针对石油勘探电法测井的三维电流场快速算法研究

A common method of oil exploration is putting a measuring instrument after drilling well in the earth. A kind of instrument consists of a few cylindrical electrodes leaking quasi-DC into the earth, and the electrical parameters of the earth can be reflected by the current distribution. If the instrument deviates from the axial of the borehole or the formation is asymmetry, then 3-D current field will be formed. Solving the 3-D current field is a forward problem, and fast calculation of the forward problem is the basis of the inverse problem to determine the formation parameters. Now 3-D finite element method was always used to solve this problem, and its speed is slow. This program will study the fast algorithm of the 3-D current field. The algorithm of combining the half numerical and half analytical numerical mode-matching method (NMM) with regional mapping finite element method will be studied for the vertical/horizontal well. A new method to solve the analytic expressions of NMM is presented, the calculation precision and stability will be improved, and the analytic expression of each layer will be unified. The numerical solution of the open boundary problem will be solved by combining the finite element method and the infinite element method, the relationship between grid distribution, stiffness matrix properties and eigenvalue will be studied, and the distribution of eigenvalue will be the optimized, which can reduce the dimension of matrix, and improve the computational efficiency. According to the defect of NMM is not suitable for deviated well, affine transformation will be researched to transform deviated well into "vertical" well, the transformation formula will be determined, and the "vertical" well will be solved with the fast NMM. The target of this project is to develop a fast calculation method, which is suitable for 3-D current field of the vertical/ horizontal well and deviated well, and this will establish the foundation for the fast calculation of the inverse problem.

大地钻井后放入测量仪器是常用的石油勘探方法，一种仪器是由多个向大地注入准直流的柱状电极构成，靠电流分布反映大地电气参数。若仪器偏心或地层不对称，则形成三维电流场，求解该电流场为正问题，其快速计算是确定地层参数逆问题的基础。目前多采用三维有限元求此类问题，速度较慢，本项目将研究三维电流场的快速算法。针对直井/水平井，研究半数值半解析的数值模式匹配法（NMM）与区域映射有限元结合的方法。提出一种递推NMM解析解的新方法，可提高计算精度与稳定性，实现各层解析式的统一化。用有限元与无限元结合的方法求开域场的数值解，研究网格分布及刚度阵性质与本征值的关系，优化本征值的分布，可降低矩阵维数，提高计算效率。针对NMM不适用于斜井的缺陷，研究将斜井变为"直井"的仿射变换，确定变换关系，对"直井"实施快速的NMM。目标是形成适用于直井/水平井和斜井模型的三维电流场的快速计算方法，为快速计算逆问题奠定基础。

数值模式匹配法（NMM法）是求解直井测井问题的一种半数值半解析方法，基于对传统NMM法解析递推式的分析，本项目提出了用新未知量的统一形式的解析解递推方法，简化了传统方法解析解的递推过程。在推得的解析式中，仍有一处正指数项，未完全消除正指数项，这使得原计划中数值解部分对本征值优化等问题的研究调整为对无限元单元形状函数的研究，在原有形状函数的基础上乘以指数型衰减因子，能更好地模拟场量的衰减特性。对不同衰减长度、不同无限元网格划分层数及不同计算方法的计算结果进行了对比分析，确定了最佳衰减长度及网格层数；与原形状函数相比，在保证计算效率的同时，可大幅度提高计算精度。本项目对仿射变换求解斜井模型的问题进行了探索性研究，在分析斜井模型特点的基础上，给出了将斜井变换为“直井”的错切变换关系式，并对变换后的拉普拉斯方程进行了推理分析，因错切变换后的拉普拉斯方程无法用分离变量法进行变量分离，故无法采用NMM法快速求解斜井模型的问题。在项目计划内容之外，本项目对本征值的影响因素进行了研究，分析总结了本征值随不同地层电阻率、侵入半径及径向网格的变化规律。本项目的研究统一了NMM法解析解形式，简化了传统的递推式；在保证计算效率的同时，数值解部分的计算精度有所提高。