2017,
Vol. 41
基于位移不连续模型的多裂缝应力干扰极限距离计算
1. 中国石油大学油气CAE技术研究中心, 山东青岛 266580;
2. 中国石油大学机电工程学院, 山东青岛 266580;
3. 西安石油大学电子工程学院, 陕西西安 710000;
4. 中国石油吉林油田分公司, 吉林松原 138000
2. 中国石油大学机电工程学院, 山东青岛 266580;
3. 西安石油大学电子工程学院, 陕西西安 710000;
4. 中国石油吉林油田分公司, 吉林松原 138000
摘要: 利用MTS试验装置对“R13区块”的致密砂岩岩心进行小试样试验, 获得“R13区块”致密砂岩的岩石力学参数(弹性模量、泊松比等)。忽略地层内水力裂缝面的不规则性, 将其简化为互相平行的光滑裂缝面。考虑到地层在裂缝面的不连续性, 将裂缝视为连续地层的内部不连续边界, 采用位移不连续理论建立边界元模型, 研究多裂缝系统周围的地层应力分布和不同簇数、不同布缝方式下的裂缝群缝间应力干扰的极限距离。结果表明, 人工多裂缝的出现明显改变了裂缝周围应力场的分布, 裂缝群簇数越多影响距离越大, 但簇数的增加对影响距离的改变逐渐趋于平缓, 呈对数增长, 而段内裂缝间距(20~30 m)的改变基本不影响应力干扰的极限距离。基于位移不连续理论的边界元模型能够准确地计算裂缝群周围地层应力场分布和裂缝群缝间应力干扰极限距离。
关键词: 多裂缝 应力干扰 位移不连续理论 极限距离
Critical distance of stress shadow of multiple-cracks based on displacement discontinuity model
1. CAE Technology Research Center, China University of Petroleum, Qingdao 266580, China;
2. College of Mechanical and Electronic Engineering in China University of Petroleum, Qingdao 266580, China;
3. College of Electronic Engineering, Xi`an Shiyou University, Xi`an 710000, China;
4. PetroChina Jilin Oilfield Company, Songyuan 138000, China
2. College of Mechanical and Electronic Engineering in China University of Petroleum, Qingdao 266580, China;
3. College of Electronic Engineering, Xi`an Shiyou University, Xi`an 710000, China;
4. PetroChina Jilin Oilfield Company, Songyuan 138000, China
Abstract: A small specimen test is conducted in MTS to get the elastic modulus and Poisson's ratio of tight sandstone core in R13 field. The irregularity of hydraulic crack surface is ignored, and the cracks are simplified as smooth and parallel to each other. Considering the discontinuity of the multi-cracks, they are treated as discontinuous boundaries of the continuous stratum. Then the displacement discontinuity method is used to construct a boundary element model, which is applicable to calculate the distribution of stress around the multi-crack system, and determine the critical distance of the crack swarms' stress shadow with different clusters and different ways of cloth sewn. The results show that the appearance of artificial multiple-cracks has significantly changed the distribution of stress field around the cracks. The more the cluster number, the longer the distance takes effect. But the influence of the cluster number increasing on the distance becomes smaller, taking on a logarithmic tendency. However, the change of crack spacing between 20 and 30 meters, in a single stage has little influence on the critical distance of stress shadow. This study indicates that the boundary element model based on the displacement discontinuity theory can accurately predict the distribution of the stress field around the multiple-cracks and the critical distance of the crack swarms' stress shadow.
Keywords: multiple-cracks stress shadow displacement discontinuity method critical distance
2 位移不连续理论基本解
3 R13区块裂缝群间应力影响范围分析
4 不同布缝方式下的应力干扰极限距离
作为低渗透油气井增产的主要措施, 水平井分段多簇压裂技术越来越受到重视。近几年川庆钻探、勘探开发研究院、吉林油田自主研发了水平井裸眼分隔器分段压裂工具, 现场成功试验百余口井, 已发展成为水平井分段压裂改造的主体技术[1]; 同时, 对水平井分段压裂技术机制的研究也成为了重要课题。大量的现场和室内试验研究表明[2-10], 油气井压裂后, 人工裂缝的存在能改变近井地带地应力的大小和方向。随着水平井压裂段数和簇数的增加, 缝间应力干扰受到重视, 合理地利用缝间应力干扰可以改变裂缝起裂方向, 提高储层改造效果, 但同时也会因为缝间的应力干扰形成局部高应力, 影响储层改造[11-13]。为优化水平井段的段间距和簇间距, 须对地层人工裂缝群周围应力场的分布进行细致研究。笔者通过对R13区块致密砂岩岩心进行力学试验, 获得R13区块岩石的力学参数, 采用位移不连续理论计算该区块分段压裂产生的裂缝群对地应力分布影响的极限距离。
1 岩心力学试验为准确获取R13区块致密砂岩的岩石力学参数, 利用MTS试验系统(图 1)对R13区块的致密砂岩岩心进行了常规力学试验。根据国际岩石力学学会(ISRM)推荐的方法, 将钻取岩心制成直径为25 mm, 高为50 mm的圆柱型试件, 如图 2所示。其端面不平行度不超过0.05 mm, 直径误差不超过0.3 mm, 尺寸测量精度为0.1 mm。根据油藏埋深地应力选取围压为35 MPa。绘制R13区块岩心应力-应变曲线如图 3所示。
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图 1 MTS岩石试验系统 Fig.1 MTS rock-testing system |
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图 2 砂岩岩心试件 Fig.2 Specimens of sandstone cores |
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图 3 砂岩岩心应力-应变曲线 Fig.3 Stress-strain curves of sandstone |
通过岩心力学试验, 对试验结果进行线性拟合, 试验结果如图 3、4所示。图中蓝色线表示试验数据, 红色直线为线性拟合结果。R13区块的岩心弹性模量为34.6 GPa, 泊松比为0.14。通过力学试验得到R13区块的岩石力学参数, 为后续裂缝应力干扰极限的研究提供了基础数据。
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图 4 径向-轴向应变曲线 Fig.4 Radial-axial strain curves |
位移不连续法是由美国学者Crouch于1973年在国际岩石力学会议上第一次提出, 并给出了位移不连续理论的二维解[14-16]。迄今为止, 已有大量的学者运用位移不连续理论进行岩石力学问题的研究[17-18]。Kresse和Wu等[19-20]采用改进的二维位移不连续法研究裂缝扩展问题。
当进行水力压裂时, 地层中会出现水力裂缝, 裂缝的上下表面间会发生相对位移和错动, 即为所说的位移不连续量。故而将裂缝的两个面视为地层内部的不连续边界。建立长为2l的不连续单元如图 5所示。裂缝面x与y方向的不连续量分别用Dx、Dy表示, 相对的单个面的位置则用u(x, 0-)、u(x, 0+)、u(y, 0-)、u(y, 0+)表示。裂缝面的位移不连续量可以表示为
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图 5 二维位移不连续单元 Fig.5 Two-dimensional displacement discontinuity element |
| $ \begin{array}{l} {D_x} = {u_x}(x, {0_-})-{u_x}(x, {0_ + }), \\ {D_y} = {u_y}(y, {0_-}) - {u_y}(y, {0_ + }), \left| x \right| \le l. \end{array} $ | (1) |
式中, 正号表示上表面, 负号表示下表面; 裂缝张开Dy为负。
Crouch和Starfied给出了计算位移和地应力的计算公式[15]如下:
| $ \begin{array}{l} {u_x} = {D_x}\left[{2\left( {1-v} \right)\frac{{\partial f}}{{\partial y}}-y\frac{{{\partial ^2}f}}{{\partial {x^2}}}} \right] - {D_y}\left[{\left( {1-2v} \right)\frac{{\partial f}}{{\partial x}}-y\frac{{{\partial ^2}f}}{{\partial x\partial y}}} \right], \\ {\rm{ }}{u_y} = {D_x}\left[{\left( {1-2v} \right)\frac{{\partial f}}{{\partial x}}-y\frac{{{\partial ^2}f}}{{\partial x\partial y}}} \right] - {D_y}\left[{2\left( {1-v} \right)\frac{{\partial f}}{{\partial y}}-y\frac{{{\partial ^2}f}}{{\partial {y^2}}}} \right]. \end{array} $ | (2) |
式中, ux为水平位移; uy为垂直位移; Dx为切向不连续位移; Dy为法向不连续位移。
f是与点的位置坐标有关的函数, 表述为
| $ \begin{array}{*{20}{l}} {f\left( {x, y} \right) = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{y\left( {{\rm{arctan}}\frac{y}{{x-a}}-{\rm{arctan}}\frac{y}{{x + a}}} \right)-} \right.}\\ {\left. {\left( {x - a} \right){\rm{ln}}\sqrt {{{\left( {x - a} \right)}^2} + {y^2}} + \left( {x + a} \right){\rm{ln}}\sqrt {{{\left( {x + a} \right)}^2} + {y^2}} } \right].} \end{array} $ | (3) |
| $ \frac{{\partial f}}{{\partial x}} = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{{\rm{ln}}\sqrt {{{\left( {x-a} \right)}^2} + {y^2}}-{\rm{ln}}\sqrt {{{\left( {x + a} \right)}^2} + {y^2}} } \right]. $ | (4) |
| $ \frac{{\partial f}}{{\partial y}} = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1-v} \right)}}\left( {{\rm{arctan}}\frac{y}{{x-a}}-{\rm{arctan}}\frac{y}{{x + a}}} \right). $ | (5) |
| $ \frac{{{\partial ^2}f}}{{\partial x\partial y}} = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{\frac{y}{{{{\left( {x-a} \right)}^2} + {y^2}}}-\frac{y}{{{{\left( {x + a} \right)}^2} + {y^2}}}} \right]. $ | (6) |
| $ \frac{{{\partial ^2}f}}{{\partial {x^2}}} =- \frac{{{\partial ^2}f}}{{\partial {y^2}}} = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{\frac{{x-a}}{{{{\left( {x-a} \right)}^2} + {y^2}}}-\frac{{x + a}}{{{{\left( {x + a} \right)}^2} + {y^2}}}} \right]. $ | (7) |
| $ \begin{array}{*{20}{l}} {\frac{{{\partial ^3}f}}{{\partial x\partial {y^2}}} =- \frac{{{\partial ^3}f}}{{\partial {x^3}}} = \frac{1}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{\frac{{{{\left( {x- a} \right)}^2}- {y^2}}}{{{{\left[{{{\left( {x-a} \right)}^2} + {y^2}} \right]}^2}}} - } \right.}\\ {\left. {\frac{{{{\left( {x + a} \right)}^2} - {y^2}}}{{{{\left[{{{\left( {x + a} \right)}^2} + {y^2}} \right]}^2}}}} \right].} \end{array} $ | (8) |
| $ \begin{array}{*{20}{l}} {\frac{{{\partial ^3}f}}{{\partial {x^3}}} =- \frac{{{\partial ^3}f}}{{\partial {x^2}\partial y}} = \frac{{2y}}{{4{\rm{ \mathit{ π} }}\left( {1- v} \right)}}\left[{\frac{{x- a}}{{{{\left[{{{\left( {x-a} \right)}^2} + {y^2}} \right]}^2}}} -} \right.}\\ {\left. {\frac{{x + a}}{{{{({{\left( {x + a} \right)}^2} + {y^2})}^2}}}} \right].} \end{array} $ | (9) |
则x与y方向的应力及剪应力表达式为
| $ \begin{array}{l} {\sigma _{xx}} = 2G{D_x}\left[{2\frac{{{\partial ^2}f}}{{\partial {y^2}}} + y\frac{{{\partial ^3}f}}{{\partial x\partial {y^2}}}} \right] + 2G{D_y}\left[{2\frac{{{\partial ^2}f}}{{\partial {y^2}}} + y\frac{{{\partial ^3}f}}{{\partial {y^3}}}} \right], {\rm{ }}\\ {\sigma _{yy}} = 2G{D_x}\left[{-y\frac{{{\partial ^3}f}}{{\partial x\partial {y^2}}}} \right] + 2G{D_y}\left[{2\frac{{{\partial ^2}f}}{{\partial {y^2}}}-y\frac{{{\partial ^3}f}}{{\partial {y^3}}}} \right], {\rm{ }}\\ {\tau _{xx}} = 2G{D_x}\left[{2\frac{{{\partial ^2}f}}{{\partial {y^2}}} + y\frac{{{\partial ^3}f}}{{\partial x\partial {y^2}}}} \right] + 2G{D_y}\left[{-y\frac{{{\partial ^3}f}}{{\partial x\partial {y^2}}}} \right]. \end{array} $ | (10) |
依据位移不连续原理, 采用R13区块的岩心试验结果, 对R13区块的裂缝间应力干扰进行了研究。
3.1 模型建立水平井每段簇数分别为2、3、4、5、6簇。图 6为多裂缝压裂过程的产层受力示意图。模拟计算中, 产层模型尺寸取800 m×400 m, 裂缝长度为100 m, 井眼位置位于裂缝中间; AB、CD加载最大水平主应力σH; AB、BC边加载最小主应力σh; 初始地层压力为p0。
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图 6 地层受力示意图 Fig.6 Diagram of strata stress |
有限元模型尺寸为600 m×1 000 m, 裂缝长度为100 m, 最大、最小水平地应力分别为24和20 MPa, 砂岩弹性模量为34.6 GPa, 泊松比为0.14, 地层孔隙压力为19 MPa, 地层孔隙度为9%。结合水平井压裂射孔工具, 簇间裂缝间距取25 m。
3.2 计算结果分析为更直观地看出地层应力改变区域, 将计算结果制成云图如图 7、8所示。结合工程实际, 当应力改变值小于5%时, 认为地层应力值未发生变化, 即为图中蓝色区域。
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图 7 不同簇数下地层最大主应力云图 Fig.7 Maximum main stress nephogram under different clusters |
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图 8 不同簇数下地层最小主应力云图 Fig.8 Minimum main stress nephogram under different clusters |
从图 7和图 8可以得出, 最大主应力和最小主应力的影响范围均随着压裂簇数的增加而增加。将图中数据提取作成曲线如图 9所示。由图 9可以看出, 在裂缝簇数小于等于4时应力影响范围随着簇数增加, 增加较快, 当裂缝簇数大于4时增加较慢; 垂直于裂缝方向的影响距离要大于平行于裂缝延伸方向的。改变簇间距(30 m)进行了相同的处理方法, 得到的曲线与图 9和图 10基本一致。
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图 9 不同压裂簇数下最大和最小主应力影响范围 Fig.9 Reach of maximum and minimum principal stresses with different fracturing cluster number |
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图 10 第一种和第二种布缝方式示意图 Fig.10 Diagrams under the first and the second decorate fractures |
针对油田现场的情况, 采取了两种布缝方式(第一种是垂直于裂缝延伸方向, 第二种是平行于裂缝延伸方向)见图 10。
根据图 10的布缝方式, 分别建立计算模型, 将计算结果列于表 1、2, 表中数据为缝长的倍数。从表中可以看出, 每个裂缝群簇数越多应力影响极限距离越长, 但增加逐渐放缓; 最大主应力要比最小主应力的影响范围大; 水平井间距应小于表中给出的极限距离, 以利于形成缝网。
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表 1 第一种布缝方式下不同簇数下缝间应力干扰极限距离 Table 1 Seam stress interference limit distance of different clusters under the first decorate fractures |
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表 2 第二种布缝方式下不同簇数下缝间应力干扰极限距离 Table 2 Seam stress interference limit distance of different clusters under the second decorate fractures |
对比这两种布缝方式的计算结果可以看出, 第一种布缝方式应力的影响极限距离明显大于第二种布缝方式下的极限距离, 因此在实际生产中建议采用第一种布缝方式, 以利于形成缝网。
5 结论(1) 由于人工裂缝的存在, 明显改变了裂缝周围地层应力场分布。采用位移不连续法得到的计算模型可以较好地计算裂缝地层应力分布。
(2) 每个裂缝群内簇数越多应力影响距离越长, 但增加逐渐放缓, 当簇数大于4时, 应力影响的极限距离增加速率接近于0。
(3) 最大主应力比最小主应力的影响范围大; 簇间距对应力干扰距离基本无影响。
(4) 第一种布缝方式的影响极限距离大于第二种布缝方式的, 实际生产中建议采用第一种布缝方式, 以利于缝网形成。
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