Pore Fluid Compressibility Research Articles

Staggered finite volume algorithms for dynamic analysis of saturated soil with low permeability and incompressible pore fluid by using OpenFOAM

Biot dynamic theory forms the basis of dynamic analysis for soil-pore fluid coupling, whose formulations can be simplified by u-p approximation for low frequency phenomena without loss of accuracy. In the past decades, fully-coupled approach and segregated method were two common strategies for Finite Volume (FV) solutions to Biot dynamic theory. However, both methods encounter problems that limit their applicability when soil permeability and pore fluid compressibility reduce to a relatively low level. Under this circumstance, fully-coupled approach requires mixed formulation to satisfy Babuška-Brezzi condition, which complicates numerical implementation. On the other hand, segregated method suffers from divergence and instability, and shows poor performance in computation efficiency. In this article, two staggered FV algorithms incorporating operator splitting technique are devoted to overcoming the aforementioned shortcomings, and they are implemented on the open-source, broadly used and versatile platform for FV calculation, OpenFOAM. Through comparing the computed results of the two algorithms with solutions in the literature, both of the proposed algorithms are shown to reasonably reflect the dynamic responses of porous media. By conducting stability analyses based on the numerical cases, the two algorithms are shown to enlarge the stability region and enable larger time steps for computation.

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

AbstractA large deformation, coupled finite‐element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain‐rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one‐dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed‐form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain‐rate, and (iii) showing differences in deformations between a single‐phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain‐rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single‐phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain‐rate loading of soft porous materials. It was discovered that explicit, adaptive time‐stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time‐stepping schemes for shock‐like loadings. Shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.

A State-Dependent Hypoplastic Bounding Surface Model for Gassy Sand

As a kind of partially saturated soil often containing dissolved gas with a high degree of saturation (S > 85%), gassy sand sediment widely exists in the marine environment all over the world. Due to the effect of gas dissolution and exsolution on the pore fluid compressibility, its stress–strain and pore pressure responses are quite different from those of common saturated and unsaturated soils when subjected to undrained loading. Since almost all the gas bubbles are occluded in the pore water of offshore gassy sand, the matric suction may be neglected, and an undrained constitutive model for gassy sand is developed based on the existing hypoplastic bounding surface model for saturated sand. Both Boyle’s and Henry’s laws are employed in the model to characterize the equilibrium behavior of the gas compressibility and solubility, and then the equivalent compressibility coefficient of the pore fluid is obtained. To avoid unrealistic volumetric expansion, the concepts of the critical state and state-dependent dilatancy stress ratio are incorporated to describe its ultimate shear strength and dilatancy characteristics, respectively. Finally, the triaxial undrained test results on gas-charged sand from Hangzhou Bay are analyzed at three initial saturation degrees of 85%, 90%, and 100%, and two effective confining pressures of 50 kPa and 200 kPa. Moreover, carbon dioxide (CO2) was selected in the test, and the samples were loose with a relative density of 30%. It is noted that a good agreement is achieved between the simulation results and the experimental data, including the influence of gas content and confining pressure on the shear dilatancy and the mean effective stress increase at the beginning of the effective stress path, among others.

An insight on the experimental volumetric behaviour of gassy soils

Induced Partial Saturation (IPS) is one of the most innovative and promising countermeasures to mitigate soil liquefaction risk. Mechanical benefits of air/gas bubbles occluded within the pore water have been studied in the last decade through undrained cyclic tests on quasi-saturated (gassy) soils, demonstrating that the increased pore fluid compressibility prevents liquefaction triggering. The greater compressibility of the air bubbles rules the volumetric strains of gassy soils during seismic shaking reducing the build up of the pore water pressure. Mele et al., (2022) verified that, at the laboratory scale, due to lower frequencies of the applied cyclic loads, a non-negligible amount of soil volumetric strains is due to dissolution of air bubbles in the water (εv,diss). The outcomes of some simple compression tests carried out on a two-phase medium made of air/water confirm that such amount cannot be correctly computed with the Henry’s law, which considers the dissolution process of air into water in the hypothesis of continuous air phase. Experimental evidences highlighted that εv,diss is mainly ruled by the continuity of air phase (linked to the chosen experimental procedure), and it exceeds the theoretical previsions when air phase is discontinuous with single bubbles occluded in the fluid phase.

A fully explicit staggered algorithm for near‐field wave propagation of fluid‐saturated porous media based on u‐p dynamic formulation

AbstractNear‐field wave propagation of the fluid‐saturated porous media (FSPM) is an important issue in soil dynamics and geotechnical earthquake engineering. In the current study, a fully explicit staggered algorithm for the near‐field wave propagation analysis of the FSPM in the time domain is developed based on the u‐p dynamic formulation. The decoupling of dynamic formulation is implemented by the diagonalization of the mass matrix and pore fluid compressibility matrix. The central difference method and Newmark constant average acceleration method are used for the solution of the displacement and velocity of solid skeleton, and the backward difference method is used for the solution of pore fluid pressure in the time domain. The effectiveness and feasibility of the matrix diagonalization are verified by the comparison with the classical numerical algorithm. The proposed algorithm is also validated by the comparison of the numerical results with the corresponding analytical solution results. Combing with the transmitting artificial boundary condition, the proposed algorithm is then applied to investigate the seismic response of a FSPM free field, and its applicability to the typical near‐field wave propagation problems of the FSPM is indicated. The stability analysis of the proposed algorithm is finally conducted, and the corresponding stability criterion is presented. For the proposed algorithm, all the dynamic variables of the FSPM are solved in an iterative way and the coupling dynamic equations need not to be solved at each time step. The computational effort can be reduced considerably, and the computational efficiency can be improved remarkably.

Influence of air entrainment and gas kinetics on liquefaction triggering during tsunami loading

Tsunamis are a significant coastal hazard that pressurise sediment pore water, which can potentially induce liquefaction, enhance erosion and scour, and undermine the stability of beach sediments supporting natural and built infrastructure in the coastal nearshore environment. Tsunami-induced pressurisation of pore water is attributed to (a) mechanical pressurisation associated with the soil skeleton's tendency to contract or expand around the soil's pore fluid under the changing overlying water weight and (b) seepage of tsunami water into and out of the sand bed. The pore fluid compressibility, which is influenced by air entrainment due to water table fluctuations caused by the tides, can significantly influence development of stabilising and destabilising hydraulic gradients during tsunami runup and drawdown, respectively. Previous work has effectively assumed a constant air content in sand beds and neglected (a) the compression and dissolution of entrained air that influences the pore fluid compressibility and (b) the interaction between saturated sediment underlying sediment with entrained air. By considering the gas kinetics influencing the pore fluid compressibility in a seepage–deformation model, the effect of different pore fluid compressibility assumptions, thickness of sediment with entrained air, initial degree of saturation and tsunami wave properties (height and duration) were studied. The thickness of sediment containing entrained air, the assumed pore fluid compressibility assumption and subtle differences in the initial degree of saturation can significantly influence the predicted maximum depth and duration of tsunami-induced liquefaction.

Simulation Analysis of Temperature Effects on the Shear Behavior of Gassy Sand

The gas in gassy sand is enclosed in the pores of soil skeleton mainly in the form of dissolved in water or free gas bubbles, which is considered to be a perfect equilibrium composed of soil particles, pore water, gas, temperature, and overlying pressure. If any of these elements changes, this balance will be destroyed, resulting in disturbance in its mechanical properties. It is generally assumed that the existence of closed gas bubbles will not affect the soil skeleton but increase the compressibility of pore fluid. Assuming that the closed gas bubbles follow Boyle’s law and Henry’s law, the relationship between pore pressure change and volume strain is established. On the basis of the constitutive model which can reflect the temperature characteristics of saturated sand proposed by the author previously, combined with the special characteristics of gassy sand, the effect of temperature effects on the shear behavior of gassy sand is preliminarily explored and analyzed. The results show that under the condition of drained shear, the increase of temperature will increase the shear strength of gassy sand, and the simulation results are in good agreement with the test data. Under the condition of undrained shear, the shear strength of loose gassy sand increases with the increase of temperature, while that of dense gassy sand decreases. Moreover, the larger the gas volume, the more obvious these phenomena tend to be. However, only qualitative simulation analysis is carried out for undrained shear, which needs further verification in combination with relevant tests.

Modelling of Static Liquefaction of Partially Saturated Non-Cohesive Soils

Static soil liquefaction is widely known to be a serious danger to the stability of structures. The phenomena governing pore water generation, which leads to liquefaction in fully saturated soils, are already quite well described. However, much less is known of these phenomena occurring in partially saturated porous media, although this, too, is an important issue in geotechnics. This study presents the application of a semi-empirical model to predict the response of partially saturated soils under undrained conditions. The model proposed is based on an incremental equation describing the pre-failure undrained response of partially saturated non-cohesive soils during monotonic shearing in a standard triaxial test. Improved differential equations taking into account pore fluid compressibility were implemented together with empirical coefficients describing soil skeleton compressibility during the unloading phase. Model coefficients were determined in triaxial compression tests. The influence of the saturation level represented by Skempton’s parameter B on the full spectrum of predicted stress paths was shown. For the analyzed saturation range, the maximum stress deviator normalized by initial mean effective stress varied from 0.38 to 1.67 for B values between 0.93 and 0.29, respectively. Model predictions were confronted with the results of triaxial tests for two types of non-cohesive soils (quartz medium sand and copper ore post-flotation industrial tailings). Good agreement between experimental data and theoretical predictions was achieved.

Unloading induced absolute negative pore pressures in a low permeable clay shale

The stability analysis of underground excavations constructed in low-permeable rocks requires a comprehensive understanding of the short- and long-term pore pressure response around the excavation. This paper presents the results of a novel in-situ experiment, which was conducted in Opalinus Clay at the Mont Terri Underground Rock Laboratory, Switzerland. The primary aim of the project was to understand unloading-induced pore pressure changes in Opalinus Clay with a focus on the possibility of the development of absolute negative pore pressures. In this experiment, the pore pressure response was initiated by overcoring a small diameter pilot borehole equipped with a pressure transducer and a full-range tensiometer. The results of the experiment showed that the stress relief was accompanied by a noticeable pore pressure drop in the core. An absolute negative pore pressure of −755 kPa was measured. To replicate the pore pressure evolution during the in-situ experiment, a 3D hydro-mechanical-coupled finite element model was developed. It is shown that the magnitude of the pore pressure drop cannot be explained by the classical poroelasticity theory for fully saturated Opalinus Clay samples. Good agreement between the experimental data and numerical results can only be achieved when unloading induced desaturation is considered. Rapid unloading may cause pore water cavitation and desaturation, which significantly increases the pore fluid compressibility. Our considerations based on the nucleation theory suggest that during unloading the required conditions for pore water cavitation in pores larger than a critical value were met.

Vertical response of a pile embedded in highly-saturated soil with compressible pore fluid and anisotropic permeability

For highly-saturated soils, the effect of air bubbles on the consolidation of soil is mainly reflected by the compressibility of pore fluid. In this paper, the effect of compressible pore fluid and anisotropic permeability on the soil mechanism is investigated based on the Biot’s consolidation theory and analytical layer element method (ALEM). The response of pile in highly-saturated soils is further investigated by employing the finite element method and theory of one-dimensional bar. The effects of compressibility of pore fluid and anisotropic parameter are investigated by means of numerical computation. The results show that the compressibility of pore fluid has a significant effect on the pile’s initial deformation while has less influence on the final displacement. The peak value of pore pressure in a highly-saturated soil with a larger compressibility is smaller and appears later. It takes less time for the pore water to completely dissipate in highly-saturated soil with a larger horizontal permeability coefficient.

Analytical Solution for 2D and 3D Lamb Problem in Saturated Soil Incorporating Effects of Compressibility of Solid and Pore Fluid

Analytical Solution for 2D and 3D Lamb Problem in Saturated Soil Incorporating Effects of Compressibility of Solid and Pore Fluid

Study on water and salt migration and deformation properties of unsaturated saline soil under a temperature gradient considering salt adsorption: Numerical simulation and experimental verification

Based on porous media theory and the principle of continuum mechanics, a mathematical model is established to describe a fully coupled process in a thermo-hydro-solute-gas-mechanical multi-field under a temperature gradient. The model considers the influence of factors, such as porosity evolution, water seepage and its evaporation–condensation, gas transport, salt adsorption/precipitation-dissolution dynamic conversion process, solid particles and pore fluid compressibility. Comsol Multiphysics is used to numerically simulate the previously mentioned multi-field coupling process. In the simulation process, some important physical parameters (such as porosity and medium density) depend on state variables (such as temperature, pore water pressure and salt content). The mathematical model and simulation results are verified through the actual measurement results of the indoor test. Thus, salt adsorption influences the heat and mass transfer process and deformation characteristics of unsaturated saline soil. The results show that the model can better reveal the water and salt transfer mechanism and deformation mechanism of unsaturated saline soil under the action of a temperature gradient, adsorption effects on the heat and mass transfer process and its deformation characteristics by affecting the porosity of the soil. Although in the process of establishing the theoretical model, the elastic constitutive model (including the effective stress principle) is used to describe the momentum conservation of the soil. Additionally, the study of salt adsorption provides theoretical preparation for further research on salt expansion in saline soil.

Influence of unconnected pores on effective stress in porous geomaterials: Theory and case study in unconventional oil and gas reservoirs

Unconventional oil and gas reservoirs are mostly characterized by unconnected pores and abnormal pore pressures. These conventional effective stress theories, such as Biot's theory, cannot display the quantitative relationship between unconnected pores' micro characteristics and the macro stress applied to the solid structure. These parameters related to the unconnected pores should be quantitatively incorporated in the effective stress formula. In this paper, two modified models were proposed to evaluate the solid framework's real effective stress and deformation. The effects of the unconnected pore pressure, unconnected porosity, and unconnected pore fluid's compressibility on effective stress and volumetric strain were theoretically investigated. It was found that the increase of unconnected pore pressure can decrease the effective stress and volumetric strain. For an abnormal high pore pressure formation, the growth of unconnected porosity can reduce the effective stress but increase the volumetric strain. The decrease of the unconnected pore fluid's compressibility can increase and decrease the effective stress and volumetric strain for abnormal low and high pore pressure reservoirs, respectively. Two case studies on the abnormal pore pressures induced by under-compaction and hydrocarbon generation were conducted. The results showed that the pore pressure exhibits significant increases when the unconnected pores and unconnected pore fluids are proactively compressed in the processes of under-compaction and hydrocarbon generation, respectively. The abnormal high pore pressures in these tight sandstone reservoirs in the Sichuan Basin, Tarim Basin, and Bohai Bay Basin can be theoretically generated when the pore space is compressed to 95%–84% of the initial pore space. The model can evaluate the effective stress and deformation for reservoirs with considerable unconnected porosities. In turn, it can also interpret the abnormal pressure generation induced by the changes of unconnected porosity and pore fluid's compressibility.

Flutter prediction of cylindrical sandwich panels with saturated porous core under supersonic yawed flow

Supersonic flutter characteristics of cylindrical sandwich panels made of a saturated functionally graded porous (FGP) core and two homogeneous face sheets are studied in this paper based on the Biot’s poroelasticity theory. The third-order shear deformation theory (TSDT) is used to model the panel and the piston theory is hired to estimate the aerodynamic pressure create by the supersonic fluid flow. First, the set of the governing equations are solved numerically via generalized differential quadrature method (GDQM) and natural frequencies are calculated for various boundary conditions. Then, convergence of the solution is confirmed and its accuracy is demonstrated by comparing the results with those reported by other authors. Finally, the effects of geometrical parameters of the panel, thickness of the porous core, porosity distribution pattern, porosity parameter (size of pores), compressibility of pore fluid and yaw angle (fluid flow direction) on the flutter boundaries of FGP sandwich panels are investigated.

Simulation of undrained quasi-saturated soil with pore pressure measurements using a discrete element (DEM) algorithm

A method is presented for using DEM to find the pore pressure, total stress, and effective stress within a granular assembly during either undrained or drained conditions, by using poroelastic principles to directly determine the pore fluid pressure. The paper considers both the saturated and quasi-saturated conditions, the latter meaning a slightly unsaturated granular material, in which isolated gas bubbles reduce the compressibility of the pore fluid but do not form liquid bridges between the particles. The presence of air increases the pore fluid compressibility and is known to improve undrained strength and liquefaction resistance and has been considered for remediation of loose, susceptible soils. Past DEM studies have approximated undrained loading as a constant-volume condition, but this approximation does not allow the simulation of complex loading sequences, even with fully saturated materials. The paper’s method allows the direct control and measurement of total stress, water pressure, and water influx. The method includes the effects of water compressibility, grain compressibility, pore air compressibility, surface tension, air solubility within the pore liquid, and vapor pressure of the water gas phase. These factors all affect the evolving sizes of the bubbles and the consequent compressibility of the air–water mixture. The resulting algorithm is applied in three examples: the laboratory saturation of a soil specimen by increasing chamber pressure and back pressure, the undrained triaxial compression of quasi-saturated specimens, and the cyclic undrained triaxial liquefaction of quasi-saturated specimens. Example simulations agree favorably with laboratory tests.

Shear deformation theories for elastic buckling of fluid-infiltrated porous plates: An analytical approach

In this paper, Reddy’s higher-order and Mindlin’s first-order plate theories are used for buckling analysis of porous rectangular plates subjected to various types of mechanical loading. The condition of the internal pores is considered to be either free of or saturated by fluid. Biot’s theory of poroelasticity is thereby employed to model the behaviour of fluid. Distribution of pores is assumed to vary through the thickness according to an asymmetric distribution. For each displacement field considered, five highly coupled partial differential equations are derived by means of variational principle. These systems of equations are first decoupled through an efficient method, and then solved analytically for Levy-type boundary conditions. Accuracy of the approach is examined by comparing the obtained results with those available in literature. Eventually, comprehensive parametric studies are provided to investigate the effects of geometrical parameters, boundary conditions, loading conditions, porosity coefficient and pore fluid compressibility on the buckling response of the system. The results suggest that a structure with higher equivalent rigidity is met, when its corresponding internal pores are saturated by fluid. The results of the current work can be considered as a benchmark for future studies.

Free vibration analysis of thick sandwich cylindrical panels with saturated FG-porous core

In this paper a numerical solution is presented for free vibration analysis of sandwich cylindrical panels made of a saturated functionally graded porous (FG-porous) core and two similar homogenous face sheets. The panel is modeled based on the third order shear deformation theory (TSDT) and stress-strain relations are considered based on the linear poroelasticity theory of Biot. Natural frequencies are calculated for various boundary conditions and a parametric study is presented to study the effects of various parameters on the natural frequencies including geometrical parameters, porosity parameter, porosity distribution pattern, thickness of the porous core and compressibility of porefluid.

A coupled CFD-DEM method with moving mesh for simulating undrained triaxial tests on granular soils

A coupled computational fluid dynamics and distinct element method (CFD-DEM) is proposed to simulate undrained triaxial tests on granular soils. In this method, the compressible volume-averaged Navier–Stokes equations of continuity and momentum with an additional state equation are used for modelling liquid flows. In addition, the volume-averaged Navier–Stokes equations are modified by including the velocity of moving grid in their convective terms to settle the problem of variable boundaries in the process of undrained triaxial compression. Then the capability of the developed method is mainly examined by comparisons with the constant volume method of simulating undrained triaxial tests, which shows that the results of the CFD-DEM method can compare favourably with those obtained from the constant volume method. Finally, the influences of strain rate, initial pore pressure and fluid compressibility on the undrained behavior of granular soils are carefully studied.

Vertical dynamic analysis of a rigid disk embedded in layered saturated soils with compressible fluid

This paper introduces an approach for modelling the dynamic behavior of a vertically loaded rigid disk embedded in layered soils by coupling the analytical layer element method (ALEM) and boundary element method (BEM). The general solutions for dynamic behavior of layered saturated soils with compressible fluid are derived via ALEM. The disk-soil interface is divided into many elements according to the disk segments by utilizing BEM. The solutions for dynamic soil-disk interaction problem are obtained by coupling BEM and ALEM, where the ALEM solution is treated as the kernel function of BEM. The effectiveness of the present method is confirmed by the comparison with existing ones. Selected examples are performed to explore the effects of excitation frequency, embedded depth of the rigid disk, compressibility of pore fluid, permeability and the stratification characteristic on the dynamic responses.

COMPACTION OF SATURATED SOILS WITH REGARD TO COMPRESSIBLE PORE FLUID AND SOIL CREEP

The paper describes the problem of consolidation of clay soil with spherical tensor and stress deviator, taking into account compressibility of pore fluid and the soil creep. In solving the problem, the differential equation is suggested for compacting cohesive soil filled with linearly compressible groundwater. It is considered that the gas volume in the groundwater does not exceed 10 %. The solution of the differential equation is based on the Fourier transform.For the numerical calculation of creep during compaction of consolidated soil an easy-touse method was used, i.e. testing of soil samples of the same monolith in the compression device, but different height. Knowing the relative vertical deformation of these samples, approximation and extrapolation methods allow to find the relative deformation at the sample height tending to zero. At almost zero sample thickness, deformation depends only on the soil creep, while the pore pressure will no longer play a role.The obtained values of the relative strain are used to select the function of the creep nucleus. It is found that the exponential function optimally describes the process of soil compaction over the entire area: from the beginning of compaction to its termination.Knowing the pressure, the stress in soil and the amount of sediment are found with a triaxial compression, taking into account its instantaneous deformation.