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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D-DC borehole resistivity measurements
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Fourier series expansion in a non-orthogonal system of coordinates for the simulation of 3D-DC borehole resistivity measurements

机译：非正交坐标系中的傅立叶级数展开，用于模拟3D-DC钻孔电阻率测量

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摘要

We describe a new method to simulate 3D borehole resistivity measurements at zero frequency (DC). The method combines the use of a Fourier series expansion in a non-orthogonal system of coordinates with an existing 2D goal-oriented higher-order self-adaptive hp-finite element algorithm. The new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells. It delivers high-accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3D simulators, since the number of Fourier modes (basis functions) needed to solve practical applications is limited (typically, below 10). Furthermore, numerical results indicate that the optimal 2D grid based on the Oth Fourier mode (also called central Fourier mode) can be employed to efficiently solve the final 3D problem, thereby, avoiding the expensive construction of optimal 3D grids. Specifically, for a challenging through-casing resistivity application, we reduce the computational time from several days (using a 3D simulator) to just 2 h (with the new method), while gaining accuracy. The new simulation method can be easily extended to different physical phenomena with similar geometries, as those arising in the simulation of 3D borehole electrodynamics and sonic (acoustics coupled with elasticity) measurements. In addition, the method is especially suited for inversion, since we demonstrate that the number of Fourier modes needed for the exact representation of the materials is limited to only one (the central mode) for the case of borehole measurements acquired in deviated wells.
机译：我们描述了一种在零频率（DC）下模拟3D钻孔电阻率测量的新方法。该方法结合了在非正交坐标系中使用傅立叶级数展开和现有的2D面向目标的高阶自适应hp有限元算法。该新方法适用于模拟在偏斜井中使用井眼测井仪采集的测量结果。它提供了高精度的仿真，并且相对于可用的3D仿真器，它可以显着降低计算复杂性，因为解决实际应用所需的傅里叶模式（基本函数）数量有限（通常低于10个）。此外，数值结果表明，可以采用基于Oth Fourier模式（也称为中央Fourier模式）的最佳2D网格有效地解决最终的3D问题，从而避免了昂贵的最佳3D网格构造。具体来说，对于具有挑战性的套管电阻率应用，我们将计算时间从几天（使用3D模拟器）减少到仅2小时（使用新方法），同时提高了准确性。这种新的模拟方法可以轻松地扩展到具有相似几何形状的不同物理现象，就像在3D钻孔电动力学和声波（声学与弹性）测量的模拟中产生的那些现象一样。另外，该方法特别适用于反演，因为我们证明了对于在偏斜井中进行井眼测量的情况，精确表示材料所需的傅里叶模式的数量仅限于一种（中央模式）。

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