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

TL;DR

The paper addresses how to select minimal periodic supercells for representative capillarity-driven Darcy flow in stationary, polydisperse granular media by connecting microstructural two-point statistics to capillary screening physics. It introduces a screened (modified-Helmholtz) cell problem under periodic boundary conditions, yielding an apparent conductivity $K_{app}$, an apparent screening parameter $\\beta_{app}$, and a decay length $\\lambda_{app}$, with a key length-scale rule based on a distribution-aware analyses using $\\phi_{\\lambda}$ and $A_{\\lambda}$. The resulting sizing rules combine a length criterion tied to the macroscopic capillary decay length and a volume criterion tied to the target variance, updated to account for grain-size distribution and screening effects. Numerical validation on a synthetic Boolean-sphere medium confirms the $A/V$ variance scaling with an explicit capillarity prefactor, demonstrates the essential low-$k$ spectral coverage, and shows the necessity of a geometric safeguard against self-interaction of large grains. The approach provides a solver-agnostic, distribution-aware procedure for reproducible supercell sizing in image-based FE/FFT simulations of capillarity in random granular media, bridging microstructure statistics and capillary physics with practical guidance for simulations.

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.

Figures (6)

• Figure 1: Sample variance of the capillarity observable $\beta_{\rm app}(Y_L)$ as a function of the inverse supercell volume $1/V=L^{-3}$ in the Boolean-sphere medium. Open circles with error bars: numerical estimates from $N_{\rm micro}=50$ realizations at ten box sizes $L\in\{5,\dots,14\}$. Dashed line: best power-law fit $\mathrm{Var}[\beta_{\rm app}(Y_L)]\approx C(1/V)^\alpha$ in log–log coordinates, with $\alpha$ close to $1$. Dotted line: reference line with slope $1$, illustrating the $A/V$ scaling with prefactor $C_{P}^{\rm cap}A$. The largest boxes lie very close to the $1/V$ trend, while the smallest boxes show a mild upward deviation attributed to insufficient low-wavenumber coverage.
• Figure 2: Radially averaged spectral density $\widehat{C}(k)$ of the two-phase indicator in the Boolean-sphere medium for ten box sizes $L\in\{5,\dots,14\}$. Top panel: full $k$-range on log–log scales, with markers only (no connecting lines) and vertical dashed lines at the smallest resolved wavenumber $k_{\min}^{\text{box}} = 2\pi/\min_\alpha L_\alpha$ (here $k_{\min}^{\text{box}} = 2\pi/L$) for each $L$. Bottom panel: zoom on the range $k\in[10^{0},k_{\max}]$ with $10^{-2}\le\widehat{C}(k)\le 10^{3}$, using the same marker style. In this zoomed view the spectra for $L\gtrsim 10$ collapse almost perfectly, while smaller boxes depart near the spectral knee, illustrating the low-wavenumber coverage condition $k_{\min}^{\text{box}}\ll k_c$ underlying the variance scaling.
• Figure 3: Effect of the ratio $L/a_{\max}$ on the capillarity observable in the Boolean-sphere medium. Top: sample mean $\overline{\beta_{\rm app}}$ versus $L/a_{\max}$ with error bars (standard error of the mean) for ten values of $L/a_{\max}\in[1,5]$. The dashed curve is an exponential fit $\beta_{\rm app}(L/a_{\max})\approx\beta_\infty + A\exp[-(L/a_{\max})/L_c]$. Bottom: sample variance $\widehat{\mathrm{Var}}[\beta_{\rm app}]$ versus $L/a_{\max}$ with error bars (standard error of the variance), together with an exponential fit of the same form. When $L/a_{\max}\approx 1$ the mean is biased and the variance is inflated by more than an order of magnitude. For $L/a_{\max}\gtrsim 4$ both quantities have essentially converged, indicating that self-interaction artifacts of the largest grains are suppressed.
• Figure 4: Representative $xy$-slices of the Boolean-sphere medium at three box sizes illustrating periodic self-interaction of the largest grains. From left to right: $L/a_{\max}\approx 1$, $2$, and $4$. In the smallest box, large grains align across opposite faces and form artificial columns, leading to anisotropic apparent response and biased $\beta_{\rm app}$. At $L/a_{\max}\approx 2$ these artifacts are reduced but still visible. At $L/a_{\max}\approx 4$ the largest grains are well separated from their periodic images, and the microstructure appears statistically isotropic, in agreement with the convergence of the mean and variance in Fig. \ref{['fig:self_interaction_beta']}.
• Figure 5: Mean wall–clock time per realization as a function of the box size $L$ for the Boolean-sphere medium. Open markers show the contributions from microstructure generation (circles), spectral FFT and radial averaging (squares), and evaluation of the capillarity observable (diamonds), together with their sum (triangles). The $y$-axis is logarithmic. Microstructure generation and the FFT dominate the total cost, while the observable evaluation is negligible. The strong increase of the total time with $L$ illustrates the practical impact of choosing the smallest supercell size compatible with the accuracy criteria developed in this work.