西南石油大学学报 (自然科学版)  2017, Vol. 39 Issue (2): 139-144

1. 中国地质大学 (武汉) 工程学院, 湖北 武汉 430000;
2. 长江大学石油工程学院, 湖北 武汉 430000;
3. 中国石化西南油气分公司工程技术研究院, 四川 德阳 618000;
4. 中国石油西南油气田公司, 四川 成都 610000;
5. "油气藏地质及开发工程"国家重点实验室西南石油大学, 四川 成都 610500

Application of Slip Lag Time in Discrete to the Calculation of Gas upward Velocity in Deep Well
HUANG Zhiqiang1,2, OU Biao3, KONG Xiangwei2 , WANG Xingyu4, LIN Yuanhua5
1. School of Engineering and Technology, Chinese Universitity of Geology (Wuhan), Wuhan, Hubei 430000, China;
2. School of Petroleum Engineering, Yangtze University, Wuhan, Hubei 430000, China;
3. Engineering and Technology Research Institute Company Southwest Oil and Gas Field, SINOPEC, Deyang, Sichuan 618000, China;
4. Southwest Oil and Gas Field Company, PetroChina, Chengdu, Sichuan 610000, China;
5. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, China
Abstract: Considering the gas slipping velocity, the flow characteristics at the stopping and starting pumps are studied, and the method of slipping lag time in discrete is proposed for calculating the speed of gas slip in deep wells. The time sliding mesh algorithm is proposed in a multiphase flow, which is used to solve for the gas slipping velocity, and the data are calculated using a cubic spline function. Corresponding calculations are performed, and a control software is designed. We applied this mathematical model to a well in Yuanba Sichuan, and obtained results that met operational requirements. The gas remains in a supercritical state in the deep well, and the gas slippage is not serious and has little effect on the gas upward velocity. When the state of the gas changes from a supercritical state into a gas near the wellhead, substantial gas slippage occurs. If the gas slippage is not considered, the error rate is up to 76.36%. It was observed that with an increase in the pump stopping time and gas monitoring time, as well as a decrease in lag time, and that of the reservoir top boundary, upward velocities of the gas tend to decrease.
Key words: gas slipping     two-phase flow     gas slipping velocity     method of slip lag time     lag time

1 模型建立

 图1 气窜发展过程示意图 Fig. 1 Process diagram of gas upward velocity

1.1 离散滑脱滞后时间模型

 $v = \frac{{{H_{{\rm{Oil}}}}-{H_{{\rm{Mud}}}}}}{{{T_{{\rm{Sta}}}}}}$ (1)

$T_{{\rm{Sta}}}$——停泵钻井液静止时间，s。

 ${T_{\rm{c}}} = \sum\limits_{i = 1}^n {\frac{{{H_{{\rm{Mud}}, i}}}}{{({H_{{\rm{Bit}}}}/{T_{{\rm{Lat}}}}) + {v_{{\rm{s}}, i}}}}}$ (2)

$v_{{\rm{s}, }i}$——$i$井段气体滑脱速度，m/s；

$H_{{\rm{Bit}}}$——钻井液循环时钻头深度，m；

$T_{{\rm{Lat}}}$——钻头位置的钻井液迟到时间，s；

$H_{{\rm{Mud}, }i}$——$i$井段长度，m。

 $\left| {{T_{\rm{c}}}-{T_{\rm{s}}}} \right|<\Delta {T_{\rm{r}}}$ (3)

$\Delta {T_{\rm{r}}}$——计算精度，s。

 $v = \dfrac{{{H_{{\rm{Oil}}}}-({H_{{\rm{Bit}}}}/{T_{{\rm{Lat}}}}) {T_{\rm{s}}}}}{{{T_{{\rm{Sta}}}}}}$ (4)
1.2 气液流动模型

 $\dfrac{\partial }{{\partial t}}\iiint\limits_{{\Omega _{\rm{m}}}} {{\rho _{\rm{m}}}{\rm{d}}\Omega } + \iint\limits_{{A_{\rm{m}}}} {{\rho _{\rm{m}}}{v_{\rm{m}}}{n_{\rm{m}}}{\rm{d}}A} = 0$ (5)

$\rho_{\rm{m}}$——气/钻井液相密度，kg/m$^3$；

$v_{\rm{m}}$——气/钻井液相速度，m/s；

${\boldsymbol{n}_m}$——钻井液相法向；

$A_{\rm{m}}$——气/钻井液相占控制体有效横截面积，m$^2$。

 $\dfrac{\partial }{{\partial t}}\iiint\limits_{{\Omega _{\rm{m}}}} {{\rho _{\rm{m}}}{v_{\rm{m}}}{\rm{d}}\Omega } + \iint\limits_{{A_{\rm{m}}}} {\rho _{\rm{m}}}v_{\rm{m}}^2{\boldsymbol{n}_{\rm{m}}}{\rm{d}}A =\\ \iint\limits_{{A_{\rm{m}}}} p{\boldsymbol{n}_{\rm{g}}}{\rm{d}}A-\iiint\limits_{{\Omega _{\rm{m}}}} {{\rho _{\rm{m}}}{\rm{g}}{\rm{d}}\Omega }-{\tau _{{\rm{m}}0}}{S_{{\rm{m}}0}}-{\tau _{{\rm{m}}1}}{S_{{\rm{m}}1}}$ (6)

${\boldsymbol{n}_{\rm{g}}}$——气相法向量，无因次；

$S_{\rm{m0}}$——气/钻井液相与裸眼井壁的接触面积，m$^2$；

$\tau_{\rm{m1}}$——气/钻井液相与套管的摩擦应力，Pa；

$S_{\rm{m1}}$——气/钻井液相与套管壁的接触面积，m$^2$；

$p$——压力，N/m$^2$；

g——重力加速度，g=9.8 m/s$^2$。

1.3 辅助模型

 $p = \dfrac{{{\rm{R}}{T_{\rm{e}}}}}{{V-b}}-\dfrac{a}{{{T^{0.5}}V(V + b)}}$ (7)

 $a = {(\sum {{y_i}a_i^{0.5}})^2}, b = \sum {{y_i}{b_i}}$ (8)

 ${a_i} = {\Omega _{\rm{a}}}{{\rm{R}}^2}T_{\rm{c}}^{2.5}/{p_{\rm{c}}}, {b_i} = {\Omega _{\rm{b}}}{\rm{R}}{T_{\rm{ct}}}/{p_{\rm{c}}}$ (9)

$\Omega_{\rm{b}}$=0.08664；

$a_i$——组分$i$的$a$值；

$b_i$——组分$i$的$b$值；

R——气体常数，R=8.314 J/(kg$\cdot$K)；

$V$——酸性气体体积，m$^3$；

$T_{\rm{e}}$——温度，K；

$T_{\rm{ct}}$——临界温度，K；

$p_{\rm{c}}$——临界压力，MPa；

$y_i$——组分的摩尔分数。

2 模型求解

 图2 气窜速度计算流程图 Fig. 2 The calculation flow chart of gas upward velocity

（1）将环空划分为$n$个网格，每个网格长度为${L_i}$；

（2）气体从网格i移动至网格$i$+1，所需用的时间为${T_{i}}= ({H}_{\rm{Oil}}/{n}) /{V_{{\rm{g}}_i}}$。

（3）利用漂移模型可求出空隙率，利用动量守恒可求出压力。

（4）假设上网格的压力与空隙率，利用式（3）中计算结果与漂移模型中的假设参数验证，可求出压力与空隙率，如收敛跳出循环，不收敛继续迭代。

 $\begin{array}{l} \frac{{(A{\rho _{\rm{g}}}{v_{{\rm{sg}}}})_{i + 1}^{n + 1} - (A{\rho _{\rm{g}}}{v_{{\rm{sg}}}})_i^{n + 1}}}{{\Delta s}} = \\ \frac{{(A{\rho _{\rm{g}}}{\phi _{\rm{g}}})_i^n}}{{2\Delta t}} + \frac{{(A{\rho _{\rm{g}}}{\phi _{\rm{g}}})_{i + 1}^n}}{{2\Delta t}} - \\ \frac{{(A{\rho _{\rm{g}}}{\phi _{\rm{g}}})_i^{n + 1} - (A{\rho _{\rm{g}}}{\phi _{\rm{g}}})_{i + 1}^{n + 1}}}{{2\Delta t}} \end{array}$ (10)

 $\dfrac{{(Av_{{\rm{sl}}} )_{i + 1}^{n + 1} - (Av_{{\rm{sl}}} )_i^{n + 1} }}{{\Delta s}} =\dfrac{{(A\phi _{\rm{l}} )_i^n + (A\phi _{\rm{l}} )_{i + 1}^n - (A\phi _{\rm{l}} )_i^{n + 1} - (A\phi _{\rm{l}} )_{i + 1}^{n + 1} }}{{2\Delta t}}$ (11)

 $(Ap)_{i + 1}^{n + 1}-(Ap)_i^{n + 1} = {K_1} + {K_2} + {K_3} + {K_4}$ (12)

 $\begin{array}{l} {K_1} = \frac{{\Delta s}}{{2\Delta t}}\left\{ {\left[ {A({\rho _{\rm{l}}}{v_{{\rm{sl}}}} + {\rho _{\rm{g}}}{v_{{\rm{sg}}}})} \right]_i^n + \left[ {A({\rho _{\rm{l}}}{v_{{\rm{sl}}}} + {\rho _{\rm{g}}}{v_{{\rm{sg}}}})} \right]_{i + 1}^n} \right.\\ \left. { - \left[ {A({\rho _{\rm{l}}}{v_{{\rm{sl}}}} + {\rho _{\rm{g}}}{v_{{\rm{sg}}}})} \right]_i^{n + 1} - \left[ {A({\rho _{\rm{l}}}{v_{{\rm{sl}}}} + {\rho _{\rm{g}}}{v_{{\rm{sg}}}})} \right]_{i + 1}^{n + 1}} \right\} \end{array}$ (13)
 $K_2 = \left[{A\left( {\dfrac{{\rho _{\rm{l}} v_{{\rm{sl}}}^2 }}{{\phi _{\rm{l}} }} + \dfrac{{\rho _{\rm{g}} v_{{\rm{sg}}}^2 }}{{\phi _{\rm{g}} }}} \right)} \right]_i^{n + 1} - \left[{A\left( {\dfrac{{\rho _{\rm{l}} v_{{\rm{sl}}}^2 }}{{\phi _{\rm{l}} }} + \dfrac{{\rho _{\rm{g}} v_{{\rm{sg}}}^2 }}{{\phi _{\rm{g}} }}} \right)} \right]_{i + 1}^{n + 1}$ (14)
 $K_3 = - \dfrac{{{\rm{g}}\Delta s}}{2}[(A\rho _{\rm{l}} )_i^{n + 1} + (A\rho _{\rm{l}} )_{i + 1}^{n + 1}]$ (15)
 $K_4 = - \dfrac{{\Delta s}}{2}\left\{ {\left[{A\left( {\dfrac{{\partial p}}{{\partial s}}} \right)} \right]_{{\rm{fr}}i}^{n + 1} + \left[{A\left( {\dfrac{{\partial p}}{{\partial s}}} \right)} \right]_{{\rm{fr}}i + 1}^{n + 1} } \right\}$ (16)

$v_{\rm{sg}}$——地层气相的表观速度，m/s；

${\phi _{\rm{l}}}$——持液率；

${\phi _{\rm{g}}}$——空隙率；

$\Delta s$——控制体长度，m；

$\Delta t$——微元时间，s；

${\rho _{\rm{g}}}$——气相密度，kg/m$^3$

${\rho _{\rm{l}}}$——液相密度，kg/m$^3$。

3 现场后效测试及应用

 图3 全烃、${{\rm{C}}_1}$及${{\rm{C}}_2}$监测数据变化趋势 Fig. 3 The monitoring data of total hydrocarbon, ${{\rm{C}}_1}$and ${{\rm{C}}_2}$

4 模型分析

4.1 井深对气体滑脱速度影响

 图4 环空气体滑脱速度 Fig. 4 Gas slippage velocity along annulus
4.2 油气层顶界位置对气窜速度影响

 图5 油气层顶界位置对气窜速度影响 Fig. 5 The influence of the reservoir top boundary on gas slippingupward velocity
4.3 静止时间对气窜速度影响

 图6 停泵钻井液静止时间对气窜速度影响 Fig. 6 The influence of static time on gas upward velocity in closing pump operations
4.4 迟到时间对气窜速度影响

 图7 迟到时间对气窜速度影响 Fig. 7 The influence of the lag time on gas upward velocity
5 结论

（1）由于油相、气相密度差异较大，油相上窜速度与气相上窜速度应分开研究。

（2）气体滑脱速度较大，对计算气窜速度影响较大，不可忽略。不考虑气体滑脱的影响，计算的气窜速度较考虑气体滑脱计算的气窜速度偏大。

（3）无论采用任何模型预测气窜速度，在钻井停泵作业中应预留一定的安全作业时间。以确保停泵过程中气体不能上窜至井口，避免钻井事故发生。

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