Polyaxial Loading Research Articles

A novel elastoplastic model for Yunnan sandstone under poly-axial loading

The lack of understanding of plastic hardening (softening) laws, especially under anisotropic stress conditions, results in inappropriate geotechnical management. Most of the yielding envelopes do not consider the effect of intermediate principal stress and the influence of Lode’s angle. In addition, the application of plastic flow rules regarding yielding surfaces compromises the softening of rock internal friction as well as the influence of Lode’s angle on the plastic potential. Moreover, the ductility to brittleness transition in the intermediate principal stress direction still requires a theoretical foundation. In this study, based on poly-axial testing results of Yunnan sandstone, we adopted a failure criterion with the intermediate principal stress proposed by Menétrey and Willam. The proposed new failure envelope was applied to capture the plastic evolution of rock samples. A plastic hardening-softening model is constructed, based on the framework of the plastic theory. The softening envelope is modified to better present the stress drop and considers the deterioration of rock internal friction in the post-peak stage of poly-axial loading. The differential of plastic potential according to the principal stresses is also modified, considering the rotation of Lode’s angle in the poly-axial loading tests. The model results were compared with laboratory testing results, which showed great consistency across 9 different loading tests (5 under triaxial stress and 4 under poly-axial stress with 22 stress-strain curves in total). The induced brittleness by the intermediate principal stress is also well captured by the proposed model.

Displacement velocity effects on rock fracture shear strengths

Triaxial shear tests are performed to assess the effects of displacement velocity and confining pressure on shear strengths and dilations of tension-induced fractures and smooth saw-cut surfaces prepared in granite, sandstone and marl specimens. A polyaxial load frame is used to apply confining pressures between 1 and 18 MPa with displacement velocities ranging from 1.15 × 10−5 to 1.15 × 10−2 mm/s. The results indicate that the shearing resistances of smooth saw-cut surfaces tend to be independent of the displacement velocity and confining pressure. Under each confinement the peak and residual shear strengths and dilation rates of rough fractures increase with displacement velocities. The sheared-off areas increase when the confining pressure increases, and the displacement rate decreases. The velocity-dependent shear strengths tend to act more under high confining pressures for the rough fractures in strong rock (granite) than for the smoother fractures in weaker rocks (sandstone and marl). An empirical criterion that explicitly incorporates the effects of shear velocity is proposed to describe the peak and residual shear strengths. The criterion fits well to the test results for the three tested rocks.

Effects of Displacement Velocity on Rock Fracture Shear Strengths under Large Confinements

Triaxial shear tests are performed to assess the effects of displacement velocity and confining pressure on shear strengths and dilations of tension-induced fractures and smooth saw-cut surfaces prepared in granite, sandstone and marl specimens. A polyaxial load frame is used to apply confining pressures between 1 and 18 MPa with displacement velocities ranging from 1.15×10-5 to 1.15×10-2 mm/s. The results indicate that the shearing resistances of smooth saw-cut surfaces tend to be independent of the displacement velocity and confining pressure. Under each confinement the peak and residual shear strengths and dilation rates of rough fractures increase with displacement velocities. The sheared-off areas increase when the confining pressure increases, and the displacement rate decreases. The velocity-dependent shear strengths tend to act more under high confining pressures for the rough fractures in strong rock (granite) than for the smoother fractures in weaker rocks (sandstone and marl). An empirical criterion that explicitly incorporates the effects of shear velocity is proposed to describe the peak and residual shear strengths. The criterion fits well to the test results for the three tested rocks.

Proppant backflow: Mechanical and flow considerations

One of the concerns of using proppant in geothermal wells, and particularly in enhanced geothermal systems, is proppant flowback. Particulate proppant maintain post-closure conductivity in hydraulically opened fractures. If that proppant is displaced from the near-wellbore region, either due to overflushing during stimulation or flowback to the wellbore at any time, the reduced fracture width chokes the injection or production. Two intermediate-scale laboratory analogs of a propped hydraulic fracture were prepared, and fluid was flowed through a normally stressed, propped fracture into a central wellbore. The tests were conducted in a polyaxial load frame. Acoustic/microseismic activity was measured during the injection programs. In one scenario—radial flow through a transverse fracture to a wellbore—the results suggest the creation of flow channels and nominally intact propped zones around the channels, maintaining fracture aperture. In the other—linear flow through a longitudinal fracture into a wellbore—there was substantially more proppant removal. The measurements have shown a greater tendency for proppant flowback in a linear flow situation (proppant movement is kinematically more restricted for radial convergent flow). The pressure gradients causing flow are exceedingly small and restraining flowback will be difficult. Convergent flow relationships could be an issue for injector wells, which will experience fluid flowback during hard shutdowns.

New artificial neural networks for true triaxial stress state analysis and demonstration of intermediate principal stress effects on intact rock strength

Simulations are conducted using five new artificial neural networks developed herein to demonstrate and investigate the behavior of rock material under polyaxial loading. The effects of the intermediate principal stress on the intact rock strength are investigated and compared with laboratory results from the literature. To normalize differences in laboratory testing conditions, the stress state is used as the objective parameter in the artificial neural network model predictions. The variations of major principal stress of rock material with intermediate principal stress, minor principal stress and stress state are investigated. The artificial neural network simulations show that for the rock types examined, none were independent of intermediate principal stress effects. In addition, the results of the artificial neural network models, in general agreement with observations made by others, show (a) a general trend of strength increasing and reaching a peak at some intermediate stress state factor, followed by a decline in strength for most rock types; (b) a post-peak strength behavior dependent on the minor principal stress, with respect to rock type; (c) sensitivity to the stress state, and to the interaction between the stress state and uniaxial compressive strength of the test data by the artificial neural networks models (two-way analysis of variance; 95% confidence interval). Artificial neural network modeling, a self-learning approach to polyaxial stress simulation, can thus complement the commonly observed difficult task of conducting true triaxial laboratory tests, and/or other methods that attempt to improve two-dimensional (2D) failure criteria by incorporating intermediate principal stress effects.

True-triaxial compressive strength of Maha Sarakham salt

True triaxial compressive strengths of the Maha Sarakham salt are experimentally determined by using a polyaxial load frame. The salt specimens with nominal dimensions of 5.4×5.4×10.8cm3 are loaded to failure with the minimum principal stress, σ3, varying from 0 to 7MPa, and the intermediate principal stress, σ2, varying from 0 to 80MPa. The major principal (axial) stress is increased at a rate of 0.5–1.0MPa/s until failure occurs. Based on the Coulomb criterion the internal friction angle determined from the triaxial loading condition (σ2=σ3) is 50°, and the cohesion is 5.0MPa. The elastic parameters of the salt tend to be independent of σ2 for the applied stress range. The effect of σ2 on the salt strength can be described best by the modified Wiebols and Cook criterion with a mean misfit of 3.5MPa. The (power law) Mogi criterion underestimates the salt strength, particularly under high σ3 values. The modified Lade and 3-D Hoek & Brown criteria overestimate the strength at all levels of σ3. The Coulomb and Hoek & Brown criteria cannot describe the salt strengths beyond the condition where σ2=σ3, as they cannot incorporate the effects of σ2. Both circumscribed and inscribed Drucker–Prager criteria severely underestimate σ1 at failure for all stress conditions, showing the largest mean misfit of 19.5MPa.

Shear strengths of sandstone fractures under true triaxial stresses

True triaxial shear tests have been performed to determine the peak shear strengths of tension-induced fractures in three Thai sandstones. A polyaxial load frame is used to apply mutually perpendicular lateral stresses (σp and σo) to the 76 × 76 × 126 mm rectangular block specimens. The normal to the fracture plane makes an angle of 59.1° with the axial (major principal) stress. Results indicate that the lateral stress that is parallel to the fracture plane (σp) can significantly reduce the peak shear strength of the fractures. Under the same normal stress (σn) the fractures under high σp dilate more than those under low σp. According to the Coulomb criterion, the friction angle decreases exponentially with increasing σp/σo ratio and the cohesion decreases with increasing σp. The lateral stress σp has insignificant effect on the basic friction angle of the smooth saw-cut surfaces. The fracture shear strengths under σp = 0 correlate well with those obtained from the direct shear tests. It is postulated that when the fractures are confined laterally by σp, their asperities are strained into the aperture, and are sheared off more easily compared to those under unconfined condition.

Effects of loading rate on strength and deformability of Maha Sarakham salt

Uniaxial and triaxial compression tests have been performed to assess the influence of loading rate on the compressive strength and deformability of the Maha Sarakham salt. The salt specimens with a nominal dimension of 5.4×5.4×5.4cm3 are compressed to failure using a polyaxial load frame. The lateral confining pressures are maintained constant at 0, 3, 7, 12, 20 and 28MPa while the axial stresses are increased at constant rates of 0.001, 0.01, 0.1, 1.0 and 10MPa/s until failure occurs. The salt elasticity and strength increase with the loading rates. The elastic (tangent) modulus determined at about 40% of the failure stress varies from 15 to 25GPa, and the Poisson's ratio from 0.23 to 0.43. The elastic parameters tend to be independent of the confining pressures. The strains induced at failure decrease as the loading rate increases. Various multiaxial formulations of loading rate dependent strength and deformability are derived. The variation of the octahedral shear stresses and strains induced at dilation and at failure with the applied shear stress rates can be best described by power relations. The distortional strain energy at dilation and at failure from various loading rates varies linearly with the mean normal stress. The proposed empirical criteria are applied to determine the safe maximum withdrawal rate of a compressed-air energy storage cavern in the Maha Sarakham salt formation. The strain energy criterion that considers both distortional and mean stress–strains at dilation tends to give the most conservative results.

Influence of Loading Rate on Deformability and Compressive Strength of Three Thai Sandstones

Uniaxial and triaxial compressive strength tests have been performed using a polyaxial load frame to assess the influence of loading rate on the strength and deformability of three Thai sandstones. The applied axial stresses are controlled at constant rates of 0.001, 0.01, 0.1, 1.0 and 10 MPa/s. The confining pressures are maintained constant at 0, 3, 7 and 12 MPa. The sandstone strengths and elastic moduli tend to increase exponentially with the loading rates. The effects seem to be independent of the confining pressures. An empirical loading rate dependent formulation of both deformability and shear strength is developed for the elastic and isotropic rocks. It is based on the assumption of constant distortional strain energy of the rock at failure under a given mean normal stress. The proposed multiaxial criterion well describes the sandstone strengths within the range of the loading rates used here. It seems reasonable that the derived loading rate dependent equations for deformability and shear strength are transferable to similar brittle isotropic intact rocks.

A Polyaxial System for Testing of Jointed Rock Mass Models

Abstract A polyaxial loading system was designed and developed at Indian Institute of Technology Delhi, India for laboratory testing of mechanical behavior of rock mass. The large-scale rock mass models of different joint geometry can be tested under polyaxial stress state simulating in situ stress conditions using this true-triaxial system. The system consists of a 1000 kN capacity vertical frame, a biaxial frame of 300 kN capacity fitted with two pairs of hydraulic jacks and platens, constant confining pressure unit for applying, monitoring, and maintaining horizontal stresses (σ2 and σ3) on specimen faces, eight-channel data acquisition system, and a personal computer to record all load and deformation data. Its working was verified by conducting true-triaxial testing on several models specimens of sand-lime blocks having three sets of orthogonal joints.

Kaiser Effect in Rocks Undergone Polyaxial Loading

The basic regularities are considered for manifesting the Kaiser effect in various regimes of polyaxial loading, which is of great importance for developing the methods of stress measurement in rocks. The correspondence between the results of experimental and theoretical investigations is shown.

Topographic characteristics of laboratory induced shear fractures

The topography of laboratory induced shear fracture surfaces of Westerly granite was studied. Three types of fracture surfaces were examined: (1) a fresh fracture from the shear failure of an intact sample under polyaxial loading (σ2 = 40 MPa > σ3 = 15 MPa); (2) a shear fracture subjected to frictional sliding of 100 μm under polyaxial loading; (3) a shear fracture subjected to frictional sliding of 800 μm under conventional triaxial loading (σ1 > σ2 = σ3 = 40 MPa). Both sliding distances are within the range of the grain size of Westerly granite. The results are represented by a power spectral method.

Complexities of rock fracture and rock friction from deformation of Westerly granite

A series of rock friction experiments has been carried out to study the complexities in rock fracture and rock friction. Intact Westerly granite samples were loaded to shear failure in a laboratory polyaxial loading apparatus. The resultant fractured samples were reloaded to cause frictional sliding. Both polyaxial loading (σ1 > σ2 > σ3 > 0) and equal confining condition (σ1 > σ2 = σ3 > 0) were used. The deformation processes were monitored by macroscopic axial stress-strain, optical holography, and ultrasonic velocity measurements.

A multi-dimensional composite model for plastic continua under polyaxial loading condition

By replacing a medium with reinforcing components oriented and distributed uniformly in a multi-dimensional space, a constitutive model is constructed. The components are extended/compressed compatibly with the strain and the resultant of load exerted on them to balance the stress. Their load-elongation relation can be determined from a conventional material test. Each component undergoes different elongation history depending on its own orientation during deformation, so that the model can simulate elasto-plastic behavior of materials under polyaxial loading conditions. The incremental constitutive matrix has been derived for application in numerical analysis and a yield criterion is also introduced. A few subsequent yield surfaces have been predicted and compared with experiments.

Orientation of hydraulic fractures in salt under hydrostatic and non-hydrostatic stresses

Laboratory experiments of hydraulic fracturing in salt have shown that the breakdown pressure does not vary significantly with the differences between the maximum and minimum stresses normal to the borehole axis. The presence of a non-hydrostatic stress in salt may be determinable from the form of the hydraulic fracture alone. A series of laboratory experiments were performed to determine the influence of deviatoric in situ stress on the form and orientation of hydraulic fractures in salt. The hydraulic-fracturing tests were run on prismatic blocks in a polyaxial loading frame over a range of stress ratios from 1 (hydrostatic) to 1.5. The fracturing oil contained a fluorescent dye that marked the fracture traces for mapping after the samples were split. The results indicate that the form of the hydraulic fracture reflects whether or not the stresses are nearly hydrostatic. For large stress ratios ( >1.5), the fractures are straight and strongly oriented. For stress ratios near one, the fractures are poorly oriented and exhibit significant branching. Thus, proximity to hydrostatic conditions can be determined from the form of the hydraulic fracture alone, regardless of ambiguities in the interpretation of pressure-time records from the fracturing experiment.

Laboratory hydraulic fracturing stress measurements in salt : In: Rock Mechanics in Productivity and Protection (paper to the 25th Symposium on Rock Mechanics, Evanston, Illinois, 25–27 June 1984) P95–102. Publ New York: AIME, 1984

This paper discusses the results of a laboratory testing program to determine the validity of hydraulic fracturing stress measurements in salt. Tests were performed on 15 cm diameter samples loaded under hydrostatic stress conditions to determine the influence of time, confining pressure, flow rate, borehole diameter, and packers on breakdown pressure. Tests were also performed in a polyaxial loading frame to determine whether or not hydraulic fracturing could be used to measure non-hydrostatic stresses. The test results indicate that the breakdown pressure is not affected by time delays of up to 64 hours between the application of load fracturing the rock. Breakdown pressures were found to fall short of predicted elastic values at higher confining pressures for the hydrostatic tests. The ratio of the horizontal stresses in non-hydrostatic tests had no effect on breakdown pressures. Flow rate and borehole diameter variations were found to have a marked effect on breakdown pressures. Shut-in pressure value, exceeded the applied minimum stresses by 10 to 60% depending on the method of determination used. The lack of a relationship between breakdown pressure and horizontal stress ratio may preclude hydraulic fracturing from being used in salt as it is conventionally applied in brittle rocks. 10more » references, 4 figures.« less

Surface deformation, crack formation, and acoustic velocity changes in pyrophyllite under polyaxial loading : Spetzler, H A; Sobolev, G A; Sondergeld, C H J Geophys Res, V86, NB2, 10 Feb 1981, P1070–1080

Surface deformation, crack formation, and acoustic velocity changes in pyrophyllite under polyaxial loading : Spetzler, H A; Sobolev, G A; Sondergeld, C H J Geophys Res, V86, NB2, 10 Feb 1981, P1070–1080

Surface deformation, crack formation, and acoustic velocity changes in pyrophyllite under polyaxial loading

Jacketed cubes of pyrophyllite (31.6 mm on an edge) with variable water content were stressed monotonically to failure under polyaxial compression (σ1 > σ2 = 2σ3). The maximum and minimum principal stresses, σ1, and σ3, were applied with pistons, and the intermediate stress σ2 with a transparent fluid. An optical window in the pressure vessel allowed in situ measurements of the σ2 face deformation by optical holography. In addition, the more conventional techniques of stress, strain, and elastic wave velocity measurements as well as optical and electron microscopy were used to study the formation and propagation of fractures. The strength ( σ1 − σ3)f of the samples increased by about 50% as σ2 was increased from 5 MPa to 100 MPa. Air dry samples were stronger than water‐soaked samples by about ∼20%. Increasing the strain rate from 2 × 10−8 to 10−6 s−1 at σ2 = 5 MPa and 30 MPa increased the strength by about 10%. At σ2 = 100 MPa this trend was reversed, and samples that were deformed at the low strain rate were stronger by approximately 5%. The crack morphology of recovered samples was studied by optical and scanning electron microscopy. Subparallel sets of fractures and en échelon fracture patterns ahead of the macrofracture were easily visible with the unaided eye when σ2 was 5 MPa or 30 MPa. However, at σ2 = 100 MPa these fracture patterns were only visible under the microscope, and the cracks appeared much thinner. The holographic observations of the σ2 face revealed the following: as σ1 was increased, broad bulges formed in a crosslike pattern along the lines of the maximum shear stress. Macrofracture initiation, which occurred in a corner, was preceded by concentrated surface deformation. As the macrofracture propagated across the sample, it deviated from the direction of the maximum shear stress. Ahead of the tip of the macrofracture and migrating with it was a pronounced bulge. In response to monotonically increasing σ1, the displacement in the σ2 direction of a point adjacent to the eventual crack plane went through a local maximum that was followed by a local minimum as the crack passed. The results of elastic wave velocity measurements in the σ2 direction were very sensitive to the spatial relationship of the macrofracture and the elastic wave travel path. However, in general, above ∼50% of (σ1 − σ3)f the velocities decreased. As the migrating bulge approached an elastic wave travel path, the velocity decrease became more pronounced. The velocity increased again as the bulge passed.

Compressibility of Unconsolidated, Arkosic Oil Sands

Abstract Presented am results of compaction of 11 unconsolidated, fine- to medium-grained, arkosic sand cores, 1-7/8 in. in diameter and 3 to 4 in. long. Direct measurements of the pore fluid pressure and bulk volume changes of each sample were made as the pore fluids were expelled. At a constant overburden (external) pressure of 3,000 psi and a temperature of 140 degrees F, the calculated bulk volume compressibilities ranged from 7.4 x 10 to 3 x 10 psi, whereas the pore volume compressibilities varied from 10 to 10 psi in the 0 to 3,000 psi effective pressure range. The void ratios in the same effective pressure range varied from 0.85 to 0.19. Compressibility increases with increasing feldspar and clay content. Compressibilities obtained when using hydrostatic loading equipment are 55 to 100 percent higher than those determined when using uniaxial compaction apparatus. Introduction Numerous investigators studied the compressibility of consolidated rocks and unconsolidated sands. The writer conducted compaction tests in a hydrostatic cell on unconsolidated producing oil sands of Pliocene age from the Los Angeles basin, Calif. There is a lack of experimental data on the compressibilities of unconsolidated sands; yet, such sands are the cause of many well completion and producing problems worldwide. The samples tested were taken from a massive sand interval, greater than 100 ft in thickness. This unit is of deep-water origin and was probably deposited by a combination of bottom currents and distant-from-the-source turbidity currents. Some streaks of very coarse-grained sand and gravel occur in the generally fine- to medium-grained massive sand interval. It is very difficult to duplicate actual reservoir conditions in the laboratory because of the various loading conditions that may exist in the reservoir. Possible loading conditions on a hypothetical Possible loading conditions on a hypothetical sediment cube are presented in Fig. 1. The first condition presented (Fig. 1A) is polyaxial loading, in which none of the three principal stresses are equal. Some investigators prefer to call this stress condition triaxial loading. Although this stress condition may represent the subsurface conditions, it is extremely difficult to duplicate in the laboratory. The second possible loading condition (Fig. 1B) is hydrostatic, in which the three principal stresses applied are equal. This type of loading probably exists during the initial stages of deposition and compaction. The third type of loading (Fig. 1C) is triaxial, in which two of the three principal stresses are equal. Although some investigators justifiably refer to it as biaxial stress, the term triaxial is strongly imbedded in the civil engineering and earth sciences literature. In the uniaxial loading condition (Fig. 1D), the applied force acts in one direction only and is perpendicular to one surface of the sample material. perpendicular to one surface of the sample material. The four faces of the cube parallel to the direction of the stress remain stationary. This arrangement can be achieved by placing the sample in a thick-walled, cylindrical chamber, the sides of which are stationary. The pressure can be applied with either one or two pistons, and the change in the volume of the sample is reflected by the change in the length of the sample. In the field of soil mechanics this method is sometimes referred to as triaxial testing. This type of loading is probably approached in an oil reservoir as the reservoir pressure is depleted as a result of production. It pressure is depleted as a result of production. It should be mentioned also that some investigators reserve the term uniaxial for cases when there is a vertical stress, but no lateral restraint and hence no lateral stress. In biaxial loading (Fig. 1E), the two principal stresses are equal, while two faces of the cube are held stationary.