A Model for the Mechanical Behaviour of Bentheim Sandstone in the Brittle Regime

— The mechanical behaviour of Bentheim sandstone, a homogeneous quartz-rich sandstone with porosity of 22.8%, was investigated by triaxial compression tests conducted on dry samples. At confining pressures up to 35 MPa, the failure mode was characterized by a typical brittle deformation regime, as the samples showed dilatancy and failed by strain softening and brittle faulting. Previous studies have shown that the mechanical behaviour and failure mode of brittle porous granular rocks are governed by the time-dependent growth of microcracks. We analyse this process using the “Pore Crack Model” based on fracture mechanics analysis. It is consistent with the microstructure of porous granular rocks since it considers the growth of axial cracks from cylindrical holes in two dimensions. These cracks grow when their stress intensity factors reach the subcritical crack growth limit. Interaction between neighbouring cracks is introduced by calculating the stress intensity factor as the sum of two terms: a component for an isolated crack and an interaction term computed using the method of successive approximations. It depends on crack length, pore radius, pore density, and applied stresses. The simulation of crack growth from cylindrical holes, associated with a failure criterion based on the coalescence of interacting cracks, is used to compare the theoretical stress at the onset of dilatancy and at macroscopic rupture to the experimental determined values. Our approach gives theoretical results in good agreement with experimental data when microstructural parameters consistent with observations are introduced.     

Acoustic Emissions Monitoring during Inelastic Deformation of Porous Sandstone: Comparison of Three Modes of Deformation

In some reservoirs, large deformations can occur during oil or gas production because of the effective stress change. For very porous rocks, these production operations can be sufficient to cause inelastic deformation and irreversible damage. Rock formations can undergo deformation by different mechanisms, including dilatancy or pore collapse. In the laboratory, it has been shown that the inelastic deformation and failure mode of porous rocks are pressure sensitive. Indeed, when subjected to an overall compressive loading, a porous rock may fail by shear localization, compaction localization, or by cataclastic compaction. Acoustic emission (AE) records provide important information to understand the failure mode of rocks: the spatial evolution of damage as well as the source mechanisms can be followed using this technique. In this paper, we present three different laboratory axisymmetric compression experiments, performed on Bleurswiller sandstone, which enable us to compare the acoustic emission signature of these three modes of deformation. Our data show that compaction localization and cataclastic compaction are characterized by similar acoustic signatures (in terms of AE sources characteristics and evolution of AE number), in comparison to the acoustic signature from shear localization. This implies similar micromechanisms involved during compaction bands formation and cataclastic compaction.     

An earthquake instability model based on faults containing high fluid-pressure compartments

It has been proposed that large strike-slip faults such as the San Andreas contain water in seal-bounded compartments. Arguments based on heat flow and stress orientation suggest that in most of the compartments, the water pressure is so high that the average shear strength of the fault is less than 20 MPa. We propose a variation of this basic model in which most of the shear stress on the fault is supported by a small number of compartments where the pore pressure is relatively low. As a result, the fault gouge in these compartments is compacted and lithified and has a high undisturbed strength. When one of these locked regions fails, the system made up of the neighboring high and low pressure compartments can become unstable. Material in the high fluid pressure compartments is initially underconsolidated since the low effective confining pressure has retarded compaction. As these compartments are deformed, fluid pressure remains nearly unchanged so that they offer little resistance to shear. The low pore pressure compartments, however, are overconsolidated and dilate as they are sheared. Decompression of the pore fluid in these compartments lowers fluid pressure, increasing effective normal stress and shear strength. While this effect tends to stabilize the fault, it can be shown that this dilatancy hardening can be more than offset by displacement weakening of the fault (i.e., the drop from peak to residual strength). If the surrounding rock mass is sufficiently compliant to produce an instability, slip will propagate along the fault until the shear fracture runs into a low-stress region. Frictional heating and the accompanying increase in fluid pressure that are suggested to occur during shearing of the fault zone will act as additional destabilizers. However, significant heating occurs only after a finite amount of slip and therefore is more likely to contribute to the energetics of rupture propagation than to the initiation of the instability.We present results of a one-dimensional dynamic Burridge-Knopoff-type model to demonstrate various aspects of the fluid-assisted fault instability described above. In the numerical model, the fault is represented by a series of blocks and springs, with fault rheology expressed by static and dynamic friction. In addition, the fault surface of each block has associated with it pore pressure, porosity and permeability. All of these variables are allowed to evolve with time, resulting in a wide range of phenomena related to fluid diffusion, dilatancy, compaction and heating. These phenomena include creep events, diffusion-controlled precursors, triggered earthquakes, foreshocks, aftershocks, and multiple earthquakes. While the simulations have limitations inherent to 1-D fault models, they demonstrate that the fluid compartment model can, in principle, provide the rich assortment of phenomena that have been associated with earthquakes.     

Shear-enhanced compaction during non-linear viscous creep of porous calcite-quartz aggregates

In this paper, we extend the previous studies of semi-brittle flow of synthetic calcite-quartz aggregates to a range of temperatures and effective pressures where viscous creep occurs. Triaxial deformation experiments were performed on hot-pressed calcite-quartz aggregates containing 10, 20 and 30 wt% quartz at confining pressure of 300 MPa, pore pressures of 50-290 MPa, temperatures of 673-1073 K and strain rates of 3.0×10⁻⁵/s, 8.3×10⁻⁵/s and 3.0×10⁻⁴/s. Starting porosity varied from 5 to 9%. We made axial and volumetric strain measurements during the mechanical tests. Pore volume change was measured by monitoring the volume of pore fluid that flows out of or into the specimen at constant pore pressure. Yield stress increased with decreasing porosity and showed a dependence on effective pressure. Thus, the yield stress versus effective pressure can be described as a yield surface with negative slope that expands with decreasing porosity and increasing strain hardening, gradually approaching the envelope of strength at 10% strain, which has a positive slope. Creep of porous rock can be modeled to first order as an isolated equivalent void in an incompressible nonlinear viscous matrix. An incremental method is used to calculate the stress-strain curve of the porous material under a constant external strain rate. The numerical simulations reproduce general trends of the deformation behavior of the porous rock, such as the yield stress decreasing with increasing effective pressure and significant strain hardening at high effective pressure. The drop of yield stress with increasing porosity is modeled well, and so is the volumetric strain rate, which increases with increasing porosity.     

Permeability-porosity Relationships in Rocks Subjected to Various Evolution Processes

— It is well known that there is no “universal” permeability-porosity relationship valid in all porous media. However, the evolution of permeability and porosity in rocks can be constrained provided that the processes changing the pore space are known. In this paper, we review observations of the relationship between permeability and porosity during rock evolution and interpret them in terms of creation/destruction of effectively and non-effectively conducting pore space. We focus on laboratory processes, namely, plastic compaction of aggregates, elastic-brittle deformation of granular rocks, dilatant and thermal microcracking of dense rocks, chemically driven processes, as a way to approach naturally occurring geological processes. In particular, the chemically driven processes and their corresponding evolution permeability-porosity relationships are discussed in relation to sedimentary rocks diagenesis.     

Compaction and Failure in High Porosity Carbonates: Mechanical Data and Microstructural Observations

We investigated systematically the micromechanics of compaction in two carbonates of porosity above 30%, Majella grainstone and Saint Maximin limestone. The composition, grain size and pore surface area of these rocks were determined. Hydrostatic compression experiments were performed under dry and wet conditions beyond the onset of grain crushing. A significant water weakening effect was observed in both rocks. A set of conventional triaxial experiments was also performed on both rocks under dry conditions at confining pressures ranging from 3 to 31 MPa. Microstructural observations were carried out on the deformed samples. The mechanical behavior of these high porosity carbonates is dominated by shear-enhanced compaction associated in most cases with strain hardening. Stress-induced cracking and grain crushing are the dominant micromechanisms of deformation in both rocks. In Majella grainstone, compactive shear bands appeared at low confinement, in qualitative agreement with the deformation bands observed in the field. At higher confining pressures, compaction localization was inhibited and homogeneous cataclastic flow developed. In Saint Maximin limestone, compaction localization was observed at all confining pressures. An increasing number of compactive shear bands at various orientations appeared with increasing strain. These new data suggest that compaction localization is important in the mechanical compaction of high porosity carbonates.     

Permeability of Triaxially Compressed Sandstone: Influence of Deformation and Strain-rate on Permeability

— The influence of differential stress on the permeability of a Lower Permian sandstone was investigated. Rock cylinders of 50 mm in diameter and 100 mm length of a fine-grained (mean grain size 0.2 mm), low-porosity (6–9%) sandstone were used to study the relation between differential stress, rock deformation, rock failure and hydraulic properties, with a focus on the changes of hydraulic properties in the pre-failure and failure region of triaxial rock deformation. The experiments were conducted at confining pressures up to 20 MPa, and axial force was controlled by lateral strain with a rate ranging from 10^?6 to 10^?7 sec^?1. While deforming the samples, permeability was determined by steady-state technique with a pressure gradient of 1 MPa over the specimen length and a fluid pressure level between 40 and 90% of the confining pressure. The results show that permeability of low-porosity sandstones under increasing triaxial stress firstly decreases due to compaction and starts to increase after the onset of dilatancy. This kind of permeability evolution is similar to that of crystalline rocks. A significant dependence of permeability evolution on strain rate was found. Comparison of permeability to volumetric strain demonstrates that the permeability increase after the onset of dilatancy is not sufficient to regain the initial permeability up to failure of the specimen. The initial permeability, which was determined in advance of the experiments, usually was regained in the post-failure region. After the onset of dilatancy, the permeability increase displays a linear dependence on volumetric strain.     

对某些与膨胀现象有关因素的定性分析

本文利用库仑-纳维（coulomb-Navier）强度理论作为地壳断层剪切破坏的判据,确定地震前震源区应力的极限状态。然后,应用断裂力学理论分析圆盘裂缝稳态扩展的条件。导出地震前稳态扩展微裂缝数目的积分表达式。以此为基础,讨论断层参数、应力场参数和岩石物理力学特性参数与膨张现象间的关系。得到的结果可用来解释某些地区震前没有膨胀现象和另一些地区震前出现强烈的膨胀现象。并且,对膨胀理论预报地震的适用范围和应用经验公式预报震级的条件,提出了一些看法。     

Magnetism of rocks and volumetric strain in uniaxial failure tests

Summary The relation between remanent magnetization and volumetric strain for gabbro samples stressed in uniaxial compression inside a near zero-field -metal shield has been examined. For samples with an induced IRM parallel to the axis of compression, remanent magnetization decreased linearly up to the onset of dilatancy. As increased stress produced additional dilatancy, the variation of remanent magnetization became nonlinear, and the stress dependence continually decreased until the rock failed. Stress cycling with the peak stress augmented for each cycle produced a continuous decrease in the zero stress value of the IRM although an appreciable amount of recovery was observed during unloading. When the sample was loaded in constant stress increments after the onset of dilatancy and held for several minutes at each level, time-dependent variations in remanent magnetization coincided with time-dependent increases in inelastic volumetric strain. In general as the inelastic creep rate increases, the rate of change in remanent magnetization increases. These results suggest that dilatancy related effects of the intensity of rock magnetization should be observed in magnetic rocks in epicentral regions prior to earthquakes and may serve as both long- and short-term precursors.     

An equivalent medium model for wave simulation in fractured porous rocks

Seismic wave propagation in reservoir rocks is often strongly affected by fractures and micropores. Elastic properties of fractured reservoirs are studied using a fractured porous rock model, in which fractures are considered to be embedded in a homogeneous porous background. The paper presents an equivalent media model for fractured porous rocks. Fractures are described in a stress‐strain relationship in terms of fracture‐induced anisotropy. The equations of poroelasticity are used to describe the background porous matrix and the contents of the fractures are inserted into a matrix. Based on the fractured equivalent‐medium theory and Biot's equations of poroelasticity, two sets of porosity are considered in a constitutive equation. The porous matrix permeability and fracture permeability are analysed by using the continuum media seepage theory in equations of motion. We then design a fractured porous equivalent medium and derive the modified effective constants for low‐frequency elastic constants due to the presence of fractures. The expressions of elastic constants are concise and are directly related to the properties of the main porous matrix, the inserted fractures and the pore fluid. The phase velocity and attenuation of the fractured porous equivalent media are investigated based on this model. Numerical simulations are performed. We show that the fractures and pores strongly influence wave propagation, induce anisotropy and cause poroelastic behaviour in the wavefields. We observe that the presence of fractures gives rise to changes in phase velocity and attenuation, especially for the slow P‐wave in the direction parallel to the fracture plane.     

Permeability Evolution During Non-linear Viscous Creep of Calcite Rocks

Permeability, storage capacity and volumetric strain were measured in situ during deformation of hot-pressed calcite aggregates containing 10, 20, and 30 wt% quartz. Both isostatic and conventional triaxial loading conditions were used. The tests were performed at confining pressure of 300 MPa, pore pressures between 50 to 290 MPa, temperatures from 673 to 873 K and strain rates of 3 × 10⁻⁵ s⁻¹. Argon gas was used as the pore fluid. The initial porosities of the starting samples varied from 5% to 9%, with higher porosity correlated to higher quartz content. Microstructural observations after the experiment indicate two kinds of pores are present: 1) Angular, crack-like pores along boundaries between quartz grains or between quartz and calcite grains and 2) equant and tubular voids within the calcite matrix. Under isostatic loading conditions, the compaction rate covaries with porosity and increases with increasing effective pressure. Most of the permeability reduction induced during compaction is irreversible and probably owes to plastic processes. As has been found in previous studies on hot-pressed calcite aggregates, permeability, k, is nonlinearly related to porosity, ϕ. Over small changes in porosity, the two parameters are approximately related as k ∝ ϕⁿ. The exponent n strongly increases as porosity decreases to a finite value (from about 4 to 6% depending on quartz content), suggesting a porosity percolation threshold. When subjected to triaxial deformation, the calcite-quartz aggregates exhibit shear-enhanced compaction, but permeability does not decrease as rapidly as it does under isostatic conditions. During triaxial compaction the exponent n only varies between 2 and 3. Non-isostatic deformation seems to reduce the percolation threshold, and, in fact, enhances the permeability relative to that at the same porosity during isostatic compaction. Our data provide constraints on the governing parameters of the compaction theory which describes fluid flow through a viscous matrix, and may have important implications for expulsion of sedimentary fluids, for fluid flow during deformation and metamorphism, and melt extraction from partially molten rocks.     

Petrophysical study of faults in sandstone using petrographic image analysis and X-ray computerized tomography

Petrographic image analysis (PIA) and X-ray computerized tomography (CT) provide local determinations of porosity in sandstone. We have investigated small faults called deformation bands in porous sandstones using these techniques. Because the petrophysical properties of the fault rock vary at a small scale (mm scale), the ability of PIA and CT to determine porosity in small volumes of rock and to map porosity distribution in two and three dimensions is crucial. This information is used to recognize the processes involved in fault development and the different kinds of microstructures associated with dilatancy and compaction. The petrophysical study of fault rock in sandstone permits one to make predictions of the hydraulic properties of a fault and thereby evaluate the sealing or fluid transmitting characteristics of faulted reservoirs and aquifers. The results of this study indicate that faulting in sandstone alters the original porosity and permeability of the host rock: the porosity is reduced by an order of magnitude and the permeability is reduced by one to more than seven orders of magnitude for faults associated with compaction.     

应力对岩石中孔隙流体压力的影响和地下水位震前异常变化机理.

在不同的围压和初始孔隙压力作用下,对孔隙岩石试件进行了不排液的三轴压缩试验,观测了轴向应变和孔隙压力随差应力的变化.试验结果表明:在低差应力下,孔隙压力随差应力的增加而增加.在中等和高差应力下孔隙压力随差应力的增加而降低.在高差应力下,孔隙压力也会随差应力的大幅度降低而升高.基于上述试验,假定地下水位的震前异常变化是地壳岩石中差应力变化的反映.根据这种假定解释了唐山大震前震中及外围地区地下水位异常变化的特征.      

Laboratory measurement of compaction-induced permeability change in porous rocks: Implications for the generation and maintenance of pore pressure excess in the crust

Permeability exerts significant control over the development of pore pressure excess in the crust, and it is a physical quantity sensitively dependent on the pore structure and stress state. In many applications, the relation between permeability and effective mean stress is assumed to be exponential and that between permeability and porosity is assumed to be a power law, so that the pressure sensitivity of permeability is characterized by the coefficient and the porosity sensitivity by the exponent . In this study, we investigate experimentally the dependence of permeability on pressure and porosity in five sandstones with porosities ranging from 14% to 35% and we review published experimental data on intact rocks, unconsolidated materials and rock fractures. The laboratory data show that the pressure and porosity sensitivities differ significantly for different compaction mechanisms, but for a given compaction mechanism, the data can often be approximated by the empirical relations. The permeabilities of tight rocks and rock joints show relatively high pressure sensitivity and low porosity sensitivity. A wide range of values for and have been observed in relation to the mechanical compaction of porous rocks, sand and fault gouge, whereas the porosity sensitivity for chemical compaction processes is often observed to be given by 3. We show that since the ratio / corresponds to the pore compressibility, the different dependences of permeability on porosity and pressure are related to the pore structure and its compressibility. Guided by the laboratory data, we conduct numerical simulations on the development of pore pressure in crustal tectonic settings according to the models ofWalder andNur (1984) andRice (1992). Laboratory data suggest that the pressure sensitivity of fault gouge is relatively low, and to maintain pore pressure at close to the lithostatic value in the Rice model, a relatively high influx of fluid from below the seismogenic layer is necessary. The fluid may be injected as vertically propagating pressure pulses into the seismogenic system, andRice's (1992) critical condition for the existence of solitary wave is shown to be equivalent to >1, which is satisfied by most geologic materials in the laboratory. Laboratory data suggest that the porosity sensitivity is relatively high when the permeability is reduced by a coupled mechanical and chemical compaction process. This implies that in a crustal layer, pore pressure may be generated more efficiently than cases studied byWalder andNur (1984) who assumed a relatively low porosity sensitivity of =2.     

Pore fluid pressure effects in anisotropic rocks: Mechanisms of induced seismicity and weak faults

Many observations and studies indicate that pore fluid pressure in the crustal rocks plays an important role in deformation, faulting, and earthquake processes. Conventional models of pore pressure effects often assume isotropic porous rocks and yield the nondeviatoric pressure effects which seem insufficient to explain diverse phenomena related to pore pressure variation, such as fluid-extraction induced seismicity and crustal weak faults. We derive the anisotropic effective stress law especially for transversely-isotropic and orthotropic rocks, and propose that the deviatoric effects of pore fluid pressure in anisotropic rocks not only affect rock effective strength but also cause variation of shear stresses. Such shear stress variations induced by either pore pressure buildup or pore pressure decline may lead to faulting instability and trigger earthquakes, and provide mechanisms for the failure of crustal weak faults with low level of shear stresses. We believe that the deviatoric effects of pore fluid pressure in anisotropic rocks are of wide application in studies of earthquake precursors and aftershocks, oil and gas reservoir characterization, enhanced oil recovery, and hydraulic fracturing.     

Mechanical compaction in heterogeneous clastic formations from plastic–poroelastic deformation principles: theory and applications

Mechanical compaction or loss of porosity due to increase in effective stress is a fundamental geological process that governs many of the rock elastic and transport parameters, all of great importance in exploring and developing subsurface reservoirs. The ability to model the compaction process enables us to improve our understanding of the seismic signature of the basin and better relate the geology of deposition to current porosity, velocity, pore pressure, and other mechanical parameters that depend on the state of compaction of the sediment. In this paper, a set of mathematical equations that can be used to model the plastic deformation associated with primary and secondary loading curves is presented. Compaction laws are posed in terms of natural strain increment formulation often used in plasticity theory to model large deformation. Laboratory and field estimates of constitutive plastic deformation relations for sand–shale mixtures are used in a numerical model that generates estimates of porosity under various pore pressures, shale content, and loading scenarios. These estimates can be used in a variety of settings to predict various basin and reservoir properties associated with different loading conditions and/or sedimentation processes.     

Mechanisms of compaction of quartz sand at diagenetic conditions

The relative contribution of cracking, grain rearrangement, and pressure solution during experimental compaction of quartz sand at diagenetic conditions was determined through electron and optical microscopy and image analysis. Aggregates of St. Peter sand (255±60 μm diameter grain size and porosity of approximately 34%) were subjected to creep compaction at effective pressures of 15, 34.5, 70, and 105 MPa, temperatures of 22 and 150°C, nominally dry or water-saturated (pore fluid pressure of 12.5 MPa) conditions, and for times up to one year. All aggregates displayed transient, decelerating creep, and volume strain rates as low as 2×10⁻¹⁰ s⁻¹ were achieved. The intensity of fracturing and degree of fragmentation increase with volume strain and have the same dependence on volume strain at all conditions tested, indicating that impingement fracturing and grain rearrangement were the main mechanisms of compaction throughout the creep phase. The increase in fracture density and decrease in acoustic emission rate at long times under wet conditions reflect an increase in the contribution of subcritical cracking. No quantitative evidence of significant pressure solution was found, even for long-term creep at 150°C and water-saturated conditions. Comparison of our findings to previous work suggests that pressure solution could become significant at temperatures or times somewhat greater than investigated here.     

孔隙压力对含随机缺陷岩石破坏过程及全部变形特征的影响

对于平面应变压缩条件下含有随机缺陷的岩样,利用FLAC研究了孔隙压力对岩样破坏过程、全部变形特征及前兆的影响。以前编写的若干FISH函数,被用于生成缺陷和计算轴向、侧向、体积应变及侧向应变与轴向应变比值的负值（计算得到的泊松比）。在峰值应力之后,密实岩石单元服从线性应变软化行为及随后的理想塑性行为,而材料缺陷呈现理想塑性行为。当孔隙压力较高时,应力-侧向应变曲线具有一平台;破坏的前兆更明显;变形后岩样的体积总是大于原始体积;在初始加栽阶段、均匀变形阶段及峰后变形阶段,由于明显的侧向膨胀,计算得到的泊松比远大于0．5。当孔隙压力较低时,在峰值应力之前,变形后岩样的体积小于原始体积,体积扩容出现于峰值应力之后,引起了负的体积应变。利用广义虎克定律,解释了平面应变弹性状态下数值结果的合理性。对岩样进行带状区域扫描后,确认随机缺陷的初始分布与岩样的最终破坏形态紧密相关。     

POROSITY AND PORE STRUCTURE FROM ACOUSTIC WELL LOGGING DATA1

Wyllie's time-average equation and subsequent refinements have been used for over 20 years to estimate the porosity of reservoir rocks from compressional (P)-wave velocity (or its reciprocal, transit time) recorded on a sonic log. This model, while simple, needs to be more convincingly explained in theory and improved in practice, particularly by making use of shear (S)-wave velocity. One of the most important, although often ignored, factors affecting elastic velocities in a rock is pore structure, which is also a controlling factor for transport properties of a rock. Now that S-wave information can be obtained from the sonic log, it may be used with P-waves to provide a better understanding of pore structure. A new acoustic velocities-to-porosity transform based on an elastic velocity model developed by Kuster and Toksöz is proposed. Employing an approximation to an equivalent pore aspect ratio spectrum, pore structure for reservoir rocks is taken into account, in addition to total pore volume. Equidimensional pores are approximated by spheres and rounded spheroids, while grain boundary pores and flat pores are approximated by low aspect ratio cracks. An equivalent pore aspect ratio spectrum is characterized by a power function which is determined by compressional-and shear-wave velocities, as well as by matrix and inclusion properties. As a result of this more sophisticated elastic model of porous rocks and a stricter theory of elastic wave propagation, the new method leads to a more satisfactory interpretation and fuller use of seismic and sonic log data. Calculations using the new transform on data for sedimentary rocks, obtained from published literature and laboratory measurements, are presented and compared at atmospheric pressure with those estimated from the time-average equation. Results demonstrate that, to compensate for additional complexity, the new method provides more detailed information on pore volume and pore structure of reservoir rocks. Examples are presented using a realistic self-consistent averaging scheme to consider interactions between pores, and the possibility of extending the method to complex lithologies and shaly rocks is discussed.     

循环加载高压流变实验中塑性应变分析

在三轴应力下的流变实验中，通过循环加载方法测出轴向应变，轴向塑性应变和弹性模量随差应力的变化，再根据多孔介质有效弹模理论，计算出岩石变形中孔隙度的变化，在假定岩石骨架不可压缩下，得到体积膨胀应变，然后通过总塑性应变与扩容机制和位错机制应变的关系，分解出位错应变，结果表明：在低围压下（＜100MPa），扩容应变占优势，并伴随着部分位错应变，在高围压下（＞100MPa），位错应变占优势，只有较小的扩容应变。