| 摘要: | 移流-延散公式(advection-dispersion equation, ADE)是傳統的地下水污染傳輸控制方程，其延散部分以費克延散(Fickian dispersion)為基礎而移流部分則建立在地下水速度場據定常平均流速。因費克延散僅能發生於均質情況而所有的含水層具有不同程度的異質性，故近來改用非費克延散(non-Fickain dispersion)理論發展新的控制方程。使用Levy motion描述非費克延散得到對時間及/或空間分數階微分項(如/,0Ctγγ 1 γ..<. ； 2 /,1Cxβββ..<.)所組成的分數階移流-延散公式(fractional advection-dispersion equation, FADE)。分數階微分具」非局部性」(nonlocal)特徵；亦即，一函數在某特定點的分數階微分與該函數其它點有長範圍的相關性。而整數階微分則僅代表函數在一特定點(局部)的微分運算，與其它點無關。自然界中非局部性特徵現象幾乎無所不在。FADE配合定常延散係數成功地詮釋與分析高偏斜及/或有長拖尾的穿透曲.，明顯地優於傳統ADE。不過有些FADE現地資料分析亦顯示FADE仍有尺度相依延散現象。此種尺度相依延散不似由非費克延散所引起，而較可能肇因自假設的定常平均流速無法反映高異質含水層的複雜流場變化。考慮到地下水溶質傳輸理論自費克延散演化至非費克延散再前進到尺度相依非費克延散，目前一項重要研究即是評量平均流速(定常或變異)對FADE的影響。本三年期研究計畫主要目的是評量定常或變異平均流速變化對尺度相依FADE的影響以增進FADE的實用性。將執行垂直水流示蹤劑試驗；垂直水流意指水流方向與疊積砂層垂直，每一砂層有不同的水力傳導係數。此種試驗的異質性相當容易建立且其水流速度分佈具與現場觀測相符的長拖尾現象。將執行不同層數、不同水力傳導係數組合的垂直水流示蹤劑試驗，所獲得之流速分佈將用於決定該異質情況的尺度因子(亦即FADE的分數階)。穿透曲.則用於調查、瞭解相關的尺度相依議題；將使用定常係數(非尺度相依)FADE 的解析解和尺度相依FADE的有限差分解進行分析。本計畫有助於瞭解定常或變異平均流速對FADE的影響，瞭解尺度相依的起因，及提升FADE在分析異質含水層溶質傳輸的實用性。 The classical solute transport governing equation in groundwater is the well-known advection-dispersion equation (ADE), which assumes Fickian dispersion and a constant mean velocity for the flow field. As Fickian dispersion can only occur in a homogeneous condition while all aquifers are heterogeneous to certain extent, non-Fickian dispersion theories have recently been invoked to develop new governing equations. Application of Levy motion to describe non-Fickian dispersion results in the fractional advection-dispersion equation (FADE), which involves the fractional-order derivative term with respect to time (e.g.,/,0Ct?????<.) and/or to space (e.g.,/,1Cx?? 2 ???<.). The fractional-order derivative is ?nonlocal? for it has long-range correlation with the function at neighboring points, while the integer-order derivative is a point-wise (local) operation. Nonlocality is ubiquitous in nature. The FADE with a constant dispersion coefficient is able to capture the highly skewed and long-tailed breakthrough curves taken from heterogeneous aquifers, a definite superiority to the classical ADE. Yet, scale dependent dispersion was also noted in a few FADE analyses. Such a scale-dependent issue is less likely caused by the underlying non-Fickian dispersion and is more probably associated with the assumed constant mean velocity that fails to reflect the irregular but structured flow filed in heterogeneous systems. As modeling of solute transport has evolved from Fickian dispersion to non-Fickian dispersion to scale-dependent non-Fickain dispersion, it is time to understand the influence of the mean velocity, constant or variable, on the FADE. The purposes of this three-year project are to evaluate the influence of the mean velocity, constant or variable, on the FADE, and to improve the applicability of FADE. We will conduct perpendicular-flow tracer tests [PFTT], in which flow is perpendicular to the sand layers of different hydraulic conductivities. In the PFTT, various patterns of heterogeneity are relatively easy to construct and their velocity distributions have long tails that mimic the field observations in heterogeneous aquifers. Tests of different numbers of layers with various hydraulic conductivity combinations will be conducted. Their velocity distributions will be employed to determine the scale indexes (i.e., the fractional orders in the FADE). The breakthrough curves measured will be analyzed, using the analytical solution to a constant coefficient FADE and the finite-difference solution to a scale-dependent FADE of both nonlocal dispersive flux and advection flux, to investigate and understand the relevant scale-dependent issues. This project will contribute to a better understanding of the influence of flow velocity on the FADE, an evaluation of the cause of the scale-dependent issues, and the enhancement of our knowledge in using the FADE to investigate solute transport in heterogeneous aquifers. 研究期間：9608 ~ 9707 |
