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Flow in bounded and unbounded pore networks with different connectivity

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read A theory for mean pressure and flow in bounded pore networks performs well when connectivity and coordination numbers are high.

desk verdict Extends non-local Darcy to bounded networks via conductivity extraction and mean-flow theory, with numerical validation holding for high-connectivity cases. read the letter →

arxiv 1907.11850 v1 pith:K5BC4UKB submitted 2019-07-27 physics.flu-dyn nlin.AO

Daniel W. Meyer , Artur Gomolinski This is my paper
classification physics.flu-dynnlin.AO
keywords porenetworksnon-localDarcylawhydraulicconductivityboundedporousmediaflowfracturecoordinationnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends a non-local version of Darcy's law, originally for unbounded pore networks, to the bounded case. It supplies a method to extract the distribution of hydraulic conductivities that drives the non-local law and then derives predictions for average pressure and flow inside finite networks. Numerical checks confirm that these predictions match direct simulations in networks whose pores have many connections; the same checks indicate that lower-connectivity networks require further adjustments to the theory.

What carries the argument

The non-local generalization of Darcy's law whose central ingredient is the extracted hydraulic conductivity distribution between pores.

What would settle it

Direct numerical solution of the pressure-flow equations on a low-connectivity bounded network whose average flow deviates markedly from the analytic prediction of the bounded-network theory.

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Extended reading notes

Core claim

The non-local Darcy formulation yields a closed theory for mean pressure and flow in bounded networks; this theory reproduces numerical network results accurately when the networks possess high connectivity and high coordination numbers, while improvements are proposed and tested for networks that fall outside that regime.

Load-bearing premise

The non-local Darcy relation derived for unbounded networks can be carried over to bounded networks without large corrections, at least when connectivity is high.

Editorial extensions

If this is right

  • Mean pressure and flow inside bounded high-connectivity networks can be obtained from the conductivity distribution alone, without solving the full discrete system.
  • The same conductivity distribution extracted from an unbounded or periodic network supplies the input needed for the bounded-network formulas.
  • Adjustments proposed for lower-connectivity networks improve the match between theory and numerical results.
  • The approach also applies to fracture networks that share the same topological structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the theory holds for high-connectivity cases, it could reduce computational cost when upscaling flow through large but locally well-connected porous samples.
  • The conductivity-extraction step might be reusable for transport problems beyond steady flow, such as solute dispersion in the same networks.
  • Testing the proposed improvements on networks with intermediate coordination numbers would clarify the range where the basic theory remains useful.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript outlines a robust method for extracting the hydraulic conductivity distribution central to a non-local generalization of Darcy's law for unbounded/periodic pore networks. It then develops a theory for mean pressure and flow in bounded networks, validates the theory's predictions against independent numerical network simulations, shows that the theory performs well for high-connectivity networks with high coordination numbers, and proposes and assesses improvements for networks with lower connectivity.

Significance. If the numerical validation holds as described, the work supplies a concrete extension of non-local Darcy-type models to bounded domains, which is relevant for pore-scale and fracture-network flow modeling. The explicit scoping of the validation to high-connectivity cases together with the assessment of proposed improvements for other regimes is a strength; the use of independent numerical simulations for validation is also positive.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'a robust method for the extraction of the hydraulic conductivity distribution' would benefit from a brief indication of what makes the method robust (e.g., independence from fitting parameters or convergence properties).
  2. [Abstract] The manuscript would be clearer if the specific form of the proposed improvements for lower-connectivity networks were summarized in one additional sentence in the abstract or introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its relevance to non-local Darcy-type models, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper outlines a method to extract the hydraulic conductivity distribution and presents a theory for mean pressure and flow in bounded networks. These are validated against independent numerical network simulations, with explicit scoping that the theory works well only for high-connectivity cases and requires improvements elsewhere. The reference to a prior non-local Darcy generalization is to external prior work and is not used as a load-bearing self-citation chain that reduces the current claims to tautology. No self-definitional steps, fitted inputs renamed as predictions, or ansatz smuggling via citation appear in the derivation chain.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Based on the abstract alone, no specific free parameters, axioms, or invented entities can be identified with certainty; the hydraulic conductivity distribution is extracted but its fitting details are not described.

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Cite this review

Pith. "Pith review of Flow in bounded and unbounded pore networks with different connectivity." pith.science (2026). https://pith.science/paper/K5BC4UKB

@misc{pith2026190711850,
  author       = {Pith},
  title        = {Pith review of: Flow in bounded and unbounded pore networks with different connectivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5BC4UKB}},
  note         = {Machine review of arXiv:1907.11850}
}
read the original abstract

This work is concerned with the intricate interplay between node or pore pressures and connection or throat conductivities in flow or pore networks. A setting similar to pore networks is given by fracture networks. Recently, a non-local generalization of Darcy's law for flow and transport in porous media was presented in the context of unbounded or periodic pore networks. In this work, we first outline a robust method for the extraction of the hydraulic conductivity distribution, which is at the heart of the non-local Darcy formulation. Second, a theory for mean pressure and flow in bounded networks is outlined. Predictions of that theory are validated against numerical network results and it is demonstrated that the theory works well for networks with high connectivity involving pores with high coordination numbers. For other networks, improvements to the outlined theory are proposed and their accuracy is assessed.

Figures

Figures reproduced from arXiv: 1907.11850 by the authors.

Figure 1
Figure 1. Sketch of a pore network in x-y-z-space. Pores are depicted as red spheres and connecting throats as black lines. The cross-sectional area of the network in y-z-directions is represented by plane C. Two slabs A and B perpendicular to the x-direction are centered at xA and xB, respectively, and have a thickness h. The network is Lx-periodic in x-direction with periodic pore copies depicted as light-red spheres. 1 Int… view at source ↗
Figure 2
Figure 2. Conductivity distributions T(s) resulting from a sandstone network of size (14mm)3 . Distributions resulting from different slab thicknesses h are depicted, i.e., h = Lm/n with Lm = 4.89 × 10−4m and n = 8, 32, and 128. and conductivity distributions were extracted by using expression (8) with different slab thicknesses h = Lm/n. The resulting distributions for n = 8, 32, and 128 are depicted in figure 2. While n = 8… view at source ↗
Figure 3
Figure 3. Conductivity distributions T(s) resulting from sandstone networks of sizes (14mm)3 (black solid) and 3mm × 30mm × 30mm (red dashed, blue dots) with throats of maximal length Lm = 4.89 × 10−4m. The distributions (black solid, blue dots) resulted from the mean-pressure-gradient setup outlined in sections 2.1 and 2.2, while (red dashed) is based on the in-/outflow slab setup proposed earlier [5]. Firstly, we apply a me… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Pore pressure statistics as a function of the mean-flow-parallel [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Pore pressure statistics as a function of the mean-flow-parallel [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Sketch of a cut pore network in x-y-z-space. The network is cut at positions xS1 and xS2 leading to a network sample S of thickness L (gray shaded block). The sample S is supplied by a liquid at the interfaces xS1 and xS2 through reservoirs R1 and R2, respectively (wit…
Figure 7
Figure 7. Figure 7: Gray-shaded integration regions in s 0 -s and x 0 -s coordinate systems. is a solution of equation (11). Insertion into equation (11) leads to Z xS2 xS1 (x 0 − xS1) Z xS1 −∞ x − x 0 xS1 − x 0 T(x 0 , x)dxdx 0 + Z xS2 xS1 (x 0 − xS2) Z ∞ xS2 x − x 0 xS2 − x 0 T(x 0 , x)…
Figure 8
Figure 8. Figure 8: Pore pressures in the sandstone network of thickness [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Normalized fluxes through sandstone networks of different thickness [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Conductivity distributions T(s) (red solid) and T 0 (s) (blue dashed) as defined through expressions (8) and (21), respectively. In panels (a) and (b), distributions resulting from the sandstone and the homogeneous network are depicted, respectively. In both networks,…
Figure 11
Figure 11. Figure 11: Pore pressures in the beadpack network of thickness [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Normalized fluxes through beadpack networks of different thickness [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Pore pressures in a homogeneous network with pore-coordination number 10 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Normalized fluxes through homogeneous networks with pore-coordination num [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]

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