Poromechanics

The loading of a granular material induces anisotropies of the particle arrangement (fabric) and of the material’s strength, incremental stiffness, and permeability. Thirteen measures of fabric anisotropy are developed, which are arranged in four categories: as preferred orientations of the particle bodies, the particle surfaces, the contact normals, and the void space. Anisotropy of the voids is described through image analysis and with Minkowski tensors. The thirteen measures of anisotropy change during loading, as determined with three-dimensional discrete element simulations of biaxial plane strain compression with constant mean stress. Assemblies with four different particle shapes were simulated. The measures of contact orientation are the most responsive to loading, and they change greatly at small strains, whereas the other measures lag the loading process and continue to change beyond the state of peak stress and even after the deviatoric stress has nearly reached a steady state. The paper implements a methodology for characterizing the incremental stiffness of a granular assembly during biaxial loading, with orthotropic loading increments that preserve the principal axes of the fabric and stiffness tensors. The linear part of the hypoplastic tangential stiffness is monitored with oedometric loading increments. This stiffness increases in the direction of the initial compressive loading but decreases in the direction of extension. Anisotropy of this stiffness is closely correlated with a particular measure of the contact fabric. Permeabilities are measured in three directions with lattice Boltzmann methods at various stages of loading and for assemblies with four particle shapes. Effective permeability is negatively correlated with the directional mean free path and is positively correlated with pore width, indicating that the anisotropy of effective permeability induced by loading is produced by changes in the directional hydraulic radius.

A finite strain multiscale hydro-mechanical model is established via an extended Hill–Mandel condition for two-phase porous media. By assuming that the effective stress principle holds at unit cell scale, we established a micro-to-macro transition that links the micromechanical responses at grain scale to the macroscopic effective stress responses, while modeling the fluid phase only at the macroscopic continuum level. We propose a dual-scale semi-implicit scheme, which treats macroscopic responses implicitly and microscopic responses explicitly. The dual-scale model is shown to have good convergence rate, and is stable and robust. By inferring effective stress measure and poro-plasticity parameters, such as porosity, Biot's coefficient and Biot's modulus from micro-scale simulations, the multiscale model is able to predict effective poro-elasto-plastic responses without introducing additional phenomenological laws. The performance of the proposed framework is demonstrated via a collection of representative numerical examples. Fabric tensors of the representative elementary volumes are computed and analyzed via the anisotropic critical state theory when strain localization occurs.

An adaptively stabilized finite element scheme is proposed for a strongly coupled hydro-mechanical problem in fluid-infiltrating porous solids at finite strain. We first present the derivation of the poromechanics model via mixture theory in large deformation. By exploiting assumed deformation gradient techniques, we develop a numerical procedure capable of simultaneously curing the multiple locking phenomena related to shear failure, incompressibility imposed by pore-fluid and/or incompressible solid skeleton, and yet produce solutions that satisfy the inf-sup condition. The template based generic programming and automatic differentiation techniques used to implement the stabilized model are also highlighted. Finally, numerical examples are given to show the versatility and efficiency of this model.

Many natural and man-made materials, such as sand, rock, concrete and bone, are multi-constituent, fluid-infiltrated porous solids. The failure of such materials is important for various engineering applications, such as CO2 sequestration, energy storage and retrieval and aquifer management as well as many other geotechnical engineering problems aimed to prevent catastrophic failures due to pore pressure build-up.

This dissertation investigates two mechanical aspects of fluid infiltrated porous media, i.e., the predictions of diffuse and localized failures of porous media and the heterogeneous microstructures developed after failures. We define failures as material conditions in which homogeneous deformation becomes unattainable.

To detect instabilities, a critical state plasticity model for sand is implemented. By seeking bifurcation points of the incremental, linearized constitutive responses, we establish local criteria that detect onsets of drained soil collapse, static liquefaction and formation of deformation bands under locally drained and undrained conditions. Fully undrained and drained triaxial compression simulations are conducted and the stability of the numerical specimens are assessed via a perturbation method.

To characterize deformation modes after failures, a multi-scale framework is designed to determine microstructural attributes from pore space extracted from X-ray tomographic images and improve the accuracy and speed of a multi-scale lattice Boltzmann/finite element hierarchical flow simulation algorithm. By comparing the microstructural attributes and macroscopic permeabilities inside and outside a compaction band formed in Aztec Sandstone, our numerical study reveals that elimination of connected pore space and increased tortuosity are the main causes that compaction bands are flow barriers.

A simplified method to analyze diffuse and localized bifurcations of sand under drained and undrained conditions is presented in this paper. This method utilizes results from bifurcation analysis and critical state plasticity theory to detect onset of pure and dilatant shear band formation, static liquefaction and drained shear failures systematically. To capture the soil collapse observed in experiments, the instability state line concept originated by Chu et al 2003 is adopted. Emphasis is given to examine how the presence of pore-fluid may facilitate or delay instability after yielding occurs. The predictions of instabilities are compared with experimental data from triaxial compression tests on Toyoura and Changi sands.

This paper presents a multi-scale analysis of a dilatant shear band using a three dimensional discrete element method and a lattice Boltzmann/finite element hybrid scheme. In particular, three dimensional simple shear tests are conducted via the discrete element method. A spatial homogenization is performed to recover the macroscopic stress from the micro-mechanical force chains. The pore geometries of the shear band and host matrix are quantitatively evaluated through morphology analyses and lattice Boltzmann/finite element flow simulations. Results from the discrete element simulations imply that grain sliding and rotation occur predominately with the shear band. These granular motions lead to dilation of pore space inside the shear band and increases in local permeability. While considerable anisotropy in the contact fabric is observed within the shear band, anisotropy of the permeability is, at most, modest in the assemblies composed of spherical grains.

Recent technology advancements on X-ray computed tomography (X-ray CT) offer a nondestructive approach to extract complex three-dimensional geometries with details as small as a few microns in size. This new technology opens the door to study the interplay between microscopic properties (e.g., porosity) and macroscopic fluid transport properties (e.g., permeability). To take full advantage of X-ray CT, we introduce a multiscale framework that relates macroscopic fluid transport behavior not only to porosity but also to other important microstructural attributes, such as occluded/connected porosity and geometrical tortuosity, which are extracted using new computational techniques from digital images of porous materials. In particular, we introduce level set methods, and concepts from graph theory, to determine the geometrical tortuosity and connected porosity, while using a Lattice Boltzmann/Finite Element scheme to obtain homogenized effective permeability at specimen-scale. We showcase the applicability and efficiency of this multiscale framework by two examples, one using a synthetic array and another using a sample of natural sandstone with complex pore structure.

An adaptively stabilized monolithic finite element model is proposed to simulate the fully coupled thermo-hydro-mechanical behavior of porous media undergoing large deformation. We first formulate a finite-deformation thermo-hydro-mechanics field theory for non-isothermal porous media. Projection-based stabilization procedure is derived to eliminate spurious pore pressure and temperature modes due to the lack of the two-fold inf-sup condition of the equal-order finite element. To avoid volumetric locking due to the incompressibility of solid skeleton, we introduce a modified assumed deformation gradient in the formulation for non-isothermal porous solids. Finally, numerical examples are given to demonstrate the versatility and efficiency of this thermo-hydro-mechanical model. Copyright © 2015 John Wiley & Sons, Ltd.

The objective of this research is to use grain-scale numerical simulations to analyze the evolution of stress anisotropy exhibited in wetted granular matters. Multiphysical particulate simulations of unsaturated granular materials were conducted to analyze how the interactions of contact force chains and liquid bridges influence the macroscopic responses under various suction pressure and loading history. To study how formation and rupture of liquid bridges affect the mechanical responses of wetted granular materials, a series of suction-controlled triaxial tests were simulated with two grain assemblies, one composed of large particles of similar sizes, another one composed of a mixture of large particles with significant amount of fines. Our results indicate that capillary stress are anisotropic in both sets of specimens, and that the stress anisotropy is more significant in granular assemblies filled with fine particles. A generalized tensorial Bishop's coefficient is introduced to analyze the connections between microstructrual attributes and macroscopic responses. Numerical simulations presented in this paper indicate that the principal values and directions of this Bishop's coefficient tensor are path dependent.

A stabilized enhanced strain finite element procedure for poromechanics is fully integrated with an elasto-plastic cap model to simulate the hydro-mechanical interactions of fluid-infiltrating porous rocks with associative and non-associative plastic flow. We present a quantitative analysis on how macroscopic plastic volumetric response caused by pore collapse and grain rearrangement affects the seepage of pore fluid, and vice versa. Results of finite element simulations imply that the dissipation of excess pore pressure may significantly affect the stress path, and thus alter the volumetric plastic responses.

Tomographic images taken inside and outside a compaction band in a field specimen of Aztec sandstone are analyzed by using numerical methods such as graph theory, level sets, and hybrid lattice Boltzmann/finite element techniques. The results reveal approximately an order of magnitude permeability reduction within the compaction band. This is less than the several orders of magnitude reduction measured from hydraulic experiments on compaction bands formed in laboratory experiments and about one order of magnitude less than inferences from two-dimensional images of Aztec sandstone. Geometrical analysis concludes that the elimination of connected pore space and increased tortuosities due to the porosity decrease are the major factors contributing to the permeability reduction. In addition, the multiscale flow simulations also indicate that permeability is fairly isotropic inside and outside the compaction band.

Bending an elastic beam leads to a complicated 3D stress distribution, but the shear and transverse stresses are so small in a slender beam that a good approximation is obtained by assuming purely uniaxial stress. In this paper, we demonstrate that the same is true for a saturated poroelastic beam. Previous studies of poroelastic beams have shown that, to satisfy the Beltrami-Michell compatibility conditions, it is necessary to introduce either a normal transverse stress or shear stresses in addition to the bending stress. The problem is further complicated if lateral diffusion is permitted. In this study, a fully coupled finite element analysis (FEA) incorporating the lateral diffusion effect is presented. Results predicted by the "exact" numerical solution, including load relaxation, pore pressure, stresses and strains, are compared to an approximate analytical solution that incorporates the assumptions of simple beam theory. The applicability of the approximate beam-bending solution is investigated by comparing it to FEA simulations of beams with various aspect ratios. For "beams" with large width-to-height ratios, the Poisson effect causes vertical deflections that cannot be neglected. It is suggested that a theory of plate bending is needed in the case of poroelastic media with large width-to-height ratios. Nevertheless, use of the approximate solution yields very small errors over the range of width-to-height ratios (viz, 1-4) explored with FEA.

As observed in reconstructed 3D images of the field compaction bands (Lenoir et al 2010), their formation may cause a significant reduction in porosity and permeability. This paper describes essential a lattice Boltzmann/finite element model that may be advantageous for calculating effective permeability of compaction bands from obscured structure in a cost-efficient way. The pressing issues include the homogenization technique and the scale effect of the effective permeability.

A fluid infiltrating porous solid is a multiphase material whose mechanical behavior is significantly influenced by the pore fluid. In particular, the diffusion, advection, capillarity, heating, cooling and freezing of pore fluid, the build-up of pore pressure and the mass exchanges among the solid and fluid constituents may all influence the stability and integrity of the solid skeleton, cause shrinkage, swelling, fracture or liquefaction. These coupling phenomena are important for numerous disciplines, including but not limited to geophysics, biomechanics, and material sciences. The objective of this course is to present the fundamental principles of poromechanics that are essential for engineering practice and to prepare students for more advanced study on porous media.

Intervertebral disc metabolic transport is essential to the functional spine and provides the cells with the nutrients necessary to tissue maintenance. Disc degenerative changes alter the tissue mechanics, but interactions between mechanical loading and disc transport are still an open issue. A poromechanical finite element model of the human disc was coupled with oxygen and lactate transport models. Deformations and fluid flow were linked to transport predictions by including strain-dependent diffusion and advection. The two solute transport models were also coupled to account for cell metabolism. With this approach, the relevance of metabolic and mechano-transport couplings were assessed in the healthy disc under loading-recovery daily compression. Disc height, cell density and material degenerative changes were parametrically simulated to study their influence on the calculated solute concentrations. The effects of load frequency and amplitude were also studied in the healthy disc by considering short periods of cyclic compression. Results indicate that external loads influence the oxygen and lactate regional distributions within the disc when large volume changes modify diffusion distances and diffusivities, especially when healthy disc properties are simulated. Advection was negligible under both sustained and cyclic compression. Simulating degeneration, mechanical changes inhibited the mechanical effect on transport while disc height, fluid content, nucleus pressure and overall cell density reductions affected significantly transport predictions. For the healthy disc, nutrient concentration patterns depended mostly on the time of sustained compression and recovery. The relevant effect of cell density on the metabolic transport indicates the disturbance of cell number as a possible onset for disc degeneration via alteration of the metabolic balance. Results also suggest that healthy disc properties have a positive effect of loading on metabolic transport. Such relation, relevant to the maintenance of the tissue functional composition, would therefore link disc function with disc nutrition.

Poroelasticity Viscoelasticity Indentation Vocal folds a b s t r a c t It is hypothesized that the bulk viscoelasticity of soft tissues is determined by two lengthscale-dependent mechanisms: the time-dependent response of the extracellular matrix (ECM) proteins at the nanometer scale and the biophysical interactions between the ECM solid structure and interstitial fluid at the micrometer scale. The latter is governed by poroelasticity theory assuming free motion of the interstitial fluid within the porous ECM structure. In a recent study (Heris, H.K., Miri, A.K., Tripathy, U., Barthelat, F., Mongeau, L., 2013. J. Mech. Behav. Biomed. Mater.), atomic force microscopy was used to measure the response of porcine vocal folds to a creep loading and a 50-nm sinusoidal oscillation. A constitutive model was calibrated and verified using a finite element model to accurately predict the nanoscale viscoelastic moduli of ECM. A generally good correlation was obtained between the predicted variation of the viscoelastic moduli with depth and that of hyaluronic acids in vocal fold tissue. We conclude that hyaluronic acids may regulate vocal fold viscoelasticity. The proposed methodology offers a characterization tool for biomaterials used in vocal fold augmentations.

A stabilized enhanced strain finite element procedure for poromechanics is fully integrated with an elasto-plastic cap model to simulate the hydro-mechanical interactions of fluid-infiltrating porous rocks with associative and non-associative plastic flow. We present a quantitative analysis on how macroscopic plastic volumetric response caused by pore collapse and grain rearrangement affects the seepage of pore fluid, and vice versa. Results of finite element simulations imply that the dissipation of excess pore pressure may significantly affect the stress path and thus alter the volumetric plastic responses.

The development of reliable experimental techniques for characterization of chemoporomechancial shale-fluid interactions is important for the design of drilling fluids that maximize shale stability. In this context, testing of millimeter-scale specimens is promising because small specimens require shorter test durations and are more readily available from offcuts of preserved core or potentially from drill cuttings or wellbore cave-in material than larger core plugs that are required for more conventional experimentation. Here we present experiments wherein we measure the axial displacement of 4 mm long by 4 mm diameter cylindrical shale specimens that are subjected firstly to a mechanical axial loading and then to an osmotic loading associated with a sudden increase in the salinity of the surrounding fluid. The response to both stages of loading is consistent with theoretical, chemoporoelastic predictions. In particular, the model predicts two types of behavior depending on the ratio between the reflection coefficient and the so-called chemomechanical coupling coefficient that quantifies the volumetric strain as a result of a change in ion content. Consistent with predictions, both monotonic shrinkage and initial shrinkage followed by partial recovery are observed in our testing campaign which includes 20 shales from a variety of geological settings. Quantitative characterization is also carried out by selecting chemoporoelastic parameter values that minimize the mismatch between the data and the model. The results show that the reflection coefficient and the chemomechanical coupling parameter are correlated with each other and with both the Cation Exchange Capacity (CEC) and the Specific Surface Area (SSA). Based on the consistency of the data from test to test and with the model, together with the fact that the key chemoporoelastic coefficients are sensibly correlated with CEC and SSA, we conclude that these millimeter-scale experiments are able to provide useful characterization for better understanding and predicting shale-fluid interactions.

Objective: To investigate the relevance of the human vertebral endplate poromechanics on the fluid and metabolic transport from and to the intervertebral disc (IVD) based on educated estimations of the poromechanical parameter values of the bony endplate (BEP). Methods: 50 micro-models of different BEP samples were generated from mCTs of lumbar vertebrae and allowed direct determination of porosity values. Permeability values were calculated by using the micro-models, through the simulation of permeation via computational fluid dynamics. These educated ranges of porosity and permeability values were used as inputs for mechano-transport simulations to assess their effect on both the distributions of metabolites within an IVD model and the poromechanical cal- culations within the cartilaginous part of the endplate i.e., the cartilage endplate (CEP). Results: BEP effective permeability was highly correlated to local variations of porosity (R2 z 0.88). Universal patterns between bone volume fraction and permeability arose from these results and from other experimental data in the literature. These variations in BEP permeability and porosity had negli- gible effects on the distributions of metabolites within the disc. In the CEP, the variability of the poro- mechanical properties of the BEP did not affect the predicted consolidation but induced higher fluid velocities. Conclusions: The present paper provides the first sets of thoroughly identified BEP parameter values that can be further used in patient-specific poromechanical studies. Representing BEP structural changes through variations in poromechanical properties did not affect the diffusion of metabolites. However, attention might be paid to alterations in fluid velocities and cell mechano-sensing within the CEP.

Poroelasticity Viscoelasticity Indentation Vocal folds a b s t r a c t It is hypothesized that the bulk viscoelasticity of soft tissues is determined by two lengthscale-dependent mechanisms: the time-dependent response of the extracellular matrix (ECM) proteins at the nanometer scale and the biophysical interactions between the ECM solid structure and interstitial fluid at the micrometer scale. The latter is governed by poroelasticity theory assuming free motion of the interstitial fluid within the porous ECM structure. In a recent study (Heris, H.K., Miri, A.K., Tripathy, U., Barthelat, F., Mongeau, L., 2013. J. Mech. Behav. Biomed. Mater.), atomic force microscopy was used to measure the response of porcine vocal folds to a creep loading and a 50-nm sinusoidal oscillation. A constitutive model was calibrated and verified using a finite element model to accurately predict the nanoscale viscoelastic moduli of ECM. A generally good correlation was obtained between the predicted variation of the viscoelastic moduli with depth and that of hyaluronic acids in vocal fold tissue. We conclude that hyaluronic acids may regulate vocal fold viscoelasticity. The proposed methodology offers a characterization tool for biomaterials used in vocal fold augmentations.

Governing equations of poroelastodynamics in time and frequency domain are derived. The continuity equation complements the momentum balance equations. After reduction for spherical symmetry (geometry and loading), the governing equations in frequency domain are solved by introducing wave potentials. The wave propagation velocities are obtained as the real parts of the characteristic equation of the coupled ODE system. Time domain solution for Dirac type boundary pressure is obtained through numerical inversion of transformed solutions. The results are compared to the solution in classical elasticity theory found in the literature.

An extension of a mathematical model for non-isothermal multiphase materials to consider the dissolution of air in liquid water and air mass sources during its desorption at lower water pressure is presented. The solid skeleton is assumed elasto-plastic; heat, water and air flows and water phase changes are taken into account. Physics of air dissolution and water cavitation in porous media is discussed. A numerical example where cavitation develops during strain localization in undrained water saturated dense sands is solved with the developed model as discretized in space and time with the Finite Element Method.

An extension of a mathematical model for non-isothermal multiphase materials to consider the dissolution of air in liquid water and air mass sources during its desorption at lower water pressure is presented. The solid skeleton is assumed elasto-plastic; heat, water and air flows and water phase changes are taken into account. Physics of air dissolution and water cavitation in porous media is discussed. A numerical example where cavitation develops during strain localization in undrained water saturated dense sands is solved with the developed model as discretized in space and time with the Finite Element Method.