本研究發展第三類注入邊界條件下二維圓柱座標移流-延散方程解析解建立以描述地表下圓柱座標系統之二維溶質傳輸情形。為建立第三類注入邊界條件二維圓柱座標移流-延散方程解析模式，採用finite Hankel轉換技巧結合Laplace轉換以求得解析解。將建立的第三類注入邊界條件解析模式與前人文獻所得到的第一類注入邊界條件解析模式做比較，以說明兩者對於溶質傳輸情形之影響。結果顯示當觀測點靠近注入邊界與地下水傳輸系統之縱向延散係數大時，兩解析解因注入端邊界之延散項差異而導致濃度穿透曲線不符合。所發展的解析模式可應用於同時決定縱向與側向延散度的二維圓柱實驗室土柱試驗或入滲追蹤劑試驗。

An exact analytical solution for two-dimensional advection-dispersion equation (ADE) in cylindrical coordinates subjected to the third-type inlet boundary condition is developed to describe the two-dimensional solute transport in a subsurface system with cylindrical geometry. The finite Hankel transform technique in combination with the Laplace transform is adopted to solve the two-dimensional ADE in cylindrical coordinates. The developed exact analytical solution is compared with the solution with first-type boundary condition available in literature to illustrate the influence of inlet boundary condition on solute transport. Results show that the significant discrepancies between breakthrough curves obtained from two analytical solutions, especially for observation point near the inlet boundary or a subsurface system large longitudinal dispersion coefficient. The developed solution is an efficient tool for simultaneous determination of the longitudinal and transverse dispersivities from a two-dimensional laboratory-scale radial column experiment or a infiltration test with a tracer.

[1] Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill, New York.
[2] Schulze-Makuch, D., 2005. Longitudinal dispersivity data and implications for scaling behavior. Ground Water, 43(3), 443–456.
[3] Bromly, M., Hinz, C., Aylmore, L. A. G., 2007, Relation of dispersivity to properties of homogeneous saturated repacked soil columns. European Journal of Soil Science, 58(1),293-301.
[4] Fetter, C. W., 1999. Contaminant Hydrogeology, second ed. Prentice Hall, Upper Saddle River.
[5] Zhang, X., Qi, X., Zhou, X., Pang, H., 2006. An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil. Journal of Hydrology, 328(3-4), 614-619.
[6] Massabò, M., Cianci, R., Paladino, O., 2006. Some analytical solutions for two-dimensional convection–dispersion equation in cylindrical geometry. Environmental Modelling and Software, 21 (5), 681-688.
[7] Batu, V., van Genuchten, M. T., 1990. First- and third-type boundary conditions in two-dimensional solute transport modeling. Water Resources Research, 26(2): 339-350.
[8] Perkins, T. K., Johnston, O. C., Hofman, R. N., 1965. Mechanics of viscous fingering in miscible systems, Society of Petroleum Engineers Journal, 5(1), 301-317.
[9] Han, N. W., Bhakta, J., Carbonell, R. G., 1985. Longitudinal and lateral dispersion in packed beds: Effect of column length and particle size distribution, AIChE Journal, 31(2), 277-288.
[10] Nishigaki, M., Sudinka, T., Sasaki, Y., Inoue, M., Moriwaki, T., 1996. Laboratory determination of transverse and longitudinal dispersion coefficients in porous media, Journal of Groundwater Hydrology, 38(1), 12-27.
[11] Cirpka, O. A., Kitanidis, P. K., 2001. Theoretical basis for the measurement of local transverse dispersion in isotropic porous media, Water Resources Research, 37(2), 243-252.
[12] Benekos, D., Cirpka, O. A., Kitanidis, P. K., 2006. Experimental determination of transverse dispersivity in a helix and a cochlea, Water Resources Research, 42(7).
[13] Massabò, M., Catania, F., Paladino, O., 2007. A new method for laboratory estimation of the transverse dispersion coefficient. Ground Water 45 (3), 339-347.
[14] Chen, J. S., Chen, C. S., Gau, H. S., Liu, C. W., 1999. A two well method to evaluate transverse dispersivity for tracer test in a radially convergent flow field, Journal of Hydrology, 223(3-4), 175-197.
[15] Chen, J. S., Liu, C. W., Liao, C. M., 2003. Two-dimensional Laplace transformed power series solution for solute transport in a radially convergent flow field, Advanced in Water Resources, 26(10), 1113-1124.
[16] Chen, J. S., 2007. Two-dimensional power series solution for non-axisymmetrical transport in a radially convergent tracer test with scale-dependent dispersion, Advances in Water Resources, 30(3), 430-438.
[17] Sneddon, I. H., 1972, The Use of Integral Transforms, McGraw-Hill,New York.
[18] Shahani, A. R., Nabavi, S. M., 2007. Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading. Applied Mathematical Modelling, (31), 1807-1818.
[19] Fried J. J., 1975. Groundwater pollution, Elsevier, New York, 330pp.
[20] 陳家洵., 1988. 調查及處理地下水污染的困難及建議, 地下水資源研討會論文集, 44-56.
[21] Visual Numerical, Inc. 1994. IMSL User’s Manual. Houston, Tex., 1, 159-161.
[22] De Hoog, F.R., J. H. Knight, and A. N. stokes, 1982. An imporved method for numerical inversion of Laplace transforms. Journal on Scientific and Statistical Computing, 3(3), 357-366.
[23] Yeh, H. D., Yang, S. Y., 2010. A new method for laboratory estimation of the transverse dispersion coefficient-discussion. Ground Water, 48(1), 16-17.
[24] Shanks, D., 1955. Non-linear transformations of divergent and slowly convergent sequences. Journal of Mathematical Physics 34, 1-42.
[25] Frippiat, C. C., Perez, P. C., Holeyman, A. E., 2008. Estimation of laboratory-scale dispersivities using an annulus-and-core device. Journal of Hydrology 362 (1-2) 57-68.
[26] Chen, J. S., 2010. Analytical model for fully three-dimensional radial dispersion in a finite-thickness aquifer. Hydrological Processes, 24(7), 934-945.