We described a groundwater model with prolate spheroid coordinates, and introduced a new parameter, namely τ the silhouette influence of the geometric under which the water flows. At first, we supposed that the silhouette influence approaches zero; under this assumption, the modified equation collapsed to the ordinary groundwater flow equation. We proposed an analytical solution to the standard version of groundwater as a function of time, space and uncertainty factor α. Our proposed solution was in good agreement with experimental data. We presented a good approximation to the exponential integral. We obtained an asymptotic special solution to the modified equation by means of the Adomian decomposition and variational iteration methods.

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