﻿ 任意夹角交叉封闭边界内平面流线计算及应用
 西南石油大学学报(自然科学版)  2018, Vol. 40 Issue (4): 116-122

Method for Computing In-plane Streamlines in Cross-sealed Boundaries of Any Angle of Tip and Its Applications
LI Gen , WU Haojun, CAI Hui, SHI Peng, OU Yinhua
CNOOC China Limited, Tianjin Branch, Tanggu, Tianjin 300459, China
Abstract: At present, to solve the function of complex potentials in cross-sealed boundaries with an analytic method requires the condition that the boundary angle of the dip equals π/n (where n is a positive integer). To extend the application of n to any real number, performing conformal transformation followed by mirror imaging is proposed. Specifically, the idea is to mirror a sealed boundary of any angle of the tip in the original plane onto a target plane by performing conformal transformation, thereby enabling the angle of the tip of the sealed boundary after mirroring to satisfy the condition that n is a positive integer. According to the property that the values of the complex potential and current of the point source (convergence) of a point in the original plane remain unchanged after conformal transformation, the complex-potential function of the corresponding point in the target plane of the point in the original plane is solved, and the resulting value is the value of the complex-potential function of the point in the original plane. An equation for computing the flow velocity of any point in flow fields was also derived. The streamline distribution in flow fields was rendered graphically by employing the contours method. In addition, the shortage of flow fields computed with the analytic method was reviewed and, correspondingly, improvements were proposed. The field application results show that an area closer to the vertex of the boundary angle of the tip has a higher streamline density, lower flow velocity, and higher degree of oil saturation.
Key words: sealed boundary     cross fault     conformal transformation     mirror imaging     streamline
0 引言

1 几何模型建立

 $z=x+{\rm i}y=r{{{\rm e}}^{{\rm i}\theta }}$ (1)
 图1 $z$平面中的夹角封闭边界 Fig. 1 The intersecting boundary in the $z$ plane

$r$—场点在极坐标的极径，m；

$\theta$—场点在极坐标辅角，rad。

2 保角变换

 $\zeta ={{z}^{\frac{{\rm{π }} }{{{\alpha }_{{0}}}}\cdot \frac{{{\alpha }_{1}}}{{\rm{π }} }}}={{z}^{\frac{{{n}_{0}}}{{{n}_{1}}}}}, {\kern 10pt} {{n}_{1}}=\dfrac{{\rm{π }} }{{{\alpha }_{1}}}$ (2)
 $\zeta =\xi +{\rm i}\eta =\rho \cdot {{{\rm e}}^{{\rm i}\vartheta }}$ (3)

$\xi$$\eta—笛卡尔坐标系下场点的横、纵坐标，m； \rho—场点极坐标极径，m； \vartheta—场点极坐标辅角，rad。 由保角变换性质，目标复平面与原始复平面内的场点坐标变换如式(4)所示，点源(汇)半径的变换如式(5)所示。  \left\{ \begin{array}{l} \rho = {r^{\frac{{{n_0}}}{{{n_1}}}}}\\ \vartheta = \dfrac{{{n_0}}}{{{n_1}}}\theta \end{array} \right. (4)  {{\rho }_{{\rm w}}}=\left| {{\zeta }}\left( {{z}_{0}} \right) \right|\cdot {{r}_{{\rm w}}} (5) 式中：\rho _{{\rm w}}$$\zeta$平面对应$z_{{\rm 0}}$的点源(汇)的半径，m；

$r_{{\rm w}}$$z平面某点源(汇)的半径，m。 若某平面封闭边界的夹角为{\rm{π }} /{{n}_{1}}，封闭边界内的点源(汇)数为m，镜像反映法对平面的边界等效后点源(汇)总数变为2m\cdot {{n}_{1}}个。使目标平面内可进行镜像处理的{{n}_{1}}的可取值很多，保角变换前后平面内点源(汇)数不变，本文取{{n}_{1}}=1，若原始平面内有m个点源(汇)，则目标平面内镜像反映后会有2m个点源(汇)。则将z平面的交叉封闭边界按式(4)映射到\zeta平面后变为直线封闭边界，边界内的单井变为直线封闭边界附近单井，根据镜像反映法可知，边界的作用可以等效为两口井：1口实井、1口虚拟井。z平面中交叉封闭边界内的区域变为\zeta平面中封闭边界的右侧区域，如图 2所示。图中，\rho _{{\rm 0}}—单井z_0映射在\zeta平面内的极径，m；\rho _{{\rm f}}—封闭边界在极坐标下的长度，m；\zeta _{{\rm 0}}—单井z_{{\rm 0}}映射在\zeta平面内的复坐标；b—井与封闭边界间距离，m。  图2 \zeta平面中的夹角封闭边界 Fig. 2 The intersecting boundary in the \xi plane 直线封闭边界附近一口井为常见形式，文献[16]给出了此情形的产量公式，如式(6)所示。  q=\dfrac{2{\rm{π }} Kh\left( {{p}_{{\rm e}}}-{{p}_{{\rm w}}} \right)}{\mu \ln \dfrac{\rho _{{\rm e}}^{2}}{2b{{\rho }_{{\rm w}}}}} (6) 式中： q—产量，m^{{\rm 3}}/s； h—油层厚度，m； \mu—黏度，mPa\cdots； K—渗透率，D； p_{{\rm e}}—供给边界上的压力，MPa； p_{{\rm w}}—井壁上的压力，MPa； \rho _{{\rm e}}—供给边界半径，m； \rho _{{\rm w}}—井半径，m。 根据保角变换性质[18]，变换前后两个复平面内对应场点的流函数、势函数及点源(汇)的流量不变。因此，z平面内产量公式如式(7)所示。  q=\dfrac{2{\rm{π }} Kh\left( {{p}_{{\rm e}}}-{{p}_{{\rm w}}} \right)}{\mu \ln \dfrac{{{\rho}_{{\rm e}}}^{{{n}_{0}}}}{2{{ {{r}_{{\rm 0}}} }^{{{n}_{0}}-1}}\cos \left( \dfrac{{{n}_{0}}}{2}{{\theta }_{{\rm 0}}} \right)\dfrac{{{n}_{0}}}{2}{{\rho}_{{\rm w}}}}} (7) 式中： \theta_{{\rm 0}}—井点极坐标辅角，rad。 3 流函数及物理参量求解 3.1 交叉封闭边界流函数计算 对于多井情况，渗流场内的复势函数符合叠加原理，如式(8)所示。流函数是复势函数中虚数部分。在u-v复平面内考虑某一源(汇)点\omega_{i}(u_{i}, v_{i})指向场点\omega(u$$v$)的径向矢量为$\lambda _{i}$，该矢量与$u$轴正向的辅角为$\varphi$，高等渗流力学中求解$\varphi$的公式见式(9)

 $W\left( \omega \right)=-\dfrac{1}{2{\rm{π }} }\sum\limits_{{}}^{{}}{{{q}_{{\rm h}i}}}\ln \left( \omega -{{\omega }_{i}} \right)$ (8)
 $\varphi =\arctan \left( \dfrac{v-{{v}_{i}}}{u-{{u}_{i}}} \right)$ (9)

$\omega$—复坐标；

$W(\omega)$$\omega平面内场点的复势函数； u$$v$—场点在笛卡尔坐标系下的横坐标和纵坐标，m；

$q_{{\rm h}i}$—点源(汇)$i$的单位厚度产量，m$^{{\rm 3}}$/s；

 $\left\{ \begin{array}{l} \varphi = {\rm{arccos}}\left( {\dfrac{{\bf {{\boldsymbol{\lambda } _0}} \cdot \bf \boldsymbol{\lambda } }}{{\left| {\bf {{\boldsymbol{\lambda } _0}} } \right| \cdot \bf {\left| \boldsymbol{\lambda } \right|} }}} \right) + {\varphi _0}, {\kern 8pt} \left( {\bf {{\boldsymbol{\lambda } _0}} \times \bf \boldsymbol{\lambda } } \right) \cdot \boldsymbol{\tau }_{\rm u} > 0\\ \varphi = {\rm{arccos}}\ {\dfrac{{\bf {{\boldsymbol{\lambda } _0}} \cdot \bf \boldsymbol{\lambda } }}{{\left| {\bf {{\boldsymbol{\lambda } _0}} } \right| \cdot \bf {\left| \boldsymbol{\lambda } \right|} }}} + {\rm{π }} + {\varphi _0}, {\kern 8pt} \left( {\bf {{\boldsymbol{\lambda } _0}} \times {\bf \boldsymbol{\lambda }} } \right) \cdot \boldsymbol{\tau }_{\rm u} < 0\\ \varphi = {\varphi _0}, {\kern 6pt} \left({ {{\boldsymbol{\lambda } _0}} \cdot \bf \boldsymbol{\lambda } }\right) > 0 \cap \left( {\bf {{\boldsymbol{\lambda } _0}} \times \bf \boldsymbol{\lambda } } \right) \cdot \boldsymbol{\tau }_{\rm u} = 0\\ \varphi = {\rm{π }} + {\varphi _0}, {\kern 8pt} \left( { {{\boldsymbol{\lambda } _0}} \cdot {\boldsymbol{\lambda }} } \right) < 0 \cap \left( {\bf {{\boldsymbol{\lambda } _0}} \times \bf \boldsymbol{\lambda } } \right) \cdot \boldsymbol{\tau }_{\rm u} = 0\\ {\rm{arccos}}~x \in \left[ {0, {\rm{π }} } \right], {\kern 8pt} x \in \left[ { - 1, 1} \right] \end{array} \right.$ (10)
 图4 $\varphi$几何关系 Fig. 4 The geometrical relationship of $\varphi$

$\boldsymbol{\lambda }$—点$\omega_{{\rm 0}}$到点$\omega$的矢量，m；

$\boldsymbol{\tau }_{\rm u}$$\tau轴正向单位矢量，m。 z平面内的主要研究区域为边界夹角顶点到单井之间的范围，保角变换后对应在\zeta 平面中为井与直线封闭边界之间的区域，对直线封闭边界作镜像反映等效后可见，该区域置于两口生产井之间的区域，如图 5所示。  图5 \zeta 平面边界等效镜像 Fig. 5 The equivalent imaging of boundary in \zeta plane 为使跳变区域在研究区域的外侧，对于封闭边界右侧的原井点\zeta$$_{{\rm 0}}$、封闭边界左侧的镜像井点$\zeta $$_{{\rm 1}}，分别取基准辅角\varphi_{{\rm 0}}=0、\varphi_{{\rm 1}}=π，计算并绘制流场图及流线分布如图 6所示。  图6 \zeta 平面两汇的流场、流线 Fig. 6 The flow field and streamline of two sinks in ζ plane \zeta 平面中点的流函数值按映射关系赋予z平面中对应点，绘制z平面的流场等值线即可知其流线分布，如图 7所示。  图7 z平面交叉封闭边界流场、流线 Fig. 7 The flow field and streamline in z plane with intersecting boundary 3.2 场点流速及过某断面流量求解 流速为某一场点所具备的属性，流量为某一截面所具有的属性。由于保角变换前后的势函数与流函数不变，根据复势函数速度定义式，原始z平面内的某一点的速度，可由目标\zeta 平面的井点进行镜像反映处理后生成的所有井点叠加的复势函数W(\zeta )对z平面内的x$$y$进行求导运算求取，推导结果如式(11)$\sim$式(15)所示。本文论述的是平面流场，厚度已经在流量中体现(为单位厚度的流量)，平面内两点间的流函数差值即为过该两点之间连线与单位厚度形成的截面流量。

 $W\left( \zeta \right)=\dfrac{-1}{2{\rm{π }} }\left[ \sum\limits_{i=1}^{2m}{{{q}_{{\rm h}i}}}\ln \left( \zeta -{{\zeta }_{i}} \right) \right]=\\{\kern 40pt}\dfrac{-1}{2{\rm{π }} }\left[ \sum\limits_{i=1}^{2m}{{{q}_{{\rm h}i}}}\ln \left( {{z}^{{{n}_{0}}/2}}-z_{i}^{{{n}_{0}}/2} \right) \right]$ (11)
 $\dfrac{{\rm d}W\left( \zeta \right)}{{\rm d}z}=\dfrac{{\rm d}W\left( \zeta \right)}{{\rm d}\zeta }\dfrac{{\rm d}\zeta }{{\rm d}z}=\\ {\kern 40pt}\dfrac{-1}{2{\rm{π }} } \dfrac{{{n}_{0}}}{2} {{r}^{{{n}_{0}}/2-1}} {{\rm e}^{{\rm i}\left( {{n}_{0}}/2-1 \right)\theta }} \sum\limits_{i=1}^{2m}{\dfrac{{{q}_{{\rm h}i}}}{{{\rho }_{i}}}}{{\rm e}^{-{\rm i}{{\vartheta }_{i}}}}$ (12)
 ${{V}_{x}}=\dfrac{1}{2{\rm{π }} } \dfrac{{{n}_{0}}}{2} {{r}^{{{n}_{0}}/2-1}} \cdot \\ {\kern 40pt} \sum\limits_{i=1}^{2m}{\dfrac{{{q}_{{\rm h}i}}\cos \left[ -{{\vartheta }_{i}}+\left( \dfrac{{n}_{0}}{2}-1 \right)\theta \right]}{{{\rho }_{i}}}}$ (13)
 ${{V}_{y}}=\dfrac{-1}{2{\rm{π }} } \dfrac{{{n}_{0}}}{2} {{r}^{{{n}_{0}}/2-1}} \sum\limits_{i=1}^{2m}{\dfrac{{{q}_{{\rm h}i}}\sin \left[ -{{\vartheta }_{i}}+\left( \dfrac{{n}_{0}}{2}-1 \right)\theta \right]}{\rho _{i}^{{}}}}$ (14)
 $V=\sqrt{V_{x}^{2}+V_{y}^{2}}$ (15)

$\rho_{i}$$\zeta 平面内场点镜像反映后i点源(汇)的极径，m； \vartheta_{i}$$\zeta$平面内镜像反映后$i$点源(汇)的辅角，rad；

$V_{{x}}$$z平面内x方向速度分量，m/s； V_{{ y}}$$z$平面内$y$方向速度分量，m/s；

$V$$z$平面内的流速，m/s。

4 算例分析

 图8 某商业软件生成的流线分布图 Fig. 8 Streamline distribution calculated by commertial simulation software

 图9 本文方法生成的流线分布图 Fig. 9 Streamline distribution calculated by this method
5 结论

(1) 利用保角变换法改进了传统利用镜像反映法求解夹角封闭边界内复势函数对边界夹角取值的限制，本方法能够处理多源(汇)的情况，相对于格林方程法只能求解区域内单源汇的局限，具有较大的优势。

(2) 流函数为辅角的函数，采用相对基准角增量的方式计算辅角，能够将流函数跳跃位置放置于非关注区域，使关注区域流函数连续。

(3) 通过对交叉封闭边界内绘制流线，能够认清流体滞留区(剩余油富集区)，为下一步挖潜提供帮助。

 [1] 许寒冰, 李相方, 石德佩, 等. 注采井网生产井含水率解析计算方法[J]. 石油学报, 2010, 31(3): 471-474. XU Hanbing, LI Xiangfang, SHI Depei, et al. An analytical method for calculating water cut of producers in injection-production pattern[J]. Acta Petrolei Sinica, 2010, 31(3): 471-474. doi: 10.7623/syxb201003022 [2] 李晓军, 张琪, 路智勇. 断块油藏水平井与直井组合井网渗流解析解及其应用[J]. 中国石油大学学报(自然科学版), 2010, 34(3): 84-88. LI Xiaojun, ZHANG Qi, LU Zhiyong. Analytic solution of horizontal-vertical composed well pattern seepage in fault-block reservoir and its application[J]. Journal of China University of Petroleum (Edition of Natural Science), 2010, 34(3): 84-88. doi: 10.3969/j.issn.1673-5005.2010.-03.018 [3] LARSEN L. A simple approach to pressure distributions in geometric shape[C]. SPE 10088-PA, 1985. doi: 10.2118/-10088-PA http://www.researchgate.net/publication/250090892_A_Simple_Approach_to_Pressure_Distributions_in_Geometric_Shapes [4] YAXLEY L M. New stablized inflow equations for rectangular and wedge shaped drainage system[C]. SPE 17082-MS, 1987. http://www.onepetro.org/general/SPE-17082-MS [5] 马水龙, 黎福长, 李星军, 等. 裂缝性油藏交叉断层条件下的压力特征[J]. 大庆石油学院学报, 1998, 22(2): 67-69. MA Shuilong, LI Fuchang, LI Xingjun, et al. Study of pressure behavior in fractured reservoir with intersecting faults[J]. Journal of Daqing Petroleum Institute, 1998, 22(2): 67-69. [6] PRASAD R K. Pressure transient analysis in the presence of two intersecting boundaries[J]. Journal of Petroleum Technology, 1975, 27(1): 89-96. doi: 10.2118/4560-PA [7] CHEN C C, RAGHAVAN R. Computing pressure distributions in wedges[J]. SPE Journal, 1995, 2(1): 24-32. doi: 10.2118/30555-PA [8] 王晓东, 穆立婷. 两任意夹角断层的井壁压力计算方法[J]. 油气井测试, 1997, 6(2): 13-16. WANG Xiaodong, MU Liting. Algorithms for pressure transient analysis in the presence of two intercept faults with an arbitrary angle[J]. Well Testing, 1997, 6(2): 13-16. [9] 赵秀才, 衣艳静, 姚军. 两夹角不渗透断层对油井压力及压力导数的影响研究[J]. 断块油气田, 2005, 12(6): 41-43. ZHAO Xiucai, YI Yanjing, YAO Jun. Study on the influence of two interacting sealing faults on oil wells pressure and its derivative[J]. Fault Block Oil & Gas Field, 2005, 12(6): 41-43. doi: 10.3969/j.issn.1005-8907.2005.06.013 [10] 卢德唐, 孔祥言. 扇形区域内有井储和表皮的井底压力[J]. 油气井测试, 1996, 5(2): 5-10. LU Detang, KONG Xiangyan. The well bottle pressure is affected with wellbore storage coefficient and skin factorin the sector area[J]. Well Testing, 1996, 5(2): 5-10. [11] CHARLES D D, PURUSHOTHAMAN R. Well test characterization of wedge-shaped, faulted reservoirs[C]. SPE 56685, 1999. doi: 10.2118/56685-MS http://dx.doi.org/10.2118%2f56685-MS [12] CHARLES D D, RIEKE H H, PURUSHOTHAMAN R. Well-test characterization of wedge-shaped faulted reservoirs[C]. SPE 72098, 2001. doi: 10.2118/72098-PA http://www.researchgate.net/publication/250089478_Well-Test_Characterization_of_Wedge-Shaped_Faulted_Reservoirs [13] 侯健, 王玉斗, 陈月明. 复杂边界条件下渗流场流线分布研究[J]. 计算力学学报, 2003, 20(3): 335-338. HOU Jian, WANG Yudou, CHEN Yueming. Research on streamline distribution of flow through porous media with complex boundary[J]. Chinese Journal of Computational Mechanics, 2003, 20(3): 335-338. doi: 10.3969/j.issn.-1007-4708.2003.03.015 [14] 何应付, 尹洪军, 刘莉, 等. 复杂边界非均质渗流场流线分布研究[J]. 计算力学学报, 2007, 24(5): 708-712. HE Yingfu, YIN Hongjun, LIU Li, et al. Research on streamline distribution of flow through heterogeneous porous media with complex boundary[J]. Chinese Journal of Computational Mechanics, 2007, 24(5): 708-712. doi: 10.3969/j.issn.1007-4708.2007.05.031 [15] 徐联玉, 尹洪军, 何应付, 等. 不渗透区域对渗流场流线分布影响研究[J]. 大庆石油地质与开发, 2006, 25(4): 57-59. XU LianYu, YIN HongJun, HE Yingfu. Effect of impermeable area on distribution of flow path in seepage flow field[J]. Petroleum Geology & Oilfield Development In Daqing, 2006, 25(4): 57-59. doi: 10.3969/j.issn.1000-3754.2006.04.020 [16] 马春晓, 尹洪军, 唐鹏飞, 等. 基于有限元的压裂水平井渗流场分析[J]. 大庆石油地质与开发, 2015, 34(2): 90-94. MA Chunxiao, YIN HongJun, Tang Pengfei, et al. Seepage field analyses of the fractured horizontal well by finite element method[J]. Petroleum Geology & Oilfield Development In Daqing, 2015, 34(2): 90-94. doi: 10.3969/J.-ISSN.1000-3754.2015.02.017 [17] 郑伟, 姜汉桥, 陈民锋, 等. 水平井注采井网合理井间距研究[J]. 西南石油大学学报(自然科学版), 2011, 33(1): 120-124. ZHENG Wei, JIANG Hanqiao, CHEN Minfeng, et al. Study on well spacing of horizontal well pattern[J]. Journal of Southwest University (Science & Technology Edition), 2011, 33(1): 120-124. doi: 10.3863/j.isan.1674-5086.2011.01.021 [18] 程林松. 高等渗流力学[M]. 北京: 石油工业出版社, 2011: 46-60. [19] 李根. 考虑重力的直线边水单井波及面积计算模型[J]. 大庆石油地质与开发, 2016, 35(2): 48-51. LI Gen. Calculating model of the swept area for the individual well with the straight edge water considering the gravity[J]. Petroleum Geology & Oilfield Development in Daqing, 2016, 35(2): 48-51. doi: 10.3969/J.ISSN.1000-3754.2016.02.008