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BEAR, J. (1972). Dynamics of Fluids in Porous Media, Ameri-can Elsevier, New York.
BEAR, J. (1979). Analysis of flow against dispersion in porous media-Comments. J. Hydrol. 40: 381-385.
CAMPOS, J. L. E. (1999). Análise numérica do transporte de contaminantes em meios porosos com reações quí-micas. Tese de Doutorado. Pontifícia Universidade Católica do Rio de Janeiro. PUC-RJ.
CASTRO, M. A. H. (1997). Uma Metodologia Numérico-Analítica para Solução de Problemas Transientes em Recursos Hídricos, XII Simpósio Brasileiro de Recur-sos Hídricos, Vitória, ES, Vol. 2, p. 313-318.
CASTRO, M. A. H. (1998). Solução de Problemas Transientes em Recursos Hídricos através da Matriz de Transfe-rência. II Simpósio de Recursos Hídricos Del Cone Sur, Santa Fé, Argentina, v. IV, p. 309-318.
CASTRO, M. A. H. (1998). Um Método Numérico-Analítico para Solução de Problemas Transientes. XI Con-gresso Brasileiro de Mecânica dos Solos e Engenha-ria Geotécnica. Brasília, Anais do XI COBRAMSEG, v. I, p. 1-8.
CHARBENEAU, R.J. (2000). Groundwater Hydraulics and Pollutant Transport. Prentice-Hall, Inc. New Jersey. COSTA, C.T. (2000). Caracterização hidrodispersiva de um
solo aluvial no semi-árido do Nordeste do Brasil. Dissertação de Mestrado. UFPE.
DANKWERTS, P. V. (1953). Continuous Flow Systems. Chem. Eng. Sci. 2: 1-13.
DAUS, A.D., FRIND, E.O., SUDICKY, E.A. (1985). Compara-tive error analisys in finite element formulation of ad-vection-dispersion equation. Advance in Water Re-sources. 8: 86-95.
FEIKE, J.L.; TORIDE, N. (1998). Analytical solutions for solute transport in finite soil columns with arbitrary initial dis-tributions. Soil Sci. Soc. Am. J. 62: 855-864.
FETTER, C.W. (1993). Contaminant Hydrogeology. Macmillan Publishing Company, 866 Third Avenue, New York, N.Y. 10022.
FREEZE, R. A., E CHEERY, J. A. (1979) Groundwater, Pren-tice-Hall, Englewood Cliffs, N. J.
ISTOK, J. D. (1989). Groundwater Modeling by the Finite Element Method, American Geophysical Union, Washington, DC, USA.
LEIJ, F.J.; TORIDE, N. (1995). Discrete time and length-averaged solutions of the advection-dispersion equa-tion. Water Resour. Res. 31: 1713-1724.
PARKER, J. C.; VAN GENUCHTEN, M. TH. (1984). Flux-average and volume-Flux-average concentration in con-tinuum approaches to solute transport. Water Re-source Research. 20: 866-872.
PARLANGE, J.Y. ; STARR,J. L.; VAN GENUCHTEN, M.TH.; BARRY, D.A.; PARKER, J.C.. (1992). Exit Condition for Miscible Displacement Experiments. Soil Sci. 153: 165-171.
ROTH, K. (1996). Lecture notes in soil physics. Institute of Soil Science, University of Hohenheim. Version 3.2. SÉGOL, G. (1993). Classic groundwater simulations: Proving
and improving numerical models. Prentice-Hall, Englewood Cliffs, NJ.
SUDICK, E.A. (1989). The Laplace transforms Galerkin Tech-nique: A time-continuous Finite Element Theory and Application to mass transport in Groundwater. Water Resour. Res., 25 (8), 1833-1846.
TORIDE, N.; LEIJ, F.J.; VAN GENUCHTEN, M. Th. (1993). A comprehensive set of analytical solutions for solute transport with first-order decay and zero-order pro-duction. Water Resour. Res. 29: 2167-2182.
VAN GENUCHTEN, M. TH.; ALVES, W.J. (1982). Analytical solutions of the one dimensional convective-dispersive solute transport equation. USDA, Agricul-tural Research Service, Tech. Bull. 1661.
YONG, R.N. et al. (1992). Principles of contaminant transport in soils. Development in geotechnical Engineering, Netherlands: Elsevier Science Publishers, 73, 327p.
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