Travel time and energy dissipation minima in heterogeneous subsurface flows

TL;DR

This paper investigates the conditions for minimizing energy dissipation and travel time in heterogeneous subsurface flows, revealing that uniform conductivity and aperture distributions optimize flow efficiency.

Contribution

It provides new theoretical insights into flow optimization in porous and fractured media using variational methods and analyzes the effects of heterogeneity on flow efficiency.

Findings

Constant conductivity minimizes advection time in porous media.

Uniform aperture distribution yields shortest advection time in fracture systems.

Small perturbations in aperture can further reduce advection time.

Abstract

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.