利用伴隨狀態法耦合水流與傳輸逆推地下水參數及未知條件

Abstract

本研究提出一個一般化逆向問題求解方法，結合了地下水模擬模式、伴隨狀態變數法、梯度法與近似牛頓法，在最小誤差平方和的架構下，最佳化估計未知參數、初始條件或邊界條件。應用伴隨狀態法推導所得之伴隨問題是一種對於參數或條件估計錯誤所造成之模擬誤差的描述，藉由狀態變數與伴隨狀態變數的積分，目標函數對應於所有未知數的梯度值可以快速獲得。 本研究以水平二維拘限含水層為考量對象，首先以一個設計過的示蹤劑試驗探討水力傳導係數(K)檢定問題。除了水頭與一階動差觀測外，再加入了地下水流速與示蹤劑零階動差的觀測。貢獻度指標分析結果指出流速僅包含觀測位置K的資訊，而水頭與動差則包含了大範圍內K 值分佈的訊息。 本研究提出不同示蹤劑濃度釋放策略，營造一個隨空間變化的零階動差場，解決零階動差二元分佈問題，且所得之零階動差其對於檢定K 的貢獻度還高於水頭與一階動差。參數檢定結果發現，水頭觀測資料描述的是地下水等勢能線的分佈，此分佈直接反應參數場大小；動差觀測描述的是流線的分佈，可以表現出流線與流線間K 值差異與的資訊。 在完整逆向問題的探討上，本研究以一個試驗設計過的抽水實驗配合最佳化演算法，同時估計蓄水係數、導水係數、初始水位、邊界水位與邊界流量。相關性分析結果顯示蓄水係數與初始水位呈高度相關；導水係數與邊界水位、邊界流量皆高度相關，這樣的相關性導致逆向問題成為非唯一解。觀測貢獻度分析發現抽水實驗初期的非穩態洩降對於蓄水係數與初始條件檢定貢獻度較高，但單一個水位觀測的貢獻度仍小於一；而抽水晚期的穩態水位觀測，則是對於檢定導水係數與邊界條件特別有效。靠近邊界的觀測井對於邊界條件特別敏感，而抽水井的洩降主要貢獻至參數檢定。在觀測充足的狀況下，逆向演算法可有效融合上述這些資訊，使得檢定結果收斂至未知數的真值。

In this paper, groundwater simulation models, adjoint state method, gradient search method, and least square algorithm are combined to formulate an efficient optimization approach to solve the groundwater inverse problem. The adjoint state method is used to evaluate effectively the gradient of objective function with respect to parameter or condition. The horizontal two-dimensional confined aquifer is the research target. First, the parameter identification is discussed through a designed tracer test. Head, flux, zeroth and first moment observations are utilized to estimate hydraulic conductivity(K) field. The moment observations at different locations contain some indirect trend information of K field. Next to head observations, they provide additional knowledge useful to parameter identification in groundwater system. Using all three kinds together, the case study demonstrates to elevate the efficiency and accuracy of solution substantially. Second, the complete inverse problem is taken into consideration. A designed pumping test is performed to simultaneously estimate aquifer parameters, initial condition, and boundary conditions in groundwater modeling. The correlation analysis shows high connection between storage coefficient and initial condition. Besides, transmissivity and boundary conditions are also highly correlated. A time series of unsteady head is needed for storage coefficient and initial condition estimation. The observation near boundary is very effective on boundary condition estimation. The observation at pumping well mostly contributes to the estimation of transmissivity.