A Laplace transform solution for tracer tests in a radially convergent flow field with upstream dispersion

When tracers are introduced into an injection borehole, noticeable concentration gradients at the injection well may cause a backward spreading of the initial plume during radially convergent tracer tests. Based on this concept a non-rigorous mathematical model is developed to estimate the effect of backward spreading. The injection well with an instantaneous slug input is treated as a mathematical source and the initial plume at the injection well is allowed to move upstream. The model assumes that advection and longitudinal dispersion are the transport mechanisms in a radially converging how field. The Laplace transform solution for solute concentration in an infinite porous medium is obtained using the method of Green's function. Breakthrough curves are computed by numerically inverting the Laplace transform solution. As compared with the solution of Moench, it is concluded that the presence of backward movement of initial plume yields a decrease of peak concentration and spreading out of breakthrough curve tails. This effect is significant for small Peclet numbers and can be neglected for large Peclet numbers. A field tracer test is carried out to demonstrate the applicability of the model. The experimental data are fitted with the type curves of this study, and those of Moench's study, to determine dispersivity and aquifer porosity. The results show that dispersivity values estimated by matching the two type curves are different. Because it neglects backward dispersion, Moench's solution overestimates the dispersivity and effective aquifer porosity.