水井捕集區（well capture zone）的制定，對地下水的水源劃定精確的水井保護區(well head protection area, WHPA)非常重要。由於自然的含水層系統的水力參數包含著不同程度的異質性，這些參數的變異性將直接地影響流場的測量與水井捕集區的制定，序率方法可量化由於自然含水層複雜的異質性以及含水層參數獲得的限制所造成的不確定性。本研究中，利用一階近似法（first-order method）之非定常性譜方法（Nonstationary spectral method, NSM）與譜方法近似解（approximate spectral method, ASM）配合反向質點追蹤法（backward particle tracking），發展出利用序率模式制定異質性受壓含水層水井捕集區。在求解水井捕集區的過程中，先利用一階近似法求解地下水流速的不確定性，並利用質點追蹤的直接傳遞概念定義在特定時間內水井捕集區所形成的不確定性區間。本研究利用數個測試例量化發展的一階近似法之精確度，其中包含有限邊界地下水流場、多井系統流場、多尺度水力傳導係數以及由於源匯項所造成的非定常性流場，並考慮不同程度的異質性(ln K 小尺度擾動量之變異數 =0.1、0.5與 1.0)，推估由於含水層異質性對制定水井捕集區所造成的影響。由本研究測試例結果顯示，在相同的水文條件下，ASM與NSM所推估出的結果與蒙地卡羅法（Monte Carlo method, MCS）近似，譜方法近似解計算效率快速，並且所推估出的結果與NSM符合。由模擬的結果得知，ASM所需要的計算機計算時間僅數秒至數分鐘，而NSM需要數分鐘至數小時，MCS則需要長達數小時甚至數週的數值計算時間。ASM法的提出，包含了解析與數值兩種序率理論的優點，提供了現實中解決序率地下水模擬問題的方法。

The delineation of well capture zones is of great importance to accurately define well head protection area (WHPA) for potential groundwater resources. Because natural aquifer systems typically involve different extent of heterogeneity in aquifer parameters, such parameter variations will then directly influence the estimations of flow fields and the delineations of well capture zones. Stochastic methods can be used to quantify uncertainties that are caused by natural complexity of aquifer heterogeneity and limited capability to obtain sufficient aquifer parameters. The ultimate results of stochastic delineation of capture zones are to contribute the remediation designs and risk analysis for contaminated aquifers. In this study, two first-order methods, including nonstationary spectral method (NSM) and approximate spectral method (ASM), associated with particle backward tracking algorithm are developed to delineate stochastic capture zones in heterogeneous confined aquifer systems. The Monte Carlo simulation (MCS) is employed for validating the developed first-order methods (i.e., ASM and NSM). To obtain a stochastic capture zone, the developed first-order methods are first employed to solve for velocity uncertainties. The concept of direct propagation of uncertainties of particle tracks is then used to define the uncertainty bandwidth of the stochastic capture zone for a specified time. In this study the developed first-order methods are first assessed to quantify the accuracy of delineated well capture zones under a variety of conditions, including bounded flow domains, multiple wells flow systems, multiple hydraulic conductivity scales, and nonstationary flows caused by complex sources and sinks in the modeling areas. Additionally, different degrees of heterogeneity (i.e., the small-scale ln K variance =0.1, 0.5, and 1.0) are considered in all modeling cases to get general insight into the effect of aquifer heterogeneity on the delineations of well capture zones. The simulation results reveal that the solutions of the NSM and ASM agree reasonably with those of MCS under same hydrogeological conditions. The ASM is efficient and the ASM solutions agree well with the full version of numerical spectral method NSM. In general the computation times for the ASM solutions are in the orders of seconds to minutes based on the computational system with i7 CPU system, while the computation times for the NSM solutions are in the orders of minutes to hours. However, the computation times for the costly MCS are in the orders of hours to possibly several weeks. The proposed ASM method has conceptually taken the advantages of analytical and numerical stochastic theories and provided an opportunity to include stochastic theories in practical groundwater modeling problems.

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