Rheology of unsaturated granular materials – RheoGranoSat

A successful modeling scheme of such complex fluids as unsaturated granular materials should be based on a good knowledge of small scale rheophysical phenomena. A successful modeling scheme of such complex fluids as unsaturated granular materials should be based on a good knowledge of small scale rheophysical phenomena. For this purpose, the project relies on original experimental characterization schemes of the mechanical behavior of unsaturated granular materials to identify different flow regimes at varying liquid saturation levels, implementing innovative tools providing access to information on their microstructure. Our project comprises three parts: a first experimental part seeks to map out the different rheological regimes in the space of control parameters, and should thus determine the density enabling shear flow to be imposed, the static yield stress or yield condition (internal friction, cohesion) of the material, and the critical shear rate and / or strain for which the viscous forces dominate the capillary forces. The second part involves the development of tools for experimental determination of the microstructure of unsaturated granular materials in the different flow regimes previously established using specific rheometers that can be inserted into imaging devices such as MRI, confocal microscopes and microtomography apparatus. The third part of the project consists grain-scale numerical simulation of the flow behavior of granular materials in different conditions, which might involve specific developments of methods. Numerical studies will aim at predicting microstructure and its evolution in terms of the history of deformation and confinement, and will be validated by the experiments. The coupling of these two last parts will thus mostly allow a description and prediction of the constitutive law for a given microstructure and containment of granular materials at different levels of saturation at different deformations. This comparison between experiments and simulations should determine which basic mechanisms and microscopic characteristics are important for the rheology.

Further use of x-ray tomography technique enabled investigations of the microstructure. It is found that, for the explored range of liquid content, samples stay homogeneous with, however, the presence of multitude complex morphologies far from simple capillary bridges. We also observed that shearing tends to reduce the number of these large liquid morphologies to the cost of simple liquid bridges. This important result seems to explain the concordance between experimental and numerical measurements. Indeed, the numerical model is restricted to the pendular state, in which the liquid phase is completely discontinuous and no coalescence occurs between liquid bridges.
Indeed, rheometric measurements on assemblies of wet polystyrene beads, in steady uniform quasistatic shear flow, for varying liquid content within the small saturation (pendular) range of isolated liquid bridges, are supplemented with a systematic study by discrete numerical simulations. The numerical results agree quantitatively with the experimental ones provided that the intergranular friction coefficient is set to the value µ ? 0.09, identified from the behaviour of the dry material. Shear resistance and solid fraction F_S are recorded as functions of the reduced pressure P^*, which, defined as P^* = d^2 s_22/F_0, compares stress s_22, applied in the velocity gradient direction, to the tensile strength F_0 of the capillary bridges between grains of diameter d, and characterizes cohesion effects. The simplest Mohr-Coulomb relation with P^*- independent cohesion c applies as a good approximation for large enough P^* (typically P^* = 2). Numerical simulations extend to different values of µ and, compared to experiments, to a wider range of P^*. The assumption that capillary stresses act similarly to externally applied ones onto the dry granular contact network (effective stresses) leads to very good (although not exact) predictions of the shear strength, throughout the numerically investigated range P^* = 0.5 and 0.05 = µ = 0.25. Thus, the internal friction coefficient µ_0^* of the dry material still relates the contact force contribution to stresses, s_12^cont=µ_0^* s_22^cont, while the capillary force contribution to stresses, s ?^cap, defines a generalized Mohr-Coulomb cohesion c, depending on P^* in general. c relates to µ_0^*, coordination numbers and capillary force network anisotropy. c increases with liquid content through the pendular regime interval, to a larger extent, the smaller the friction coefficient. The simple approximation ignoring capillary shear stress s_12^cap (referredto as the Rumpf formula) leads to correct approximations for the larger saturation range within the pendular regime, but fails to capture the decrease of cohesion for smaller liquid contents.

The numerical results agree quantitatively with the experimental ones provided that the intergranular friction coefficient is set to the value µ ? 0.09, identified from the behaviour of the dry material. This agreement applies to the common region (experiment & numerical simulations) in space of three parameters P^*, the inertial number I and the liquid content.
The work of developing a numerical method coupling «discrete elements« (DEM) for the motion of solid grains and «Boltzmann lattice method« (LBM) for the unsaturated liquid has proved technically to be formidable. We are faced with several difficulties: it is necessary to simulate two fluids to correctly model the interfacial tension; the conservation of the wetting liquid volume must be well ensured; the modeling of the shape of the grains on a discrete lattice leads to systematic errors, except to refine the lattice to a degree such that the computational volume, with a representative number of solid grains, becomes prohibitive.
The LBM treatment adopted by Lhassan Amarsid for the unsaturated liquid is a complete multiphase approach, that of Rothman and Keller, which involves a calculation of the pressure in the gas phase, the viscosity of which remains negligible. This approach gives good results in the model situations tested, for which the grains are fixed. Thus, the shape of the meniscus between two grains, depending on the distance which separates them and the liquid volume, conforms to the analytical or numerical results of the literature. However, the method remains subject to inaccuracies which for the time being seem prohibitive when the grains are in motion, with discontinuities of non-physical forces associated with the crossing of the steps of the LBM network by the interfaces. These problems related to digital technology could not be resolved during the limited time of Lhassan Amarsid's postdoc, and we are keeping the resulting codes in reserve pending further development to overcome the technical obstacles encountered.

1. Badetti M., Fall A, Chevoir F. and Roux J.-N. ‘Shear strength of wet granular materials: macroscopic cohesion and effective stress’, (hal-01980048). European Physical Journal E (2018) 41:68, doi.org/10.1140/epje/i2018-11677-8
2. Badetti. M, Fall A., Chevoir F. et al. ‘Rheology and microstructure of unsaturated wet granular materials: Experiments and simulations’. (hal-01980091), Journal of Rheology 62, 1175 (2018); doi.org/10.1122/1.5026979
3. Waguaf, L. et al. ‘Nuclear Magnetic Resonance volumetric antenna’, European Conference on Antennas and Propagation, Krakow, Poland, 2019, hal-02268825
4. Marhabaie, S. et al. ‘A Bird-Cage Coil for MRI Studies of Unsaturated Granular Materials’ European Conference on Antennas and Propagation, Copenhague, Danmark, 2020, hal-02909150
5. S. Deboeuf and A. Fall, ‘Unsaturated wet granular flows over a rough incline: frictional and cohesive rheology’, hal-03266242

We propose to work on unsaturated granular materials and in particular on their rheological behavior. We seek to determine the mechanical behavior in the solid and fluid regimes of granular materials in the presence of a non-saturating liquid, which acts through its viscosity and through capillary effects. Such materials, intermediate between dry granular assemblies and highly concentrated suspensions, will be studied experimentally at different scales, from the microstructure to macroscopic behavior. Our goal is to establish the foundations of an understanding of capillary and/or viscous phenomena involved in these materials. A successful modeling scheme of such complex fluids as unsaturated granular materials should be based on a good knowledge of small scale rheophysical phenomena. To this end, the project relies on original experimental characterization schemes of the mechanical behavior of these materials to identify different flow regimes at varying liquid contents, implementing innovative tools providing access to information on their microstructure. The project comprises three parts: a first experimental part (i) seeks to map out the different rheological regimes in the space of control parameters, and should thus determine the density enabling shear flow to be imposed, the yield condition (internal friction, cohesion) of the material, and the critical shear rate and/or strain for which the viscous forces dominate the capillary forces. The second part (ii) involves the development of tools for experimental determination of the microstructure of unsaturated granular materials in the different flow regimes previously established using specific rheometers that can be inserted into imaging devices such as MRI, confocal microscopy and microtomography devices. The third part (iii) of the project consists of grain-scale numerical simulations of granular materials in different deformation and flow conditions, which might involve specific developments. Numerical studies will aim at predicting microstructure and its evolution in terms of the history of deformation and confinement, and will be confronted to experiments. The coupling of parts (ii) and (iii) will lead to the formulation of constitutive laws based on the microstructure, depending on saturation and deformation history. Such microscopic investigations are necessary for two reasons. First, before any continuum modeling, one needs to control density, water content and shear rate homogeneity. Slowly deformed granular materials are prone to shear banding instabilities, which capillary cohesion might favor; water content heterogeneities might couple to those shear-bands. Then, the synergy of experiments and simulations should determine the essential rheophysical mechanisms (rearrangements, formation of clusters…)
Possible applications belong to the field of civil and environmental engineering (mixing of cementitious materials, asphalt, unsaturated soils ...) but may also pertain to other industrial domains (food, pharmacy ...), as well as to geomechanics (landslides, rock joints, fault zones).