tensor while trading transport efficiency against storage capacity.

What carries the argument

The weakly solvable nonlinear parabolic system obtained by upscaling macroscopic particle transport against microscopic moving-boundary problems for non-contacting solid cores

If this is right

  • The effective dispersion tensor decreases as local deposition increases the solid fraction.
  • Storage capacity rises while transport efficiency falls, producing a measurable trade-off.
  • The model accounts for aggregation, fragmentation, and detachment at the pore scale.
  • Two-scale finite-element simulations can track how microstructure evolution feeds back into macroscopic coefficients.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coupling structure could be adapted to regimes where cores contact and topology changes appear, though new analytical tools would be required.
  • Results on the dispersion-storage trade-off may guide parameter choices in filtration or groundwater transport models.
  • Long-time asymptotics might reveal whether the non-clogging approximation still predicts eventual macroscopic blockage.

Load-bearing premise

Neighbouring solid cores never touch, so the phase boundaries evolve without topology changes or complete pore blockage.

What would settle it

A numerical run in which two adjacent cores grow until contact occurs and the computed solution ceases to exist or the existence proof breaks down.

Figures

Figures reproduced from arXiv: 2604.09637 by Adrian Muntean, Christos Nikolopoulos, Michael Eden.

Figure 1
Figure 1. Figure 1: Schematic representation of the two-scale geometry with the macro￾scopic domain Ω and the different perforated microstructures Y f (x1) and Y f (x2) associated with points x1, x2 ∈ Ω. The precise working assumptions for the micro￾scopic geometries are given within section 2. Our initial motivation was to understand the behaviour of colloidal particles in heterogeneous soils [18, 4, 6], but very similar set… view at source ↗
Figure 2
Figure 2. Figure 2: A schematic representation of the growth setup we have in mind. Fig￾ure (a) shows the tubular neighborhood of thickness at least σ ∗ for one possible microstructure (linked to x1 ∈ Ω). Figure (b) depicts the initial setup (σ = 0), (c) the geometry after growth (σ1 > 0), and (d) after shrinkage (σ2 < 0). For each i = 1, ..., N, let ui : ST × Ω → [0, ∞) (we set u = (u1, ..., uN )) denote the molar concentrat… view at source ↗
Figure 3
Figure 3. Figure 3: Numerical solution of the cell problem (equations eqs. (2.2d) to (2.2f)) and specifically for w1 with S being: (a) a circle; (b) an ellipse with its long axis forming a 150o angle with the x − axis; (c) an ellipse with its long axis forming a 135o angle with the x-axis; (d) an ellipse with axis ratio 0.5 and with his long axis on the x−axis. At the initial moment (σ = 0), the ellipse axes are Ra(0) = 0.01 … view at source ↗
Figure 4
Figure 4. Figure 4: Entries of the diffusion tensor with respect to the parameter σ in the case that the initial shape is (a) a circle (b) an ellipse with its long axis forming a 30o angle with the x-axis, (c) an ellipse with its long axis forming a 45o angle with the x-axis and (d) a bean shaped curve. point. We use a finite element method to solve the two-dimensional version of the field equations (2.2a) together with (4.9)… view at source ↗
Figure 5
Figure 5. Figure 5: Time frames presenting the evolution of the long axis of the ellipse in the cell. Clogging is apparent as the distribution of Ra increases and reaches its maximum value through￾out the domain. At the end of the simulation, all the domain becomes clogged. In this case, the evolution of clogging seems to smooth out the singularity of the macroscopic domain. We notice that a similar behaviour occurs for other… view at source ↗
Figure 6
Figure 6. Figure 6: Time frames presenting the evolution of the long axis of the ellipse in the cell. exactly the same as in the previous simulations. We observe that the inflow through the boundary [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Time frames presenting the evolution of the long axis of the ellipse in the cell in the case of an initial non-uniform distribution. creates clogging areas in front of the aforementioned ”barriers” since we have increased accumu￾lation of colloids in these areas. Additionally, the convex corners are susceptible to clogging while the concave one is much less inclined to clog [PITH_FULL_IMAGE:figures/full_f… view at source ↗
read the original abstract

We study a reaction-diffusion model posed on two distinct spatial scales that accounts for diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles within a porous material. In this model, the macroscopic transport of the particles is described by an effective equation whose transport coefficients are determined by cell problems posed on the underlying pore scale. The internal pore geometry can change over time due to deposition or detachment of colloidal particles. We represent the evolving microstructure as solid cores whose phase boundaries can grow or shrink over time. As deposition progresses, neighbouring growing cores may come into contact, leading to local clogging of the pore space. We investigate how such evolving microstructures influence the effective transport and storage properties of porous layers. We establish basic analytical results concerning the weak solvability of the resulting multiscale evolution problem, which takes the form of a strongly non-linear parabolic system, in the non-clogging regime. For the numerical approximation of weak solutions we propose a two-scale finite element discretization. Numerical experiments illustrate how local clogging affects the effective dispersion tensor and quantify the resulting trade-off between transport efficiency and storage capacity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper develops a two-scale reaction-diffusion model for colloidal deposition in porous media, where macroscopic transport coefficients are obtained from cell problems on an evolving pore-scale geometry represented by growing solid cores. It claims weak solvability of the resulting nonlinear parabolic system in the non-clogging regime (no topology changes or pore blockage) and proposes a two-scale finite-element discretization whose numerical experiments illustrate the impact of local clogging on the effective dispersion tensor and the transport-storage trade-off.

Significance. If the weak solvability result can be made rigorous while remaining inside the non-clogging regime, the work supplies a mathematically grounded framework for linking microscale deposition dynamics to macroscopic effective properties, which is relevant to filtration, contaminant transport, and porous-media engineering. The two-scale numerical scheme offers a practical tool for exploring parameter regimes where clogging begins to matter.

major comments (1)
  1. [Existence result for the weak formulation (abstract and §3)] The central existence theorem for the nonlinear multiscale parabolic system is stated only in the non-clogging regime on a fixed interval [0,T]. No a priori estimate or stopping-time argument is supplied that guarantees a uniform positive lower bound on the distance between neighbouring phase boundaries, so it is unclear whether the solution remains inside the assumed regime for the full time horizon or whether the result is merely local-in-time (or requires smallness conditions on initial data and reaction rates).
minor comments (1)
  1. [Numerical discretization section] The description of the two-scale finite-element scheme would benefit from an explicit statement of how the moving phase boundaries are tracked or approximated inside the cell problems at each macro time step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the insightful comments. We respond to the major comment as follows and have incorporated revisions to improve clarity.

read point-by-point responses

  1. Referee: [Existence result for the weak formulation (abstract and §3)] The central existence theorem for the nonlinear multiscale parabolic system is stated only in the non-clogging regime on a fixed interval [0,T]. No a priori estimate or stopping-time argument is supplied that guarantees a uniform positive lower bound on the distance between neighbouring phase boundaries, so it is unclear whether the solution remains inside the assumed regime for the full time horizon or whether the result is merely local-in-time (or requires smallness conditions on initial data and reaction rates).

    Authors: We thank the referee for highlighting this point. Theorem 3.1 establishes the existence of weak solutions on any fixed interval [0,T] under the explicit hypothesis that the microstructure remains non-clogging throughout [0,T], i.e., that the distance between neighbouring phase boundaries stays strictly positive. This assumption is part of the theorem statement and ensures that the cell problems retain a fixed topology. The result is therefore global on [0,T] whenever the non-clogging condition holds; it is not merely local in time. We do not supply an a priori lower bound on the inter-boundary distance that is uniform in T or independent of the data, nor do we construct a stopping time to obtain a maximal existence interval. In the revised manuscript we have added a clarifying paragraph after Theorem 3.1 that states the conditional character of the result, notes that a stopping-time argument could be used to define the maximal time of existence within the non-clogging regime, and observes that smallness assumptions on the initial data and reaction rates would guarantee global-in-time non-clogging but are not required for the conditional statement we prove. This revision makes the scope of the theorem fully explicit without changing its statement. revision: partial

Circularity Check

0 steps flagged

No circularity: existence result derived from model equations under explicit regime assumption

full rationale

The central claim is a weak solvability theorem for the nonlinear parabolic multiscale system, obtained via standard analytical methods (e.g., Galerkin approximation, a priori estimates, compactness) applied to the given PDEs. The non-clogging regime is an explicit modeling restriction stated upfront, not derived from or equivalent to the solution itself. No fitted parameters are renamed as predictions, no self-citations carry the load-bearing uniqueness or ansatz steps, and the numerical experiments are presented separately as illustrations. The derivation chain remains self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the modeling choice of representing the microstructure by solid cores whose boundaries move according to deposition and detachment, together with the explicit restriction to the non-clogging regime; no free parameters or new physical entities are introduced beyond standard diffusion and reaction coefficients.

axioms (2)
  • domain assumption The evolving microstructure can be represented as solid cores whose phase boundaries grow or shrink over time due to deposition or detachment.
    This representation is invoked to define the moving-boundary cell problems that determine the effective macroscale coefficients.
  • ad hoc to paper The system remains in the non-clogging regime so that neighbouring cores do not merge and fully block pores.
    The solvability analysis is stated to hold only under this restriction.

pith-pipeline@v0.9.0 · 5508 in / 1305 out tokens · 50344 ms · 2026-05-15T07:35:58.929132+00:00 · methodology

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Reference graph

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