Minimal advective travel time along arbitrary streamlines of porous media flows: The Fermat–Leibnitz–Bernoulli problem revisited

Travel time of marked fluid particles along arbitrary streamlines in arbitrary porous streamtubes is estimated from below based on the Cauchy–Bunyakovskii (Schwartz) and Jensen inequalities. In homogeneous media the estimate is strict and expressed through the length of the streamline, hydraulic conductivity, porosity and the head fall. The minimum is attained at streamlines of unidirectional flow. The bounds for heterogeneous soils, non-Darcian flows and unsaturated media are also written. If such bounds are attained the corresponding trajectories become brachistochrones. For example, in a two-layered aquifer and seepage perpendicular to the layers there is a unique conductivity–porosity ratio which makes a broken streamline brachistocronic. Similarly, if conductivities of two layers are fixed there is a unique incident angle between flow in one medium and the interface which makes a refracted streamline brachistocronic.