山區地下水之質量傳輸研究

Mass Transport In Mountain Groundwater

王靖翔(Ching-Hsiang Wang)

指導教授 ： 黃良雄

國立臺灣大學/工學院/土木工程學研究所/碩士(2020年)

https://doi.org/10.6342/NTU202003603

摘要

地下水分布面積廣大，且水平距離尺度遠大於垂向深度尺度，若使用三維模式難以兼顧計算量以及準確度。因此本研究延續郭遠錦(2004)之三維濃度變量拆解技巧，將三維變量拆解為二維水平變量(垂向積分平均)以及一維垂向變量兩者之和，再經過二維及一維迭代計算以模擬三維移流及延散現象。山區地形因為板塊擠壓，土層之地質分布較平地崎嶇複雜，因此本研究延續王元亨(2013)之 轉換技巧，將複雜的山區地形轉換為各層厚度為1之水平土層。

　　本研究先將三維卡氏座標控制方程式轉換成 座標之控制方程式，並且將三維濃度變量拆解為二維水平變量與一維垂向變量之和。三維控制方程式垂向積分求得二維控制方程式，三維控制方程式減去二維控制方程式為一維控制方程式，再將二維變量及一維變量迭代計算最後加總求得三維濃度。在濃度垂向計算上，王元亨(2013)假設濃度垂向上為二次函數分布，與解析解的指數函數不符；郭遠錦(2004)在垂向上使用解析法並搭配Duhamel’s theorem積分，雖精準但效率較差。本研究在濃度垂向計算上採用有限差分法，精確度與效率在王元亨(2013)與郭遠錦(2004)之模式之間取得平衡。

　　本模式針對解析解和斜坡進行驗證與應用，其模擬結果顯示模式掌握平地或斜坡之延散及移流現象。透過模擬非均質土層，模擬濃度傳入不同延散係數土層之行為，與郭遠錦(2004)與王元亨(2013)之模式比對趨勢一致，顯示模式合理性。最後探討濃度由含水層傳入阻水層時產生邊界層的現象，透過不同虛擬分層厚度進而了解邊界層濃度分布差異。

關鍵字

山區質量傳輸模式 ； 有限解析法 ； σ轉換

並列摘要

The groundwater distribution is large and the horizontal distance scale is much larger than the vertical depth scale, which makes it difficult to consider both computational accuracy and calculation. Therefore, this study continued the decompose technique of Yuan-Ching Kuo (2004), decomposing the three-dimensional concentration variables into the sum of the horizontal variables (vertical integration average) and the vertical variables. The model simulates three-dimensional advection-dispersive phenomena by computing two-dimensional horizontal variables and one-dimensional vertical variation iteratively. The topography in the mountainous areas is undulating due to plate extrusion, cause the geological distribution of the mountainous is more complex than that in the flatlands. Thence, this study continued the σ-transform technique of Yuan-Heng Wang (2013), transforming the complex mountainous terrain into a horizontal terrain with a thickness of 1 in each layer.

　　In this study, we first transformed the three-dimensional concentration government equation in Cartesian coordinate system into sigma coordinate system, and decomposed three-dimensional concentration variables into the sum of the horizontal variables (vertical integration average) and the vertical variables. Second, we integral three-dimensional government equation into two-dimensional horizontal government equation. The one-dimensional vertical government equation is the three-dimensional government equation minus the two-dimensional government equation. Finally we computed two-dimensional horizontal variables and one-dimensional vertical variables iteratively and sum up to get three-dimensional concentration variable. On vertical computation of mass, Yuan-Heng Wang (2013) assumed the mass distribution on vertical direction satisfied quadratic polynomial function, which is different from the exact solution of exponential function. Yuan-Ching Kuo (2004) solved the vertical variables by analytical method with Duhamel’s theorem integration. It was very accurate but time costing. In this study, we use uniform grid finite difference method on vertical computation, in order to made a balance of the efficient of Yuan-Heng Wang’s (2013) method and the accuracy of Yuan-Ching Kuo’s (2004) method.

　　The mountain mass transport model is vertified by analytical solution, and tested by slope case. The result show that can simulate adcective-dispersive phenomena in horizontal and slope terrain. Furthermore, simulated a heterogeneous field transport case, and compared the results with Yuan-Ching Kuo (2004) and Yuan-Heng Wang (2013) to show the reasonability of this model. Finally, discuss the boundary layer phenomenon when concentration transport from aquifer into aquitqrd by applying virtual layer technique.

並列關鍵字

the mountain mass transport model ； finite analytical method ； sigma transform

參考文獻

Bear, J. , “Hydraulic of Groundwater, McGraw-Hill” ,New York,1982.

Ozisik, M. N. , “Heat Conduction”, Wiley, New York,1980.

Chen, C.J., Naseri-Neshat, H., and Ho, K. S., “Finite Analytic Numerical Solution of Heat Transfer in Two-Dimensional Cavity Flow”, J. Nume. Heat Transfer, 4,pp.179-197, 1981.

Chen, C. J., and Chen, H.C, “Finite Analytical Numerical Method for unsteady Two-Dimensional Navier-Stokes equation”, Journal of Computation Physics, 53, pp.209-226, 1984.

Tsai, T.L., Yang, J.C., and Huang, L.H. ”Characteristics Method Using Cubic–Spline Interpolation for Advection–Diffusion Equation.” J. Hydraul. Eng., 130(6), pp. 580-585, 2004.