考虑表面扩散的实际状态页岩气表观渗透率新模型
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摘要
为了表征实际状态下的页岩气在纳米孔隙中的运移,考虑黏滞流、Knudsen扩散以及表面扩散机制,推导出页岩气表观渗透率模型。以224组试验数据为依据,优选所建模型中压缩因子及黏度的计算方法,并通过实际试验数据系统验证所建模型的准确性。通过定义黏滞流、Knudsen扩散以及表面扩散渗透率比重,计算各机制的贡献率。分析压力、孔隙半径等因素的影响。结果表明:所建模型能够准确预测页岩气的表观渗透率,误差为3.02%,优于现有模型;优选出的DK模型和Sutton模型计算压缩因子和黏度误差分别为0.53%和2.7%,压力和孔隙半径对表观渗透率影响最大;Langmuir最大吸附量、Langmuir压力以及等量吸附热主要影响表面扩散渗透率比重;在低压和高压条件下,各因素对表观渗透率及比重的影响表现出不同的变化规律;孔隙半径为1 nm、压力分别为0.1和20 MPa时,表面扩散对渗透率的贡献率达75%和30%,说明低孔隙半径或低压条件下表面扩散现象不可忽略。
In this paper,the shale gas was assumed as a real gas,and a new apparent permeability model for gas flowing in a single nanopore was derived,in which the viscous flow and Knudsen diffusion of free gas,and surface diffusion of adsorbed gas were considered. Models with high accuracy for calculating the compression factor and viscosity of the shale were selected based on 224 sets of experimental data,and the accuracy of the apparent permeability model was verified in comparison with the experimental data. The permeability ratios induced by the viscous flow,the Knudsen diffusion and the surface diffusion to the total apparent permeability were defined to measure the contributions of the three flow mechanisms to the total permeability. Moreover,the effects of temperature,pore radius,Langmuir adsorption capacity,Langmuir pressure and isosteric adsorption heat on the apparent permeability were analyzed along with the effects of the viscous flow,Knudsen diffusion and surface diffusion at low and high pressure conditions. The results indicate that the error of the proposed model is 3. 02%,which is less than the existing models. It is reasonable to use the DK model and the Sutton model to calculate the Z-factor and the viscosity of the shale gas,with average relative errors of 0. 53% and 2. 7%. Pressure,temperature and pore radius can have significant influences on the apparent permeability of the shale gas,while the Langmuir adsorption capacity,the Langmuir pressure and isosteric adsorption heat mainly affect the surface diffusion. The ratios of the permeability induced by the surface diffusion to the total apparent permeability are 75% and 30%,respectively,when the pore radius is 1 nm at pressure of 0. 1 MPa and 20 MPa. So the surface diffusion cannot be ignored within small pores or at low-pressure conditions.
In this paper,the shale gas was assumed as a real gas,and a new apparent permeability model for gas flowing in a single nanopore was derived,in which the viscous flow and Knudsen diffusion of free gas,and surface diffusion of adsorbed gas were considered. Models with high accuracy for calculating the compression factor and viscosity of the shale were selected based on 224 sets of experimental data,and the accuracy of the apparent permeability model was verified in comparison with the experimental data. The permeability ratios induced by the viscous flow,the Knudsen diffusion and the surface diffusion to the total apparent permeability were defined to measure the contributions of the three flow mechanisms to the total permeability. Moreover,the effects of temperature,pore radius,Langmuir adsorption capacity,Langmuir pressure and isosteric adsorption heat on the apparent permeability were analyzed along with the effects of the viscous flow,Knudsen diffusion and surface diffusion at low and high pressure conditions. The results indicate that the error of the proposed model is 3. 02%,which is less than the existing models. It is reasonable to use the DK model and the Sutton model to calculate the Z-factor and the viscosity of the shale gas,with average relative errors of 0. 53% and 2. 7%. Pressure,temperature and pore radius can have significant influences on the apparent permeability of the shale gas,while the Langmuir adsorption capacity,the Langmuir pressure and isosteric adsorption heat mainly affect the surface diffusion. The ratios of the permeability induced by the surface diffusion to the total apparent permeability are 75% and 30%,respectively,when the pore radius is 1 nm at pressure of 0. 1 MPa and 20 MPa. So the surface diffusion cannot be ignored within small pores or at low-pressure conditions.
引文
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[7]ISMAIL M I,HORNE R N.Modeling adsorption of gases in nanoscale pores using grand canonical monte carlo simulation techniques[R].SPE 170948,2014.
[8]BESKOK A,KARNIADAKIS G E.A model for flows in channels,pipes,and ducts at micro and nano scales[J].Microscale Thermophysical Engineering,1999,3:43-77.
[9]JAVADPOUR F,FISHER D,UNSWORTH M.Nanoscale gas flow in shale gas sediments[J].Journal of Canadian Petroleum Technology,2007,46(10):55-61.
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[11]KLINKENBERG L J.The permeability of porous media to liquids and gases[J].Socar Proceedings,1941,2(2):200-213.
[12]JONES F O,OWENS W W.A laboratory study of lowpermeability gas sands[J].Journal of Petroleum Technology,1980,32(9):1631-1640.
[13]FLORENCE F A,RUSHING J,NEWSHAM K E,et al.Improved permeability prediction relations for low permeability sands[R].SPE 107594,2007.
[14]ERTEKIN T,KING G R,SCHWERER F C.Dynamic gas slippage:a unique dual mechanism approach to the flow of gas in tight formations[R].SPE 12045,1986.
[15]MICHEL G G,SIGAL R F,CIVAN F,et al.Parametric investigation of shale gas production considering nano-scale pore size distribution,formation factor,and non-Darcy flow mechanisms[R].SPE 147438,2011.
[16]SAKHAEE-POUR A,BRYANT S L.Gas permeability of shale[R].SPE 146944,2012.
[17]CIVAN F.Effective correlation of apparent gas permeability in tight porous media[J].Transport in Porous Media,2009,82(2):375-384.
[18]秦积舜,李爱芬.油层物理学[M].东营:中国石油大学出版社,2006.
[19]DO D D,WANG K.A new model for the description of adsorption kinetics in heterogeneous activated carbon[J].Carbon,1998,36(10):1539-1554.
[20]HWANG S T,KAMMERMEYER K.Surface diffusion in microporous media[J].The Canadian Journal of Chemical Engineering,1966,44(2):82-89.
[21]MENGAL S A,WATTENBARGER R A.Accounting for adsorbed gas in shale gas reservoirs[R].SPE 141085,2011.
[22]熊伟,郭为,刘洪林,等.页岩的储层特征以及等温吸附特征[J].天然气工业,2012,32(1):113-116.XIONG Wei,GUO Wei,LIU Honglin,et al.Shale reservoir characteristics and isothermal adsorption properties[J].Natural Gas Industry,2012,32(1):113-116.
[23]张志英,杨盛波.页岩气吸附解吸规律研究[J].实验力学,2012,27(4):492-497.ZHANG Zhiying,YANG Shengbo.On the adsorption and desorption trend of shale gas[J].Journal of Experimental Mechanics,2012,27(4):492-497.
[24]SUZUKI M,FUJII T.Concentration dependence of surface diffusion coefficient of propionic acid in activated carbon particles[J].AICh E Journal,1982,28(3):380-385.
[25]HIGASHI K,ITO H,OISHI J.Surface diffusion phenomena in gaseous diffusion(I)[J].Journal of the Atomic Energy Society of Japan/Atomic Energy Society of Japan,1963,5(10):846-853.
[26]YANG R T,FENN J B,HALLER G L.Modification to the Higashi model for surface diffusion[J].AICh E Journal,1973,19(5):1052-1053.
[27]KAPOOR A,YANG R T.Surface diffusion on energetically heterogeneous surfaces[J].AICh E Journal,1989,35(10):1735-1738.
[28]CHEN Y D,YANG R T.Concentration dependence of surface diffusion and zeolitic diffusion[J].AICh E Journal,1991,37(10):1579-1582.
[29]DO D D.A model for surface diffusion of ethane and propane in activated carbon[J].Chemical Engineering Science,1996,51(17):4145-4158.
[30]郭亮,彭晓峰,吴占松.甲烷在成型纳米活性炭中的吸附动力学特性[J].化工学报,2008,59(11):2726-2732.GUO Liang,PENG Xiaofeng,WU Zhansong.Dynamical characteristics of methane adsorption on monolith nanometer activated carbon[J].Journal of Chemical Industry and Engineering,2008,59(11):2726-2732.
[31]KAST W,HOHENTHANNER C R.Mass transfer within the gas-phase of porous media[J].International Journal of Heat and Mass Transfer,2000,43(5):807-823.
[32]COOPER S M,CRUDEN B A,MEYYAPPAN M,et al.Gas transport characteristics through a carbon nanotubule[J].Nano Letters,2004,4(2):377-381.
[33]杨胜来,魏俊之.油层物理学[M].北京:石油工业出版社,2004.
[34]AHMED T.Reservoir engineering handbook[M].Houston:Gulf Publishing Company,2000.
[35]HEIDARYAN E,SALARABADI A,MOGHADASI J.A novel correlation approach for prediction of natural gas compressibility factor[J].Journal of Natural Gas Chemistry,2010,19(2):189-192.
[36]HEIDARYAN E,MOGHADASI J,RAHIMI M.New correlations to predict natural gas viscosity and compressibility factor[J].Journal of Petroleum Science and Engineering,2010,73(1/2):67-72.
[37]HEIDARYAN E,ESMAEILZADEH F,MOGHADASI J.Natural gas viscosity estimation through corresponding states based models[J].Fluid Phase Equilibria,2013,354:80-88.
[38]SANJARI E,LAY E N,PEYMANI M.An accurate empirical correlation for predicting natural gas viscosity[J].Journal of Natural Gas Chemistry,2011,20:654-658.
[39]JARRAHIAN A,HEIDARYAN E.A simple correlation to estimate natural gas viscosity[J].Journal of Natural Gas Science and Engineering,2014,20:50-57.
[40]HEIDARYAN E,MOGHADASI J,SALARABADI A.A new and reliable model for predicting methane viscosity at high pressures and high temperatures[J].Journal of Natural Gas Chemistry,2010,19(5):552-556.
[41]LONDONO F E,ARCHER R A,BLASINGAME T A.Correlations for hydrocarbon-gas viscosity and gas density—validation and correlation of behavior using a largescale database[R].SPE 75721,2005.
[42]HEIDARYAN E,JARRAHIAN A.Natural gas viscosity estimation using density based models[J].The Canadian Journal of Chemical Engineering,2013,91(6):1183-1189.
[43]CARR N L,KOBAYASHI R,BURROWS D B.Viscosity of hydrocarbon gases under pressure[J].Journal of Petroleum Technology,1954,6(10):47-55.
[44]卢焕章.石油化工基础数据手册[M].北京:化学工业出版社,1982.
[45]ROY S,RAJU R,CHUANG H F,et al.Modeling gas flow through microchannels and nanopores[J].Journal of Applied Physics,2003,93(8):4870-4879.
[2]周尚文,薛华庆,郭伟,等.川南龙马溪组页岩核磁渗透率新模型研究[J].中国石油大学学报(自然科学版),2016,40(1):56-61.ZHOU Shangwen,XUE Huaqing,GUO Wei,et al.A new nuclear magnetic resonance permeability model of shale of Longmaxi Formation in southern Sichuan Basin[J].Journal of China University of Petroleum(Edition of Natural Science),2016,40(1):56-61.
[3]SWAMI V,CLARKSON C R,SETTARI A.Non Darcy flow in shale nanopores:do we have a final answer?[R].SPE 162665,2012.
[4]张磊,姚军,孙海,等.利用格子Boltzmann方法计算页岩渗透率[J].中国石油大学学报(自然科学版),2014,38(1):87-91.ZHANG Lei,YAO Jun,SUN Hai,et al.Permeability calculation in shale using lattice Boltzmann method[J].Journal of China University of Petroleum(Edition of Natural Science),2014,38(1):87-91.
[5]孙海,姚军,张磊,等.基于孔隙结构的页岩渗透率计算方法[J].中国石油大学学报(自然科学版),2014,38(2):92-98.SUN Hai,YAO Jun,ZHANG Lei,et al.A computing method of shale permeability based on pore structures[J].Journal of China University of Petroleum(Edition of Natural Science),2014,38(2):92-98.
[6]TOKUMASU T,MATSUMOTO Y.Dynamic molecular collision(DMC)model for rarefied gas flow simulations by the DSMC method[J].Physics of Fluids,1999,11(7):1907-1920.
[7]ISMAIL M I,HORNE R N.Modeling adsorption of gases in nanoscale pores using grand canonical monte carlo simulation techniques[R].SPE 170948,2014.
[8]BESKOK A,KARNIADAKIS G E.A model for flows in channels,pipes,and ducts at micro and nano scales[J].Microscale Thermophysical Engineering,1999,3:43-77.
[9]JAVADPOUR F,FISHER D,UNSWORTH M.Nanoscale gas flow in shale gas sediments[J].Journal of Canadian Petroleum Technology,2007,46(10):55-61.
[10]JAVADPOUR F.Nanopores and apparent permeability of gas flow in mudrocks(shales and siltstone)[J].Journal of Canadian Petroleum Technology,2009,48(8):16-21.
[11]KLINKENBERG L J.The permeability of porous media to liquids and gases[J].Socar Proceedings,1941,2(2):200-213.
[12]JONES F O,OWENS W W.A laboratory study of lowpermeability gas sands[J].Journal of Petroleum Technology,1980,32(9):1631-1640.
[13]FLORENCE F A,RUSHING J,NEWSHAM K E,et al.Improved permeability prediction relations for low permeability sands[R].SPE 107594,2007.
[14]ERTEKIN T,KING G R,SCHWERER F C.Dynamic gas slippage:a unique dual mechanism approach to the flow of gas in tight formations[R].SPE 12045,1986.
[15]MICHEL G G,SIGAL R F,CIVAN F,et al.Parametric investigation of shale gas production considering nano-scale pore size distribution,formation factor,and non-Darcy flow mechanisms[R].SPE 147438,2011.
[16]SAKHAEE-POUR A,BRYANT S L.Gas permeability of shale[R].SPE 146944,2012.
[17]CIVAN F.Effective correlation of apparent gas permeability in tight porous media[J].Transport in Porous Media,2009,82(2):375-384.
[18]秦积舜,李爱芬.油层物理学[M].东营:中国石油大学出版社,2006.
[19]DO D D,WANG K.A new model for the description of adsorption kinetics in heterogeneous activated carbon[J].Carbon,1998,36(10):1539-1554.
[20]HWANG S T,KAMMERMEYER K.Surface diffusion in microporous media[J].The Canadian Journal of Chemical Engineering,1966,44(2):82-89.
[21]MENGAL S A,WATTENBARGER R A.Accounting for adsorbed gas in shale gas reservoirs[R].SPE 141085,2011.
[22]熊伟,郭为,刘洪林,等.页岩的储层特征以及等温吸附特征[J].天然气工业,2012,32(1):113-116.XIONG Wei,GUO Wei,LIU Honglin,et al.Shale reservoir characteristics and isothermal adsorption properties[J].Natural Gas Industry,2012,32(1):113-116.
[23]张志英,杨盛波.页岩气吸附解吸规律研究[J].实验力学,2012,27(4):492-497.ZHANG Zhiying,YANG Shengbo.On the adsorption and desorption trend of shale gas[J].Journal of Experimental Mechanics,2012,27(4):492-497.
[24]SUZUKI M,FUJII T.Concentration dependence of surface diffusion coefficient of propionic acid in activated carbon particles[J].AICh E Journal,1982,28(3):380-385.
[25]HIGASHI K,ITO H,OISHI J.Surface diffusion phenomena in gaseous diffusion(I)[J].Journal of the Atomic Energy Society of Japan/Atomic Energy Society of Japan,1963,5(10):846-853.
[26]YANG R T,FENN J B,HALLER G L.Modification to the Higashi model for surface diffusion[J].AICh E Journal,1973,19(5):1052-1053.
[27]KAPOOR A,YANG R T.Surface diffusion on energetically heterogeneous surfaces[J].AICh E Journal,1989,35(10):1735-1738.
[28]CHEN Y D,YANG R T.Concentration dependence of surface diffusion and zeolitic diffusion[J].AICh E Journal,1991,37(10):1579-1582.
[29]DO D D.A model for surface diffusion of ethane and propane in activated carbon[J].Chemical Engineering Science,1996,51(17):4145-4158.
[30]郭亮,彭晓峰,吴占松.甲烷在成型纳米活性炭中的吸附动力学特性[J].化工学报,2008,59(11):2726-2732.GUO Liang,PENG Xiaofeng,WU Zhansong.Dynamical characteristics of methane adsorption on monolith nanometer activated carbon[J].Journal of Chemical Industry and Engineering,2008,59(11):2726-2732.
[31]KAST W,HOHENTHANNER C R.Mass transfer within the gas-phase of porous media[J].International Journal of Heat and Mass Transfer,2000,43(5):807-823.
[32]COOPER S M,CRUDEN B A,MEYYAPPAN M,et al.Gas transport characteristics through a carbon nanotubule[J].Nano Letters,2004,4(2):377-381.
[33]杨胜来,魏俊之.油层物理学[M].北京:石油工业出版社,2004.
[34]AHMED T.Reservoir engineering handbook[M].Houston:Gulf Publishing Company,2000.
[35]HEIDARYAN E,SALARABADI A,MOGHADASI J.A novel correlation approach for prediction of natural gas compressibility factor[J].Journal of Natural Gas Chemistry,2010,19(2):189-192.
[36]HEIDARYAN E,MOGHADASI J,RAHIMI M.New correlations to predict natural gas viscosity and compressibility factor[J].Journal of Petroleum Science and Engineering,2010,73(1/2):67-72.
[37]HEIDARYAN E,ESMAEILZADEH F,MOGHADASI J.Natural gas viscosity estimation through corresponding states based models[J].Fluid Phase Equilibria,2013,354:80-88.
[38]SANJARI E,LAY E N,PEYMANI M.An accurate empirical correlation for predicting natural gas viscosity[J].Journal of Natural Gas Chemistry,2011,20:654-658.
[39]JARRAHIAN A,HEIDARYAN E.A simple correlation to estimate natural gas viscosity[J].Journal of Natural Gas Science and Engineering,2014,20:50-57.
[40]HEIDARYAN E,MOGHADASI J,SALARABADI A.A new and reliable model for predicting methane viscosity at high pressures and high temperatures[J].Journal of Natural Gas Chemistry,2010,19(5):552-556.
[41]LONDONO F E,ARCHER R A,BLASINGAME T A.Correlations for hydrocarbon-gas viscosity and gas density—validation and correlation of behavior using a largescale database[R].SPE 75721,2005.
[42]HEIDARYAN E,JARRAHIAN A.Natural gas viscosity estimation using density based models[J].The Canadian Journal of Chemical Engineering,2013,91(6):1183-1189.
[43]CARR N L,KOBAYASHI R,BURROWS D B.Viscosity of hydrocarbon gases under pressure[J].Journal of Petroleum Technology,1954,6(10):47-55.
[44]卢焕章.石油化工基础数据手册[M].北京:化学工业出版社,1982.
[45]ROY S,RAJU R,CHUANG H F,et al.Modeling gas flow through microchannels and nanopores[J].Journal of Applied Physics,2003,93(8):4870-4879.