Rock Damage and Fluid Transport, Part I

Fracture toughness measurements under static loading conditions have been carried out in Barre and Lac du Bonnet granites. An advanced AE technique has been adopted to monitor real-time crack initiation and propagation around the principal crack in these tests to understand the processes of brittle failure under tension and related characteristics of the resulting fracture process zone. The anisotropy of Mode I fracture toughness has been investigated along specific directions. Microcrack density and orientation analysis from thin section studies have shown these characteristics to be the primary cause of the observed variation in fracture toughness, which is seen to vary between 1.14 MPa.(m)^1/2 and 1.89 MPa.(m)^1/2 in Barre granite. The latter value represents the case in which the crack is propagated at right angles to the main set of microcracks. The creation of a significant fracture process zone surrounding the propagating main crack has been confirmed. Real-time imaging of the fracture process and formation of fracture process zone by AE techniques yielded results in very good agreement with those obtained by direct optical analysis.

Crack damage results in a decrease of elastic wave velocities and in the development of anisotropy. Using non-interactive crack effective medium theory as a fundamental tool, we calculate dry and wet elastic properties of cracked rocks in terms of a crack density tensor, average crack aspect ratio and mean crack fabric orientation from the solid grains and fluid elastic properties. Using this same tool, we show that both the anisotropy and shear-wave splitting of elastic waves can be derived. Two simple crack distributions are considered for which the predicted anisotropy depends strongly on the saturation, reaching up to 60% in the dry case. Comparison with experimental data on two granites, a basalt and a marble, shows that the range of validity of the non-interactive effective medium theory model extends to a total crack density of approximately 0.5, considering symmetries up to orthorhombic. In the isotropic case, Kachanov’s (1994) non-interactive effective medium model was used in order to invert elastic wave velocities and infer both crack density and aspect ratio evolutions. Inversions are stable and give coherent results in terms of crack density and aperture evolution. Crack density variations can be interpreted in terms of crack growth and/or changes of the crack surface contact areas as cracks are being closed or opened respectively. More importantly, the recovered evolution of aspect ratio shows an exponentially decreasing aspect ratio (and therefore aperture) with pressure, which has broader geophysical implications, in particular on fluid flow. The recovered evolution of aspect ratio is also consistent with current mechanical theories of crack closure. In the anisotropic cases—both transverse isotropic and orthorhombic symmetries were considered—anisotropy and saturation patterns were well reproduced by the modelling, and mean crack fabric orientations we recovered are consistent with in situ geophysical imaging.

Our results point out that: (1) It is possible to predict damage, anisotropy and saturation in terms of a crack density tensor and mean crack aspect ratio and orientation; (2) using well constrained wave velocity data, it is possible to extrapolate the contemporaneous evolution of crack density, anisotropy and saturation using wave velocity inversion as a tool; 3) using such an inversion tool opens the door in linking elastic properties, variations to permeability.

Acoustic emissions (AE), compressional (P), shear (S) wave velocities, and volumetric strain of Etna basalt and Aue granite were measured simultaneously during triaxial compression tests. Deformation-induced AE activity and velocity changes were monitored using twelve P-wave sensors and eight orthogonally polarized S-wave piezoelectric sensors; volumetric strain was measured using two pairs of orthogonal strain gages glued directly to the rock surface. P-wave velocity in basalt is about 3 km/s at atmospheric pressure, but increases by > 50% when the hydrostatic pressure is increased to 120 MPa. In granite samples initial P-wave velocity is 5 km/s and increases with pressure by < 20%. The pressure-induced changes of elastic wave speed indicate dominantly compliant low-aspect ratio pores in both materials, in addition Etna basalt also contains high-aspect ratio voids. In triaxial loading, stress-induced anisotropy of P-wave velocities was significantly higher for basalt than for granite, with vertical velocity components being faster than horizontal velocities. However, with increasing axial load, horizontal velocities show a small increase for basalt but a significant decrease for granite. Using first motion polarity we determined AE source types generated during triaxial loading of the samples. With increasing differential stress AE activity in granite and basalt increased with a significant contribution of tensile events. Close to failure the relative contribution of tensile events and horizontal wave velocities decreased significantly. A concomitant increase of double-couple events indicating shear, suggests shear cracks linking previously formed tensile cracks.

New observations of fracture nucleation are presented from three triaxial compression experiments on intact samples of Westerly granite, using Acoustic Emission (AE) monitoring. By conducting the tests under different loading conditions, the fracture process is demonstrated for quasistatic fracture (under AE Feedback load), a slowly developing unstable fracture (loaded at a ‘slow’ constant strain rate of 2.5 × 10)⁻⁶ /s) and an unstable fracture that develops near instantaneously (loaded at a ‘fast’ constant strain rate of 5 × 10)⁻⁵ /s). By recording a continuous ultrasonic waveform during the critical period of fracture, the entire AE catalogue can be captured and the exact time of fracture defined. Under constant strain loading, three stages are observed: (1) An initial nucleation or stable growth phase at a rate of ∼ 1.3 mm/s, (2) a sudden increase to a constant or slowly accelerating propagation speed of ∼ 18 mm/s, and (3) unstable, accelerating propagation. In the ∼ 100 ms before rupture, the high level of AE activity (as seen on the continuous record) prevented the location of discrete AE events. A lower bound estimate of the average propagation velocity (using the time-to-rupture and the existing fracture length) suggests values of a few m/s. However from a low gain acoustic record, we infer that in the final few ms, the fracture propagation speed increased to 175 m/s. These results demonstrate similarities between fracture nucleation in intact rock and the nucleation of dynamic instabilities in stick slip experiments. It is suggested that the ability to constrain the size of an evolving fracture provides a crucial tool in further understanding the controls on fracture nucleation.

We test the hypothesis that the general trend of P-wave and S-wave sonic log velocities and resistivity with depth in the pilot hole of the KTB site Germany, can be explained by the progressive closure of the compliant porosity with increasingly effective pressure. We introduce a quantity θ_c characterizing the stress sensitivity of the mentioned properties. An analysis of the downhole measurements showed that estimates of the quantitiy θ_c for seismic velocities and electrical formation factor of the in situ formation coincide. Moreover, this quantity is 3.5 to 4.5 times larger than the averaged stress sensitivity obtained from core samples. We conclude that the hypothesis mentioned above is consistent with both data sets. Moreover, since θ_c corresponds approximately to the inverse of the effective crack aspect ratio, larger in situ estimates of θ_c might reflect the influence of fractures and faults on the stress sensitivity of the crystalline formation in contrast to the stress sensitivity of the nearly intact core samples. Finally, because the stress sensitivity is directly related to the elastic nonlinearity we conclude that the elastic nonlinearity (i.e., deviation from linear stress-strain relationship i.e., Hooke’s law) of the KTB rocks is significantly larger in situ than in the laboratory.

Time delays associated with processes leading to a failure or stress relaxation in materials and earthquakes are studied in terms of continuum damage mechanics. Damage mechanics is a quasiempirical approach that describes inelastic irreversible phenomena in the deformation of solids. When a rock sample is loaded, there is generally a time delay before the rock fails. This period is characterized by the occurrence and coalescence of microcracks which radiate acoustic signals of broad amplitudes. These acoustic emission events have been shown to exhibit power-law scaling as they increase in intensity prior to a rupture. In case of seismogenic processes in the Earth’s brittle crust, all earthquakes are followed by an aftershock sequence. A universal feature of aftershocks is that their rate decays in time according to the modified Omori’s law, a power-law decay. In this paper a model of continuum damage mechanics in which damage (microcracking) starts to develop when the applied stress exceeds a prescribed yield stress (a material parameter) is introduced to explain both laboratory experiments and systematic temporal variations in seismicity.

This paper reviews the use of the cracked Chevron-notched Brazilian disc (CCNBD) for fracture toughness testing. Theoretical and experimental backgrounds of the method are described. Some issues regarding the current development (i.e., recalibration) of the specimen geometry are presented and discussed. A number of geometries related to the CCNBD proposed recently for fracture toughness testing of rock are then introduced and commented on.

Apparent fracture toughness in Mode I of microcracking materials such as rocks under confining pressure is analyzed based on a cohesive crack model. In rocks, the apparent fracture toughness for crack propagation varies with the confining pressure. This study provides analytical solutions for the apparent fracture toughness using a cohesive crack model, which is a model for the fracture process zone. The problem analyzed in this study is a fluid-driven fracture of a two-dimensional crack with a cohesive zone under confining pressure. The size of the cohesive zone is assumed to be negligibly small in comparison to the crack length. The analyses are performed for two types of cohesive stress distribution, namely the constant cohesive stress (Dugdale model) and the linearly decreasing cohesive stress. Furthermore, the problem for a more general cohesive stress distribution is analyzed based on the fracture energy concept. The analytical solutions are confirmed by comparing them with the results of numerical computations performed using the body force method. The analytical solution suggests a substantial increase in the apparent fracture toughness due to increased confining pressures, even if the size of the fracture process zone is small.

This paper discusses the applicability of the tension-softening model in the determination of the fracture toughness of rocks, where the fracture toughness evaluated based on the tension-softening model is compared with the crack growth resistance deduced from laboratory-scale hydraulic fracturing tests. It is generally accepted that the fracture process is dominated by the growth of a fracture process zone for most types of rocks. In this study, the J-integral based technique is employed to determine the fracture toughness of Iidate granite on the basis of the tension-softening model, where compact tension specimens of different dimensions were tested in order to examine the specimen size effect on the measured fracture toughness. It was shown that the tension-softening relation deduced from the J-integral based technique allowed us to determine the specimen size independent fracture toughness K _c of Iidate granite.Laboratory-scale hydraulic fracturing tests were performed on cubic specimens (up to a 10 m sized specimen), where cyclic pressurization was conducted using a rubber-made straddle packer to observe the extent of the hydraulically induced crack. The experimental results of pressure and crack length were then used to construct the crack growth resistance curve based on the stress intensity factor K. The crack growth resistance obtained from the hydraulic fracturing tests was observed to initially increase and then level off, giving a constant K value for a long crack extension stage. The plateau K value in the crack growth resistance curve was found to be in reasonable agreement with the fracture toughness K _c deduced from the tension-softening relation. It was demonstrated that the tension-softening model provides a useful tool to determine the appropriate fracture toughness of rocks, which may be applicable for the analysis of the process of large-scale crack extension in rock masses.

An experimental procedure for testing dynamic tensile strength and elastic modulus of rock materials at high strain rate loading is presented in this paper. In our test the split Hopkinson pressure bar (SHPB) was used to diametrally impact the Brazilian disc (BD) and flattened Brazilian disc (FBD) specimens of marble. A tensile strain rate of about 45 1/s was achieved at the center of the specimen. In order to improve the accuracy of the analysis, the initiation time difference between the strain waves acting on the two flat ends of the FBD specimen was treated properly. Typical failure modes corresponding to different loading conditions were observed. It was verified with a finite-element simulation that the equilibrium condition was established in the specimen before its failure. This numerical simulation validates the experimental procedure and also proves the suitability of formulation for the basic equations.

This paper reviews the efforts made in the last 100 years to characterize the effect of the intermediate principal stress σ ₂ on brittle fracture of rocks, and on their strength criteria. The most common theories of failure in geomechanics, such as those of Coulomb, and Mohr, disregard σ ₂ and are typically based on triaxial testing of cylindrical rock samples subjected to equal minimum and intermediate principal stresses (σ ₃=σ ₂). However, as early as 1915 Böker conducted conventional triaxial extension tests (σ ₁=σ ₂) on the same Carrara marble tested earlier in conventional triaxial compression by von Kármán that showed a different strength behavior. Efforts to incorporate the effect of σ ₂ on rock strength continued in the second half of the last century through the work of Nadai, Drucker and Prager, Murrell, Handin, Wiebols and Cook, and others. In 1971 Mogi designed a high-capacity true triaxial testing machine, and was the first to obtain complete true triaxial strength criteria for several rocks based on experimental data. Following his pioneering work, several other laboratories developed equipment and conducted true triaxial tests revealing the extent of σ ₂ effect on rock strength (e.g., Takahashi and Koide, Michelis, Smart, Wawersik). Testing equipment emulating Mogi’s but considerably more compact was developed at the University of Wisconsin and used for true triaxial testing of some very strong crystalline rocks. Test results revealed three distinct compressive failure mechanisms, depending on loading mode and rock type: shear faulting resulting from extensile microcrack localization, multiple splitting along the σ ₁ axis, and nondilatant shear failure. The true triaxial strength criterion for the KTB amphibolite derived from such tests was used in conjunction with logged breakout dimensions to estimate the maximum horizontal in situ stress in the KTB ultra deep scientific hole.

Stress changes within and around a depleting petroleum reservoir can lead to reservoir compaction and surface subsidence, affect drilling and productivity of oil wells, and influence seismic waves used for monitoring of reservoir performance. Currently modeling efforts are split into more or less coupled geomechanical (normally linearly elastic), fluid flow, and geophysical simulations. There is evidence (from e.g. induced seismicity) that faults may be triggered or generated as a result of reservoir depletion. The numerical technique that most adequately incorporates fracture formation is the DEM (Discrete Element Method). This paper demonstrates the feasibility of the DEM (here PFC; Particle Flow Code) to handle this problem. Using an element size of 20 m, 2-D and 3-D simulations have been performed of stress and strain evolution within and around a depleting reservoir. Within limits of elasticity, the simulations largely reproduce analytical predictions; the accuracy is however limited by the element size. When the elastic limit is exceeded, faulting is predicted, particularly near the edge of the reservoir. Simulations have also been performed to study the activation of a pre-existing fault near a depleting reservoir.

An understanding of the formation of shear fractures is important in many rock engineering design problems. Laboratory experiments have been performed to determine the Mode II fracture toughness of Mizunami granite rock samples using a cylindrical ‘punch-through’ testing device. In this paper we attempt to understand and interpret the experimental results by numerical simulation of the fundamental shear fracture initiation and coalescence processes, using a random array of displacement discontinuity crack elements. It is found that qualitative agreement between the experimental and numerical results can be established, provided that shear-like micro-scale failure processes can be accommodated by the failure initiation rules that are used in the numerical simulations. In particular, it is found that the use of an exclusively tension-driven failure initiation rule does not allow the formation of macro-shear structures. It is apparent, also, that further investigation is required to determine how consistent rules can be established to link micro-failure criteria to equivalent macro-strength and toughness properties for a macro-shear slip surface.

Upscaling is a major issue regarding mechanical and transport properties of rocks. This paper examines three issues relative to upscaling. The first one is a brief overview of Effective Medium Theory (EMT), which is a key tool to predict average rock properties at a macroscopic scale in the case of a statistically homogeneous medium. EMT is of particular interest in the calculation of elastic properties. As discussed in this paper, EMT can thus provide a possible way to perform upscaling, although it is by no means the only one, and in particular it is irrelevant if the medium does not adhere to statistical homogeneity. This last circumstance is examined in part two of the paper. We focus on the example of constructing a hydrocarbon reservoir model. Such a construction is a required step in the process of making reasonable predictions for oil production. Taking into account rock permeability, lithological units and various structural discontinuities at different scales is part of this construction. The result is that stochastic reservoir models are built that rely on various numerical upscaling methods. These methods are reviewed. They provide techniques which make it possible to deal with upscaling on a general basis. Finally, a last case in which upscaling is trivial is considered in the third part of the paper. This is the fractal case. Fractal models have become popular precisely because they are free of the assumption of statistical homogeneity and yet do not involve numerical methods. It is suggested that using a physical criterion as a means to discriminate whether fractality is a dream or reality would be more satisfactory than relying on a limited data set alone.