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研究生： 王士榮
Wang, Shih-Jung
論文名稱： 序率孔彈性模式之研究
The Study on Stochastic Poroelastic Model
指導教授： 徐國錦
Hsu, Kuo-Chin
學位類別： 博士
Doctor
系所名稱： 工學院 - 資源工程學系
Department of Resources Engineering
論文出版年： 2009
畢業學年度： 97
語文別： 中文
論文頁數： 167
中文關鍵詞： 倒轉行為 、水壓變化 、地層下陷 、變形效應 、不確定性 、土體變形 、蒙地卡羅法 、序率孔彈性理論 、一階二動差法
外文關鍵詞： Reverse behavior, Deformation effect, Subsidence, Change in pore water pressure, Soil deformation, Uncertainty, Monte Carlo simulation, First-order second-moment method, Stochastic poroelastic theory
相關次數： 點閱：147  下載：5 
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    • 固體與流體間的交互作用為水文學中重要的物理機制之一，並廣泛應用於土壤力學、岩石力學或是水文地質學等領域中，而孔彈性理論為目前常用來解釋土體與水體間耦合作用之物理模式。然而，目前孔彈性理論幾乎僅在定率模式中做探討，而現地採樣之不確定性與資料之不完全性，造成定率模式之應用受到限制。序率方法採用統計的概念，設定參數或邊界條件為隨機變數來探討變數間之平均行為與共變異行為，不僅可以獲得定率式之結果，還可以可用來估算變數之相關性與不確定性，提供更多孔彈性耦合系統中之資訊。本研究為首次採用計算效率較高之一階二動差序率方法（FOSM）來建構FOSM序率孔彈性模式，同時並採用序率方法中常用之蒙地卡羅法（MC）來建構MC序率孔彈性模式，再藉由孔彈性參數受到變形效應下之變化建構出非線性模式，最後以一維度模式展現序率孔彈性理論之應用。
    模式驗證中，FOSM序率孔彈性模式在頂端載重案例中之平均值解，與共變異函數在側向水流案例中，皆與文獻中之解析解和數值解一致，確認本研究所建構之FOSM序率孔彈性模式之適用性。敏感度分析中，影響模式行為較顯著之參數為水力傳導係數與楊氏模數，而泊松比對模式之影響與楊氏模數相似，孔隙率則無顯著影響。頂端載重與底端排水之邊界作用對土體造成之影響不同，而楊氏模數對頂端載重與底端排水作用下之位移影響較顯著，而水力傳導係數則對底端排水作用下之位移與孔隙水壓變化之影響皆顯著。
    在地層下陷問題之分析上，頂端載重與底端排水邊界條件對土體沉陷行為之影響不同，而同時施加頂端載重與底端排水時，初始行為主要受到頂端載重之影響而後段之行為則主要受到底端排水之影響，且其平均行為符合線性疊加原理。受到已知邊界之影響，位移與孔隙水壓變化之變異函數在已知邊界為零，且隨著已知邊界距離之增加而增加；頂端載重作用下，最終土體趨近於靜態穩定，因此共變異函數會趨近於零；底端排水作用下，最終土體趨近於動態穩定，因此共變異函數會趨近於定值，其不確定性來自於土體中水流之影響。
    在變形效應作用下，孔隙率、水力傳導係數與楊氏模數之變化率結果皆顯示，頂端載重加底端排水之作用小於同時施加頂端載重與底端排水作用之結果，表示同時施加兩種邊界驅動力時，其綜合作用之效應大於任何單獨驅動力之作用。平均總沉陷量、孔隙水壓變化與頂端水流通量在無變形效應下符合線性疊加原理，而受到變形效應之影響成為非線性行為。單向壓密試驗之參數擬合結果顯示，水力傳導係數、楊氏模數與泊松比之參數推估與使用土體介於粉砂到黏土間之特性相當。而沉陷量與孔隙水壓變化之變異數解中，土體在壓密初始時間僅在靠近兩邊界端處產生排水與沉陷，因此造成孔隙水壓變化之最大變異數發生在靠近邊界處且為對稱；而受到底部固定位移邊界之影響，沉陷量之最大變異數發生在頂端位置。
    在倒轉效應之模擬中，變異函數 可代表孔隙水壓變化與位移相對於平均值之相關性，在FOSM與MC數值模式中，抽水時 在淺層且短時間內，為孔隙水壓下降而位移亦減少（膨脹）之倒轉行為；而排水作用停止的模擬中，同樣可以觀察到Rhade效應，且當水力傳導係數或楊氏模數較小時，倒轉行為較為顯著。倒轉行為的發生和空間與時間之尺度有關，當空間尺度較小或時間尺度較小時，較大的水力傳導係數或楊氏模數所建立的模式中，將同樣可以觀察到倒轉行為。

    The interaction between solid deformation and fluid flow is an important mechanism in hydrogeology. The behavior is widely applied to soil mechanics, rock mechanics, and hydrogeology fields. Poroelastic theory is a commonly used physical model, which uses displacement and change in pore water pressure as the variables to explain the coupled behavior between soil deformation and water flow. The traditional poroelastic theory is treated as the deterministic model; however, material properties usually show strong variations in nature and the heterogeneity may strongly affect the system. In this study stochastic approach was applied to explore the statistic behavior and estimate the uncertainty. The first-order second-moment (FOSM) stochastic method was used to construct the stochastic poroelastic model. The commonly used stochastic method – Monte Carlo simulation (MC) was also constructed to validate the results of FOSM. The nonlinear effect on the parameters change caused by deformation was taken into consideration. Both mean and covariance behavior of variables were evaluated in the stochastic poroelastic model under different scenarios. The numerical model was verified and showed the perfectly match with the analytical and numerical solutions. Parameter sensitivity was analyzed and showed that hydraulic conductivity and Young’s modulus play different roles under different driving forces.
    The application of the stochastic poroelastic model to the subsidence problems shows that the dynamic behavior of a coupled flow-stress system is more complex than a system that does not consider the deformation of porous media. Loading effects on deformation and pore pressure are instantaneous while the effect of discharge takes time to propagate from the boundary through the whole domain. Both loading and discharge boundary conditions can significantly affect the uncertainty of the system response. In the scenario combining loading on the top boundary and discharge at the bottom boundary, the mean total settlement and the average flux satisfy the relationship of superposition to be the sum of the separated effects of loading on the top boundary and discharge at the bottom boundary, but the variances do not.
    The results of deformation effect show that the maximum changes of porosity, hydraulic conductivity, and Young’s modulus are nonlinear in three examined scenarios. While hydraulic conductivity increases the time for the system to reach steady state, Young’s modulus decreases the time. The coupled system that considers the deformation effect takes longer to reach the final state, especially for the second-moment statistics. The nonlinear stochastic poroelastic model is applied to the laboratory consolidation experiment to estimate the parameters in the consolidation test. Besides the mean estimation of parameter, the model also provides the uncertainty analysis in the simulation.
    The reserve phenomenon of aquifer due to groundwater pumping and its ceases was investigated. The results show that while the expansion of soil deformation and the reduction in the change of pore water pressure are not significant in the system’s mean behavior, the second moment solutions were able to well capture the reverse phenomena that are analogous to the Noordbergum and Rhade effects shown in the deterministic system. The adopted one-dimensional FOSM method can serve as a tool to analyze the reverse phenomenon occurring in heterogeneous porous media, where the monitoring screen is adjacent to the pumped layer but at a distance from the pumping well. The best monitoring location is found to be close to mid depth of the observing area.

    目 錄 頁次 摘要　　　…………………………………………………………………………… I ABSTRACT　　……………………………………………………………………… III 誌謝　　　…………………………………………………………………………… V 目錄　　　…………………………………………………………………………… VI 表目錄　　…………………………………………………………………………… IX 圖目錄　　…………………………………………………………………………… X 符號表　　…………………………………………………………………………… XVI 第一章 緒論　…………………………………………………………………… 1 　　1.1 研究動機與目的　……………………………………………………… 1 　　1.2 前人研究　………………………………………………………………… 3 　　　　1.2-1 孔彈性理論之發展與應用　…………………………………… 3 　　　　1.2-2 序率理論之發展與應用　　…………………………………… 4 　　1.3 研究步驟與流程　………………………………………………………… 6 第二章 理論介紹　……………………………………………………………… 11 　　2.1 Terzaghi壓密理論　………………………………………………………… 11 　　2.2 孔彈性理論　……………………………………………………………… 15 　　2.3 序率理論　………………………………………………………………… 18 第三章 序率孔彈性理論模式建構　…………………………………………… 23 　　3.1 異質場孔彈性理論控制方程式推導　…………………………………… 23 　　3.2 FOSM序率孔彈性理論之建構　…………………………………………… 25 　　　　3.2-1 序率孔彈性理論控制方程式　………………………………… 25 　　　　3.2-2 初始與邊界條件拆解　　……………………………………… 29 　　3.3 參數之變形效應　　……………………………………………………… 36 　　　　3.3-1 參數因土體變形而變化之關係式　……………………………… 36 　　　　3.3-2 參數變化之分析　……………………………………………… 40 第四章 一維度序率孔彈性模式建構　　……………………………………… 45 　　4.1 FOSM序率方法　　……………………………………………………… 45 　　　　4.1-1 一維度FOSM序率孔彈性控制方程式　………………………… 45 　　　　4.1-2 初始與邊界條件　……………………………………………… 47 　　4.2 數值方法　　　…………………………………………………………… 48 　　4.3 Thomas演算法　…………………………………………………………… 52 　　4.4 序率孔彈性模式求解方法　　…………………………………………… 53 　　　　4.4-1 FOSM序率孔彈性模式求解　……………………………………… 53 　　　　4.4-2 MC序率孔彈性模式求解　　……………………………………… 55 第五章 模式基本分析　………………………………………………………… 59 　　5.1 模式之驗證與比較　　………………………………………………… 59 　　　　5.1-1 頂端載重狀態之平均值解比較　………………………………. 59 　　　　5.1-2 無變形水流條件下之變異函數解比較　……………………… 65 　　　　5.1-3 MC與FOSM之結果比較　………………………………………… 68 　　5.2 敏感度分析　……………………………………………………………… 75 第六章 地層下陷行為之探討與分析　………………………………………… 83 　　6.1 地層下陷研究現況　……………………………………………………… 83 　　6.2 邊界驅動模式之設定　…………………………………………………… 84 　　6.3 三種邊界驅動模式之結果　　…………………………………………… 86 　　　　6.3-1 案例1：頂端載重且底端不排水 ………………………………… 86 　　　　6.3-2 案例2：頂端不載重且底端排水 ……………………………… 89 　　　　6.3-3 案例3：頂端載重且底端排水 ……………………………… 92 　　6.4 三種邊界驅動模式之探討　　…………………………………………… 94 第七章 非線性模式之分析與應用　　………………………………………… 99 　　7.1 變形效應研究現況　……………………………………………………… 99 　　7.2 非線性MC序率孔彈性模式分析　…………………………………… 100 　　　　7.2-1 變形效應模式之設定　………………………………………… 100 　　　　7.2-2 變形效應模式之結果與比較　………………………………… 102 　　　　7.2-3 變形效應在邊界驅動模式中之探討　………………………… 114 　　7.3 非線性模式在單向壓密試驗之應用　…………………………………… 119 　　　　7.3-1 非線性模式與單向壓密試驗　………………………………… 119 　　　　7.3-2 單向壓密試驗　………………………………………………… 120 　　　　7.3-3 單向壓密試驗模式之設定　　………………………………… 122 　　　　7.3-4 單向壓密試驗模式之結果與討論　　………………………… 126 第八章 位移與水壓之倒轉行為分析　　……………………………………… 135 　　8.1 抽水引致倒轉行為之研究現況　………………………………………… 135 　　8.2 倒轉效應模式之設定　…………………………………………………… 137 　　8.3 倒轉效應模式之結果與討論　　………………………………………… 139 　　　　8.3-1 抽水引致一維度序率孔彈性模式之倒轉行為　……………… 139 　　　　8.3-2 倒轉效應之參數敏感度分析　………………………………… 144 　　　　8.3-3 Rhade效應　　……………………………………………………… 148 第九章 結論與建議　…………………………………………………………… 151 　　9.1 結論　……………………………………………………………………… 151 　　　　9.1-1 序率孔彈性模式之建構、驗證與敏感度分析　………………… 151 　　　　9.1-2 地層下陷行為之探討與應用　………………………………… 152 　　　　9.1-3 非線性模式之分析與應用　　………………………………… 152 　　　　9.1-4 位移與水壓倒轉行為分析　　………………………………… 153 　　9.2 建議　……………………………………………………………………… 154 參考文獻　　　……………………………………………………………………… 155 個人簡歷與著作　…………………………………………………………………… 163

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