J.-N Roux

A numerical toolkit to understand the mechanics of partially saturated granular materials

The mechanisms by which a wetting, non-saturating liquid bestows macroscopic cohesion and strength to a granular material are usually not accessible to micromechanical investigations for saturations exceeding the pendular regime of isolated menisci, easily studied by discrete element models. The "JFM-Rapids" paper (vol. 762, R5, 2015) by Delenne, Richefeu and Radjaï, exploiting a multiphase Lattice Boltzmann approach, pioneers the simulation of the micromorphology and of the mechanical effects on grains of an interstitial liquid, in equilibrium with its vapour, for the whole saturation range. Interestingly, in accordance with some experiments and phenomenological models, the results suggest that the mechanical effect of capillary forces is maximized for some intermediate saturation level (near 40% in the model), well beyond the pendular range. In general, the proposed simulation technique opens the way to many studies of partially saturated granular assemblies, for different saturation or imbibition processes and histories.

Mixtures of granular materials with some amount of interstitial liquid [START_REF] Herminghaus | Dynamics of wet granular matter[END_REF][START_REF] Mitarai | Wet granular materials[END_REF] are ubiquitous in our everyday environment and activities as well as in engineering applications [START_REF] Mitchell | Fundamentals of Soil Behavior[END_REF] and industrial processes [START_REF] Litster | The science and engineering of granulation process[END_REF]. The quasi-static, solid-like behaviour of wet granular materials is considerably influenced by the capillary pressure p c = p a -p w , the difference between liquid pressure p w and the (larger) gas pressure, p a , entailing additional forces on the solid grains, compared to the dry (or gas-saturated) case, and endowing such materials with cohesion and strength. In general, capillary pressure p c decreases for growing saturation S, defined as as the ratio of the wetting liquid volume to the total interstitial volume, although p c -S curves are not uniquely defined, but depend on saturation history and process. Pressure uniformity within continuous liquid domains require complex-shaped, constant curvature liquid-vapour interfaces, as [START_REF] Scheel | Morphological clues to wet granular pile stability[END_REF] visualized by X-ray microtomography.

A recurring issue in the literature on macroscopic modeling of partially saturated granular media, mostly motivated by soil mechanics applications, [START_REF] Mitchell | Fundamentals of Soil Behavior[END_REF][START_REF] Lu | A closed-form equation for effective stress in unsaturated soil[END_REF] is the possibility of dealing with capillary effects in terms of "effective stresses", or corrected stresses that, if applied to the sole granular skeleton, would produce the same mechanical behaviour. In a saturated medium, such an effective stress tensor σ is very successfully defined by the celebrated Terzaghi formula σ ij = σ ij -pδ ij , in terms of total applied stress σ and interstitial fluid pressure p (compressive stresses are counted positive here). The simplest proposed generalization to non-saturated cases, which is

σ ij = σ ij -p a δ ij + Sp c δ ij , (1.1)
coinciding with the Terzaghi formula with a volume-averaged interstitial pressure, is, at best, satisfactory as a rough approximation [START_REF] Lu | A closed-form equation for effective stress in unsaturated soil[END_REF]. Appropriate macroscopic modelling of capillary effects, through the effective stress concept or by other means, is still actively debated. Discrete element modeling (DEM) provides useful insights in the microscopic origins of macroscopic granular material mechanics, but relies on models for intergranular forces, ignoring the continuous interstitial fluid. Its use to model partially saturated grain packs [START_REF] Richefeu | Shear strength properties of wet granular materials[END_REF][START_REF] Scholtès | Micromechanics of granular materials with capillary effects[END_REF]) is limited to the so-called pendular regime of small saturations, for which the wetting liquid is confined within isolated bridges joining neighbouring grains. Such pairs are attracted by a capillary force F 0 = πγd for two spheres of diameter d joined by a small meniscus of a perfectly wetting liquid with surface tension γ, and those tensile forces (which have little dependence with meniscus volume) simply shift the values of normal contact forces in the granular skeleton. Beyond the pendular regime, though (above S 5% for nearly equal-sized spherical grains according to [START_REF] Scheel | Morphological clues to wet granular pile stability[END_REF], the determination of the morphology adopted by the liquid phase and of the mechanical effects of capillary forces escapes simple DEM approaches.

The contribution by Delenne et al., relying on a two phase lattice Boltzmann method (LBM) apt to provide a complete description of the spatial distribution of the liquid phase throughout the complete saturation range 0 ≤ S ≤ 1, offers a way to overcome this limitation, and shows that coupled LBM-DEM studies are a particularly promising tool to explore the micromechanics of partially saturated granular materials, and discuss the foundations of macroscopic modelling schemes.

Overview: salient results of the LBM study

Specifically, the study carried out by Delenne and his coauthors 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 inter granular menisci, then in more extended liquid clusters as those menisci merge, as shown in Fig. 1. Compared to previous LBM studies of fluid mixtures within porous media [START_REF] Sukop | Invasion percolation of single component, multiphase fluids with lattice Boltzmann models[END_REF][START_REF] Huang | Relative permeabilities and coupling effects in steady-state gasliquid flow in porous media: a lattice-Boltzmann study[END_REF], 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 3D models (save for the possibility of both fluid phases simultaneously percolating in 3D).

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. 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 onto the granular skeleton may be attempted -this amounts to a reevaluation of the third term of the r.h.s. of Eq. (1.1). For given total stress transmitted through the medium, σ, 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 σ . 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) wet by the liquid. These pressure terms are equal to the product of the fastly decreasing capillary pressure p c (S) by either S itself or by a wet perimeter fraction growing with S, present 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 µ * , a positive capillary contribution to shear strength equal to -µ * 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 sketched in Fig. 2). The effects on contact network stability is unlikely to coincide with those of a global pressure change.

Anyway, one should bear in mind that the use of Eq. (1.1), or variants thereof, are 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.

Perspectives

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 succion curve p c (S) and its hysteresis. Grain shape [START_REF] Azéma | Packings of irregular polyhedral particles: Strength, structure, and effects of angularity[END_REF]) should also affect the geometry of the solid-liquid interface and the imbibition process. But the most promising continuation of this work should of course 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, likely confirming the optimal effect on strength of saturations in the 0.4 to 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 to deal 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.

Figure 1 .

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

Figure 2 .

 2 Figure 2. Net capillary forces on grains, as calculated separately in each wet region of their perimeters.