  We establish a number of results concerning conditions for minimum energy
dissipation and advective travel time in porous and fractured media. First, we
establish a pair of converse results concerning fluid motion along a streamline
between two points of fixed head: 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 with a given average conductivity is achieved when the conductivity
is constant. Finally, 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 small perturbations in aperture away from
uniformity always act to reduce advection time.
