二維徑向土柱實驗配合數學模式可推估縱向及側向延散係數，在現有的相關研究中，數學模式常假設土柱長度為無窮大，此假設較不符合實際情況。本研究發展在有限長土柱之二維圓柱座標系統移流－延散方程式之解析解，以描述圓柱形之二維溶質傳輸，分別利用finite Hankel轉換與通用積分轉換方法（GITT）求得解析解，所發展的解析解與Laplace轉換有限差分法（LTFD）進行驗證，驗證結果兩者十分吻合。因為執行積分轉換後的解析解包含finite Hankel與GITT逆轉換級數，當Peclet number較大與延散度比很小時，需要比較高的累加次數才能夠讓解收斂。和前人發展的解析解比較並對邊界條件的影響進行討論，Peclet number較小時若以無窮大邊界的數學模式進行推估可能會低估延散係數。發展的解析解進一步可應用於分析土柱實驗所得濃度穿透曲線，推估孔隙率、縱向與側向延散係數。

　　Two-dimensional radial column experiment combine with mathematical models to estimate longitudinal and transverse dispersion coefficient. Existing research of mathematical models assumed infinite length of column, but this assumption is not conforming to the actual circumstances. In this research, we developed a analytical solution of the two-dimensional advection-dispersion equations in cylindrical coordinates with a finite soil column for describing solute transport in two-dimensional cylindrical geometry. This analytical solution is obtained in the use of finite Hankel transform and generalized integral transform technique (GITT). Development of analytical solution is verified with Laplace transform finite difference (LTFD). This analytical solution consists of two infinite series expansions after the finite Hankel and GITT inverse transforms. When Peclet number larger and the dispersion ratio is very small, it needs more the number of summed terms for solution convergence. We compared with the previous development of analytical solution, and discussed the effect of different boundary conditions. When the Peclet number are smaller, the concentration of infinite boundary will be lower. If the analytical solutions with infinite boundary condition are used to estimate the solute transport parameters, and therefore underestimate the dispersion coefficients. We further used to analyze the concentration breakthrough curve of column test and estimated porosity, longitudinal and transverse dispersion coefficients.

[1] Sauty, J. P., 1980. An analysis of hydrodispersive transfer in aquifer. Water Resources Research, 16(1), 145-158.
[2] Dirk, S. M., 2005. Longitudinal Dispersivity data and implications for scaling behavior. Ground Water, 43(3), 443-456.
[3] Ogata, A., 1970. Theory of dispersion in a granular medium, Geological Survey Professional Paper, 441-Ι.
[4] Chen, J. S., C. S. Chen, H. S. Gau and C. W. Liu, 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.
[5] Chen, J. S., C.W. Liu and C. M. Liao, 2002, A novel analytical power series solution for solute transport in a radially convergent flow field, Journal of Hydrology, 266(1-2), 120-138.
[6] Chen, J. S., C. W. Liu, and C. M. Liao, 2003, Two-dimensional Laplace transformed power series solution for solute transport in a radially convergent flow field, Advances in Water Resources, 26(10), 1113-1124.
[7] Bear, J., 1961. On the tracer form of dispersion in porous media. Journal of Geophysical Research, 66(4), 1185-1197.
[8] Bear, J., 1961. Some experiments in dispersion. Journal of Geophysical Research, 66(8), 2455-2467.
[9] Grane, F. E., G. H. F. Gardner, 1961. Measurement of transverse dispersion in granular media. Journal of Chemical and Engineering Data, 6 (2), 283-287.
[10] Pichens, J. F., G. E. Grisak, 1981. Scale-dependent dispersion in a stratified granular aquifer, Water Resources Research, 17(4), 1191-1211.
[11] Robbins, G.A., 1989. Methods for determining transverse dispersion coefficients of porous media in laboratory column experiments. Water Resources Research, 25(6), 1249-1258.
[12] Cirpka, O. A., P.K. Kitanidis, 2001, Theoretical basis for the measurement of local transverse dispersion in isotropic porous media, Water Resources Research, 37(2), 243-252.
[13] Benekos, I.D., O. A. Cirpka, and P.K. Kitanidis, 2006. Experimental determination of transverse dispersivity in a helix and a cochlea. Water Resources Research, 42, W07406. doi:10.1029/2005/WR00471.
[14] Massabò M., F. Catanial, and O. Paladino, 2007, A new method for laboratory estimation of the transverse dispersion coefficient, Ground Water, 45(3), 339-347.
[15] Frippiat, C. C., P. C. Pérez, and A. E. Holeyman, 2008, Estimation of laboratory-scale dispersivities using an annulus-and-core device. Journal of Hydrology, 362, 57-68.
[16] Massabò, M., R. Cianci, and O. Paladino, 2006, Some analytical solutions for two-dimensional convection equation in cylindrical geometry, Environmental Modelling & Software, 21(5), 681-688.
[17] Zhang, X., X. Qi, X. Zhou, and H. Pang, 2006, An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil, Journal of Hydrology, 328, 614-619.
[18] Sneddon, I. H., 1972, The use integral transforms, McGraw-Hill, New York.
[19] Selim, H. M., R. S. Mansell, Analytical solution of the equation of reactive solutes through soils, Water Resources Research, 12 (1976), 528-12532.
[20] Cleary, R. W., Adrian, D. D., 1972, Analytical solution of convective- dispersive equation for cation adsorption in soils. Soil Science Society of America Proceedings, 37, 197-199.
[21] Ozisik, M.N., Heat Conduction, Wiley, New York, 1980.
[22] Cotta, R.M., Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton, FL, 1993.
[23] Pérez Guerrero, J. S., L. C. G. Pimentel, T. H. Skaggs, M. Th. van Genuchten, 2009. Analytical solution of the advection-diffusion transport equation using a change-of-variable and integral transform technique. International Journal of Heat and Mass Transfer, 52, 3297-3304.
[24] Yah, H. D., S. Y. Yang, 2010. A New Method for Laboratory Estimation of the Transverse Dispersion Coefficient-Discussion. Ground Water, 48(1), 16-17.
[25] Fetter, C.W., 1999. Contaminant Hydrogeology, second ed. Prentice Hall, Upper Saddle River.
[26] Sudicky, E.A., 1989. The Laplace transform Galerkin technique: a time -continuous finite element theory and application to mass transport in groundwater. Water Resources Research, 25(8), 1833-1846.
[27] Moridis, G.J., D.L. Reddel, 1991. The Laplace transform finite difference method for simulation of flow through porous media. Water Resources Research, 27(8), 1873-1884.