自由含水層參數的數值求法

Abstract

自由含水層中,水平及垂直方向的水力傳導係數,蓄水係數及比出水,是四 個主要的水文地質參數. 傳統上,求解代表自由水層的參數時,以圖解法居 多. 圖解法係利用現地觀測井抽水數據,配合相關的標準曲線以推求參數 值,然而,圖解法具有費時而且精度不高的缺點. 本研究的目的,是要發展 一個數值方法,分析觀測井的抽水洩降數據,以推求自由水層的四個水文地 質參數.研究中,首先分析Neuman在1974年所提的部份貫穿井在自由含水層 中的水位洩降解析式,此解析解是一個含零到無限遠的機分式子,其中包含 Bessel函數J0(X)與無限級數乘積的項. 求解此積分問題時,用直接累加結 果與最後累加項比值的絕對值,做為判定收斂的條件,需要耗費大量的計算 機時間,而且積分的結果精度較不易掌握,特別是無因次時間較小的情形 下,更顯著. 當無窮級數項與Bessel函數的數值大小,以零為中心,呈正負 交互振盪,且絕對值為單調緩慢遞減的情形時,應用Longman方法做數值計 算,與直接累加法比較,可得到快速收斂與良好的精度控制.在求取四個水 文地質參數時,採用最小二乘方法,將觀測洩降與Neuman自由含水層解析式 的預測洩降差值的平方和,分別對四個未知參數做偏微分並令其值等於零, 得到四個非線性的方程組,此時後,誤差平方的總和會為最小,所導得的聯 立方程組,再以牛頓法解出未知的四個水文參數值.本論文所發展的方法與 圖解法,將同樣分析二組實際的抽水數據,並針對三個誤差式值加以比較, 討論,結果顯示,本論文所發展的方法,有高精度及快速收斂的優點. A method using the nolinear least-squares and Newton's method is proposed to identify the parameters from pumping test in homogeneous and anisotropic unconfined aquifer. Based on the least-squares approach, the sumof the squares of difference between the observed drawdowns and predicteddrawdown using the estimated parameters is minimizied. The Newton's methodand Guass elimination are used to solve the system of nonlinear least- squaresequations for the unknow aquifer parameters.Evaluation of Neuman's analytical solution for flow toward a well in an unconfined aquifer commonly requires large amounts of computation time. The integrand of Neuman's solution contains a product of the Bessel oscillatoryfuntion J0(X) and an infinite series. Large computation time is require for evaluation the dimensionless drawdown when the dimensionless time is small.The Longman's method is effective to evaluate a function or an infinite serieswhich shows altermative oscillation and the absolute values of each term aredecreasing slowly and monotonously. It is found that the Longman's method isa very effective approach to evaluate the Neuman's dimensionless drawdownwhen ZD = 0 and/or ts is small.Comparison of the predicted results between the proposed method and the graphical method when analyzing various data are presented. The proposedmethod can achieve quick convergence and good accuracy. It can be applied tothe cases of partial penetration of pumped and/or observed wells where the graphical method is only siutable for analyzing the aquifer parameters under thecondition of fully penetrating well.