页岩气试井分析中Bessel函数溢出问题的解决方法

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In(x)和Kn(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In(x),当x＞650,I0(x)与I1(x)部分计算机出现浮点数溢出,高于10215.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I0与I1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I0与I1计算过程中都乘以e-(redu),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I0与I1,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 In(x) and Kn(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 In(x),floating-point overflow occurs in I0(x) and I1(x) in the case of x＞650,and it is higher than 10215.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,I0 and I1 are likely to overflow,and it is difficult to plot the early typical curves.Based on theoretical studies,Type Ⅰ and Ⅱ modified Bessel functions are recombined for calculation,and I0 and I1 are multiplied with e-(redu) 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 I0 and I1 so as to obtain complete typical curves,otherwise calculation overflow will happen.