Theory Of Linear Poroelasticity Research Articles

Simulation of mechanical behaviour of the proximal femur as a poroelastic solid using particles

Full understanding of the mechanical behaviour of living bone is the key to solving many important problems of modern orthopaedics and arthroplasty. An important role in the study of the mechanical behaviour of living bone belongs to the development and use of comprehensive numerical models. In this work, we model the mechanical behaviour of the proximal femur as a 3D poroelastic solid consisting of the interior cancellous part and the outer cortical part. For simplicity, both parts are assumed isotopic linear poroelastic material. However, they have different values of porosity, fluid content and elastic properties, which are taken from literature. For computer simulation, we used the so-called movable cellular automaton method, which is a representative of simply deformed discrete elements i.e. computational particle mechanics. The method allows simulating dynamics of the elastic skeleton deformation and viscous fluid flow in the skeleton pores according to Biot’s theory of linear poroelasticity. Using the model developed we study the mechanical behaviour of the proximal part of the femur in compression with different rate of loading and different permeability. The results obtained for both saturated and drained bones are discussed.

Swelling and shrinking in prestressed polymer gels: an incremental stress-diffusion analysis.

Polymer gels are porous fluid-saturated materials which can swell or shrink triggered by various stimuli. The swelling/shrinking-induced deformation can generate large stresses which may lead to the failure of the material. In the present research, a nonlinear stress-diffusion model is employed to investigate the stress and the deformation state arising in hydrated constrained polymer gels when subject to a varying chemical potential. Two different constraint configurations are taken into account: (i) elastic constraint along the thickness direction and (ii) plane elastic constraint. The first step entirely defines a compressed/tensed configuration. From there, an incremental chemo-mechanical analysis is presented. The derived model extends the classical linear poroelastic theory with respect to a prestressed configuration. Finally, the comparison between the analytical results obtained by the proposed model and a particular problem already discussed in literature for a stress-free gel membrane (one-dimensional test case) will highlight the relevance of the derived model.

Feasibility of in-situ evaluation of soil void ratio in clean sands using high resolution measurements of Vp and Vs from DPCH testing

For engineering purposes, the density of soil is often expressed in terms of void ratio. Void ratio is a key parameter in critical state soil mechanics and governs our understanding of soil compressibility, permeability, and shear strength. Typically, in-situ void ratio is evaluated based on laboratory measurements on high-quality, samples of soil. While undisturbed sampling of clayey soils is commonplace in geotechnical practice, high-quality samples of granular soils are difficult and expensive to obtain. Hence, void ratio of granular soils is typically estimated using empirical relationships to in-situ measurements from penetration testing. Even for clean sands there is significant variability in these estimates, and for mixed-grain soils the applicability and performance of the empirical relationships is quite uncertain. Therefore, a more reliable way of measuring in-situ void ratio is needed. This paper examines the feasibility of evaluating in-situ soil void ratio based on the theory of linear poroelasticity and the propagation velocity of compression and shear waves (i.e., vp and vs, respectively) through fluid-saturated porous materials. Specifically, soil void ratio is evaluated via a porosity relationship that is a function of vp, vs, and four additional parameters describing the physical properties of the soil (i.e., the Poisson's ratio of the soil skeleton, the bulk modulus and mass density of water, and the mass density of the solid soil particles). In this study, the effectiveness of using high-resolution vp and vs measurements from direct-push crosshole testing to estimate in-situ void ratios is investigated at ten, predominantly clean-sand case history sites in Christchurch, New Zealand, via a comparison with void ratio estimates developed from penetration testing measurements. The seismic-based void ratio estimates are shown to be particularly sensitive to vp, requiring a measurement error of less than 2%. Given the need to make such precise in-situ measurements of vp, the DPCH method is believed to show particular promise for enabling accurate, seismic-based estimates of void ratio in the future.

ОБ ОДНОЙ МОДЕЛИ РАЗБУХАНИЯ ГЛИНИСТОГО СЛАНЦА ВОКРУГ ЦИЛИНДРИЧЕСКОЙ СКВАЖИНЫ

This paper proposes a modified version of the linear theory of poroelasticity, described by the three elastic parameters applied to the shale swelling with aqueous electrolyte. It is assumed that the shale behaves as an isotropic, ideal ionic membrane, and in this case the swelling depends only on the total stress and on the chemical potential of water in the pores of the rock. An analysis of the plane strain around the wellbore has been made.

Poroelastic modeling of cutting rock in pressurized condition

Cutting rock in pressurized conditions is essentially different as compared to the ambient conditions. Under pressurized condition, cutter is not only fragmenting the rock matrix, but also driving pore fluid to flow forward. Due to the coupling of stresses and pore pressure, the pore pressure change caused by cutter will affect effective stresses in the rock and eventually affect mechanical specific energy (MSE). The change in stresses and pore pressure caused by cutter movement are the poroelastic effects in cutting process.To study pressurized cutting process, we developed a cutting model based on the theory of linear poroelasticity. The model can predict coupled stresses and pore pressure in the rock and give a better understanding of the poroelastic effect in the cutting process. To obtain the desired solution, a Fourier Transform is employed. After the analytic solution in Fourier space is obtained, a Discrete Fourier Transform is employed to numerically invert the solution back to the real space. The Mohr-Coulomb failure criterion is also introduced to determine failure of the rock ahead of the cutter. Eventually, the model can predict cutting forces and MSE in pressurized cutting process. The model is also compared to the published experimental data and a good consistency between model and experiment results was found.The influence of hydrostatic pressure and the existence of cuttings on MSE are studied and poroelastic effect during cutting process is also discussed. The results in this paper can be applied to bit-rock interaction in well drilling.

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

Dynamic analysis of a rectangular porous plate resting on an elastic foundation using high-order shear deformation theory

This research deals with the dynamic analysis of a rectangular plate made of porous materials. The porous plate is subjected to a dynamic transverse load and is resting on a Pasternak foundation. Linear poroelasticity theory is used to obtain the Biot formulation of the constitutive equations for the porous material. Also, the Young modulus and density of the porous plate vary in the transverse direction versus the porosity of the plate. Tennessee marble is the porous material that used in this paper. Reddy’s third-order shear deformation theory with five unknowns, the energy method, and Hamilton’s principle are applied to derive the equations of motion of the porous plate. These equations are solved by a differential quadrature method as a numerical method due to five coupled large equations. A detailed numerical study indicates the significant effects of aspect ratio, thickness ratio, boundary conditions, elastic medium, load intensity, and porosity on deflection of the porous plate. Results of this study can be useful to design of pneumatic conveying, handling, and control systems.

A Computational Study of Vocal Fold Dehydration During Phonation.

While vocal fold dehydration is often considered an important factor contributing to vocal fatigue, it still remains unclear whether vocal fold vibration alone is able to induce severe dehydration that has a noticeable effect on phonation and perceived vocal effort. A three-dimensional model was developed to investigate vocal fold systemic dehydration and surface dehydration during phonation. Based on the linear poroelastic theory, the model considered water resupply from blood vessels through the lateral boundary, water movement within the vocal folds, water exchange between the vocal folds and the surface liquid layer through the epithelium, and surface fluid accumulation and discharge to the glottal airway. Parametric studies were conducted to investigate water loss within the vocal folds and from the surface after a 5-min sustained phonation under different permeability and vibration conditions. The results showed that the dehydration generally increased with increasing vibration amplitude, increasing epithelial permeability, and reduced water resupply. With adequate water resupply, a large-amplitude vibration can induce an overall systemic dehydration as high as 3%. The distribution of water loss within the vocal folds was non-uniform, and a local dehydration higher than 5% was observed even under conditions of a low overall systemic dehydration (<1%). Such high level of water loss may severely affect tissue properties, muscular functions, and phonations characteristics. In contrast, water loss of the surface liquid layer was generally an order of magnitude higher than water loss inside the vocal folds, indicating that the surface dehydration level is likely not a good indicator of the systemic dehydration.

Free Vibration of Embedded Porous Plate Using Third-Order Shear Deformation and Poroelasticity Theories

This research aims at studying free vibration of rectangular plate made of porous materials in whichY-foam,G-foam, andCoustoneare used and compared with each other. To obtain the Biot formulation of the constitutive equations for a porous material, linear poroelasticity theory is used. Young modulus and density of porous plate are different in transverse direction versus porosity. In order to increase the accuracy of results in comparison with classical plate and first-order shear deformation theories, Reddy’s theory was utilized in this research. Besides, five coupled equations of motion have been studied using Hamilton’s principle and are solved by differential quadrature method (DQM). Detailed results of this study show the significant effect of aspect ratio, thickness ratio, boundary conditions, and porosity on dimensionless frequency and deflection of porous plate. Results of this study can contribute to the design of pneumatic conveying, handling, and control systems.

A computational study of vocal fold dehydration during phonation

Vocal fold dehydration during phonation is investigated in a continuum model of the vocal folds. Based on the linear poroelastic theory, the model simulates water movement inside the vocal folds during phonation, water exchange between the vocal folds and the surface mucosal layer through the epithelium layer, surface water accumulation and loss to the glottal airway, and water resupply from blood through the lateral boundary. Parametric studies are conducted to investigate water loss within the vocal folds after 5 minutes of phonation at different voice conditions. The results show that with normal water resupply from the blood, water loss within the vocal folds increases with greater vibration amplitude and higher epithelium permeability. At very large vibration amplitudes, the water loss within the vocal folds can be as high as 3%, which may severely affect body functions and possibly phonation characteristics and vocal effort. Reduced water resupply from blood further increases the degree of dehydration. In contrast, water loss in the surface mucosal layer is an order of magnitude higher than water loss within the vocal folds, indicating the surface dehydration level is likely not a good indicator of the degree of dehydration within the vocal folds.

Large Deformations of a Soft Porous Material

Compressing a porous material will decrease the volume of the pore space, driving fluid out. Similarly, injecting fluid into a porous material can expand the pore space, distorting the solid skeleton. This poromechanical coupling has applications ranging from tissue mechanics to hydrogeology. The classical theory of linear poroelasticity captures this coupling by combining Darcy's law with Terzaghi's effective stress and linear elasticity in a linearized kinematic framework. Linear poroelasticity is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. The well-known theory of large-deformation poroelasticity combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework. This theory has been used extensively in biomechanics to model large elastic deformations in soft tissues, and in geomechanics to model large elastoplastic deformations in soils. Here, we first provide an overview and discussion of this theory with an emphasis on the physics of poromechanical coupling. We present the large-deformation theory in an Eulerian framework to minimize the mathematical complexity, and we show how this nonlinear theory simplifies to linear poroelasticity under the assumption of small strain. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of two uniaxial model problems: Fluid outflow driven by an applied mechanical load (the consolidation problem) and compression driven by a steady fluid throughflow. We use these problems to explore the steady and dynamical errors associated with the linear model in both situations, as well as the impact of introducing a deformation-dependent permeability.

2014 Drucker Medal Paper: A Derivation of the Theory of Linear Poroelasticity From Chemoelasticity

The purpose of this brief paper is to present a new derivation of Biot's theory of linear poroelasticity (Biot, M., 1935, “Le Probleḿe de la Consolidation des Matiéres Argileuses Sous une Charge,” Ann. Soc. Sci. Bruxelles,B55, pp. 110–113; Biot, M., 1941, “General Theory of Three-Dimensional Consolidation,” J. Appl. Phys., 12, pp. 155–164; and Biot, M., and Willis, D., 1957, “The Elastic Coefficients of the Theory of Consolidation,” J. Appl. Mech., 24, pp. 594–601) in a modern thermodynamically consistent fashion, and show that it may be deduced as a special case of a more general theory of chemoelasticity.

A nonlinear, transient finite element method for coupled solvent diffusion and large deformation of hydrogels

Hydrogels are capable of coupled mass transport and large deformation in response to external stimuli. In this paper, a nonlinear, transient finite element formulation is presented for initial boundary value problems associated with swelling and deformation of hydrogels, based on a nonlinear continuum theory that is consistent with classical theory of linear poroelasticity. A mixed finite element method is implemented with implicit time integration. The incompressible or nearly incompressible behavior at the initial stage imposes a constraint to the finite element discretization in order to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition for stability of the mixed method, similar to linear poroelasticity as well as incompressible elasticity and Stokes flow; failure to choose an appropriate discretization would result in locking and numerical oscillations in transient analysis. To demonstrate the numerical method, two problems of practical interests are considered: constrained swelling and flat-punch indentation of hydrogel layers. Constrained swelling may lead to instantaneous surface instability for a soft hydrogel in a good solvent, which can be regulated by assuming a stiff surface layer. Indentation relaxation of hydrogels is simulated beyond the linear regime under plane strain conditions, in comparison with two elastic limits for the instantaneous and equilibrium states. The effects of Poisson’s ratio and loading rate are discussed. It is concluded that the present finite element method is robust and can be extended to study other transient phenomena in hydrogels.

Leaf hydraulics I: Scaling transport properties from single cells to tissues

In leaf tissues, water may move through the symplast or apoplast as a liquid, or through the airspace as vapor, but the dominant path remains in dispute. This is due, in part, to a lack of models that describe these three pathways in terms of experimental variables. We show that, in plant water relations theory, the use of a hydraulic capacity in a manner analogous to a thermal capacity, though it ignores mechanical interactions between cells, is consistent with a special case of the more general continuum mechanical theory of linear poroelasticity. The resulting heat equation form affords a great deal of analytical simplicity at a minimal cost: we estimate an expected error of less than 12%, compared to the full set of equations governing linear poroelastic behavior. We next consider the case for local equilibrium between protoplasts, their cell walls, and adjacent air spaces during isothermal hydration transients to determine how accurately simple volume averaging of material properties (a ‘composite’ model) describes the hydraulic properties of leaf tissue. Based on typical hydraulic parameters for individual cells, we find that a composite description for tissues composed of thin walled cells with air spaces of similar size to the cells, as in photosynthetic tissues, is a reasonable preliminary assumption. We also expect isothermal transport in such cells to be dominated by the aquaporin-mediated cell-to-cell path. In the non-isothermal case, information on the magnitude of the thermal gradients is required to assess the dominant phase of water transport, liquid or vapor.

Kinetics of swelling under constraint

Swellable elastomers are used to seal flow channels in oilfield operations. After sealing, the elastomers are constrained triaxially, and a contact load builds up between the elastomers and surrounding rigid materials. For these applications, the ability to predict the evolution of the contact load is important. This work introduces an experimental setup to measure the contact load as a function of time. The experimental data are well represented by a simple time-relaxation equation derived from the linear poroelastic theory, enabling a determination of the effective diffusivity of solvent inside the elastomers.

Mechanical Stresses Induced by Evaporation in Consolidated Colloidal Suspensions of Hard Particles. Poroelasticity Theory Versus Experiments

Drying of colloidal suspensions often yields intriguing, complex and generally undesirable crack patterns. Despite a consensus on the main underlying physical mechanisms that lead to their formation, quantitative prediction seems at present out of reach. Notably, a prerequisite for this is the determination of the mechanical stress field. Here, our point of view is to provide a sound and well-established continuum mechanics model that can be used safely and easily by a large community and particularly by the fracture mechanics one. We show by comparison with beam deflection experiments, that (i) the use of Biot’s linear poroelasticity theory allows for the prediction of the stresses that lead to the crack formation and that (ii) the setup allows for the identification of some poroelastic constants. This evidenced that the hypothesis of incompressible constituents is reasonable for the nanolatex suspensions considered here, greatly simplifying the constitutive equations and reducing the number of material constants from four to two.

Mathematical problems in the coupled linear theory of bone poroelasticity

This paper concerns with the quasi-static coupled linear theory of bone poroelasticity for materials with double porosity and some basic results of the classical theories of elasticity and thermoelasticity are generalized. The system of equations of this theory is based on the equilibrium equations, conservation of fluid mass, the effective stress concept and Darcy’s law for material with double porosity. The Green’s formulas in the considered theory are obtained, the formulas of Somigliana type integral representations of regular vector and regular (classical) solutions are presented. The uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. The single-layer, double-layer and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral method) and the theory of singular integral equations.

Generalized poroelastic wellbore problem

SUMMARYThis paper presents a novel analytical solution to the transient, z‐dependent, and asymmetric problem of an infinite wellbore drilled into a fluid‐saturated porous medium. The formulations are based on Biot's linear theory of poroelasticity, in which the dependency of poroelastic field variables to spatial coordinates as well as time domain is considered in the most general form. This gives flexibility to the solution in cases that cannot be analyzed using the conventional plane strain or symmetric models. One such case is when calculating the stress variations around an inclined wellbore where the far‐field stresses are acting over a finite vertical section. The results of our solution to this case with a three‐dimensional state of far‐field stress are used to analyze the stability of inclined wellbores passing through abnormally stressed formations.The presented solution is capable of finding expressions for fundamental solutions with stress or flow boundary conditions at the wellbore. These solutions are here adopted to analyze the pressure disturbances generated by multiprobe formation tester, a standard wireline device that is designed for downhole fluid sampling as well as estimating the directional permeabilities of subsurface earth formations. A comparison with the conventional solution for the relevant pressure diffusion equation indicates that the poroelastic effect is relatively significant in relation to the transient response of the pore pressure. Further, it is shown that the finite dimensions of sink probe would, to a great extent, contribute to the formation's pore pressure variations at its immediate proximity. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Mathematical Modeling of Tidal Effects in Groundwater

This paper presents a special use of the linear poroelasticity theory to describe tidally induced groundwater oscillations. Two models of oscillation inducing mechanism make use of this theory to predict groundwater level fluctuations. The numerical solutions of both models are presented and compared with well water level measurement obtained in Police Basin, north-east Bohemia.

A fluid pressure and deformation analysis for geological sequestration of carbon dioxide

We present a fluid pressure and deformation analysis for geological sequestration of carbon dioxide based on a simplified hydro-mechanical model. This model includes the geomechanical part that relies on the theory of linear elasticity, while the fluid flow is based on the Darcy's law. Two parts are coupled together using the linear poroelasticity theory. For a typical geological sequestration in a semi-infinite geometry with diminishing pressure and deformation fields at infinity, the Helmholtz decomposition can be applied to the displacement vector. Hence, the flow equation can be decoupled from the equation of linear elasticity. Solutions for fluid pressure were obtained for this typical scenario and solutions for ground deformation were obtained using the method of Green's function. Finally, solutions were compared against numerical results using a finite element method for aquifers with two different thicknesses. General agreement can be obtained between analytical and numerical solutions. The model is useful in estimating the temporal and spatial variation of fluid pressure and the mechanical deformation during the entire injection period.