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

(14) |

(15) |

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*(*g _{1}*,

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

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

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

10/23/2004