Evaluation of rock strength criteria for wellbore stability analysis

Polyaxial strength test data of five rocks are used to examine the Mohr–Coulomb, Drucker–Prager, modified Lade, Mogi–Coulomb and three-dimensional (3D) Hoek–Brown criteria regarding their ability, with parameters determined based on the triaxial compression test data, to represent the rock behavior under polyaxial stress states. Then the five strength criteria, with parameters determined based on the triaxial compression test data, are used to analyze wellbore stability of both vertical and inclined boreholes. The results show that the Mohr–Coulomb criterion under-predicts the polyaxial strength and estimates the highest minimum mud pressure required for wellbore stability while the Drucker–Prager criterion over-predicts the polyaxial strength and estimates the lowest minimum mud pressure. The modified Lade, Mogi–Coulomb and 3D Hoek–Brown criteria, with parameters based on triaxial test data, can either over-predict or under-predict the polyaxial strength. The over-prediction of the modified Lade criterion can be very large, and thus it may be unsafe to use it to estimate the minimum mud pressure. Both the over-prediction and under-prediction are relatively small for the 3D Hoek–Brown and Mogi–Coulomb criteria. Therefore, the 3D Hoek–Brown and Mogi–Coulomb criteria are recommended for wellbore stability analysis.     

Applicability of Mohr–Coulomb and Hoek–Brown strength criteria to the dynamic strength of brittle rock

A series of dynamic uniaxial and triaxial compression, uniaxial tension and unconfined shear tests were conducted on the Bukit Timah granite of Singapore. The results are analyzed in this paper in order to examine the validity and applicability of the Mohr–Coulomb and the Hoek–Brown criteria to the rock material strength properties subjected to dynamic loads. The study indicates that rock material strength under dynamic loads can be approximately described by the Mohr–Coulomb criterion, at low confining pressure range. The change of strength is primarily due to the variation of cohesion with loading rate. The rock material strength under dynamic loads is better described by the Hoek–Brown criterion. Assessment of the Hoek–Brown criterion shows that the uniaxial compressive strength increases with increasing loading rate, and the parameter m appears unaffected by the loading rate.     

Stability charts for rock slopes based on the Hoek–Brown failure criterion

This paper uses numerical limit analysis to produce stability charts for rock slopes. These charts have been produced using the most recent version of the Hoek–Brown failure criterion. The applicability of this criterion is suited to isotropic and homogeneous intact rock, or heavily jointed rock masses. The rigorous limit analysis results were found to bracket the true slope stability number to within ±9% or better, and the difference in safety factor between bound solutions and limit equilibrium analyses using the same Hoek–Brown failure criterion is less than 4%. The accuracy of using equivalent Mohr–Coulomb parameters to estimate the stability number has also been investigated. For steep slopes, it was found that using equivalent parameters produces poor estimates of safety factors and predictions of failure surface shapes. The reason for this lies in how these equivalent parameters are estimated, which is largely to do with estimating a suitable minor principal stress range. In order to obtain better equivalent parameter solutions, this paper proposes new equations for estimating the minor principal stress for steep and gentle slopes, which can be used to determine equivalent Mohr–Coulomb parameters.     

Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto-plastic rock masses

We present a novel methodology for estimation of equivalent Mohr–Coulomb strength parameters that can be used for design of supported tunnels in elasto-plastic rock masses satisfying the non-linear empirical Hoek–Brown failure criterion. We work with a general adimensional formulation of the Hoek–Brown failure criterion in the space of normalized Lambe's variables for plane stress, and we perform linearization considering the stress field in the plastic region around the tunnel. The procedure is validated using analytical solutions to a series of benchmark test cases. Numerical solutions are also employed to validate the procedure in cases for which analytical solutions are not available. Results indicate that the stress field in the plastic region around the tunnel, as well as the linearization method employed and the quality of the rock mass, has a significant impact on computed estimates of equivalent Mohr–Coulomb strength parameters. Results of numerical analyses also show that our proposed linearization method can be employed to estimate loads and moments on the tunnel support system. We recommend the equating model responses (EMR) method to compute equivalent Mohr–Coulomb strength parameters when the tunnel support pressure is accurately known, and we further show that our newly introduced linearization method can be employed as an alternative to the best fitting in the existing stress range (BFe) and best fitting in an artificial stress range (BFa) methods, providing performance estimates that are generally better than estimates of the BFe and BFa methods when differences with the response of the Hoek–Brown rock mass are of engineering significance (say more than 10%).     

Upper bound solution for ultimate bearing capacity with a modified Hoek–Brown failure criterion

Conventional calculations of ultimate bearing capacity are formulated in terms of a linear Mohr–Coulomb (MC) failure criterion. However, experimental data shows that the strength envelops of almost all types of rocks are nonlinear over the wide range of normal stresses. In this paper, the strength envelope of rock masses is considered to follow a modified Hoek–Brown failure criterion that is a nonlinear failure criterion. Two different kinds of techniques are used to develop the ultimate bearing capacity in the framework of limit analysis in plasticity.The first technique is the generalized tangential technique proposed by the authors. Based on a multi-wedge translation failure mechanism, a generalized tangential technique is used to formulate the bearing capacity problem as a classical optimization problem where the objective function, which is to be minimized with respects to the parameters of failure mechanism and the location of tangency point, corresponds to the dissipated power. The minimum solution is obtained by optimization. Using the technique, the effects of rock weight under the base of the footing and surcharge load can be considered. The second technique, “tangential” line technique, was originally used to analyze slope stability with a nonlinear failure criterion. In order to assess the validity of the proposed method, the “tangential” line technique is extended to evaluate the bearing capacity factor with the nonlinear failure criterion, where the effects of self-weight and surcharge load on the bearing capacity cannot considered. The second technique, however, has to utilize the previously calculated ultimate bearing capacity factor with a linear MC failure criterion. Numerical results are compared and presented for practical use in rock engineering.     

Stability analysis of vertical boreholes using the Mogi–Coulomb failure criterion

A main aspect of wellbore stability analysis is the selection of an appropriate rock failure criterion. The most commonly used criterion for brittle failure of rocks is the Mohr–Coulomb criterion. This criterion involves only the maximum and minimum principal stresses, σ₁ and σ₃, and therefore assumes that the intermediate stress σ₂ has no influence on rock strength. As the Mohr–Coulomb criterion ignores the strengthening effect of the intermediate stress, it is expected to be too conservative in estimating the critical mud weight required to maintain wellbore stability. Recently, Al-Ajmi and Zimmerman [Relationship between the parameters of the Mogi and Coulomb failure criterion. Int J Rock Mech Min Sci 2005;42(3):431–39.] developed the Mogi–Coulomb failure criterion, and showed that it is reasonably accurate in modelling polyaxial failure data from a variety of rocks. We then develop a model for the stability of vertical boreholes, using linear elasticity theory to calculate the stresses, and the fully-polyaxial Mogi–Coulomb criterion to predict failure. Our model leads to easily computed expressions for the critical mud weight required to maintain wellbore stability.     

Rock characterisation facility (RCF) shaft sinking — numerical computations using FLAC

Using FLAC and FLAC^3D, numerical computations were carried out to simulate a comprehensive pump test at Sellafield and to predict the hydro-mechanical consequences sinking of a shaft at that site might bring about. Transient discharge into the borehole could not be reproduced; however, the reproduction of the post-transient flow was good. Shaft sinking was modelled with FLAC^3D. The mechanical as well as the hydrogeological effects of the planned shaft on its rock surrounding, both near and far field, were typical and did not point to any difficulties that might be encountered had the shaft been constructed. By using new failure criterion, it was possible to capture the excavation damage zone by the simulations. It could not be captured by more established failure criteria such as Mohr–Coulomb or Hoek and Brown. Considering the assumptions made to set up the simulations, no coupling between the mechanical and the fluid flow computations was deemed necessary.     

Discussion on “A generalized three-dimensional failure criterion for rock masses”

There are many methods to construct true triaxial strength criteria for rocks.Jaiswal and Shrivastva(2012)proposed a strength criterion,named J–S criterion,in the deviatoric plane,which provides nearly the same misfts for true triaxial test data as the exponential criterion.It is diffcult to calculate the strength at given2and3using the J–S criterion,and the multiple solutions to the nonlinear equation may induce confusion and mistake.Strength envelopes in deviatoric planes are not geometric similar;therefore,true triaxial test data cannot be grouped in the mean stress to check strength criteria in the deviatoric plane.     

Measurements of induced stress and strength in the near-field around a tunnel and associated estimation of the Mohr–Coulomb parameters for rock mass strength

The state of induced stress measured by the compact conical-ended borehole overcoring technique in the immediate roof of an approach tunnel excavated under high rock stress is described. During the measurements, core disking was observed. An X-ray Computed Tomography (CT) scanner was used to select strain data uninfluenced by the core disking; then the induced rock stress was estimated from selected strain data. From these results, it is shown that the non-destructive investigation using X-ray CT is effective for visualization of the fracturing within cores and the selection of strains measured during overcoring. Furthermore, the Mohr–Coulomb failure criterion parameters for the rock mass were estimated by comparing the measured stresses with the shear strength of in situ rock and the uniaxial compressive strength determined in laboratory tests.     

A modified Hoek–Brown failure criterion for anisotropic intact rock

The Hoek–Brown criterion parameters (σ_ci, m_i and s) are significantly influenced by the strength anisotropy of intact rock. In the present study, the criterion was modified by incorporating a new parameter (k_β) to account for the effect of strength anisotropy, thus being able to determine the strength of intact anisotropic rock under loading in different orientations of the plane of anisotropy. The range of the parameter (k_β) for the rocks tested has been analytically investigated by carrying out triaxial tests, in different orientations of the foliation plane. The proposed modification was studied for metamorphic rocks (gneiss, schist, marble), but could also be applied to other rock types exhibiting “inherent” anisotropy, e.g. sedimentary as well as igneous rocks. The proposed modified criterion is intended for use for prediction of strength of intact rock, but can also be extended to rock masses.     

An equivalent plasticity theory for jointed rock masses

A non-representative volume element (NRVE) approach to equivalent rock mass properties shows that the form of the elastic–plastic constitutive equations is the same for homogeneous material elements and multiple-material elements, subsequently homogenized. Thus the average stress and strain increments in an arbitrary jointed rock mass volume are related by {dσ}=([C*_ep]){dε} where σ is effective stress. The equivalent elastic-plastic properties matrix [C*_ep] is the sum of an equivalent elastic moduli matrix [C^*] and a plastic ‘correction’ matrix [C*_p, as usual. However, there are no equivalent plastic potentials Y^* or yield functions, failure criteria F^* or strengths. The equivalent elastic-plastic properties are constructed from the elastic moduli and strengths of the rock mass joints, the intact rock between and strain influence functions that relate local to overall average strains. Numerical examples that simulate laboratory-like tests on jointed rock cubes illustrate the approach.     

Non-dilatant deformation and failure mechanism in two Long Valley Caldera rocks under true triaxial compression

We conducted laboratory rock strength experiments in two ultra-fine-grained brittle rocks, hornfels and metapelite, which together are the major constituent of the Long Valley Caldera (California, USA) basement in the 2025–2996 m depth range. Both rocks are banded, and have very low porosity. Uniaxial compression tests at different orientations with respect to banding planes reveal that while the hornfels compressive strength is nearly isotropic, the metapelite possesses distinct anisotropy. Conventional triaxial tests in these rocks reveal that their respective strengths in a specific orientation increase approximately linearly with confining pressure. True triaxial compression experiments in specimens oriented at a consistent angle to banding, in which the magnitudes of the least (σ₃) and the intermediate (σ₂) principal stresses are different but kept constant during testing while the maximum principal stress is increased until failure, exhibit a behavior unlike that previously observed in other rocks under similar testing conditions. For a given magnitude of σ₃, compressive strength σ₁ does not vary significantly in both Long Valley rock types, regardless of the applied σ₂, suggesting little or no intermediate principal stress effect. Strains measured in all three principal directions during loading were used to obtain plots of σ₁ versus volumetric strain. These are consistently linear almost to the point of rock failure, suggesting no dilatancy. The phenomenon was corroborated by SEM inspection of failed specimens that showed no microcrack development prior to the emergence of one through-going shear failure plane steeply dipping in the σ₃ direction. The strong dependency of compressive strength on the intermediate principal stress in other crystalline rocks was found to be related to microcrack initiation upon dilatancy onset, which rises with increased σ₂ and retards the failure process. We infer that strength independence of σ₂ in the Long Valley rocks derives directly from their non-dilatant deformation.     

Experimental investigation of mechanical properties of bedded salt rock

Because of salt cavern utilization for liquid, gas and solid waste storage, salt rock mechanical properties are needed for assessments of facility, stability and safety. Bedded salt deposits are widespread and used as much or more than diapiric salt bodies as storage facility hosts, but experimental data on the mechanical properties of bedded salt rock with impurities are far less common than data available on relatively pure diapiric salt rocks. Through laboratory uniaxial and triaxial compression experiments on rock salt (halite), interlayers (anhydrite) and bedded composite specimens (anhydrite–halite and mudstone–halite), differences in mechanical properties of the various lithologies are explored. In the composite specimens, the weakest or the most deformable component governs the behavior. Also, the properties of bedded composite lithology specimens tend to be in between the property ranges of the “pure” lithologies. The elastic modulus of the bedded salt rock increases from 5.3 to 24.1 GPa with an increase in the confining stress from 0 to 15 MPa, with some evidence of sample damage. The ductile transition for halite at the strain rates used is at about σ₃10 MPa.With increasing σ₃, the anhydrite–halite composite lithology deformation showed strain hardening and a strong trend to ductile behavior as the halite bands tended to dominate the behavior. Strain incompatibility effects exist along interfaces between creeping and non-creeping phases in anhydrite–halite composite lithologies. Mudstone–halite rocks tended to be extremely weak, compared with all other specimens.     

A Hoek–Brown criterion with intrinsic material strength factorization

Probably the most common failure criterion for rock masses is the Hoek–Brown (HB) failure criterion. The HB criterion is an empirical relation that extrapolates the strength of intact rock to that of rock masses. For design purposes, the HB criterion is often fitted using equivalent Coulomb failure lines. However, equivalent Mohr–Coulomb (MC) shear strength parameters cannot yield the same failure characteristics as the HB criterion. The curvilinear HB criterion automatically accommodates changing stress fields; the MC criterion does not. The extended HB criterion proposed in this paper provides a solution to this problem by incorporating an intrinsic material strength factorization scheme. The original HB criterion is additionally enhanced by adopting the spatial mobilized plane (SMP) concept, first introduced by Matsuoka and Nakai (MN). The SMP concept accounts for the experimentally proven, influence of intermediate principal stresses on failure, which is disregarded in the original HB criterion. A small set of examples provided at the end of the article gives a good indication of the merits of using the extended HB criterion in practical applications.     

A nonlinear strength criterion for rock-like materials based on fracture mechanics

A nonlinear strength criterion for rock-like materials is developed in this paper. Taking α as an angle of micro-failure orientation in rock-like materials, a formulation between α and load is derived from a mixed-mode fracture criterion based on linear elastic fracture mechanics. According to micro-failure experimental phenomena of rock-like materials, a failure characteristic parameter under triaxial compression condition is chosen, which is relevant to confining pressure and is an invariant. A theoretical nonlinear strength criterion is also derived, which is exactly in the same mathematical form as the original Hoek–Brown empirical strength criterion. In addition, it is also found that the coefficient m in the Hoek–Brown criterion has physical meaning which is related to the ratio between the uniaxial compressive strength and the uniaxial tensile strength.     

Fracture toughness study for a brittle rock subjected to mixed mode I/II loading

A semi-disk specimen containing an angled edge crack has been used in the past for conducting fracture tests on a brittle rock named Johnstone [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212]. The test specimen is appropriate for investigating brittle fracture when the rock samples are subjected to the combined effects of tension and shear along the crack line. However, the experimental results reported in Lim, Johnston, Choi, Boland [Fracture testing of a soft rock with semi-circular specimens under three-point bending. Part 2—mixed mode. Int J Rock Mech Min Sci Geomech Abstr 1994b;31(3):199–212.] are inconsistent with all of the well-known theoretical criteria available for predicting mixed mode brittle fracture. In this paper, a modified criterion is used to provide accurate predictions for the reported experimental results. The modified criterion makes use of a three-parameter model (based on K_I, K_II and T) for describing the crack tip stresses. It is shown that the non-singular stress term T has a significant role when the rock fracture tests are conducted on the semi-disk specimens.     

Shear fracture (Mode II) of brittle rock

Mode II fracture initiation and propagation plays an important role under certain loading conditions in rock fracture mechanics. Under pure tensile, pure shear, tension- and compression-shear loading, the maximum Mode I stress intensity factor, K_Imax, is always larger than the maximum Mode II stress intensity factor, K_IImax. For brittle materials, Mode I fracture toughness, K_IC, is usually smaller than Mode II fracture toughness, K_IIC. Therefore, K_Imax reaches K_IC before K_IImax reaches K_IIC, which inevitably leads to Mode I fracture. Due to inexistence of Mode II fracture under pure shear, tension- and compression-shear loading, classical mixed mode fracture criteria can only predict Mode I fracture but not Mode II fracture. A new mixed mode fracture criterion has been established for predicting Mode I or Mode II fracture of brittle materials. It is based on the examination of Mode I and Mode II stress intensity factors on the arbitrary plane θ,K_I(θ) and K_II(θ), varying with θ(−180°θ+180°), no matter what kind of loading condition is applied. Mode I fracture occurs when (K_IImax/K_Imax)<1 or 1<(K_IImax/K_Imax)<(K_IIC/K_IC) and K_Imax=K_IC at θ_IC. Mode II fracture occurs when (K_IImax/K_Imax)>(K_IIC/K_IC) and K_IImax=K_IIC at θ_IIC. The validity of the new criterion is demonstrated by experimental results of shear-box testing.Shear-box test of cubic specimen is a potential method for determining Mode II fracture toughness K_IIC of rock since it can create a favorable condition for Mode II fracture, i.e. K_IImax is always 2–3 times larger than K_Imax and reaches K_IIC before K_Imax reaches K_IC. The size effect on K_IIC for single- and double-notched specimens has been studied for different specimen thickness B, dimensionless notch length a/W (or 2a/W) and notch inclination angle α. The test results show that K_IIC decreases as B increases and becomes a constant when B is equal to or larger than W for both the single- and double-notched specimens. When a/W (or 2a/W) increases, K_IIC decreases and approaches a limit. The α has a minor effect on K_IIC when α is within 65–75°. Specimen dimensions for obtaining a reliable and reproducible value of K_IIC under shear-box testing are presented. Numerical results demonstrate that under the shear-box loading condition, tensile stress around the notch tip can be effectively restrained by the compressive loading. At peak load, the maximum normal stress is smaller than the tensile strength of rock, while the maximum shear stress is larger than the shear strength in the presence of compressive stress, which results in shear failure.     

Influence of intermediate principal stress on rock fracturing and strength near excavation boundaries—Insight from numerical modeling

The influence of the intermediate principal stress on rock fracturing and strength near excavation boundaries is studied using a FEM/DEM combined numerical tool. A loading condition of σ₃=0 and σ₁≠0, and σ₂≠0 exists at the tunnel boundary, where σ₁, σ₂, and σ₃, are the maximum, intermediate, and minimum principal stress components, respectively. The numerical study is based on sample loading testing that follows this type of boundary stress condition. It is seen from the simulation results that the generation of tunnel surface parallel fractures and microcracks is attributed to material heterogeneity and the existence of relatively high intermediate principal stress (σ₂), as well as zero to low minimum principal stress (σ₃) confinement. A high intermediate principal stress confines the rock in such a way that microcracks and fractures can only be developed in the direction parallel to σ₁ and σ₂. Stress-induced fracturing and microcracking in this fashion can lead to onion-skin fractures, spalling, and slabbing in shallow ground near the opening and surface parallel microcracks further away from the opening, leading to anisotropic behavior of the rock. Hence, consideration of the effect of the intermediate principal stress on rock behavior should focus on the stress-induced anisotropic strength and deformation behavior of the rocks. It is also found that the intermediate principal stress has limited influence on the peak strength of the rock near the excavation boundary.     

True-triaxial strength criteria for rock

This paper reviews some strength criteria which include the role of the intermediate principal stress, and proposes a new criterion. Strength criteria of the form σ_oct=f_N(σ_oct), such as Drucker–Prager, represent a rotation surface in the principal stress space, symmetric to the line σ₁=σ₂=σ₃ in the meridian plane. Because σ_oct=f_N(σ_oct) must fit the pseudo-triaxial compressive strength, it will have a non-physical outcome for triaxial extension. Mogi's criteria, σ_oct=g₁(σ_m,2) and σ_max=g₂(σ_β) are able to fit experimental data reasonably well, but the prediction of strength is not good and sometimes problematic. Strength criterion with the form λ(σ₁, σ₂, σ₃)=F[η(σ₁, σ₂, σ₃)], or a curve of two variables which can be decided by fitting pseudo-triaxial experimental data, is not expected to describe the strength under various stress states, no matter how high the correlation coefficient of λ and η is, or how low the misfit of the equation λ=F(η) is, as these seemingly good correlations usually result from the dominant influence of the maximum principal stress in the metrics of λ and η. The intermediate principal stress may improve the strength of rock specimen, but its influence will be restricted by σ₃. Also when σ₂ is high enough to cause failure in the σ₂–σ₃ direction, the strength will decrease with the increasing σ₂. The new strength criterion with exponent form has just such a character, and gives much lower misfits than do all seven criteria discussed by Colmenares and Zoback [Colmenares LB, Zoback MD. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int J Rock Mech Min Sci 2002;39:695–729].     

2D and 3D finite element analysis of underground openings in an inhomogeneous rock mass

The finite element analysis of the underground openings excavated for Koyna hydroelectric project, Maharashtra, India, has been conducted. 2D and 3D models have been developed assuming that the rock mass obeys Drucker–Prager failure criterion. The computed deformations and the stress distribution, around these openings, have been compared with the in situ measurements. The study reveals that the 2D elasto-plastic analysis underestimates the deformations. On the other hand, the 3D elasto-plastic analysis yields results, which compare reasonably well with the in situ measurements. The effect of weak zones in the rock mass and creation of multiple cavities in the nonhomogeneous rock mass has also been considered in the analyses. Such a study is found to be very helpful for evaluating the stability of underground openings when extensive realistic input data is available for nonhomogeneous rock mass.