Capillarity-Driven Consolidation

• Capillarity-driven consolidation is the process where capillary forces, originating from surface tension and meniscus curvature, actively reorganize fluid distribution and pore structure in porous or granular media.
• Methodologies involve pore-scale measurements, Darcy flow descriptions, and continuum models to quantify capillary pressures, transport coefficients, and deformation behaviors across diverse materials.
• The consolidation mechanism can lead to either irreversible densification or counterintuitive expansion, as capillary stresses mediate bond formation and structural reorganization in complex media.

Capillarity-driven consolidation refers to processes in which surface tension, meniscus curvature, and interfacial-energy minimization reorganize a particulate, porous, or deformable medium so that fluid distribution, pore geometry, strength, or local density change over time. In the literature, the term spans true densification of partially saturated or drying granular bodies, capillarity-mediated deformation of compliant porous structures, and mechanistically related analogs such as interfacial particle recruitment or capillary contraction of slender fluid filaments. The common feature is that capillary forces are not merely boundary conditions: they provide a thermodynamic driving force for transport, force transmission, structural rearrangement, and, in some systems, irreversible bond formation (Melnikov et al., 2016, Nasouri et al., 2018, Delenne et al., 2010, Roubíček, 2019).

1. Fundamental driving mechanisms

At the pore scale, capillary loading is set by curvature. In nanoporous silicon, drying under controlled humidity imposes a liquid potential $P_{\text{liq}}-P_0=\Psi_{\text{ext}}$ and reaches negative liquid pressures of order $-10^2$ MPa; the measured capillary pressure plateau is $\Psi_c=-76\pm5\ \text{MPa}$, with an inferred pore radius $r_{p,c}=1.7\pm0.2\ \text{nm}$. In that framework, capillary suction is a mechanically operative pore-pressure load, not merely a formal thermodynamic quantity, and the relevant transport and transient timescales are governed by Darcy flow and poroelastic diffusion with $C=\kappa/(\phi\chi)$ (Vincent et al., 2015).

In rigid mesoporous silica, spontaneous imbibition follows a Darcy-scale Lucas–Washburn description over times ranging from seconds to 10 days, while the capillary pressure scale remains $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$. The same analysis shows that transport coefficients are controlled not only by pore size, porosity, and tortuosity but also by an immobile adsorbed boundary layer, quantified by a mean slip length $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$. This identifies a recurrent feature of capillarity-driven consolidation problems: the hydraulically active pore space can be smaller than the geometrically accessible pore space, so stress transmission and mass redistribution need not scale with nominal pore size alone (Gruener et al., 2018).

For deforming continua, capillarity can be formulated directly in the actual configuration. A large-strain Cahn–Hilliard model writes the capillary contribution to free energy as $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$, which yields the chemical potential

$$\mu = \partial_\zeta \varphi(\nabla y,\zeta) - \operatorname{div}\!\Big(\kappa (\nabla y)^{-1}(\nabla y)^{-T}\nabla \zeta\Big)$$

and a Korteweg-like stress in momentum balance. This places capillarity-driven consolidation within a broader chemo-mechanical class in which concentration gradients simultaneously drive transport and load the skeleton mechanically (Roubíček, 2019).

2. Granular connectivity, bridge mechanics, and densification resistance

In partially saturated granular media, capillarity-driven consolidation requires more than isolated menisci; it requires connectivity of the liquid structure. A percolation study on random close packings of monodisperse spheres at $\phi \approx 0.63$ defines bridge existence by $-10^2$0 and finds a percolation threshold $-10^2$1, with correlation-length exponent $-10^2$2, consistent with ordinary three-dimensional percolation. The direct result is geometric rather than mechanical, but it establishes the onset of sample-spanning capillary pathways as a prerequisite for long-range suction continuity and capillary force transmission (Cárdenas-Barrantes et al., 2016).

At lower saturations in the pendular regime, the dominant mechanism is bridge rupture and local redistribution. A grain-scale shear model assumes bridge rupture at $-10^2$3, followed by strictly local redistribution of liquid to neighboring contacts, and shows that the wetness field obeys a continuum law $-10^2$4 with $-10^2$5 in units of $-10^2$6. Under homogeneous shear, an initially Gaussian liquid profile spreads diffusively as $-10^2$7 with $-10^2$8 per unit shear strain. Crucially, the wet region becomes denser as the cohesion number $-10^2$9 increases, because capillary cohesion lowers the local shear rate and suppresses dilation; this is a direct capillarity-driven compaction mechanism, albeit under shear rather than classical one-dimensional consolidation (Mani et al., 2012).

At arbitrary saturation, a fluid-particle model that resolves bridges, menisci, trimers, pentamers, heptamers, and larger clusters shows how capillary morphology modifies triaxial response. Liquid exchange through wetting films is represented by

$\Psi_c=-76\pm5\ \text{MPa}$0

and triaxial tests over $\Psi_c=-76\pm5\ \text{MPa}$1 show a very large increase in strength from the dry state to $\Psi_c=-76\pm5\ \text{MPa}$2, a maximum around $\Psi_c=-76\pm5\ \text{MPa}$3, and an internal friction angle that remains essentially unchanged at $\Psi_c=-76\pm5\ \text{MPa}$4. In Mohr–Coulomb terms, capillarity acts mainly through the cohesion $\Psi_c=-76\pm5\ \text{MPa}$5, not through a major change in frictional geometry. For consolidation-oriented interpretation, this means capillary suction can strongly increase resistance to rearrangement and collapse while the liquid-cluster network continuously reorganizes under deformation (Melnikov et al., 2016).

3. Deformation of compliant porous bodies

Capillarity-driven consolidation is especially explicit when the confining solid is compliant. In poroelastocapillary rise between two permeable sheets, capillary pressure pulls the sheets together while liquid permeation softens the wetted region. For paper-like sheets, the reported Young’s modulus drops from $\Psi_c=-76\pm5\ \text{MPa}$6 to $\Psi_c=-76\pm5\ \text{MPa}$7 upon wetting, and the coupled rise–bending problem is governed by the elastocapillary number $\Psi_c=-76\pm5\ \text{MPa}$8 and Bond number $\Psi_c=-76\pm5\ \text{MPa}$9. The system passes through three geometric regimes—no contact, touching at the lower end, and finite coalescence—and the criterion $r_{p,c}=1.7\pm0.2\ \text{nm}$0 marks the deformation-controlled timescale. Wetting-induced softening causes earlier coalescence and a significantly lower final liquid capture, so the pore space collapses under capillary load rather than remaining a passive conduit (Nasouri et al., 2018).

In soft porous media undergoing drainage, capillary loading appears directly in continuum force balance. A Multiphase Darcy–Brinkman–Biot framework gives

$r_{p,c}=1.7\pm0.2\ \text{nm}$1

so the term $r_{p,c}=1.7\pm0.2\ \text{nm}$2 localizes capillary stress at the invasion front. The capillary fracturing number

$r_{p,c}=1.7\pm0.2\ \text{nm}$3

separates subcritical capillary compaction from capillary failure. When $r_{p,c}=1.7\pm0.2\ \text{nm}$4, a consolidation-like regime with distributed deformation is possible; when $r_{p,c}=1.7\pm0.2\ \text{nm}$5, capillary entry stresses exceed structural resistance and front-localized fracture emerges (Carrillo et al., 2020).

A decisive complication is that capillarity does not always cause net contraction. During water imbibition into mesoporous Vycor glass, high-resolution dilatometry shows a square-root-of-time expansion and, after top breakthrough, an additional abrupt length increase. The strain of a filled representative volume is written as

$r_{p,c}=1.7\pm0.2\ \text{nm}$6

so a contractile Laplace term competes with an expansive Bangham term due to surface-stress release. In this system the Bangham effect dominates: the fully filled sample expands by about $r_{p,c}=1.7\pm0.2\ \text{nm}$7, with a measured jump of $r_{p,c}=1.7\pm0.2\ \text{nm}$8 compared with a predicted $r_{p,c}=1.7\pm0.2\ \text{nm}$9. This is a direct warning against equating capillarity-driven consolidation with contraction; in nanoporous solids, wetting can produce net expansion rather than densification (Sanchez et al., 2023).

4. Drying, cementation, and post-capillary strengthening

Drying-driven consolidation introduces a second stage in which capillary bonds transform into solid bonds. In granular packings of glass beads or Ventoux sand wetted by saturated NaCl brine and prepared in the pendular state, the global crystallization index

$C=\kappa/(\phi\chi)$0

organizes the mechanical response. Three regimes are reported. In regime I, up to about $C=\kappa/(\phi\chi)$1, the compressive yield stress remains at its capillary value $C=\kappa/(\phi\chi)$2. In regime II, cementation starts at the surface and a front propagates inward, producing nonlinear strengthening. In regime III, all bonds are partially cemented and the strength increases approximately linearly with $C=\kappa/(\phi\chi)$3, reaching values as large as 35 times the initial capillary strength as $C=\kappa/(\phi\chi)$4. The micromechanical upscaling relation

$C=\kappa/(\phi\chi)$5

reduces, for typical monodisperse packings, to $C=\kappa/(\phi\chi)$6. Capillarity here provides the initial integrity, but the strongest consolidation-like strengthening occurs only after crystallization converts liquid bridges into cemented bonds (Delenne et al., 2010).

In polycrystalline materials, capillarity-driven consolidation is often coupled to post-densification grain-boundary migration rather than continuing pore collapse. A nonlinear grain-growth model treats grain-face advance as a two-dimensional nucleation process and obtains

$C=\kappa/(\phi\chi)$7

with $C=\kappa/(\phi\chi)$8 and $C=\kappa/(\phi\chi)$9. This replaces the classical linear law $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$0 by an exponential barrier law and explains growth, stagnation, and abnormal grain growth in terms of temperature, interfacial energy, step free energy, grain size, and grain-size distribution. In a consolidation context, this is not densification itself, but it governs how capillarity continues to reshape a polycrystalline microstructure after densification has occurred (Hu et al., 2017).

5. Interfacial restructuring and other analog mechanisms

Some capillarity-driven processes closely resemble consolidation without constituting true bulk densification. In the inverse Saffman–Taylor configuration, water displacing air in a Hele–Shaw cell is normally stable, yet partially wettable hydrophilic particles on the walls are captured by the advancing meniscus because adsorption lowers interfacial energy. Once the interface becomes saturated with particles, the front destabilizes into fingers in order to create more interfacial area per displaced volume. At low capillary number, $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$1, the measured critical radius agrees with an interfacial-area balance; in a rectangular channel the selected finger width satisfies $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$2. This is best interpreted as capillarity-driven interfacial accumulation and packing with instability-induced restructuring, not as demonstrated volumetric consolidation of a porous particle bed (Bihi et al., 2016).

A comparable broadening of terminology occurs in free-surface rheology. For entangled polymer solutions, capillarity-driven filament thinning exhibits an early tube-reorientation regime, a brief intermediate elasto-capillary regime, and a finite-extensibility regime near pinch-off, with an apparent extensional relaxation time that approaches $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$3 at large entanglement number $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$4 and satisfies $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$5 (Du et al., 2022). For weakly rate-thickening fluids, a late self-similar regime is governed by a quadratic thinning law

$p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$6

with a distinct geometric correction factor $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$7 (Du et al., 2022). These are capillarity-driven contraction problems, but they concern localization of a fluid thread rather than consolidation of a particulate or porous solid.

6. Mathematical formulations, representativity, and limits

Several mathematical frameworks isolate the transport side of capillarity-driven consolidation. A one-dimensional thin-film model for two superposed fluid layers in a porous medium,

$p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$8

proves global existence of nonnegative weak solutions when capillarity is the sole driving mechanism. The model rigorously captures capillary redistribution, degeneracy when one layer vanishes, and mass conservation, but it does not include skeleton deformation or effective-stress coupling, so it is a hydraulic submodel rather than a full consolidation theory (Matioc, 2011).

At equilibrium, capillary meniscus geometry already constrains the forces available for consolidation-like loading. In two dimensions, the capillarity equation $p_c = \frac{2\sigma \cos\theta}{r_{\rm L}}$9 admits a first integral $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$0, with attracting solutions for $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$1 carrying a normalized force $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$2 and repelling solutions for $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$3 bounded by $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$4 and $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$5 (Bhatnagar et al., 2016). For moving interfaces, a potential-theory treatment of the capillarity-driven Hele–Shaw problem reduces the free boundary to a quasilinear parabolic evolution $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$6, proves local well-posedness in $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$7 for $\bar b=-(1.4\pm0.3)\ \mathrm{nm}$8, instantaneous smoothing, and exponential stability of circular equilibria after factoring out the neutral directions associated with area and center of mass conservation (Matioc et al., 21 Aug 2025).

Representative simulation of capillarity-screened Darcy flow in rigid random granular media introduces a distinct scale problem. A screened local equation,

$\bar b=-(1.4\pm0.3)\ \mathrm{nm}$9

with $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$0, homogenizes to apparent conductivity $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$1, screening parameter $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$2, and decay length $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$3. Because screening suppresses high-wavenumber microstructural fluctuations, the paper defines a capillarity-weighted volume fraction $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$4 and a screened integral range $$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$5, then derives the sizing rule

$$\frac{\kappa}{2}\det \nabla y\,|(\nabla y)^{-T}\nabla \zeta|^2$$6

together with a volume criterion based on target coefficient of variation. For capillarity-driven consolidation, this provides a statistically controlled route for selecting the pore-scale domain used to extract hydraulic submodels from random granular microstructures (Tantardini et al., 19 Sep 2025).

A recurrent limitation across the literature is that many capillarity-controlled studies establish prerequisites, analogs, or submodels rather than irreversible densification itself. Percolation analyses quantify when connected capillary pathways appear; thin-film and Hele–Shaw analyses resolve capillarity-driven transport and interface motion; interfacial fingering experiments reveal packing-limited front restructuring; and nanoporous imbibition shows that wetting can induce expansion rather than contraction. It is therefore misleading to identify every capillarity-controlled restructuring process with consolidation. The most precise usage reserves capillarity-driven consolidation for cases where capillary stresses or capillarity-mediated transport alter the mechanical state of a deformable skeleton, the density of a granular assembly, or the permanence of interparticle bonds, while related interfacial and hydraulic phenomena remain essential mechanistic reference points.