在页岩气试井分析中Bessel函数溢出问题的解决方法
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摘要
Bessel函数在页岩气藏多段压裂水平井的试井解释理论中发挥着非常重要的作用,但变形Bessel函数I_n(x)和K_n(x)在自变量趋于无限小或无限大时,使用多项式逼近的方法求解往往会溢出,难以得到典型曲线致使试井解释困难。为此,从理论上剖析了当储集层渗透率非常低时在无因次时间低于10~(-6)时变形Bessel函数都会溢出的原因。结果表明:对于第一类Bessel函数I_n(x),当x>650,I_0(x)与I_1(x)部分计算机出现浮点数溢出,高于10~(215)。页岩气试井理论中根据无因次时间的定义,渗透率越小,压裂裂缝半长越大,测试时间越短,根据Laplace反演I_0与I_1的值非常容易溢出,难以绘制出早期典型曲线。在理论研究的基础上,通过对第一、二类变形Bessel函数的重新组合计算,将I_0与I_1计算过程中都乘以,保证了变形Bessel函数在计算过程中不会溢出。结论认为,大长度水平井或大型压裂井模型中,由于水平段长度或裂缝长度越大,无因次时间相应的就越短、Laplace变量就越大,必须用该方法处理Bessel函数的I_0与I_1,才能获得完整的典型曲线,否则计算就会溢出。
Bessel functions play an important role in well testing interpretation theory of multi-stage fracturing horizontal shale-gas wells. However, the solution derived from the polynomial approximation method tends to overflow when the independent variables of the modified Bessel functions I_n(x) and K_n(x) approach infinitely small or large values. As a result, it is difficult to obtain typical curves and interpret well testing. In this paper, the reasons were analyzed theoretically why all the modified Bessel functions are overflowing when reservoir permeability is very low and dimensionless time is less than 10~(–6). It is shown that as for Bessel function I_n(x), floating-point overflow occurs in I_0(x) and I_1(x) in the case of x>650, and it is higher than 10~(215). According to the definition of the dimensionless time described in the shale gas well testing theory, the lower the permeability and the longer the induced fracture half length, the shorter the testing time. According to Laplace inversion, I_0 and I_1 are likely to overflow, and it is difficult to plot the early typical curves. Based on theoretical studies, TypeⅠ and Ⅱ modified Bessel functions are recombined for calculation, and I_0 and I_1 are multiplied with in the process of calculation. Thus, no overflow emerges while the modified Bessel functions are calculated. It is concluded that in the model of large-length horizontal wells or large-scale fracturing wells, the longer the horizontal sections or fractures are, the shorter the corresponding dimensionless time is and the greater the Laplace variable is, so it is necessary to take advantage of this method to process Bessel functions I_0 and I_1 so as to obtain complete typical curves, otherwise calculation overflow will happen.
Bessel functions play an important role in well testing interpretation theory of multi-stage fracturing horizontal shale-gas wells. However, the solution derived from the polynomial approximation method tends to overflow when the independent variables of the modified Bessel functions I_n(x) and K_n(x) approach infinitely small or large values. As a result, it is difficult to obtain typical curves and interpret well testing. In this paper, the reasons were analyzed theoretically why all the modified Bessel functions are overflowing when reservoir permeability is very low and dimensionless time is less than 10~(–6). It is shown that as for Bessel function I_n(x), floating-point overflow occurs in I_0(x) and I_1(x) in the case of x>650, and it is higher than 10~(215). According to the definition of the dimensionless time described in the shale gas well testing theory, the lower the permeability and the longer the induced fracture half length, the shorter the testing time. According to Laplace inversion, I_0 and I_1 are likely to overflow, and it is difficult to plot the early typical curves. Based on theoretical studies, TypeⅠ and Ⅱ modified Bessel functions are recombined for calculation, and I_0 and I_1 are multiplied with in the process of calculation. Thus, no overflow emerges while the modified Bessel functions are calculated. It is concluded that in the model of large-length horizontal wells or large-scale fracturing wells, the longer the horizontal sections or fractures are, the shorter the corresponding dimensionless time is and the greater the Laplace variable is, so it is necessary to take advantage of this method to process Bessel functions I_0 and I_1 so as to obtain complete typical curves, otherwise calculation overflow will happen.
引文
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[8]朱维耀,亓倩,马千,邓佳,岳明,刘玉章.页岩气不稳定渗流压力传播规律和数学模型[J].石油勘探与开发,2016,43(2):261-267.Zhu Weiyao,Qi Qian,Ma Qian,Deng Jia,Yue Ming&Liu Yuzhang.Unstable seepage modeling and pressure propagation of shale gas reservoirs[J].Petroleum Exploration and Development,2016,43(2):261-267.
[9]赵金洲,许文俊,李勇明,蔡坤赤,徐苗.低渗透油气藏水平井分段多簇压裂簇间距优化新方法[J].天然气工业,2016,36(10):63-69.Zhao Jinzhou,Xu Wenjun,Li Yongming,Cai Kunchi&Xu Miao.A new method for cluster spacing optimization of multi-cluster staged fracturing in horizontal wells of low-permeability oil and gas reservoirs[J].Natural Gas Industry,2016,36(10):63-69.
[10]李勇明,陈曦宇,赵金洲,申峰,乔红军.水平井分段多簇压裂缝间干扰研究[J].西南石油大学学报(自然科学版),2016,38(1):76-83.Li Yongming,Chen Xiyu,Zhao Jinzhou,Shen Feng&Qiao Hongjun.The effects of crack interaction in multi-stage horizontal fracturing[J].Journal of Southwest Petroleum University(Science&Technology Edition),2016,38(1):76-83.
[11]郭建春,尹建,赵志红.裂缝干扰下页岩储层压裂形成复杂裂缝可行性[J].岩石力学与工程学报,2014,33(8):1589-1596.Guo Jianchun,Yin Jian&Zhao Zhihong.Feasibility of formation of complex fractures under cracks interference in shale reservoir fracturing[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(8):1589-1596.
[12]曾顺鹏,张国强,韩家新,袁彬,王彦鹏,冀政.多裂缝应力阴影效应模型及水平井分段压裂优化设计[J].天然气工业,2015,35(3):55-59.Zeng Shunpeng,Zhang Guoqiang,Han Jiaxin,Yuan Bin,Wang Yanpeng&Ji Zheng.Model of multi-fracture stress shadow effect and optimization design for staged fracturing of horizontal wells[J].Natural Gas Industry,2015,35(3):55-59.
[2]Fair PS&Simmons JF.Novel well testing applications of Laplace transform deconvolution[C]//SPE Annual Technical Conference and Exhibition,4-7 October 1992,Washington DC,USA.DOI:http://dx.doi.org/10.2118/24716-MS.
[3]Roumboutsos A&Stewart G.A direct deconvolution or convolution algorithm for well test analysis[C]//SPE Annual Technical Conference and Exhibition,2-5 October 1988,Houston,Texas,USA.DOI:http://dx.doi.org/10.2118/18157-MS.
[4]Daviau F,Mouronval G,Bourdarot G&Curutchet P.Pressure analysis for horizontal wells[J].SPE Formation Evaluation,1985,3(4):716-724.
[5]Goode PA&Thambynayagam RKM.Pressure drawdown and buildup analysis of horizontal wells in anisotropic media[J].SPE Formation Evaluation,1987,2(4):683-697.
[6]Bourgeois M&Horne RN.Well test model recognition using Laplace space type curves[J].SPE Formation Evaluation,1993,8(1):17-25.
[7]李笑萍,张大为.数理方法与试井数学模型[M].北京:石油工业出版社,1993:28-30.Li Xiaoping&Zhang Dawei.Mathematical methods and test model[M].Beijing:Petroleum Industry Press,1993:28-30.
[8]朱维耀,亓倩,马千,邓佳,岳明,刘玉章.页岩气不稳定渗流压力传播规律和数学模型[J].石油勘探与开发,2016,43(2):261-267.Zhu Weiyao,Qi Qian,Ma Qian,Deng Jia,Yue Ming&Liu Yuzhang.Unstable seepage modeling and pressure propagation of shale gas reservoirs[J].Petroleum Exploration and Development,2016,43(2):261-267.
[9]赵金洲,许文俊,李勇明,蔡坤赤,徐苗.低渗透油气藏水平井分段多簇压裂簇间距优化新方法[J].天然气工业,2016,36(10):63-69.Zhao Jinzhou,Xu Wenjun,Li Yongming,Cai Kunchi&Xu Miao.A new method for cluster spacing optimization of multi-cluster staged fracturing in horizontal wells of low-permeability oil and gas reservoirs[J].Natural Gas Industry,2016,36(10):63-69.
[10]李勇明,陈曦宇,赵金洲,申峰,乔红军.水平井分段多簇压裂缝间干扰研究[J].西南石油大学学报(自然科学版),2016,38(1):76-83.Li Yongming,Chen Xiyu,Zhao Jinzhou,Shen Feng&Qiao Hongjun.The effects of crack interaction in multi-stage horizontal fracturing[J].Journal of Southwest Petroleum University(Science&Technology Edition),2016,38(1):76-83.
[11]郭建春,尹建,赵志红.裂缝干扰下页岩储层压裂形成复杂裂缝可行性[J].岩石力学与工程学报,2014,33(8):1589-1596.Guo Jianchun,Yin Jian&Zhao Zhihong.Feasibility of formation of complex fractures under cracks interference in shale reservoir fracturing[J].Chinese Journal of Rock Mechanics and Engineering,2014,33(8):1589-1596.
[12]曾顺鹏,张国强,韩家新,袁彬,王彦鹏,冀政.多裂缝应力阴影效应模型及水平井分段压裂优化设计[J].天然气工业,2015,35(3):55-59.Zeng Shunpeng,Zhang Guoqiang,Han Jiaxin,Yuan Bin,Wang Yanpeng&Ji Zheng.Model of multi-fracture stress shadow effect and optimization design for staged fracturing of horizontal wells[J].Natural Gas Industry,2015,35(3):55-59.