Abstract: An exact analytical solution and an approximate early time solution are presented for the problem of advection and dispersion of a tracer in a steady,spherically symmetrical,divergent flow field in a porous formation.The porous formation is assumed to be semi-infinite.However,for an aquifer with finite thickness,the solution is also applicable if some conditions are satisfied.These are defined in terms of a characteristic length Lc and dimensionless number Nb,and are based on the error in pressure head between the solutions for semiinfinite and finite thickness aquifers.The partial differential equation describing the trans,port process is a confluent hypergeometric equation which may be solved by variable substitution and Laplace transform,the solution is expressed with parabolic cylindrical functions.The tracer breakthrough at a fixed point is shown to assymmetrical.The solution can be used to analyse tracer tests which use a short-screened well to create an approximately spherical flow field in a thick formation, and so provides an alternative design of test when fully penetrating boreholes are not appropriate.

Key words: analytical model, formation evaluation, tracer, equation, LapIace transform

摘要： An exact analytical solution and an approximate early time solution are presented for the problem of advection and dispersion of a tracer in a steady,spherically symmetrical,divergent flow field in a porous formation.The porous formation is assumed to be semi-infinite.However,for an aquifer with finite thickness,the solution is also applicable if some conditions are satisfied.These are defined in terms of a characteristic length Lc and dimensionless number Nb,and are based on the error in pressure head between the solutions for semiinfinite and finite thickness aquifers.The partial differential equation describing the trans,port process is a confluent hypergeometric equation which may be solved by variable substitution and Laplace transform,the solution is expressed with parabolic cylindrical functions.The tracer breakthrough at a fixed point is shown to assymmetrical.The solution can be used to analyse tracer tests which use a short-screened well to create an approximately spherical flow field in a thick formation, and so provides an alternative design of test when fully penetrating boreholes are not appropriate.

关键词: analytical model, formation evaluation, tracer, equation, LapIace transform