Propagation of a plane-strain hydraulic fracture with a fluid lag: Early-time solution

This paper studies the propagation of a plane-strain fluid-driven fracture with a fluid lag in an elastic solid. The fracture is driven by a constant rate of injection of an incompressible viscous fluid at the fracture inlet. The leak-off of the fracturing fluid into the host solid is considered negligible. The viscous fluid flow is lagging behind an advancing fracture tip, and the resulting tip cavity is assumed to be filled at some specified low pressure with either fluid vapor (impermeable host solid) or pore-fluids infiltrating from the permeable host solid. The scaling analysis allows to reduce problem parametric space to two lumped dimensionless parameters with the meaning of the solid toughness and of the tip underpressure (difference between the specified pressure in the tip cavity and the far field confining stress). A constant lumped toughness parameter uniquely defines solution trajectory in the parametric space, while time-varying lumped tip underpressure parameter describes evolution along the trajectory. Further analysis identifies the early and large time asymptotic states of the fracture evolution as corresponding to the small and large tip underpressure solutions, respectively. The former solution is obtained numerically herein and is characterized by a maximum fluid lag (as a fraction of the crack length), while the latter corresponds to the zero-lag solution of Spence and Sharp [Spence, D.A., Sharp, P.W., 1985. Self-similar solution for elastohydrodynamic cavity flow. Proc. Roy. Soc. London, Ser. A (400), 289–313]. The self-similarity at small/large tip underpressure implies that the solution for crack length, crack opening and net fluid pressure in the fluid-filled part of the crack is a given power-law of time, while the fluid lag is a constant fraction of the increasing fracture length. Evolution of a fluid-driven fracture between the two limit states corresponds to gradual expansion of the fluid-filled region and disappearance of the fluid lag. For small solid toughness and small tip underpressure, the fracture is practically devoid of fluid, which is localized into a narrow region near the fracture inlet. Corresponding asymptotic solution on the fracture lengthscale corresponds to that of a crack loaded by a pair of point forces which magnitude is determined from the coupled hydromechanical solution in the fluid-filled region near the crack inlet. For large solid toughness, the fluid lag is vanishingly small at any underpressure and the solution is adequately approximated by the zero-lag self-similar large toughness solution at any stage of fracture evolution. The small underpressure asymptotic solutions obtained in this work are sought to provide initial condition for the propagation of fractures which are initially under plane-strain conditions.     

Fluid velocity based simulation of hydraulic fracture—a penny shaped model. Part II: new,accurate semi-analytical benchmarks for an impermeable solid

In the first part of this paper a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off was created for a penny-shaped crack. The power-law rheological model of fluid was assumed and the final scheme was capable of tackling both the viscosity and toughness dominated regimes of crack propagation. The obtained solutions were shown to achieve a high level of accuracy. In this paper simple, accurate, semi-analytical approximations of the solution are provided for the zero leak-off case, for a wide range of values of the material toughness and parameters defining the fluid rheology. A comparison with other results available in the literature is undertaken.     

Fluid velocity based simulation of hydraulic fracture: a penny shaped model—part I: the numerical algorithm

In the first part of this paper, a universal fluid velocity based algorithm for simulating hydraulic fracture with leak-off, previously demonstrated for the PKN and KGD models, is extended to obtain solutions for a penny-shaped crack. The numerical scheme is capable of dealing with both the viscosity and toughness dominated regimes, with the fracture being driven by a power-law fluid. The computational approach utilizes two dependent variables; the fracture aperture and the reduced fluid velocity. The latter allows for the application of a local condition of the Stefan type (the speed equation) to trace the fracture front. The obtained numerical solutions are carefully tested using various methods, and are shown to achieve a high level of accuracy.     

On the toughening of brittle materials by grain bridging: Promoting intergranular fracture through grain angle, strength, and toughness

The structural reliability of many brittle materials such as structural ceramics relies on the occurrence of intergranular, as opposed to transgranular, fracture in order to induce toughening by grain bridging. For a constant grain boundary strength and grain boundary toughness, the current work examines the role of grain strength, grain toughness, and grain angle in promoting intergranular fracture in order to maintain such toughening. Previous studies have illustrated that an intergranular path and the consequent grain bridging process can be partitioned into five distinct regimes, namely: propagate, kink, arrest, stall, and bridge. To determine the validity of the assumed intergranular path, the classical penetration/deflection problem of a crack impinging on an interface is re-examined within a cohesive zone framework for intergranular and transgranular fracture. Results considering both modes of propagation, i.e., a transgranular and intergranular path, reveal that crack-tip shielding is a natural outcome of the cohesive zone approach to fracture. Cohesive zone growth in one mode shields the opposing mode from the stresses required for cohesive zone initiation. Although stable propagation occurs when the required driving force is equivalent to the toughness for either transgranular or intergranular fracture, the mode of propagation depends on the normalized grain strength, normalized grain toughness, and grain angle. For each grain angle, the intersection of single path and multiple path solutions demarcates “strong” grains that increase the macroscopic toughness and “weak” grains that decrease it. The unstable transition to intergranular fracture reveals that an increasing grain toughness requires a growing region of the transgranular cohesive zone be near the cohesive strength. The inability of the body to provide the requisite stress field yields an overdriven and unstable configuration. The current results provide restrictions for the achievement of substantial toughening through intergranular fracture.     

Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

基于GMTS准则的岩石复合型断裂机理研究

使用国际岩石力学协会规定的半圆盘岩石试件,加工不同倾角的直裂纹试样,通过三点弯曲加载试验得到不同I-II复合比断裂的断裂韧性和初始断裂角．传统裂纹扩展准则忽视了常数项即T应力及更高阶项的影响,导致该扩展准则的理论预测结果存在较大缺陷,本文通过考虑常数项,建立广义最大周向应力准则(GMTS)．在此基础上,分别采用传统的裂纹扩展准则和考虑T应力的裂纹扩展准则预测不同复合比裂纹的断裂韧性和初始扩展角,然后对比理论预测结果和实验结果．分析可得：常数项即T应力对断裂的临界应力强度因子和初始断裂角的影响是不可忽略的,且II型断裂占比较大时影响更大,广义最大周向应力准则预测值与实验测试结果之间的误差最小．     

Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture

We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width. By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that are consistent with linear elastic fracture mechanics. Submitted to Journal of Elasticity on February 5, 2007. Re-submitted with revisions on May 30, 2007.     

应力波载荷作用下线弹性断裂过程的动态分析方法研究

利用Hopkinson单压杆实验装置 ,对材料的线弹性动态断裂特性进行了研究 ,建立了应力波载荷作用下动态裂纹起裂及扩展过程的动态分析方法 ,采用该方法可同时测得材料的动态裂纹起裂时间、断裂韧性和裂纹扩展速度。40Cr钢三点弯曲试样的实验结果表明 :该钢的动态裂纹扩展过程主要是减速过程 ,在2 2 5TPam /s的加载速率下 ,起裂时间为 2 8 0 0 s,最大裂纹扩展速度为 478 91m/s ,动态断裂韧性为6 3 12MPam。     

Simultaneous initiation and growth of multiple radial hydraulic fractures from a horizontal wellbore

Multi-stage fracturing is the current preferred method of completion of horizontal wells in unconventional hydrocarbon reservoirs. Its core component consists in simultaneously initiating and propagating an array of hydraulic fractures. We develop a numerical model for the initiation and growth of an array of parallel radial hydraulic fractures. The solution accounts for fracture growth, coupling between elastic deformation and fluid flow in the fractures, elastic stress interactions between fractures and fluid flow in the wellbore. We also take into account the presence of a local pressure drop (function of the entering flow rate) at the connection between the well and the fracture, i.e., a choke-like effect due to current well completion practices, also referred to as entry friction. The partitioning of the fluid into the different fractures at any given time is part of the solution and is a critical indicator of simultaneous (balanced fluid partitioning) versus preferential growth. We validate our numerical model against reference solutions and a laboratory experiment for the initiation and growth of a single radial hydraulic fracture. We then investigate the impact of stress interaction on preferential growth of a subset of fractures in the array. Our results show that a sufficiently large local entry friction provides a strong feedback in the system and thus can counteract elastic stress interaction between fractures, thereby ensuring simultaneous growth. We propose a dimensionless number capturing the competition between stress interaction and local entry friction. This dimensionless number is a function of rock properties, fracture spacing and injection parameters. We verify that it captures the transition from the case of simultaneous growth (entry friction larger than interaction stress) to the case of preferential growth of some fractures (interaction stress larger than entry friction). We also discuss the implication of these results for multi-stage fracturing engineering practices.     

Similarity solution of a penny-shaped fluid-driven fracture in a zero-toughness linear elastic solid

This note deals with the problem of a penny-shaped hydraulic fracture propagating in an impermeable elastic solid. Growth of the fracture is driven by injection of an incompressible Newtonian fluid at the center of the fracture. The solution is restricted to the so-called viscosity-dominated regime where it can be assumed that the solid has zero toughness. The paper describes the construction of a semi-analytical similarity solution, which incorporates the known singularity of the fluid pressure at the center of the fracture and at the tip and which is based on series expansions of the fracture opening and fluid pressure in terms of Jacobi polynomials.     

冲击载荷作用下黑砂岩动态断裂参数的分形修正

基于分形理论研究了偏折裂纹扩展路径对动载荷作用下黑砂岩的动态断裂力学参数的测试误差影响作用,采用传统的分离式霍普金森压杆（split Hopkinson pressure bar, SHPB）实验装置对修正侧开单裂纹半孔板（improved single cleavage semi-circle specimen, ISCSC）试样进行动态冲击实验,随后采用裂纹扩展计进行裂纹起裂时间与裂纹扩展速度等动态断裂力学参数测试,采用分形理论对测试的裂纹扩展速度与动态应力强度因子进行修正,利用实验-数值法对黑砂岩的动态断裂韧度进行计算。研究结果表明,ISCSC构型构件能够有效应用于岩石材料动态裂纹扩展行为的研究,并发生了止裂现象,经分形修正的裂纹扩展速度与动态断裂韧度更接近实际裂纹动态扩展情况,修正前后得到黑砂岩材料的裂纹扩展速度误差为33.51%,动态断裂韧度最大误差为7.68%,说明利用分形理论对动态断裂韧度等动态断裂参数计算更合理。     

Self-similar problems of propagation of an extended hydrofracture in a permeable medium

The propagation of an extended hydrofracture in a permeable elastic medium under the influence of an injected viscous fluid is considered within the framework of the model proposed in [1, 2]. It is assumed that the motion of the fluid in the fracture is turbulent. The flow of the fluid in the porous medium is described by the filtration equation. In the quasisteady approximation and for locally one-dimensional leakage [3] new self-similarity solutions of the problem of the hydraulic fracture of a permeable reservoir with an exponential self-similar variable are obtained for plane and axial symmetry. The solution of this two-dimensional evolution problem is reduced to the integration of a one-dimensional integral equation. The asymptotic behavior of the solution near the well and the tip of the fracture is analyzed. The difficulties of using the quasisteady approximation for solving problems of the hydraulic fracture of permeable reservoirs are discussed. Other similarity solutions of the problem of the propagation of plane hydrofractures in the locally one-dimensional leakage approximation were considered in [3, 4] and for leakage constant along the surface of the fracture in [5–7].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 91–101, March–April, 1992.     

ANALYSIS OF TWO DISSIMILAR FUNCTIONALLY GRADED STRIPS CONTAINING INTERFACE CRACK UNDER PLANE DEFORMATION

In this paper the plane elasticity problem of two bonded dissimilar functionally graded strips containing an interface crack is studied.The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain.The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically.Numerical results show that fracture toughness of materials can be greatly improved by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

Multi-scale experimental analysis of rate dependent metal–elastomer interface mechanics

A remarkable high fracture toughness is sometimes observed for interfaces between materials with a large elastic mismatch, which is reported to be caused by the fibrillar microstructure appearing in the fracture process zone. In this work, this fibrillation mechanism is investigated further to investigate how this mechanism is dissipating energy. For that purpose, thermoplastic urethane (TPU)-copper interfaces are delaminated at various rates in a peel test experimental setup. The fracture process zone is visualized in situ at the meso-scale using optical microscopy and at the micro-scale using Environmental Scanning Electron Microscopy (ESEM). It is shown that the geometry of the fracture process zone is insensitive to the delamination rate, while the interface traction scales logarithmically with the rate. This research has revealed that, the interface roughness is shown to be pivotal in initiating the fibrillation delamination process, which facilitates the high fracture toughness. The multi-scale experimental approach identified two mechanisms responsible for this high fracture toughness. Namely, the viscous dissipation of the TPU at the high strain levels occurring in the fibrils and the loss of stored elastic energy which is disjointed from the propagation due to the size of the process zone.     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.     

Moving singularities in elasto-diffusive solids with applications to fracture propagation

Solutions are derived for steady-state motion of a singularity class, which includes point sources and dislocations, through a medium in which the elastic stress-field can evolve with time due to the diffusion of an internal second-phase species, such as a pore-fluids and lattice impurity concentrations in a crystalline solid, or the transfer of heat. The technique is to integrate the known influence functions for a stationary singularity. Attention is focused on the most tractable aspect, namely the stress field on the trajectory of motion: this suffices for simulation of growing shear and tensile fractures (e.g. in a porous fluid-saturated solid). Continuous densities of fluid sources and point discontinuities (dislocations) are suitably distributed (as determined by solving the resulting singular integral equations) to satisfy solid stress and fluid pressure or flow conditions on the fracture surfaces. Alternative methods for finding the complete dislocation influence function are discussed and comparisons with existing source solutions are made. Substantial stabilization effects are found in fracture propagation.     

Rate effects on toughness in elastic nonlinear viscous solids

A micromechanics-based constitutive relation for void growth in a nonlinear viscous solid is proposed to study rate effects on fracture toughness. This relation is incorporated into a microporous strip of cell elements embedded in a computational model for crack growth. The microporous strip is surrounded by an elastic nonlinear viscous solid referred to as the background material. Under steady-state crack growth, two dissipative processes contribute to the macroscopic fracture toughness—the work of separation in the strip of cell elements and energy dissipation by inelastic deformation in the background material. As the crack velocity increases, voids grow in the strain-rate strengthened microporous strip, thereby elevating the work of separation. In contrast, the energy dissipation in the background material decreases as the crack velocity increases. In the regime where the work of separation dominates energy dissipation, toughness increases with crack velocity. In the regime where energy dissipation is dominant, toughness decreases with crack velocity. Computational simulations show that the two regimes can exist in certain range of crack velocities for a given material. The existence of these regimes is greatly influenced by the rate dependence of the void growth mechanism (and the initial void size) as well as that of the bulk material. This competition between the two dissipative processes produces a U-shaped toughness-crack velocity curve. Our computational simulations predict trends that agree with fracture toughness vs. crack velocity data reported in several experimental studies for glassy polymers and rubber-modified epoxies.     

页岩气藏压裂缝网扩展数值模拟

为探究页岩气藏水力压裂复杂裂缝网络的形成机理,开展了缝网扩展的数值模拟研究.考虑应力阴影和天然裂缝作用,建立了井筒和裂缝中流体流动模型,利用位移不连续方法求解应力与位移不连续量,然后构建了压力与裂缝宽度的迭代方程,并采用牛顿迭代法求解.通过比较数值解经典模型解析解,验证了模型和迭代解法的正确性.多簇裂缝同步扩展时裂缝间距越小,压裂液分配到各条裂缝越不均匀,靠近井筒跟部的裂缝的分流量越大,从而裂缝宽度越大;考虑天然裂缝作用时,逼近角越小或者应力各向异性越弱,水力裂缝越容易发生转向扩展,裂缝网络越复杂.      

An investigation of hydraulic fracturing. Part I: one-phase flow model

The prediction of the growth of a hydraulic fracture in an oil bearing formation based on the injection rate of fluid is valuable in applications of the waterflood technique in secondary oil recovery. In this paper, the problem of hydraulic fracture growth is studied under the assumption of uniform distribution of pressure in the fracture and unidirectional permeating flow in an infinitely large isothermal linearly elastic porous medium saturated with a one-phase incompressible fluid. The condition of plane strain is imposed in the study. A comparison of the constant fracture toughness criterion based on the asymptotic value for large crack growth with the crack tip ductility criterion for an ideally plastic solid under plane strain and small-scale yielding conditions indicates that the effect of ductility of rock on the crack growth is so small that the steady state value of the energy release rate can be reached within a short period of crack growth. Thus we can employ the constant fracture toughness criterion in our study. The analysis includes the effects of both fracture volume increase and leak-off of fluid from the surface of the fracture. A nonlinear singular integro-differential equation can be formulated for the quasi-static hydraulic fracture growth under a prescribed injection rate. It is solved numerically by a modified fourth order Runge-Kutta method.     

采用压缩单裂纹圆孔板确定岩石动态起裂、扩展和止裂韧度

爆炸、冲击、地震等人为或自然灾害不可避免,经常造成大量土木工程设施的破坏,因此岩石在动态载荷作用下的行为受到特别关注.岩石动态断裂韧度是评价岩石抵抗裂纹动态起裂、扩展和止裂性能的材料参数,开展岩石动态断裂韧度测试方法的研究对相关理论基础和实验技术的要求较高.岩石动态断裂韧度分为动态起裂、动态扩展、动态止裂三种,虽然关于动态起裂和动态扩展的研究已有一些成果,对岩石动态止裂的研究仍是一个难题,至今几乎无人问津.研究表明,在分离式霍普金森压杆撞击压缩单裂纹圆孔板岩石试样的I型动态断裂试验中,动态起裂、扩展、止裂的全过程可以由黏贴在压缩单裂纹圆孔板试样上的裂纹扩展计监测,岩石的动态起裂、扩展、止裂韧度可以用实验-数值-解析法确定.特别值得一提的是首次测出了岩石的动态止裂韧度.裂纹扩展计信号还显示,压缩单裂纹圆孔板在止裂后,停止的裂纹还会再次动态起裂、扩展并超出裂纹扩展计的检测范围.从能量的角度分析了动态止裂的过程,指出测试动态止裂韧度时要注意的一些问题.结果显示,岩石动态起裂韧度和动态扩展韧度分别随动态加载率和裂纹扩展速度的增大而增大,岩石动态起裂韧度略大于动态止裂韧度.