Biot's Consolidation Equations Research Articles

3-D numerical study of offshore tripod wind turbine pile foundation on wave-induced seabed response

Investigation analysis of wave-seabed-structure interaction (WSSI) is a pre-requisite in every marine engineering design and construction due to the huge failure of offshore structures every year, resulting in financial losses. In this study, an integrated 3-D numerical model is established to investigate the wave-induced oscillatory seabed response around the offshore substructure foundation of the tripod support pile. In the integrated numerical model, the Reynolds-Average Navier-Stokes (RANS) equations with k−ε turbulence closure model for the mean fluid flow is applied as the governed equation for the fluid computation, while the Biot consolidation equations are applied as the governing equation for the porous seabed substructure foundation model. The integrated numerical model is validated against physical experimental data from other previous works; to demonstrate that the present numerical model has the capability and capacity of simulating the WSSI around the offshore tripod pile structure foundation, which indicated good results. The results from the wave-induced liquefaction around the tripod support pile show that liquefaction depth at the offshore area and the lateral area of the upstream axillary pile leg tends to be larger than the two axillary pile legs at the lee area.

The Biot–Stokes coupling using total pressure: Formulation, analysis and application to interfacial flow in the eye

We consider a multiphysics model for the flow of Newtonian fluid coupled with Biot consolidation equations through an interface, and incorporating total pressure as an unknown in the poroelastic region. A new mixed-primal finite element scheme is proposed solving for the pairs fluid velocity–pressure and displacement–total poroelastic pressure using Stokes-stable elements, and where the formulation does not require Lagrange multipliers to set up the usual transmission conditions on the interface. The stability and well-posedness of the continuous and semi-discrete problems are analysed in detail. Our numerical study is framed in the context of applicative problems pertaining to heterogeneous geophysical flows and to eye poromechanics. For the latter, we investigate different interfacial flow regimes in Cartesian and axisymmetric coordinates that could eventually help describe early morphologic changes associated with glaucoma development in canine species.

Influence and evaluation of saturated soil shield construction on existing highway

Abstract In order to study the shield construction under an existing highway, the initial displacement and the excess pore water pressure solution from Biot consolidation equation were used to derive the analytical solutions of the vertical deformation and pore water pressure of saturated soil caused by the frontal friction and side friction of the cutterhead. In addition, by introducing the layered method and combining it with other theoretical analytical equations, the expressions of total vertical deformation and total pore water pressure caused by tunnel excavation with different overlying materials were obtained. The simulation results on an engineering project showed that the angle α between the direction of the road and the propulsion axis of the shield had a significant influence on the surface settlement of the road. When the angle α was increased, the settlement curve had higher variation; but the variation would not exceed the maximum settlement value above the shield axis. When the road was perpendicular to the shield axis, a critical point of the road bulging and settlement was formed above the incision. Due to the grinding resistance of the cutterhead, the pore water pressure under the roadbed was distributed asymmetrically on both sides of the shield axis.

Time-dependent interaction between superstructure, raft and layered cross-anisotropic viscoelastic saturated soils

By introducing the elastic-viscoelastic correspondence principle and the integral transform technique, the extended precise integral solution to Biot's consolidation equation is derived for cross-anisotropic viscoelastic saturated soils based on the Merchant model. A displacement-time solution for a slowly changing load is obtained based on the above solution in the Laplace transformed domain, which is taken as the kernel function for the boundary element method (BEM). The Mindlin plate model is established by 8-node isoparametric finite element method (FEM), and the substructure condensation technique is employed to couple the stiffness matrices of the superstructure and the plate. With the BEM-FEM coupling method, a semi-analytical and semi-numerical solution is proposed for the interaction between layered cross-anisotropic viscoelastic saturated soils and raft foundation considering the stiffness contributions of the superstructure. Numerical examples are performed to study the influences of viscoelastic parameters, cross-anisotropic parameters, raft thickness and superstructure stiffness on the time-dependent behavior of the settlement and bending moment for the raft.

Numerical Investigation of Stone Column Improved Ground for Mitigation of Liquefaction

Installation of a stone column in loose soil is one of the robotics ground enhancement techniques that reduce the risk liquefaction and related ground deformation. However, the experimental investigation is costly and it is time-consuming to check the feasibility of this technique for the evaluation of ground subsidence and fluctuations in excess pore water pressure concerning different area ratios of a high modulus column. This paper presents a numerical investigation into shallow foundations underlain by a stone column as a liquefaction countermeasure. The Biot's consolidation equations are coupled with stress equilibrium equations in finite element modeling for cyclic loading conditions. The critical state model based on generalized plasticity theory was applied to model the nonlinear behavior of soil. The unknown displacement and pore pressure are evaluated using code written in FORTRAN 90. The granular column installation effect is also considered to predict displacements and excess pore pressure (EPP) at each time step. With the introduction of a granular column, considerable reductions were observed in displacements and EPP. The results underline the effectiveness of granular column installation as an effective measure in the mitigation of liquefaction.

Numerical analysis of seabed dynamic response in vicinity of mono-pile under wave-current loading

Pile foundations have been widely used in offshore engineering. In this study, a three-dimensional numerical model was used to investigate the seabed response around a mono-pile under wave-current loading. Reynolds-averaged Navier-Stokes equations were used to simulate the flow field, and Biot's consolidation equations were used for simulating the response of a porous seabed. The pore water pressure within soil and the effective stress along the depth of the seabed were simulated for various current velocities, with currents traveling either along or against the wave. Results indicate that the current has a significant effect on the effective stress and the pore water pressure distributions, which increases with the current velocity, and that the current traveling against the wave increases the liquefaction depth of the porous seabed.

2D Numerical Study of the Stability of Trench under Wave Action in the Immersing Process of Tunnel Element

The evaluation of the trench stability under the action of ocean waves is an important issue in the construction of an immersed tunnel. In this study, a two-dimensional coupling model of a wave-seabed-immersed tunnel is proposed for the dynamic responses of a trench under wave action in the immersing process of tunnel elements. The porous seabed is characterized by Biot consolidation equations. The k − ε model and RANS equation are adopted to achieve the flow field simulation, and the level set method (LSM) is used to capture the free surface between the water and air. The proposed numerical model is verified using the experimental data and analytical results. Then, the transient liquefaction and shear failure in the vicinity of the trench are discussed at two different conditions, namely, after the foundation groove is excavated and after the tunnel element is placed. The pore pressure amplitude on the weather side slope is demonstrated to be significantly smaller than that on the lee side slope. Also, the distribution of the surrounding flow field and pressure field change dramatically after the tunnel element is settled, leading to the significant changes of seabed stability.

Attenuation mechanisms in fractured fluid‐saturated porous rocks: a numerical modelling study

ABSTRACTSeismic attenuation mechanisms receive increasing attention for the characterization of fractured formations because of their inherent sensitivity to the hydraulic and elastic properties of the probed media. Attenuation has been successfully inferred from seismic data in the past, but linking these estimates to intrinsic rock physical properties remains challenging. A reason for these difficulties in fluid‐saturated fractured porous media is that several mechanisms can cause attenuation and may interfere with each other. These mechanisms notably comprise pressure diffusion phenomena and dynamic effects, such as scattering, as well as Biot's so‐called intrinsic attenuation mechanism. Understanding the interplay between these mechanisms is therefore an essential step for estimating fracture properties from seismic measurements. In order to do this, we perform a comparative study involving wave propagation modelling in a transmission set‐up based on Biot's low‐frequency dynamic equations and numerical upscaling based on Biot's consolidation equations. The former captures all aforementioned attenuation mechanisms and their interference, whereas the latter only accounts for pressure diffusion phenomena. A comparison of the results from both methods therefore allows to distinguish between dynamic and pressure diffusion phenomena and to shed light on their interference. To this end, we consider a range of canonical models with randomly distributed vertical and/or horizontal fractures. We observe that scattering attenuation strongly interferes with pressure diffusion phenomena, since the latter affect the elastic contrasts between fractures and their embedding background. Our results also demonstrate that it is essential to account for amplitude reductions due to transmission losses to allow for an adequate estimation of the intrinsic attenuation of fractured media. The effects of Biot's intrinsic mechanism are rather small for the models considered in this study.

Wave induced instantaneously-liquefied soil depth in a non-cohesive seabed

Instantaneous liquefaction can be induced by the upward seepage force generated in a sandy seabed under wave troughs. The wave-induced pore-pressure distribution in the non-cohesive seabed with an instantaneously-liquefied layer is investigated analytically. Based on the analytical solution to Biot's consolidation equations for the response of a poro-elastic bed to water waves, the expressions of liquefied soil depth are derived with modifying the criterion for instantaneous liquefaction, which is verified with existing offshore field observations and multi-scale numerical simulations. Analytical investigation indicates that, in the instantaneously-liquefied soil layer, the buoyant weight of soil is essentially balanced by upward seepage force into a quasi-static state. Underneath the fully liquefied layer, the effective stress can be remarkably reduced. An excess pore-pressure ratio is then proposed to quantitatively evaluate the instantaneous liquefaction degree and its corresponding influential depth. Parametric studies indicate that, for a certain excess pore-pressure ratio, the influential depth decreases with increasing saturation degree, permeability of soil or water depth, but increases with increasing wave height. The influential depth of a non-cohesive seabed under waves increases significantly with reducing the threshold value of excess pore-pressure ratio, which may have much influence on the stability design for submarine structures.

Numerical modelling of liquefaction in loose sand deposits subjected to ocean waves

The failure of marine structures is often attributed to liquefaction in loose sand deposits that are subjected to ocean waves. In this study, a two-dimensional integrated numerical model is developed to characterize the liquefaction behaviours of loosely deposited seabed foundations under various types of ocean waves. In the present model, Reynolds-Averaged Navier–Stokes (RANS) equations are used to simulate the surface wave motion, and Biot's consolidation equations are used to link the solid-pore fluid interactions in a porous medium. A poro-elasto-plastic solution is used to reproduce foundation behaviour under cyclic shearing. Unlike previous investigations, both oscillatory and residual soil responses were considered; they are coupled in an instantaneous approach. Verification of the model results to the previous centrifugal wave tests is carried out, obtaining fairly good agreement. Numerical examples show that foundation behaviour under various types of wave loading, particularly standing waves or a solitary wave, embodies a completely two-dimensional process in terms of residual pore pressure development. The parametric studies demonstrate that liquefaction caused by the build-up of pore pressures is more likely to occur in loosely deposited sand foundations with poor drainage and under large waves.

A simplified quasi-static analysis of wave-induced residual liquefaction of seabed around an immersed tunnel

Evaluation of wave-induced soil response is particularly important for coastal geotechnical engineers involved in the foundation design for marine structures such as immersed tunnels, breakwaters, platforms etc. In this study, a finite element model for the wave-induced soil response around an immersed tunnel is developed, in which the seabed is treated as a porous medium and characterized by Biot's consolidation equations, while the tunnel is considered as a single-phase medium and behaves under a linear elastic law. A new conceptual pore-pressure buildup model is developed to investigate the wave-induced residual liquefaction around the tunnel due to the accumulation of pore pressures under cyclic shearing. The consolidation analysis of seabed foundation under the gravitational forces including the body forces of the structure is pre-assessed and incorporated. It is demonstrated from numerical results that the development of residual pore pressure is asymmetric in nature around the tunnel, which may result in the uneven liquefaction depth and deformation in the seabed near the bottom of the tunnel, thus compromising the stability of the structure. Parametric studies show that wave-induced residual liquefaction is more likely to occur in the shallow seabed with lower relative density and permeability under the wave loading with higher wave height and longer period.

Numerical study on loosely deposited foundation behavior around a composite breakwater subject to ocean wave impact

In this study, an integrated model is developed to investigate the potential risk for failure of caisson-type breakwaters caused by liquefaction in its loosely deposited sand foundations. In the present model, the Volume-Averaged Reynolds Averaged Navier-Stokes (VARANS) equations are used for solving flow inside and outside the porous media, while Biot's consolidation equation is used for linking the soil skeleton-pore fluids interactions. A new developed poro-elastoplastic model is implemented to reproduce the behavior of loose sand foundation under cyclic shearing. Verification of the present numerical framework includes (i) comparison with the laboratory experiments in terms of water surface levels around the structure, and the fluctuating component of the pore water pressure inside the rubble mound and seabed foundation; and (ii) comparison with the geotechnical centrifugal wave tests for the wave-induced residual component of pore water pressures in a porous seabed. Based on the present model, the hydrodynamic process and the associated dynamic foundation behavior involved in the interactions between the ocean waves, a composite breakwater and its loose sand foundations was performed and analyzed. Particular focus is put on the residual pore pressure development and the resulting liquefaction around the structure under combined actions of both wave and structure rocking motions. Based on the numerical examples presented, under combined actions of both wave and caisson motions, liquefaction starts on both edges of the breakwater and spreads sideways until reaching an equilibrium state where the maximum liquefaction depth can be observed. The liquefied zone around the breakwater is asymmetric in nature, and the maximum depth of soil liquefaction appears at the seaward side of the structure. Furthermore, the parametric study indicates that the maximum liquefaction depth around the composite breakwater increases with increasing wave steepness (H/L0) and relative water depth (d/L0) and decreasing soil permeability (ks).

Study on aging of single pile in soft soil foundation

In this paper, the single pile under load is analyzed by using the Biot consolidation equation and the modified method of the coupling of the Komala - Huang model. The consolidation of soil is simulated by Biot consolidation theory, and the whole process of the deformation of single pile is presented in the paper by means of the modified Komala - Huang model. Some useful conclusions are obtained by analyzing the variation of the pore pressure and the settlement of the soil under the action of load.

Cyclic characterization of wave-induced oscillatory and residual response of liquefiable seabed

The paper presents a robust modeling method for a fully coupled effective stress analysis of the wave-induced liquefaction scenarios, based on the Biot consolidation theory. In this context, emphasis is placed on the implementation of a well-calibrated cyclic soil model and an empirical shear-volume coupling equation, which links the increment of volumetric strain per cycle of wave with the shear strain occurring during that particular cycle. Unlike most of the previous investigations using the amplitude of shear stress over the wave period (or the phase-resolved oscillatory shear stresses) as the source term for calculating the residual pore pressure, in this study, the source term is related to the rate of plastic volumetric deformation and implemented into the Biot consolidation equation to consider both generation and partial dissipation of excess pore pressure during wave propagation. Then, the proposed modeling method is incorporated into a dynamic finite difference analysis procedure. Model calibrations are carried out in terms of individual soil element behavior and seabed response under progressive waves, respectively. Overall good agreement demonstrates the reliability of the modeling method for the prediction of wave-induced seabed response. Finally, the paper highlights the potentials of proposed modeling framework to simulate the basic features of wave-induced seabed response (i.e., the coupling of progressive buildup of pore pressure and cyclic softening of soil skeleton), by means of numerical examples. The obtained results show that the cyclic behavior of liquefiable seabed under wave actions can be well captured by the proposed model.

Numerical analysis of road pavement thermal deformability, based on Biot viscoelastic model of porous medium

Abstract The following study presents numerical calculations for establishing the impact of temperature changes on the process of distortion of bi-phase medium represented using Biot consolidation equations with Kelvin–Voigt rheological skeleton presented, on the example of thermo-consolidation of a pavement of expressway S17. We analyzed the behavior of the expressway under the action of its own weight, dynamic load caused by traffic and temperature gradient. This paper presents the application of the Biot consolidation model with the Kelvin–Voigt skeleton rheological characteristics and the influence of temperature on the deformation process is taken into account. A three-dimensional model of the medium was created describing the thermal consolidation of a porous medium. The 3D geometrical model of the area under investigation was based on data obtained from the land surveying and soil investigation of a 200 m long section of the expressway and its shoulders.

Parametric study of the wave-induced residual liquefaction around an embedded pipeline

Wave-induced liquefaction in a porous seabed around submarine pipeline may cause catastrophic consequences such as large horizontal displacements of pipelines on the seabed, sinking or floatation of buried pipelines. Most previous studies in relation to the wave and seabed interactions with embedded pipeline dealt with the wave-induced instaneous seabed response and possible resulting momentary liquefaction (where the soil is liquefied instantaneously during the passage of a wave trough), using theory of poro-elasticity. Studies for the interactions between a buried pipeline and a soil undergoing build-up of pore pressure and residual liquefaction have been comparatively rare. In this paper, this complicated process was investigated by using a new developed integrated numerical model with RANS (Reynolds averaged Navier–Stokes) equations used for governing the incompressible flow in the wave field and Biot consolidation equations used for linking the solid–pore fluid interactions in a porous seabed with embedded pipeline. Regarding the wave-induced residual soil response, a two-dimensional poro-elastoplastic solution with the new definition of the source term was developed, where the pre-consolidation analysis of seabed foundation under gravitational forces including the body forces of a pipeline was incorporated. The proposed numerical model was verified with laboratory experiment to demonstrate its accuracy and effectiveness. The numerical results indicate that residual liquefaction is more likely to occur in the vicinity of the pipeline compared to that in the far-field. The inclusion of body forces of a pipeline in the pre-consolidation analysis of seabed foundation significantly affects the potential for residual liquefaction in the vicinity of the pipeline, especially for a shallow-embedded case. Parametric studies reveal that the gradients of maximum liquefaction depth with various wave and soil characteristics become steeper as pipeline burial depth decreases.

Numerical study of wave-induced soil response in a sloping seabed in the vicinity of a breakwater

In this study, a mathematical integrated model is developed to investigate the wave-induced sloping seabed response in the vicinity of breakwater. In the present model, the wave model is based on the Volume-Averaged/Reynolds Averaged Navier–Stokes (VARANS) equations, while Biot's consolidation equation is used to govern the soil model. The influence of turbulence fluctuations on the mean flow with respect to the complicated interaction between wave, sloping seabed and breakwater are obtained by solving the Volume-Averaged k−ϵ model. Unlike previous investigations, the phase-resolved absolute shear stress is used as the source of accumulation of residual pore pressure, which can link the oscillatory and residual mechanisms simultaneously. Based on the proposed model, parametric studies regarding the effects of wave and soil characteristics as well as bed slopes on the wave-induced soil response in the vicinity of breakwater are investigated. Numerical results indicate that wave-induced seabed instability is more likely to occur in a steep slope in the case of soil with low relative density and low permeability under large wave loadings. It is also found that, the permeability of breakwater significantly affect the potential for liquefaction, especially in the region below the breakwater.

Effect of Variation of Temperature Field on the Process of Thermal Consolidation of Tailings Pond “Zelazny Most”

Abstract The following study presents numerical calculations for establishing an impact of temperature changes on the process of distortion of bi–phase medium. The Biot consolidation equations with Kelvin–Voigt rheological skeleton were used for that purpose. The process was exemplified by thermal consolidation of post floatation dump “–elazny Most”. We analyzed the behavior of the landfill under the action of its own weight, forces of floating filtration and temperature gradient. Values of certain effective parameters of model were obtained during laboratory tests on material obtained from the landfill. The remaining data for mediums with similar characteristics were taken from literature. The results obtained from the stress state in the landfill allow the magnitude of plasticity potential to be specified based on known strength criteria. Change in the value sign of the plasticity potential clearly testifies to the emergence of an area of plasticity of material from landfill, however, this does not indicate the loss of stability of this hydrotechnical structure.

Large scale micro finite element analysis of 3D bone poroelasticity

In this paper, a solver for poroelasticity problems related to osteoporotic human bones is discussed. Osteoporosis is a major health problem that compromises the integrity of bones. A good understanding of the disease requires an accurate simulation of the physics. For that purpose, a finite element solver based on Biot's consolidation equations has been developed. A mixed formulation is used to discretize the geometries taken from medical imaging. The resulting indefinite linear systems are solved by Krylov space methods supplemented by variants of Schur complement-based block preconditioners.

Numerical model of electro-osmotic consolidation in clay

Electro-osmotic consolidation is a unique soil improvement technique. A physics-based model for electro-osmotic consolidation is proposed, in which Biot's consolidation equation is coupled with the electro-osmotic flow and control equation for an electrical field, and the displacement of soil mass and pore-water pressure can be obtained. Furthermore, the non-linear variation of the mechanical and electrical properties of the soil mass during consolidation is incorporated into the model. The numerical solution agrees well with the analytical solution. The developed model can predict the ground displacement and provide useful data for the design of an electro-osmosis treatment system.