地球物理学报 ›› 2016, Vol. 59 ›› Issue (8): 3110–3120.doi: 10.6038/cjg20160833

• 应用地球物理学 • 上一篇    下一篇

轴对称地层中高分辨率阵列侧向测井信赖域反演法

潘克家1,2, 汤井田3, 杜华坤3, 蔡志杰4   

  1. 1. 中南大学数学与统计学院, 长沙 410083;
    2. 油气藏地质及开发工程国家重点实验室(成都理工大学), 成都 610059;
    3. 中南大学有色金属成矿预测教育部重点实验室, 地球科学与信息物理学院, 长沙 410083;
    4. 复旦大学数学科学学院, 上海 200433
  • 收稿日期:2015-05-16 修回日期:2016-05-24 出版日期:2016-08-05
  • 通讯作者: 杜华坤,男,1977年生,云南镇雄人,博士,中南大学讲师,主要从事地球物理数据处理与正反演研究.E-mail:hkdu2008@126.com E-mail:hkdu2008@126.com
  • 作者简介:潘克家,男,1981年生,湖南宁乡人,2004年毕业于中国石油大学,2009年在复旦大学获得理学博士学位,现为中南大学副教授,主要从事地电磁场正反演研究.E-mail:pankejia@hotmail.com
  • 基金资助:
    国家自然科学基金项目(41474103,41204082),国家863计划项目(2014AA06A602),湖南省自然科学基金项目(2015JJ3148)和成都理工大学油气藏地质及开发工程国家重点实验室资助项目(PLC201305)联合资助.

Trust region inversion algorithm of high-resolution array lateral logging in axisymmetric formation

PAN Ke-Jia1,2, TANG Jing-Tian3, DU Hua-Kun3, CAI Zhi-Jie4   

  1. 1. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
    2. State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation(Chengdu University of Technology), Chengdu 610059, China;
    3. Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education/School of Geosciences and Info-Physics, Central South University, Changsha 410083, China;
    4. School of Mathematical Sciences, Fudan University, Shanghai 200433, China
  • Received:2015-05-16 Revised:2016-05-24 Online:2016-08-05

摘要: 本文研究轴对称地层中高分辨率阵列侧向测井(HRLA)的多参数信赖域反演方法.首先对前期HRLA的有限元正演方法进行改进,提出基于叠加原理和并行直接稀疏求解器PARDISO的快速正演方案,更适合于反演计算.将HRLA反演问题转化为非线性最小二乘问题,利用信赖域算法求解.为提高反演速度,推导了目标函数对优化参数偏导数的具体计算公式.对典型地层模型,与已有文献中Jacobi预条件共轭梯度法(JCG)计算结果比较,发现PARDIDO比JCG快10倍以上,验证了本文正演程序的正确性与高效性.利用信赖域算法求解了电阻率四参数反演和传统的三参数反演.研究结果表明:并行直接稀疏求解器PARDISO能有效求解此类HRLA正演问题,对6次不同探测深度的测井模拟,所形成的有限元刚度矩阵完全相同,只须进行一次矩阵分解,大大加快了正反演的速度.信赖域算法收敛速度快,且具有全局收敛性. HRLA的信赖域反演结果几乎不依赖于初值的选取,从较差初值出发仍能得到满意的反演结果.另外信赖域算法抗噪能力比较强,即使对测井数据添加信噪比为10dB(甚至5dB)的高斯白噪声,仍能通过反演得到较为准确的地层参数.

关键词: 阵列侧向测井, 信赖域算法, 反演, 全局收敛, PARDISO

Abstract: The high-resolution laterolog array (HRLA) tool can simultaneously provide six curves of logging responses with different depths of investigation, which give abundant information for the inversion of geological formation parameters. This paper proposes a trust region algorithm to recover several parameters in the axisymmetric formation from HRLA data, including the mud-invasion radius, mud resistivity and original formation resistivity.#br#Firstly, we propose an accurate and efficient method to solve the forward problem in order to obtain the logging responses of HRLA tool in the axisymmetric formation, where the superposition principle is used to simplify the calculation of resistivity logging responses, and the parallel direct solver PARDISO is employed to solve the linear system with multiple right-hand sides resulting from finite element approximation. The decomposition of the stiffness matrix is needed only once when using PARDISO to solve these linear systems, which is cheap in terms of work and fast in time. Secondly, the problem of identifying formation parameters is transformed into a nonlinear least squares problem that can be solved by the trust region algorithm.#br#In order to speed up the optimization process, we derive an explicit formula for the partial derivative of the objective function with respect to each resistivity parameter. Finally, for a typical formation model we present resistivity four-parameter and traditional three-parameter inversion results by using different initial guesses.In order to test the validity and effectiveness of the proposed method, we design a typical formation model with borehole radius rh=0.1016 m, invasion radius ri=0.4016 m, permeable formation thickness H=2 m, borehole mud resistivity Rm=1 Ωm, surrounding rock resistivity Rs=10 Ωm and original formation resistivity Rt=500 Ωm. Comparing the forward modeling result with the result in the existing literature, we find that the maximum relative error between the two results is only 0.01%, which verifies the correctness of the forward algorithm program. It takes less than half a second to perform forward modeling with 24881 grid points using PARDISO on a personal laptop. The computational time of the Jacobi conjugate gradient (JCG) method is more than ten times of that of PARDISO, and PARDISO provides a more accurate solution than the JCG. We use the trust region algorithm to invert formation parameters with four different initial values, one of which is far away from the real model. The inversion results show that the method can converge to the true model quickly even with very poor initial guesses. When using noisy data mixed with Gaussian white noise with signal-to-noise ratio of 10dB or even 5dB, the trust inversion algorithm still gives satisfactory results.#br#Trust region algorithm combined with PARDISO is successfully used to multi-parameter inversion of HRLA data. Numerical results show that the trust region inversion algorithm converges fast, and has a global convergence property. Another advantage of the proposed method is the robustness to noise.

Key words: Array lateral logging, Trust region algorithm, Inversion, Global convergence, PARDISO

中图分类号: 

  • P631
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