NUMERICAL EXAMPLES

Examples shown in Figures 1-3 make use of thermal conductivity and electrical formation factor data from Asaad (1955). Three different sandstones (labelled B, C, D) were studied by Asaad, and several different sets of experiments were performed on each. The Figures show data from experiments B30, C10, C20, and D10. I plot both the new formation factor bounds (FF) and the Hashin-Shtrikman bounds (HS) based on volume fraction information. A selection of the data is displayed in all three cases. Electrical formation factor measurements were made on all three samples (F₁^B = 12.0, F₁^C = 23.0, F₁^D = 33.0). Frame formation factor can be determined from measurements of thermal conductivity when the pores are evacuated. But a value of effective grain thermal conductivity must be found. Asaad (1955) solved this problem -- using an extrapolation method -- assuming that a certain geometric mean approximation (which is just a straight line on a log-log plot) when fit to the data would then give an accurate estimate of the point at which $G(g_1 = g_2^{\rm eff},g_2) \simeq g_2^{\rm eff}$. Results displayed as they are here on the log-log plots in Figs. 2 and 3 show that Asaad's method is in fact quite accurate for all these data. Then, $F_2^{\rm eff} \simeq g_2^{\rm eff}/G(0,g_2)$, and I find F₂^B = 13.5, F₂^C = 15.9, F₂^D = 3.72. Measured porosity values were $\phi^B = 0.220$, $\phi^C = 0.158$, $\phi^D = 0.126$.