Terzaghi's Effective Stress Principle Research Articles

Multiscale modeling of compaction bands in saturated high-porosity sandstones

We propose a hierarchical multiscale modeling approach to simulate the hydro-mechanical coupling behavior in saturated high-porosity sandstone, with a particular focus on the occurrence and transition of compaction bands. In this multiscale approach, we use the finite element method (FEM) to solve the global boundary value problem for a sandstone, and the discrete element method (DEM) to solve the local representative volume elements (RVEs) embedded at the Gauss integration points of the FE mesh to provide the effective material constitutive responses required for FEM. The global governing equations for a coupled hydro-mechanical problem are solved with the standard u–p formulation, where the total stress is decomposed into the pore pressure and the effective stress which is directly derivable from the DEM solution of the RVEs, according to the seminal Terzaghi's effective stress principle. By choosing proper RVEs to represent the typical microstructure of high-porosity sandstone, we perform both drained and undrained simulations on saturated sandstone specimens to examine localized compactive failures developed therein. We identify that both mean effective stress and porosity control critically the initiation and development of failure modes in sandstone. Under undrained shear, our simulations show that a sandstone specimen may first develop a compaction band followed by a gradual transition to a shear-enhanced compaction band and even shear band, with the build-up of excessive pore pressure and hence drop of mean effective stress. In the specimen, regions experiencing localized volumetric contraction show relatively higher pore pressure than other regions and act as the source of flux flow. The Darcy flux peaks during the formation of compaction band and decreases gradually in the following stages. To distinguish the different failure stages with distinct physical attributes, the ratio of deviatoric strain to volumetric strain is adopted as a useful indicator signifying the underlying microstructural mechanisms for different deformation bands. This study signifies the importance of considering pore pressure and mean effective stress in fluid-involved engineering operations, e.g. aquifer management, hydrocarbon extraction and CO2 geological sequestration.

A new effective stress constitutive equation for cemented porous media based on experiment and derivation

The stress-strain of porous media is totally different from that of continuous solid materials. Terzaghi effective stress principle, as the cornerstone of soil mechanics, fits well for stress-strain of soils but fails to fit for cemented porous media. Many researchers are dedicated to propose a more comprehensive effective stress principle. By combining previous effective stress equations proposed by Skempton, Bishop, and Lade & de Boer, a more accurate and comprehensive effective stress constitutive equation is derived. A homogeneous coefficient ω, a stress concentration coefficient β, and a cemented coefficient λ for effective stress principle of all types of pore media are proposed. Appropriate experiments including three types of porous media, namely soils, rocks, and ceramics are designed. The strains of soils, rocks, and ceramics are tested in different confining pressure and fluid pressure. The results show that (a) the strain of soils is mostly caused by intergranular strain and λ is nearly 0, (b) the strain of ceramics is mainly caused by grains strain and λ is more close to 1 − φ, (c) the strain of rocks is combined with grains strain and intergranular strain and 0 < λ < 1 − φ. Three strain patterns of porous media are controlled by the cemented coefficient and the stress conditions. Demonstrated by theoretical derivation and experimental test, the new effective stress can be widely used to all types of porous media.

Confined compressive strength model of rock for drilling optimization

Abstract The confined compressive strength (CCS) plays a vital role in drilling optimization. On the basis of Jizba's experimental results, a new CCS model considering the effects of the porosity and nonlinear characteristics with increasing confining pressure has been developed. Because the confining pressure plays a fundamental role in determining the CCS of bottom-hole rock and because the theory of Terzaghi's effective stress principle is founded upon soil mechanics, which is not suitable for calculating the confining pressure in rock mechanics, the double effective stress theory, which treats the porosity as a weighting factor of the formation pore pressure, is adopted in this study. The new CCS model combined with the mechanical specific energy equation is employed to optimize the drilling parameters in two practical wells located in Sichuan basin, China, and the calculated results show that they can be used to identify the inefficient drilling situations of underbalanced drilling (UBD) and overbalanced drilling (OBD).

A Two-Scale Computational Model of pH-Sensitive Expansive Porous Media.

We propose a new two-scale model to compute the swelling pressure in colloidal systems with microstructure sensitive to pH changes from an outer bulk fluid in thermodynamic equilibrium with the electrolyte solution in the nanopores. The model is based on establishing the microscopic pore scale governing equations for a biphasic porous medium composed of surface charged macromolecules saturated by the aqueous electrolyte solution containing four monovalent ions [Formula: see text]. Ion exchange reactions occur at the surface of the particles leading to a pH-dependent surface charge density, giving rise to a nonlinear Neumann condition for the Poisson-Boltzmann problem for the electric double layer potential. The homogenization procedure, based on formal matched asymptotic expansions, is applied to up-scale the pore-scale model to the macroscale. Modified forms of Terzaghi's effective stress principle and mass balance of the solid phase, including a disjoining stress tensor and electrochemical compressibility, are rigorously derived from the upscaling procedure. New constitutive laws are constructed for these quantities incorporating the pH-dependency. The two-scale model is discretized by the finite element method and applied to numerically simulate a free swelling experiment induced by chemical stimulation of the external bulk solution.

A stress-displacement solution for a pressure tunnel with impermeable liner in elastic porous media

CONSIDERING THE EFFECTS OF THE CHANGES OF THE PORE WATER PRESSURE AROUND THE OPENING, THE CONSTRUCTION SEQUENCE OF THE TUNNEL AND THE INTERACTION BETWEEN THE LINER AND THE SURROUNDING GEOMATERIAL ON THE MECHANICAL RESPONSE OF THE TUNNEL AND IN CONJUNCTION WITH THE ANALYSES OF THE CONTINUITY AND BOUNDARY CONDITIONS FOR THE STRESS AND DISPLACEMENT AND HYDRAULIC CONDITIONS, THE ELASTIC PROBLEM FOR A DEEP PRESSURE TUNNEL WITH IMPERMEABLE LINER IN A SATURATED ELASTIC POROUS MEDIA THAT OBEYS TERZAGHIRS EFFECTIVE STRESS PRINCIPLE IS INVESTIGATED. THE INFLUENCES OF THE RELATIVE LINER THICKNESS AND RIGIDITY AND THE RELATIVE DISTANCE OF THE POINT UNDER INVESTIGATION TO THE TUNNEL AXIS ON THE STRESS-DISPLACEMENT FIELDS FOR VARIOUS COMBINATIONS OF THE MECHANICAL AND GEOMETRIC PARAMETERS ARE EVALUATED AND DISCUSSED.

Analysis of Economic Loss Caused by Ground Material Subsidence

Ground subsidence is one of the environmental geological problems today which arises in the process of world urbanization and can't be ignored. It is related closely to the local economic activities, and causes great hazard to the sustainable socio-economic development. Therefore, this problem has been paid much attention to by the whole society for a long time. The author of this paper analyzes the urban ground subsidence mechanism by adopting Terzaghi's effective stress principle and one-dimensional consolidation theory, and puts forward measures to prevent and control the ground subsidence.

Analytical and numerical study of the effect of water pressure on the mechanical response of cylindrical lined tunnels in elastic and elasto-plastic porous media

This paper revisits the classical problem of excavating a deep cylindrical tunnel in a saturated elastic or elasto-plastic porous ground that obeys Terzaghi's effective stress principle. A generalized form of the classical two-dimensional analytical solution by Lamé for the problem of excavating a cylindrical tunnel in elastic medium subject to axi-symmetric loading is presented. The solution considers the long-term (i.e., post-transient) mechanical effect of changes in pore pressure due to drainage of the ground around the opening on the mechanical response of the tunnel. The proposed solution together with the analytical solution for axi-symmetric loading of an annular ring representing a liner are used to explain the fundamental differences in support loading obtained for various hydro-mechanical conditions. These conditions involve excavating the tunnel with a permeable liner, with and without removal of water from the tunnel, excavating the tunnel with an impermeable liner, with subsequent drainage of water in the ground around the tunnel, etc. The results obtained with the analytical solution are compared with results obtained with the hydro-mechanical option in a commercial finite difference code. The finite difference approach is also used to obtain results and explain the behavior of the tunnel under different hydro-mechanical conditions when the ground is assumed to behave elasto-plastically.

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.

On the dissipation of negative excess porewater pressure induced by excavation in soft soil

Unloading induces negative excess porewater pressure in soil mass around a foundation pit during excavation. In this work, the dissipation rule of negative excess porewater pressure after excavation was studied. Analytical formulas for calculating the negative excess porewater pressures and the effective stresses were derived based on one-dimensional consolidation theory and Terzaghi's effective stress principle. The influence of the dissipation of negative excess porewater pressure on earth pressure inside and outside a foundation pit and the stability of the retaining structure were analyzed through a numerical example. It was indicated that the dissipation of negative excess porewater pressure is harmful to the stability of the retaining structure and that rapid construction can make full use of the negative porewater pressure.

On the dissipation of negative excess porewater pressure induced by excavation in soft soil

Unloading induces negative excess porewater pressure in soil mass around a foundation pit during excavation. In this work, the dissipation rule of negative excess porewater pressure after excavation was studied. Analytical formulas for calculating the negative excess porewater pressures and the effective stresses were derived based on one-dimensional consolidation theory and Terzaghi's effective stress principle. The influence of the dissipation of negative excess porewater pressure on earth pressure inside and outside a foundation pit and the stability of the retaining structure were analyzed through a numerical example. It was indicated that the dissipation of negative excess porewater pressure is harmful to the stability of the retaining structure and that rapid construction can make full use of the negative porewater pressure.

Macroscopic and Micromechanical Approaches to the Modelling of the Osmotic Swelling in Clays

The osmotic swelling in clays has been extensively studied at the physico-chemical scale. The present paper addresses the question of the modelling of this phenomenon from the mechanical point of view. First, the classical macroscopic thermodynamic framework for saturated porous continua is extended in order to take into account the solid-salt interaction through the concept of macroscopic activity coefficient of the salt. The micromechanical approach then incorporates this interaction through the concept of swelling pressure which is used for describing the internal forces in the fluid phase at the microscopic scale. The results of a physico-chemical theory for the solid-salt interaction, such as the e.d.l. theory, can be introduced in both approaches. Each of them leads to the identification of a deviation, of chemical origin, to Terzaghi's effective stress principle. Besides, the micromechanical approach allows us to clearly differentiate the mechanical and the chemical parts of clay materials elasticity.

A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle

The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress principle of Terzaghi and Fillunger. In the context of a first density-gradient theory for a saturated solid we interpret the porewater pressure as a Lagrange multiplier in the expression for the deformation energy, assuring that the saturation constraint is verified. We prove that this saturation pressure is distributed among the constituents according to their respective volume fraction (Delesse law) only if they are both true density-preserving. We generalize the Delesse law to the case of compressible constituents. If a material-dependent effective stress contribution is to arise, it is, in general, nonvanishing simultaneously in both the solid and fluid constituents. Moreover, saturation pressure, effective stresses and compressibility constitutive equations determine the exchange volume forces. In a theoretical formulation without non-isotropic strain measures, second density-gradient effects must be incorporated, not only to accommodate for the equilibrium-solid-shear stress and the interaction among neighboring solid-matrix pores, but also to describe internal capillarity effects. The earlier are accounted for by a dependence of the thermodynamic energy upon the density-gradient of the solid, while the latter derives from a mixed density-gradient dependence. Examples illustrate the necessity of these higher gradient effects for properly posed boundary value problems describing the mechanical behaviour of the disturbed rock zone surrounding salt caverns. In particular, we show that solid second-gradient strains allow for the definition of the concept of static permeability, which is distinct from the dynamic Darcy permeability.

Modified Darcy's law, Terzaghi's effective stress principle and Fick's law for swelling clay soils

Abstract Governing equations often used in soil mechanics and hydrology include the classical Darcy's law, Terzaghi's effective stress principle, and the classical Fick's first law. It is known that the classical forms of these relations apply only to non-swelling, granular materials. In this paper, we summarize recent generalizations of these results for swelling porous media obtained using hybrid mixture theory (HMT) by the authors. HMT is a methodical procedure for obtaining macroscopic constitutive restrictions which are thermodynamically admissible by exploiting the entropy inequality for spatially-averaged properties. HMT applied to the modeling of swelling clay particles, viewed as clusters of adsorbed water and clay minerals, produces additional terms necessary to account for the physico-chemical forces between the adsorbed water and clay minerals or, more generally, for swelling colloids. New directions for modeling consolidation of swelling clays are proposed based on our view of clay particles as a two-phase system.

A Multiscale Theory of Swelling Porous Media: II. Dual Porosity Models for Consolidation of Clays Incorporating Physicochemical Effects

A three-scale theory of swelling clay soils is developed which incorporates physico-chemical effects and delayed adsorbed water flow during secondary consolidation. Following earlier work, at the microscale the clay platelets and adsorbed water (water between the platelets) are considered as distinct nonoverlaying continua. At the intermediate (meso) scale the clay platelets and the adsorbed water are homogenized in the spirit of hybrid mixture theory, so that, at the mesoscale they may be thought of as two overlaying continua, each having a well defined mass density. Within this framework the swelling pressure is defined thermodynamically and it is shown to govern the effect of physico-chemical forces in a modified Terzaghi's effective stress principle. A homogenization procedure is used to upscale the mesoscale mixture of clay particles and bulk water (water next to the swelling mesoscale particles) to the macroscale. The resultant model is of dual porosity type where the clay particles act as sources/sinks of water to the macroscale bulk phase flow. The dual porosity model can be reduced to a single porosity model with long term memory by using Green's functions. The resultant theory provides a rational basis for some viscoelastic models of secondary consolidation.

Analysis of induced seismicity for stress field determination and pore pressure mapping

The focal mechanisms of some one hundred microseismic events induced by various water injections have been determined. Within the same depth interval, numerous stress measurements have been conducted with the HTPF method. When inverted simultaneously, the HTPF data and the focal plane solutions help determine the complete stress field in a fairly large volume of rock (about 15×106 m3). These results demonstrate that hydraulically conductive fault zones are associated with local stress heterogeneities. Some of these stress heterogeneities correspond to local stress concentrations with principal stress magnitudes much larger than those of the regional stress field. They preclude the determination of the regional stress field from the sole inversion of focal mechanisms. In addition to determining the regional stress field, the integrated inversion of focal mechanisms and HTPF data help identify the fault plane for each for each of the focal mechanisms. These slip motions have been demonstrated to be consistent with Terzaghi's effective stress principle and a Coulomb friction law with a friction coefficient ranging from 0.65 to 0.9. This has been used for mapping the pore pressure in the rock mass. This mapping shows that induced seismicity does not outline zones of high flow rate but only zones of high pore pressure. For one fault zone where no significant flow has been observed, the local pore pressure has been found to be larger than the regional minimum principal stress but no hydraulic fracturing has been detected there.

Numerical analysis of sedimentation and consolidation by the moving finite element method

AbstractTraditionally, sedimentation and self‐weight consolidation have been viewed as physically distinct processes requiring separate treatment. Relatively recently, Pane and Schiffman1 and also Philip and Smiles2 have suggested that the two processes may be described by a single partial differential equation, essentially that of Gibson et al.3 The former suggests a modification of Terzaghi's effective stress principle while the paper by Philip and Smiles suggests that a suitable modelling of material properties is sufficient. We have adopted the latter approach by allowing for the compressibility of the material in question to change abruptly from finite values to infinity in the so‐called transition region which delineates that portion of space where effective stress is zero from that where effective stress in non‐zero. This procedure gives rise to serious difficulties when trying to solve the governing partial differential equation numerically. These difficulties are circumvented by using a relatively new numerical technique known as the Moving Finite Element (MFE) method. The MFE method is especially effective in solving problems having solutions that characteristically exhibit shock‐like structure. The modelling of sedimentation and self‐weight consolidation from a single governing model is an ideal candidate for MFE due to the abrupt, almost discontinuous change in void ratio displayed in the transition region.

Viscoplastic deformations of coarse-rubbly soils subject to cyclic effects

1. Under a cyclic loading, the strength of coarse-rubbly soils (over the entire investigated range of variation in the static stressed state and stress amplitudes) remains virtually constant as compared with a static loading. 2. Additional soil compaction, which depends on the level of the static stressed state, the amplitude of the event, and the soil density, and also additional development of plastic shear strains take place under deviator cyclic loading. 3. Stabilization of volume and shear strains in the sublimiting state under cyclic effects occurs over a prolonged period as computed for thousands of load cycles; the major portion of the deformation, however, is accumulated during the first several tens of cycles. 4. The failure of a saturated coarse-rubbly soil under cyclic loading occurs in conformity with Terzaghi's effective-stress principle.

Limitations to the Use of Effective Stresses in Partly Saturated Soils

Synopsis Terzaghi's effective stress principle may be stated in the form of two propositions:— (i) Changes in volume and shearing strength of a soil are due exclusively to changes in effective stress. (ii&gt; The effective stress σ′ in a soil is defined as the excess of the total applied stress σ over the pore pressure u. The validity of this principle for fully saturated soils is now well established and has proved invaluable in the prediction of the behaviour of such soils. In recent years workers have attempted to extend the principle to cover the case of partly saturated soils and to date most of this work has been concerned with deriving a suitable form of effective stress equation, Up till now the validity of the first proposition, upon which the main practical significance of the effective stress principle relies, does not appear to have been questioned. In this Paper the results of a series of oedometer and all-round compression tests on partly saturated and fully saturated soils are presented. The soils used range from a silty sand to a silty clay. The results, together with additional experimental data derived mainly from oedometer tests on wind-blown sands and expansive clays, indicate that most soils, from sands right thorugh to clays exhibit behaviour which, below a critical degree of saturation, cannot be accounted for purely on the basis of effective stress changes. In sands the critical degree of saturation is below 50%. In clays, however, the critical degree of saturation appears to be as high as 90%. An explanation for the observed behaviour of partly saturated soils is offered. It is suggested that the structural changes resulting from a change of pressure deficiency in a soil are very different from those resulting from an equivalent change in applied stress. Although the experimental work described in the Paper is confined mainly to considerations of volume change the conclusions drawn apply equally well to shear strength. The practical significance of the results of the invetigation are discussed briefly. Le principe de tension effective de Terzaghi peut s'énoneer sous la forme de deux propositions:— (i) Les changements en volume et en effort de cisaillement d'un sol sont dûs exclusive-ment aux changement de tension effective. (ii) La tension effective σ′ dans un sol est déterminée d'après l'excédent de la tension appliqué totale σ sur la pression interstitielle u. La validité du principe pour des sols compléte-ment saturés est maintenant bien établie et s'est montrée inestimable pour la prédiction de la manière dont ces sols se comportent. Ces dernières années des chercheurs ont essayé de développer ce principe pour qu'il comprenne la cas des sols partiellement saturés et jusqu'à présent la plupart de ce travail a eu pour but de développer une forme convenable d'équation de la première proposition, sur laquelle se base la principale signification pratique du principe de tension effective, ne semble pas avoir été mise en doute. Dans cet article on présente les résultats d'un série d'essais œ dométrique et d'essais complets de compression sur des sols partiellement saturés et complètement saturés. Les sols employés varient du sable limoneux à l'argile limoneuse. Les résul-tats, avec aussi les renseignements expérimentaux additionnels derivés principalement des essais œdométriques sur des sables éoliens et des argiles expansibles, indiquent que la plupart des sols, depuis les sables jusqu'aux argiles montrent qu'ils se comportent, au dessous d'un degré critifque de saturation, d'une manière qui ne peut pas s'expliquer simplement sur la base des changments de tension effective. Pour les sables le degré critique de saturation est inférieur à 50%. Per contre pour les argiles le degré critique de saturation semble atteindre 90%. On donne une explication sur la manière dont se comporte les sols partiellement saturés qui ont été observés. On suggère que les changements de structure résultant d'un changement de manque de pression dans un sol sont très différents de ceux qui résultent d'un changement équivalent dans la tension appliqueée. Malgré que le travail expéri-mental décrit dans l'article soit limité principalement aux considérations de changement de volume les conclusions tirées s'appliquent aussi bien à la résistance au cisaillement. On expose une brève discussion de la signification pratique des résultats de l'enquête.