焦石坝地区页岩Cole-Cole模型参数的快速计算方法

doi:10.13809/j.cnki.cn32-1825/te.2020.05.006

摘要/Abstract

摘要：

利用不同频率的电源下测量的岩心复电阻率,计算Cole-Cole模型中的4个模型参数（ρ₀、m、τ和c）,是岩心复电阻率频谱实验中重要的非线性寻优问题。然而,在焦石坝地区,参数τ很小,参数c变化范围大,使得最小二乘法寻优程序难以收敛。针对该问题,对寻优程序的初始值预估算法和最优迭代步长进行了研究。改进后的最小二乘法寻优程序,收敛速度明显加快,并且很好地避免了原有程序中参数τ值超出定义域而产生的计算错误的问题。研究表明,根据Cole-Cole模型函数的极值性质和渐近性质,可以直接求解参数τ,并且可以得出其余3个模型参数很好的迭代初始值。此外,将最小二乘法误差函数的梯度计算公式展开为泰勒级数,可以获得最优迭代步长,大幅提高算法收敛速度。

关键词: 标准Cole-Cole模型, 最小二乘法, 梯度下降算法, 迭代初始值预估, 最优步长, 龙马溪组页岩

Abstract:

Calculation of 4 parameters, ρ₀, m, τ and c, of Cole-Cole model by complex rock core resistivity data measured with multi-frequency is an important nonlinear optimization problem in rock core complex resistivity experiments. However, in Jiaoshiba Zone, a very small τ and c with big value range cause the difficulty of the convergence of optimization program. Therefore, the initial value determination and optimized iteration step have been studied. The improved program performances well on the speed of convergence, and avoids the calculation failure caused by out-range of the parameter τ. The study indicates that according to the maximum property and asymmetry property, the parameter τ can be obtained directly, and good iteration initial values of the rest 3 parameters can be obtained either. Meanwhile, by transforming the error function to be Tyler series, the optimized iteration step can be obtained and the convergence speed can be upgraded significantly.

Key words: Cole-Cole model, least square method, gradient descent algorithm, iteration initial value estimation, optimized iteration step, shale in Longmaxi Formation

中图分类号:

P315.69

刘智颖,张柏桥,许巍等. 焦石坝地区页岩Cole-Cole模型参数的快速计算方法[J]. 油气藏评价与开发, 2020, 10(5): 42-48.

Zhiying LIU,Baiqiao ZHANG,Wei XU, et al. A fast computing method for rock electric parameters of Cole-Cole model in Jiaoshiba Zone[J]. Reservoir Evaluation and Development, 2020, 10(5): 42-48.

图/表 9

图1

表1

表2

程序1的参数及计算结果"

样号	总迭代次数	相对误差/%	初始值	计算结果
2	不收敛
18	不收敛
23	19 394	9.07	c^（0）=0.5 τ^（0）=1.0×10^-6 m^（0）=0.9 $ρ 00$ =4 000	c=0.796 784 τ=3.213 45×10^-6 m=0.999 83 ρ₀=4 003.354 93

表2

图2

表3

程序1的运行过程"

样号	迭代次数	相对误差	当前迭代值
样号	迭代次数	相对误差	$ρ 0 k m k c k τ k$
23	3 000	593.4 %	4 314.495 3	0.897 643	0.698 743	1.029 5×10^-6
	8 000	164.72 %	4 394.180 4	0.958 395	0.768 959	3.019 4×10^-6
	15 000	43.61 %	4 009.352 6	0.999 875	0.719 845	3.579 9×10^-6

表3

表4

程序2的参数及计算结果"

样号	总迭代次数	相对误差/%	初始值	计算结果
2	不收敛
18	不收敛
23	8 394	9.56	c^（0）=0.5 τ^（0）=1.0×10^-6 m^（0）=0.9 $ρ 00$ =4 000	c=0.795 344 2 τ=3.228 42×10^-6 m=0.992 713 56 ρ₀=4 002.566 89

表4

表5

程序2的运行过程"

样号	迭代次数	相对误差/%	当前迭代值
样号	迭代次数	相对误差/%	$ρ 0 k m k c k τ k$
23	2 000	763.8	4 108.325 6	0.901 436	0.704 852	2.094 9×10^-6
	6 000	108.4	4 287.649 5	0.980 784	0.746 424	3.787 4×10^-6
	8 000	10.65	4 003.906 3	0.999 763	0.726 441	3.706 5×10^-6

表5

表6

程序3的参数及计算结果"

样号	总迭代次数	相对误差/%	初始值	计算结果
2	12	7.57	c^（0）=0.8 m^（0）=1.001 $ρ 00$ =893.5	c=0.786 648 66 τ=1.074 09×10^-6 m=0.999 987 3 ρ₀=593.384 774
18	19	6.45	c^（0）=0.8 m^（0）=0.809 9 $ρ 00$ =90.3	c=0.826 65 τ= 3.580 51×10^-7 m=0.902 873 43 ρ₀=84.846 30
23	8	9.23	c^（0）=0.8 m^（0）=0.917 61 $ρ 00$ =4 457.3	c=0.798 434 τ=3.222 04×10^-6 m=0.999 830 ρ₀=4 002.943 1

表6

表7

程序3的运行过程"

样号	迭代次数	相对误差/%	当前迭代值
样号	迭代次数	相对误差/%	$ρ 0 k m k c k τ k$
2	0	10.76	893.500 00	1.001 000 0	0.800 000	1.074 09×10^-6
	5	8.73	649.488 10	0.999 424 0	0.790 821
	10	7.57	593.302 60	0.999 988 0	0.786 605
18	0	7.03	90.300 00	0.809 900 0	0.800 000	3.580 51×10^-7
	8	6.51	85.348 23	0.902 727 6	0.826 130
	16	6.44	84.828 41	0.902 808 3	0.826 650
23	0	9.37	4 457.300 0	0.917 600 0	0.800 000	3.222 04×10^-6
	3	9.24	4 138.187 3	0.960 949 5	0.810 452
	6	9.23	4 002.996 7	0.999 829 7	0.798 430

表7

参考文献 16

[1]	PELTON W H, WARD S H, HALLOF P G , et al. Mineral discrimination and removal of inductive coupling with multi-frequency IP[J]. Geophysics, 1978,43(3):588-609.
[2]	黄涛, 余刚, 王绪本, 等. 页岩岩芯复电阻率频散特征主要影响因素的实验研究[J]. 物探化探计算技术, 2016,38(2):167-174.
	HUANG T, YU G, WANG X B, et al. Experiment research of the main influence factors of shale sample complex resistivity dispersion characteristics[J]. Computing Techniques For Geophysical And Geochemical Exploration, 2016,38(2):167-174.
[3]	DIAS C A . Developments in a model to describe low-frequency electrical polarization of rocks[J]. Geophysics, 2000,65(2):437-451 doi: 10.1190/1.1444738
[4]	徐凤姣, 谢兴兵, 周磊, 等. 时域电磁法在我国南方富有机质页岩勘探中的可行性分析[J]. 石油物探, 2016,55(2):294-302.
	XU F J, XIE X B, ZHOU L, et al. The feasibility of TDEM in the exploration for rich organic shale reservoir in Southern China[J]. Geophysical Prospectiong for Petroleum, 2016,55(2):294-302.
[5]	DIAS C A . A non-grounded method for measuring electrical induced polarization and conductivity: ph.D. thesis[D]. Berkeley: University of California, Berkeley, 1968.
[6]	XU W, KE S Z, YIN C F , et al. Measurement and inversion of complex resistivity spectra for different core samples[C]// paper SEG-2014-0469 presented at the 2014 SEG Annual Meeting, 26-31 October 2014, Denver, Colorado, USA.
[7]	高雷阜, 周庆. 多项式优化问题极小值数量及最优值下界分析[J]. 辽宁工程技术大学学报(自然科学版), 2018,37(3):669-672.
	GAO L F, ZHOU Q. Number of minimum value and lower bound of optimal value for polynomial optimization problem[J]. Journal of Liaoning Technical University(Natural Science), 2018,37(3):669-672.
[8]	向葵, 胡文宝, 严良俊, 等. 川黔地区页岩复电阻率的频散特性[J]. 石油地球物理勘探, 2014,49(5):1013-1019.
	XIANG K, HU W B, YAN L J, et al. Complex resistivity dispersion characteristics of shale samples in Sichuan and Guizhou area[J]. OGP, 2014,49(5):1013-1019.
[9]	肖占山, 赵云生, 赵宝成, 等. 基于岩石电性参数频散特性的储层参数评价方法[J]. 物探与化探, 2019,43(5):1105-1110.
	XIAO Z S, ZHAO Y S, ZHAO B C, et al. The evaluation method of reservoir parameters based on the rock’s electrical parameter dispersion properties[J]. Geophysical and Geochemical Exploration, 2019,43(5):1105-1110.
[10]	MADDEN T R, CANTWELL T . Induced polarization: A review [C]// presented at the Mining Geophysics（SEG）, 1967, Tulsa, Oklahoma, USA. USA.
[11]	李斐, 郝卫峰, 王文睿, 等. 非线性病态问题解算的扰动分析[J]. 测绘学报, 2011,40(1):5-9.
	LI F, HAO W F, WANG W R, et al. The perturbation analysis of nonlinear ill-conditioned solution[J]. Acta Geodaetica et Cartographica Sinica, 2011,40(1):5-9.
[12]	李鹏飞, 严良俊, 余刚. 南方页岩岩芯的复电阻率测试与分析[J]. 工程地球物理学报, 2014,11(3):383-386.
	LI P F, YAN L J, YU G. The southern shale core complex resistivity testing and analysis[J]. Journal of engineering geophysics, 2014,11(3):383-386.
[13]	赵迪斐, 郭英海, 朱炎铭, 等. 龙马溪组页岩气复电阻率勘探的理论依据[J]. 非常规油气, 2016,3(3):15-20.
	ZHAO D F, GUO Y H, ZHU Y M, et al. The theoretical basis of shale gas exploration with complex resistivity in Longmaxi Formation[J]. Uncoventional Oil & Gas, 2016,3(3):15-20.
[14]	李春梅, 王翠方, 段雪峰. 求解一类矩阵迹极小问题的非线性共轭梯度法[J]. 数学杂志, 2020,40(3):323-331.
	LI C M, WANG C F, DUAN X F. The nonlinear conjugate gradient method for solving a class of the matrix trace minimization problem[J]. Journal of Mathematics, 2020,40(3):323-331.
[15]	刘红英, 张奇业. 多元二次函数极值的研究[J]. 高等数学研究, 2015,18(2):5-7.
	LIU H Y, ZHANG Q Y. Necessary and sufficient conditions of quadratic function minimum[J]. Studies in College Mathmatics, 2015,18(2):5-7.
[16]	王晓明, 朱一心. 关于二次型零点的注记[J]. 大学数学, 2020,36(3):23-28.
	WANG X M, ZHU Y X. On the rank of zeros of a quadratic form[J]. College Mathematics, 2020,36(3):23-28.

样号	组	取心深度/m	层位	岩性描述	钻切方向	岩心规格/cm
2	龙马溪	3 566.60	9	黑色页岩	垂直层理	柱样：?24.48×36.55
18	龙马溪	3 636.60	3	黑色页岩	垂直层理	柱样：?24.48×15.93
23	龙马溪	3 558.65	浊积砂	泥质砂岩	无层理	柱样：?24.93×84.62