Analysis of refracturing in horizontal wells: Insights from the poroelastic displacement discontinuity method

In this paper, a fully coupled 2‐dimensional poroelastic displacement discontinuity method is used to investigate the refracturing process in horizontal wells. One of the objectives of refracturing is to access new reserves by adding new hydraulic fractures in zones that were bypassed in the initial fracturing attempt. Pore pressure depletion in the vicinity of old fractures directly affects the state of stress and eventually the propagation of newly created hydraulic fractures. Thus, a poroelastic analysis is required to identify guidelines for the refracturing process, in particular to understand the extension of the pore pressure depletion, and eventually, the orientation of new as well as old fractures. We propose a fully coupled approach to model the whole process of child fracture propagation in a depleted area between 2 parent fractures in the same wellbore. This approach omits the need of using multistep workflow that is regularly used to model the process. The maximum tensile stress criterion (σ criterion) is used for hydraulic fracture propagation. The proposed method is verified using available analytical solutions for total stress and pore pressure loading modes on a line fracture in drained and undrained conditions. Then, test cases of multifractured horizontal wells are studied to calculate the time evolution of the stress and pore pressure fields around old fractures and to understand the effect of these fields on the propagation path of newly created fractures. Finally, the effect of the pore pressure depletion on the propagation path of the newly created fractures in the bypassed area of the same wellbore is studied. The results show that the depleted areas around old fractures are highly affected by the extent and severity of the stress redistribution and pore pressure depletion. It is observed that a successful creation of new fractures may only happen in certain time frames. The results of this study provide new insights on the behavior of newly created fractures in depleted zones. They also clarify the relationship between stress change and pore pressure depletion in horizontal wells.     

Large‐scale poroelastic fractured reservoirs modeling using the fast multipole displacement discontinuity method

An effective approach to modeling the geomechanical behavior of the network and its permeability variation is to use a poroelastic displacement discontinuity method (DDM). However, the approach becomes rather computationally intensive for an extensive system of cracks, particularly when considering coupled diffusion/deformation processes. This is because of additional unknowns and the need for time‐marching schemes for the numerical integration. The Fast Multipole Method (FMM) is a technique that can accelerate the solution of large fracture problems with linear complexity with the number of unknowns both in memory and CPU time. Previous works combining DDM and FMM for large‐scale problems have accounted only for elastic rocks, neglecting the fluid leak‐off from the fractures into the matrix and its influence on pore pressure and stress field. In this work we develop an efficient geomechanical model for large‐scale natural fracture networks in poroelastic reservoirs with fracture flow in response to injection and production operations. Accuracy and computational performance of the proposed method with those of conventional poroelastic DDM are compared through several case studies involving up to several tens of thousands of boundary elements. The results show the effectiveness of the FMM approach to successfully evaluate field‐scale problems for the design of exploitation strategies in unconventional geothermal and petroleum reservoirs. An example considering faults reveals the impact of reservoir compartmentalization because of sealing faults for both geomechanical and flow variables under elastic and poroelastic rocks. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical procedure for simulation of hydraulically-driven fracture propagation in poroelastic media

A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described. The method uses a partitioned solution procedurè to solve a finite element approximation to problems described by the theory of poroelasticity, in conjunction with a finite difference approximation for modelling fluid flow along the fracture. An equilibrium fracture model based on a generalized, Dugdale–Barenblatt concept is used to determine the fracture dimensions. An important feature is that the fracture length is a natural product of the solution algorithm. Two example problems verify the accuracy of the numerical procedure and a third example illustrates a fully-coupled simulation of fracture propagation. Photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena.     

A stress and displacement discontinuity element method for elastic transversely anisotropic rock

The paper presents closed‐form solutions for stress and displacement influence functions for stress discontinuity (SD) and displacement discontinuity (DD) elements, for a two‐dimensional plane‐strain elastic, transversely anisotropic medium. The solutions for SD elements are based on Kelvin's problem and for DD elements on the concept of dipoles. Stress and displacement influence functions are derived for the following elements: constant SD, linear SD, constant DD, linear DD, square root DD, parabolic DD, constant DD surface, and linear DD surface elements. The formulations are incorporated into FROCK, a hybridized boundary element method code, and are validated by providing comparisons between the results from FROCK and the finite element code ABAQUS. A limited parametric analysis shows the effects of slight anisotropy on the stress field around the tip of a crack and of the orientation of the crack with respect to the axes of elastic symmetry. Copyright © 2014 John Wiley & Sons, Ltd.     

模拟水压致裂的另一种二维FDEM-flow方法

可考虑岩石本身渗透性的FDEM-flow方法充分利用原FEMDEM中节理单元和三角形单元独特的拓扑连接关系,将节理单元作为流体流动的天然通道,基于立方定律来表征流体在岩石本身及裂隙中的流动,而岩石本身渗透性则通过标定合适节理单元张开度来表征。用一个含解析解自由面稳态渗流算例初步验证了本文渗流算法及标定岩石本身渗透率方法的正确性。最后计算了一个水力压裂算例,验证了该FDEM-flow方法可同时考虑岩石本身的渗透性和裂隙的渗透性,用简单的纯裂隙渗流完成了对复杂问题的处理。研究结果表明:修改后的FDEM-flow方法能够很好地再现压裂过程中流体压力的分布及裂隙的扩展,可简洁地求解水力压裂这一复杂力学的问题,适用范围更广。为模拟页岩气开采、干热岩地热获取中的水力压裂问题提供新的求解工具。     

水力裂缝穿越非连续面扩展时的断裂过程研究

非连续面发育是非常规油气储层的显著地质特征之一,水力裂缝能否穿越非连续面扩展会关系到压裂的改造效果。为研究水力裂缝穿越非连续面扩展时断裂过程区（fracture process zone,简称FPZ）发育特征,采用自主设计的可视化压裂试验装置对含预制摩擦界面的砂岩平板试件开展水力压裂试验。基于数字图像相关法实时监测了水力裂缝正交穿越界面扩展过程中的位移及应变场特征。试验结果表明,水力裂缝穿越界面扩展之前,断裂过程区已经开始跨越界面发育;裂缝能否穿越界面扩展在FPZ的初始发育阶段已经注定,不受FPZ内应力软化过程影响。基于Renshaw-Pollard准则建立了考虑FPZ边界范围的裂缝穿越非连续面扩展准则,并通过前人及文中试验数据进行了可靠性验证。相比而言,改进准则更准确地考虑了裂缝前端线弹性断裂力学的适用范围。研究发现FPZ长宽比对裂缝穿越界面扩展准则有显著影响,相同条件下,FPZ长宽比越大,裂缝正交穿越界面扩展所需要的摩擦系数下限值越小。     

黏弹性节理岩体的位移不连续模型研究

选用三参数标准线性固体作为岩石本构,提出了一种考虑岩体黏弹性的位移不连续模型;根据一维黏弹性波的特征线法,推导了节理处质点速度、应力和应变递推公式。首先,基于分离式霍普金森压杆（SHPB）对砂层进行试验,得到其应力-应变关系,并换算出砂层节理的法向刚度;接着,通过一维强间断黏弹性波的波速公式、高频波衰减系数以及任一频率下的衰减系数,确定出数值算法中的三参数。最后,基于自制的摆锤装置,探讨了一维应力波在节理岩体中的传播规律,试验中以两根长1 000 mm、直径为68.50 m的岩杆作为入射和透射杆,以3 mm砂层模拟节理。试验和数值结果吻合度良好,进一步验证了该方法的可靠性。     

Fully coupled hydraulic fracture simulation using the improved element partition method

The improved element partition method (IEPM) is a newly developed fracture simulation approach. IEPM allows a fracture to run across an element without introducing extra degrees of freedom. It can also simulate any number of fractures in a prescribed mesh without remeshing. In this study, the IEPM is extended to hydraulic fracture simulation. First, the seepage and volumetric storage matrix of a cracked element are derived using virtual nodes (the intersection points of a crack with element edges). Subsequently, the fully coupled hydromechanical equation is derived for this cracked element. To eliminate the extra degrees of freedom (virtual nodal quantities), the water pressure and displacement of the virtual nodes are associated with their adjacent nodes through least squares interpolation. Finally, the fully coupled equation in terms of nodal quantities is obtained. The verification cases validate the method. By using this method, the field-scale hydraulic fracturing process is well simulated. The proposed approach is simple and efficient for field-scale hydraulic fracture simulation.     

Phase field modeling of hydraulic fracture propagation in transversely isotropic poroelastic media

Acta Geotechnica - This paper proposes a phase field model (PFM) for describing hydraulic fracture propagation in transversely isotopic media. The coupling between the fluid flow and displacement...     

Numerical simulation of particle plugging in hydraulic fracture by element partition method

Hydraulic fracturing (HF) treatment often involves particle migration and is applied for propping or plugging fractures. Particle migration behaviors, e.g., bridging, packing, and plugging, significantly affect the HF process. Hence, it is crucial to effectively simulate particle migration. In this study, a new numerical approach is developed based on a coupled element partition method (EPM). The EPM is used to model natural and hydraulic fractures, in which a fracture is allowed to propagate across an element, thereby avoiding remeshing in fracture simulations. To characterize the water flow process in a fracture, a fully hydromechanical coupled equation is adopted in the EPM. To model particle transportation in fractures with water flow, each particle is treated as a discrete element. The particles move in the fracture as a result of being dragged by fluid. Their movement, contact, and packing behaviors are simulated using the discrete element method. To reflect the plugging effect, an equivalent aperture approach is proposed. Using this method, the particle migration and its effect on water flow are well simulated. The simulation results show that this method can effectively reproduce particle bridging, plugging, and unblocking in a hydraulic fracture. Furthermore, it is demonstrated that particle plugging significantly affects water flow in a fracture and hence the propagation of hydraulic fracture. This method provides a simple and feasible approach for the simulation of particle migration in a hydraulic fracture.     

Modeling hydraulic fracture propagation in permeable media with an embedded strong discontinuity approach

The paper presents an embedded strong discontinuity approach to simulate single hydraulic fracture propagation in the poroelastic medium under plane-strain conditions. The method enriches the strain field with the discontinuous deformation mode and allows the fracture to be modeled inside elements. The Mode-I fracture initiation and propagation are described by the trilinear cohesive law, which is implemented by the penalty method. The enhanced permeability inside the fractured elements is dependent on the fracture aperture. Hydraulic fracture propagation is driven by the high pressure gradient near the fracture. Fluid transfer between the fracture and bulk rock is automatically captured within the poroelastic framework. The numerical framework is verified by the comparisons with the asymptotic analytical solutions for single hydraulic fracture propagation.     

Fast multipole displacement discontinuity method (FM‐DDM) for geomechanics reservoir simulations

The displacement discontinuity method (DDM) is frequently used in geothermal and petroleum applications for modeling the behavior of fractures in linear‐elastic rocks. The DDM requires O(N²) memory and O(N³) floating point operations (where N is the number of unknowns) to construct the coefficient matrix and solve the linear system of equations by direct methods. Therefore, the conventional implementation of the DDM is not computationally efficient for very large systems of cracks, often limiting its application to small‐scale problems. This work presents an approach for solving large‐scale fracture problems using the fast multipole method (FMM). The approach uses both the DDM and a kernel‐independent version of the FMM along with a preconditioned generalized minimal residual algorithm to accelerate the solution of linear systems of equations using desktop computers. Using the fundamental solutions for constant displacement discontinuity in a two‐dimensional elastic medium, several numerical examples involving fracture networks representing fractured reservoirs are treated. Numerical results show good agreement with analytical solutions and demonstrate the efficiency of the FMM implementation of the DDM for large‐scale simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Three-dimensional elastic analysis of cracks with the g2 constant displacement discontinuity method

A new triangular element was created that could be used for the improvement of the accuracy of the constant displacement discontinuity method (CDDM). This element is characterized by three degrees of freedom in the three-dimensional space as in the classical CDDM approach. The element is based on strain gradient elasticity theory that accounts for the difference of the average value of stress with the local stress at surfaces with large curvature (eg, crack borders, corners, and notches) in elastic bodies. The new element is characterized by a strain gradient term in addition to the two Lamè constants that gives a more representative value of the stresses at the centroid of crack edge elements compared with the classical elasticity solution and thus an accurate stress intensity factor. In this approach, special crack border elements with square-root radius dependent displacements and numerical integrations are avoided. The extra strain gradient term is calibrated once only on the analytical solution for the penny-shaped crack. In a verification stage, the accuracy of the computational algorithm for the elliptic and rectangular crack problems is demonstrated. Then, the algorithm, which also accounts for crack closure in compression, is applied for the modeling of crack propagation and crack interaction in uniaxial tension and compression loading. It is illustrated that the numerical predictions are in accordance with experimental evidence pertaining to uniaxial compression of transparent precracked specimens in the lab.     

A finite difference method for earthquake sequences in poroelastic solids

Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth’s subsurface) is a major, new hazard in the USA, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here, we present a finite difference method for 2D linear poroelasticity with rate-and-state friction faults, accounting for spatially variable properties. Semi-discrete stability and accuracy are proven using the summation-by-parts, simultaneous-approximation-term (SBP-SAT) framework for discretization and boundary condition enforcement. Convergence rates are verified using the method of manufactured solutions and comparison to the analytical solution to Mandel’s problem. The method is then applied to study fault slip triggered by fluid injection and diffusion through high-permeability fault damage zones. We demonstrate that in response to the same, gradual forcing, fault slip can occur in either an unstable manner, as short-duration earthquakes that radiate seismic waves, or as stable, aseismic, slow slip that accumulates over much longer time scales. Finally, we use these simulation results to discuss the role of frictional and elastic properties in determining the stability and nature of slip.     

Analytical influence functions for displacement discontinuity and point force elements

The analytical influence functions for the elements of a hybridized displacement discontinuity and indirect boundary element method to model fracture propagation are presented. The influence functions are exact and in closed form and, thus, have significant advantages over their numerical counterparts.     

Explicit versus implicit front advancing schemes for the simulation of hydraulic fracture growth

The proper computation of the time evolution of the fracture front is the main challenge of three-dimensional (3D) hydraulic fracture growth simulation. We discuss explicit and implicit variants of a hydraulic fracture propagation scheme based on a level set representation of the fracture. Such a scheme couples a finite discretization of the governing equations and the near-tip hydraulic fracture asymptotes. We benchmark the accuracy, robustness, and stability of these different front advancing schemes on a number of test cases. Our results indicate a large computational gain of the explicit scheme at the expense of a slightly less accurate solution (few percent less accuracy over few time steps) when crossing heterogeneities. The predictor corrector scheme combines at least an approximately 25% computational gain while retaining the stability and accuracy of the fully implicit version of the scheme in all cases.     

An embedded fracture modeling framework for simulation of hydraulic fracturing and shear stimulation

A numerical modeling framework is described that is able to calculate the coupled processes of fluid flow, geomechanics, and rock failure for application to general engineering problems related to reservoir stimulation, including hydraulic fracturing and shear stimulation. The numerical formulation employs the use of an embedded fracture modeling approach, which provides several advantages over more traditional methods in terms of computational complexity and efficiency. Specifically, the embedded fracture modeling strategy avoids the usual requirement that the discretization of the fracture domain conforms to the discretization of the rock volume surrounding the fractures. As fluid is exchanged between the two domains, conservation of mass is guaranteed through a coupling term that appears as a simple source term in the governing mass balance equations. In this manner, as new tensile fractures nucleate and propagate subject to mechanical effects, numerical complexities associated with the introduction of new fracture control volumes are largely negated. In addition, the ability to discretize the fractures and surrounding rock volume independently provides the freedom to choose an acceptable level of discretization for each domain separately. Three numerical examples were performed to demonstrate the utility of the embedded fracture model for application to problems involving fluid flow, mechanical deformation, and rock failure. The results of the numerical examples confirm that the embedded fracture model was able to capture accurately the complex and nonlinear evolution of reservoir permeability as new fractures propagate through the reservoir and as fractures fail in shear.     

改进的数值流形法在水力劈裂中的应用

数值流形法在非连续变形分析领域具有独特优势。结合裂纹尖端场函数的基本概念,分析了水力劈裂破坏问题,模拟了水力劈裂破坏过程,避免了扩展有限元中的阶跃函数和水平集概念。为了避免裂纹尖端在单元内部不同位置而产生误差,对裂纹尖端附近一定范围内的每一个物理覆盖附加奇异覆盖函数。选取一个算例比较分析了内水压力对应力强度因子的影响,当考虑裂纹面内压时,定量分析比较了各因素对应力强度因子的影响大小,并应用于分支裂纹水力劈裂破坏。计算结果表明,改进后的计算结果与解析解相吻合。未考虑裂纹面内压,误差往往较大。考虑裂纹面内压后,随着裂纹长度的增加,误差逐渐减小;随着网格密度的增加,误差也逐渐减小。分支裂纹的渐进破坏结果表明该改进方法的可行性,具有较大的实际应用价值。     

基于ABAQUS平台的水力裂缝扩展有限元模拟研究

随着扩展有限元理论的深入研究,利用扩展有限元方法模拟水力压裂具有了一定的可操作性。相比于常规有限元方法,XFEM方法具有计算结果精度高和计算量小的优点。但是,如何模拟射孔孔眼、如何模拟流体与岩石相互作用以及分析水力裂缝的扩展规律仍然是难题。以研究水力压裂裂缝扩展规律为目的,建立了岩石多孔介质应力平衡方程、流体渗流连续性方程和边界条件。通过有限元离散化方法对耦合方程矩阵进行处理。通过富集函数定义初始裂缝（射孔孔眼）,选择最大主应力及损伤变量D分别作为裂缝起裂和扩展判定准则,利用水平集方法模拟水力裂缝扩展过程。数值模拟结果显示：增加射孔方位角、压裂液排量和减小水平地应力差,起裂压力上升;黏度对起裂压力无明显影响。增加射孔方位角、压裂液排量、黏度和减小水平地应力差值有助于裂缝宽度的增加。增加水平地应力差值、压裂液排量和减小射孔方位角以及压裂液黏度有助于裂缝长度增加,反之亦然。基于ABAQUS的水力裂缝扩展有限元法可对不同井型和诸多储层物性参数及压裂施工参数进行分析,且裂缝形态逼真,裂缝面凹凸程度清晰,结果准确。此研究可作为一种简便有效研究水力压裂裂缝扩展规律的方法为油田水力压裂设计与施工提供参考与依据。     

Influence functions for stress and displacement discontinuity elements in an anisotropic medium

Influence functions, that permit us to determine stresses and displacements at an arbitrary point in an infinite, homogeneous, linear elastic, anisotropic medium due to different three-dimensional (3-D) stress or displacement discontinuities distributed on infinite, flat, band-type elements, are presented. Any straight-line segment on the band, which is perpendicular to its infinite side, has the same distribution of the discontinuities. Along with the functions, their Taylor series approximations are also provided. The last can be useful to analyse stresses and displacements at points distant from the elements. The functions allow us to avoid procedures of numerical integration in the Indirect Boundary Element Method and/or the Displacement Discontinuity Method computer codes that are able to solve complete plane-strain problems with 3-D boundary conditions for an elastic, anisotropic medium. © 1997 by John Wiley & Sons, Ltd.