Bounds on various transport coefficients in heterogeneous media have been heavily studied now for over forty years (Hashin and Shtrikman, 1962; Milton, 2002; Torquato, 2002). One of the more interesting developments in this area has been the introduction of rigorous methods for developing bounds on complex constants (closed curves in the complex plane), especially the dielectric constant and conductivity of heterogeneous media (Bergman, 1978, 1980; Milton, 1980, 1981; Bergman, 1982; Korringa and LaTorraca, 1986; Stroud et al., 1986). These methods represent a great technical achievement in this field, but they nevertheless can sometimes be difficult to apply to real data since they require high precision and strong consistency among the data used in computing the bounds. In some cases it would be helpful for applications if some simpler and perhaps more robust methods and results were available.
In this short paper I consider the question of whether it is possible to make use of the analytical methods in a different way to find bounds on transport coefficients. I will limit discussion here to real coefficients, taking thermal conductivity as our main example, but the results apply equally well to other transport coefficients including electrical conductivity and fluid permeability (Berryman, 1992). Furthermore, the resulting bounds depend only on commonly measured quantities in porous media called formation factors (Archie, 1942; Korringa and LaTorraca, 1986), and they show no unusual sensitivity to measurement errors or any need for careful checking of consistency relations among the measurements.