Biot Coefficient Research Articles

Stress-induced crystal preferred orientation in the poromechanics of in-pore crystallization

A non-hydrostatic stress field affects the orientation of crystals growing in the pore network of an elastic porous medium. The hypothesis of a hydrostatic state of stress within the crystal has been implicitly made in the recent extension of poromechanics to in-pore crystalization ( Coussy, 2006). This underlying hypothesis is revisited on a small-scale conceptual model based on Eshelby's problem and shows that chemo-mechanical equilibrium requires that the crystal adapts its shape and orientation to the far-field stress, therefore resulting at equilibrium in a hydrostatic state of stress within the crystal. The optimum crystal shape as a function of the far-field stress is consistently investigated, highlighting limiting cases. The small scale model allows to understand the macroscopic effects associated with deviatoric stresses in the poromechanics of in-pore crystallization. Moreover, it provides the building block for an up-scaling of the macroscopic tangent poroelastic properties, which depend on both the current crystal saturation and the state of stress. A dilute micromechanical scheme illustrates the variation of the macroscopic Biot's coefficient tensor as a function of deviatoric stresses. A simple configuration akin to a potential laboratory experiment finally illustrates the strong induced anisotropy of the crystallization induced macroscopic strain when deviatoric stresses are applied to the material prior to crystallization.

Geomechanical analysis of the Naylor Field, Otway Basin, Australia: Implications for CO 2 injection and storage

A geomechanical assessment of the Naylor Field, Otway Basin, Australia has been undertaken to investigate the possible geomechanical effects of CO 2 injection and storage. The study aims to evaluate the geomechanical behaviour of the caprock/reservoir system and to estimate the risk of fault reactivation. The stress regime in the onshore Victorian Otway Basin is inferred to be strike–slip if the maximum horizontal stress is calculated using frictional limits and DITF (drilling induced tensile fracture) occurrence, or normal if maximum horizontal stress is based on analysis of dipole sonic log data. The NW–SE maximum horizontal stress orientation (142°N) determined from a resistivity image log is broadly consistent with previous estimates and confirms a NW–SE maximum horizontal stress orientation for the Otway Basin. An analytical geomechanical solution is used to describe stress changes in the subsurface of the Naylor Field. The computed reservoir stress path for the Naylor Field is then incorporated into fault reactivation analysis to estimate the minimum pore pressure increase required to cause fault reactivation (Δ P p). The highest reactivation propensity (for critically-oriented faults) ranges from an estimated pore pressure increase (Δ P p) of 1 MPa to 15.7 MPa (estimated pore pressure of 18.5–33.2 MPa) depending on assumptions made about maximum horizontal stress magnitude, fault strength, reservoir stress path and Biot's coefficient. The critical pore pressure changes for known faults at Naylor Field range from an estimated pore pressure increase (Δ P p) of 2 MPa to 17 MPa (estimated pore pressure of 19.5–34.5 MPa).

Unified microporomechanical approach for mechanical behavior and permeability of misaligned unidirectional fiber reinforcement

The microporomechanical approach (via homogenization schemes) has been used and combined with triaxial tests to verify the Biot theory for the perfectly straight unidirectional fiber assembly in a previous paper [Tran T, Binetruy C, Comas-Cardona S, Abriak NE. Microporomechanical behaviour of perfectly straight unidirectional fiber material: theoretical and experimental. Compos Sci Technol 2009;69:199–206.]. The comparison of theoretical and experimental results is in good agreement, i.e. the Biot coefficients are clearly lower than one for densely packed fiber array. This result will be developed in this article in the case where the fibers are not perfectly straight but in misalignment (unidirectional fiber assembly in localized contact). Furthermore, within the same theoretical framework, the transverse compression modulus and the hydraulic permeability will be also estimated for the fiber reinforcement of double-scale porosity. The homogenization schemes used in this article are the self-consistent and the one proposed by Mori–Tanaka. The estimated and, when possible, bibliographical results for different types of fibrous materials (carbon, kevlar and glass fibers) are compared and show good agreement.

Poroelastic behaviour of a water-saturated limestone

This paper presents the results of an experimental study carried out on a porous, water-saturated limestone. First, a set of poroelastic parameters is obtained from a wide range of experiments (drained, undrained, pore pressure loading and unjacketed tests) conducted on the same sample at the same stress state. A “null” test, inspired from unsaturated soil testing, is added to confirm the previous results, in particular the value of the Biot coefficient. Then, the variation of the measured parameters between samples is investigated. Finally, the influence of the stress state is studied in the framework of semilinear poroelasticity. Several expressions are compared to model the evolution of the drained bulk modulus as a function of Terzaghi's effective stress.

Pore-Pressure-Coefficient Anisotropy Measurements for Intrinsic and Induced Anisotropy in Sandstone

SummaryIn this study, we determine experimentally the effect of inherent and stress-induced anisotropy on stiffness components, elastic moduli, and Biot's pore-pressure coefficients (PPCs) for Lyons outcrop Colorado sandstone, which exhibits a clear transverse isotropic rock structure. Both dynamic and quasistatic methods were used under a nonhydrostatic state of stress to perform the measurements on dry core samples. Our assumption of apparent transverse anisotropy was confirmed initially with acoustic velocity measurements and at a later stage in the loading with experimental transverse anisotropic failure analysis. The objective of this study is to identify and isolate the effect of stress-induced anisotropy from the inherent transverse anisotropy on the measured stiffness components, elastic moduli, and Biot's PPCs. The effect of stress-induced anisotropy appears to have significant control on measured stiffness components, elastic moduli, and Biot's PPCs in comparison to the inherent-transverse-anisotropy effect. Our work shows that the stiffness components, Mij and thus the computed elastic moduli, are highly influenced by the stress-induced anisotropy, especially the off-diagonal stiffness components, M12 and M13, where the increase in their magnitudes from the dynamic measurements before failure is determined to be 100 and 81%, respectively. The difference in the magnitude between the axial and lateral Biot's PPCs in line with bedding planes and perpendicular to them is measured to be 24 and 16% from the quasistatic and dynamic methods, respectively; whereas, the effect of stress-induced anisotropy reduced the dynamic average magnitude of the Biot's PPCs along the bedding planes and transverse to these planes by 63% across a stress range of 145 MPa.

Biot's coefficient as an indicator of strength and porosity reduction: Calcareous sediments from Kerguelen Plateau

Chalk develops as a result of diagenesis of pelagic calcareous ooze. In a newly deposited ooze sediment, porosity ranges from 60% to 80% but porosity reduces with burial. We studied how different porosity reduction mechanisms change the strength of these deep sea carbonate-rich sediments and effect Biot's coefficient, β. In calcareous ooze, β is one. Mechanical compaction reduces porosity, but only leads to a minor decrease in β. Recrystallization renders particles smoother, but does not lead to reduction in β unless it gives rise to pore stiffening cementation. Pore stiffening cementation causes β to fall, even when porosity remains constant. Biot's coefficient correlates with strength-indicating properties: compressional and shear modulus, oedometer modulus, yield strength, strain from direct loading and creep strain. Our data indicate that β may be used for predicting the diagenetic process involved in porosity reduction and strengthening of chalk during burial diagenesis.

On the effective stress in unsaturated porous continua with double porosity

Using mixture theory we formulate the balance laws for unsaturated porous media composed of a double-porosity solid matrix infiltrated by liquid and gas. In this context, the term ‘double porosity’ pertains to the microstructural characteristic that allows the pore spaces in a continuum to be classified into two pore subspaces. We use the first law of thermodynamics to identify energy-conjugate variables and derive an expression for the ‘effective’, or constitutive, stress that is energy-conjugate to the rate of deformation of the solid matrix. The effective stress has the form σ ¯ = σ + B p ¯ 1 , where σ is the total Cauchy stress tensor, B is the Biot coefficient, and p ¯ is the mean fluid pressure weighted according to the local degrees of saturation and pore fractions. We identify other emerging energy-conjugate pairs relevant for constitutive modeling of double-porosity unsaturated continua, including the local suction versus degree of saturation pair and the pore volume fraction versus weighted pore pressure difference pair. Finally, we use the second law of thermodynamics to determine conditions for maximum plastic dissipation in the regime of inelastic deformation for the unsaturated two-porosity mixture.

Effect of heat-treatment and hydrostatic loading upon the poro-elastic properties of a mortar

This contribution aims at identifying experimentally the poro-elastic properties of a cement-based material under different levels of confining pressure, and after a heat-treatment up to 400 °C. The model material used is a normalized mortar, with a (W/C) ratio of 0.5. After a given heating/cooling cycle, drained bulk modulus K b, solid matrix bulk modulus K s and Biot's coefficient b are measured at different confining pressure levels (with a maximum of 25 MPa). Results show that under drained conditions, mortar stress–strain relationship evolves with increasing heat-treatment temperature from linear elastic with brittle failure (up to 105 °C heat treatment) to plastic and ductile (from 200 °C and above). Plastification testifies of material degradation under gradual confining pressure. At the microstructure scale, this is attributed to thermal damage after heat treatment above 105 °C, which consists mainly in various micro-cracking. This leads to easier failure of solid skeleton bridges (or trabecules), and to pore network collapse. Concomitantly to this, at given confining pressure P c, secant drained bulk modulus K b decreases monotonously, for heat-treatment temperatures above 105 °C. On the opposite, at given heat-treatment temperature above 105 °C, secant drained bulk modulus K b increases when confining pressure is increased. This testifies of a solid matrix rigidification in the elastic domain, and it is attributed to increased skeleton compactness linked with pore network collapse. This is directly attributable to heat treatment followed by confinement. At given confining pressure P c, matrix bulk modulus K s and Biot's coefficient b increase with heat-treatment temperature above 105 °C. The increase in b means that mortar becomes less and less cohesive and more and more of a granular nature. Moreover, Biot's coefficient b and solid matrix bulk modulus K s are independent of confining pressure P c for intact mortar, whereas they decrease for heat-treated mortars when P c increases. From literature analysis alone, it was quite unexpected that after heat treatment, K s should vary under confinement. This is interpreted as the closure, under confining pressure, of micro-void connections and of micro-cracks created by heat-treatment. Therefore, increasing confinement induces more and more occluded pores in the solid matrix, whereby K s diminishes.

The solid phase stress tensor in porous media mechanics and the Hill–Mandel condition

An assessment of the stress tensors used currently for the modeling of partially saturated porous media is made which includes concepts like total stress, solid phase stress, and solid pressure. Thermodynamically constrained averaging theory is used to derive the solid phase stress tensor. It is shown that in the upscaling procedure the Hill conditions are satisfied, which is not trivial. The stress tensor is then compared to traditional stress measures. The physical meaning of two forms of solid pressure and of the Biot coefficient is clarified. Finally, a Bishop–Skempton like form of the stress tensor is obtained and a form of the total stress tensor that does not make use of the effective stress concept.

Microporomechanical behavior of perfectly straight unidirectional fiber assembly: Theoretical and experimental

The theory of poromechanics, most often used for macroscopically isotropic granular material, is applied to unidirectional fiber assemblies. When the deformation of the constituting fibers of those assemblies is taken into account during undrained forced compression, the Biot coefficients are expected to be lower than one. In the case of transversely isotropic materials, two independent Biot coefficients exist, one according to the fiber length direction and another relative to the plane perpendicular to that direction. The present paper aims at predicting the Biot coefficients for perfectly straight unidirectional (UD) fiber assemblies using micromechanical methods: dilute, Mori–Tanaka and Ponte Castañeda–Willis estimates. An experimental procedure, based on both uniaxial and triaxial material testing machines, is also detailed and applied to rubber fiber assemblies. The theoretical and experimental Biot coefficient results are compared and show a good agreement.

A framework for the construction of state surfaces of unsaturated soils in the elastic domain

This paper deals with volume changes of an unsaturated soil in the elastic domain. Two constitutive relations are established in the general framework of unsaturated porous media mechanics. They express the variations of the void ratio e and the water content w with respect to the mean net stress (σ − pg*) and the suction s. Associated with validated experimental results, these relations give access to the construction of the state surface e = f((σ − pg*), s) in the elastic domain. This surface shows two domains: the first, near saturation, is based on the hypothesis of a Biot's coefficient bw equal to 1; the second, for high suctions, is based on an experimental value of bw. Next, a preliminary construction of the state surface Gsw = g((σ − pg*), s) is proposed for a given soil. Although restricted to the elastic domain, these constructions give a new understanding of the suction role in important problems related to overconsolidation, shrinkage, or swelling of unsaturated soils.

A coupled thermo-hydro-mechanical-damage model for concrete subjected to moderate temperatures

This study focuses on the concrete behavior subjected to moderate temperatures, with a particular emphasis on the transient thermo-hydric stage. A simplified coupled thermo-hydro-mechanical model is developed with the assumption that the gaseous phase is composed uniquely of vapor. Estimations of the mechanical parameters, Biot coefficient and permeability as a function of damage and saturation degree are provided by applying effective-medium approximation schemes. The isotherm adsorption curves are supposed to depend upon both temperature and crack-induced porosity. The effects of damage and parameters linked to transfer (in particular the adsorption curves) on the concrete structure response in the transient phase of heating are then investigated and evaluated. To this aim, the model is applied to the simulation of concrete cylinders with height and diameter of 0.80 m subjected to heating rates of 0.1 and 10 °C/min up to 160 °C. The numerical results are analyzed, commented and compared with experimental ones in terms of water mass loss, temperatures and gas pressures evolutions. A numerical study indicates that some parameters have a greater influence on the results than others, and that certain coupling terms in the mass conservation equation of water may be neglected.

Elastic behaviour of North Sea chalk: A well‐log study

ABSTRACTWe present two different elastic models for, respectively, cemented and uncemented North Sea chalk well‐log data. We find that low Biot coefficients correlate with anomalously low cementation factors from resistivity measurements at low porosity and we interpret this as an indication of cementation. In contrast, higher Biot coefficients and correspondingly higher cementation factors characterize uncemented chalk for the same (low) porosity. Accordingly, the Poisson's ratio–porosity relationship for cemented chalk is different from that of uncemented chalk.We have tested the application of the self‐consistent approximation, which here represents the unrelaxed scenario where the pore spaces of the rock are assumed to be isolated, and the Gassmann theory, which assumes that pore spaces are connected, as tools for predicting the effect of hydrocarbons from the elastic properties of brine‐saturated North Sea reservoir chalk. In the acoustic impedance–Poisson's ratio plane, we forecast variations in porosity and hydrocarbon saturation from their influence on the elastic behaviour of the chalk. The Gassmann model and the self‐consistent approximation give roughly similar predictions of the effect of fluid on acoustic impedance and Poisson's ratio, but we find that the high‐frequency self‐consistent approach gives a somewhat smaller predicted fluid‐saturation effect on Poisson's ratio than the low‐frequency Gassmann model. The Gassmann prediction for the near and potentially invaded zone corresponds more closely to logging data than the Gassmann prediction for the far, virgin zone. We thus conclude that the Gassmann approach predicts hydrocarbons accurately in chalk in the sonic‐frequency domain, but the fluid effects as recorded by the acoustic tool are significantly affected by invasion of mud filtrate.The amplitude‐versus‐angle (AVA) response for the general North Sea sequence of shale overlying chalk is predicted as a function of porosity and pore‐fill. The AVA response of both cemented and uncemented chalk generally shows a declining reflectivity coefficient versus offset and a decreasing normal‐incidence reflectivity with increasing porosity. However, for the uncemented model, a phase reversal will appear at a relatively lower porosity compared to the cemented model.

How depositional texture and diagenesis control petrophysical and elastic properties of samples from five North Sea chalk fields

Chalk samples from Dan, Tyra, South Arne, Valhall and Ekofisk fields were collected from hydrocarbon-bearing intervals in the Cretaceous Tor and Palaeogene Ekofisk formations in the central North Sea. The samples were compared with respect to stable isotope ratios, lithotype, texture, porosity, permeability, capillary entry pressure, as well as the dynamic elastic Biot's coefficient and Poisson's ratio. The depositional texture and present grain-size distribution were quantified by petrographic image analysis. Oxygen isotope ratio and Biot's coefficient were used as indicators of cementation. Porosity varies more than 20 porosity units within each hydrocarbon field and is controlled by three parameters: (1) sorting as expressed by Dunham texture, so that mudstones tend to have highest porosity and packstones the lowest; (2) sorting of the carbonate mud, where a mixture of clay-size chalk particles and silicates tend to reduce porosity; and (3) by pore-filling cementation. The relative significance of these parameters varies with field and formation. The presence of chalk clasts as an indicator of re-deposited chalk seems to have no relationship to porosity. Permeability and capillary entry pressure depend on porosity and mineral content as expressed in specific surface. Prediction of permeability and capillary entry pressure may be aided by information on carbonate content or on Poisson's ratio.

About drying effects and poro-mechanical behaviour of mortars

The drying of mortar and concrete generally affects their elastic properties and strength. Such property variations are of interest, since they affect the durability of structures designed with these materials. Previous studies, conducted on two mortars with different W/C ratio, showed that drying such material led to decreases in their elastic modulus and their Poisson ratio. Although generally attributed to micro-cracking, these decreases are nevertheless consistent with simple poro-mechanical effects i.e. transition from undrained behaviour toward drained behaviour. The main objective of this experimental study was therefore to state whether or not pore fluid pressure played a direct role in the variation of elastic properties measured on saturated or dried material. Poro-mechanical measurements of drained and undrained bulk moduli, Biot and Skempton coefficients were carried out. They actually revealed that poro-mechanical coupling effects could explain a part of the differences observed between elastic properties. Ethanol was chosen as the pore fluid to avoid possible unexpected chemical effects on the materials tested (often observed with water). To conclude about the saturation effects, a complementary experimental procedure was conducted in two steps: compression tests were performed on previously dried samples, which were then re-saturated with ethanol before being mechanically tested again. The results obtained are unambiguous and showed that poro-mechanical effects are not involved in the change in elastic properties. On an other hand, the major roles played by microcracks induced by drying and by capillary suction are highlighted: (a) The decrease in elastic properties, observed as drying occurs, is shown to be due to induced micro-cracking; (b) the increase in capillary pressure, brought about by drying, leads to an increase in the mortar failure strength.

Investigation of the undrained poroelastic response of sandstones to confining pressure via laboratory experiment, numerical simulation and analytical calculation

Abstract To describe the poroelastic behaviour of sandstones, two factors have to be considered: the grain structure and the pore volume included. Changes in these two factors through diagenetic processes, tectonic loading or other forces lead to different results. Often external stresses induce a compaction of the rock and, therefore, a reduction of pore volume and an increased fluid pressure. Under undrained conditions, the largest pore pressure response can be observed. Besides the Biot coefficient, the Skempton coefficient ( B ) is one of the most important variables of elastic rock deformation, as it describes the pore pressure change related to the acting stresses. This study shows three ways of determining the Skempton coefficient and gives evidence of its pressure dependence. First, the undrained poroelastic response of a Bentheimer sandstone sample to confining pressure change was measured. Second, a thin-section micrograph was transferred into a finite-element model, including a discretization of the grain structure and the pore space. Finally, the Skempton coefficient of a linear elastic hollow sphere was calculated to prove the laboratory experiment and the numerical simulation.

Analysis of the solid phase stress tensor in multiphase porous media

AbstractConservation equations for mass, momentum, energy, and entropy are formulated for the phases and interfaces of a three‐phase system consisting of a solid and two immiscible fluids. The microscale equations are averaged to the macroscale by integration over a representative elementary volume. Thermodynamic statements for each of the phases and interface entities are also formulated at the microscale and then averaged to the macroscale. This departure from most uses of thermodynamics in macroscale analysis ensures consistency between models and parameters at the two scales. The expressions for the macroscale rates of change of internal energy are obtained by differentiating the derived forms for energy and making use of averaging theorems. These thermodynamic expressions, along with the conservation equations, serve as constraints on the entropy inequality. A linearization of the resulting equations is employed to investigate the theoretical origins of the Biot coefficient that relates the hydrostatic part of the total stress tensor to the normal force applied at the solid surface by the pore fluids. The results here are placed in the context of other formulations and expressions that appear in the literature. Copyright © 2006 John Wiley & Sons, Ltd.

Rock salt dilatancy boundary from combined acoustic emission and triaxial compression tests

This paper presents an experimental investigation of rock salt dilatancy boundary based on combined acoustic emission and triaxial compression tests carried out on the rock salt samples from the Asse Salt Mine, Germany. The experimental results were evaluated to determine the dilatancy boundary under specified stress, stress loading rate and pore pressure. Pore volume changes caused by deviatoric stresses were measured during triaxial compression tests. The dilatancy boundary was then determined from the maximum compression on a stress–strain curve which separates the compression and dilatancy regions. Variations in acoustic emissions that occur during microcrack development under triaxial compression were recorded with ten sensors mounted on the outer surface of the cylindrical samples. A spontaneous increase observed in the cumulative number of acoustic events is consistent with the dilatancy boundary determined from the minimum pore volume. It is confirmed that the dilatancy boundary depends on both stress loading rate and pore pressure. The dilatancy boundary slightly decreases with increasing the loading rate, but increases with increasing the minor normal stress. High pore pressures accelerate the dilatancy. A Biot coefficient of 0.25 for Asse rock salt was determined from the dilatancy boundary. An analysis of two-dimensional (2D) fractal dimensions determined for several samples shows that the samples with smaller dimensions have slightly higher dilatancy boundaries. The dilatancy boundary values of Asse rock salt are lower than those reported for crystalline rocks, although both rock types show similar dilatancy behaviour.

Pétrofabrique et propriétés mécaniques des argilites

The petrofabric of mudstones influences their thermal, hydraulic and mechanical properties. Measurements of thermal diffusivity, permeability and velocity of ultrasonic waves show that the anisotropy of these properties in this rock is rather low. The diminution of Biot coefficient with axial stress cannot be explained only by the diminution of porosity even by considering an incompressible matrix to 1% for an axial stress varying from 8 to 24 MPa corresponds. The origin of the diminution could be explained by the loss of connection of part of the porous system, hitherto connected, under the effect of an increasing axial stress. An isotropic elastoplastic model is a reasonable approach for the description of the mechanical behaviour of mudstone. However, it is useful to take into account the damage by microfissuring, which could not be neglected and must be considered coupled with plasticity. To cite this article: F. Homand et al., C. R. Geoscience 338 (2006).

Explicit use of the Biot coefficient in predicting shear‐wave velocity of water‐saturated sediments

ABSTRACTPredicting the shear‐wave (S‐wave) velocity is important in seismic modelling, amplitude analysis with offset, and other exploration and engineering applications. Under the low‐frequency approximation, the classical Biot–Gassmann theory relates the Biot coefficient to the bulk modulus of water‐saturated sediments. If the Biot coefficient under in situ conditions can be estimated, the shear modulus or the S‐wave velocity can be calculated. The Biot coefficient derived from the compressional‐wave (P‐wave) velocity of water‐saturated sediments often differs from and is less than that estimated from the S‐wave velocity, owing to the interactions between the pore fluid and the grain contacts. By correcting the Biot coefficients derived from P‐wave velocities of water‐saturated sediments measured at various differential pressures, an accurate method of predicting S‐wave velocities is proposed. Numerical results indicate that the predicted S‐wave velocities for consolidated and unconsolidated sediments agree well with measured velocities.