Velocity variations in carbonate rocks due to dual porosity and wave-induced fluid flow

Stiffness variations in carbonates may be described as resulting from different concentrations of flat compliant pores or cracks, which can have a significant effect on the effective stiffness and acoustic properties (e.g., velocities and attenuations) of dry as well as saturated carbonates, although they carry extremely little porosity. As shown in this paper, the effects of dual porosity and wave-induced fluid flow or pore pressure communication may also play a significant role. On the basis of a previously published T-matrix approach to model the effective viscoelastic properties of cracked porous media, we illustrate the (frequency-dependent) effects of wave-induced fluid flow (mainly squirt flow) or pore pressure communication for a model structure consisting of a mixture of fluid-saturated porous grains and fluid-saturated cavities (vugs, etc.) that are embedded in a solid matrix associated with carbonates. We assume that the pores within the porous grains are decoupled from the pores in the solid matrix (and possibly saturated with different fluids) but that each pore system at the micro and/or mesoscale may or may not be connected. For each of four different connectivity models, we present numerical results for four different cases of microstructure (that emphasize the importance of cracks and flat compliant pores). Our numerical results indicate that the velocity and attenuation spectra of carbonates vary significantly, even when the crack density and all other volume concentrations are constant.     

Effective elastic properties of randomly fractured soils: 3D numerical experiments

This paper is concerned with numerical tests of several rock physical relationships. The focus is on effective velocities and scattering attenuation in 3D fractured media. We apply the so‐called rotated staggered finite‐difference grid (RSG) technique for numerical experiments. Using this modified grid, it is possible to simulate the propagation of elastic waves in a 3D medium containing cracks, pores or free surfaces without applying explicit boundary conditions and without averaging the elastic moduli. We simulate the propagation of plane waves through a set of randomly cracked 3D media. In these numerical experiments we vary the number and the distribution of cracks. The synthetic results are compared with several (most popular) theories predicting the effective elastic properties of fractured materials. We find that, for randomly distributed and randomly orientated non‐intersecting thin penny‐shaped dry cracks, the numerical simulations of P‐ and S‐wave velocities are in good agreement with the predictions of the self‐consistent approximation. We observe similar results for fluid‐filled cracks. The standard Gassmann equation cannot be applied to our 3D fractured media, although we have very low porosity in our models. This is explained by the absence of a connected porosity. There is only a slight difference in effective velocities between the cases of intersecting and non‐intersecting cracks. This can be clearly demonstrated up to a crack density that is close to the connectivity percolation threshold. For crack densities beyond this threshold, we observe that the differential effective‐medium (DEM) theory gives the best fit with numerical results for intersecting cracks. Additionally, it is shown that the scattering attenuation coefficient (of the mean field) predicted by the classical Hudson approach is in excellent agreement with our numerical results.     

龙马溪组页岩数字岩心动态法弹性等效数值建模

数字岩心是计算岩石弹性性质的一类常用方法.龙马溪组页岩具有多矿物构成、复杂微结构和强非均质等特征,常规岩石物理的弹性等效解析建模局限性较大,目前流行的静态数值等效建模方法的精度有限.本文基于高分辨率的页岩数字岩心数据,采用多阈值分割方法将数字岩心分解为黏土、石英、孔隙、TOC、长石类、黄铁矿类等六种矿物类型;利用矿物组分等效模量法计算各类矿物的弹性模量;采用二元函数分水岭方法表征不同压力下的岩石孔隙变形和颗粒接触关系变化;通过取向分布函数(ODF)定量分析矿物颗粒展布造成的各向异性特征.最后基于Biot孔弹方程,采用不分裂卷积完全匹配层(CPML)旋转交错网格有限差分法模拟不同压力下弹性波在数字岩心中的传播.以未加压的数字岩心为参考模型,计算不同压力下弹性波走时的平均时间差,进而估算各压力点的数字岩心等效速度.与该岩心样品的超声实验测量速度比较,动态法数值计算结果略偏高,据此校正数值计算过程中表征岩石微结构及颗粒接触关系随压力变化的二元函数,有效改善动态法弹性等效数值建模精度.     

Attenuation mechanisms in fractured fluid-saturated porous rocks: a numerical modelling study

Seismic attenuation mechanisms receive increasing attention for the characterization of fractured formations because of their inherent sensitivity to the hydraulic and elastic properties of the probed media. Attenuation has been successfully inferred from seismic data in the past, but linking these estimates to intrinsic rock physical properties remains challenging. A reason for these difficulties in fluid-saturated fractured porous media is that several mechanisms can cause attenuation and may interfere with each other. These mechanisms notably comprise pressure diffusion phenomena and dynamic effects, such as scattering, as well as Biot's so-called intrinsic attenuation mechanism. Understanding the interplay between these mechanisms is therefore an essential step for estimating fracture properties from seismic measurements. In order to do this, we perform a comparative study involving wave propagation modelling in a transmission set-up based on Biot's low-frequency dynamic equations and numerical upscaling based on Biot's consolidation equations. The former captures all aforementioned attenuation mechanisms and their interference, whereas the latter only accounts for pressure diffusion phenomena. A comparison of the results from both methods therefore allows to distinguish between dynamic and pressure diffusion phenomena and to shed light on their interference. To this end, we consider a range of canonical models with randomly distributed vertical and/or horizontal fractures. We observe that scattering attenuation strongly interferes with pressure diffusion phenomena, since the latter affect the elastic contrasts between fractures and their embedding background. Our results also demonstrate that it is essential to account for amplitude reductions due to transmission losses to allow for an adequate estimation of the intrinsic attenuation of fractured media. The effects of Biot's intrinsic mechanism are rather small for the models considered in this study.     

Harmonic wave diffraction by two circular cavities in a poroelastic formation

This study investigates the dynamic interaction of time harmonic plane waves with a pair of parallel circular cylindrical cavities of infinite length buried in a boundless porous elastic fluid-saturated medium. The novel features of Biot dynamic theory of poroelasticity along with the appropriate wave field expansions, the pertinent boundary conditions, and the translational addition theorems for cylindrical wave functions are employed to develop a closed-form solution in the form of infinite series. The analytical results are illustrated with numerical examples in which two empty cavities are insonified by a fast compressional or a shear wave at end-on incidence. The basic dynamic field quantities such as the hoop stress amplitude and the radial displacement of the elastic frame are evaluated and discussed for representative values of the parameters characterizing the system. The effects of the proximity of the two cavities, the incident wave frequency and type are examined. Particular attention has been focused on multiple scattering interactions in addition to the slow wave coupling effects which is known to be the primary distinction of the scattering phenomenon in poroelasticity from the classical elastic case. Limiting case involving two empty cylindrical cavities in an elastic solid is considered and excellent agreement with a well-known solution is established.     

Analytical exploration of γ-function explicit method for pseudodynamic testing of nonlinear systems

It has been well studied that the γ-function explicit method can be effective in providing favorable numerical dissipation for linear elastic systems. However, its performance for nonlinear systems is unclear due to a lack of analytical evaluation techniques. Thus, a novel technique is proposed herein to evaluate its efficiency for application to nonlinear systems by introducing two parameters to describe the stiffness change. As a result, the numerical properties and error propagation characteristics of the γ-function explicit method for the pseudodynamic testing of a nonlinear system are analytically assessed. It is found that the upper stability limit decreases as the step degree of nonlinearity increases; and it increases as the current degree of nonlinearity increases. It is also shown that this integration method provides favorable numerical dissipation not only for linear elastic systems but also for nonlinear systems. Furthermore, error propagation analysis reveals that the numerical dissipation can effectively suppress the severe error propagation of high frequency modes while the low frequency responses are almost unaffected for both linear elastic and nonlinear systems.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

A new energy density calculation method for reservoir delineation using seismic low-frequency signal

In comparison to high-frequency signals, low-frequency seismic signals suffer less from scattering and intrinsic attenuation during wave propagation, penetrate deeper strata and thus can provide more energy information related to the hydrocarbon reservoirs. Based on the asymptotic representation for the frequency-dependent reflections in the fluid-saturated pore-elastic media, we first derive a novel equation of the reservoir energy density and present an efficient workflow to calculate the reservoir energy density using low-frequency seismic data. Then, within a low-frequency range (from 1 to 30 Hz), we construct an objective function to determine the optimal frequency, using the energy densities calculated from the post-stack seismic traces close to the wells. Next, we can calculate the reservoir energy density using the instantaneous spectra of optimal frequency at the low-frequency end of the seismic spectrum. Tests on examples for synthetic and field data demonstrate that the proposed reservoir energy density can produce high-quality images for the fluid-saturated reservoirs, and it produces less background artefacts caused by elastic layers. This method provides a new way to detect the location of hydrocarbon reservoirs and characterize their spatial distribution.     

Effect of Corrugation and Reinforcement on the Dispersion of SH-wave Propagation in Corrugated Poroelastic Layer Lying over a Fibre-reinforced Half-space

The presence of porosity and reinforcement in a medium is an important factor affecting seismic wave propagation and plays vital role in many geophysical prospects. Also, the presence of salt and ore deposits, mountains, basins, mountain roots, etc. is responsible for the existence of corrugated boundary surfaces of constituent layers. Such facts brought motivation for the present paper which deals with the propagation of SH-wave in a heterogeneous fluid-saturated poroelastic layer with corrugated boundaries lying over an initially stressed fibre-reinforced elastic halfspace. Closed form of dispersion relation has been obtained and is found in well agreement to classical Love wave equation for isotropic case. The effect of corrugation, wave number, undulation, position parameter, horizontal compressive/tensile initial stress and heterogeneity on phase velocity has been analysed through numerical computation and graphical illustration. Moreover, comparative study exploring the effect of presence and absence of reinforcement in half-space on dispersion curve is the major highlight of the current study.     

A triple porosity scheme for fluid/solid substitution: theory and experiment

Quantifying the effects of pore-filling materials on elastic properties of porous rocks is of considerable interest in geophysical practice. For rocks saturated with fluids, the Gassmann equation is proved effective in estimating the exact change in seismic velocity or rock moduli upon the changes in properties of pore infill. For solid substance or viscoelastic materials, however, the Gassmann theory is not applicable as the rigidity of the pore fill (either elastic or viscoelastic) prevents pressure communication in the pore space, which is a key assumption of the Gassmann equation. In this paper, we explored the elastic properties of a sandstone sample saturated with fluid and solid substance under different confining pressures. This sandstone sample is saturated with octadecane, which is a hydrocarbon with a melting point of 28°C, making it convenient to use in the lab in both solid and fluid forms. Ultrasonically measured velocities of the dry rock exhibit strong pressure dependency, which is largely reduced for the filling of solid octadecane. Predictions by the Gassmann theory for the elastic moduli of the sandstone saturated with liquid octadecane are consistent with ultrasonic measurements, but underestimate the elastic moduli of the sandstone saturated with solid octadecane. Our analysis shows that the difference between the elastic moduli of the dry and solid-octadecane-saturated sandstone is controlled by the squirt flow between stiff, compliant, and the so-called intermediate pores (with an aspect ratio larger than that of compliant pore but much less than that of stiff pores). Therefore, we developed a triple porosity model to quantify the combined squirt flow effects of compliant and intermediate pores saturated with solid or viscoelastic infill. Full saturation of remaining stiff pores with solid or viscoelastic materials is then considered by the lower embedded bound theory. The proposed model gave a reasonable fit to the ultrasonic measurements of the elastic moduli of the sandstone saturated with liquid or solid octadecane. Comparison of the predictions by the new model to other solid substitution schemes implied that accounting for the combined effects of compliant and intermediate pores is necessary to explain the solid squirt effects.     

Scattering and refracting of plane strain wave by a cylindrical inclusion in fluid-saturated soils

The problems involving scattering and refracting of plane strain wave by a cylinder in fluid-saturated soils are studied in this paper. The theoretical solutions for the amplitudes of scattering and refracting are derived on the basis of modified Biot’s theory for two-phase medium in engineering, in which the compressibility of soil grains and the viscous coupling effect between the skeletal frame and the pore fluid are taken into account. Moreover, the amplitude equations of potential function are presented when the fluid-saturated soil cylindrical inclusion is degenerated into cylindrical cavity, rigid cylinder, single phase elastic cylinder and single phase fluid cylinder.     

流体饱和多孔介质波动方程的有限元解法

讨论了流体饱和多孔介质中波传播问题的有限元解法,首先在Ｂｉｏｔ理论的基础 上,概述了数学问题的提法,然后提出了一种新型简便的人工边界上的无反射边界条件,同时 给出了有人工边界时流体饱和多孔介质波动方程的有限元计算公式．数值试验的结果表明, 本文提出的无反射边界条件和数值方法均很有效．     

Computation of elastic properties of 3D digital cores from the Longmaxi shale

The dependence of elastic moduli of shales on the mineralogy and microstructure of shales is important for the prediction of sweet spots and shale gas production. Based on 3D digital images of the microstructure of Longmaxi black shale samples using X-ray CT, we built detailed 3D digital images of cores with porosity properties and mineral contents. Next, we used finite-element (FE) methods to derive the elastic properties of the samples. The FE method can accurately model the shale mineralogy. Particular attention is paid to the derived elastic properties and their dependence on porosity and kerogen. The elastic moduli generally decrease with increasing porosity and kerogen, and there is a critical porosity (0.75) and kerogen content (ca. ≤3%) over which the elastic moduli decrease rapidly and slowly, respectively. The derived elastic moduli of gas- and oil-saturated digital cores differ little probably because of the low porosity (4.5%) of the Longmaxi black shale. Clearly, the numerical experiments demonstrated the feasibility of combining microstructure images of shale samples with elastic moduli calculations to predict shale properties.     

Surface waves in a micropolar elastic solid containing voids

The present paper investigates the effect of voids on the propagation of surface waves in a homogeneous micropolar elastic solid medium which contains a distribution of vacuous pores (voids). The general theory for surface wave propagation in micropolar elastic media containing voids has been presented. Particular cases of surface waves (Rayleigh’s, Love’s and Stoneley’s) in micropolar media which contain vacuous pores have been deduced from the above general theory. Discussions have been made in each case to highlight the effect of voids and micropolar character of the material medium separately. Their joint effect has also been studied in details. Modulation of Rayleigh wave velocity has been studied numerically. It is observed that Love waves are not affected by the presence of voids.     

数字岩心逆建模理论下的储层参数定量预测方法

数字岩心微观孔隙结构十分复杂,有限元模拟物性参数与弹性参数之间关系是非线性的,直接反演其物性参数准确度低、稳定性差.本文发展了一种数字岩石物理逆建模方法,实现了基于数字岩心的储层参数有效预测.从数字岩心基函数的构建出发,基于有限元方法,计算了一系列具有等间距物性参数值(孔隙度、泥质含量和含水饱和度)的数字岩心弹性参数(体积模量、剪切模量和密度),通过插值算法建立了数字岩心弹性参数三维数据集,从而实现了弹性模量的有限元数值解的快速构建;然后搜索弹性参数的单值等值面,通过等值面的空间交会得到交点,完成储层参数预测.测试结果表明:基于数字岩心逆建模理论的储层参数预测结果与实际模型一致,具有可行性,并且可以通过增加插值点数目提高预测的准确性;孔隙度和泥质含量预测结果稳定性很好,而含水饱和度对噪声的加入较为敏感.     

Poroelastic finite-difference modeling for ultrasonic waves in digital porous cores

Scattering attenuation in short wavelengths has long been interesting to geophysicists. Ultrasonic coda waves, observed as the tail portion of ultrasonic wavetrains in laboratory ultrasonic measurements, are important for such studies where ultrasonic waves interact with small-scale random heterogeneities on a scale of micrometers, but often ignored as noises because of the contamination of boundary reflections from the side ends of a sample core. Numerical simulations with accurate absorbing boundary can provide insight into the effect of boundary reflections on coda waves in laboratory experiments. The simulation of wave propagation in digital and heterogeneous porous cores really challenges numerical techniques by digital image of poroelastic properties, numerical dispersion at high frequency and strong heterogeneity, and accurate absorbing boundary schemes at grazing incidence. To overcome these difficulties, we present a staggered-grid high-order finite-difference (FD) method of Biot’s poroelastic equations, with an arbitrary even-order (2L) accuracy to simulate ultrasonic wave propagation in digital porous cores with strong heterogeneity. An unsplit convolutional perfectly matched layer (CPML) absorbing boundary, which improves conventional PML methods at grazing incidence with less memory and better computational efficiency, is employed in the simulation to investigate the influence of boundary reflections on ultrasonic coda waves. Numerical experiments with saturated poroelastic media demonstrate that the 2L FD scheme with the CPML for ultrasonic wave propagation significantly improves stability conditions at strong heterogeneity and absorbing performance at grazing incidence. The boundary reflections from the artificial boundary surrounding the digital core decay fast with the increase of CPML thicknesses, almost disappearing at the CPML thickness of 15 grids. Comparisons of the resulting ultrasonic coda Q _sc values between the numerical and experimental ultrasonic S waveforms for a cylindrical rock sample demonstrate that the boundary reflection may contribute around one-third of the ultrasonic coda attenuation observed in laboratory experiments.     

随机弹性介质中地震波散射衰减分析(英文)

地震波衰减一直是许多学科研究的热点,因为可以反映介质的特性。导致地震波衰减的因素很多,如:传播过程中由于能量扩散导致的几何衰减,固体岩石内部晶粒间相对滑移导致的摩擦衰减,岩石结构不均匀引起的地震波散射衰减。本文主要从统计的观点出发,通过多次数值模拟的方法研究纵波散射在随机弹性介质中所引发的衰减。首先用随机理论建立了二维空间随机弹性介质模型,然后用错格伪谱法的数值方法模拟了波在随机介质中的传播,再通过波场中虚拟检波器的记录,用谱比法估计了弹性波在随机介质中的散射衰减。不同非均匀程度随机弹性介质中的数值结果表明:介质不均匀程度越高,散射衰减越大;在散射体尺寸小于波长的前提下,不同散射体尺寸的计算结果说明:散射体尺寸越大,弹性波衰减越明显。最后提出了一种不均匀孔隙介质中流体流动衰减的方法。通过对随机孔隙介质中地震波的总衰减和散射衰减分别进行了计算,并定量得出了随机孔隙介质中流体流动衰减,结果表明:在实际地震频段下,当介质不均匀尺度101米量级时,散射衰减比流体流动衰减要大,散射衰减是地震波在实际不均匀岩石孔隙介质中衰减的主要原因。     

Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid

The modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.