SECOND DERIVATION

Another derivation of the same bounds may provide additional insight into their significance.

Again starting from (4), this time I will go directly to the integral term and start making approximations to it. First, consider  

$$\int_0^\infty \frac{dx {\cal G}(x)}{\frac{1}{g_1} + \frac{x}... ...g_1}{g_2}} \ge g_1\int_0^\infty \frac{dx {\cal G}(x)}{1 + x},$$ (14)

where the inequality holds whenever $g_1 \le g_2$. Then, similarly, I have  

$$\int_0^\infty \frac{dx {\cal G}(x)}{\frac{1}{g_1} + \frac{x}... ...{g_1} + x} \le g_2\int_0^\infty \frac{dx {\cal G}(x)}{1 + x},$$ (15)

again whenever $g_1 \le g_2$. I can then make use of the identity in sumrule (12) to replace the integral on the far right in both of these expressions. And, finally, applying (14) to (4) gives exactly the lower bound (6), while applying (15) to (4) gives exactly the upper bound (9). All the same comments about reversal of the sense of the inequalities applies here if instead $g_1 \ge g_2$. So, the final result is again (13).

This derivation has the advantage that it is clear from the inequalities (14) and (15) exactly what approximations have been made in each case to arrive at the bounds on G(g₁,g₂).

 

gBasaadsmlgwg Figure 1 Comparison of the formation factor bounds (FF$^\pm$), the Hashin-Shtrikman bounds (HS$^\pm$), and thermal conductivity data from Asaad (1955). Data are for sandstone sample B.

 

gCasaadlglgwg Figure 2 Comparison of the formation factor bounds (FF$^\pm$), the Hashin-Shtrikman bounds (HS$^\pm$), and thermal conductivity data from Asaad (1955). Data are for sandstone sample C, including two distinct data sets.

 

gDasaadlglgwg Figure 3 Comparison of the formation factor bounds (FF$^\pm$), the Hashin-Shtrikman bounds (HS$^\pm$), and thermal conductivity data from Asaad (1955). Data are for sandstone sample D.