Capillarity in Stationary Random Granular Media: Distribution-Aware Screening and Quantitative Supercell Sizing

Christian Tantardini; Fernando Alonso-Marroquin

arxiv: 2509.21350 · v4 · submitted 2025-09-19 · ❄️ cond-mat.soft

Capillarity in Stationary Random Granular Media: Distribution-Aware Screening and Quantitative Supercell Sizing

Christian Tantardini , Fernando Alonso-Marroquin This is my paper

Pith reviewed 2026-05-18 16:13 UTC · model grok-4.3

classification ❄️ cond-mat.soft

keywords capillaritygranular mediasupercell sizingscreened flowpolydisperse grainstwo-point statisticsDarcy flowrepresentative volume

0 comments

The pith

A framework uses microstructure statistics and a screened integral range to select minimal supercells for representative capillarity simulations in random granular media.

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

The paper develops a quantitative method to find the smallest periodic supercell size that reliably captures average capillary-screened flow through packs of randomly placed grains of varying sizes. It begins with two-point statistics that describe grain arrangement and models capillarity as a screening process that damps long-range fluctuations. This produces a weighted volume fraction and an adapted integral range that keeps variance units consistent while matching known limits when screening disappears. From these quantities the authors obtain explicit rules: one limiting the shortest cell edge by correlation length, decay length and grain size quantile, and another setting cell volume to meet a target coefficient of variation. The result is a reproducible way to choose simulation domains for image-based solvers of coarse-grained Darcy flow.

Core claim

Treating capillarity as a modified-Helmholtz problem with phase-dependent transport under periodic boundaries yields an apparent conductivity, apparent screening parameter and macroscopic decay length via homogenization. A distribution-aware polydispersity treatment then introduces a capillarity-weighted volume fraction and a screened analogue of the integral range that preserves variance units and recovers classical descriptors in the appropriate limits. These descriptors produce two sizing rules: a length criterion on the shortest cell edge set by microstructural correlation length, macroscopic decay length and a high quantile of grain size, together with a volume criterion that connects a

What carries the argument

The screened analogue of the integral range, which adapts the classical statistical descriptor to the low-pass filtering effect of capillarity while preserving variance units and recovering ordinary limits when screening vanishes.

If this is right

Reproducible supercell selection for finite-element or fast-Fourier-transform solvers of screened Darcy flow.
Direct coupling of two-point covariance information to the capillary decay length and apparent conductivity.
Explicit length criterion controlled by microstructural correlation length, macroscopic decay length and grain-size quantile.
Volume criterion that links target coefficient of variation to the screened integral range and phase contrast.

Where Pith is reading between the lines

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

The same screened-range construction could be tested on other linear screened transport problems such as diffusion-reaction or electrostatics in heterogeneous media.
Direct comparison of predicted supercell sizes against experimental image data would show whether the coefficient-of-variation target is reached in practice.
The approach suggests a route to sizing domains for media whose grain-size distribution varies slowly in space.

Load-bearing premise

A screened analogue of the integral range can be defined so that it preserves variance units and recovers classical descriptors when screening is absent.

What would settle it

Generate an ensemble of supercells sized by the proposed length and volume rules, compute the apparent conductivity for each, and check whether the observed coefficient of variation stays below the chosen target; a systematic excess would falsify the sizing criterion.

Figures

Figures reproduced from arXiv: 2509.21350 by Christian Tantardini, Fernando Alonso-Marroquin.

read the original abstract

We develop a quantitative framework to determine the minimal periodic supercell required for representative simulations of capillarity-screened Darcy flow in stationary random, polydisperse granular media. The microstructure is characterized by two-point statistics (covariance and spectral density) that govern finite-size fluctuations. Capillarity is modeled as a screened, modified-Helmholtz problem with phase-dependent transport under periodic boundary conditions; periodic homogenization yields an apparent conductivity, an apparent screening parameter, and a macroscopic capillary decay length. Because screening imparts a spatial low-pass response, we introduce a distribution-aware treatment of polydispersity consisting of a capillarity-weighted volume fraction and a screened analogue of the integral range that preserves variance units and recovers classical descriptors in the appropriate limits. These descriptors lead to two sizing rules: (i) a length criterion on the shortest cell edge controlled by a microstructural correlation length, the macroscopic decay length, and a high quantile of grain size; and (ii) a volume criterion that links the target coefficient of variation to the screened integral range and the phase contrast. The framework couples statistical microstructure information to capillary response and yields reproducible, distribution-aware supercell selection for image-based finite-element or fast-Fourier-transform solvers. The resulting criteria are therefore intended for representativity of the coarse-grained screened response, rather than for isolated nonlinear pore-scale events.

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.

Desk Editor's Note private letter to a colleague

The paper gives concrete supercell sizing rules for screened capillary flow simulations in polydisperse granular media by extending the integral range concept, but the claim that this screened version preserves variance units needs explicit checks.

read the letter

Hi colleague, the main thing to know is that this work supplies two practical rules for picking the smallest periodic supercell that still represents the coarse-grained screened Darcy response in random polydisperse grain packs. It does so by introducing a capillarity-weighted volume fraction and a screened analogue of the integral range built from the two-point covariance after the modified-Helmholtz operator. The rules tie cell-edge length to correlation length, macroscopic decay length, and a high grain-size quantile, and they tie volume to target coefficient of variation, the screened range, and phase contrast. This is meant for image-based FEM or FFT solvers that need reproducible averages rather than isolated pore events. What is new is the distribution-aware treatment that keeps the variance scaling and recovers the classical integral range when screening goes to zero. The periodic homogenization step for apparent conductivity and screening parameter is standard and applied cleanly here. The paper does a good job showing how screening acts as a spatial low-pass filter and how that changes the sizing logic compared with unscreened cases. The soft spot is the assumption that the screened range preserves variance units without higher-order corrections. In polydisperse media the grain-size distribution enters both the covariance and the local screening length, so it is not automatic that the homogenized fluctuations scale exactly as the classical case. The stress-test concern lands here: nothing in the basic homogenization guarantees the proportionality holds once phase-dependent capillarity is active. If the full paper contains numerical validations or a derivation that rules out those corrections, the framework strengthens; otherwise that step remains the part to watch. This is for researchers who run representative-volume simulations of granular or porous media with capillary screening, especially in hydrology or materials contexts. A reader who needs reproducible supercell choices for such solvers will get direct value from the criteria. I would send it to peer review. The ideas are targeted and build on established tools, so referees can usefully test the variance claim and the practical rules.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a quantitative framework to determine minimal periodic supercell sizes for representative simulations of capillarity-screened Darcy flow in stationary random polydisperse granular media. Microstructure is characterized via two-point covariance and spectral density; capillarity is modeled as a phase-dependent screened (modified-Helmholtz) problem under periodic boundary conditions. Periodic homogenization produces apparent conductivity, apparent screening parameter, and macroscopic decay length. A distribution-aware treatment introduces a capillarity-weighted volume fraction and a screened analogue of the integral range that is asserted to preserve variance units and recover classical descriptors in appropriate limits. These yield a length criterion (shortest cell edge controlled by microstructural correlation length, macroscopic decay length, and high quantile of grain size) and a volume criterion (target coefficient of variation linked to screened integral range and phase contrast). The criteria are intended for image-based FE or FFT solvers to ensure representativity of the coarse-grained screened response.

Significance. If the central assumptions hold and the framework is validated, the work would supply explicit, reproducible, distribution-aware supercell sizing rules that couple statistical microstructure descriptors directly to capillary response. This could improve efficiency and consistency of simulations in soft-matter and porous-media modeling by moving beyond ad-hoc cell sizes while recovering classical limits.

major comments (1)

[Abstract (definition of screened integral range and volume criterion)] The volume criterion (Abstract) rests on the claim that the screened analogue of the integral range 'preserves variance units and recovers classical descriptors in the appropriate limits.' In polydisperse media the grain-size distribution enters both the two-point covariance and the local screening length; nothing in the periodic-homogenization step is shown to guarantee that the fluctuation statistics of the homogenized tensor remain proportional to this length scale rather than acquiring contrast-dependent higher-order corrections. This assumption is load-bearing for the central sizing rule and requires an explicit derivation or targeted numerical test.

minor comments (1)

[Abstract] The abstract states that the criteria are 'intended for representativity of the coarse-grained screened response, rather than for isolated nonlinear pore-scale events,' but does not indicate whether any supporting numerical validations, error analyses, or comparisons against direct simulations appear in the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the single major comment below with a point-by-point response, providing additional context from the derivation while agreeing to strengthen the presentation through revision.

read point-by-point responses

Referee: The volume criterion (Abstract) rests on the claim that the screened analogue of the integral range 'preserves variance units and recovers classical descriptors in the appropriate limits.' In polydisperse media the grain-size distribution enters both the two-point covariance and the local screening length; nothing in the periodic-homogenization step is shown to guarantee that the fluctuation statistics of the homogenized tensor remain proportional to this length scale rather than acquiring contrast-dependent higher-order corrections. This assumption is load-bearing for the central sizing rule and requires an explicit derivation or targeted numerical test.

Authors: We appreciate the referee drawing attention to this foundational assumption. The screened integral range is introduced in Section 3.2 by integrating the two-point covariance of the stationary random medium against the Green's function of the modified-Helmholtz operator that encodes the capillarity screening; this construction is designed to retain the variance units of the classical integral range while incorporating the low-pass filtering effect of screening. The classical descriptors are recovered analytically in the limits of vanishing screening (decay length to infinity) and vanishing correlation length relative to screening length, as stated in the text. Polydispersity enters through the two-point covariance computed from the explicit grain-size distribution and through the capillarity-weighted volume fraction that averages the phase-dependent screening lengths. The periodic homogenization in Section 4 yields the apparent conductivity and screening parameter, after which the volume criterion applies the standard two-point statistical estimate for the coefficient of variation of the apparent quantities, now using the screened integral range in place of the unscreened one. We acknowledge, however, that the manuscript does not contain an expanded derivation isolating potential contrast-dependent higher-order corrections to the fluctuation scaling, nor a dedicated numerical verification across a range of phase contrasts. We will add both an explicit appendix derivation and a targeted ensemble test (comparing predicted versus measured coefficients of variation for several polydispersities and contrasts) in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's framework applies standard periodic homogenization to a screened modified-Helmholtz problem and augments classical two-point statistics with a new screened integral-range analogue explicitly constructed to recover known limits. The volume and length criteria follow directly from this construction plus the target coefficient of variation, without any parameter being fitted to the final supercell size or the target result being presupposed in the definition of the screened range. No load-bearing self-citations, self-definitional loops, or renaming of known results appear in the provided derivation steps; the approach remains self-contained against external benchmarks of homogenization theory and spatial statistics.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The framework rests on periodic homogenization of the modified-Helmholtz problem and the validity of two-point statistics for capturing finite-size fluctuations in polydisperse media.

free parameters (1)

high quantile of grain size
Controls the length criterion for shortest cell edge in the sizing rules.

axioms (2)

domain assumption Microstructure is characterized by two-point statistics (covariance and spectral density) that govern finite-size fluctuations.
Basis for linking microstructure to capillary response under periodic boundary conditions.
domain assumption Screening imparts a spatial low-pass response allowing distribution-aware treatment of polydispersity.
Justifies introduction of capillarity-weighted volume fraction and screened integral range.

invented entities (1)

screened analogue of the integral range no independent evidence
purpose: Preserves variance units for polydisperse media and recovers classical descriptors under capillarity screening.
New statistical descriptor developed to handle phase-dependent transport in the sizing criteria.

pith-pipeline@v0.9.0 · 5774 in / 1323 out tokens · 116568 ms · 2026-05-18T16:13:17.530457+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

screened analogue of the integral range that preserves variance units and recovers classical descriptors
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

periodic homogenization yields an apparent conductivity, an apparent screening parameter, and a macroscopic capillary decay length

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Representative-volume sizing in finite cylindrical computed tomography by low-wavenumber spectral convergence
cond-mat.soft 2026-01 conditional novelty 6.0

A method using axial detrending and low-wavenumber spectral convergence determines REV sizes of approximately 93 mm diameter and 83 mm height in Thalassinoides-bearing rock CT data.

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages · cited by 1 Pith paper

[1]

Torquato,Random Heterogeneous Materi- als: Microstructure and Macroscopic Properties (Springer, 2002)

S. Torquato,Random Heterogeneous Materi- als: Microstructure and Macroscopic Properties (Springer, 2002)

work page 2002
[2]

S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke,Stochastic Geometry and Its Applica- tions, 3rd ed. (Wiley, 2013)

work page 2013
[3]

D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes. Volume I: Ele- mentary Theory and Methods(Springer, 2003)

work page 2003
[4]

Lu and S

B. Lu and S. Torquato,Local volume fraction fluc- tuations in heterogeneous media, The Journal of Chemical Physics93, 3452–3459 (1990)

work page 1990
[5]

Quintanilla and S

J. Quintanilla and S. Torquato,Local volume frac- tion fluctuations in random media, The Journal of Chemical Physics106, 2741–2751 (1997)

work page 1997
[6]

Torquato,Hyperuniform states of matter, Physics Reports745, 1–95 (2018)

S. Torquato,Hyperuniform states of matter, Physics Reports745, 1–95 (2018)

work page 2018
[7]

de Gennes, F

P.-G. de Gennes, F. Brochard-Wyart, and D. Qu´ er´ e,Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves(Springer, 2004)

work page 2004
[8]

L. D. Landau and E. M. Lifshitz,Fluid Mechan- ics, 2nd ed., Course of Theoretical Physics, Vol. 6 (Pergamon Press, 1987)

work page 1987
[9]

Bensoussan, J.-L

A. Bensoussan, J.-L. Lions, and G. Papanico- laou,Asymptotic Analysis for Periodic Struc- tures, AMS Chelsea Publishing No. 374 (Amer- ican Mathematical Society, 2011)

work page 2011
[10]

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and In- tegral Functionals(Springer, 1994)

work page 1994
[11]

Moulinec and P

H. Moulinec and P. Suquet,A numerical method for computing the overall response of nonlinear composites with complex microstructure, Com- puter Methods in Applied Mechanics and Engi- neering157, 69–94 (1998)

work page 1998
[12]

M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans,Multi-scale computational homoge- nization: Trends and challenges, Journal of Com- putational and Applied Mathematics234, 2175– 2182 (2010)

work page 2010
[13]

C. L. Y. Yeong and S. Torquato,Reconstructing random media, Physical Review E57, 495–506 (1998)

work page 1998
[14]

Y. Jiao, S. Torquato, and F. H. Stillinger,A superior descriptor of random textures and its predictive capacity, Proceedings of the National Academy of Sciences106, 17634–17639 (2009)

work page 2009
[15]

Kanit, S

T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin,Determination of the size of the repre- sentative volume element for random composites: Statistical and numerical approach, International Journal of Solids and Structures40, 3647–3679 (2003)

work page 2003
[16]

Ostoja-Starzewski,Material spatial random- ness: From statistical to representative volume el- ement, Probabilistic Engineering Mechanics21, 112–132 (2006)

M. Ostoja-Starzewski,Material spatial random- ness: From statistical to representative volume el- ement, Probabilistic Engineering Mechanics21, 112–132 (2006)

work page 2006
[17]

Wiener,Generalized harmonic analysis, Acta Mathematica55, 117–258 (1930)

N. Wiener,Generalized harmonic analysis, Acta Mathematica55, 117–258 (1930)

work page 1930
[18]

Khintchine,Korrelationstheorie der sta- tion¨ aren stochastischen prozesse, Mathematische Annalen109, 604–615 (1934)

A. Khintchine,Korrelationstheorie der sta- tion¨ aren stochastischen prozesse, Mathematische Annalen109, 604–615 (1934)

work page 1934
[19]

A. M. Yaglom,Correlation Theory of Stationary and Related Random Functions, Springer Series in Statistics, Vol. I: Basic Results (Springer, 1987)

work page 1987
[20]

Hill,Elastic properties of reinforced solids: Some theoretical principles, Journal of the Me- chanics and Physics of Solids11, 357–372 (1963)

R. Hill,Elastic properties of reinforced solids: Some theoretical principles, Journal of the Me- chanics and Physics of Solids11, 357–372 (1963)

work page 1963
[21]

M. P. Allen and D. J. Tildesley,Computer Simula- tion of Liquids, 2nd ed. (Oxford University Press, 2017). [22]GROMACS Reference Manual(2023), see sec- tions on periodic boundary conditions and minimum-image convention; cutoff must be smaller than half the shortest box vector

work page 2017
[22]

P. A. Kralchevsky and K. Nagayama,Capillary interactions between particles at a liquid inter- face, Advances in Colloid and Interface Science 85, 145–192 (2000)

work page 2000
[23]

M. C. Leverett,Capillary behavior in porous solids, Transactions of the AIME142, 152–169 (1941)

work page 1941
[24]

Chavent and J

G. Chavent and J. Jaffr’e,Mathematical Mod- els and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media, Studies in Math- ematics and Its Applications, Vol. 17 (North- Holland, Amsterdam, 1986)

work page 1986
[25]

Winkelmann, L

M. Winkelmann, L. Krien, and D. Vollhardt,So- lution to the modified helmholtz equation for ar- 12

work page
[26]

Finite-size variance obeys Var[P(Y)]≈(C P A)/V

Microstructure statistics (Section§II B) Estimate correlation lengthξand integral rangeAfromC(r) or bC(k). Finite-size variance obeys Var[P(Y)]≈(C P A)/V. Refs:Ain Eq. (19); variance and 1/Vscaling Eqs. (29)–(33), generic form Eq. (31)

work page
[27]

Refs: Eq

Low-wavenumber coverage (Section§II C) Use the latticeK(Y) and requirek min = 2π/min(L α)≪k c; practically min(L α)≳c ξ ξ. Refs: Eq. (44), criterion Eq. (47), practical rule Eq. (48)

work page
[28]

Refs:a max in Eq

Geometric safeguard for polydispersity (Section§II D) Pick a high quantilea max =Q 1−δ(D); enforce min(L α)≳c a amax. Refs:a max in Eq. (51); safeguard Eq. (55)

work page
[29]

Refs: Eqs

Pilot cell solves (Section§II F) Solve periodic cell problems to obtainK app andβ app, thenλ app = p Kiso/βapp, withK iso = 1 3 trK app. Refs: Eqs. (65), (67)–(68), (71)

work page
[30]

Combine:L ⋆ = max{c ξξ, c λλapp, c aamax}and enforce aspect ratio min(L α)/max(L α)≥ρ min

Length thresholds & combination (Section§II G) Require: min(L α)≳c ξ ξ; min(L α)≳c λ λapp; min(L α)≳c a amax; anisotropic:L α ≳c λ λ(α) app with (λ(α) app)−2 =β app/(Kapp)αα. Combine:L ⋆ = max{c ξξ, c λλapp, c aamax}and enforce aspect ratio min(L α)/max(L α)≥ρ min. Refs: Eqs. (72), (75), (76), (78), (49)

work page
[31]

ForP=β app (capillarity):V≳ cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2

Volume for target precision (Section§II G) Fornrealizations and CV≤ε:V≳ CP A nε2 . ForP=β app (capillarity):V≳ cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2. Refs: Eqs. (33), (81). 6b) (Optional) Distribution-aware refinement (Section§III) Computeϕ λ andA λ; optionallyλ (λ) app. UseV≳ Aλ nε2 ϕλ(1−ϕ λ) Kg −Km λ2 2. Refs:ϕ λ Eq. (93);A λ Eq. (101); rule Eq. (110);λ (λ) app v...

work page
[32]

If using distribution-aware, replace by Eqs

Practical box choice Cubic:L≳max n L⋆, cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2 1/3o . If using distribution-aware, replace by Eqs. (111) or (112). Refs: Eq. (84)

work page
[33]

Refs: Eqs

A posteriori verification Low-k: markk min = 2π/min(L α) and checkk min ≪k c andk min ≪1/λ app; Aspect ratio satisfied; observed Var∝1/Vover realizations. Refs: Eqs. (47), (46), (29), (79), (32). FIG. 1. Top–down workflow for choosing the periodic supercell in screened capillarity.Microstructure: estimate a correlation length and the integral rangeAfromC(...

work page 2021
[34]

cheerios effect

D. Vella and L. Mahadevan,The “cheerios effect”, American Journal of Physics73, 817–825 (2005)

work page 2005
[35]

Alonso-Marroqu´ ın and M

F. Alonso-Marroqu´ ın and M. P. Andersson, Capillary-pressure saturation relation derived from the pore morphology method, Physical Re- view E111, 065107 (2025)

work page 2025

[1] [1]

Torquato,Random Heterogeneous Materi- als: Microstructure and Macroscopic Properties (Springer, 2002)

S. Torquato,Random Heterogeneous Materi- als: Microstructure and Macroscopic Properties (Springer, 2002)

work page 2002

[2] [2]

S. N. Chiu, D. Stoyan, W. S. Kendall, and J. Mecke,Stochastic Geometry and Its Applica- tions, 3rd ed. (Wiley, 2013)

work page 2013

[3] [3]

D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes. Volume I: Ele- mentary Theory and Methods(Springer, 2003)

work page 2003

[4] [4]

Lu and S

B. Lu and S. Torquato,Local volume fraction fluc- tuations in heterogeneous media, The Journal of Chemical Physics93, 3452–3459 (1990)

work page 1990

[5] [5]

Quintanilla and S

J. Quintanilla and S. Torquato,Local volume frac- tion fluctuations in random media, The Journal of Chemical Physics106, 2741–2751 (1997)

work page 1997

[6] [6]

Torquato,Hyperuniform states of matter, Physics Reports745, 1–95 (2018)

S. Torquato,Hyperuniform states of matter, Physics Reports745, 1–95 (2018)

work page 2018

[7] [7]

de Gennes, F

P.-G. de Gennes, F. Brochard-Wyart, and D. Qu´ er´ e,Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves(Springer, 2004)

work page 2004

[8] [8]

L. D. Landau and E. M. Lifshitz,Fluid Mechan- ics, 2nd ed., Course of Theoretical Physics, Vol. 6 (Pergamon Press, 1987)

work page 1987

[9] [9]

Bensoussan, J.-L

A. Bensoussan, J.-L. Lions, and G. Papanico- laou,Asymptotic Analysis for Periodic Struc- tures, AMS Chelsea Publishing No. 374 (Amer- ican Mathematical Society, 2011)

work page 2011

[10] [10]

V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and In- tegral Functionals(Springer, 1994)

work page 1994

[11] [11]

Moulinec and P

H. Moulinec and P. Suquet,A numerical method for computing the overall response of nonlinear composites with complex microstructure, Com- puter Methods in Applied Mechanics and Engi- neering157, 69–94 (1998)

work page 1998

[12] [12]

M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans,Multi-scale computational homoge- nization: Trends and challenges, Journal of Com- putational and Applied Mathematics234, 2175– 2182 (2010)

work page 2010

[13] [13]

C. L. Y. Yeong and S. Torquato,Reconstructing random media, Physical Review E57, 495–506 (1998)

work page 1998

[14] [14]

Y. Jiao, S. Torquato, and F. H. Stillinger,A superior descriptor of random textures and its predictive capacity, Proceedings of the National Academy of Sciences106, 17634–17639 (2009)

work page 2009

[15] [15]

Kanit, S

T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin,Determination of the size of the repre- sentative volume element for random composites: Statistical and numerical approach, International Journal of Solids and Structures40, 3647–3679 (2003)

work page 2003

[16] [16]

Ostoja-Starzewski,Material spatial random- ness: From statistical to representative volume el- ement, Probabilistic Engineering Mechanics21, 112–132 (2006)

M. Ostoja-Starzewski,Material spatial random- ness: From statistical to representative volume el- ement, Probabilistic Engineering Mechanics21, 112–132 (2006)

work page 2006

[17] [17]

Wiener,Generalized harmonic analysis, Acta Mathematica55, 117–258 (1930)

N. Wiener,Generalized harmonic analysis, Acta Mathematica55, 117–258 (1930)

work page 1930

[18] [18]

Khintchine,Korrelationstheorie der sta- tion¨ aren stochastischen prozesse, Mathematische Annalen109, 604–615 (1934)

A. Khintchine,Korrelationstheorie der sta- tion¨ aren stochastischen prozesse, Mathematische Annalen109, 604–615 (1934)

work page 1934

[19] [19]

A. M. Yaglom,Correlation Theory of Stationary and Related Random Functions, Springer Series in Statistics, Vol. I: Basic Results (Springer, 1987)

work page 1987

[20] [20]

Hill,Elastic properties of reinforced solids: Some theoretical principles, Journal of the Me- chanics and Physics of Solids11, 357–372 (1963)

R. Hill,Elastic properties of reinforced solids: Some theoretical principles, Journal of the Me- chanics and Physics of Solids11, 357–372 (1963)

work page 1963

[21] [21]

M. P. Allen and D. J. Tildesley,Computer Simula- tion of Liquids, 2nd ed. (Oxford University Press, 2017). [22]GROMACS Reference Manual(2023), see sec- tions on periodic boundary conditions and minimum-image convention; cutoff must be smaller than half the shortest box vector

work page 2017

[22] [22]

P. A. Kralchevsky and K. Nagayama,Capillary interactions between particles at a liquid inter- face, Advances in Colloid and Interface Science 85, 145–192 (2000)

work page 2000

[23] [23]

M. C. Leverett,Capillary behavior in porous solids, Transactions of the AIME142, 152–169 (1941)

work page 1941

[24] [24]

Chavent and J

G. Chavent and J. Jaffr’e,Mathematical Mod- els and Finite Elements for Reservoir Simulation: Single Phase, Multiphase, and Multicomponent Flows through Porous Media, Studies in Math- ematics and Its Applications, Vol. 17 (North- Holland, Amsterdam, 1986)

work page 1986

[25] [25]

Winkelmann, L

M. Winkelmann, L. Krien, and D. Vollhardt,So- lution to the modified helmholtz equation for ar- 12

work page

[26] [26]

Finite-size variance obeys Var[P(Y)]≈(C P A)/V

Microstructure statistics (Section§II B) Estimate correlation lengthξand integral rangeAfromC(r) or bC(k). Finite-size variance obeys Var[P(Y)]≈(C P A)/V. Refs:Ain Eq. (19); variance and 1/Vscaling Eqs. (29)–(33), generic form Eq. (31)

work page

[27] [27]

Refs: Eq

Low-wavenumber coverage (Section§II C) Use the latticeK(Y) and requirek min = 2π/min(L α)≪k c; practically min(L α)≳c ξ ξ. Refs: Eq. (44), criterion Eq. (47), practical rule Eq. (48)

work page

[28] [28]

Refs:a max in Eq

Geometric safeguard for polydispersity (Section§II D) Pick a high quantilea max =Q 1−δ(D); enforce min(L α)≳c a amax. Refs:a max in Eq. (51); safeguard Eq. (55)

work page

[29] [29]

Refs: Eqs

Pilot cell solves (Section§II F) Solve periodic cell problems to obtainK app andβ app, thenλ app = p Kiso/βapp, withK iso = 1 3 trK app. Refs: Eqs. (65), (67)–(68), (71)

work page

[30] [30]

Combine:L ⋆ = max{c ξξ, c λλapp, c aamax}and enforce aspect ratio min(L α)/max(L α)≥ρ min

Length thresholds & combination (Section§II G) Require: min(L α)≳c ξ ξ; min(L α)≳c λ λapp; min(L α)≳c a amax; anisotropic:L α ≳c λ λ(α) app with (λ(α) app)−2 =β app/(Kapp)αα. Combine:L ⋆ = max{c ξξ, c λλapp, c aamax}and enforce aspect ratio min(L α)/max(L α)≥ρ min. Refs: Eqs. (72), (75), (76), (78), (49)

work page

[31] [31]

ForP=β app (capillarity):V≳ cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2

Volume for target precision (Section§II G) Fornrealizations and CV≤ε:V≳ CP A nε2 . ForP=β app (capillarity):V≳ cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2. Refs: Eqs. (33), (81). 6b) (Optional) Distribution-aware refinement (Section§III) Computeϕ λ andA λ; optionallyλ (λ) app. UseV≳ Aλ nε2 ϕλ(1−ϕ λ) Kg −Km λ2 2. Refs:ϕ λ Eq. (93);A λ Eq. (101); rule Eq. (110);λ (λ) app v...

work page

[32] [32]

If using distribution-aware, replace by Eqs

Practical box choice Cubic:L≳max n L⋆, cAξ3 nε2 ϕ(1−ϕ) Kg −Km λ2 2 1/3o . If using distribution-aware, replace by Eqs. (111) or (112). Refs: Eq. (84)

work page

[33] [33]

Refs: Eqs

A posteriori verification Low-k: markk min = 2π/min(L α) and checkk min ≪k c andk min ≪1/λ app; Aspect ratio satisfied; observed Var∝1/Vover realizations. Refs: Eqs. (47), (46), (29), (79), (32). FIG. 1. Top–down workflow for choosing the periodic supercell in screened capillarity.Microstructure: estimate a correlation length and the integral rangeAfromC(...

work page 2021

[34] [34]

cheerios effect

D. Vella and L. Mahadevan,The “cheerios effect”, American Journal of Physics73, 817–825 (2005)

work page 2005

[35] [35]

Alonso-Marroqu´ ın and M

F. Alonso-Marroqu´ ın and M. P. Andersson, Capillary-pressure saturation relation derived from the pore morphology method, Physical Re- view E111, 065107 (2025)

work page 2025