中文摘要：

采用一维自适应hp有限元方法求解ECOS过套管电阻率的改进型传输线方程.该方法从粗网格开始迭代,通过对当前粗网格的单元尺寸（h）和网格上的插值函数的阶数（p）进行适当细化,不断寻找当前粗网格拟最优的细化网格.自适应hp策略基于参考解的投影型插值后验误差估计,不依赖具体问题且十分有效,能够适应介质连续变化情况.使用该方法计算了金属套管电导率异常、水泥环和围岩等因素间断和连续变化时,过套管电阻率测井响应,分析了相应的响应规律.数值计算表明,自适应hp有限元方法能够使用较小的自由度获得较高的计算精度.

英文摘要：

A one-dimensional adaptive-hp finite element method is used to solve the modified transmission equation for the ECOS through-casing resistivity.This method iterates from an initial coarse grid to seek a new quasi-optimal hp-refined mesh, by appropriately h refinements and p refinement of current coarse grid.The adaptive-hp strategy of this method is guided by the projection-based interpolation of reference solution over the finer grid of the current coarse grid.This strategy is problem independent and is very efficient.The favorable advantage of this method is that it can incorporate the medium with continuous variations, and we use it to evaluate the responses of through-casing resistivity in the environment where the metal casing conductivity, cement sheaths resistivity and surrounding rock resistivity vary discontinuously or continuously and to analyze corresponding effects.Numerical results illustrate that the adaptive-hp finite element method can achieve more highly-accurate simulation with the less number of degree of freedom.