三維地層下陷之解析解研究

Abstract

目前有關於地層下陷之分析方法，多著重於數值計算模式，對於應用數學解析模式則較少討論，尤以荷重引致地層下陷之研究更是不多。因此，依據曾鈞敏(2009)推導出含孔隙水壓力與土體位移之三維非耦合方程式，並發現其型式與史托克斯流(Stokes’ flow)相似之研究成果，本研究擬建立抽水與荷重引致地層下陷控制方程式，並就控制方程式之特性尋找適當之解析解，最後以案例分析說明。 本研究因抽水與荷重引致地層下陷控制方程式之型式與史托克斯流(Stokes’ flow)相似，故似可以邊界積分法求解，但由於壓力與位移的量階(order)相差太多，易造成數值運算的不穩定，故無法使用邊界積分法求解地層下陷之問題。 對於求解抽水問題時，點抽水源可視為點涵(point sink)作用，故壓力為源流奇異點，可採用奇異解法(singularity method)之映射方法(image method)求解；然而，位移項並無奇異點可與壓力對應，故無法以映射法求解位移；因此，本研究結合映射法與參考Happel and Brenner(1983)中提及利用分離變數法(separation of variables)求解三維史托克斯流於圓柱座標下之解析解方法，即複合法，成功求解出抽水行為下之壓力場與位移場。 對於求解荷重問題時，假設荷重強度為點力(point force)作用，故位移項可視為Stokeslet奇異點；對應壓力則為Dipole型式，因此可使用奇異解法之映射法搭配適當邊界條件，求出壓力場與位移場；此外，關於Stokeslet於剛性邊界之映射解，可引用Blake(1971)以Stokeslet映射無滑動剛性邊界之結果。 最後分別以非拘限有限深含水層與拘限有限深含水層兩個案例說明，由計算結果可知當抽水與荷重引致地層下陷時，其壓力與位移以及垂直位移量與水平位移量的量階差，以作為後續研究以數值方法計算之參考。

On landsubsidence analysis, application of numerical model is more than analystical model, and the study of landsubsidence due to loding is less disscussion. Therefore, according to the complete uncoupled equation using only pore pressure and soil displacement of Tseng’s(2009), we construct the landsubsidence governing equation due to groundwater overpumping and loding, then find the appropriate analytical solution and illustrate the result with cases. Because the three-dimensional uncoupled equation is very similar to Stokes’ flow, it seems that we can use boundary integral element method to slove equation. However, the order of pore water pressure is more than soil displacement, hence that will cause computational instability. In the landsubsidence due to overpumping problem, water being pumped underground is a point sink, and the pressure singularity is mass source, hence it can sloved by the image method of singularity method. But the displacement can not use image method. Consequently, we combine image method with separation of variables method which sloved three-dimensional Stokes’ flow in cylindrical coordinates by Happel and Brenner(1983) to successfully solve the governing equation, and it’s called multiple method. In the landsubsidence due to loading problem, the load strength is a point force, hence the displacement singularity is a stokeslet and pressure singularity is a dipole, so it is sloved by the image method of singularity method. Besides, in this study we can apply the result of image system for a stokeslet in a no-slip boundary of Blake(1971). We illustrate result with the two cases of finite depth unconfined aquifer and finite depth confined aquifer. From thres result, in the landsubsidence due to groundwater overpumping and loading problem, we understand the amount of order of pressure and displacement and vertical displacement and horizontal displacement. Therefore, it is a useful reference for numerical model.