Biot's Consolidation Research Articles

Dr Olivier Coussy, Scientist and Engineer of the Mechanics and Physics of Porous Materials (1953–2010)

Dr Olivier Coussy, Scientist and Engineer of the Mechanics and Physics of Porous Materials (1953–2010)

A fully coupled 3-D mixed finite element model of Biot consolidation

The numerical solution to the Biot equations of 3-D consolidation is still a challenging task because of the ill-conditioning of the resulting algebraic system and the instabilities that may affect the pore pressure solution. Recently new approaches have been advanced based on mixed formulations. In the present paper a fully coupled 3-D mixed finite element model is developed with the aim at alleviating the pore pressure numerical oscillations at the interface between materials with different permeabilities. A solution algorithm is implemented that takes advantage of the block structure of the discretized problem. The proposed model is verified against well-known analytical solutions and successfully experimented with in realistic applications of soil consolidation.

Comparison between Coupled and Uncoupled Consolidation Analysis of a Rigid Sphere in a Porous Elastic Infinite Space

Linear consolidation analyses are usually treated either by means of Terzaghi-Rendulic uncoupled theory or Biot's consolidation theory. In this note, the problem of consolidation displacements around an axially loaded sphere was considered. It is demonstrated that both the uncoupled analysis and the coupled analysis give the same governing equation for pore fluid pressure dissipation with time. A simplified procedure for deriving transient strain components is illustrated. A general solution for time-dependent displacements is obtained using uncoupled consolidation analysis. Close agreement is evident between the new approximate uncoupled analysis solution and the existing coupled analysis solution with a maximum error of less than 0.5%.

Numerical simulation of geofluid focusing and penetration due to hydraulic fracture

Ore body formation and mineralization are the consequence of transport, focusing and mixing processes of geofluids from the deep earth into the upper crust of the earth. These processes largely depend on geological structures, particularly heterogeneity such as cracks and faults. Both geological stress and geofluid-induced hydraulic pressure can create and alter the heterogeneity, thus changing the transport, focusing and mixing patterns. As a preliminary study, this paper investigates the intrusion and focusing of geofluids through simulation of the hydraulic fracturing process. First, the deformation of porous media and geofluid flow are coupled through the Biot's consolidation theory to consider both the elastoplasticity of porous media and the deformation induced evolution of permeability. Second, the crack induced anisotropy of stiffness and permeability is described, on the basis of a smeared theory and an equivalent anisotropic medium, to establish a criterion for tensile crack initiation and propagation at an element level. Finally, the resulting nonlinear equations are solved by a coupled algorithm of the meshless method and the finite element method (FEM). Numerical examples indicate that the proposed method is able to simulate the stress-induced crack propagation, geofluid intrusion and focusing in fluid-saturated porous media. It thus provides a potential tool for simulating the detailed mineralization process associated with the geological stress and geofluid intrusion.

Probabilistic Analysis of Coupled Soil Consolidation

Coupled Biot consolidation theory was combined with the random finite-element method to investigate the consolidation behavior of soil deposits with spatially variable properties in one-dimensional (1D) and two-dimensional (2D) spaces. The coefficient of volume compressibility ( mv ) and the soil permeability (k) are assumed to be lognormally distributed random variables. The random fields of mv and k are generated by the local average subdivision method which fully takes account of spatial correlation, local averaging, and cross correlations. The generated random variables are mapped onto a finite-element mesh and Monte Carlo finite-element simulations follow. The results of parametric studies are presented, which describe the effect of the standard deviation, spatial correlation length, and cross correlation coefficient on output statistics relating to the overall “equivalent” coefficient of consolidation. It is shown that the average degree of consolidation defined by excess pore pressure and settlemen...

Embankments on Soft Soil Reinforced with Stone Columns: Numerical Analysis and Proposal of a New Design Method

A parametric study of an embankment on soft soils reinforced with stone columns is performed using a computer program based on the finite element method. The cylindrical unit cell formulation is used by modeling one column and its surrounding soft soil with confined axisymmetric behaviour. The computer program incorporates the Biot consolidation theory (coupled formulation of the flow and equilibrium equations) with constitutive relations simulated by the p–q–θ critical state model. The following parameters are analysed: the replacement area ratio, the deformability of the column material, the thickness of the soft soil, the deformability of the fill and the friction angle of the column material. Based on the results of this study, a new design method is proposed, relating the settlement improvement factor to the two factors that revealed major influence: the replacement area ratio and the ratio between the deformability of the soft soil and the deformability of the column material.

Viscoplastic constitutive approach for rate-sensitive structured clays

This paper extends an existing elastic–viscoplastic (EVP) constitutive model using a state-dependent viscosity parameter to describe the engineering response of undisturbed structured clay. The term structure refers to the effects of fabric and weak cementation bonds between clay particles. The extended constitutive model is coupled with the Biot consolidation theory and is formulated to describe the intrinsic or unstructured response of clay using overstress viscoplasticity, an elliptical cap yield surface, Drucker–Prager failure envelope, and a hardening law from critical state theory. The clay structure is mathematically accounted for by assuming that the initial fluidity of structured clay at yield and failure is very low and that the fluidity increases with increasing plastic strain. This process is usually referred to as “destructuration.” The formulation is evaluated using Saint-Jean-Vianney (SJV) clay by comparing calculated and measured behaviour during consolidated isotropically undrained triaxial compression, triaxial creep, and constant rate-of-strain Ko′-consolidation tests. The comparisons indicate that the EVP constitutive model can describe most of the rate-sensitive behaviour of SJV clay during both drained and undrained laboratory tests involving either constant-volume shear or predominantly volumetric compression of the soil skeleton.

Two formulations for dynamic response of a cylindrical cavity in cross‐anisotropic porous media

AbstractTwo formulations for calculating dynamic response of a cylindrical cavity in cross‐anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u–w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd.

A local boundary integral-based meshless method for Biot's consolidation problem

Traditional numerical techniques such as FEM and BEM have been successfully applied to the solutions of Biot's consolidation problems. However, these techniques confront some difficulties in dealing with moving boundaries. In addition, pre-designing node connectivity or element is not an easy task. Recently, developed meshless methods may overcome these difficulties. In this paper, a meshless model, based on the local Petrov–Galerkin approach with Heaviside step function as well as radial basis functions, is developed and implemented for the numerical solution of plane strain poroelastic problems. Although the proposed method is based on local boundary integral equation, it does not require any fundamental solution, thus avoiding the singularity integral. It also has no domain integral over local domain, thus largely reducing the computational cost in formulation of system stiffness. This is a truly meshless method. The solution accuracy and the code performance are evaluated through one-dimensional and two-dimensional consolidation problems. Numerical examples indicate that this meshless method is suitable for either regular or irregular node distributions with little loss of accuracy, thus being a promising numerical technique for poroelastic problems.

Time-Dependent Deformation Behavior of Jiangsu Marine Clay

In this article, the mechanical behavior of a Jiangsu marine clay was investigated by drained triaxial tests, traixial rheological tests, and one-dimensional compression and swelling tests. A visco-plastic model, the Bingham model combining two yield surfaces model, was proposed to describe the time-dependent deformation behaviors of the marine clay. The governing equation of Biot's consolidation theory for the visco-plastic soil is solved using a finite element code which incorporates the visco-plastic model. Using the finite element method, settlements of a typical embankment on the Lianxu expressway in China are calculated. Settlement calculations using the visco-plastic model are in agreement with the measured settlements in the field. The results demonstrate that the visco-plastic model is appropriate for calculating the visco-plastic deformations of Jiangsu marine clay. Theoretical and experimental studies show that the visco-plastic deformation of Jiangsu marine clay is substantial.

Numerical simulation of two‐dimensional flow and compaction during the consolidation of laminated composites

AbstractA new special finite element formulation and code are developed to simulate the two‐dimensional flow and compaction process during the autoclave processing of complex fiber‐reinforced thermosetting composite laminates. The numerical model is based on the Biot's consolidation principle, the fluid continuity equation, and Darcy's law. The simulation is performed for Hercules AS4/3501‐6 laminates and the predicted results are in good agreement with the results available in the literature. The laminate thickness and fiber volume fraction distribution of cured laminates are investigated from both the experiments and the simulations for T700S/5228 laminates. The simulations and experimental results are in good agreement for all the studied cases. A significantly uneven degree of consolidation along the laminate thickness direction is observed, and it is necessary to optimize the cure cycle to get uniformly compact and good quality laminates. POLYM. COMPOS., 2008. © 2008 Society of Plastics Engineers

Cut Slopes in Clayey Soils: Consolidation and Overall Stability by Finite Element Method

When a cut slope in a saturated clay is undertaken, a transient water flow occurs and stress transferences from the water to the soil skeleton take place in time (consolidation). Mainly in strongly overconsolidated clays, these stress transferences may determine swelling of soil and therefore reduction of its shear strength in time. However, the lowering of the water level associated to the cut increases effective mean stress, which may therefore counterbalance the above-mentioned effect. In the paper, the behaviour of a cut slope in an overconsolidated clay is analysed by a finite element program that incorporates the Biot consolidation theory (coupled analysis), with constitutive relations simulated by the p–q–θ critical state model. In addition, the variation in time of the overall stability is assessed with a computer program that uses the finite element results and formulations of the critical state soil mechanics. In order to achieve a more complex geotechnical interpretation of the problem, the analysis in time of the excess pore pressures, effective stresses, displacements and stress levels is also presented. Finally, comparisons of stability results are analysed by changing some parameters, namely the problem geometry (weight of excavated soil) and the over-consolidation ratio of the clay.

Identifying parameters controlling soil delayed behaviour from laboratory and in situ pressuremeter testing

AbstractThe aim of this paper is to present a methodology for identifying the soil parameters controlling the delayed behaviour from laboratory and in situ pressuremeter tests by using an elasto‐viscoplastic model (EVP‐MCC) based on Perzyna's overstress theory and on the elasto‐plastic Modified Cam Clay model. The influence of both the model parameters and the soil permeability was studied under the loading condition of pressuremeter tests by coupling the proposed model equations with Biot's consolidation theory. On the basis of the parametric study, a methodology for identifying model parameters and soil permeability by inverse analysis from three levels of constant strain rate pressuremeter tests was then proposed and applied on tests performed on natural Saint‐Herblain clay. The methodology was validated by comparing the optimized values of soil parameters and the values of the same parameters obtained from laboratory test results, and also by using the identified parameters to simulate other tests on the same samples. The analysis of the drainage condition and the strain rate effect during a pressuremeter test demonstrated the coupled influence of consolidation and viscous effects on the test results. The numerical results also showed that the inverse analysis procedure could successfully determine the parameters controlling the time‐dependent soil behaviour. Copyright © 2008 John Wiley & Sons, Ltd.

Wave scattering in cross-anisotropic porous media around the cavities and inclusions

A method is presented for evaluating dynamic response of a cylindrical cavity or inclusion in an infinite cross-anisotropic porous medium. The basis of the method is the Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u - w formulation of Biot's equations are solved. The stress, displacement, and pore pressure fields are evaluated in the vicinity of the cylinder in complex plane. Also the effects of many parameters ( k h / k v , E h / E v , G v / E v , ν hh ) on the responses are evaluated by numerical examples.

Numerical stabilization of Biot's consolidation model by a perturbation on the flow equation

AbstractIn this paper a stabilized finite element scheme for the poroelasticity equations is proposed. This method, based on the perturbation of the flow equation, allows us to use continuous piecewise linear approximation spaces for both displacements and pressure, obtaining solutions without oscillations independently of the chosen discretization parameters. The perturbation term depends on a parameter which is established in terms of the mesh size and the properties of the material. In the one‐dimensional case, this parameter is shown to be optimal. Some numerical experiments are presented indicating the efficiency of the proposed stabilization technique. Copyright © 2008 John Wiley & Sons, Ltd.

Two-Phase Shrinking Porous Media Drying: A Modeling Approach Including Liquid Pressure Gradients Effects

This article deals with the drying modeling of a saturated deformable porous media (solid + fluid). This modeling takes into account the mechanical behavior of the product as well as its initial conditions such as its moisture content and geometry. The effect of fluid pressure gradient on the fluid transfer is described by using Darcy's law, and the fluid compressibility is characterized by its bulk modulus of elasticity. The solid phase is supposed to be incompressible and the porous media permanently diphasic (ideal shrinkage hypothesis). The description of the strong thermo-hydro-mechanical coupling that takes place within the material during drying is taken into account in the light of Biot's consolidation theory by adopting Terzaghi's effective stress principle. The numerical resolution is performed by FEM and ALE methods. A comparison between experimental results (convective drying of an alumina gel) and numerical results improves the ability of this modeling to describe thermo-hydro-mechanical coupling. It further highlights a current lack of precision in the determination of the thermophysical parameters values that control the model.

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

AbstractFinite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 John Wiley & Sons, Ltd.

Analysis of A Braced Excavation In Soft Soils Considering The Consolidation Effect

The studies of excavations in soft clayey soils are normally based on undrained total stress analyses. A better approach consists of taking into account the effects of consolidation during the excavation-bracing process and after the completion of the construction by means of coupled finite element analyses in effective stresses. In this paper, the geotechnical behaviour of a braced excavation in the soft soils of San Francisco (USA) is analysed, both during and after the construction period. Numerical analyses are performed with a finite element program, which incorporates the Biot consolidation theory (coupled formulation of the water flow and equilibrium equations) and soil constitutive relations simulated by the p-q-θ critical state model. Numerical results are compared with field results.

A poroelastic model of transcapillary flow in normal tissue

Starling's seminal work on the absorption of fluids from connective tissue spaces (and Starling's hypothesis that the energy for transcapillary flow lies in the difference between hydrostatic and osmotic pressures across the capillary wall) has long formed the basis of much of experimental physiology. Related recent experimental evidence points to a more active role of the interstitium in controlling interstitial fluid pressure (IFP) which has significant implications for clinical oncology. In light of these considerations, it is clearly of importance to reconsider the relationship between IFP and transcapillary transport, in addition to the regulation of IFP in normal tissue. In this paper, we adopt the Michel–Weinbaum viewpoint on the locality of Starling forces and model the capillary wall as a poroelastic solid using Biot's consolidation theory. However, the incorporation of the Michel–Weinbaum hypothesis requires an extension of Darcy's law to include the effects of oncotic pressure in the mechanism of filtration through the capillary wall. A unique feature of the model of transcapillary flow developed here is its ability to predict the stress and strain distribution across the capillary wall, which to our knowledge has not been attempted previously. We are optimistic that, with rapidly advancing technological capabilities, experimentalists will soon be able to test many of the model predictions.

A new mixed finite element method for poro‐elasticity

AbstractDevelopment of robust numerical solutions for poro‐elasticity is an important and timely issue in modern computational geomechanics. Recently, research in this area has seen a surge in activity, not only because of increased interest in coupled problems relevant to the petroleum industry, but also due to emerging applications of poro‐elasticity for modelling problems in biomedical engineering and materials science. In this paper, an original mixed least‐squares method for solving Biot consolidation problems is developed. The solution is obtained via minimization of a least‐squares functional, based upon the equations of equilibrium, the equations of continuity and weak forms of the constitutive relationships for elasticity and Darcy flow. The formulation involves four separate categories of unknowns: displacements, stresses, fluid pressures and velocities. Each of these unknowns is approximated by linear continuous functions. The mathematical formulation is implemented in an original computer program, written from scratch and using object‐oriented logic. The performance of the method is tested on one‐ and two‐dimensional classical problems in poro‐elasticity. The numerical experiments suggest the same rates of convergence for all four types of variables, when the same interpolation spaces are used. The continuous linear triangles show the same rates of convergence for both compressible and entirely incompressible elastic solids. This mixed formulation results in non‐oscillating fluid pressures over entire domain for different moments of time. The method appears to be naturally stable, without any need of additional stabilization terms with mesh‐dependent parameters. Copyright © 2007 John Wiley & Sons, Ltd.