Fracture Tensor Component Research Articles

Development of New Three-Dimensional Rock Mass Strength Criteria

Two new three-dimensional rock mass strength criteria are developed in this paper by extending an existing rock mass strength criterion. These criteria incorporate the effects of the intermediate principal stress, minimum principal stress and the anisotropy resulting from these stresses acting on the fracture system. In addition, these criteria have the capability of capturing the anisotropic and scale dependent behavior of the jointed rock mass strength by incorporating the effect of fracture geometry through the fracture tensor components. The new criteria are proposed after analyzing 284 numerical modeling results of the polyaxial, triaxial and biaxial compression tests conducted on the jointed rock blocks having one or two joint sets by the PFC3D software. Some of these simulation results were compared with experimental results to validate the developed PFC3D model that was used for numerical modeling of jointed blocks. In this research to have several samples with the same properties a synthetic rock material that is made out of a mixture of gypsum, sand and water was used. Altogether, 12 joint systems were chosen; some of them had one joint set and the rest had two joint sets. Joint sets have different dip angles varying from 15° to 45° at an interval of 15° with dip directions of 30° and 75° for the two joint sets. Each joint set also has three persistent joints with the joint spacing of 42 mm in a cubic sample of size 160 mm. The minimum and intermediate principal stress combination values were chosen based on the uniaxial compressive strength (UCS) value of the modeled intact synthetic rock. The minimum principal stress values were chosen as 0, 0.2, 0.4 and 0.6 of the UCS. For each minimum principal stress value, the intermediate principal stress value varies starting at the minimum principal stress value and increasing at an interval of 0.2 of the UCS until it is slightly lower than the strength of the sample under the biaxial loading condition with the same minimum principal stress value. To express the new rock mass strength criteria, it was also necessary to determine the intact rock strengths under the same confining stress combinations mentioned earlier. Therefore, the intact rock was also modeled for all three compression tests and the intact rock strengths were found for 33 different minimum and intermediate principal stress combinations.

Seismic characterization of reservoirs with asymmetric fractures

Variations of reflection amplitude with offset and azimuth are sensitive to the presence of natural and induced fractures. We tested inversion for fracture compliance matrix components from wide azimuth noisy synthetic PP-reflection data. The model was a fractured reservoir with monoclinic symmetry, formed by two sets of vertical asymmetric fractures embedded in a VTI background. The fractures (joints) were assumed to be vertical, with noncircular shape and/or with asymmetric shear compliance. Results of synthetic inversion showed that an incorrect assumption about fracture shear symmetry (e.g., treating asymmetric fractures as rotationally invariant fractures) can cause considerable error in estimation of the fracture compliance matrix. Components of effective second- and fourth-rank fracture compliance matrices for a medium with monoclinic symmetry (which takes into account layering and multiple fracture sets) can be used as attributes related to the characteristics of the fractured medium. Monte Carlo simulation was used to test the effect of uncertainties in the a priori information (about background VTI parameters of unfractured rock), as they affect inversion for these attributes. According to this analysis, the direction of fast shear-wave polarization was inverted robustly in the Monte Carlo simulation. Although the average values of the components of fracture compliance matrices obtained from Monte Carlo simulation were in agreement with the actual values used for forward modeling, individual values obtained in Monte Carlo simulation were sensitive to uncertainties in the background properties in general. Because the elastic properties of background VTI media without fractures (or other azimuthally variable features) do not cause azimuthal changes in reflection coefficient variation with offset, simultaneous inversion for background properties and fracture tensor components require additional constraints.

Seismic characterization of naturally fractured reservoirs using amplitude versus offset and azimuth analysis

ABSTRACTP‐wave seismic reflection data, with variable offset and azimuth, acquired over a fractured reservoir can theoretically be inverted for the effective compliance of the fractures. The total effective compliance of a fractured rock, which is described using second‐ and fourth‐rank fracture tensors, can be represented as background compliance plus additional compliance due to fractures. Assuming monoclinic or orthotropic symmetry (which take into account layering and multiple fracture sets), the components of the effective second‐ and fourth‐rank fracture compliance tensors can be used as attributes related to the characteristics of the fractured medium. Synthetic tests indicate that using a priori knowledge of the properties of the unfractured medium, the inversion can be effective on noisy data, with S/N on the order of 2. Monte Carlo simulation was used to test the effect of uncertainties in the a priori information about elastic properties of unfractured rock. Two cases were considered with Wide Azimuth (WAZ) and Narrow Azimuth (NAZ) reflection data and assuming that the fractures have rotationally invariant shear compliance. The relative errors in determination of the components of the fourth‐rank tensor are substantially larger compared to the second‐rank tensor, under the same assumptions.Elastic properties of background media, consisting in horizontal layers without fractures, do not cause azimuthal changes in the reflection coefficient variation with offset. Thus, due to the different nature of these properties compared to fracture tensor components (which cause azimuthal anomalies), simultaneous inversion for background isotropic properties and fracture tensor components requires additional constraints.Singular value decomposition (SVD) and resolution matrix analysis can be used to predict fracture inversion efficacy before acquiring data. Therefore, they can be used to determine the optimal seismic survey design for inversion of fracture parameters. However, results of synthetic inversion in some cases are not consistent with resolution matrix results and resolution matrix results are reliable only after one can see a consistent and robust behaviour in inversion of synthetics with different noise levels.

REV and its properties on fracture system and mechanical properties, and an orthotropic constitutive model for a jointed rock mass in a dam site in China

Fracture data available for one of the rock masses (limestone) in the dam site of Yujian River Reservoir were used to build and validate a stochastic 3-D fracture network model, and to perform a REV and equivalent continuum study in 3-D. A number of relations are developed in the paper between the rock mass mechanical parameters and fracture tensor components in 3-D. Based on the mechanical parameter values obtained in every 45° direction in 3-D, the principal parameter values, principal directions and tensors are developed for rock mass mechanical parameters to represent the REV block size properties. An incrementally linear elastic, orthotropic constitutive model is suggested to represent the equivalent continuum pre-failure mechanical behavior of the jointed rock mass by incorporating the effect of joint geometry network by the fracture tensor components.

A new rock mass failure criterion for biaxial loading conditions

To simulate brittle rocks, a mixture of glastone, sand and water was used as a model material. Thin galvanized sheets of thickness 0.254 mm were used to create joints in blocks made out of the model material. To investigate the failure modes and strength, both the intact material blocks as well as jointed model material blocks of size 35.6 × 17.8 × 2.5 cm having different joint geometry configurations were subjected to uniaxial and biaxial compressive loadings. A new intact rock failure criterion is proposed at the 3-D level. This criterion is validated for biaxial loading through laboratory experimental results obtained on intact model material blocks. Results obtained from both the intact and jointed model material blocks are used to develop a strongly non-linear new rock mass failure criterion for biaxial loading. In this failure criterion, the fracture tensor component is used to incorporate the directional effect of fracture geometry system on jointed block strength. The failure criterion shows the important role, the intermediate principal stress plays on rock mass strength.

Experimental and Numerical Simulations of Jointed Rock Block Strength under Uniaxial Loading

To simulate brittle rocks, a mixture of sand, plaster of paris, and water was used as a model material. Thin galvanized sheets were used to create joints in blocks made out of the model material. To investigate the failure modes and strength, 30 × 12.5 × 8.6 cm jointed model material blocks having different joint geometry configurations were subjected to uniaxial compressive loading. Results indicated three failure modes: (1) tensile failure through intact material; (2) combined shear and tensile failure or only shear failure on joints; and (3) mixed failure of the above two modes depending on the joint geometry. The fracture tensor component in a certain direction quantifies the directional effect of the joint geometry, including number of fracture sets, fracture density, and probability distributions for size and orientation of these fracture sets. Results obtained from the experiments were used to develop a strongly nonlinear relation between the fracture tensor component and the jointed block strength...

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.

Effect of Block Size and Joint Geometry on Jointed Rock Hydraulics and REV

The effect of joint geometry parameters and the size of jointed rock block on the hydraulic properties of jointed rock, including the equivalent continuum behavior, was investigated through numerical experimentation. The chance to reach equivalent continuum behavior for a rock mass having a certain joint configuration increases with the increase of block size. The REV (representative elementary volume) size to show hydraulic equivalent continuum behavior for a jointed rock system was found to depend on the orientation of the joint sets; the REV size seems to decrease with increasing joint density and joint size. The REV does not seem to exist for some rock masses having joint systems with low relative orientation angles (systems with two joint sets) and low densities. The average block permeability (K0) value at the REV size for a joint system increases with increase in joint size and joint density. The equivalent continuum behavior of a joint system can be expressed with respect to a cut-off value for the first invariant of fracture tensor (F0). The block size corresponding to the aforementioned cut-off F0 (between 10 and 30 for the joint systems investigated in this study) can be considered to provide the REV size for a given joint configuration. A threshold value for F0 (a joint network having two perpendicular joint sets produced 2.75) can be used to find the chance for a given joint network to have nonzero block permeability. A strong power functional relation seems to exist between the directional permeability and the fracture tensor component for the connected joint configuration when rock blocks contain minor discontinuities.

Relations between Fracture Tensor Parameters and Jointed Rock Hydraulics

The fracture tensor is a combined measure of joint geometry parameters. The possible relations between the properties of the fracture tensor and the hydraulic properties of jointed rock including the equivalent continuum behavior are investigated in this paper through numerical experiments performed at the two-dimensional level incorporating joint networks with two joint sets. The possibility of a nonzero average block permeability and the equivalent continuum behavior of a joint system can be expressed with the help of the first invariant of fracture tensor (\iF\d0). A threshold value of \iF\d0 (to achieve nonzero block permeability) between 3 and 6 and a cutoff value of \iF\d0 (to achieve equivalent continuum behavior) between 10 and 30 were obtained for the joint geometry configurations studied in the investigation. Both the threshold value and the cutoff value of \iF\d0 were found to depend on the relative orientation of joint sets. Joint systems with low relative orientation angles (for systems with only two joint sets) and low densities do not show hydraulically equivalent continuum behavior. A nonlinear relation seems to exist between the directional permeability and the fracture tensor component for the connected joint configuration when rock blocks contain only minor discontinuities. For rock blocks containing major discontinuities, this relation becomes linear.

A physical model study of jointed rock mass strength under uniaxial compressive loading

Abstract Jointed rock mass strength is investigated through physical modeling using model material blocks subjected to only uniaxial compressive loading. Jointed model material blocks exhibited different failure modes depending on the joint configurations. Orientation of the joint sets played a significant role related to the modes of failure. It was possible to obtain a strong non-linear relation between the jointed model mass strength and the fracture tensor component to cover the strengths resulting from all the different failure modes observed in the investigation. The fracture tensor component was used to obtain the combined effect of number of joint sets, joint density, and distributions of size and orientation of the joint sets in a chosen direction. Needed future research are suggested to generalize the promising rock mass strength criterion obtained in the performed research.

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

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.