Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions

This paper presents an analysis of the propagation of a penny-shaped hydraulic fracture in an impermeable elastic rock. The fracture is driven by an incompressible Newtonian fluid injected from a source at the center of the fracture. The fluid flow is modeled according to lubrication theory, while the elastic response is governed by a singular integral equation relating the crack opening and the fluid pressure. It is shown that the scaled equations contain only one parameter, a dimensionless toughness, which controls the regimes of fracture propagation. Asymptotic solutions for zero and large dimensionless toughness are constructed. Finally, the regimes of fracture propagation are analyzed by matching the asymptotic solutions with results of a numerical algorithm applicable to arbitrary toughness.     

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.     

含水合物粉质黏土压裂成缝特征实验研究

水力压裂技术是一种重要的油气井增产、增注措施,已经广泛应用于页岩油气等非常规资源的商业开采中.目前对于粉质黏土水合物沉积物的水力压裂成缝能力尚不清楚.本文采用南海水合物沉积层的粉质黏土制备沉积物试样,并与实验室配制的粉细砂土沉积物对比,分析粉质黏土沉积物的水力成缝能力及主控因素.实验结果表明含水合物和冰的沉积物破裂压力较高,这与粉质黏土沉积物特殊的应力-应变特征和渗透性有关.当沉积物应变高于6%时,试样强度迅速上升,呈现应变强化的特征,对水力拉伸裂缝的扩展具有一定的阻碍作用.粉质黏土沉积物粒径细小,渗透性差,难以通过渗透作用传递压力,提高了沉积层的破裂压力.此外,粉质黏土水合物沉积层裂缝扩展存在明显延迟效应,说明裂缝扩展受到流体压力和热应力的共同影响.适当延长注入时间,保持流体与沉积层充分接触,会起到分解水合物、降低破裂压力的作用.该研究成果有利于深入理解水力裂缝在水合物沉积层中的扩展规律,对探索压裂技术在水合物沉积层开发中的应用具有重要意义.     

Crack tip behavior in near-surface fluid-driven fracture experiments

This Note presents experimental results for the near-tip fracture opening of fluid-driven fractures. The effect of fluid viscosity, quantified by a dimensionless parameter, was varied between tests. The tip region closely followed the classical square-root behavior from linear elastic fracture mechanics when the viscosity parameter was small. Conversely, when the viscosity parameter was of order one and the lag between the fluid-filled region and the fracture front accounted for less than 30% of the fracture, the tip region behaved according to a known intermediate asymptotic solution which results from the solid/fluid coupling. To cite this article: A.P. Bunger et al., C. R. Mecanique 333 (2005).     

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.     

Problem of propagation of a gas fracture in a porous medium

The problem of gas fracture formation in a porous medium is investigated. An inertialess viscous polytropic gas flow along the fracture is considered. The assumption of small fracture width with respect to the height and length makes it possible to adopt the vertical plane cross-section hypothesis on the basis of which the dependence of the gas pressure inside the fracture on its width can be reduced to a linear law. Initially, the soil surrounding the fracture is soaked with oil-bearing fluid. During fracturing the reservoir gas penetrates into the soil mass and displaces the fluid. A closed system of equations, which describes the evolution of the fracture opening, the depth of gas penetration into the reservoir, and the gas velocities inside the fracture, is constructed. The limiting regimes of gas seepage into the surrounding reservoir are considered and a one-parameter family of self-similar solutions of the system is given for each. The asymptotics of the solution in the neighborhood of the fracture nose is investigated and analytic expressions for the fracture length are obtained. The solution of the problem of gas fracture is compared with the hydraulic fracturing problem in an analogous formulation within the framework of the plane cross-section hypothesis.     

Simulation of Pressure Transient Behavior for Asymmetrically Finite-Conductivity Fractured Wells in Coal Reservoirs

Based on Fick’s law in matrix and Darcy flow in cleats and hydraulic fractures, a new semi-analytical model considering the effects of boundary conditions was presented to investigate pressure transient behavior for asymmetrically fractured wells in coal reservoirs. The new model is more accurate than previous model proposed by Anbarci and Ertekin, SPE annual technical conference and exhibition, New Orleans, 27–30 Sept 1998 because new model is expressed in the form of integral expressions and is validated well through numerical simulation. (1) In this paper, the effects of parameters including fracture conductivity, coal reservoir porosity and permeability, fracture asymmetry factor, sorption time constant, fracture half-length, and coalbed methane (CBM) viscosity on bottomhole pressure behavior were discussed in detail. (2) Type curves were established to analyze both transient pressure behavior and flow characteristics in CBM reservoir. According to the characteristics of dimensionless pseudo pressure derivative curves, the process of the flow for fractured CBM wells was divided into six sub-stages. (3) This paper showed the comparison of transient steady state and pseudo steady state models. (4) The effects of parameters including transfer coefficient, wellbore storage coefficient, storage coefficient of cleat, fracture conductivity, fracture asymmetry factor, and rate coefficient on the shape of type curves were also discussed in detail, indicating that it is necessary to keep a bigger fracture conductivity and fracture symmetry for enhancing well production and reducing pressure depletion during the hydraulic fracturing design.     

上覆单相弹性层饱和地基上弹性基础的竖向振动分析

基于Biot动力控制方程,运用Fourier积分变换技术,并按照混合边值条件和连续条件建立了上覆单相弹性层饱和地基上弹性基础竖向振动的对偶积分方程.利用正交多项式将对偶积分方程化简,得到了动力柔度系数随无量纲频率b0的变化关系曲线,从而得到了上覆单相弹性层饱和地基上弹性基础的竖向振动规律.数值分析结果表明,对于弹性基础,当弹性基础的挠曲刚度较大时,发现弹性基础的竖向振动特性与刚性基础的类同,可忽略挠曲刚度对竖向振动的影响,且当无量纲频率较小的时候,动力柔度系数Cv随着无量纲频率b0的变化而发生显著的变化,但当无量纲频率b0较大的时候,动力柔度系数Cv受无量纲频率的影响较小,甚至基本上不受影响.当弹性基础的挠曲刚度较小时,随着挠曲刚度的减小,弹性基础的竖向振动将发生显著的变化,动力柔度系数Cv的实部和虚部的绝对值均变大.     

多孔饱和半空间上弹性圆板垂直振动的积分方程

应用新的方法求解多孔饱和固体的动力基本方程－Ｂｉｏｔ波动方程,首先把Ｂｉｏｔ波动方程化为仅有土骨架位移和孔隙水压力的偏微分方程组,并且逐次解耦方法（不引入位移势函数）求解此偏微分方程组,然后按混合边值条件建立多孔饱和半空间上弹性圆板垂直振动的对偶积分方程,用Ａｂｅｌ变换化对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。文中考虑两种孔隙流体的表面边界条件：（ａ）半空间表面（包括圆板与半空间的接触面）是     

三维压裂缝网不稳定压力半解析求解方法

受地应力及压裂工艺影响, 大斜度井水力压裂缝网展布复杂, 缝网中存在不同倾斜方向、不同展布形态及不同贯穿程度的压裂缝. 本文通过将裂缝面离散为若干矩形微元实现裂缝形态有效表征, 将渗流过程划分为基质向裂缝流动及裂缝向井筒流动两阶段, 采用有限差分方法构建离散裂缝面内不稳定渗流数值解, 结合封闭边界面源函数及叠加原理构建基质内不稳定渗流解析解, 耦合裂缝内流动数值解与基质内流动解析解, 求解了三维压裂缝网不稳定压力. 基于积分中值定理提出了点源、特殊线源代替面源求解基质内渗流的求解方法, 分析了该方法的可行性及适用条件, 在保证模型精度的同时提升了计算效率. 研究表明, 在基质内采用点源函数面积分求解面源的方法可准确求解三维压裂缝网井底压力动态但计算效率极低, 基于积分中值定理的点源、特殊线源近似面源求解方法可大大提升计算效率, 且在裂缝微元划分较为精细(微元无因次边长小于0.15)时可取得较高精度, 基于该模型分析了裂缝导流能力、裂缝倾角、裂缝高度及裂缝段间距对压裂大斜度井典型试井曲线的影响.      

An asymptotic solution for fluid production from an elliptical hydraulic fracture at early-times

The early-time transient flow during the start-up of fluid production from a porous medium by a well intersected by a vertical elliptical hydraulic fracture is studied using an asymptotic analysis. The analysis is focused on the situation of practical interest where the fracture conductivity is high so that production from the fracture dominates. The first three terms in a short-time asymptotic expansion for the production rate during constant-pressure production, and for the well-pressure during constant-rate production, are obtained. It is shown that the fracture tip starts to influence the production rate only when the dimensionless time is increased to the square of the reciprocal of the dimensionless fracture conductivity. The asymptotic results also show that geometric factors of an elliptical fracture introduce non-negligible corrections to the so-called bilinear flow in the early times, which were previously erroneously associated with the effect of the fracture tip.     

Experiment and simulation of slurry flow in irregular channels to understand proppant transport in complex fractures

Slurry flow and proppant placement in irregular fractures are crucial to evaluate hydraulic fracturing stimulation but need to be better understood. This study aims to investigate how irregular fracture affects proppant transport and distribution using laboratory experiments and micro-scale numerical models. The unresolved method of the computational fluid dynamics (CFD) and the discrete element method (DEM) considers Saffman lift force, Magnus force, and virtual mass force to accurately capture the frequent interaction between proppant and slickwater. Experimental results validated the reliability of the optimized CFD-DEM model and calibrated primary parameters. The effects of crack height and width, bending angle, and distance between the crack and inlet on particle distribution were studied. The results indicated that the improved numerical method could rationally simulate proppant transport in fractures at a scale factor. The small crack height causes downward and upward flows, which wash proppant to the fracture rear and form isolated proppant dunes. A wider region in the fracture is beneficial to build up a large dune, and the high dune can hinder particle transport into the fracture rear. When the crack is close to the inlet, the primary fracture without proppants will close to hinder gas production. The smaller the bending angle, the smaller the proppant dune. A regression model can precisely predict the dune coverage ratio. The results fundamentally understand how complex fractures and natural cracks affect slurry flow and proppant distribution.     

围压下泥岩断裂韧性测试与解释方法

根据围压条件下的断裂力学理论 ,采用双翼裂纹厚壁圆筒 ,对泥岩进行了不同围压和不同泥质含量条件下的人工岩样断裂韧性测试。建立了有限元方法解释断裂韧性的数值模型。在进行大量实验测试的基础上 ,通过回归分析表明 ,断裂韧性与围压、泥质含量有较好的线性统计关系。解决了水力压裂设计中断裂韧性参数的准确获取问题 ,有利于提高压裂设计、预测的精度     

Finite Element study of fracture initiation in flaws subject to internal fluid pressure and vertical stress

Hydraulic fracturing is a method used routinely in oil and gas exploitation and in engineered geothermal systems. While used frequently, there are many aspects of hydraulic fracturing, such as the direction of propagation of the newly-created fractures, which are not very well understood. Even though it is known that the local stress field plays a fundamental role in the orientation of the new fractures, there may be other factors, such as the geometry of the existing fractures and the magnitude of the hydraulic pressure applied, that may play a major role in the path that a new fracture follows when pressurized.The main goal of this study is to numerically analyze the effect of the ratio between a vertical load, or stress, and the hydraulic pressure applied in existing flaws on the stress field in the vicinity of the flaw tips. For that purpose, a double flaw geometry 2a-30-30 was modeled in the Finite Element code ABAQUS, and different vertical loads and internal flaw pressures were applied to the model. The variation of the maximum principal stresses and maximum shear stresses around the flaw tips were analyzed and related to fracture initiation.The study showed that the ratio between the water pressure applied in the flaws and the vertical load/stress (WP/VL) plays a crucial role in the magnitude and shape of the stress field around a flaw tip, and therefore in the location of tensile and shear fracture initiation. As WP/VL increases, the location of initiation of new tensile fractures shifts from the upper face of the studied flaw towards the region right ahead of the flaw tip; simultaneously, the location of initiation of new shear fractures shifts from the region ahead of the flaw tip to the upper face of the analyzed tip.     

混凝土靶体计及压实效应的球腔动态膨胀模型和侵彻计算

建立了一混凝土靶体的球腔动态膨胀的侵彻模型。靶体的静水压力一体积应变关系考虑了不同的密实混凝土模型。也研究了混凝土作为多孔材料在高压下的压实效应。混凝土有拉伸强度极限值,其剪切强度与压力相关为Mohr—Coulomb定律。球腔动态膨胀模型考虑球对称空腔在无限介质里从零半径开始以常速度膨胀,产生一塑性区和一弹性区。当膨胀速度足够小时,由于介质的低拉伸强度,塑性区和弹性区之间可存在一径向裂纹区。可由解析方法获得空腔表面压力和弹塑性区界面传播速度随空腔表面扩张速度变化的曲线,再由球腔动态膨胀模型所得的空腔表面压力分布用于计算混凝土靶体的刚性杆弹侵彻深度。本文模型的计算结果与经验公式,不可压缩和线性可压缩混凝土模型的进行了比较。     

页岩气高效开采的可压裂度和射孔簇间距预测

页岩储层的可压裂性是影响页岩气产量的关键因素. 本文基于断裂力学理论, 以高围压下岩石层理弱面的剪切破坏为主要研究对象, 依据岩石抗拉强度和层理弱面抗剪强度的比值关系, 首先提出了可压裂度的概念, 给出了无量纲的定性曲线图, 涵盖了岩石脆性矿物质含量, 粘性主导和韧性主导裂缝尖端流体压强、射孔簇分布间距的综合地质与工程因素. 接着提出了一种新的表征高围压下页岩可压裂度的无量纲参数, 在保证达到充分解吸附的最小压裂间距前提下, 依据该参数可计算水平井压裂中的射孔簇间距, 可作为工程参考指标. 本文将断裂力学理论结合水力压裂高效开采页岩气工程, 具有力学理论意义和工程应用前景.      

Fluid flow in a porous medium containing partially closed fractures

The model of Snow, in which a fracture is represented by two parallel channel walls, has frequently been used to study the flow of fluid in fractured reservoirs. Although this model gives important insight into the flow in fractures, very few naturally occurring fractures have smooth parallel faces. In this paper, a simple model of partially contacting and en-echelon fractures frequently found in geological materials is presented. In this model, a fracture is viewed as a planar region where separation and contact zones both exist. To analyse the fluid flow in a porous medium containing fractures of this type, a planar array of periodically spaced fracture segments is analysed. The flow through a single fracture is deduced by taking the limit as the spacing between neighbouring fractures becomes large. The hydraulic conductivity parallel to the fractures is found to be the parallel combination of the conductivity of the porous matrix and the system of parallel fractures, the individual fracture conductance being a series combination of the hydraulic conductance of the separation and contact zones. This interpretation enables the conductance of the contact zones to be evaluated and the results to be generalised to the case in which the material in the contact regions has a hydraulic conductivity different to that of the matrix. This may arise, for example, from grain-size reduction during fracturing or may result from a partial mineralisation or cementation of the fracture.     

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.     

Crack Tip Equation of Motion in Dynamic Gradient Damage Models

We propose in this contribution to investigate the link between the dynamic gradient damage model and the classical Griffith’s theory of dynamic fracture during the crack propagation phase. To achieve this main objective, we first rigorously reformulate two-dimensional linear elastic dynamic fracture problems using variational methods and shape derivative techniques. The classical equation of motion governing a smoothly propagating crack tip follows by considering variations of a space-time action integral. We then give a variationally consistent framework of the dynamic gradient damage model. Owing to the analogies between the variational ingredients of these two models and under some basic assumptions concerning the damage band structuration, one obtains a generalized Griffith criterion which governs the crack tip evolution within the non-local damage model. Assuming further that the internal length is small compared to the dimension of the body, the previous criterion leads to the classical Griffith’s law through a separation of scales between the outer linear elastic domain and the inner damage process zone.     

On the theory of seepage waves of pressure in a fracture in a porous permeable medium

Seepage pressure waves in fractures in a porous permeable medium are studied. The effects of the reservoir and fracture porosity and permeability, the fracture width, and the rheological properties of the saturating fluid on the perturbation dynamics in the fracture are analyzed. It is shown that in porous permeable reservoirs, fractures are wave channels through which low-frequency fluctuations of borehole pressure propagate. Accurate solutions are obtained which describe the evolution of pressure fields in a fracture with an instantaneous change in the borehole pressure by a constant value. Based on these solutions, dependences of the fluid flow rate on time and interface pressure are determined.