兩條平行河川間自由水層因抽水所引起的三維地下水流通解

Abstract

本研究發展一個數學模式，用以描述在兩條平行河川之間的自由含水層，因抽水井所引起的三維地下水流之解析解。所建構的數學模式，包含一個新的地下水流控制方程式，使所發展的解析解可適用於三種抽水井，包含垂直井、水平井、及輻射收集井；接著採用自由液面方程式，描述液面因抽水而產生的洩降，並以第三類邊界條件代表兩條平行河川的低透水性河床。應用積分轉換的方法，推導出此數學模式的水力水頭解析解，此解由含特徵函數的三階無限級數所組成，其特徵值的計算需用到尋根法；我們提出一個解析公式用來求適當的初始猜值，配合牛頓法可有效率地求出特徵值。依據達西定律和水頭解析解，可導出描述河川滲入率(stream depletion rate, SDR)的解析解，此解析解對現地的水平井或輻射收集井的預測，可與現地問題量測的結果相吻合。依據解析解的模擬結果，我們得到以下的結論：未受壓含水層中的重力排水對SDR有顯著的影響，在抽水中期，SDR不隨抽水時間增加而增加，若忽略水頭垂直方向的變化，會顯著地高估SDR，此結論和紐西蘭Doyleston附近的現地實驗結果一致。此外，輻射收集井的側臂分布對於洩降有顯著的影響，在河川滲入行為發生之前，最大洩降位於在輻射井的中心；當河水開始滲入含水層，最大洩降開始遠離井中心並往內陸方向移動。

This thesis develops a mathematical model for describing three-dimensional groundwater flow induced from a vertical well, horizontal well or radial collector well (RC well) in an unconfined aquifer bounded by two parallel streams. A new governing equation with a sink term standing for the well is presented. A simplified free surface equation is used to describe the depletion of water table in the aquifer. The third-type boundary condition is employed for the boundary condition at the interface where a low-permeability streambed is connected to the aquifer. The aquifer we concern is of finite extent; therefore, the head solution of the model, derived by integral transforms, can be expressed in terms of an infinite series with eigenvalues requiring a root-finding scheme such as Newton method. An analytical expression is developed to give initial guesses for the eigenvalues. The solution for stream depletion rate (SDR) describing filtration rate from the streams is acquired based on Darcy’s law and the head solution. The present solution is applied to predict the hydraulic head near a horizontal well or a RC well for the real-world cases. The predicted results are reasonable when compared with the field observed data. With the aid of the present solution, we have found that the gravity drainage of an unconfined aquifer has significant effects on temporal SDR. The curve of temporal SDR tends to be flat due to the gravity drainage during the middle period of pumping time. The vertical groundwater flow described by the free surface equation should be used even for the case of a fully-penetrating well. The SDR will be overestimated if neglecting the vertical flow in the model. Such a result is confirmed by the comparison of SDR predicted from the present solution with that taken from a field SDR experiment executed near Doyleston in New Zealand. Additionally, lateral configurations of a RC well have significant effects on spatial drawdown distributions. The largest drawdown occurs right at the center of a RC well before the filtration and moves landward once the filtration starts to recharge the aquifer.