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

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：21

、訪客IP：3.143.9.115

姓名

林建興(Jian-hsing Lin) 查詢紙本館藏

畢業系所

應用地質研究所

論文名稱

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)

相關論文

★ 微水試驗以兩階段式方法推估薄壁因子與含水層水力導數	★ 受負薄壁效應影響微水實驗參數推估方法
★ 單井循環流水力實驗之理論改進與發展	★ 地表下NAPL監測技術-薄膜擴散採樣器之發展
★ 水文地層剖析儀與氣壓式微水試驗儀調查淺層含水層水力傳導係數之研究	★ Evaluation and management of groundwater resource in Hadong area of Vietnam using groundwater modeling
★ 時間分數階傳輸模式對反應性示蹤劑砂箱實驗之分析	★ 利用雙封塞微水試驗推估裂隙含水層水力傳導係數
★ 多深度微水試驗之測試段長度對水力傳導係數影響	★ 時間分數階徑向發散流場傳輸模式與單一裂隙示蹤劑試驗分析
★ 含水層下邊界對於斜井雙極水流試驗影響	★ 大傾角裂隙岩層抽水試驗用雙孔隙率模式分析
★ 裂隙岩層的水流流通面積對跨孔雙封塞微水試驗資料分析之影響	★ 有效井管半徑模式與有限厚度模式對薄壁效應多深度微水試驗之比較
★ 非受壓含水層之三維斜井捕集區解析解	★ 利用分布參數方法發展傾斜裂隙岩層抽水試驗雙孔隙率模式

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

[檢視]

[下載]

本電子論文使用權限為同意立即開放。
已達開放權限電子全文僅授權使用者為學術研究之目的，進行個人非營利性質之檢索、閱讀、列印。
請遵守中華民國著作權法之相關規定，切勿任意重製、散佈、改作、轉貼、播送，以免觸法。

摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 非反應性示蹤劑
★ 時間分數階階數
★ 冰磧裂隙

關鍵字(英)

★ fractured till
★ nonreactive tracer
★ fractional-in-time

論文目次

英文摘要 i
中文摘要 ii
目錄 iii
圖目錄 iv
表目錄 vii
符號說明 viii
第一章緒論 1
1-1 研究背景 1
1-1-1 傳統移流-延散方程 9
1-1-2 非費克行為 17
1-2 研究目的 21
第二章時間分數階模式建立與驗證 22
2-1 假設及數學模式建立 22
2-2 修正型態的冪律記憶函數及時間分數階模式解 24
2-3 有限差分法推導及其驗證 25
2-3-1 分數階有限差分法推導 25
2-3-2 時間分數階移流數值模式驗證 35
2-3-2 傳統一維溶質傳輸模式驗證 38
2-3-3 其他分數階型態之數值模式驗證 42
第三章現地資料分析與結果 49
3-1 視擴散係數的時間變化 49
3-2 實驗場址介紹 55
3-3 實驗資料分析 60
第四章結論與建議 77
參考文獻 79
附錄A 時間分數階移流解析解推導 85
附錄B Fox H-函數及其他分數階解析解推導過程 86

參考文獻

蔣曉芸、徐明瑜 (2004) 污染源濃度分佈分數階模型及其解，山東大學學報(理學版)，第39卷，第3期，第37-41頁。
常福宣、吳吉春、薛禹群、戴水漢 (2005) 多孔隙溶質運移問題中的分數彌散，水動力學研究與進展，第20卷，第1期，第50-55頁
Adams, E. E., and L. W. Gelhar (1992), Field study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis, Water Resour. Res., Vol 28(12), pp.3293-3307.
Bear, J. 1972. Dynamics of Fluids in Porous Media. New York: Elsevier.
Benson, D. A. (1998), The fractional advection-dispersion equation: Development and application, Ph.D. thesis, Univ. of Nev., Reno.
Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000a), Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6), 1403– 1412.
Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000b), The fractional-order governing equation of Levy motion, Water Resour. Res., 36(6), 1413– 1423.
Berkowitz, B., J. Bear, and C. Braester (1988), Continuum models for contaminant transport in fractured porous formations, Water Resour. Res., Vol 24, pp.1225-1236.
Berkowitz, B., J. Klafer, R. Metzler, and H. Scher (2002), Physical pictures of transport in heterogeneous media:Advection-dispesion, random walk and fractional derivative formulations, Water Resour. Res., 38(10), 1191, doi:10.1029/2001WR001030.
Berkowitz, B., A. Cortis, M. Dentz, and H. Scher (2006), Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys., Vol 44, RG2003.
Berkowitz, B., S. Emmanuel, and H. Scher (2008), Non-Fickian transport and multiple-rate mass transfer in porous media, Water Resour. Res., Vol 44, W03402, doi:10.1029/2007WR005906.
Broholm, K., Nilsson, B., Sidle, R.C., Arvin, E., (2000). Transport and biodegradation of creosote compounds in clayey till, a field experiment. J. Contam. Hydrol. 41, 239–260.
Bromly, M., and C. Hinz (2004), Non-Fickian transport in homogeneous unsaturated repacked sand, Water Resour. Res., Vol 40, W07402.
Caputo, M. (1967), Linear models of dissipation whose Q is almost frequency
independent-II, Geophys. J. R. Astron. Soc., 13, 529– 539.
Caputo, M., and F. Mainardi (1971), A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91, 134– 147.
Coats, K. H., and B. D. Smith (1964), Dead-end pore volume and dispersion in porous media, Soc. Pet. Eng. J., Vol 4(1), pp.73–84.
Deng, Z.-Q., V. P. Singh, and L. Bengtsson (2004), Numerical solution of fractional advection-dispersion equation , J. Hydraul. Eng.,Vol 130(5),pp.422–431.
Dentz, M., and B. Berkowitz (2003), Transport behavior of a passive solute in continuous time random walks and multirate mass transfer, Water Resour. Res., 39(5), 1111, doi:10.1029/2001WR001163.
Eggleston, J., and S. Rojstaczer (1998), Identification of large-scale hydraulic conductivity trends and the influence of trends on contaminant transport, Water Resour. Res., Vol 34(9), pp.2155–2168.
Endo, H. K., J. C. S. Long, C. R. Wilson, and P. A. Witherspoon (1984), A model for investigating mechanical transport in fracture networks, Water Resour. Res., Vol 20, pp.1390-1400, 1984.
Farrell, J. and M. Reinhard (1994), Desorption of halogenated organics from model solids, sediments, and soil under unsaturated conditions, 2. Kinetics. Environ. Sci. Technol., 28(1), 63-72.
Fetter, C. W., Jr. (1994), Applied Hydrogeology. 3d ed. New York: Prentice Hall, Inc. 691 P.
Fredericia, J. (1990), Saturated hydraulic conductivity of clayey tills and the role of fractures, Nord. Hydrol., 21, 119-132.
Freeze, R. Allen, and John A. Cherry. (1979), Groundwater. Englewood Cliffs, NJ. : Prentice Hall , 604 PP.
Guan, J., F. J. Molz, Q. Zhou, H. H. Liu, and C. Zheng (2008), Behavior of the mass transfer coefficient during the MAD-2 experiment: New insights, Water Resour. Res., 44, W02423, doi: 10.1029/2007WR006120.
Haggerty, R., and S. M. Gorelick (1995), Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res., Vol 31(10), pp.2383-2400.
Haggerty, R., S. A. McKenna, and L. C. Meigs (2000), On the late-time behavior of tracer test breakthrough curves, Water Resour. Res., 36(11), 3467 – 3479.
Haggerty, R., and S.W. Fleming, L.C. Meigs and S. A. Mckenna (2001), “Tracer tests in a fractured dolomite: 2. Analysis of mass transfer in single-well injection-withdrawal tests ”, Water Resour. Res., 37(5), 1129-1142.
Haggerty, R., C. F. Harvey, C. Freiherr von Schwerin, and L. C. Meigs (2004), What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results, Water Resour.Res., 40, W01510, doi:10.1029/2002WR001716.
Huang, G. H., Q. Z. Huang, and H. B. Zhan (2006), Evidence of onedimensionalscale-dependent fractional advection-dispersion, J. Contam. Hydrol., Vol 85, pp.53– 71.
Jorgensen, P. R., and J. Fredericia (1992), Migration of nutrients, pesticides and heavy metals in fractured clayey till, Gøotechnique, 42, 67-77.
Levy, M., and B. Berkowitz (2003), Measurement and analysis of non-Fickian dispersion in heterogeneous porous media, J. Contam. Hydrol., Vol 64, pp.203-226.
Lynch, V. E., B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks (2003), Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., Vol 192, pp.406– 421.
Mainardi, F. G. Pagnini, R.K. Saxena (2005), The Fox H functions in fractional diffusion, J. Comput. Appl. Math. 178 321–331.
Metzler, R., and J. Klafter (2000), The random walk’s guide to anomalous diffusion: A fractional dynamic approach, Phys. Rep., Vol 339, pp.1 – 77.
Meerschaert MM, Scheffler HP. (2004), Limit theorems for continuous time random walks with infinite mean waiting times. J Appl Probab ;41(3):623–38.
Meerschaert MM, Tadjeran C (2004). Finite difference approximations for fractional advection–dispersion flow equations. J Comput Appl Math ;172:65–77.
Meerschaert MM, Tadjeran C (2006). Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math ;56:80–90.
Miller, K.S., Ross, B., (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.
Neuman, S. P. (1995), On Advective Transport in Fractal Permeability and velocity Fields, Water Resour.Res., Vol 31(6), pp.1455–1460.
Oberhettinger, F., and L. Badii (1973), Tables of Laplace Transforms, Springer-Verlag, Berlin Heidelberg New York.
Oldham and Spanier, K.B. Oldham and J. Spanier (1974), The Fractional Calculus, Academic press, New York .
Podlubny, I. Podlubny (1999), Fractional Differential Equations, Academic Press, San Diego.
Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. (1993), Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, pp. 21-22.
Schumer, R., D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft (2001), Eulerian derivation for the fractional advection-dispersion equation, J. Contam. Hydrol., 48, 69–88.
Schumer, R., D. A. Benson, M. M. Meerschaert, and B. Baeumer (2003),Fractal mobile/immobile solute transport, Water Resour. Res., 39(10),1296, doi:10.1029/2003WR002141.
Sidle, C.R., Nilsson, B., Hansen, M., Fredericia, J., (1998). Spatially varying hydraulic and solute transport characteristics of a fractured till determined by field tracer tests, Funen, Denmark. Water Resour. Res. 34_10., 2515–2527.
Starr, R. C., R. W. Gillham, and E.A. Sudicky (1985), Experimental investigation of solute transport in stratified porous media: 2. The reactive case, Water Resour. Res., 21(7), 1043-1050.
Sudicky, E. A., and E. O. Frind (1982), Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures, Water Resour. Res., Vol 18, pp.1634-1642.
Tang, D. H., E. O. Frind, and E. A. Sudicky (1981), Contaminant transport in fractured porous media: Analytical solution for a single fracture, Water Resour. Res., Vol 17, pp.555-564.
Zhang, X., J. W. Crawford, L. K. Deeks, M. I. Stutter, A. G. Bengough, and I. M. Young (2005), A mass balance based numerical method for the fractional advection-dispersion equation: Theory and application, Water Resour. Res., 41, W07029, doi:10.1029/2004WR003818.
Zhang, Y., D. A. Benson, M. M. Meerschaert, and E. M. LaBolle (2007a), Space-fractional advection-dispersion equations with variable parameters:Diverse formulas, numerical solutions, and application to the Macrodispersion Experiment site data, Water Resour. Res., Vol 43,W05439.
Zhang, Y., D. A. Benson, and B. Baeumer (2008), Moment analysis for spatiotemporal fractional dispersion, Water Resour. Res., Vol44, W04424, doi:10.1029/2007WR006291.
Zheng, C., and J. J. Jiao (1998), Numerical simulation of tracer tests in a heterogeneous aquifer , J. Environ. Eng., Vol 124(6), pp. 510– 516.

指導教授

陳家洵(Chia-shiun Chen)

審核日期

2009-7-17

推文