王健, 陈浩, 王秀明

1 中国科学院声学研究所声场与声信息国家重点实验室， 北京　100190 2 中国科学院大学， 北京　100049



用于固体矿床多分量感应测井响应模拟的矢量有限元法

王健1,2, 陈浩1, 王秀明1

1 中国科学院声学研究所声场与声信息国家重点实验室， 北京100190 2 中国科学院大学， 北京100049

摘要本文开发了基于非结构化四面体网格的三维矢量有限元法，实现了固体矿床井眼中多分量感应测井响应的数值模拟，并分析了多分量感应测井仪器在复杂矿床模型中的响应特征.本文通过采用几何因子背景场，有效地避免了源的奇异性问题；同时，在井眼边界采用非均质网格并用Gauss-Legendre积分计算四面体单元的等效电导率.利用LU分解求解线性方程组，实现了一次网格划分多点的数值计算，提高了计算效率，从而实现快速连续的多分量感应测井模拟.非结构化的四面体网格确保了该方法可以模拟实际问题中所能遇到的复杂的矿体模型.基于水平三层分层和径向分层模型，验证了算法在各向同性和各向异性介质中的可靠性.我们还以三个不同的矿床模型为例，研究了多分量感应测井仪的不同分量的探测特性，结果表明，结合九个分量的信息，可以探测矿体的深度，也可以识别矿体的方位和走向，为精确地描述矿体的三维分布特征打下了基础.

关键词矢量有限元； 多分量感应测井； 固体矿床

Due to the characteristic of solid deposit, such as wide variety, complex distribution, we use the vector finite element method to simulate the response of multicomponent induction instrument, which has an advantage for medium with complex structure and boundary based on unstructured mesh. Moreover, the method could avoid defects of nodal finite element method. Specifically, geometry factor background field is used to eliminate the singularities of source efficiently. Meanwhile, heterogeneous grids are introduced at the boundary of the borehole and the equivalent conductivities of the tetrahedron elements are calculated by using Gauss-Legendre integral. To take advantage of the LU decomposition to solve the linear system of equations, we could obtain multipoint numerical results for each meshing. It greatly improves the efficiency. So as to ensure a rapid and successive simulation of the multicomponent induction logging.

It has been tested and verified that this method is valid and reliable in the isotropic and anisotropic medium based on the horizontal three-layered and radial model. In the dipping layered formation, all nine components are studied. TheXXandYYresponses are more sensitive to the dipping angle than theZZwith sharp dipping angle; the cross-componentsXZandZXare influenced by border and anisotropy. The apparent conductivity curves ofXZandZX-components show horns in the opposite direction. While in the spherical and cylindrical ore body model, we show azimuthal sensitivity of all nine components of apparent conductivity. Circle lines for response ofZZ-component confirm that the lack of azimuthal sensitivity of the conventionalZZ.The remaining eight components do show azimuthal sensitivity, but four of them (XX,XY,YY,YX) have the period of 180°, while the other four (XZ,YZ,ZX,ZY) have the period of 360°.It is quite obvious that having any combination of measurement from the first group will not allow a resolution between left and right because of the 180°symmetry. The pairs of signals from the second group (ZX,ZY) or (XZ,YZ) uniquely define the azimuthal direction of the ore body.

We develop a three dimensional vector finite element program for multicomponent induction logging based on unstructured tetrahedral mesh in frequency domain, then we use it to simulate the response of multicomponent instrument in solid deposit. The numerical results show that it is able to identify anisotropy of ore body as same as in oil and gas reservoir. TheZZ-component indicates the vertical conductivity, and theXXandYY-components indicate the horizontal conductivity. The cross-componentsXZ,ZX,YZ,ZYhave the period of 360°, therefore provide azimuthal information by which you could laid a foundation to accurately describe the distribution of ore deposit.

1引言

自从Doll(Doll,1949)提出了几何因子理论以来，感应测井在油气储层……