Elastic properties of saturated porous rocks with aligned fractures

Elastic properties of fluid saturated porous media with aligned fractures can be studied using the model of fractures as linear-slip interfaces in an isotropic porous background. Such a medium represents a particular case of a transversely isotropic (TI) porous medium, and as such can be analyzed with equations of anisotropic poroelasticity. This analysis allows the derivation of explicit analytical expressions for the low-frequency elastic constants and anisotropy parameters of the fractured porous medium saturated with a given fluid. The five elastic constants of the resultant TI medium are derived as a function of the properties of the dry (isotropic) background porous matrix, fracture properties (normal and shear excess compliances), and fluid bulk modulus. For the particular case of penny-shaped cracks, the expression for anisotropy parameter ε has the form similar to that of Thomsen [Geophys. Prospect. 43 (1995) 805]. However, contrary to the existing view, the compliance matrix of a fluid-saturated porous-fractured medium is not equivalent to the compliance matrix of any equivalent solid medium with a single set of parallel fractures. This unexpected result is caused by the wave-induced flow of fluids between pores and fractures.     

Frequency‐dependent anisotropy of porous rocks with aligned fractures

Naturally fractured reservoirs are becoming increasingly important for oil and gas exploration in many areas of the world. Because fractures may control the permeability of a reservoir, it is important to be able to find and characterize fractured zones. In fractured reservoirs, the wave‐induced fluid flow between pores and fractures can cause significant dispersion and attenuation of seismic waves. For waves propagating normal to the fractures, this effect has been quantified in earlier studies. Here we extend normal incidence results to oblique incidence using known expressions for the stiffness tensors in the low‐ and high‐frequency limits. This allows us to quantify frequency‐dependent anisotropy due to the wave‐induced flow between pores and fractures and gives a simple recipe for computing phase velocities and attenuation factors of quasi‐P and SV waves as functions of frequency and angle. These frequency and angle dependencies are concisely expressed through dimensionless velocity anisotropy and attenuation anisotropy parameters. It is found that, although at low frequencies, the medium is close to elliptical (which is to be expected as a dry medium containing a distribution of penny‐shaped cracks is known to be close to elliptical); at high frequencies, the coupling between P‐wave and SV‐wave results in anisotropy due to the non‐vanishing excess tangential compliance.     

流体饱和度对裂缝-孔隙岩石频变纵波速度及其各向异性的影响

流体饱和度会改变裂缝性储层的纵波速度,从而影响地层速度的频散特性及各向异性程度,导致储层地震响应特征复杂,储层预测多解性强,流体识别难度大.本文根据多相流体饱和裂缝性储层的特点,借助于Norris和KG模型,建立部分饱和裂缝-孔隙等效介质模型,给出频变地震波速度随流体饱和度变化的精确关系式.数值模拟结果表明,当气、水两相共存时,随着含水饱和度的增加,高频段纵波相速度逐渐增大,各向异性程度逐渐减小;低频段纵波相速度逐渐减小,各向异性程度不变;相速度频散及其各向异性程度逐渐增强.组合已有的孔隙弹性理论模型,对实验室人工裂缝-孔隙砂岩岩样的纵波速度进行拟合,计算得到的曲线与实验室测量散点值吻合度较高,表明组合模型在给定参数下的有效性.该研究能够为多相流体饱和裂缝性储层的地震响应特征分析奠定扎实的理论基础,为提高储层预测的确定性和流体识别的准确性提供可靠的理论依据.

     

Modelling frequency-dependent seismic anisotropy in fluid-saturated rock with aligned fractures: implication of fracture size estimation from anisotropic measurements

Measurements of seismic anisotropy in fractured rock are used at present to deduce information about the fracture orientation and the spatial distribution of fracture intensity. Analysis of the data is based upon equivalent-medium theories that describe the elastic response of a rock containing cracks or fractures in the long-wavelength limit. Conventional models assume frequency independence and cannot distinguish between microcracks and macrofractures. The latter, however, control the fluid flow in many subsurface reservoirs. Therefore, the fracture size is essential information for reservoir engineers. In this study we apply a new equivalent-medium theory that models frequency-dependent anisotropy and is sensitive to the length scale of fractures. The model considers velocity dispersion and attenuation due to a squirt-flow mechanism at two different scales: the grain scale (microcracks and equant matrix porosity) and formation-scale fractures. The theory is first tested and calibrated against published laboratory data. Then we present the analysis and modelling of frequency-dependent shear-wave splitting in multicomponent VSP data from a tight gas reservoir. We invert for fracture density and fracture size from the frequency dependence of the time delay between split shear waves. The derived fracture length matches independent observations from borehole data.     

Formulation of hydrodynamic pressures in cracks due to earthquakes in concrete dams

A new analytical development of the seismic hydrodynamic pressure inside pre-existing cracks on the upstream face of concrete dams is presented. The finite control volume approach is utilized to derive an expression for the seismic hydrodynamic pressure using the continuity principle and the linear momentum theorem for the fluid inside the crack. The derived pressure expression is a function of the relative crack-opening acceleration and velocity. The acceleration and velocity terms are then recast in the form of added mass and damping matrices which can then be included at the nodes inside the discrete crack of a finite element model. This procedure linearizes the solution of the problem. A dam, 55 m high and having an initial crack opening of 2 mm at the base or near the crest and subjected to two different accelerograms, is analysed. For high-frequency ground motion, the seismic hydrodynamic pressure inside the crack, at the base of the dam, appears to be 50 per cent higher than the corresponding hydrostatic pressure.     

Frequency-dependent anisotropy due to meso-scale fractures in the presence of equant porosity

Fractured rock is often modelled under the assumption of perfect fluid pressure equalization between the fractures and equant porosity. This is consistent with laboratory estimates of the characteristic squirt-flow frequency. However, these laboratory measurements are carried out on rock samples which do not contain large fractures. We consider coupled fluid motion on two scales: the grain scale which controls behaviour in laboratory experiments and the fracture scale. Our approach reproduces generally accepted results in the low- and high-frequency limits. Even under the assumption of a high squirt-flow frequency, we find that frequency-dependent anisotropy can occur in the seismic frequency band when larger fractures are present. Shear-wave splitting becomes dependent on frequency, with the size of the fractures playing a controlling role in the relationship. Strong anisotropic attenuation can occur in the seismic frequency band. The magnitude of the frequency dependence is influenced strongly by the extent of equant porosity. With these results, it becomes possible in principle to distinguish between fracture- and microcrack-induced anisotropy, or more ambitiously to measure a characteristic fracture length from seismic data.     

Formation of cracks due to moving source over a thick crust

Summary In this paper the problem of a point source of stress moving over the surface of a thick aelotropic plate resting of a rigid foundation has been considered. Following the method ofAleksandrov & Vorovich (1960) the stress componentsZ _x andZ _z have been expanded in series of ascending powers of 1/h when the source velocity is less than (c ₄₄/)^1/2. When the velocity exceeds (c ₄₄/)^1/2 it has been shown that two cracks are produced in different directions and their successive reflections at the upper and lower surface are also obtained.     

Elastic wave propagation and scattering in prestressed porous rocks

Poro-acoustoelastic theory has made a great progress in both theoretical and experimental aspects, but with no publications on the joint research from theoretical analyses, experimental measurements, and numerical validations. Several key issues challenge the joint research with comparisons of experimental and numerical results, such as digital imaging of heterogeneous poroelastic properties, estimation of acoustoelastic constants, numerical dispersion at high frequencies and strong heterogeneities, elastic nonlinearity due to compliant pores, and contamination by boundary reflections. Conventional poroacoustoelastic theory, valid for the linear elastic deformation of rock grains and stiff pores, is modified by incorporating a dualporosity model to account for elastic nonlinearity due to compliant pores subject to high-magnitude loading stresses. A modified finite-element method is employed to simulate the subtle effect of microstructures on wave propagation in prestressed digital cores. We measure the heterogeneity of samples by extracting the autocorrelation length of digital cores for a rough estimation of scattering intensity. We conductexperimental measurements with a fluid-saturated sandstone sample under a constant confining pressure of 65 MPa and increasing pore pressures from 5 to 60 MPa. Numerical simulations for ultrasound propagation in the prestressed fluid-saturated digital core of the sample are followed based on the proposed poro-acoustoelastic model with compliant pores. The results demonstrate a general agreement between experimental and numerical waveforms for different stresses, validating the performance of the presented modeling scheme. The excellent agreement between experimental and numerical coda quality factors demonstrates the applicability for the numerical investigation of the stress-associated scattering attenuation in prestressed porous rocks.     

Dispersion and attenuation of elastic waves due to multiple scattering from cracks

Multiple scattering from cracks is considered in the two-dimensional plane-strain condition. It is assumed that identical cracks are distributed uniformly in space and that the effective waves propagate normal to the crack surfaces. Then, the apparent dispersion and attenuation are calculated as functions of frequency for three independent modes of wave propagation: SV, P and SH.The calculated results show that, in each case, the attenuation coefficient Q^?1 takes a peak value when the wavelength is nearly twice the crack width, while phase velocity has a maximum deviation from the intrinsic value at a frequency lower than the peak frequency for Q^?1.     

Attenuation and dispersion ofSH waves due to scattering by randomly distributed cracks

The effect of randomly distributed cracks on the attenuation and dispersion ofSH waves is theoretically studied. If earthquake ruptures are caused by sudden coalescence of preexisting cracks, it will be crucial for earthquake prediction to monitor the temporal variation of the crack distribution. Our aim is to investigate how the property of crack distribution is reflected in the attenuation and dispersion of elastic waves.We introduce the stochastic property, in the mathematical analysis, for the distributions of crack location, crack size and crack orientation. The crack size distribution is assumed to be described by a power law probability density (p(a) a ^– fora _minaa _max according to recent seismological and experimental knowledge, wherea is a half crack length and the range 13 is assumed. The distribution of crack location is assumed to be homogeneous for the sake of mathematical simplicity, and a low crack density is assumed. The stochastic property of each crack is assumed to be independent of that of the other cracks. We assume two models, that is, the aligned crack model and the randomly oriented crack model, for the distribution of crack orientation. All cracks are assumed to be aligned in the former model. The orientation of each crack is assumed to be random in the latter model, and the homogeneous distribution is assumed for the crack orientation. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.We observe the following features common to both the aligned crack model and the randomly oriented crack model. The attenuation coefficientQ ^–1 decays in proportion tok ^–1 in the high frequency range and its growth is proportional tok ² in the low frequency range, wherek is the intrinsic wave number. This asymptotic behavior is parameter-independent, too. The attenuation coefficientQ ^–1 has a broader peak as increases and/ora _min/a _max decreases. The nondimensional peak wave numberk _p a _max at whichQ ^–1 takes the peak value is almost independent ofa _min/a _max for =1 and 2 while it considerably depends ona _min/a _max for =3. The phase velocity is almost independent ofk in the rangeka _max<1 and increases monotonically ask increases in the rangeka _max>1. While the magnitude ofQ ^–1 and the phase velocity considerably depend on the orientation of the crack in the aligned crack model, the above feature does not depend on the crack orientation.The accumulation of seismological measurements suggests thatQ ^–1 ofS waves has a peak at around 0.5 Hz. If this observation is combined with our theoretical results onk _p a _max, the probable range ofa _max of the crack distribution in the earth can be estimated for =1 or 2. If we assume 4 km/sec as theS wave velocity of the matrix medium,a _max is estimated to range from 2 to 5 km. We cannot estimatea _max in a narrow range for =3.     

含混合裂隙、孔隙介质的纵波衰减规律研究

地下多孔介质中的孔隙类型复杂多样,既有硬孔又有扁平的软孔.针对复杂孔隙介质,假设多孔介质中同时含有球型硬孔和两种不同产状的裂隙(硬币型、尖灭型裂隙),当孔隙介质承载载荷时,考虑两种不同类型的裂隙对于孔隙流体压力的影响,建立起Biot理论框架下饱和流体情况含混合裂隙、孔隙介质的弹性波动方程,并进一步求取了饱和流体情况下仅由裂隙引起流体流动时的含混合裂隙、孔隙介质的体积模量和剪切模量,随后,在此基础上讨论了含混合裂隙、孔隙介质在封闭条件下地震波衰减和频散的高低频极限表达式.最后计算了给定模型的地震波衰减和频散,发现地震波衰减曲线呈现"多峰"现象,速度曲线为"多频段"频散.针对该模型分析讨论了渗透率参数、裂隙纵横比参数以及流体黏滞性参数对于地震波衰减和频散的影响,表明三个参数均为频率控制参数.     

Elastic anisotropy and its influencing factors in organic-rich marine shale of southern China

Shale is observed to have strong anisotropy due to its unique mineralogy and microstructure, and this anisotropy property has significant impact on seismic and well-log data. The organic-rich marine shale in the southern and eastern Sichuan Basin is one of the most important shale-gas reservoir formations in China. To investigate the elastic anisotropy of this shale and its influencing factors, we performed ultrasonic velocity measurements, X-ray diffraction analysis, rock-eval pyrolysis and vitrinite reflectance measurement on the samples from the Upper Ordovician Wufeng Formation and the Lower Silurian Longmaxi Formation. The experimental results show the that:(1) the velocity anisotropy of the Wufeng-Longmaxi(WL) shale varies from 10% to 50%, and most samples have strong anisotropy;(2) the P-and S-wave anisotropy parameters(Thomsen's εand γ) increase with clay contents, but this relationship can be greatly affected by the clay orientation index;(3) organic matter content(OMC) is found to have little influence in seismic anisotropy for the over mature WL shale, whereas the OMC determines the magnitude of anisotropy of immature/mature shales(e.g. the Bakken shale or the Bazhenov shale) according to the published literatures, because organic matters in shales of different maturity have different morphologies and distributions;(4) the OMC of WL shale has positive correlation with quartz content, and this weakens the correlation between OMC and the magnitude of anisotropy to a certain extent. The results of this study provide an important rock-physics basis and data support for seismic anisotropy exploration, quantitative interpretation and resource evaluation of the organic-rich marine shales in southern China.     

The attenuation of stress waves in fluid saturated porous rock

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周期性层状含孔隙、裂隙介质模型纵波衰减特征

地震波在含孔隙、裂隙斑块饱和介质传播过程中会诱发多个尺度孔隙流体流动而产生衰减和速度频散.在含有宏观尺度“Biot流”和介观尺度“局域流”衰减诱导机制的周期性层状孔隙介质模型基础上,引入了微观尺度硬币型和尖灭型裂隙“喷射流”的影响,构建了周期性层状含孔隙、裂隙介质模型.利用双解耦弹性波动方程的方法数值计算了该模型地震频带的纵波衰减和速度频散并与周期性层状孔隙介质模型做了对比研究.分析了该模型在不同裂隙参数（裂隙密度、裂隙纵横比）及裂隙体积含量下的纵波衰减和频散特征,裂隙密度越高对于纵波衰减和频散的影响越大,裂隙纵横比越小,由裂隙引起的纵波衰减部分向高频段移动,裂隙体积含量越少,纵波衰减先降低后小幅增加再降低,频散速度增加,并逐渐接近于周期性层状孔隙介质模型的纵波衰减和频散速度曲线.最后研究了周期性层状含孔隙、裂隙介质模型有效平面波模量的高低频极限以及流固相对位移在该模型中的分布特征.     

A laboratory simulation of rock breakdown due to freeze-thaw in a maritime Antarctic environment

Results of freeze-thaw simulations on three large blocks of quartz-micaschist are presented. Three types of water to ice phase change were identified from temperature and ultrasonic measurements. It is suggested that the type of phase change results from a particular combination of rock moisture content, solute concentration, freeze amplitude, and rate of fall of temperature. The temperature at which ice thawed inside the rock (?0.7 to ?1.9°C) was also found, and this indicates the possibility of freeze-thaw effects without positive temperatures. Approximately 80 per cent of the water that will freeze under natural conditions, in the Maritime Antarctic environment under study, appears to have done so by ?6°.     

Research on dependence of resistivity changing anisotropy on microcracks extending in rock with experiment

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The development of magnetic susceptibility anisotropy through crystallographic preferred orientation in a calcite rock

High-field torque-meter measurements of diamagnetic susceptibility anisotropy of a suite of samples of Carrara marble, axially shortened by amounts up to 50% at (1.5–3.0) · 10⁸ Pa confining pressure and at 20–500°C (mainly 400°C), have been compared with optical measurements of preferred crystallographic orientation. A revised value for the susceptibility anisotropy of calcite has been obtained from studies of single crystals, and it has been shown to be almost independent of the state of intracrystalline plastic strain. From the measured anisotropy of calcite, quantitative comparison of optical and magnetic fabric measurements is possible. It is found that these measurements agree and the implications of the observed progressive development of fabric intensity with strain are discussed.     

P‐ and S‐wave anisotropy to characterise and quantify damage in media: laboratory experiment using synthetic sample with aligned microcracks

Damage characterisation in solid media is studied in this work through ultrasonic measurements. A synthetic three‐dimensional printed sample including a system of horizontally aligned microcracks is used. In contrast to other manual fabrication methods presented in the literature, the construction process considered here ensures a better control and accuracy of size, shape, and spatial distribution of the microcrack network in the synthetic sample. The acoustic measurements were conducted through a specific device using triple acoustic sensors, which allows capturing at each incident direction three wave modes. The evolution of the ultrasonic velocities with respect to incident angle accounted for the damage‐induced anisotropy. The experimental results are then compared with some well‐known effective media theories in order to discuss their potential use for the following studies. Finally, we highlighted and compared the accuracy of these theories used for inversion procedure to quantify damage in the medium.     

Comment on the effect of anisotropy on the onset of convection in a porous medium

The effect of anisotropy on the onset of convection in a saturated porous medium is discussed. In particular, the case of time-dependent density-driven convection is examined. The applicability of the value of an equivalent Rayleigh number as the criterion for the onset of convection is discussed.