A new model for predicting the critical liquid-carrying velocity in inclined gas wells

WANG Wujie,CUI Guomin,WEI Yaoqi,PAN Jie

Table 1 Calculation formulas of geometric parameters under different phase interfaces.

Geometric parameters Concave interface Convex interface Flat interface ${{\delta }_{\text{l}}}\left( \varphi \right)$ ${{\delta }_{\text{l}}}\left( \varphi \right)=R-\overline{OE}$ ${{\delta }_{\text{l}}}\left( \varphi \right)=R-\overline{OE}$ ${{\delta }_{\text{l}}}\left( \varphi \right)=R-\overline{OE}$ $\overline{OE}\text{ }\ \ (\varphi =0)$ $\overline{OE}\text{=}{{R}_{1}}-\overline{O{{O}_{1}}}\ \text{ }\ \ (\varphi =0)$ $\overline{OE}\text{=}-\left( {{R}_{1}}-\overline{O{{O}_{1}}} \right)\ \ \ \text{ }\ \ (\varphi =0)$ $\overline{OE}=R\cos {{\varphi }_{0\text{P}}}\ \ \text{ }\ \ (\varphi =0)$ $\overline{OE}\text{ }\ \ (\varphi \ne 0)$ $\overline{OE}=\frac{\sin \left( \varphi -\theta \right){{R}_{1}}}{\sin \varphi }\ \ \ \text{ }\ \ (\varphi \ne 0)$ $\overline{OE}=\frac{-\sin \left( \varphi -\theta \right){{R}_{1}}}{\sin \varphi }\ \ \ \text{ }\ \ (\varphi \ne 0)$ $\overline{OE}=\frac{R\cos {{\varphi }_{0\text{P}}}}{\cos \varphi }\ \ \ \ \text{ }\ \ (\varphi \ne 0)$ $\overline{O{{O}_{1}}}$ $\overline{O{{O}_{1}}}=-\frac{R\sin \left( {{\varphi }_{\text{PA}}}-{{\varphi }_{0}} \right)}{\sin {{\varphi }_{\text{PA}}}}$ $\overline{O{{O}_{1}}}=\frac{R\sin \left( {{\varphi }_{\text{PA}}}-{{\varphi }_{0}} \right)}{\sin {{\varphi }_{\text{PA}}}}$ ${{R}_{1}}$ ${{R}_{1}}=-\frac{R\sin {{\varphi }_{0}}}{\sin {{\varphi }_{\text{PA}}}}$ ${{R}_{1}}=\frac{R\sin {{\varphi }_{0}}}{\sin {{\varphi }_{\text{PA}}}}$ θ $\sin \theta =\frac{\overline{O{{O}_{1}}}\sin \varphi }{{{R}_{1}}}$ $\sin \theta =\frac{\overline{O{{O}_{1}}}\sin \varphi }{{{R}_{1}}}$ ε $\frac{1-\varepsilon }{\varepsilon }=\frac{\text{ }\!\!\pi\!\!\text{ }-{{\varphi }_{0}}+\frac{1}{2}\sin \left( 2{{\varphi }_{0}} \right)-{{\left( \frac{\sin {{\varphi }_{0}}}{\sin {{\varphi }_{\text{PA}}}} \right)}^{2}}\left[ \text{ }\!\!\pi\!\!\text{ }-{{\varphi }_{\text{PA}}}+\frac{1}{2}\sin \left( 2{{\varphi }_{\text{PA}}} \right) \right]}{{{\varphi }_{0}}-\frac{1}{2}\sin \left( 2{{\varphi }_{0}} \right)+{{\left( \frac{\sin {{\varphi }_{0}}}{\sin {{\varphi }_{\text{PA}}}} \right)}^{2}}\left[ \text{ }\!\!\pi\!\!\text{ }-{{\varphi }_{\text{PA}}}+\frac{1}{2}\sin \left( 2{{\varphi }_{\text{PA}}} \right) \right]}$ $\frac{1-\varepsilon }{\varepsilon }=\frac{\text{ }\!\!\pi\!\!\text{ }-{{\varphi }_{0\text{P}}}+\frac{1}{2}\sin \left( 2{{\varphi }_{0\text{P}}} \right)}{{{\varphi }_{0\text{P}}}-\frac{1}{2}\sin \left( 2{{\varphi }_{0\text{P}}} \right)}$