利用傅立葉及拉普拉斯轉換推求單一水平井的水力水頭解析解

Abstract

本篇論文發展出一數學模式，針對一個與河川平行的單一水平井，描述在井抽水時自由含水層中水位的分佈。分別利用傅立葉正弦、傅立葉及拉普拉斯轉換方法，對控制方程式及相關的邊界條件、初始條件進行轉換，求得在拉普拉斯域的解；接著，利用Bromwich積分法對拉普拉斯域解作逆轉，最後依序進行傅立葉、傅立葉正弦作逆轉而求得時間域的解。本文將自由含水層的上層邊界條件，設定為自由液面方程式，但忽略方程式裡的二次微分項。所求得的解，透過達西定律可以推導得河川袪水率的方程式，若河川遠離至不受抽水影響處，則此解析解可用來描述無限延伸的自由含水層之水位分佈。此外，若是把比出水量設定為零，則此解析解可以描述受壓含水層抽水洩降的分佈。

This thesis develops a mathematical model for describing the head distribution in an unconfined aquifer with a single pumping horizontal well parallel to a fully penetrating stream. The upper boundary of the aquifer is represented by a free water surface equation in which the second-order differential terms are neglected. The Laplace-domain solution of the model is developed by applying Fourier sine, Fourier and Laplace transforms to the governing equation as well as the associated initial and boundary conditions. The time-domain solution is obtained after taking the inverse Laplace transform using the Bromwich integral method and inverse Fourier and Fourier sine transforms. Based on the solution and Darcy’s law, the equation representing the stream depletion rate is then derived. The solution can be used to simulate head distributions in an aquifer infinitely extending in horizontal direction if the well is located far away from the stream. In addition, the solution can also be used to simulate head distributions in confined aquifers if the specific yield is set zero.