定水頭與定流量試驗洩降解之研究

Abstract

Theis方程式可用來估算抽水條件下洩降隨著空間或時間的變化，亦可根據洩降觀測值推估含水層的參數。物理上，在抽水試驗的初期，已知井位的洩降觀測值會隨時間而變化，然後隨著時間的增加而趨於穩定。但是在數學上，Theis方程式在時間很大時，並不能回復到穩態的Thiem 方程式。此外，被廣泛應用的Thiem方程式，卻不適用於距離為零或無限遠時的情況。另一方面，由於徑向流場的洩降解，具有複雜且不易計算的特性，因此適用於時間很小或很大的近似解，可符合工程簡易計算的需求。在地下水相關的研究中，利用拉普拉斯域變數很小相當於時間很大(small p large t, SPLT) 的關係，可自拉普拉斯域解得到適用於時間很大的近似解。然而，Chen and Stone [1993]的研究指出，SPLT方法應用在推求定水頭試驗的井緣流量，會得到錯誤的近似解。本研究的目的，是推導定水頭與定流量試驗在不同邊界條件下的暫態洩降解，並討論穩態解與Thiem 方程式的關係，以及驗證Chen and Stone [1993]的推導。研究的結果顯示，地下水流系統必須是在有限區域的條件下，暫態洩降解才會在時間很大時，回復到Thiem 方程式，此結果符合質量平衡的概念。因此，在應用Thiem方程式時，討論井半徑為零或距離為無限遠的不合理情況，是沒有意義的。本研究亦證明，定水頭試驗的井緣流量經由SPLT方法，可以得到正確的近似解。

Theis equation is a non-equilibrium equation which can be used to predict the drawdown distribution during pumping or analyze drawdown data in determining the aquifer parameters. Physically, the aquifer drawdown changes with time at the early stage of pumping and approaches a constant value after a long period of pumping. However, the Theis equation can not reduce to Thiem equation mathematically when time approaches infinity. Also, the Thiem equation is not valid if the well radius approaches zero or the outer boundary goes to infinity. The main objectives of this dissertation are to derive the steady-state drawdown solution from the transient solutions of the constant-head and the constant-flux tests and to explain the use of mass balance concept in obtaining the steady-state solution. The result indicates that a flow system of a finite domain and a well of finite diameter are the necessary conditions for obtaining the Thiem equation from the transient solutions. While Thiem equation is employed, it implies that the problem has to be addressed within a region instead of zero well radius and/or infinite outer boundary. In addition, the dimensionless times criterion required to approximate the solutions of finite domain by the infinite-domain solution and the Thiem equation are also presented. An approximate solution is useful for practical applications if the corresponding analytical solution is complicated and difficult to accurately evaluate. The second objective of this dissertation is to examine the algorithm for obtaining a large-time solution by using the Laplace transforms and the well-known “small p – large t” relationship. In the past, the relationship was commonly applied to the Laplace domain solution in developing a large-time solution in the groundwater area. However, Chen and Stone [1993] pointed out that the use of this relationship might fail to obtain a correct solution of the wellbore flux for the constant-head test problem. This dissertation also shows that the relationship of small p versus large t is appropriate for obtaining a large-time solution of the transient constant-head test through the detailed mathematical development.