滲漏含水層考慮彎曲效應之地下水流半解析解

Abstract

傳統地下水理論假設含水層總應力為一定值，且忽略受壓含水層在抽水時所伴隨產生的力學行為，但實際抽水時，水層結構可能會像平板一樣彎曲。本研究目的為探討定流量抽水下，推導滲漏受壓含水層受到彎曲效應影響所產生洩降分佈的半解析解。本研究利用所發展出的半解析解，探討滲漏受壓含水層系統在定抽水率下，洩降分佈受到彎曲及滲漏效應的影響。藉由建立數學模式，以描述滲漏受壓含水層系統承受滯水層中彎曲效應影響的洩降分佈。此模式包括三個耦合的控制方程式，分別是描述受壓水層暫態洩降分佈方程式、滲漏水層暫態洩降分佈方程式、及基於板殼理論所描述水層彎曲效應的方程式；接著，針對此三個方程式，應用拉普拉斯（Laplace）和漢可爾（Hankel）轉換方法求得模式的半解析解，最後，使用修正Crump方法進行數值逆轉推求時間域的結果。此新解在忽略彎曲或滲漏效應下，可分別簡化至拉普拉斯域的Hantush (1960) 解和時間域的Wang et al. (2004) 解。本研究利用所推得的解作分析，顯示抽水試驗初期含水層洩降受到彎曲效應的影響，而試驗晚期則受到含水層滲漏因子的影響較為明顯。敏感度分析結果顯示，含水層壓縮性只在試驗初期敏感，造成含水層釋放出的水量較傳統地下水理論估計值小。因含水層洩降分佈在試驗晚期對滯水層水力傳導係數較具敏感度，導致相較於無滲漏受壓含水層具有較小的含水層洩降分佈。此外，洩降對含水層的厚度及水力傳導係數具有最高的敏感度。

Traditionally, the groundwater flow induced by pumping often assumes that the total stress of aquifer maintains constant all the time and the aquitard mechanical behavior is negligible. The formation of the aquifer system may actually bend like a plate when subject to pumping. This study aims at developing a new semi-analytical model for investigating the impacts of aquitard bending and leakage rate on the drawdown of the confined aquifer due to a constant-rate pumping in a leaky confined aquifer (LCA) system. A mathematical model is first build for delineating the LCA drawdown distribution subject to bending effect emerged in the confining unit. This model contains three governing equations, delineating the transient behavior of drawdown distributions in the aquitard and the confined aquifer and the vertical displacement based on a thin plate theory of small-deflection in response to aquitard bending. The semi-analytical solution is found by employing the methods of the Laplace transform and Hankel transform. The corresponding time-domain result of the derived solution is established by applying the modified Crump method. Without considering the bending effect, this new solution can reduce to the Hantush solution in Laplace domain (Hantush, M.S., 1960. Modification of the theory of leaky aquifers. Journal of Geophysical Research 65(11), 3713-3725). On the other hand, this solution reduces to the Wang et al. solution (Wang, X.S., Chen, C.X., Jiao, J.J., 2004. Modified Theis equation by considering the bending effect of the confining unit. Advances in Water Resources 27(10), 981-990.) in time domain when neglecting the leakage effect. The results predicted from the present solution show that the aquifer drawdown will be influenced by the bending effect at early time and the leakage effect at late time. The results of sensitivity analysis indicate that the skeleton compression of the aquifer is sensitive only at early time, causing less amount of water released from pumped aquifer than that predicted by the traditional groundwater theory. The aquifer drawdown is more sensitive to aquitard’s hydraulic conductivity at late time, leading smaller drawdown than non-leaky one. Additionally, both the hydraulic conductivity and thickness of the aquifer are the most sensitive parameters to the predicted drawdown.