改善未飽和層水流傳輸模式質量守恆與數值擴散問題之研究

Abstract

本研究之主要目的係改善未飽和層水流傳輸模式質量守恆與數值擴散問題，進而以有限差分法建立一維未飽和層水流的運動模式，並探討其模式的數值特性。本文分別採用Celia（1990）的混合型理查氏方程式（mixed form Richards equation）、及Smolarkiewicz（1983）的簡單正定運移法（simple positive-definite advection scheme）建立質量守恆及反數值擴散之控制方程式，特別處理未飽和與飽和緩衝帶之連續性問題後，針對時間及空間上不同差分形式的水力傳導係數、數學與數值特性、時間間距與網格的大小，探討其在模式中的適用性及影響性，並根據Touma and Vauclin（1986）之入滲實驗資料，及Los Alamos國家試驗室所做的排水試驗資料，進行模式測試及驗證。 數值試驗的結果顯示，先用Celia’ form質量守恆差分法求解線性化理查氏方程式，再利用簡單正定運移法經過兩次以上反數值擴散修正後，可以明顯得到反數值擴散效果，並且隨著反數值擴散修正次數的增加，其質量愈趨近於守恆。不同的水力傳導係數平均形式會對入滲鋒的傳遞速度產生極大的影響；不同的土壤水力特性型態，其較適用之水力傳導係數平均型式也不同。除調和平均型水力傳導係數平均型式外，算術平均、幾何平均及上風型水力傳導係數平均型式在濕鋒傳遞的描述上表現相當，因不同的土壤水力特性型態而有略微之差異。利用數值正解與均方根誤差檢定模式時間間距與網格大小之較佳組合。Haverkamp et al.（1977）土壤試驗資料檢定分析得知較佳時間與網格之組合為0.003 hr、1.0 cm，而Touma and Vaclin （1986）之實例驗證亦得到相同之較佳組合。時間間距及格網大小在濕鋒傳遞效應明顯時，對數值擴散的效應也較強烈。變動時間間距在濕鋒傳遞的推估上有放大誤差的效應，不過可藉由最大時間間距的控制，提高其收斂性，並維持準確度。 在模式的應用與驗證時發現土壤漸濕過程中（入滲試驗），未飽和層水流的傳遞特性比擴散效應強，數值的收斂性較差，迭代的次數較多；漸乾過程中（排水試驗），傳遞效應相對較弱，數值的收斂性較佳。藉 值的數值測試可以得到可容許之空間間距範圍，入滲試驗中較佳之空間間距值相對於排水試驗值為小。

The main purpose of study is to improve mass conservation and numerical diffusion problem for water flow model in unsaturated zone. The 1-D model is solved by finite difference method, and its numerical characteristic is also investigated. This study adopts Celia’s (1990) mixed Richards’ equation and Smolarkiewicz’s (1983) simple positive-definite advection scheme to formulate the mass conservation anti-diffusion equations. After the special treatment of the continuity problem in the unsaturated and saturated buffer zone, this study aims at different finite difference forms of the hydraulic conductivity coefficient in the temporal and spatial domain, mathematical and numerical characteristic, and time interval and grid size. The applicability and influence of those factors for the model are analyzed, and the test and verification of the model are executed by using data from Touma and Vauclin’s (1986) infiltration experiment and Los Almas National Laboratory’s drainage experiment. Numerical results show that anti-diffusion can be achieved by first using Celia’s mass-conservation finite difference method for the linearlized Richards’equation, and then using simple positive-definite advection scheme more than twice to current the numerical diffusion. In addition, the mass is more conserved as the correction times of anti-diffusion increase. Different Forms of the averaged hydraulic conductivity coefficient will significantly affect the advection velocity of the infiltration front; different soils have their respectively suitable averaged forms of the hydraulic conductivity coefficient. Arithmetical average, geometric average, and upstream weighting average of the hydraulic conductivity coefficients have almost the same performance of describing the advection of the infiltration front, but with a slight difference due to different soil composition. The choice of the suitable time step and grid size of the model is on basis of exact solution and root-mean-square error test. Throuth the composition of Havercamp et al.’s (1977) and Touma and Vauclin’s (1986) experimental data shows that and is the better choice. When the advection of the infiltration front is significant, the selection of and has large affect on the numerical diffusion. Variable and will increase the error of estimating the advection of the infiltration front, but the error can be reduced by selecting allowable maximum . From the application and verification study of model, it can be found that advection process is more important than diffusion process when the soil is getting wet (infiltration experiment) and more iterations are required to quarantee the numerical convergence. On the other hand, the advection affect is weaker when the soil is getting dry (drainage experiment), and the model has better performance of convergence. Through the test on the peclet number, the allowable can be obtained. It can be seen that allowable used in the infiltration experiment is smaller than that in the drainage experiment.