定水頭部分貫穿井混合邊界值問題之研究

Abstract

含水層試驗通常為用來瞭解水層水文地質參數的重要工作。與水層試驗有關常用到的解析解，絕大多數是考慮井內的濾管(井篩)為全開的情況，即濾管貫穿整個水層的厚度。在濾管是全開的假設下，水層內的流況可視為水平流，在這種情況下，描述水層內水流的數學方程式不含垂直方向的分量，故較容易求得解析解。若考慮水層為部份貫穿，則數學上，定水頭試驗井會形成混合邊界，即在井緣同時存在兩種邊界，在井篩上為定水頭邊界，井篩外為不透水邊界。此混合邊界值問題，無法使用傳統的積分轉換方法求解，因而變得非常難解。在過去文獻中，對於此類混合邊界問題，皆將井緣的邊界條件作簡化以推求半解析解或解析解，然而這樣的解，計算得井緣附近的水位與流速場值，都會有數值誤差。 本研究利用Laplace及finite Fourier cosine transform將偏微分方程轉換成常微分方程式，接著代入井緣的邊界條件裡，形成一組dual或triple series equations，最後解得方程式裡的未知係數，即可計算在侷限含水層中的洩降值。

The mathematical model describing the aquifer response to a constant head test performed at a fully penetrating well can be easily solved by the conventional integral transform technique. The Dirichlet-type condition should be chosen as the boundary condition along the screen for such a test well. However, the boundary condition for a test well with partial penetration must be considered as a mixed-type condition. Generally, the Dirichlet condition is prescribed along the well screen and the Neumann type no-flow condition is specified over the unscreened part of the test well. The model for such a mixed boundary problem in a confined aquifer system of infinite radial extent and finite vertical extent is solved by the method of dual or triple series equations. This approach provides analytical results for the drawdown in the partially penetrating well and the well discharge along the screen. The semi-analytical solutions are particularly useful for the practical applications from the computational point of view.