Skempton Coefficient Research Articles

The Impact of Size-Dependent and Stress-Dependent Fracture Properties on the Biot and Skempton Coefficients of Fractured Rocks

The Impact of Size-Dependent and Stress-Dependent Fracture Properties on the Biot and Skempton Coefficients of Fractured Rocks

An extended continuum-mechanics standard linear solid rheology for fluid-saturated porous rock

SUMMARY In a large body of rock-physics research, seismic wave velocity dispersion and attenuation in fluid-saturated porous rock are studied by constructing analytical or numerical models for time- or frequency-dependent dynamic (effective, or viscoelastic) moduli. A key and broadly used model of such kind is the Zener's, or the standard linear solid (SLS). This model is qualitatively successful in explaining many field and laboratory observations and serves as the key element of many generalizations such as the Burgers model for plastic deformations or the generalized SLS explaining band-limited or near-constant seismic attenuation. However, as a physical model of fluid-saturated porous rock, the SLS has several major limitations: disregard of inertial effects, absence of secondary wave modes and lack of key physical parameters such as porosity and Skempton coefficients. Grainy and porous rock is an unconsolidated material in which the effective density is frequency-dependent, and its effects on wave velocities may exceed those of the dynamic modulus. To overcome these limitations of the empirical SLS, we propose a rigorous rheologic model based on classical continuum mechanics and called the extended SLS, or eSLS. This rheology explains the available attenuation and dispersion observations equally well, but it is also close to Biot's model, honours all poroelastic relations, includes inertial effects, and reveals several new physical properties of the material. Detailed comparison of the eSLS and Biot's models gives a physical-mechanism-based classification of wave-induced fluid flow (WIFF) phenomena. In this classification, the so-called ‘global-scale’ flows occur in Biot's type structures within the material, whereas the ‘local-scale’ WIFF occurs in eSLS-type structures. Combining Biot's and eSLS models gives a broad class of rheologies for linear anelastic phenomena within rock with a single type of porosity. The model can be readily generalized to multiple porosities and different types of internal variables, such as describing squirt flows, wetting or thermoelastic effects. Modelling is conducted with relatively little effort, using a single matrix equation similar to a mechanical form of the standard SLS. By combining the eSLS and Biot's models, observations of dynamic-modulus dispersion and attenuation can be inverted for macroscopic mechanical properties of porous materials.

Bending analysis of FGSP nanoplate resting on elastic foundation by using nonlocal quasi-3D theory

In this paper, the bending response of a functionally graded saturated porous (FGSP) nanoplate resting on the Pasternak elastic foundation is analyzed within the framework of quasi-3D higher-order shear deformation theory (quasi-3D HSDT) for the first time. The material properties are presumed to change gradually along the thickness direction following three patterns of porosity distribution: uniform, non-uniform symmetric, and asymmetric. The theory of poroelasticity developed by Biot is utilized in modeling the stress-strain relationships for the saturated condition. Moreover, the nanoscale effects of the structures are considered by Eringen's nonlocal elasticity theory. The governing equations are derived by using the principle of minimum potential energy according to quasi-3D HSDT, which ensures transverse shear stress-free on the upper and lower surfaces of the nanoplate. Based on the obtained closed-form solution, the impacts of the porosity distribution patterns, porosity coefficient, Skempton coefficient, geometrical parameters, elastic foundation, and nonlocal parameters on the bending behavior have been explored. According to the findings, when the pores are saturated by the fluid, the plate stiffness increases. Additionally, increasing the values of the nonlocal parameter for FGSP nanoplates leads to an increase in deflection and stresses. Finally, the present study quantitatively reveals the size-dependent effects of a saturated porous medium.

Novel procedure for measuring the saturation of the pore-pressure system in piezocone tips

Uncertainty in the measurement of the cone penetration test with pore-water pressure measurement (i.e. piezocone testing CPTU) can be caused by an equipment malfunction, poor calibration and maintenance of sensors, but also by a lack of saturation of the piezocone. The standards allow for many different options in terms of filters, saturation fluids and methodologies. The novel device described herein aims at measuring in a very simple manner the degree of saturation of the piezocone prior to testing. Because the quality of pore-water pressure measurements can currently only be assessed a posteriori – that is, after the test has been completed, or at best while testing, the methodology proposed herein has the potential to provide quality benchmarks for CPTU. The tool measures a parameter inspired by the well-established Skempton coefficient B, which is routinely used to estimate the sample saturation prior to laboratory testing. Furthermore, by performing the same measurement after a CPTU test, it is possible to assess whether saturation was lost during cone penetration. The paper describes a set of preliminary results that show a clear correlation between the saturation degree of filters and/or of the overall pore-pressure measurement system and the B parameter proposed, thus providing a proof of concept in laboratory settings.

Stress‐Induced Anisotropic Poroelasticity in Westerly Granite

AbstractWe measured poroelastic properties of cracked granite under triaxial conditions, at elevated confining pressure and a range of differential stresses. Skempton's coefficients and undrained Young's modulus and Poisson's ratio were determined directly by recording in situ fluid pressure during rapid cycles of axial and radial stress. Drained properties were measured both statically and dynamically at ultrasonic frequencies. At a given confining pressure, increasing differential stress leads to the development of elastically transverse isotropy, with symmetry axis aligned with the compression axis. Skempton's coefficients are also anisotropic, with larger changes in pore pressure in response to radial stress (coefficient Bx) than to axial stress steps (Bz). The anisotropy in the Skempton coefficients increases with increasing differential stress, with Bz decreasing and Bx slightly increasing. The evolution of static moduli and the Skempton coefficients is well approximated by Gassmann's equation using dry moduli obtained from ultrasonic measurements. Simplified predictions of the Skempton coefficients based on crack density tensors inverted from dynamic data also shows acceptable agreement with direct observations. Perfect quantitative agreement is not reached, due to the imprecision of our dynamic measurements, model simplifications, and inherent differences between static moduli obtained using stress steps of several MPa stress and dynamic ultrasonic stress oscillations.

Thermo-elastic buckling behaviors of advanced fluid-infiltrated porous shells integrated with GPLs-reinforced nanocomposite patches

In the existing paper, thermo-mechanical buckling performance of three-layered advanced fluid-infiltrated porous shells that are embedded between two GPLs-reinforced composite layers is investigated. The influence of the fluids in the pores is investigated using Skempton coefficient. Imperfection types and ratio effects are also considered. Moreover, for the face layers, Graphene nanoplatelets (GPLs) weight fraction and dispersion patterns are studied. After deriving the governing equations employing virtual displacement method, they are solved analytically via Navier’s approach. Finally, the impact of different factors on the critical buckling loads (CBLs) and temperatures (CBTs) of cylindrical, spherical, and doubly-curved shells is assessed.

Evaluating a Poroelastic Model via Pore Pressure Signals in Seafloor Sediments

AbstractBelow a certain threshold of wave energy for a given depth, unconsolidated sediment should respond to wave action poroelastically. To this end, a generalized simple poroelastic model was applied to in situ data collected in coastal sediments near Oceanside, California. Pore pressure data were recorded by sensors attached to cylindrical unexploded ordinance surrogates, and water pressure sensors on bottom moorings. Orientation data were collected from inertial motion units attached to the surrogates, which were placed on the seabed and subsequently shallow buried during a high‐energy event on 13 February 2021. The model is characterized by two parameters. A consolidation coefficient controls the depth of diffusive flow, and Skempton's coefficient represents the ratio of the original signal into components transmitted by the sediment matrix and the pore fluid. The model assumes that the pore pressure signal is coherent with the seafloor signal; spectral coherence above 0.75 was used to select the frequency range from 0.05 to 0.2 Hz for the analysis. Spectral analysis methods were then applied to obtain averaged auto spectral densities of the pore pressure and bottom mooring signals over the coherent frequency band. Attenuation between sensor pairs was calculated. The results yielded burial depths over time for the surrogates that agreed with changes in significant wave heights, liquefaction, and recovery depths.

Why an indirect estimate of the unjacketed pore modulus may not work

The unjacketed pore modulus is a measure of changes in pore volume due to the variation in pore pressure for constant Terzaghi effective pressure. In the oil and gas industry, this parameter is commonly used in linking the undrained bulk modulus of a porous saturated rock to the drained bulk modulus and the properties of the fluid and solid components through Gassmann’s equation. For an ideal porous rock, the unjacketed pore modulus should be identical to the solid bulk modulus of the major mineral constituent. While direct measurements confirm this prediction, previous indirect estimates suggest that this parameter may have values close to the bulk modulus of the fluid rather than the bulk modulus of the grains. These indirect results demonstrate that the estimate of the undrained bulk modulus, due to substitution of different fluids in the rock, can change considerably if the unjacketed pore modulus is equal to the bulk modulus of the fluid. In this paper, we first briefly provide the results of the direct approach and then present laboratory measurements of Skempton’s pore pressure coefficient for a quartz arenite sandstone and use these data to indirectly estimate the unjacketed pore modulus. We show that the difficulties associated with the measurement of the Skempton coefficient clearly contribute to the underestimation from the indirect approach.

Chemical compaction and fluid segregation in piston cylinder experiments

Fluid segregation is a ubiquitous process in deep-seated rocks, mainly driven by buoyancy and gravitational compaction, which occur in drained systems until permeability decreases and the system become practically undrained. The mechanism and rate of fluid segregation following this stage are poorly constrained, despite the importance of these factors for fluid distribution and the physical and chemical properties in the Earth. To this end, we performed sintering experiments of quartzite, with H2O–CO2 fluids of 1.9%–18.0% added volume fraction using a piston-cylinder apparatus at 900°C under nominally isotropic pressure of 1 GPa. The subsequent chemical redistribution of silica and fluids resulted in capsule-scale fluid segregation (CFS) and the formation of dense quartzite (∼0.3%) within 192 h in pure H2O systems. Comparative experiments showed that dissolution/precipitation was not caused by the temperature gradient. Instead, we considered the fluid pressure difference within the experimental capsule between fluid-rich and fluid-poor domains caused by the different response against small pressure oscillation during experiments. Fluid-rich rock is elastically “soft” (i.e., having larger Skempton's coefficient) compared to fluid-poor rock to increase pore fluid pressure under a compressional stress change. Therefore, more silica dissolves in fluid-rich domains with higher solubility, while it precipitates in fluid-poor domains through diffusive transport, expanding porosity contrast. This chemical compaction in the capsule scale is effective as long as elasticity dominates matrix viscosity because the pressure difference relaxes with time. The calculated relaxation time of quartzite at the experimental condition was comparable to the intervals of the oil pressure addition which produced stress change. A one-dimensional model for porosity evolution showed that time-averaged SiO2 diffusivity during CFS was much larger than the grain boundary diffusivity of tightly sintered quartz aggregates, but was comparable to the diffusivity of SiO2 in aqueous fluid with ∼0.1 vol% fluid fraction. During metamorphism where the fluid pressure difference may be maintained for longer duration, the spontaneous silica cementation may play a role for formation of syn-metamorphic veins and control of the amount of pore fluids transported to deep Earth.

Thermo-Poromechanical Properties of Pierre II Shale

During the injection of carbon dioxide (CO2) for CO2 capture and storage (CCS) operations, the near-well (including casing, cement, and rock around it) can undergo several thermal loadings. These loadings can significantly increase or decrease the pore pressure and can thus lead to mechanical failure of the cement sheath and rock formation. When these failures appear in the caprock, they can compromise the integrity of the storage site. The understanding of thermo-mechanical behaviour of a potential caprock shale is, therefore, of great importance for the success of CCS operations. In this paper, experiments were performed on Pierre II shale, under confining and initial pore pressures comparable to field conditions. A 60 °C loading amplitude (between 30 and 90 °C) was applied on the shale material both under undrained and drained conditions. The results, analysed within the framework of anisotropic thermo-poro-elasticity, highlight the anisotropic behaviour of the thermal expansion coefficients, as well as of the Skempton coefficient. The thermal pressurization coefficient was also evaluated and showed a potential pore pressure change as high as 0.11 MPa/°C.

Geometrically Nonlinear Study of Functionally Graded Saturated Porous Plates Based on Refined Shear Deformation Plate Theory and Biot’s Theory

This research presents the geometrically nonlinear investigation of functionally graded saturated porous material (FGSPM) plate under undrained conditions. In conjunction with von Karman’s nonlinearity, the refined shear deformation plate theory (RSDPT) is implemented to model the FGSPM plate. The effective material characteristics of the saturated porous plate change constantly in the thickness direction. The pores of the saturated porous plate are examined in fluid-filled conditions. Thus, the constitutive equations are established using Biot’s linear poroelasticity theory. The governing equations are developed by combining a nonlinear finite element technique with Hamilton’s principle. Then, the direct iterative approach is utilized to extract the geometrically nonlinear numerical results. The emphasis is placed on exploring the effects of numerous parameters such as Skempton coefficient, volume fraction grading index, porosity volume index, porosity distributions, and boundary conditions during the extensive numerical analyses on the linear frequency, large amplitude frequencies, and nonlinear central deflections of the FGSPM plate. It is evident from the investigation that saturated fluid in the pores substantially impacts the nonlinear deflection and vibration behavior of the FGSPM plate.

Three-Dimensional Buckling Analysis of Functionally Graded Saturated Porous Rectangular Plates under Combined Loading Conditions

The present work studies the buckling behavior of functionally graded (FG) porous rectangular plates subjected to different loading conditions. Three different porosity distributions are assumed throughout the thickness, namely, a nonlinear symmetric, a nonlinear asymmetric and a uniform distribution. A novel approach is proposed here based on a combination of the generalized differential quadrature (GDQ) method and finite elements (FEs), labeled here as the FE-GDQ method, while assuming a Biot’s constitutive law in lieu of the classical elasticity relations. A parametric study is performed systematically to study the sensitivity of the buckling response of porous structures, to different input parameters, such as the aspect ratio, porosity and Skempton coefficients, along with different boundary conditions (BCs) and porosity distributions, with promising and useful conclusions for design purposes of many engineering structural porous members.

Empirical Formula for Dynamic Biot Coefficient of Sandstone Samples from South-West of Poland

In this research, two empirical correlations have been introduced to calculate the dynamic Biot coefficients of low-porosity and high-porosity sandstone samples from two open pit mines located in South-West Poland. The experiments were conducted using an acoustic velocity measurement apparatus. Under the undrained condition, firstly, the confining pressure was increased in increments of 200 psi, and the values of pore pressure and dynamic elastic modulus were recorded. This experiment was continued until the Skempton coefficient remained in the range of 0.98–1. Secondly, an experiment on the same sample was conducted under drained conditions, and the corresponding dynamic elastic moduli were calculated. Then, using the calculated dynamic elastic moduli, the dynamic Biot coefficient was determined for each sample under different confining pressure. Finally, two empirical correlations were formulated for each sandstone category. The results demonstrate that, as the confining pressure increases, the Biot coefficient decreases from 0.79 to 0.50 and from 0.84 to 0.45 for low-porosity and high-porosity samples, respectively. Furthermore, as the porosity increases, the sandstone behavior increasingly approaches that of soil. The empirical correlations can be used for sandstone formations with the same porosity in projects where there is not a measurement method for the Biot coefficient.

On the higher-order thermal vibrations of FG saturated porous cylindrical micro-shells integrated with nanocomposite skins in viscoelastic medium

Since the multi-layered structures are widely used nowadays, and due to interesting applications of cylindrical shells, this study is dedicated to analyzing free vibrational behaviors of functionally graded saturated porous micro cylindrical shells with two nanocomposite skins. Based on Biot's assumptions, constitutive relations for the core are presented and effective properties of the skins are determined via the rule of mixture. A sinusoidal theory is used to capture the shear deformation effects, and to account for the scale effects, the modified couple stress theory is employed which suggests a material length-scale parameter for predicting the results in small-dimension. With the aid of extended form of Hamilton's principle for dynamic systems, differential equations of motion are extracted. Fourier series functions are used to obtain natural frequencies and after validating them, a set of parametric studies are carried out. The results show the significant effects of porosity and Skempton coefficient, pores placement patterns, CNTs addition and distribution patterns, temperature variations, material length-scale parameter and viscoelastic medium on the natural frequencies of the microstructure. The outcomes of this work could be used to design and manufacture more reliable micro cylindrical structures in thermo-dynamical environments.

Dynamic analysis of functionally graded rotating thick truncated cone made of saturated porous materials

In the current article, dynamic behavior of functionally graded saturated porous rotating thick truncated cone is investigated for the first time. It is assumed that porosity distributions along the thickness are (1) symmetric nonlinear, (2) nonlinear asymmetric and (3) uniform distributions. The Biot poroelastic law instead of Hooke’s law is used to model the constitutive equations. Also, 2D axisymmetric theory of elasticity instead of simple shell theories is used for the governing motion equations. Graded finite element and Newmark methods have been used to solve the governing motion equations. First, natural frequencies of stationary functionally graded saturated porous cone are obtained, and then by considering the effect of centrifugal force and spin softening effect, dynamic response of porous cone in an undrained condition for different variables such as porosity and Skempton coefficients, rotational velocity, semi vertex angle of the cone and different distribution of porosity has been investigated. Obtained results denote that by increasing Skempton coefficient, natural frequencies are increased, and natural frequencies of PUD distribution are lower than other distributions. Also, rotational velocity and semi-vertex angle have significant effect on the displacement and stress field.

Axisymmetric Thermal Stresses of a Piezoelectric Poroelastic Circular Solid Plate

Abstract This paper presents the steady-state thermoelasticity solution for a circular solid plate is made of an undrained porous piezoelectric hexagonal material symmetry of class 6 mm. The porosities of the plate vary through the thickness; thus, material properties, except Poisson's ratio, are assumed as exponential functions of axial variable z in cylindrical coordinates. Having axisymmetric general form, external thermal and electrical loads are acted on the plate and the piezothermoelastic behavior of the plate is investigated. Using a full analytical method based on Bessel Fourier's series and separation of variables, the governing partial differential equations are derived. A formulation is given for the displacements, electric potential, thermal stresses, and electric displacements resulting from prescribed the general form of thermal, mechanical, and electric boundary conditions. Finally, the application of the derived formulas is illustrated by an example for a cadmium selenide solid, the results of which are presented graphically. Also, the effects of material property indexes, the porosity, and Skempton coefficients are discussed on the displacements, thermal stresses, electrical potential function, and electric displacements.

An experimental investigation on the poroelastic response of a water-saturated limestone to hydrostatic compression

This contribution addresses an experimental study on the poroelastic behavior of a water-saturated limestone. Samples come from Sarvak limestone, a major hydrocarbon-producing reservoir in Iran. Three back-saturated intact specimens were subjected to hydrostatic compression with different boundary pressure conditions. A coherent workflow from testing to data interpretation was established to efficiently deal with possible sources of uncertainty and provide accurate results. Obtained results revealed that drained and undrained bulk moduli, as well as the Biot and Skempton coefficients, strongly depend on the applied Terzaghi effective stress. Conversely, calculated values for the unjacketed solid modulus \( {K}_s^{\prime } \) were found to be constant in the range of applied stresses. Variations of the poroelastic moduli with the effective stress were modeled and used to extract depth-dependent poroelastic properties of the studied lithotype along a well. Besides, evaluated poroelastic moduli were utilized to verify the linear theory of poroelasticity. The indirectly calculated unjacketed pore modulus \( {K}_s^{\prime \prime } \) appeared to be highly variable with changes in the effective stress and possesses even negative values when the Skempton coefficient is lower than a critical value. A comparison between unjacketed bulk moduli and the bulk deformation modulus of the main mineral constituents Ks also disclosed that they are basically uncorrelated, and the assumption of ideal porous rock, \( {K}_s^{\prime \prime }={K}_s^{\prime }={K}_s \), does not hold in general for the examined rock samples. Nevertheless, consideration of the ideality assumption in the theory of poroelasticity may result in reliable estimates for the Skempton coefficient in certain circumstances, particularly at high effective stresses.

Teleseismic waves reveal anisotropic poroelastic response of wastewater disposal reservoir

Connecting earthquake nucleation in basement rock to fluid injection in basal, sedimentary reservoirs, depends heavily on choices related to the poroelastic properties of the fluid-rock system, thermo-chemical effects notwithstanding. Direct constraints on these parameters outside of laboratory settings are rare, and it is commonly assumed that the rock layers are isotropic. With the Arbuckle wastewater disposal reservoir in Osage County, Oklahoma, high-frequency formation pressure changes and collocated broadband ground velocities measured during the passing of large teleseismic waves show a poroelastic response of the reservoir that is both azimuthally variable and anisotropic; this includes evidence of static shifts in pressure that presumably relate to changes in local permeability. The azimuthal dependence in both the static response and shear coupling appears related to tectonic stress and strain indicators such as the orientations of the maximum horizontal stress and faults and fractures. Using dynamic strains from a nearby borehole strainmeter, we show that the ratio of shear to volumetric strain coupling is ~0.41 which implies a mean Skempton's coefficient of <inline-formula><tex-math id="M2">\begin{document}$ A = 0.24 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="SI6250-Andrew_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="SI6250-Andrew_M2.png"/></alternatives></inline-formula> over the plausible range of the undrained Poisson's ratio. Since these observations are made at relatively low confining pressure and differential stress, we suggest that the hydraulically conductive fracture network is a primary control on the coupling between pore pressure diffusion and elastic stresses in response to natural or anthropogenic sources.

Large deflection analysis of functionally graded saturated porous rectangular plates on nonlinear elastic foundation via GDQM

In the current study, large deflection analysis of a functionally graded saturated porous (FGSP) rectangular plate subjected to transverse loading which is located on a nonlinear three-parameter elastic foundation is provided. The constitutive law for the porous materials is written based on Biot's model which considers the effect of fluids within the pores. The mechanical properties of the plate are changed through its thickness according to different functions which are called porosity distributions. The shear deformation effects are taken into account, accordingly, the first-order shear deformation theory (FSDT) is used to describe the displacement components of the plate. Employing the Minimum total potential energy principle and calculus of variation, the governing equations, and associated boundary conditions are extracted. A generalized differential quadrature method (GDQM) is used to solve them for various boundary conditions. The results for the simpler state are validated with the previously published works and then the effects of different parameters on the deflection of the plate are investigated. It is seen increasing the porosity and Skempton coefficient, enhances and reduces the deflection of the structure, respectively. The results of this study may help to design and manufacture more reliable engineering structures that are exposed to loads.

Rock Damage and Aquifer Property Estimation from Water Level Fluctuations in Wells Induced by Seismic Waves: A Case Study in X10 Well, Xinjiang, China

There is a coupling relationship between surrounding rock stress, deformation, and fracture evolution, especially in the microdynamics of the crust caused by mining activities and earthquakes. Previous research has investigated many cases regarding the coseismal water level responses and proposed a method to calculate the aquifer parameters by tidal analysis. However, to date, measurement of the degree of rock damage in the field has not been reported. Quantifying the fracture characteristics is essential for accurate evaluation of rock stability. This study has analyzed the relationship between the seismograms and hydroseismograms in response to the Mw 7.8 Solomon Islands earthquake and the Mw 7.8 Kaikōura earthquake, both events occurring in 2016. The calculated and measured changes in water level in the X10 well were fitted in order to study the relationships among the volumetric strain, the deviatoric strain, and the oscillations in the pore pressure. Then, we further estimate the degree of rock damage and the hydraulic characteristics of the aquifer. The results showed that the values for the rock damage parameter, 0.662 &lt; αD &lt; 0.754, and the Skempton coefficient, −0.100 &lt; A &lt; 0.026, estimated for the Solomon Islands earthquake signified higher damage and dilatancy in the X10 well. Also, the respective values for the parameters, 0.293 &lt; αD &lt; 0.363 and 0.226 &lt; A &lt; 0.251, calculated for the Kaikōura earthquake signified a lower degree of rock damage. It is concluded that the changes in the pore pressure were influenced by both the volumetric strain and the deviatoric strain. The degree of rock damage and the hydraulic properties of the aquifer estimated from the water level fluctuations in the wells which were induced by the seismic waves represent the actual aquifer characteristics.