P.H.S.W. Kulatilake

New peak shear strength criteria for anisotropic rock joints

Abstract In general, roughness profiles of rock joints consist of non-stationary and stationary components. At the simplest level, only one parameter is sufficient to quantify non-stationary joint roughness. The average inclination angle I, along with the direction considered for the joint surface, is suggested to capture the non-stationary roughness. Most of the natural rock joint surface profiles do not belong to the self similar fractal category. However they may be modelled by self-affine fractals. Using a new term called specific length, it is shown that even though the fractal dimension D is a useful parameter, it alone is insufficient to quantify the stationary roughness of non-self similar profiles. Also, it is shown why contradictory results for the estimation of D of non-self similar profiles appear in the literature. To estimate D accurately for non-self similar profiles, it seems necessary to use scales of measurement less thanthe crossover length of the profile. Because the crossover dimension of joint roughness profiles can be extremely small, in practice it may be quite difficult to measure roughness at scales of less than the crossover dimension and thus to estimate D accurately. To overcome the aforementioned problems, it is suggested to combine D with a parameter which is negatively correlated to D and also has the potential to compensate for the errors caused by an inaccurate D, and to use the combined parameter to quantify stationary roughness in practice. Four new strength criteria which take the following general form are suggested for modelling the anisotropic peak shear strength of rock joints at low normal effective stresses (0–0.4 times unconfined compressive strength): τ = σ tan φ + a(SRP) c log 10 σ J σ d + I where σ, τ, σJ, φ and SRP denote, respectively, the effective normal stress on the joint, peak shear strength, joint compressive strength, basic friction angle, and the stationary roughness parameter. The following four options are suggested to represent the term a(SRP)c: az2′c, aKdbDc, aKsbDc or aKνbDc. Joint roughness data should be used to estimate the roughness parameters I, z2′, kd, Ks, Kν and D in different directions on the joint surface. Parameter D reflects the rate of change in length in response to a change in the scale of measurement r. Because z2′, Kd, Ks and Kν are scale-dependent parameters, they can be used to model the scale effect. The coefficients a, b, c and d in the strength criteria should be determined by performing regression analysis on experimental shear strength data. In practice, to allow for modelling uncertainties, the new equations should be used with a factor of safety of about 1.5.

Estimation of mean trace length of discontinuities

SummaryTrace lengths of discontinuities observed on finite exposures are biased due to sampling errors. These errors should be corrected in estimating mean trace length. A technique, which takes into account the sampling errors, is proposed for estimating the mean trace length on infinite, vertical sections from the observations made on finite, rectangular, vertical exposures. The method is applicable to discontinuities whose orientation is described by a probability distribution function. The method requires that the numbers of discontinuities with both ends observed, one end observed, and both ends censored be known. The lengths of observed traces and the density function of trace length are not required. The derivation assumes that midpoints of traces are uniformly distributed in the vertical plane. Also independence between trace length and orientation is assumed. Data on a Pennsylvania shale in Ohio, U. S. A., were used as an example.

Physical and particle flow modeling of jointed rock block behavior under uniaxial loading

Abstract Laboratory experiments and numerical simulations, using Particle Flow Code (PFC3D ), were performed to study the behavior of jointed blocks of model material under uniaxial loading. The effect of joint geometry parameters on the uniaxial compressive strength of jointed blocks was investigated and this paper presents the results of the experiments and numerical simulations. The fracture tensor component in a given direction is used to quantify the combined directional effect of joint geometry parameters including joint density, orientation and size distributions, and the number of joint sets. The variation of the uniaxial compressive strength of the jointed blocks of the model material with the fracture tensor component was investigated. Both the laboratory experiments and the numerical simulations showed that the uniaxial block strength decreases in a nonlinear manner with increasing values of the fracture tensor component. It was observed that joint geometry configuration controls the mode of failure of the jointed blocks and three modes of failure were identified, namely (a) tensile splitting through the intact material, (b) failure by sliding along the joint plane and/or by displacement normal to the joint plane and, (c) mixed mode failure involving both the failure mechanisms in (a) and (b). It has also been shown that with careful parameter calibration procedures, PFC3D could be used to model the strength behavior of jointed blocks of rock.

JOINT NETWORK MODELLING WITH A VALIDATION EXERCISE IN STRIPA MINE, SWEDEN

Abstract In this paper, eight joint geometry modelling schemes are suggested and applied to a set of Stripa mine joint data to build 3-D joint networks to a granitic rock mass. These modelling schemes include investigations for statistical homogeneity of the rock mass, corrections for sampling biases, and applications of stereological principles, to estimate 3-D joint geometry parameters from 1-D or 2-D joint geometry parameter values. Results show the possibility of obtaining different estimates for both joint size and joint intensity parameters through these different schemes. This indicates the importance of performing validation studies for developed joint geometry modelling schemes. Validation procedures developed and performed indicated the need to try out different schemes in modelling joint geometry parameters in order to establish realistic 3-D joint geometry modelling schemes which provide good agreement with field data during verification. It is important to realize that different types of joint geometry modelling schemes are needed to model joint networks in different types of rock formations.

Effect of finite size joints on the deformability of jointed rock in three dimensions

Abstract A numerical decomposition technique, which has emerged from a linking between joint geometry modelling and generation schemes and a distinct element code (3DEC), is used to evaluate the effects of joint geometry parameters of finite size joints on the deformability properties of jointed rock at the three-dimensional (3-D) level. Variation of deformability parameters of jointed rock with joint geometry parameters such as joint density, joint size/block size and joint orientation, are shown through 3-D plots. Relations are developed between deformability properties of jointed rock and fracture tensor parameters. An incrementally linear elastic, orthotropic constitutive model is suggested to represent the pre-failure mechanical behaviour of jointed rock. This constitutive model has captured the anisotropic, scale-dependent behaviour of jointed rock. In this model, the effect of the joint geometry network in the rock mass is incorporated in terms of fracture tensor components. Some insight is given related to estimation of Representative Elementary Volumes (REVs) and REV property values with respect to deformability properties of jointed rock.

Requirements for accurate quantification of self affine roughness using the roughness-length method

Abstract Self-affine fractals have the potential to represent rock joint roughness profiles. Fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D , input standard deviation, σ , and data density, d , were generated. For different values of the input parameter of the roughness–length method (window length, w ), D and another associated fractal parameter A were calculated for the aforementioned profiles. The calculated D was compared with the D used for the generation to determine the accuracy of calculated D . Suitable ranges for w were estimated to produce accurate D (within ±10% error) for the generated profiles. The results showed that to obtain reliable estimates for fractal parameters of a natural rock joint profile, it is necessary to choose a unit for the profile length to satisfy a data density ( d ) greater than or equal to 5.1. For roughness profiles having 5.1≤ d ≤51.23 and 1.2≤ D ≤1.7, w values between 2.5% and 10% of the profile length were found to be highly suitable to produce accurate fractal parameter estimates. It is recommended to use at least seven w values between the estimated minimum and maximum suitable w values in estimating fractal parameters of a natural rock joint profile. It was found that σ and a global trend of a roughness profile have no effect on calculated D . The estimated A was found to increase with both D and σ . The parameter D captures the auto-correlation and A captures the amplitude of a roughness profile at different scales. Therefore, the parameters D and A are recommended to use with the roughness–length method in quantifying rock joint roughness. In addition, at least one more parameter is required to quantify the global trend of a roughness profile, if it exist; in many cases just the inclination or declination angle of the roughness profile in the direction considered would be sufficient to estimate the global trend. Calculated cross-over lengths (segment length of a profile at which a self-affine profile becomes self-similar) for the profiles investigated were found to be extremely small (less than 0.6% of the profile length) indicating that laser profilometers are required to make roughness measurements at interval lengths comparable to the cross-over lengths of the natural rock joint profiles. To calculate rock joint roughness parameters accurately using the self-similar techniques, it is necessary to have roughness measurements made at interval lengths comparable to the cross-over length of the profile. This indicate clearly the difficulty of using self-similar techniques such as the divider method in estimating rock joint roughness accurately.

Stochastic fracture geometry modeling in 3-D including validations for a part of Arrowhead East Tunnel, California, USA

Abstract Eight-hundred and fifty nine fractures of a gneissic rock mass were mapped using 16 scanlines placed on steep rock exposures that were within 300 m of a tunnel alignment before the tunnel excavation. These data were analyzed using the software package FRACNTWK to find the number of fracture sets that exist in the rock mass, 3-D fracture frequency for each set and the probability distributions of orientation, trace length, fracture size in three dimensions (3-D) and spacing for each of the fracture sets. In obtaining these distributions corrections were applied for sampling biases associated with orientation, trace length, size and spacing. Developed stochastic 3-D fracture network for the rock mass was validated by comparing statistical properties of observed fracture traces on the scanlines with the predicted fracture traces on similar scanlines. The one-dimensional (1-D) fracture frequency of the rock mass in all directions in 3-D was calculated and is presented in terms of a stereographic plot. The 1-D fracture frequency along the tunnel alignment direction was predicted to be about 6.5 fractures/m before the tunnel excavation. This prediction was found to be in excellent agreement with the observed values obtained about 1 year later during the tunnel excavation. This was another validation conducted for the developed 3-D fracture network.

Use of the distinct element method to perform stress analysis in rock with non-persistent joints and to study the effect of joint geometry parameters on the strength and deformability of rock masses

SummaryTo use the distinct element method, it is necessary to discretize the problem domain into polygons in two dimensions (2 D) or into polyhedra in three dimensions (3 D). To perform distinct element stress analysis in a rock mass which contains non-persistent finite size joints, it is necessary to generate some type of fictitious joints so that when they are combined with the non-persistent joints, they discretize the problem domain into polygons in 2 D or into polyhedra in 3 D. The question arises as to which deformation and strength parameter values should be assigned to these fictitious joints so that they behave as intact rock. In this paper, linear elastic, perfectly-plastic constitutive models with the Mohr-Coulomb failure criterion, including a tension cut-off, were used to represent the mechanical behaviour of both intact rock and fictitious joints. It was found that, by choosing the parameter values for the constitutive models as given below, it is possible to make the fictitious joints behave as intact rock, in a global sense.a)For both the intact rock and the fictitious joints, the same strength parameter values should be used.b)A joint shear stiffness (JKS) value for fictitious joints should be chosen to produce a shear modulus/JKS ratio (G/JKS) between 0.008 and 0.012 m.c)A joint normal stiffness/JKS ratio (JKN/JKS) between 2 and 3 should be chosen. The most appropriate value to choose in this range may be the Youngs modulus/G value (E/G) for the particular rock. Some examples are given in the paper to illustrate how to use the distinct element method to perform stress analysis of rock blocks which contain non-persistent joints and to study the effect of joint geometry parameters on strength and deformability of rock masses.

Box fractal dimension as a measure of statistical homogeneity of jointed rock masses

Abstract A written computer programme to estimate the box fractal dimension (DB) is verified by estimating DB of the triadic Koch curve for which the theoretical D is known. The influence of a number of input parameters of the box-counting method on the accuracy of estimated DB is evaluated using the same Koch curve. The employed size range of the applied box networks was found to be the parameter which has the strongest influence on the accuracy of estimated DB. This indicated the importance of finding the range of self-similarity or self-affinity for the object considered to select the proper range for the box sizes and, in turn, to obtain accurate estimates of DB. By calculating DB for different block sizes sampled from three generated two-dimensional joint patterns, it is shown that DB can capture the combined effect of joint-size distribution and joint density on the statistical homogeneity of rock masses. The spatial variation of DB along a 350 m stretch of a tunnel in the shiplock area of the Three Gorges dam site is computed using the joint data mapped on the walls and the roof of the tunnel. This spatial variation of DB is used, along with the visual geological evaluation of the joint trace maps of the tunnel, in making decisions about statistical homogeneity of the rock mass around the tunnel. The results obtained on statistically homogeneous regions were found to be quite similar to the results obtained from a previous statistical homogeneity investigation which incorporated the effect of number of joint sets and their orientation distributions, but not the spatial variation of DB. It is recommended that the spatial variation of DB is used, along with the results of other methods such as contingency table analysis and equal area plots, which incorporate the effect of joint orientation distribution, in addition to the geology of the site, in determining the statistically homogeneous regions of jointed rock masses.

Requirements for accurate quantification of self-affine roughness using the variogram method

Abstract Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, input standard deviation, σ, and data density, d, were generated. For different values of the input parameter of the variogram method (lag distance, h), D and another associated fractal parameter Kv were calculated for the aforementioned profiles. It was found that σ has no effect on calculated D. The estimated Kv was found to increase with D, σ and d according to the equation Kv = 2.0 × 10−5d0.35σ1.95D14.5. The parameter Kv seems to have potential to capture the scale effect of roughness profiles. Suitable ranges for h were estimated to obtain computed D within ±10% of the D used for the generation and also to satisfy a power functional relation between the variogram and h. Results indicated the importance of removal of non-stationarity of profiles to obtain accurate estimates for the fractal parameters. It was found that at least two parameters are required to quantify stationary roughness; the parameters D and Kv are suggested for use with the variogram method. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists; at the basic level, the mean inclination/declination angle of the surface in the direction considered can be used to represent the non-stationarity. A new concept of feature size range of a roughness profile is introduced in this paper. The feature size range depends on d, D and σ. The suitable h range to use with the variogram method to produce accurate fractal parameter values for a roughness profile was found to depend on both d and D. It is shown that the feature size range of a roughness profile plays an important role in obtaining accurate roughness parameter values with both the divider and the variogram methods. The minimum suitable h was found to increase with decreasing d and increasing D. Procedures are given in this paper to estimate a suitable h range for a given natural rock joint profile to use with the variogram method to estimate D and Kv accurately for the profile.