有限自由含水層中水平井抽水引起的地下水流解析模式

Abstract

本研究建構一個數學模式，描述於自由含水層內因水平井抽水引起的三維地下水流。考慮含水層水平向四個邊界呈現矩形，分別包含兩條平行的河川和兩個平行的不透水邊界；而垂直向厚度有限，水層底部不透水、頂部為自由液面。模式包括代表井抽水的點沉項，自由液面由一階方程式來描述其水位的變化，而河川有低滲透性河床，由Robin邊界條件代表。應用雙重積分轉換、拉普拉斯轉換、和餘數定理，可求得模式的水頭解析解；接著，從點沉項的位置，沿著水平井對水頭解析解作積分，所得結果除以井長度，可獲得水平井抽水引起的水頭分佈解析解。透過達西定律與水頭解析解，可導出河川袪水率的解析解，以描述井抽水產生的河川滲流量。本研究也進行水頭對參數改變的敏感度分析，參數包含水平與垂直向的水力傳導係數、河床滲透性、比出水率、比產水率、及水平井的長度和深度。此外，我們提出一個量化條件，指出水層中水平井引起的三維和二維水流的適用範圍。此外，針對河川袪水率，我們也提出另一個量化條件，以判斷可忽略垂直流的情況。當滿足上述兩個條件時，過去的二維水流模式，即可用來估計地下水位變化和河川袪水率。

This study develops a mathematical model for describing three-dimensional (3-D) groundwater flow due to pumping at a horizontal well in a finite unconfined aquifer bounded by two parallel streams and two no-flow boundaries. The 3-D transient flow equation of the hydraulic head has a point sink term representing the pumping rate of the well. A first-order free surface equation is used to describe water table change, and the Robin boundary condition is chosen to represent the flow rate across the streambed. The point-sink solution of the model is obtained based on the methods of double-integral transform and Laplace transform. The solution of the head distribution is then developed by integrating the point-sink solution along the well and dividing the result by the well length. On the basis of Darcy’s law and the head distribution along the streams, the solution for the stream depletion rate (SDR) can also be developed. With the aid of the head solution, the sensitivity analysis can be then applied to explore the behavior of the hydraulic head in response to the change of a specific parameter such as horizontal and vertical hydraulic conductivities, streambed permeability, specific storage, specific yield, well length and well depth. A quantitative criterion is provided to identify the region where the groundwater flow is 3-D or 2-D without the vertical component. In addition, another criterion is also given to allow the neglect of the effect of vertical flow on the SDR. Conventional 2-D flow models are capable of predicting accurate head and SDR if satisfying these two criteria.