地球物理学报  2016, Vol. 59 Issue (8): 3121-3132 PDF

1. 中国石油大学(华东)理学院, 青岛 266580;
2. 山东省高校新能源物理与材料科学重点实验室(中国石油大学), 青岛 266580

Effect of tool eccentricity on responses of array induction logging tools in cylindrical multilayered formations
WEI Bao-Jun1,2, WANG Cheng-Yuan1, CHANG Xin-Li1, CAO Jing-Qiang1
1. College of Science, China University of Petroleum, Qingdao 266580, China;
2. Key Laboratory of New Energy Physics & Material Science in Universities of Shandong(China University of Petroleum), Qingdao 266580, China
Abstract: The effect of eccentricity on the responses of electromagnetic well-logging tools in a borehole is of great value for precisely interpreting the measured data. For real electromagnetic well-logging tools such as array induction-logging tools, usually both a metal mandrel and an insulating protection layer are included in the tools. Only the existence of the metal mandrel was considered for simulating and analyzing the effect of tool eccentricity in conventional analytical methods, and the influence of the insulating protection layer has been neglected. Since the insulating protection layer has occupied part of the space of the drilling fluid, its influence on the responses should also be considered whether the tool is centric or eccentric in a borehole. An analytical algorithm was developed that can simulate the responses of centric or eccentric electromagnetic well-logging tools with both a metal mandrel and an insulating protection layer in a borehole. The algorithm was based on the dyadic Green's functions of cylindrical multilayered medium, in which the coils are supposed to be positioned in the 0-th layer, i.e. the insulating protection layer. The borehole is taken as the 1-th layer in the algorithm, which was used to be taken as the 0-th layer where the coils are supposed to be positioned in conventional analytical methods. Thus the new model adopted is more suitable for describing real electromagnetic well-logging tools, whose responses can not be directly simulated by conventional analytical methods. In the new developed analytical algorithm, the continuity condition of the tangential electromagnetic fields across the boundary between the insulating protection layer and the borehole fluid was used, thus the fields in the borehole can be expressed by the fields in the insulating layer. The relationship between the electromagnetic fields under the cylindrical coordinates relative to the borehole and to the tool was then used and finally the new algorithm was developed. The responses of centric and eccentric array induction logging tools in a borehole were simulated by the algorithm, and the difference between the responses of the tool with or without an insulating protection layer was compared. The observations include:(1) Given the borehole diameter and the conductivity of the drilling fluid and formation, the relationship between the receiver coil's response and the eccentric distance was obtained for different arrays. The results showed that the effect of the insulating layer on the tool's response is small and can be neglected when the fluid's conductivity is relatively low, whether the transmitter-receiver spacing is long or short. The effect is big for the condition of relatively high fluid's conductivity, and the shorter the transmitter-receiver spacing the bigger the effect, the bigger the eccentric distance the bigger the effect. (2) Given the conductivity of the drilling fluid and formation, the relationship among the coil's geometric factor and the borehole's diameter was obtained for different arrays with the tool being in the borehole center. Since the tool was positioned in the borehole center, the effect of the insulating layer was relatively small. (3) Given the conductivity of the drilling fluid and formation, the relationship among the coil's geometric factor, borehole's diameter and eccentricity ratio for different arrays was obtained. The results can be used to analyze the radial detection depth with different borehole diameter and eccentricity ratio. From all the results we can conclude that it is better to consider all the specific constructions of the tool and coils when simulating the responses of electromagnetic well-logging tools in a borehole, and it is better to adopt a model as much as closer to real tools, thus the simulated results will be much more precise..
Key words: Array induction logging      Eccentricity      Dyadic Green's functions      Cylindrical multilayered formations      Metal mandrel      Insulating layer
1 引言

2 基本理论 2.1 均匀介质中谱域并矢Green函数

 (1a)

 (1b)

 (2a)

 (2b)

 (3a)

 (3b)

 (4a)

 (4b)

 (4c)

ρ分量和φ分量均可用z分量表示.当ρ≤ρ″时，只需将式(3)至式(4)中的Hankel函数Hν(1)(kρρ)换成Bessel函数Jν(kρρ)，将Jν(kρρ″)换成Hν(1)(kρρ″)即可.相应地，式(3)中相关矩阵定义为Jν，φ(kρρ)Jν，ρ(kρρ).

 (5a)

 (5b)

 (5c)

 (6a)

 (6b)

 (6c)

2.2 柱状成层介质中谱域并矢Green函数

 (7a)

 (7b)

 (7c)

 (8a)

 (8b)

 (8c)

2.3 仪器在井眼中偏心时的谱域并矢Green函数

 图 1 井眼内仪器偏心示意图 Fig. 1 Illustration of eccentric borehole geometry

 (9a)

 (9b)

 (9c)

 (10a)

 (10b)

 (10c)

 (11a)

 (11b)

 (11c)

 (12a)

 (12b)

 (12c)

 (13a)

 (13b)

 (13c)

 (14a)

 (14b)

 (14c)

 (15a)

 (15b)

 (16a)

 (16b)

 (17a)

 (17b)

 (17c)

 (18a)

 (18b)

 (18c)

 (19)

 (20)

 (21)

 (22a)

 (22b)

 (22c)

2.4 阵列感应测井仪器响应的计算

 (23)

 (24)

 (25)

 (26)

3 数值算例及讨论

 图 2 泥浆电导率σ1=1.0 S·m-1、地层电导率σ2=0.01 S·m-1时接收线圈响应随偏心距离的变化关系 (a)第一组；(b)第二组；(c)第三组. Fig. 2 Relationship between receiver coil′s response and eccentric distance when the drilling fluid′s conductivity σ1=1.0 S·m-1 and the formation′s conductivity σ2=0.01 S·m-1
 图 3 泥浆电导率σ1=0.001 S·m-1、地层电导率σ2=1.0 S·m-1时接收线圈响应随偏心距离的变化关系 (a)第一组；(b)第二组；(c)第三组. Fig. 3 Relationship between receiver coil′s response and eccentric distance when the drilling fluid′s conductivity σ1=0.001 S·m-1 and the formation′s conductivity σ2=1.0 S·m-1

 图 4 仪器在井眼中居中时线圈系几何因子随井径变化关系 A1n—第一组线圈无绝缘层，A1y—第一组线圈有绝缘层，其余类推. Fig. 4 Relationship between coil′s geometric factor and borehole diameter with tool being in the borehole center

 图 5 第一组线圈系几何因子随井径及偏心率的变化关系 (a)无绝缘层；(b)有绝缘层. Fig. 5 Relationship between coil′s geometric factor，borehole diameter and eccentricity ratio for the first coils
 图 6 第二组线圈系几何因子随井径及偏心率变化关系 (a)无绝缘层；(b)有绝缘层. Fig. 6 Relationship between coil′s geometric factor，borehole diameter and eccentricity ratio for the second coils
4 结论

(1) 本文基于柱状成层介质的并矢Green函数得到解析表达式，可高效模拟电磁测井仪器在井眼中偏心时的电磁场分布及响应，并且在模拟时既可考虑金属心轴的存在也可考虑线圈系所处绝缘保护层的存在.由于在推导过程中假设线圈系位于仪器内部的绝缘层内而非仪器外侧的井眼中，所采用模型更接近于仪器实际结构.

(2) 对于相对高泥浆电导率情况而言，绝缘层对线圈系响应的影响明显，且仪器偏心距越大影响越大，线圈距越短影响越大.对于相对低泥浆电导率情况而言，无论线圈距长短，线圈系所处绝缘层对仪器响应的影响均较小，可忽略不计.相同泥浆电导率情况下线圈距越短偏心对线圈系响应的影响越大，同一组线圈系在相对高泥浆电导率井眼中偏心对其响应的影响更大.

 Abramowitz M, Stegun I A. andbook of Mathematical Functions. New York: Dover Publications, 1970 . Dai Z D, Lu S. Dyadic Green's Functions in Electromagnetic Theory . (in Chinese) Wuhan: Wuhan University Press, 1996 . Gianzero S C. 1978. Effect of sonde eccentricity on responses of conventional induction-logging tools. IEEE Transactions on Geoscience Electronics , 16(4): 332–339. Hagiwara T, Banning E J, Ostermeier R M, et al. 2005. Effects of mandrel, borehole, and invasion for tilt-coil antennas.//SPE Reservoir Evaluation & Engineering. SPE. Hue Y K, Teixeira F L. 2006. Analysis of tilted-coil eccentric borehole antennas in cylindrical multilayered formations for well-logging applications. IEEE Trans. Antennas Propag. , 54(4): 1058–1064. LovellJ R, ChewW C. 1987. Response of a point source in a multicylindrcally layered medium. IEEE Trans. Geosci. Remote Sens , GE-25(6): 850–858. Lovell J R, Chew W C. 1990. Effect of tool eccentricity on some electrical well-logging tools. IEEE Trans. Geosci. Remote Sens. , 28(1): 127–136. Wang H N, So P, Yang S W, et al. 2008. Numerical modeling of multicomponent induction well-logging tools in the cylindrically stratified anisotropic media. IEEE Trans. Geosci. Remote Sens. , 46(4): 1134–1147. Wei B J. 2007. Response and calibration of a new logging-while-drilling resistivity tool. Chinese Journal of Geophysics (in Chinese) , 50(2): 632–641. Wei B J, Tian K, Zhang X, et al. 2010. Evaluating influence of eccentricity on response of electromagnetic wave resistivity logging-while-drilling by vector eigenfunction expansion formulae for dyadic Green's functions. Journal of China University of Petroleum (Edition of Natural Science) (in Chinese) , 34(5): 57–62. Wei B J, Ou Y F, Wu Y, et al. 2012. Simulation of tilted coil's response in cylindrically stratified media and its application for electromagnetic propagation measurement-while-drilling. Journal of China University of Petroleum (Edition of Natural Science) (in Chinese) , 36(5): 72–79. Wei B J, Wang C Y, Yu Y M, et al. 2015. Simulating multi-component induction logging tool's calibration by vector eigenfunctionexpansion formulae for dyadic Green's functions. Journal of China University of Petroleum (Edition of Natural Science) (in Chinese) , 39(2): 39–47. Yu Y M, Xiao J Q, Wei B J, et al. 2014. Evaluating influence of metal mandrel on response of multi-component induction logging by vector eigenfunction expansion formulae for dyadic Green's functions. Journal of China University of Petroleum (Edition of Natural Science) (in Chinese) , 38(4): 57–64. 戴振铎, 鲁述. 电磁理论中的并矢格林函数. 武汉: 武汉大学出版社, 1996 . 魏宝君. 2007. 一种新型随钻电阻率测井仪器的响应和刻度. 地球物理学报 , 50(2): 632–641. 魏宝君, 田坤, 张旭, 等. 2010. 用并矢Green函数的矢量本征函数展开式评价偏心对随钻电磁波电阻率测井响应的影响. 中国石油大学学报:自然科学版 , 34(5): 57–62. 魏宝君, 欧永峰, 武杨, 等. 2012. 柱状成层介质中倾斜线圈响应的模拟及其在电磁波传播随钻测量中的应用. 中国石油大学学报:自然科学版 , 36(5): 72–79. 魏宝君, 王成园, 俞燕明, 等. 2015. 用并矢Green函数的矢量本征函数展开式模拟多分量感应测井仪器的刻度. 中国石油大学学报:自然科学版 , 39(2): 39–47. 俞燕明, 肖加奇, 魏宝君, 等. 2014. 用并矢Green函数的矢量本征函数展开式评价金属心轴对多分量感应测井响应的影响. 中国石油大学学报:自然科学版 , 38(4): 57–64.