Travel time and energy dissipation minima for potential flows in heterogeneous geologic media

We establish a number of results concerning conditions for minimum energy dissipation and advective travel time along path lines for potential flows in heterogeneous porous and fractured media, governed respectively by Darcy's law and the cubic law. First, we establish a pair of converse results concerning fluid motion along a streamline between two points of fixed head in porous media in arbitrary dimensions: the minimal advective time is achieved under conditions of constant energy dissipation, and minimal energy dissipation is achieved under conditions of constant velocity along the streamline (implying homogeneous conductivity in the vicinity of the streamline). We also show directly by means of variational methods that minimum advection time along a streamline in one dimension with a given average conductivity is achieved when the conductivity is constant. Next, we turn our attention to minimum advection time and energy dissipation in parallel and sequential fracture systems governed by the cubic law, for which fracture cross section and conductivity are intimately linked. We show that, as in porous domains, flow partitioning between different pathways always acts to minimize system energy dissipation. Finally, we consider minimum advection time as a function of aperture distribution in a sequence of fracture segments. We show that, for a fixed average aperture, a uniform-aperture system displays the shortest advection time. However, we also show that any sufficiently small perturbations in aperture away from uniformity always act to reduce advection time.