油气钻井岩屑沉降阻力计算模型
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摘要
油气钻井过程中,破碎的岩屑在井筒钻井液中存在着自由沉降的现象,为了防止和避免岩屑沉积造成沉砂卡钻等井下安全事故,需要研究岩屑颗粒的沉降规律、预测岩屑沉降的末速度。为此,基于Stokes定律和Newton-Rittinger模型,提出了黏性阻力占比系数与压差阻力占比系数的概念,应用最小二乘法对实验数据回归得到阻力占比系数方程,分别推导出岩屑颗粒在牛顿流体与幂律流体中沉降时非斯托克斯区域的阻力计算模型,并通过该模型依据沉积实验数据对岩屑颗粒的沉降末速度进行计算和分析。研究结果表明:(1)岩屑颗粒在幂律流体中沉降时,所受到的黏性阻力和压差阻力不仅与颗粒雷诺数相关,而且还与流性指数及稠度系数相关;(2)岩屑颗粒在牛顿流体中沉降,当颗粒雷诺数小于2.944 6时黏性阻力大于压差阻力,当颗粒雷诺数大于2.944 6时压差阻力大于黏性阻力;(3)颗粒雷诺数小于1.11时岩屑沉降主要考虑黏性阻力,颗粒雷诺数介于1.11~500时岩屑沉降受到黏性阻力与压差阻力的共同作用,颗粒雷诺数大于500时压差阻力在岩屑沉降中占主导作用。结论认为,借助于该计算模型,当钻井液为牛顿流体时,可以预测颗粒雷诺数介于0~105的岩屑沉降末速度;当钻井液为幂律流体时,可以预测颗粒雷诺数介于0~105、流性指数介于0.062 3~1的岩屑沉降末速度;上述范围能够满足钻井工程中对于岩屑沉降速度进行预测的需求。
In the process of oil and gas well drilling, the broken cuttings settle freely in the drilling fluid in the wellbore. To avoid downhole accidents caused by the cuttings settlement, therefore, it is necessary to study the cuttings settlement laws and predict the terminal cuttings settlement velocity. In this paper, a concept of resistance scale between viscosity and differential pressure was put forward based on the Stokes law and Newton-Rittinger model. Then, the equation of resistance scale was obtained by regressing the experiment data using the least square method. Finally, the resistance scale calculation model was derived for non-Stokes zone during the cuttings settlement in Newtonian fluids and power-law fluids. Furthermore, based on experimental data, the terminal cuttings settling velocity was calculated by using the new models. The following results were obtained. First, the viscosity and differential pressure on the cuttings during their settlement in power-law fluids are not only related to the particle Reynolds number, but to the flow behavior index and consistency index. Second, during the cuttings settlement in Newtonian fluids, the viscosity is higher than the differential pressure if the particle Reynolds number is less than 2.944 6. Third, the differential pressure is higher than the viscosity if the particle Reynolds number is higher than 2.944 6. Fourth, when the particle Reynolds number is lower than 1.11, the viscosity plays a dominant role in cuttings settlement; when the number is 1.11–500, cuttings settlement is under the joint effect of viscosity and differential pressure; when the number is higher than 500, the differential pressure is dominant. In conclusion, this calculation model can be used to predict the terminal settlement velocity of cuttings with a particle Reynolds number of 0-105 when the drilling fluid is a Newtonian fluid, and the terminal settlement velocity of cuttings with a particle Reynolds number of 0-105 and a flow behavior index of 0.062 3-1 when the drilling fluid is a power-law fluid. These mentioned ranges can satisfy the drilling engineering requirements on the prediction of terminal cuttings settlement velocity.
In the process of oil and gas well drilling, the broken cuttings settle freely in the drilling fluid in the wellbore. To avoid downhole accidents caused by the cuttings settlement, therefore, it is necessary to study the cuttings settlement laws and predict the terminal cuttings settlement velocity. In this paper, a concept of resistance scale between viscosity and differential pressure was put forward based on the Stokes law and Newton-Rittinger model. Then, the equation of resistance scale was obtained by regressing the experiment data using the least square method. Finally, the resistance scale calculation model was derived for non-Stokes zone during the cuttings settlement in Newtonian fluids and power-law fluids. Furthermore, based on experimental data, the terminal cuttings settling velocity was calculated by using the new models. The following results were obtained. First, the viscosity and differential pressure on the cuttings during their settlement in power-law fluids are not only related to the particle Reynolds number, but to the flow behavior index and consistency index. Second, during the cuttings settlement in Newtonian fluids, the viscosity is higher than the differential pressure if the particle Reynolds number is less than 2.944 6. Third, the differential pressure is higher than the viscosity if the particle Reynolds number is higher than 2.944 6. Fourth, when the particle Reynolds number is lower than 1.11, the viscosity plays a dominant role in cuttings settlement; when the number is 1.11–500, cuttings settlement is under the joint effect of viscosity and differential pressure; when the number is higher than 500, the differential pressure is dominant. In conclusion, this calculation model can be used to predict the terminal settlement velocity of cuttings with a particle Reynolds number of 0-105 when the drilling fluid is a Newtonian fluid, and the terminal settlement velocity of cuttings with a particle Reynolds number of 0-105 and a flow behavior index of 0.062 3-1 when the drilling fluid is a power-law fluid. These mentioned ranges can satisfy the drilling engineering requirements on the prediction of terminal cuttings settlement velocity.
引文
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[9]Cheng Niansheng.Simplified settling velocity formula for sediment particle[J].Journal of Hydraulic Engineering,1997,123(2):149-152.
[10]Guo Junke.Logarithmic matching and its applications in computational hydraulics and sediment transport[J].Journal of Hydraulic Research,2002,40(5):555-565.
[11]Shah SN,EI Fadili Y&Chhabra RP.New model for single spherical particle settling velocity in power law(visco-inelastic)fluids[J].International Journal of Multiphase Flow,2007,33(1):51-66.
[12]Camenen B.Simple and general formula for the settling velocity of particles[J].Journal of Hydraulic Engineering,2007,133(2):229-233.
[13]Kao DTY&Hwang ALY.Determination of particle settling velocity in Heterogeneous suspensions and its effects on energy loss prediction in solid-liquid freight pipelines[J].Journal of Powder and Bulk Solids Technology,1980,4(1):31-40.
[14]桂夏辉,李延锋,刘炯天,王永田,曹亦俊.液固流化床内颗粒沉降特性试验研究[J].煤炭学报,2010,35(8):1374-1379.Gui Xiahui,Li Yanfeng,Liu Jiongtian,Wang Yongtian&Cao Yijun.Study on settlement characteristic of the grain in fluidized bed[J].Journal of China Coal Society,2010,35(8):1374-1379.
[15]Chhabra RP.Steady non-Newtonian flow about a rigid sphere[J].Encyclopedia of Fluid Mechanics,1986,1:983-1033.
[16]Chhabra RP.Motion of spheres in power law(viscoinelastic)fluids at intermediate Reynolds numbers,a unified approach[J].Chemical Engineering and Processing:Process Intensification,1990,28(2):89-94.
[17]Dallon DS.A drag-coefficient correlation for spheres settling in Ellis fluids[D].Salt Lake City,UT:University of Utah,1967.
[18]Lali AM,Khare AS,Joshi JB&Nigam KDP.Behaviour of solid particles in viscous non-Newtonian solutions:Settling velocity,wall effects and bed expansion in solid–liquid fluidized beds[J].Powder Technology,1989,57(1):39-50.
[19]Prakash S.Experimental evaluation of terminal velocity in non-Newtonian fluids in the turbulent region[J].Indian Chemical Engineer,1983,25:1-4.
[20]Shah SN.Proppant settling correlations for non-Newtonian fluids under static and dynamic conditions[J].Society of Petroleum Engineers Journal,1982,22(2):164-170.
[21]Shah SN.Proppant settling correlations for non-Newtonian fluids[J].SPE Production Engineering,1986,1(6):446-448.
[22]Shahi S&Kuru E.An experimental investigation of settling velocity of natural sands in water using particle image shadowgraph[J].Powder Technology,2015,281:184-192.
[23]Shahi S&Kuru E.Experimental investigation of the settling velocity of spherical particles in power-law fluids using particle image shadowgraph technique[J].International Journal of Mineral Processing,2016,153:60-65.
[24]Terfous A,Hazzab A&Ghenaim A.Predicting the drag coefficient and settling velocity of spherical particles[J].Powder Technology,2013,239:12-20.
[25]Elgaddafi R,Ahmed R,George M&Growcock F.Settling behavior of spherical particles in fiber-containing drilling fluids[J].Journal of Petroleum Science and Engineering,2012,84-85:20-28.
[26]Elgaddafi R,Ahmed R&Growcock F.Settling behavior of particles in fiber-containing Herschel Bulkley fluid[J].Powder Technology,2016,301:782-793.
[27]Bro?ek M&Surowiak A.Argument of separation at upgrading in the jig[J].Archives of Mining Sciences,2010,55(1):21-40.
[28]Surowiak A&Bro?ek M.Methodology of calculation the terminal settling velocity distribution of spherical particles for high values of the Reynold's number[J].Archives of Mining Sciences,2014,59(1):269-282.
[29]Kelessidis VC.Terminal velocity of solid spheres falling in Newtonian and non-Newtonian liquids[J].Tech.Chron.Sci.,2003,24(1/2):43-54.
[30]Kelessidis VC&Mpandelis G.Measurements and prediction of terminal velocity of solid spheres falling through stagnant pseudoplastic liquids[J].Powder Technology,2004,147(1/3):117-125.
[31]Ford JT,Oyeneyin MB,Gao EH,Williamson RS&Peel LC.The formulation of milling fluids for efficient hole cleaning:An experimental investigation[C]//European Petroleum Conference,25-27 October 1994,London,UK.DOI:https://dx.doi.org/10.2118/28819-MS.
[32]Miura H,Takahashi T,Ichikawa J&Kawase Y.Bed expansion in liquid–solid two-phase fluidized beds with Newtonian and non-Newtonian fluids over the wide range of Reynolds numbers[J].Powder Technology,2001,117(3):239-246.
[33]Reynolds PA&Jones TER.An experimental study of the settling velocities of single particles in non-Newtonian fluids[J].International Journal of Mineral Processing,1989,25(1/2):47-77.
[2]Stokes GG.On the effect of the internal friction of fluids on the motion of pendulums[J].Transactions of the Cambridge Philosophical Society,1851,9:8-106.
[3]Rittinger PR.Lehrbuch der aufbereitungskunde[M].Berlin:Ernst&Kern,1867.
[4]孙玉波.重力选矿[M].北京:冶金工业出版社,1982.Sun Yubo.Gravity beneficiation[M].Beijing:Metallurgical Industry Press,1982.
[5]姚书典.重选原理[M].北京:冶金工业出版社,1992.Yao Shudian.Gravity selection principle[M].Beijing:Metallurgical Industry Press,1992.
[6]谢广元.选矿学[M].徐州:中国矿业大学出版社,2001.Xie Guangyuan.Mineral processing[M].Xuzhou:China University of Mining and Technology Press,2001.
[7]钱宁,万兆惠.泥沙运动力学[M].北京:科学出版社,1983.Qian Ning&Wan Zhaohui.Mechanics of sediment transport[M].Beijing:Science Press,1983.
[8]Peden JM&Luo Yuejin.Settling velocity of variously shaped particles in drilling and fracturing fluids[J].SPE Drilling Engineering,1987,2(4):337-343.
[9]Cheng Niansheng.Simplified settling velocity formula for sediment particle[J].Journal of Hydraulic Engineering,1997,123(2):149-152.
[10]Guo Junke.Logarithmic matching and its applications in computational hydraulics and sediment transport[J].Journal of Hydraulic Research,2002,40(5):555-565.
[11]Shah SN,EI Fadili Y&Chhabra RP.New model for single spherical particle settling velocity in power law(visco-inelastic)fluids[J].International Journal of Multiphase Flow,2007,33(1):51-66.
[12]Camenen B.Simple and general formula for the settling velocity of particles[J].Journal of Hydraulic Engineering,2007,133(2):229-233.
[13]Kao DTY&Hwang ALY.Determination of particle settling velocity in Heterogeneous suspensions and its effects on energy loss prediction in solid-liquid freight pipelines[J].Journal of Powder and Bulk Solids Technology,1980,4(1):31-40.
[14]桂夏辉,李延锋,刘炯天,王永田,曹亦俊.液固流化床内颗粒沉降特性试验研究[J].煤炭学报,2010,35(8):1374-1379.Gui Xiahui,Li Yanfeng,Liu Jiongtian,Wang Yongtian&Cao Yijun.Study on settlement characteristic of the grain in fluidized bed[J].Journal of China Coal Society,2010,35(8):1374-1379.
[15]Chhabra RP.Steady non-Newtonian flow about a rigid sphere[J].Encyclopedia of Fluid Mechanics,1986,1:983-1033.
[16]Chhabra RP.Motion of spheres in power law(viscoinelastic)fluids at intermediate Reynolds numbers,a unified approach[J].Chemical Engineering and Processing:Process Intensification,1990,28(2):89-94.
[17]Dallon DS.A drag-coefficient correlation for spheres settling in Ellis fluids[D].Salt Lake City,UT:University of Utah,1967.
[18]Lali AM,Khare AS,Joshi JB&Nigam KDP.Behaviour of solid particles in viscous non-Newtonian solutions:Settling velocity,wall effects and bed expansion in solid–liquid fluidized beds[J].Powder Technology,1989,57(1):39-50.
[19]Prakash S.Experimental evaluation of terminal velocity in non-Newtonian fluids in the turbulent region[J].Indian Chemical Engineer,1983,25:1-4.
[20]Shah SN.Proppant settling correlations for non-Newtonian fluids under static and dynamic conditions[J].Society of Petroleum Engineers Journal,1982,22(2):164-170.
[21]Shah SN.Proppant settling correlations for non-Newtonian fluids[J].SPE Production Engineering,1986,1(6):446-448.
[22]Shahi S&Kuru E.An experimental investigation of settling velocity of natural sands in water using particle image shadowgraph[J].Powder Technology,2015,281:184-192.
[23]Shahi S&Kuru E.Experimental investigation of the settling velocity of spherical particles in power-law fluids using particle image shadowgraph technique[J].International Journal of Mineral Processing,2016,153:60-65.
[24]Terfous A,Hazzab A&Ghenaim A.Predicting the drag coefficient and settling velocity of spherical particles[J].Powder Technology,2013,239:12-20.
[25]Elgaddafi R,Ahmed R,George M&Growcock F.Settling behavior of spherical particles in fiber-containing drilling fluids[J].Journal of Petroleum Science and Engineering,2012,84-85:20-28.
[26]Elgaddafi R,Ahmed R&Growcock F.Settling behavior of particles in fiber-containing Herschel Bulkley fluid[J].Powder Technology,2016,301:782-793.
[27]Bro?ek M&Surowiak A.Argument of separation at upgrading in the jig[J].Archives of Mining Sciences,2010,55(1):21-40.
[28]Surowiak A&Bro?ek M.Methodology of calculation the terminal settling velocity distribution of spherical particles for high values of the Reynold's number[J].Archives of Mining Sciences,2014,59(1):269-282.
[29]Kelessidis VC.Terminal velocity of solid spheres falling in Newtonian and non-Newtonian liquids[J].Tech.Chron.Sci.,2003,24(1/2):43-54.
[30]Kelessidis VC&Mpandelis G.Measurements and prediction of terminal velocity of solid spheres falling through stagnant pseudoplastic liquids[J].Powder Technology,2004,147(1/3):117-125.
[31]Ford JT,Oyeneyin MB,Gao EH,Williamson RS&Peel LC.The formulation of milling fluids for efficient hole cleaning:An experimental investigation[C]//European Petroleum Conference,25-27 October 1994,London,UK.DOI:https://dx.doi.org/10.2118/28819-MS.
[32]Miura H,Takahashi T,Ichikawa J&Kawase Y.Bed expansion in liquid–solid two-phase fluidized beds with Newtonian and non-Newtonian fluids over the wide range of Reynolds numbers[J].Powder Technology,2001,117(3):239-246.
[33]Reynolds PA&Jones TER.An experimental study of the settling velocities of single particles in non-Newtonian fluids[J].International Journal of Mineral Processing,1989,25(1/2):47-77.