利用時間分數階移流模式對非反應性示蹤劑在裂隙介質的分析; Use of a Fractional-in-Time Advective Model to Analyze Nonreactive Tracer Test Data in a Fractured Formation

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题名:	利用時間分數階移流模式對非反應性示蹤劑在裂隙介質的分析;Use of a Fractional-in-Time Advective Model to Analyze Nonreactive Tracer Test Data in a Fractured Formation
作者:	林建興;Jian-hsing Lin
贡献者:	應用地質研究所
关键词:	非反應性示蹤劑;時間分數階階數;冰磧裂隙;fractured till;nonreactive tracer;fractional-in-time
日期:	2009-07-09
上传时间:	2009-09-22 09:59:42 (UTC+8)
出版者:	國立中央大學圖書館
摘要:	非費克溶質傳輸在現地和實驗室尺度的多孔隙介質與裂隙地質構造都可觀測到。量測得到的濃度穿透曲線與移流-延散方程式(Advective-dispersion equation, ADE)的計算比較，特徵包括濃度提早出現及長時間下有拖尾現象，無法使用傳統移流-延散方程式(ADE)來處理。使用連續時間隨機步行方法(Continuous time random walk, CTRW)成功分析在冰磧裂隙發達的非反應性示蹤劑氯化鈉實驗。在長時間極限下，CTRW相當於時間分數階模式，實際上是否能使用時間分數階模式來分析同樣的氯化鈉實驗資料，是我們感興趣的。於是建立了時間分數階模式，研究中修改了Schumer et al.[2003]冪律記憶函數型態，加入時間尺度有效率係數：F[Tγ-1]；其中γ為時間分數階階數(0<γ<1)。加入F值使得孔隙比為無因次。模式中的解是經由拉普拉司對時間的轉換並利用拉普拉司數值逆轉方法得到。驗證拉普拉司數值逆轉換準確性，推導出有限差分法，時間上採用Caputo微分形式。γ值由濃度穿透曲線下的冪律拖尾求得，F為套配參數。利用不同γ與F值，時間分數階模式成功分析九筆氯化鈉實驗資料。也證實了實驗資料利用時間分數階模式與CTRW是一樣好。 Non-Fickian solute transport has been observed at field and laboratory scales in a wide variety of porous and fractured geological formations. Characterized by an early arrival and prolonged late-time tail in the breakthrough, the non-Fickian transport cannot be analyzed with the models based on conventional advective-dispersion equation (ADE). There is a tracer test in a highly structured fractured till formation, of which the experimental data of nonreactive sodium chloride (NaCl) can be successfully analyzed using the continuous time random walk (CTRW) approach. As the CTRW is equivalent to the fractional-in-time approach under the limit of large time, it is of practical interest to know whether or not a fractional-in-time model can be developed to analyze the same set of NaCl data. The fractional-in-time model developed herein is a ramification of that of Schumer et al. [2003], in which the power-law memory function is modified by including a time scaling effective rate coefficient, F in[Tγ-1], where 0<γ<1 is the order of the fractional time derivative. Including this F makes the porosity ratio involved in the model dimensionless, as should be. The solutions of this model are determined using the Laplace transform with respect to time and a numerical Laplace inversion technique. Validity of this numerically inverted solutions is verified by making use of a finite-difference scheme, where the fractional-in-time derivative is formulated using the Caputo definition. The value of γ can be estimated using the power-law tail exhibited by the breakthrough, while F is evaluated as a fitting parameter. A total of nine sets of NaCl experimental data are successfully analyzed with the fractional-in-time model with different values of γ and F. It is proven that the experiment data can be equally well matched by the fractional-in-time model as well as the CTRW model.
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