高阶自适应有限元三维直流电阻率正演方法及其在沁水盆地煤气层压裂监测中的应用

doi:10.13810/j.cnki.issn.1000-7210.2021.01.024

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摘要自适应有限元解的精度主要受单元大小和形函数阶数的影响，为了得到高精度的有限元解，需要将网格自适应加密到足够的密度或者在网格中应用较高阶数的形函数，但这会大大增加计算时间，同时消耗大量计算资源。为提高有限元解的精度并节约计算资源，应用h型自适应加密算法并结合高阶形函数实现了三维直流电阻率模型的正演。在程序实现过程中，通过一维多项式的张量积生成三维空间中任意阶数的形函数，应用Kelly后验误差估计指导网格自适应加密。数值算例表明，形函数阶数p=3时，算法具有较高的精度，并且比p=1、2时有限元解的收敛更快，即实现了用最少的自由度个数得到最精确的有限元解。最后通过对沁水盆地南部某煤气层压裂监测区的三维合成数据模型进行直流电阻率模拟，验证了程序的有效性。

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关键词 ：直流电阻率法, 有限元正演, 高阶形函数, 自适应

Abstract：Considering the fact that the accuracy of an adaptive finite element solution is mainly affected by the cell size (h) and the order of a shape function (p), we should adaptively refine the meshes to be dense enough or apply a higher-order shape function in the meshes to acquire a high-accuracy finite element solution. However, this will greatly increase the time burden and require sufficient me-mory space of the computer. In light of these problems, in this paper, we combine an h-adaptive refinement algorithm with a higher-order shape function for the forward modeling of a 3D DC resistivity model. In the program, we use the tensor pro-duct of 1D polynomials to generate a shape function of any order in the 3D space and apply Kelly posteriori error estimation to guide the adaptive refinement of meshes. Numerical examples show that in the case of p=3, our program has high accuracy, and the convergence of the finite element solutions is faster than that in the case of p=1 or 2. This means that the most accurate finite element solution with the fewest degrees of freedom can be obtained when the order of the shape function is 3. Finally, the simulation of 3D DC resistivity is performed on the 3D modeling of a coalbed fracturing monitoring region in the southern Qinshui Basin, verifying the effectiveness of our program.

Key words： DC resistivity method finite element based forward modeling high-order shape function adaptive

收稿日期: 2020-03-11

PACS:

P631

基金资助:本项研究受国家自然科学基金项目“三维电阻率法与张量可控源电磁法联合聚焦反演及其在活断层研究中的应用探索”（U1704128）、“基于几何多重网格和无矩阵方法的三维大地电磁法自适应有限元正演研究”（41904078）、国家重点研发计划课题“面向深部资源勘查的重磁、电磁处理解释软件研发”（2018YFC0603602）及河南省瓦斯地质与瓦斯治理重点实验室基金项目“三维电磁法联合聚焦反演在煤层瓦斯抽采监测中的应用研究”联合资助。

通讯作者: 秦策,河南省焦作市山阳区世纪大道2001号河南理工大学物理与电子信息学院,454150。Email:ce.qin@hpu.edu.cn E-mail: ce.qin@hpu.edu.cn

作者简介: 赵宁,博士,副教授,博士生导师,1982年出生;2005年获河南理工大学信息与计算科学专业学士学位,2008年获成都理工大学信号与信息处理专业硕士学位,2014年获成都理工大学固体地球物理专业博士学位;2017~2019年在东方地球物理公司做博士后研究;现就职于河南理工大学物理与电子信息学院,主要从事电磁探测技术正反演的研究与应用。

引用本文:

赵宁, 黄明卫, 申亚行, 陶德强, 秦策. 高阶自适应有限元三维直流电阻率正演方法及其在沁水盆地煤气层压裂监测中的应用[J]. 石油地球物理勘探, 2021, 56(1): 209-216. ZHAO Ning, HUANG Mingwei, SHEN Yahang, TAO Deqiang, QIN Ce. Forward modeling of 3D DC resistivity based on high-order adaptive finite element and its application in Qinshui Basin. Oil Geophysical Prospecting, 2021, 56(1): 209-216.

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