改进叠加原理求解低渗储层关井阶段压力响应
|
摘要
流体在低渗储层中的流动受流固界面作用影响,是低速非达西渗流。拟启动压力梯度常用于描述非达西渗流,但其在关井阶段的作用机制尚不明确。直接应用传统压降叠加原理的关井阶段压力响应计算结果已被证明和数值计算所得规律相反。针对上述矛盾,基于拟启动压力梯度,建立了低渗储层关井阶段试井数学模型。该模型是非齐次的二阶线性微分方程。根据其解的可叠加性,证明和提出了改进的压降叠加原理,即低渗储层关井阶段井底压力变化等于一个考虑拟启动压力梯度时变化值与一个不考虑拟启动压力梯度时变化值的叠加。计算结果显示,开井过程拟启动压力梯度使压力-压力导数曲线后期上翘,作用机制类似于"封闭"边界;关井过程拟启动压力梯度使压力-压力导数曲线后期下掉,作用机制类似于"恒压"边界。计算得到的曲线变化趋势与文献中数值计算结果吻合,验证了改进压降叠加原理的正确性。改进的压降叠加原理解决了传统压降叠加原理在非达西渗流压力恢复测试解释的错误,使得通过压力恢复测试准确识别储层平均拟启动压力梯度成为可能。
The fluids flow in the low permeability reservoir as the low velocity non-Darcy flow, which has a nonlinear section in the small pressure gradient range. The corresponding flow equations are nonhomogeneous or non-linear, leading to the traditional pressure superposition principle is improper anymore. Specifically, its change law of threshold pressure gradient(TPG) on build-up well test typical curves is contrast to the results from numerical method. By introducing pseudo TPG, the established well test mathematical model is a linear system. According to its additivity, we propose a new pressure superposition principle and prove it mathematically. The new principle shows that there are two pressure drop resulting in the total change of bottom hole pressure(BHP) in shut-in period, which are pressure drop in a real well with pseudo TPG and that in a virtual well without pseudo TPG, respectively. Based on the characteristic curves, we also observe that TPG makes the curves value to increase in production or injection process, and to decrease in buildup process, which is same with change law from numerical calculation in literature. In production or injection process, it is resulted from a closed moving-boundary. In buildup process, it is resulted from a supply moving-boundary. The new pressure superposition principle improve the accuracy of pressure interpretation in buildup process, giving exact TPG values to evaluate the flow state.
The fluids flow in the low permeability reservoir as the low velocity non-Darcy flow, which has a nonlinear section in the small pressure gradient range. The corresponding flow equations are nonhomogeneous or non-linear, leading to the traditional pressure superposition principle is improper anymore. Specifically, its change law of threshold pressure gradient(TPG) on build-up well test typical curves is contrast to the results from numerical method. By introducing pseudo TPG, the established well test mathematical model is a linear system. According to its additivity, we propose a new pressure superposition principle and prove it mathematically. The new principle shows that there are two pressure drop resulting in the total change of bottom hole pressure(BHP) in shut-in period, which are pressure drop in a real well with pseudo TPG and that in a virtual well without pseudo TPG, respectively. Based on the characteristic curves, we also observe that TPG makes the curves value to increase in production or injection process, and to decrease in buildup process, which is same with change law from numerical calculation in literature. In production or injection process, it is resulted from a closed moving-boundary. In buildup process, it is resulted from a supply moving-boundary. The new pressure superposition principle improve the accuracy of pressure interpretation in buildup process, giving exact TPG values to evaluate the flow state.
引文
[1]TALEGHANI A D,OLSON J E.How natural fractures could affect hydraulic-fracture geometry[J].SPEJournal 2014,19(1):161-171.
[2]LUO P,LUO W G,LI S.Effectiveness of miscible and immiscible gas flooding in recovering tight oil from Bakken reservoirs in Saskatchewan,Canada[J].Fuel,2017,208:626-636.
[3]HE J M,LIN C,LI X,ZHANG Y X,CHEN Y.Initiation,propagation,closure and morphology of hydraulic fractures in sandstone cores[J].Fuel,2017,208:65-70.
[4]HUANG W B,LU S F,HERSI O S,WANG M,DENGS W,LU R J.Reservoir spaces in tight sandstones:Classification,fractal characters,and heterogeneity[J].Journal of Natural Gas Science&Engineering,2017,46:80-92.
[5]SIRIPATRACHAI N,ERTEKIN T,JOHNS R T.Compositional simulation of hydraulically fractured tight formation considering the effect of capillary pressure on phase behavior[J].SPE Journal,2017,22(4):1046-1063.
[6]WANG S J,HUANG Y Z,CIVAN F.Experimental and theoretical investigation of the Zaoyuan field heavy oil flow through porous media[J].Journal of Petroleum Science&Engineering,2006,50(2):83-101.
[7]MILLER R J,LOW P F.Threshold gradient for water flow in clay systems[J].Soil Science Society of America Journal,1963,27(6):605-609.
[8]PASCAL H.Nonsteady flow through porous media in the presence of a threshold gradient[J].Acta Mechanica1981,39(3-4):207-224.
[9]KUTI?EK M.Non-darcian flow of water in soils-laminar region:A review[J].Developments in Soil Science,1972,2:327-340.
[10]蔡明金,贾永禄,王永恒,白宇,聂仁仕.低渗透双重介质油藏垂直裂缝井压力动态分析[J].石油学报,2008,29(5):723-726.CAI Mingjin,JIA Yonglu,WANG Yongheng,BAI Yu,NIE Shiren.Dynamic pressure analysis on wells with vertical fractures in low-permeability dual-porosity reservoir[J].Acta Petrolei Sinica,2008,29(5):723-726.
[11]GUO J J,ZHANG S,ZHANG L H,QING H R,LIUQ G.Well testing analysis for horizontal well with consideration of threshold pressure gradient in tight gas reservoirs[J].Journal of Hydrodynamics,2012,24(4):561-568.
[12]李道伦,杨景海,查文舒,王磊,卢德唐.叠加原理不能求解含启动压力梯度渗流方程[J].西南石油大学学报(自然科学版),2015,37(4):81-89.LI Daolun,YANG Jinghai,ZHA Wenshu,WANGLei,LU Detang.Unsuitability of using superposition principle to solve equations incorporated with threshold pressure gradient[J].Journal of Southwest Petroleum University(Naturnal Science Edition),2015,37(4):81-89.
[13]DIWU P X,LIU T J,YOU Z J,JIANG B Y,ZHOUJ.Effect of low velocity non-Darcy flow on pressure response in shale and tight oil reservoirs[J].Fuel,2018,216,398-406.
[14]李爱芬,刘敏,张少辉,姚军.特低渗透油藏渗流特征实验研究[J].西安石油大学学报(自然科学版),2008,23(2):35-39.LI Aifen,LIU Min,ZHANG Shaohui,YAO Jun.Experimental study on the percolation characteristic of extra low-permeability reservoir[J].Journal Xi’an Shiyou University(Naturnal Science Edition),2008,23(2):35-39.
[15]时宇,杨正明,黄延章,姚军.低渗透储层非线性渗流模型研究[J].石油学报,2009,30(5):731-734.SHI Yu,YANG Zhengmin,HUANG Yanzhang,YAOJun.Study on non-linear seepage flow model for lowpermeability reservoir[J].Acta Petrolei Sinica,2009,30(5):731-734.
[16]LI D L,ZHA W S,LIU S F,WANG L,LU D T.Pressure transient analysis of low permeability reservoir with pseudo threshold pressure gradient[J].Journal of Petroleum Science&Engineering,2016,147:308-316.
[17]PRADA A,CIVAN F.Modification of Darcy’s law for the threshold pressure gradient[J].Journal of Petroleum Science&Engineering,1999,22(4):237-240.
[18]王晓冬,郝明强,韩永新.启动压力梯度的含义与应用[J].石油学报,2013,34(1):188-191.WANG Xiaodong,HAO Mingqiang,HAN Yongxin.Implication of the threshold pressure gradient and its application[J].Acta Petrolei Sinica,2013,34(1):188-191.
[19]STEHFEST H.Algorithm 368:Numerical inversion of Laplace transforms[D5][J].Communications of the Acm,1970,13(1):47-49.
[20]DIWU Pengxiang,LIU Tongjing,JIANG Baoyi,WANGRui,YANG Peidie,YANG Jiping,WANG Zhaoming.Well test mathematical model for fractures network in tight oil reservoirs[J].IOP Conference Series:Earth and Environmental Science,2018,121(5):1-7.
[21]VOLLER V,CROSS M.Accurate solutions of moving boundary problems using the enthalpy method[J].International Journal of Heat&Mass Transfer,1981,24(3):545-556.
[22]SCHNEIDERS L,HARTMANN D,MEINKE M,S C H R?D E R W.Anaccuratemoving boundary formulation in cut-cell methods[J].Journal of Computational Physics,2013,235:786-809.
[23]廖新维,沈平平.现代试井分析[M].北京:石油工业出版社,2002.LIAO Xinwei,SHEN Pingping.Modern well test analysis[M].Beijing:Petroleum Industry Press,2002.
[2]LUO P,LUO W G,LI S.Effectiveness of miscible and immiscible gas flooding in recovering tight oil from Bakken reservoirs in Saskatchewan,Canada[J].Fuel,2017,208:626-636.
[3]HE J M,LIN C,LI X,ZHANG Y X,CHEN Y.Initiation,propagation,closure and morphology of hydraulic fractures in sandstone cores[J].Fuel,2017,208:65-70.
[4]HUANG W B,LU S F,HERSI O S,WANG M,DENGS W,LU R J.Reservoir spaces in tight sandstones:Classification,fractal characters,and heterogeneity[J].Journal of Natural Gas Science&Engineering,2017,46:80-92.
[5]SIRIPATRACHAI N,ERTEKIN T,JOHNS R T.Compositional simulation of hydraulically fractured tight formation considering the effect of capillary pressure on phase behavior[J].SPE Journal,2017,22(4):1046-1063.
[6]WANG S J,HUANG Y Z,CIVAN F.Experimental and theoretical investigation of the Zaoyuan field heavy oil flow through porous media[J].Journal of Petroleum Science&Engineering,2006,50(2):83-101.
[7]MILLER R J,LOW P F.Threshold gradient for water flow in clay systems[J].Soil Science Society of America Journal,1963,27(6):605-609.
[8]PASCAL H.Nonsteady flow through porous media in the presence of a threshold gradient[J].Acta Mechanica1981,39(3-4):207-224.
[9]KUTI?EK M.Non-darcian flow of water in soils-laminar region:A review[J].Developments in Soil Science,1972,2:327-340.
[10]蔡明金,贾永禄,王永恒,白宇,聂仁仕.低渗透双重介质油藏垂直裂缝井压力动态分析[J].石油学报,2008,29(5):723-726.CAI Mingjin,JIA Yonglu,WANG Yongheng,BAI Yu,NIE Shiren.Dynamic pressure analysis on wells with vertical fractures in low-permeability dual-porosity reservoir[J].Acta Petrolei Sinica,2008,29(5):723-726.
[11]GUO J J,ZHANG S,ZHANG L H,QING H R,LIUQ G.Well testing analysis for horizontal well with consideration of threshold pressure gradient in tight gas reservoirs[J].Journal of Hydrodynamics,2012,24(4):561-568.
[12]李道伦,杨景海,查文舒,王磊,卢德唐.叠加原理不能求解含启动压力梯度渗流方程[J].西南石油大学学报(自然科学版),2015,37(4):81-89.LI Daolun,YANG Jinghai,ZHA Wenshu,WANGLei,LU Detang.Unsuitability of using superposition principle to solve equations incorporated with threshold pressure gradient[J].Journal of Southwest Petroleum University(Naturnal Science Edition),2015,37(4):81-89.
[13]DIWU P X,LIU T J,YOU Z J,JIANG B Y,ZHOUJ.Effect of low velocity non-Darcy flow on pressure response in shale and tight oil reservoirs[J].Fuel,2018,216,398-406.
[14]李爱芬,刘敏,张少辉,姚军.特低渗透油藏渗流特征实验研究[J].西安石油大学学报(自然科学版),2008,23(2):35-39.LI Aifen,LIU Min,ZHANG Shaohui,YAO Jun.Experimental study on the percolation characteristic of extra low-permeability reservoir[J].Journal Xi’an Shiyou University(Naturnal Science Edition),2008,23(2):35-39.
[15]时宇,杨正明,黄延章,姚军.低渗透储层非线性渗流模型研究[J].石油学报,2009,30(5):731-734.SHI Yu,YANG Zhengmin,HUANG Yanzhang,YAOJun.Study on non-linear seepage flow model for lowpermeability reservoir[J].Acta Petrolei Sinica,2009,30(5):731-734.
[16]LI D L,ZHA W S,LIU S F,WANG L,LU D T.Pressure transient analysis of low permeability reservoir with pseudo threshold pressure gradient[J].Journal of Petroleum Science&Engineering,2016,147:308-316.
[17]PRADA A,CIVAN F.Modification of Darcy’s law for the threshold pressure gradient[J].Journal of Petroleum Science&Engineering,1999,22(4):237-240.
[18]王晓冬,郝明强,韩永新.启动压力梯度的含义与应用[J].石油学报,2013,34(1):188-191.WANG Xiaodong,HAO Mingqiang,HAN Yongxin.Implication of the threshold pressure gradient and its application[J].Acta Petrolei Sinica,2013,34(1):188-191.
[19]STEHFEST H.Algorithm 368:Numerical inversion of Laplace transforms[D5][J].Communications of the Acm,1970,13(1):47-49.
[20]DIWU Pengxiang,LIU Tongjing,JIANG Baoyi,WANGRui,YANG Peidie,YANG Jiping,WANG Zhaoming.Well test mathematical model for fractures network in tight oil reservoirs[J].IOP Conference Series:Earth and Environmental Science,2018,121(5):1-7.
[21]VOLLER V,CROSS M.Accurate solutions of moving boundary problems using the enthalpy method[J].International Journal of Heat&Mass Transfer,1981,24(3):545-556.
[22]SCHNEIDERS L,HARTMANN D,MEINKE M,S C H R?D E R W.Anaccuratemoving boundary formulation in cut-cell methods[J].Journal of Computational Physics,2013,235:786-809.
[23]廖新维,沈平平.现代试井分析[M].北京:石油工业出版社,2002.LIAO Xinwei,SHEN Pingping.Modern well test analysis[M].Beijing:Petroleum Industry Press,2002.