﻿ 不同注采方式的流场计算模型及物模验证
 西南石油大学学报(自然科学版)  2018, Vol. 40 Issue (2): 129-134

1. 中国石油化工股份有限公司石油勘探开发研究院, 北京 海淀 100083;
2. 中国石油大学(北京)石油工程学院, 北京 昌平 102249

Flow Field Calculation Model and Verification by Physical Model in Different Injection Production Well Pattern
YE Shuangjiang1 , JIANG Hanqiao2
1. Research Institute of Petroleum Exploration and Production, SINOPEC, Haidian, Beijing 100083, China;
2. College of Petroleum Engineering, China University of Petroleum(Bejing), Changping, Beijing 102249, China
Abstract: Regarding the problems on flow field distribution with different injection-production schemes (single injection and single production of vertical well and horizontal well, single injection and single production of horizontal well, and dual injection and single production of horizontal well), the study on mathematical calculation models of flow field and physical simulation with different injection-production schemes is carried out. Both conformal transformation and the mirror image reflection theory are applied to establish the mathematical model of the flow field for different injection-production schemes and deduce the mathematical expressions for the potential function and stream function. Thus, the inner equipotential lines and streamline distribution diagrams of different injection-production schemes are obtained. The computational results of the calculation model of the flow field for different injection-production schemes are compared and verified through a physical simulation experiment of water-oil displacement with different injection-production schemes. The results show that the calculation results of the calculation model for flow fields in different injection-production schemes are basically consistent with the physical simulation experiment results. The inner flow field for the single injection and single production of vertical and horizontal wells shows a certain linear flow, and both the water-driving wave and area are increased compared to the conventional injection-production schemes of vertical well. The inner injection-production cross-well streamline of single injection and single production in horizontal wells presents a linear flow. The area controlled by the streamline is larger; the displacement effect is better in comparison with single injection and single production of vertical and horizontal wells. The streamline near the heel end and toe end of the inner horizontal production well in dual injection and single production of a horizontal well is relatively dense; that near the mid position of the horizontal production well is relatively sparse, being easy to form a lag oil zone.
Key words: injection-production schemes     conformal transformation     potential function     stream function     physical simulation

1 不同注采方式流场计算模型 1.1 直井水平井一注一采注采方式流场计算模型

 $\left\{ \begin{array}{l} x=L{\rm ch}~\xi \cos\eta \\ y=L{\rm sh}~\xi \sin\eta \end{array} \right.$
 图1 直井水平井一注一采示意图 Fig. 1 A schematic map of one vertical injector and one horizontal producer

$z$平面的流动变为$w$平面的已知流动，对应于$w$平面上的是宽度为$\pi$的半无限长带状地层，排油坑道位于纵轴上，长度为$\pi$，在(${{\xi }_{0}}, {{\eta }_{0}}$)处有一口注入直井。

 图4 水平井一注一采示意图 Fig. 4 A schematic map of one vertical well injection and one horizontal well production

 ${{\phi }_{2}}=\dfrac{1}{4\pi L} \sum\limits_{i=1}^{2}{q_{i}{'}}({{A}_{4}}+{{A}_{5}}+{{A}_{6}})$ (8)
 ${{A}_{4}}=\left[{\rm arccot} \dfrac{x-\left( {{x}_{{\rm w}i}}+L \right)}{ y-{{y}_{{\rm w}i}} }-{\rm arccot} \dfrac{x-\left( {{x}_{{\rm w}i}}-L \right)}{ y-{{y}_{{\rm w}i}} } \right]\cdot \\ {\kern 40pt} \dfrac{y-{{y}_{{\rm w}i}}}{L}$ (9)
 ${{A}_{5}}= \dfrac{x-{{x}_{{\rm w}i}}}{2L} \ln \dfrac{{{\left[x-\left( {{x}_{{\rm w}i}}-L \right) \right]}^{2}}+{{\left( y-{{y}_{{\rm w}i}} \right)}^{2}}}{{{\left[x-\left( {{x}_{{\rm w}i}}+L \right) \right]}^{2}}+{{\left( y-{{y}_{{\rm w}i}} \right)}^{2}}}$ (10)
 ${A_6} = \dfrac{1}{2}\ln \left[{{{\left( {x-{x_{{\rm{w}}i}}{\rm{ + }}L} \right)}^2} + {{\left( {y-{y_{{\rm{w}}i}}} \right)}^2}} \right] +\\ {\kern 40pt} \dfrac{1}{2}\ln \left[{{{\left( {x-{x_{{\rm{w}}i}}-L} \right)}^2} + {{\left( {y-{y_{{\rm{w}}i}}} \right)}^2}} \right]$ (11)

 ${{\varphi }_{2}}=\dfrac{1}{2\pi L} \sum\limits_{i=1}^{2}{{{q}_{i}{'}}}({{B}_{1}}+{{B}_{2}}+{{B}_{3}})$ (12)
 ${{B}_{1}}=\left[x-\left( {{x}_{{\rm w}i}}+L \right) \right]{\rm arccot} \dfrac{\left[x-\left( {{x}_{{\rm w}i}}+L \right) \right]}{y-{{y}_{{\rm w}i}}}$ (13)
 ${{B}_{2}}=-\left[x-\left( {{x}_{{\rm w}i}}-L \right) \right]{\rm arccot} \dfrac{\left[x-\left( {{x}_{{\rm w}i}}-L \right) \right]}{y-{{y}_{{\rm w}i}}}$ (14)
 ${{B}_{3}}=\dfrac{ y-{{y}_{{\rm w}i}} }{2}\ln \dfrac{{{\left[x-\left( {{x}_{{\rm w}i}}+L \right) \right]}^{2}}+{{\left( y-{{y}_{{\rm w}i}} \right)}^{2}}}{\left[x-\left( {{x}_{{\rm w}i}}-L \right) \right]+{{\left( y-{{y}_{{\rm w}i}} \right)}^{2}}}$ (15)

 图5 水平井一注一采等势线 Fig. 5 Equipotential line pattern for one horizontal injection and one horizontal well production
 图6 水平井一注一采流线分布图 Fig. 6 Stream line pattern for one horizontal injection and one horizontal well production

1.3 水平井两注一采注采方式流场计算模型

 图7 水平井两注一采示意图 Fig. 7 A schematic map of two horizontal injection and one horizontal production

 ${{\phi }_{3}}=\dfrac{1}{4\pi L}\times \sum\limits_{i=1}^{3}{{q_{i}{''}}}({{A}_{4}}+{{A}_{5}}+{{A}_{6}})$ (16)
 ${{\varphi }_{3}}=\dfrac{1}{2\pi L}\times \sum\limits_{i=1}^{3}{{q_{i}{''}}}({{B}_{1}}+{{B}_{2}}+{{B}_{3}})$ (17)

 图8 水平井两注一采等势线 Fig. 8 Equipotential line pattern for two horizontal injection and one horizontal well production
 图9 水平井两注一采流线分布图 Fig. 9 Stream line pattern for two horizontal injection and one horizontal well production

2 不同注采方式流场计算模型物理模拟验证

2.1 三维平板模型实验方法

 图10 不同注采方式流场物理模拟实验流程 Fig. 10 Flow chart of different injection production well pattern flow field experiment
2.2 实验结果及分析

 图11 不同注采方式物理模拟流场分布图 Fig. 11 Flow field of different injection production well pattern by experiment

3 结论

(1) 基于保角变换和镜像反映原理，建立并求解了不同注采方式流场计算数学模型，绘制了不同注采方式内部等势线和流线分布图。

(2) 不同注采方式内部流场平板模型物理模拟实验，直观反映了不同注采方式内部流场分布形态，此验证了流场计算模型的正确性。

(3) 直井水平井一注一采内部流场表现出一定的线性流，水驱波及面积较常规直井注采方式有所增大。水平井一注一采内部注采井间流线呈线性流，与直井水平井一注一采相比，流线控制的面积进一步增大，驱替效果更好。水平井两注一采内部水平生产井跟端和趾端附近流线较为密集，水平生产井中部位置附近流线相对稀疏易形成滞油区。

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