2. Overview

Specifically, the study carried out by Delenne et al. (Reference Delenne, Richefeu and Radjaï2015) uses a two-dimensional (2D) model of an isotropically compressed model granular assembly (care being taken nevertheless to allow for pore-space phase percolation despite the continuous contact network). While the vapour pressure stays fixed below the saturating value, a gradual increase of the liquid content is enforced, resulting in a steady decrease of capillary pressure (as $p_{w}$ increases). The liquid distribution adapts itself as saturation grows, approximately satisfying equilibrium requirement, which involves motions and rearrangements of liquid nuclei, first in intergranular menisci, then in more extended liquid clusters as those menisci merge, as shown in figure 1. Compared to previous LBM studies of fluid mixtures within porous media (Sukop & Or Reference Sukop and Or2003; Huang & Lu Reference Huang and Lu2009), the choice of the condensation route to saturation enables the study of more representative homogeneous states, with no gradient effect on the liquid phase distribution and morphology. After the pendular regime ends ( $S=0.15$ ), menisci are observed to merge into a set of liquid clusters with varying sizes. Those clusters, in turn, gradually coalesce, the percolation threshold of the liquid domain occurring near $S=0.6$ . Those phenomena should be qualitatively unchanged between 2D and three-dimensional (3D) models (save for the possibility of both fluid phases simultaneously percolating in 3D).

Figure 1. Aspects of liquid distribution at growing saturation $S$ .

One major original contribution of the study is the evaluation of Laplace pressures as exerted on grains for varying $S$ . More precisely, Delenne et al. (Reference Delenne, Richefeu and Radjaï2015) actually evaluate the average pressure increase or decrease within the solid grain phase, with respect to the situation at vanishing liquid saturation, with constant interstitial vapour pressure $p_{a}$ . Upon investigating the capillary effects on the mechanics of the granular skeleton, this latter value may be fixed to zero, for simplicity. As the grains are kept fixed in this study, a direct measurement of the strength (say, the resistance to shear) of the granular skeleton is not possible. Yet, an effective stress approach to the effect of interstitial fluids on the granular skeleton may be attempted; this amounts to a re-evaluation of the third term of the right-hand side of (1.1). For given total stress transmitted through the medium, ${\bf\sigma}$ , a negative contribution to the pressure in the grains will tend, by compensation, to increase the average contact pressure stabilizing the contact network – the isotropic part of tensor ${\bf\sigma}^{\prime }$ . The negative capillary contribution to the pressure in the grains, due to the capillary pressure pulling on their surface in contact with the liquid phase, denoted as $p_{p}^{-}$ , can be directly calculated in the LBM approach, from the grain perimeter length (in this 2D case) wetted by the liquid. These pressure terms, which are equal to the product of the fast decreasing capillary pressure $p_{c}(S)$ with either $S$ itself or with a wet perimeter fraction growing with $S$ , reach a maximum for some intermediate saturation (near 0.4 for $Sp_{c}$ , near 0.6 for $-p_{p}^{-}$ ). A direct application of the effective stress concept would imply, in a grain pack with internal friction coefficient ${\it\mu}^{\ast }$ , a positive capillary contribution to shear strength equal to $-{\it\mu}^{\ast }p_{p}^{-}$ . However, the effective stress principle is not expected to apply in such a naive form, as the net capillary effects for intermediate saturations will produce a disordered force field acting on the grains in various directions, as is apparent in the figure by the title, which shows the net capillary forces on the grains, as calculated separately in each wetted region of their perimeters. The effects on contact network stability should differ from those of a global pressure change. Anyway, one should bear in mind that the use of (1.1), or variants thereof, is reported to be qualitatively correct in predicting a maximum capillary contribution to shear strength for some saturation level well above the upper limit of the pendular regime. Qualitatively, the strong capillary pressure decrease as a function of $S$ is compensated by the increase of the grain surface in contact with the wetting liquid.

3. Future

The simplest immediate developments of this work would explore, in a fixed grain pack, the effects of varying solid fraction, coordination number and network anisotropy, on the suction curve $p_{c}(S)$ and its hysteresis. But the most promising continuation of this work should involve a full-fledged LBM–DEM coupled model, in which the granular network is allowed to readjust to capillary effect. Much awaited direct investigations of cohesion and strength of model unsaturated granular materials should ensue, probably confirming the optimal effect on strength of saturations in the 0.4–0.6 range. The authors are also planning to develop a 3D model, which should eventually enable a critical review of effective stress concepts for partially saturated materials. As the LBM approach is naturally capable of dealing with fluid flow, a whole range of poromechanical phenomena in partially saturated granular materials will become available for numerical studies at the microscopic scale, as well as processes combining imbibition and mixing.