在島嶼含水層系統受雙重潮汐效應影響的水頭變動之分析

Abstract

本論文發展出一個可描述在海島地形的含水層中，受雙重潮汐影響致使地下水頭變動的數學模式。其上層邊界條件以一含有滲漏項的簡化自由液面方程式表示，且此方程式可廣泛地表達受壓、非受壓、滲漏受壓含水層的上層邊界條件。此模式之閉合解由兩個級數組成，以直接傅立葉法與有限傅立葉正弦轉換推導所得，其中一個級數可透過特殊的圍線路徑積分與殘值定理簡化成閉合形式。若地形條件為寬闊的海島，則此解預測的地下水頭，與受單邊潮汐影響之含水層的解相同。本研究發現，即使上邊界為低滲透性的弱透水層，位於其下的受壓水層仍會有顯著的垂直流。另外，以此解析解預測澳洲花園島的振幅係數與相位差，與觀測所得之五十七天水位變動的現地數據相當契合。

A general mathematical model is developed for describing head fluctuations in an aquifer of oceanic islands subject to a dual tide effect. The upper boundary condition of the aquifer is represented by an equation combining the simplified free surface equation with a leakage term. Such an equation is considered as a general expression representing the upper boundary condition of confined, unconfined, or leaky confined aquifers. The closed-form solution of the model represented by two series terms is then developed by the direct Fourier method and finite Fourier sine transform. One of the series can further reduce to a closed-form expression by means of contour integral and residue theorem. Under the condition that the width of the island is very large, this solution can give the predicted head almost the same as that of the solution for an aquifer subject to a single tide effect. It is found that the presence of an upper aquitard may result in significant vertical flow in the lower confined aquifer even if the aquitard permeability is low. The amplitude coefficient and phase shift predicted from the solution subject to the dual tide effect agree well with the ones estimated from 57-day head fluctuation data observed at Garden Island, Australia.