Grain-scale analysis of proppant crushing and embedment using calibrated discrete element models

Acta Geotechnica - Proppant crushing and embedment in hydraulically-induced fractures is a major drawback to the recovery of unconventional oil/gas and geothermal energy production. This study...     

Coupled flow network and discrete element modeling of injection-induced crack propagation and coalescence in brittle rock

We present a numerical analysis on injection-induced crack propagation and coalescence in brittle rock. The DEM network coupling model in PFC is modified to capture the evolution of fracture geometry. An improved fluid flow model for fractured porous media is proposed and coupled with a bond-based DEM model to simulate the interactions among cracks induced by injecting fluid in two nearby flaws at identical injection rates. The material parameters are calibrated based on the macro-properties of Lac du Bonnet granite and KGD solution. A grain-based model, which generates larger grains from assembles of particles bonded together, is calibrated to identify the microscopic mechanical and hydraulic parameters of Lac du Bonnet granite such that the DEM model yields a ratio between the compressive and tensile strength consistent with experiments. The simulations of fluid injection reveal that the initial flaw direction plays a crucial role in crack interaction and coalescence pattern. When two initial flaws are aligned, cracks generally propagate faster. Some geometrical measures from graph theory are used to analyze the geometry and connectivity of the crack network. The results reveal that initial flaws in the same direction may lead to a well-connected crack network with higher global efficiency.

     

岩石强度和应变速率对水平挤压变形影响的离散元模拟

离散元模拟是一种新的动力学模拟方法,它可以提供模型内部肉眼不能辨别的微小形变信息,生动的给出构造形变的几何特征与动力学过程,是我们深入了解地质构造形成演化历史、理解岩石变形机制、准确解译地震数据的良好方法.本文基于离散元理论构建模型,用C 和Java语言编写程序加以实现,模拟挤压条件下盐上褶皱和推覆带构造的形成、演化过程,通过四组对比实验的比较分析,详细研究了岩性强弱和推覆速度对盐相关挤压构造演化的控制作用.研究结果表明1)沿积压方向,断层的年龄由老到新,约靠近推覆前端的断层越新.在挤压形变过程中,一旦有新的主要断层形成,之前的老断层即停止活动;2)脆性地层发育较多的断层,主断层贯穿整个地层,韧性地层中断层数量较少,断层位移小,只切断部分地层;3)脆性地层形变过程中地层厚度基本不变,显示地层内部应变较小,韧性地层的厚度有明显变化,显示地层内部应变较强;4)在岩性相似的务件下,挤压速率幔的系统断层发育较多,并且形变传播较远.     

A novel random discrete element analysis of rock fragmentation

This paper presents a novel probabilistic approach of random discrete element analysis (RDEA) to investigate the mechanism of rock fragmentation under uniaxial compression. This model combines the advantages of both random field theory and discrete element method in characterizing the spatial variation and uncertainty of microscopic material properties. The numerical results reveal that the stress-strain curves of a group of tests can match well the general trend of the experimental data, with the mean uniaxial compressive strength (UCS) of 10.18 MPa and the mean Young modulus of 1.73 GPa. The coefficient of variation (COV) for the rock samples is much lower than that of the initial random fields of particles because of the averaging effect of microscopic material property in obtaining the bulk values. The rock fragmentation is initiated by the breakage of weak particles within the rock mass, and it develops rapidly as the vertical loading stress approaches the UCS. The final damage zone resides dominantly in the weak region of the rock sample, and the distribution of material property coefficients follows a similar beta distribution as the corresponding initial random field. Rock samples with persistent “pillar-like” structures of strong particles can effectively resist the normal compression, resulting in high rock strengths. The traditional DEM simulation with a set of constant material properties can only represent one extreme realization of random field, which could significantly overestimate the rock strength. The proposed RDEA approach can effectively capture the uncertainty and complex interactions of rock fragmentation in a more realistic and reliable way.     

Development of a GPGPU-parallelized hybrid finite-discrete element method for modeling rock fracture

The hybrid finite-discrete element method (FDEM) is widely used for engineering applications, which, however, is computationally expensive and needs further development, especially when rock fracture process is modeled. This study aims to further develop a sequential hybrid FDEM code formerly proposed by the authors and parallelize it using compute unified device architecture (CUDA) C/C++ on the basis of a general-purpose graphics processing unit (GPGPU) for rock engineering applications. Because the contact detection algorithm in the sequential code is not suitable for GPGPU parallelization, a different contact detection algorithm is implemented in the GPGPU-parallelized hybrid FDEM. Moreover, a number of new features are implemented in the hybrid FDEM code, including the local damping technique for efficient geostatic stress analysis, contact damping, contact friction, and the absorbing boundary. Then, a number of simulations with both quasi-static and dynamic loading conditions are conducted using the GPGPU-parallelized hybrid FDEM, and the obtained results are compared both quantitatively and qualitatively with those from either theoretical analysis or the literature to calibrate the implementations. Finally, the speed-up performance of the hybrid FDEM is discussed in terms of its performance on various GPGPU accelerators and a comparison with the sequential code, which reveals that the GPGPU-parallelized hybrid FDEM can run more than 128 times faster than the sequential code if it is run on appropriate GPGPU accelerators, such as the Quadro GP100. It is concluded that the GPGPU-parallelized hybrid FDEM developed in this study is a valuable and powerful numerical tool for rock engineering applications.     

Discrete element modeling of tool‐rock interaction I: rock cutting

Tool‐rock interaction processes can be classified as indentation or cutting depending on the direction of motion of the tool with respect to the rock surface. The modes of failure induced in the rock by an indenting or a cutting tool can be ductile and/or brittle. The ductile mode is associated with the development of a damage zone, whereas the brittle mode involves the growth of macrocracks. This is the first part of a series of two papers concerned with an analysis of the cutting and the indentation processes based on using the discrete element method. In this paper, numerical simulations of the cutting process are conducted to reproduce the transition from a ductile to a brittle failure mode with increasing depth of cut, which is observed in experiments. The numerical results provide evidence that the critical depth of cut d _* controlling the failure mode transition is related to the characteristic length ? = (K_Ic ∕ σ_c)² with K_Ic denoting the material toughness and σ_c its unconfined compressive strength. The nature of frictional contact between the cutter face and the rock in the ductile failure mode is also examined. It is shown that the inclination of the total cutting force is controlled by a multi‐directional flow mechanism ahead of the cutter that is related to the formation of a wedge of failed material, intermittently adhering to the cutter. As a result, the inclination of the total cutting force varies with the rake angle of the cutter and cannot be considered an intrinsic measure of the interfacial friction between the cutter and the rock. Copyright © 2012 John Wiley & Sons, Ltd.     

基于离散元法的节理岩体边坡稳定性分析

节理岩体边坡的稳定性在很大程度上取决于节理的强度及其分布形式。由于节理岩体边坡的失稳破坏具有大变形和非连续的特点,因此,离散单元法成为研究节理岩体边坡破坏机制的最有效方法之一。通过采用离散元软件PFC2D进行数值模拟,对完整岩石及节理的力学性能进行研究,并建立含密集节理的岩体边坡模型,讨论了节理连通率对边坡破坏形式的影响。结果表明,节理岩体边坡的失稳破坏是一个渐进的过程;在多组节理密集分布的岩体边坡中,连通率越大,其稳定性越差;随着连通率的减小,边坡的破坏形式由大范围的滑坡转变为局部崩塌的形式     

危岩水力劈裂分析的扩展有限元法

危岩是三峡库区典型的地质灾害类型之一,而主控结构面受荷断裂扩展是危岩发育成灾的关键核心。将危岩主控结构面类比为宏观裂纹,利用扩展有限元法在模拟裂纹扩展方面的优势,基于考虑裂纹面水压力作用的虚功原理推导出了采用扩展有限元法分析水力劈裂问题的控制方程,给出了危岩主控结构面水力劈裂问题的扩展有限元实现方法,对重庆万州太白岩危岩主控结构面的水力劈裂进行了数值模拟分析。计算结果表明：暴雨是威胁危岩稳定性的最敏感因素,随着裂隙水压力上升,裂端拉应力会急剧升高,危岩的稳定性降低;I型裂纹扩展是危岩主要的结构面扩展形式,结构面一旦发生开裂,将处于非稳定扩展状态。     

Intersecting dilated convex polyhedra method for modeling complex particles in discrete element method

This paper describes a new method for representing concave polyhedral particles in a discrete element method as unions of convex dilated polyhedra. This method offers an efficient way to simulate systems with a large number of (generally concave) polyhedral particles. The method also allows spheres, capsules, and dilated triangles to be combined with polyhedra using the same approach. The computational efficiency of the method is tested in two different simulation setups using different efficiency metrics for seven particle types: spheres, clusters of three spheres, clusters of four spheres, tetrahedra, cubes, unions of two octahedra (concave), and a model of a computer tomography scan of a lunar simulant GRC‐3 particle. It is shown that the computational efficiency of the simulations degrades much slower than the increase in complexity of the particles in the system. The efficiency of the method is based on the time coherence of the system, and an efficient and robust distance computation method between polyhedra as particles never intersect for dilated particles. © 2014 The Authors. International Journal for Numerical and Analytical Methods in Geomechanics published by John Wiley & Sons Ltd.     

西藏樟木后山危岩崩塌颗粒离散元数值分析

崩塌危岩是高陡岩质边坡浅表层破坏常见的动力地质灾害方式之一,它突发性强,随机性大,速度快,发生猛烈,一直是边坡工程勘察中的重点。对西藏樟木后山高陡边坡危岩进行了现场勘察,分析研究了危岩崩塌体的分布范围。利用PFC3D模拟了在降雨件下危岩崩塌危岩的运动轨迹、速度及不同摩擦系数崩塌体的堆积特征。计算分析结果表明,崩塌危岩运动路径受地形地貌影响较大,与地面发生碰撞,多次改变行进方向,简化二维计算可能得出错误结论;坡面摩擦系数影响崩塌危岩的堆积形态,摩擦系数越小,堆积体位置越远,运动状态易表现流态性状。坡面摩擦系数取1.2时,模拟结果与实际较为吻合;崩塌危岩主要危害的范围为陡崖下部武警二营至樟木沟口一带。利用PFC3D模拟危岩崩塌运动可以初步确定崩塌体三维堆积形态及影响范围,为防灾减灾工程设计提供有益参考。     

Three-dimensional discrete element simulation of indirect tensile behaviour of a transversely isotropic rock

This paper presents the development of a three-dimensional discrete element model using flat-joint and smooth-joint contact models to investigate the effect of anisotropy on the tensile behaviour of slate, a transversely isotropic rock, under Brazilian testing from both macro and microscales. The effect of anisotropy is further realised by exploring the influence of foliation orientations (β and ψ) on the tensile strength, fracture pattern, microcracking and stress distribution of the transversely isotropic rock. The variation of tensile strength with foliation orientation is presented. The cross-weak-plane fracture growth observed in laboratory is reproduced, and the criterion for which to form is also given from the aspect of foliation orientation. Furthermore, the proportional variations of microcracks well account for the effects of foliation orientation on the tensile strength and failure pattern. Finally, it is found that the existence of weak planes increases both the heterogeneity and the anisotropy of stress distributions within the transversely isotropic rock, with the degree of influence varying with the foliation orientation.     

Polygonal grain‐based distinct element modeling for mechanical behavior of brittle rock

The microstructure of rock was numerically reproduced by a polygonal grain‐based model, and its mechanical behavior was examined by performing the uniaxial compression test and Brazilian tests via the Universal Distinct Element Code. The numerical results of the model demonstrated good agreement with the experimental results obtained with rock specimens in terms of the stress–strain behavior, strength characteristics, and brittle fracture phenomenon. An encouraging result is that the grain‐based model‐Universal Distinct Element Code model can reproduce a low ratio of tensile to compressive strength of 1/20 to 1/10 without the need for an additional process. This finding is ascribed to the fact that the geometrical features of polygons can effectively capture the effects of angularity, finite rotation, and interlocking of grains that exist in reality. A numerical methodology to monitor the evolution of micro‐cracks was developed, which enabled us to examine the progressive process of the failure and distinguish the contribution of tensile cracking to the process from that of shear cracking. From the observations of the micro‐cracking process in reference to the stress–strain relation, crack initiation stress, and crack damage stress, it can be concluded that the failure process of the model closely resembles the microscopic observations of rock. We also carried out a parametric study to examine the relationships between the microscopic properties and the macroscopic behavior of the model. Depending on the micro‐properties, the model exhibited a variety of responses to the external load in terms of the strength and deformation characteristics, the evolution of micro‐cracks, and the post‐peak behavior. Copyright © 2016 John Wiley & Sons, Ltd.     

基于细观试验和离散元法的盐岩力学特性

当前盐岩的宏观力学模型通常是唯象模型,不能很好地解释盐岩受力变形破坏的真正物理基础。盐岩是由于化学沉积而形成的矿物集合体,是一种主要由NaCl和少量杂质组成的多晶体,其变形机制主要由晶粒与晶界的力学特性控制。通过扫描电镜（SEM）,获得盐岩晶粒的微细观结构特征,采用分子动力学方法和纳米压痕技术,确定盐岩晶粒和晶界的微细观力学参数;将盐岩晶粒作为块体,基于Voronoi多边形技术,建立盐岩的微细观数值模型;利用离散元方法,对盐岩试件在单轴压缩和直剪条件下的宏观力学行为进行了数值模拟。数值模拟结果与宏观力学试验结果吻合度高,表明基于盐岩微细观晶粒结构特征并结合离散元数值模拟的方法能够较好地研究盐岩的宏观力学性能及其材料物理基础。     

Numerical simulation of drained triaxial test using 3D discrete element modeling

A discrete element modeling of granular material was carried out using a 3D spherical discrete model with a rolling resistance, in order to take into account the roughness of grains. The numerical model of Labenne sand was generated, and the desired porosity was obtained by a radius expansion method. Using numerical triaxial tests the micro-mechanical properties of the numerical material were calibrated in order to match the macroscopic response of the real material. Numerical simulations were carried out under the same conditions as the physical experiments (porosity, boundary conditions and loading). The pre-peak, peak and post-peak behavior of the numerical material was studied. The calibration procedure revealed that the peak stress of the sand sample does not only depend on local friction parameters but also on the rolling resistance. The larger the value of the applied rolling resistance, the higher the resulting stress peak. Furthermore, the deformational response depends strongly on local friction. The numerical results are quantitatively in agreement with the laboratory test results.     

Dimensionless input parameters in discrete element modeling and assessment of scaling techniques

A set of dimensionless input parameters were defined for DEM using a characteristic time which is a function of density and elastic modulus of particles and an arbitrary characteristic length. Dimensionless strain rate and mass damping ratio are inversely proportional to the characteristic time, and stress is normalized by elastic modulus to give dimensionless stress. It was demonstrated that the response of a model in the dimensionless scale is invariant with the choice of density, elastic modulus and the characteristic length if dimensionless strain rate and mass damping ratio are kept constant. Small time step is a prohibitive aspect of DEM. Scaling techniques are widely employed to enlarge the time step. Using the dimensionless scheme, it was learned that density scaling is equivalent to the use of a higher strain rate, and stiffness scaling results in a higher strain rate and an elevated stress state in the dimensionless scale.     

Relating discrete element method parameters to rock properties using classical and micropolar elasticity theories

Micro–macro relations for discrete element method (DEM) media are derived using both classical and micropolar elasticity theories. The DEM media are classified into two main categories: dense packing, and loose packing. For both categories, relations for Young modulus (E), Poisson's ratio (ν) to represent static behaviors, and wave velocities (P‐wave and S‐wave) to represent dynamic behaviors are derived using the internal DEM parameters (k_n, k_s) and compared with values obtained from static and dynamic numerical tests. Whereas the dynamic behaviors for the two categories and the static behaviors for the dense packing match the analytical relations, the static behavior for the loose packing does not. Micropolar elasticity theory is also used to study the behaviors of the DEM media, where it is shown that if element rotation is included, DEM media behave according to linear elasticity theory. However, if element rotation is constrained, asymmetrical stresses arise in the DEM media, and a new expression is derived for the S‐wave, which allows it, under certain conditions, to travel faster than the P‐wave. Copyright © 2011 John Wiley & Sons, Ltd.     

颗粒离散元岩石模型的颗粒尺寸效应研究

颗粒尺寸是影响颗粒离散元模型宏观力学性质与计算效率的一个重要因素。为充分考虑由模型随机性导致的模拟结果的不确定性,利用统计学方法对模型的颗粒尺寸效应进行研究。整体检验结果表明：特征长度比L/R的改变对模型力学参数（峰值强度、弹性模量、泊松比及峰值应变）与破坏特征参数（黏结破裂率）的总体分布位置均有显著性影响,且各参数的变异系数会随着L/R的减小而增大。进一步的多重比较结果表明：当L/R≥125时,L/R对峰值强度、弹性模量、泊松比及峰值应变的总体分布位置均无显著影响;当L/R≥79时,相邻3个粒径水平的黏结破坏率总体分布位置无显著性差异;随着L/R的减小,模型损伤程度增加,破坏模式由整体剪切破坏转向局部损伤引起的失稳破坏,最终失去模拟岩石材料的效力。最后,综合各项力学参数的统计学检验结果、模型破坏模式及计算效率,岩石模型颗粒的特征长度比取L/R=200较为合适。     

Kinematics and internal deformation of granular slopes: insights from discrete element modeling

The kinematics and internal deformation of a failure mass during the flow-like moving off a slope were monitored and quantified with the particle flow method in this study. Two kinds of cases were investigated, noncohesive and cohesive granular slopes. Three different internal friction angles and cohesive strengths were considered to systematically investigate their effect on the kinematics and internal deformation of the failure mass. We analyzed the movement within the failure mass and concluded that the mass moves downwards in an undulating pattern. The slope surface topography changes from a straight line to curved lines with slope breaks in a convex geometry. In addition, dilatation within the failure mass, which deforms internally and heterogeneously, is strongly dependent on its mechanical properties. A larger mass moves downslope, and the mass moves faster and further in the model with lower internal friction and cohesion. The internal friction and cohesion have a positive impact on porosity and two-dimensional (or volumetric in 3D) strain within the failure mass.