石油油井在鑽井、探勘、植入鋼套管、固井過程中，油井形成圓柱多層結構。傳統油井超聲波探勘技術，會將測井儀器置於圓柱多層結構內，超聲波轉能器 (ultrasonic transducer) 發射訊號產生彈性波於地質結構中傳遞，同時其亦可接收反射訊號，藉此了解地層結構的狀態並確認固井水泥層的完善。本論文的目的是協助完成與美國Halliburton 新加坡子公司的建教合作計畫中關於圓柱多層結構中的彈性波 (elastic waves) 場型的基礎分析，進而幫助Halliburton公司在石油探勘、固井方面所需的彈性波模擬技術有所提升。
吾人首先回顧彈性力學中的基本物理量，包含應力 (stress)、應變 (strain)、及剛度張量 (stiffness tensor)。藉由描述連續介質的廣義虎克定律 (Hooke's law) 與牛頓第二運動定律 (Newton's second law of motion)，吾人可獲得於連續介質中傳遞的彈性波的控制方程式：速度-應力耦合一階偏微分方程式 (velocity-stress coupled first-order PDE)。此控制方程式可進一步推導出彈性波所滿足之向量波動方程式，其解包含兩種擾動模態：P波與S波。
接著吾人將採用時域有限差分法 (FDTD, finite-difference time-domain method) 將控制方程式離散化以在 MATLAB 平台上進行數值模擬。各個未知物理量的安排採用標準的錯置網格 (staggered layout)。FDTD的模擬將在標準矩形網格與圓柱形網格上實現。藉著將圓柱形網格的計算結果內插至矩形網格的取樣位置上，吾人可更嚴謹地交錯驗證這兩種網格下模擬結果的正確性。在選取適當的模擬參數下，矩形網格與圓柱形網格的模擬結果近乎一致，並對模擬結果所得的應力場 (stress field) 與速度場 (velocity field) 的場型作圖，藉此了解彈性波在圓柱多層結構中的場型分佈。

In oil field exploration, various logging devices are put into the borehole structure during and after the drilling process and also after the well cementation job to check for the hydraulic isolation between different fluid layers. Ultrasonic transducers transmit acoustic signals and produce elastic waves in the geological structure. The reflected and scattered signals are then received and processed at the same time. By the sonic and ultrasonic measuring techniques, we may understand the compositions and orientations of the geological structure and confirm the isolation quality of the cementing layer. This work is part of the four-year cooperative education research program “Modeling of borehole ultrasonic measurement” between National Sun Yat-sen University and Halliburton Far East Pte Ltd.
We begin with the review of the basic physics of elasticity, including definitions of stress, strain, and stiffness tensors. For continuous media, we may apply the Hooke's law to linearly relate the strain and the stress tensors. This is followed by Newton's second law of motion to obtain, VS-PDEs, the first-order (in time and space), velocity-stress coupled partial differential equations for elastic wave propagation in the continuum. These control equations can be shown to be equivalent to the standard second-order vector wave equations for elastic waves and the solutions are well known to include both compressional (P) waves and shear (S) waves.
We use the FDTD method to discretize the first-order VS PDEs and perform numerical simulations on the MATLAB platform. The arrangement of unknown quantities is based on the standard staggered grid (in both space and time) layout. Simulations in FDTD will be implemented on standard rectangular grid in both the Cartesian and polar coordinate systems. The calculations in the cylindrical mesh are then mapped into a rectangular grid for cross verification with the calculations done in the Cartesian mesh. By selecting the appropriate simulation parameters, simulation results in rectangular grid and in cylindrical grid are nearly identical. We plot, from these simulation results, for both types of the stress and velocity components. From these results we are able to gain clear physical pictures regarding the distribution, propagation and scattering of the elastic waves in a cylindrical multilayer structure.

論文審定書 .................................................................................................... i
誌謝 ............................................................................................................... ii
中文摘要 ...................................................................................................... iii
英文摘要 ...................................................................................................... iv
目錄 ............................................................................................................... v
圖次 ............................................................................................................ viii
表次 ............................................................................................................... x
第一章 緒論 ............................................................................................... 1
1.1 研究動機 ........................................................................................................... 1
1.2 研究方法與架構 ............................................................................................... 1
第二章 彈性波波動方程式推導 ............................................................... 4
2.1 彈性力學物理概念 ........................................................................................... 4
2.1.1 應力與應變之定義 ................................................................................. 4
2.1.2 廣義虎克定律 ....................................................................................... 10
2.1.3 連續介質中的牛頓第二運動定律 ....................................................... 12
2.2 速度-應力耦合一階偏微分方程式推導 ....................................................... 15
2.2.1 二維直角坐標系下的形式 ................................................................... 15
2.2.2 二維圓柱座標系下的形式 ................................................................... 17
2.3 彈性波波動方程式推導 ................................................................................. 20
第三章 時域有限差分法 (FDTD) 簡介與實現 ................................... 23
3.1 時域有限差分法之概述 ................................................................................. 23
3.2 時域有限差分法之實現說明 ......................................................................... 26
3.2.1 FDTD的布局與離散化 ...................................................................... 26
3.2.2 不連續界面的處理方法 ....................................................................... 28
3.2.3 時間步與空間步的參數最佳化 ........................................................... 29
3.2.4 計算邊界的處理方法 ........................................................................... 32
第四章 均勻介質中的彈性波場型計算 ................................................. 33
4.1 二維矩形網格FDTD的模擬 ........................................................................ 34
4.1.1 矩形網格的布局與方程式離散化 ...................................................... 34
4.1.2 矩形網格的模擬結果 .......................................................................... 39
4.2 二維圓柱形網格FDTD的模擬 .................................................................... 42
4.2.1 圓柱形網格的布局與方程式離散化 .................................................. 42
4.2.2 圓柱形網格的模擬結果 ...................................................................... 47
4.3 等間距柱形網格轉換為等間距矩形網格 ..................................................... 50
4.3.1 二維內差法說明 ................................................................................... 50
4.3.2 圓柱形網格與矩形網格的交互驗證 ................................................... 50
第五章 圓柱多層二維結構中的彈性波場型計算................................. 53
5.1 圓柱多層結構與FDTD模擬 ........................................................................ 53
5.1.1 圓柱多層結構 ....................................................................................... 53
5.1.2 FDTD 模擬 ........................................................................................... 54
5.2 圓柱形網格FDTD的彈性波場型模擬結果與討論 .................................... 57
5.2.1 圓柱多層結構—水泥層無裂縫 ........................................................... 57
5.2.2 圓柱多層結構—水泥層中存有裂縫 ................................................... 63
5.2.3 圓柱多層結構—軌跡比較圖 ............................................................... 68
第六章 結論 ............................................................................................. 72
參考文獻 ..................................................................................................... 74

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