Transmission of Elastic P-waves across Single Fractures with a Nonlinear Normal Deformational Behavior

Summary This paper presents a theoretical study on normally incident elastic P-wave transmission across single dry fractures with a nonlinear normal deformational behavior. The effects of nonlinear fracture normal behavior on P-wave transmission are examined without the mixture of fracture shear behavior. The linear displacement discontinuity model for wave propagation across fractures is extended to a nonlinear model – the hyperbolic elastic model (BB model). Numeric solutions of magnitudes of transmission (|T _non|) and reflection (|R _non|) coefficients, for normally incident P-wave transmission across the nonlinear deformable fractures, are obtained and related to the closure behavior of fractures. Parametric studies are conducted to acquire an insight into the effects of the nonlinear fracture normal deformation on P-wave transmission, in terms of initial normal stiffness and the ratio of current maximum closure to maximum allowable closure of the fractures, as well as the incident wave amplitude and frequency. Comparisons between the linear and nonlinear models are presented. It is shown that, |T _lin| and |R _lin| for the linear model are special solutions of |T _non| and |R _non| for the nonlinear model, when the incident wave amplitude is so low that the current maximum closure of fracture incurred during the wave transmission is much smaller, relative to the maximum allowable closure. In addition, the nonlinear fracture behavior gives rise to a phenomenon of higher harmonics during the wave transmission across the fracture. The higher harmonics contribute to the increase of |T _non| from |T _lin|.     

A Numerical Study on Wave Transmission Across Multiple Intersecting Joint Sets in Rock Masses with UDEC

This paper presents a numerical study on wave transmission across jointed rock masses with UDEC, where multiple intersecting joint sets exist. The capability of UDEC of studying wave transmission across rock joints is validated through comparison with analytical solutions and experimental data. Through parametric studies on wave transmission across jointed rock masses, it is found that joint mechanical and spatial parameters including joint normal and shear stiffnesses, nondimensional joint spacing, joint spacing ratio, joint intersecting angle, incident angle, and number of joint sets together determine the wave transmission. And for P wave incidence, compared with other parameters, joint normal stiffness, nondimensional joint spacing, and joint intersecting angle have more significant effects on wave transmission. The physical reasons lying behind those phenomena are explained in detail. Engineering applications and indications of the modeling results are also mentioned.     

Study of wave attenuation across parallel fractures using propagator matrix method

Stress wave attenuation across fractured rock masses is a great concern of underground structure safety. This paper presents an analytical study on wave attenuation across parallel fractures at arbitrary incidence angles, where multiple reflections occurring between fractures are taken into account. Combined with displacement discontinuous model, plane wave analysis and propagator matrix method are applied to develop relations between the first layer and the nth layer with respect to potential amplitudes or displacements and stresses in matrix form. With initial and boundary conditions for different scenarios, potential amplitudes in any layer or displacements and stresses at any point can be obtained by solving corresponding matrixes. After parametric studies, it is found that parameters including incidence angle, normalized fracture stiffness, number of fractures, and fracture spacing have obvious effects on wave attenuation across parallel fractures. Copyright © 2011 John Wiley & Sons, Ltd.     

Validating the ability of the discontinuous deformation analysis method to model normal P‐wave propagation across rock fractures

The effects of fractures on wave propagation problems are increasingly abstracting the attention of scholars and engineers in rock engineering field. This study aims to fully validate the ability of discontinuous deformation analysis (DDA) to model normal P‐wave propagation across rock fractures. The effects of a single fracture and multiple parallel fractures are all tested. The results indicate that DDA can accurately reflect the fracture effects, including the fractures stiffness, the fracture spacing and the fracture number, and the effects of incident wave frequency on one‐dimensional P‐wave propagation problems. Thus, DDA is able to deal well with normal incident P‐wave propagation problems. Copyright © 2017 John Wiley & Sons, Ltd.     

基于离散颗粒模型的岩体结构面动-静态刚度系数对比的数值模拟研究

岩体工程计算分析中结构面刚度系数是至关重要的力学参数,计算分析的精度和可靠程度与这个参数密不可分,结构面刚度系数取值仍然是一个难点。岩体中应力波传播至结构面处将会发生反射和透射现象,利用应力波透射系数可反演结构面动态刚度系数。本文从细观力学角度运用颗粒离散元方法,开发分段线性接触模型及应力波吸收边界模型,开展宏观岩体中应力波传播的模拟,结合准静态压缩试验模拟,研究了较为平直的岩体结构面分别在不同正应力条件下的动、静态刚度系数的变化特征。模拟结果表明:(1)利用C++语言开发的分段线性接触模型很好地实现了结构面非线性变形特征的模拟;(2)基于颗粒离散元方法能够准确地反映岩体中应力波传播规律,应力波通过不同刚度结构面的透射系数与理论解一致;(3)在离散颗粒模型中加入黏滞吸收边界条件很好地实现了在有限尺寸模型中远场应力波传播模拟;(4)在岩体模型中结构面接触部位运用分段线性接触模型,通过模拟应力波传播与单轴压缩试验分别获得了一致性较好的结构面动、静态刚度系数,结构面动/静态刚度系数之比值约为1.0。本文对岩体中结构面刚度的测试和取值具有重要的指导意义。     

Numerical study on S-wave transmission across a rough,filled discontinuity

This paper presents a numerical simulation of S-wave propagation across a rough, filled discontinuity using the universal distinct element code (UDEC). The ability of UDEC to simulate a stress wave across a smooth and planar discontinuity filled with an elastic material is validated through comparisons with analytical solutions. Next, the effect of the plastic deformation of the fill on the wave propagation is investigated. The model is extended to further study S-wave propagation across a filled discontinuity with rough interfaces, which is described using a sawtooth. The transmission coefficient defined by the energy is used to measure the wave attenuation. Finally, a parametric study is conducted to investigate the influences of the filled thickness, asperity angle, and incident amplitude on the transmission waves and transmission coefficients. The asperity angle and filled thickness together determine the transmitted waveform and transmission coefficient. The transmitted wave may be cut off when the incident wave amplitude exceeds a threshold value. The transmission coefficient decreases with a different trend with the incident wave amplitude increasing when the asperity angle varies. Compared with planar discontinuity, a filled discontinuity with rough interfaces is more sensitive to the amplitude of the incident wave. The causes of these phenomena are analyzed in detail. In addition, the deformation of the fill material is strongly related to the wave attenuation.     

Dynamic Model of Fracture Normal Behaviour and Application to Prediction of Stress Wave Attenuation Across Fractures

Summary.  The purpose of this paper is to establish a dynamic constitutive model of fracture normal behaviour, based on laboratory tests of artificial fractures cast by cement mortar. A series of tests are systematically carried out under quasi-static (10⁻¹ MPa/s) up to highly dynamic (10³ MPa/s) monotonic loading conditions. The normal stress-fracture closure response is measured at different loading rates. Based on the measured curves, a nonlinear (hyperbolic) dynamic model of fracture normal behaviour, termed as dynamic BB model, is proposed. The dynamic model is modified from the existing BB model of static normal behaviour of fractures by taking into account the loading-rate effect. Two important dynamic parameters of fractures, FSC _d (dynamic fracture stiffness constant, which describes the incremental ratio of dynamic initial stiffness) and FCC _d (dynamic fracture closure constant, which describes the decremental ratio of dynamic maximum allowable closure), are identified. They indicate the quantitative degree of loading-rate effect on fracture normal behaviour subjected to dynamic loads. For practical application, the new model is incorporated into the Universal Distinct Element Code (UDEC) and subsequently, UDEC modelling of normally incident P-wave transmission across single fractures with the dynamic BB model is conducted. Wave transmission coefficient is obtained for various combinations of fracture dynamic parameters, as well as different wave amplitudes and frequencies. The numerical results show that wave transmission coefficient for a fracture with the dynamic BB model is greater than that for a fracture with the static BB model. In addition, a fracture with higher values of FSC _d and FCC _d leads to higher transmission (lower attenuation). Author’s address: J. Zhao, Ecole Polytechnique Federale de Lausanne (EPFL), Rock Mechanics Laboratory, 1015 Lausanne, Switzerland     

Permeability tensor and representative elementary volume of fractured rock masses

Based on a simulation of three-dimensional fracture networks and a superposition principle of liquid dissipation energy for fractured rock masses, a model of the fracture permeability tensor is proposed. An elastic constitutive model of rock fractures, considering fracture closure and dilation during shearing, is also proposed, based on the dilation angle of the fracture. Algorithms of flow-path searching and calculation of the effective flow coefficients for fracture networks are presented, together with a discussion on the influence of geometric parameters of the fractures (trace length, spacing, aperture, orientation and the number of fracture sets) on magnitude, anisotropy of hydraulic permeability and the size of a representative elementary volume (REV). The anisotropy of hydraulic permeability of fractured rock masses is mainly affected by orientation and the number of fracture sets, and the REV size is mainly influenced by trace length, spacing and the number of fracture sets. The results of studies on REV size and the influence of in-situ stress on hydraulic conductivity of the rock mass on the slope of Jinping-I hydropower station, China, are presented using the developed models and methods. The simulation results agreed well with the results obtained from field water-pressure measurements, with an error of less than 10 %.     

A Numerical Investigation of Fracture Infilling and Spacing in Layered Rocks Subjected to Hydro-Mechanical Loading

In order to better understand opening-mode fracture initiation and propagation perpendicular to the bedding plane at depth in sedimentary rocks, a series of two-dimensional (2D) numerical simulations is conducted. First, the stress states between two adjacent fractures for a typical three-layer model with pre-assigned fractures are simulated. Second, the same three-layer model without pre-assigned fractures is adopted to study the initiation and propagation of fractures in layered rocks. Numerical results show that infilling fractures grow more easily from flaws located near the interface than from those in the middle of the fractured layer. Flaws can begin to propagate to form a complete infilling fracture when the size of the flaws exceeds half of the thickness of the central layer. Under different overburden stress conditions and internal fluid pressure, the numerically obtained ratio of the critical fracture spacing to layer thickness varies between 0.465 and 0.833. This range encompasses the often-cited ratios of spacing to layer thickness in the literature for well-developed fracture sets. In addition, both the fracture pattern and the critical value of the fracture spacing to layer thickness ratio are strongly dependent on the heterogeneous characteristics of the central layer. In cases with a relatively homogeneous central layer, more interface fractures occur, and the interface delamination evidently influences the fracture saturation.     

页岩气藏缝网压裂裂缝间距优化研究

页岩气藏一般具有低孔隙、低渗透等特征,对其实施缝网压裂是高效开发页岩气的最佳途径。采用位移不连续法建立线弹性二维均质地层诱导应力场分布数学模型,通过水平应力差异系数对顺序压裂和交替压裂的裂缝间距进行优化研究。结果表明,水平应力差异系数受到裂缝净压力、裂缝缝长、原地应力场等因素的影响;裂缝净压力越大、缝长越长,水平应力差异系数越小;随着与裂缝距离的增加,水平应力差异系数呈现先减小后增加的趋势,因此,存在后续裂缝形成复杂网络的最佳裂缝间距;顺序压裂裂缝间距不宜过大,且后续压裂裂缝间距应适当减小;交替压裂裂缝间距最优时,缝间水平应力差异系数最小,对中间裂缝形成缝网最有利。     

弹性纵波垂直通过非线性结构面的傅里叶级数分析法

利用傅里叶级数计算任意函数形式的弹性纵波垂直通过非线性结构面的透射波和反射波的速度波形。采用双曲模型描述结构面的变形特性,基于位移不连续模型,结合波振面处动量守恒定律,推导了弹性纵波垂直通过非线性结构面的基本方程。假设应力波在含结构面岩体中传播时,结构面的存在不改变应力波波形函数的最小正周期,运用傅里叶级数理论和周期延拓方法,得到了任意函数形式的弹性纵波垂直入射时透射波和反射波速度波形的傅里叶级数解,并验证了傅里叶级数解是合理的。利用傅里叶级数解,分析了单一频率的正弦谐波入射至结构面时,透射波中各阶谐波的振幅和相位与谐波阶数的关系。研究结果表明,各阶谐波的振幅与阶数呈负指数关系衰减,前7阶谐波振幅的衰减指数为谐波阶数的二次函数,当谐波阶数大于7时,衰减指数为谐波阶数的一次函数;各阶谐波的相位与谐波阶数呈线性关系。     

Geologic Stochastic Modeling and Connectivity Assessment of Fracture Systems in the Boston Area

Summary The MIT stochastic model for representing fracture patterns has been applied to the Boston area. The model is a geometric-mechanical model, which means that underlying mechanisms are geometrically represented. In this paper the model is calibrated by comparing the fracture spacing and trace length simulated by the model with data collected by Billings (1976) in a number of tunnels in the Boston area. Following this, geometric fracture connectivity is investigated. This is done by recording in the model simulations, the number of interconnected fractures in so called sub-networks and by recording the geometric extents of these sub-networks. The results indicate limited horizontal but large vertical connectivity.     

含裂隙煤层的地震记录模拟

采用线性滑移(Linear Slip Deformation)裂隙介质模型、各向异性弹性波模拟方法,对含裂隙煤层进行了2D2C(二维二分量)单炮记录模拟与分析。结果表明:煤层含裂隙后与不含裂隙时相比,反射时差的变化已不足以识别,但由于波速的改变会使煤层顶底界面的波阻抗发生变化,从而引起波的动力学特征的改变,AVO现象尤其明显。当裂隙为非垂直时,炮点两侧的反射波能量和振幅出现不对称现象。x分量和z分量的特征明显不同,转换波主要记录在x分量上,横波在x分量上有不可忽视的能量。因此,应用各向异性理论、利用多分量转换波研究煤层裂隙的特征是可能的。     

Evaluation of CBM reservoir fracture characteristics by azimuthal AVA inversion

Exploitation of coalbed methane (CBM) is directly affected by the strike and density of fractures, particularly because fractures can act as conduits for water and gas (after desorption). To realize the azimuthal amplitude versus angle (AVA) inversion for a CBM reservoir, we treated reservoirs whose fractures are aligned vertically as being equivalent to a horizontal transverse isotropic medium and simplified the P-P wave reflection coefficient formula to an approximately linear equation. Then, we adopted a modified damping least squares algorithm to solve the azimuthal AVA inversion problem. With edited macro-bins, corrected residual moveout, and normalized amplitude, the azimuthal gathers exhibited unambiguous azimuthal AVA characteristics. The #3 CBM reservoir in the Qinshui Basin was selected for a case study. The density and strike of the reservoir fractures were obtained and analyzed, and the reservoir was divided into three classes according to fracture density. Finally, in comparison with the gas production data, we confirmed that the inversion results and reservoir classification were both reasonable and credible.     

龟裂系数法在安达地区火山岩储层裂缝研究中的应用

根据安达地区火山岩储层的实际情况,利用钻井、测井等资料综合分析预测火山岩储层的裂缝发育情况,运用纵波波速的比值——"龟裂系数"方法对该区裂缝发育频率进行了分析、计算,从而总结了安达地区火山岩储层裂缝的平面展布特征。结论表明,安达地区火山岩储层的裂缝较为发育,裂缝参数特征必然影响该区天然气的勘探和生产。     

裂缝参数对考虑滑脱效应页岩气水平井产能的影响

低渗透页岩气藏中,气体渗流时会受滑脱效应的影响。建立了考虑滑脱效应的气、水两相页岩气藏渗流数学模型,并建立了理想地质模型,采用数值模拟方法,研究了水力压裂的不同裂缝参数对水平井产能的影响。模拟结果表明:裂缝条数、长度和间距是影响页岩气井产能的重要参数,而裂缝宽度和渗透率对产能的影响相对较弱;页岩气井的产能随着裂缝条数和裂缝长度的增加而增大;水平井的水平段长度及裂缝条数一定时,可通过增大裂缝间距来减少裂缝间的相互干扰。      

基于qp 波及qs 波衰减系数检测裂缝的方法

裂缝储层预测是当前国内外油气勘探中的重点和难点,而裂缝储层预测最关键的问题是裂缝发育带的检测。基于qp 波、qsh 波、qsv 波的衰减系数对裂缝发育带的检测,利用Hudson 理论模型经数值计算,分别研究了裂缝密度、入射角及频率对qp 波、qsh 波、qsv 波衰减系数的影响。理论模型试算表明,频率和裂缝密度是影响衰减系数的主要参数,而裂缝中充填物不同时,衰减系数也表现出很大的差异,这为利用qp 波、qsh 波、qsv 波衰减属性检测裂缝发育带提供了理论依据。     

利用测井资料建立地层剪破裂模型

利用测井资料建立地层剪破裂模型是对现有张破裂的有效补充,有利于完善钻井、压裂等石油工程中的破裂机理。运用FMI（地层微电阻率扫描成像）测井资料能识别钻井诱导缝和剪破裂的方位、数量和密度。本文通过对地层剪破裂的理论研究和影响因素分析,利用内摩擦系数与矿物含量之间的非线性假设,建立了地层剪破裂的临界剪应力模型。研究发现,地层中的黏土矿物含量、内摩擦系数、内聚强度和应力大小是诱发剪破裂的重要因素。利用内摩擦系数（或内摩擦角）与泥质体积分数之间的统计回归模型简化了地层剪破裂模型,并增强了模型的适用性。实际资料表明,地层发生剪破裂时的临界剪应力与钻井诱导缝数量之间呈反比关系,理论研究与实际资料具有良好的一致性。     

浅埋煤层开采地面塌陷裂缝规律及防治方法

陕北浅埋煤层大规模、高强度开采诱发了严重的地面塌陷,造成大面积土地损毁、水土流失和植被死亡,导致表生环境出现退化。为掌握浅埋煤层开采地面塌陷裂缝发育规律,明晰其机理,提出适宜的治理恢复措施,实现“煤?水?生态”的协调发展,以陕北张家峁井田和柠条塔井田为研究区,采用实地调查、模拟实验和理论分析相结合的方法开展了浅埋煤层开采地面塌陷规律及防治方法研究。结果表明:浅埋煤层开采地表裂缝呈“O”型展布,静态发育特征与采高和地形地貌呈正相关关系、与采深呈负相关关系,且同一工作面切眼附近地表裂缝发育程度最高、巷道次之、面内最低;地表裂缝具有“先开后(半)合”和“只开不合”2种活动特征,整体活动时间为4~9 d,活动期间裂缝初始开裂宽度与最大发育宽度呈线性正相关关系,与稳定宽度呈线性和指数2种正相关关系;黄土沟壑区下坡段开采地表裂缝活动与表土块体的稳定性系数有关,而稳定系数与坡角呈负相关的一次幂函数,与主裂缝间距呈正相关的一次幂函数。上坡段开采坡体裂缝“先开后(半)合”活动受岩块倒转和坡体滑移双重控制,面内沟底裂缝“先开后合”的活动特征受关键岩层运移控制。研究提出了黄土沟壑区沟底贯通型裂缝“裂缝填充+沟道恢复”、坡体裂缝 “裂缝充填+微地形改造”的治理方法和风沙滩地塌陷区的“三圈”修复模式。研究成果在陕北安山煤矿和柠条塔煤矿进行了应用,效果良好。      

平面交叉裂隙非线性渗流模型参数人工神经网络预测

岩体裂隙网络渗流广泛存在于地下工程中,对地下工程的建设和运行安全具有重要的影响。因此,研究裂隙网络渗流有着重要的理论和实际意义。本文根据立方定律和Forchheimer方程推导所得的交叉裂隙渗流模型,运用数值模拟和人工神经网络方法,对平面交叉裂隙渗流模型非线性参数与模型几何条件的关系进行探究。通过数值模拟,获得了平面交叉裂隙非线性渗流模型的参数;运用人工神经网络遗传算法,探究了交叉裂隙几何条件与交叉裂隙渗流模型中非线性系数之间的关系,证明了平面交叉裂隙非线性渗流模型适用于描述交叉裂隙渗流规律,验证了神经网络方法预测非线性系数的可行性和准确性。同时,还对比分析了运用拟合数值表达式和人工神经网络两种方法的特点。