1998 Research Symposium
Micromechanics and Flow
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ABSTRACTS Session II - Micromechanics and FracturingPorosity-Dependent Mechanisms of Failure Around Boreholes in Brea Sandstone
We have conducted laboratory-drilling experiments in Berea sandstone blocks of two different porosities subjected to truly triaxial far-field stress conditions representing depths reaching five kilometers. Depending on the simulated in situ stress conditions applied to the blocks, drilling led to borehole failure (breakouts). We then used a petrographic microscope to study thin sections of the drilled boreholes and inferred from them the mechanism of failure that resulted in the formation of breakouts. The observed behavior throws new light on the mechanical behavior of poorly consolidated sandstones and on a possible source of sand production. In the well-consolidated Berea sandstone (17% porosity) failure took the form of a ‘V’-shaped breakout occurring along the s h springline, similar to observations in granite and other crystalline rocks. This shape results from successive failure of rock flakes formed by dilatant extensile microcracks subparallel to s H direction. The observed microcracks preceding breakout failure are mainly intergranular (they propagate with greater ease in the much weaker rock matrix), with intermittent intragranular grain-splitting cracks. An entirely different failure mechanism was observed in the more poorly consolidated Berea sandstone (22.5% porosity). In this rock borehole failure occurred at the same locations as above (along the s h springline), but surprisingly the final shape of the breakout is best described as a narrow linear fracture or slot perpendicular to s H direction. At first, this behavior appears to be counterintuitive. Our interpretation of the mechanism forming this type of breakout is that it is related to the reduced cementing matrix material between the grains as inferred from the higher porosity. This also implies weak bonding between the quartzitic grains. Under sufficient far-field stresses, failure that first occurs in the much weaker rock matrix at the points of high stress concentration around the hole, partially or totally debonds the quartz grains. The debonded and exposed grains, assisted perhaps by the circulating drilling fluid, spall off in non-dilatant fashion, creating a higher stress concentration behind them. This in turn facilitates further disintegration of the cementing matrix, more non-dilatant grain debonding, and additional grains being removed along the s h spring line. The rugged free surface of the fracture-shaped breakout shows whole quartz grains left intact during this failure process, supporting the assertion that failure is limited to that of the matrix material. The final shape of breakouts in this sandstone and the inferred fracturing mechanism are very similar to the non-dilatational behavior observed in poorly-sintered glass bead bricks by Bessinger et al. (1997). This newly observed behavior could be a source of sand production, and may have major significance in studying instability problems in oil wells intersecting poorly consolidated sandstones. Micromechnaics of Poorly Consolidated Media
There are significant differences between the macroscopic mechanical properties (including static and dynamic deformation and strength) of well consolidated, indurated clastic rock and cohesionless granular material, and these differences arise from different micromechanical processes. Poorly consolidated rocks have intermediate properties. This work is focused on understanding the transition in micromechanical processes between cohensionless and well consolidated rock. The approach is to perform a variety of mechanical measurements on poorly consolidated samples which vary, first of all, in the amount of cohesion between grains. Qualitative and quantitative relationships are sought between micromechanical parameters and macroscopic properties. Correlations are also sought between macroscopic properties, for example, between seismic wave velocities and attenuation and other properties such as fracture toughness or compressive strength. Previous work has shown that under differential compressive stress, extensile crack growth accompanied by frictional dissipation fundamentally affects the macroscopic stress-strain strength properties of well consolidated granular rock. The extensile cracks form at the grain scale and are oriented parallel to the maximum compressive stress direction. Around a wellbore, where the major compressive stress is a hoop stress, extensile cracks form parallel to the surface of the wellbore, leading to typically observed wellbore failure features. In cohesionless material, grain sliding, rotation and rearrangement are dominate micromechanical processes under a large range of differential stress. In poorly consolidated rock it might be supposed that there is a transition between behavior dominated by grain sliding and rearrangements, and extensile crack growth. Experimental results indicate that the character of borehole breakouts may be significantly different if grains kinematically move instead of fracturing. Seismic wave propagation has been studied in saturated granular media under changing compressive stress, where the compressive stress can be considered as providing an effective cohesion between grains. As uniaxial stress was increased a monotonic increase in P-wave velocity was observed. However, quite different trends were observed in the amplitude and frequency content of the pulses. At low stress the behavior was best explained by a model characterizing the sample as a suspension of solid particles in a liquid. As stress is increased there is a transition to a condition in which energy is primarily propagating through the granular skeleton with the stiffness of the grain contacts having a first order effect on the amplitude and frequency content, as well as the velocity of the P-wave. Finally, to provide some preliminary observations of fracture in poorly consolidated media, three point bending tests were performed on sintered glass bead samples. Values of Young's modulus, E, were obtained from acoustic velocity measurements. Both Klc and E for the lightly sintered glass bead material were very low compared to rock data. The data support intuition that values of Klc for materials will go to zero as E approaches zero. Visual observation during the tests suggest that the process of failure in these very weak samples was similar to that in stronger rock. Results imply that the principles of fracture mechanics can be applied to crack propagation in granular media as long as there is non zero bond strength between grains. 3D Pore Scale Geometry of Sandstones, Basalts and Fractures
We have developed a computational package, 3DMA, to use as a tool to provide analysis of the geometry of the pore and grain phases of digitized three dimensional images. Current analyses for each phase include determination of volume fraction, specific surface area, 2-point spatial covariance, disconnected volume distribution, the medial axis (1D skeleton), path length and tortuosity distribution, coordination number, throat size and nodal pore volume distributions. We have applied this analysis to a variety of porous media images including rocks (sandstones, basalts, carbonates), glass micro-bead packs, cellulose fiber networks, biological images (neurons) and two-fluid mixtures. Medial axis analysis is a central component of the package. It provides a one dimensional skeleton enabling efficient searching and geometrical characterization as well as a graph for the application of graph theoretic network tools. I will present the results of our analyses for Fontainebleau sandstone, vesiculated basalt, and fractures in Harcourt granite and Carrara marble. Micromechanical Studies of Fracture
Preliminary work aimed at developing a micromechanical model for the deformation and failure of porous rocks will be described. A detailed description of the microstructure and mineral distribution in specimens of Berea sandstone has been obtained by high-resolution x-ray tomography. An efficient finite-element model has been developed that uses these x-ray tomographs for structural input; stress and strain fields in samples of more than 10 million elements can be determined using this model. Fracture can be modeled by the progressive failure of elements under high tensile loads; local tensile stresses develop in microstructural models, even if the macroscopic loads are compressive. An example of predicted and measured fracture patterns will be given. Strain Localization in Compacting Rock
Mollema and Antonellini (1996) recognized a new type of geologic structure, which they called compaction bands, in porous sandstones from the Navajo formation. These bands were tabular zones of crushed material that have undergone no shear, just pure compressional deformation and a consequent reduction in porosity. Haimson and Song (1997) found features that would seem to be identical to compaction bands in a laboratory study of borehole breakouts. Lajtai (1974) also reported what appear to be compaction bands in association with shear cracks in laboratory models. Because of the potential deleterious effects on the permeability of porous reservoir rocks, I used theory and experiment to investigate the possibility of producing compaction bands in Castlegate sandstone that had about 25% porosity. I applied the strain localization theory of Rudnicki and Rice (1975) to the problem of inhomogeneous compaction and found that the theory predicts that bands of pure compaction can occur for certain combinations of Poisson's ratio and dilatancy factor. Some features that may indeed be compaction bands were formed experimentally. The agreement between theory and experiment, at first sight, seems somewhat inconclusive because compaction bands and thick shear bands sometimes occur in the same specimen. But when subtle relations between constitutive properties of Castlegate sandstone and the predictions of the strain localization theory are considered, the interrelation between theory, field, and experiment can be explained.
An experimental and modeling approach has been pursued in testing the applicability of a bifurcation theory of shear-localization developed by Rudnicki and Rice (RR) to the process of shear deformation in dilating rocks. The modeling approach uses the results of triaxial testing to determine the constitutive parameters required by the RR formulation. A model was defined with sufficient flexibility to capture the major features of the triaxial tests, hardening, softening, pressure-dependence and shear-strain-driven dilatancy. Using a simplex fitting technique, the model parameters were determined that best fit all of the data. This approach avoids over-emphasizing any one data set, as in common in the use of uniaxial test results to determine fitting parameters. Fitting the stress-strain data and then differentiating the fitted analytical expressions, eliminated much of the numerical noise that differentiating the experimental results would have introduced. Having derived expressions for the plastic tangent modulus, slope of the yield surface and dilatancy parameter as a function of mean stress and plastic shear strain, it was then possible to calculate the results for any loading path for comparison with experiments. In addition to stress and strain, the criterion for localization could be calculated for a general stress path. Results for Tennessee marble under plane strain were in good agreement with the calculated results. Determining that localization has occurred is in general difficult using measures of strain that average over the entire sample. Using acoustic emissions it is possible to observe the localization process as the location of acoustic emissions evolves from a more or less random cloud of events to a well-defined planar feature. Results for Gosford sandstone under triaxial and plane strain show that localization occurs in the softening regime (post-peak) for triaxial test conditions and in the hardening regime (pre-peak) for plane strain, confirming a key prediction of the RR localization theory. Frequency-Dependent Acoustic Anisotropy from Fractures - Modeling †Visiting scientist at Lawrence Berkeley National Laboratory; on leave from Schlumberger-Doll Research, Ridgefield, CT Fractures in rock are localized planes of compliance. When fractures are aligned in near parallel sets, as often occurs in nature, this ordered compliance can give rise to anisotropy in the elastic properties. The conventional approach for determining the elastic anisotropy of such a medium is based on a static (i.e., zero frequency) approximation. The static description predicts directionally-dependent wave velocities and polarizations of the quasi-compressional and the two quasi-shear waves. The static approach, however, does not predict frequency-dependent wave phenomena such as attenuation, dispersion, and fracture guided waves that have been observed in laboratory and field measurements in fractured rock. One of the objectives of this research is to quantify the effects of wave frequency, fracture stiffness, and fracture spacing on the velocities and amplitudes of waves propagating in fractured rock. To investigate the frequency-dependent characteristics of seismic waves arising from a set of multiple, parallel fractures, we have formulated an analytic expression for elastic wave propagation in a medium composed of an infinite series of parallel fractures. The solution uses Floquet theory to impose periodic boundary conditions on the stress and displacement fields. Fractures are modeled as displacement-discontinuity boundaries. The equation was solved numerically to obtain slowness and wave surfaces for a range of frequencies and fracture stiffnesses. Analysis of this fully dynamic effective medium theory for fractured rock reveals that deviations from the static effective medium velocity can occur, even when the wavelength is much larger than the fracture spacing, provided that the fracture stiffnesses are low. The Floquet analysis also predicts pass-stop band transmission and reflection for waves propagating at oblique incidence to the fractures. This behavior was examined using a one-dimensional propagator code and a broadband source. When the entire waveform including the coda (i.e., the multiply reflected waves) is used to compute the spectrum, half-wavelength, inter-fracture resonances produce a transmission coefficient with alternating regions of complete transmission and of complete reflection. However, when the spectrum is computed from just the first coherent pulse, the transmission coefficient shows a strong low-pass filtering that results from reflection losses. These results suggest that the presence of fractures can be inferred from the low-pass filtering of the transmitted first coherent arrival and, possibly, from the full waveform transmission coefficient, provided that the source generates wavelengths that are on the order of the fracture spacing. Frequency Dependent Acoustic Anisotropy from Fractures: Experimental Many reservoirs include multiple, near-planar discontinuities such as joints and fractures that occur on multiple length scales. The presence of a set or sets of fractures produces an anisotropy of material properties. Traditionally, the wave propagation through a fractured rock has been modeled using effective medium theories that use a static approximation to develop analytic expressions for the elastic moduli of a population of microcracks or planar fractures. The effective medium approach assumes that a fracture will reduce the modulus of the rock, which in turn reduces the seismic velocity. The effect of the fracture is distributed throughout the bulk and the discreteness of the fracture is lost. Hence, a seismic reduction in velocity is observed, but the location of the cause of the reduction is lost. An alternative method for analyzing the seismic reponse of a fracture that retains the discreteness of the effect of the fracture is the displacement discontinuity or non-welded contact theory. Several investigators have modeled a fracture as a non-welded contact which is assumed to have negligible thickness compared to the seismic wavelength. The non-welded contact is represented as an interface across which stresses are continuous but displacements are not. This pureley elastic theory gives rise to frequency dependent transmission and reflection coefficients as well as frequency dependent velocities. Field and laboratory measurements have shown that a fracture behaves as a low-pass filter that attenuates the wave by removing the high-frequency components of the signal and produces a frequency-dependent time delay. The displacement discontinuity boundary conditions have also been used to derive plane wave transmission and reflection coefficients, group time delays, interface waves, and guided love waves. In this study, experimentally measured compressional wave anisotropy caused by the presence of a single fracture and multiple parallel fractures is compared to theoretical predictions based on the displacement discontinuity theory. The comparison between the theoretically predicted transmission coefficients and the experimentally determined values enabled us to investigate the robustness of the displacement discontinuity theory for obliquely incident compressional waves on a single fracture and on multiple parallel fractures. The analytical solution for plane-wave propagation across a displacement discontinuity predicts: (1) that at high frequencies the group time delay is not as sensitive to changes in stiffness as transmitted amplitudes; (2) that the transmission coefficient decreases in magnitude with increasing frequency; (3) that the transmission coefficient decreases for glancing angles of incidence to the fracture; (4) that a discontinuity in the value of the transmission coefficient should occur at the orientation when the trajectory of a wave crosses additional fractures; and (5) that single values of normal and tangential fracture specific stiffness can be used to fit all angles of incidence for all frequencies. These predictions are examined in light of the experimental data. While the displacement discontinuity theory is able to fit the measured compressional wave transmission coefficients well for most angles of incidence, discrepancies between theory and experiment still exist. The deviations for glancing angles of incidence and for angles of incidence near the transition from multiple fractures to a single fracture are not completely understood. The plane-wave solution used in this analysis does not account for the partitioning of energy into interface waves, i.e., waves that propagate along the fracture and which depend on fracture specific stiffness. A full numerical method needs to be applied to compare oblique incidence wave propagation across a fracture to accurately assess energy partitioning by a fracture or sets of fracture. In addition, further effort is needed to explore the frequency dependent fracture stiffness. If the frequency dependence gives an indication of the distribution of asperities or contact regions in a fracture, this information could be used to predict the hydraulic properties of fractures.
This talk touches on methodologies devised to help understand/predict the mechanical behavior of solids from the viewpoint of a metallurgist / materials scientist. The various approaches in this area that presently exist can be categorized into essentially five groups, each group distinguished from the others by a characteristic length scale range, namely: electronic structure (to about 0.15 nm), atomistic simulations (to about 15 nm), micro-scale modeling (to 15,000 nm), mesoscale modeling (to 5 x 10^6 nm), and continuum plasticity (to ?). The focus of each of these groups will be described briefly and it will be noted that, generally, within each category there exists sophisticated, highly predictive capabilities. The principal challenge remains, however, to link these categories so that an intelligent and intelligible transfer of information is able to take place among the groups. Currently, in the bulk of the metals fabrication industry, there is no need for this linkage. For example, empirical constitutive relations developed for structural steels together with 3D finite element models are usually adequate for predicting (with satisfactory accuracy) the outcomes of most complex forming operations. Similar applications involving more complex materials (e.g., complicated by anisotropy) are also commonplace. However, new materials continue to be developed and recently, some of these materials have strength levels that are so high that new forms of plasticity and failure mechanisms have appeared. In addition, exotic applications, e.g. involving very small sizes, operation at very high temperature or other extreme environments, are creating a new set of demands on manufacturing processes. In such circumstances, the development of an understanding of behavior and predictive capability is best served (economically) by linking activity over the entire hierarchy of length scales. We will give some examples of results of the atomistic and micro-scale modeling and briefly describe the kind of information that needs to move up the length scale chain from smallest to largest. |
