estimation of borrow material properties have not been established when 

 the borrow zone is heterogeneous. Hence special attention must be given 

 when suitable borrow material from homogeneous deposits cannot be found. 



The relationship between the critical ratio and the relative diver- 

 gence between the phi moments of native and borrow materials is shown in 

 Figure 5-3, The horizontal axis is a dimensionless measure of the relative 

 difference between borrow and native phi means. It is computed as the 

 borrow phi mean minus the native phi mean divided by the native phi stand- 

 ard deviation„ The vertical axis (plotted on a logarithmic scale) is the 

 ratio of borrow phi standard deviation to native phi standard deviation. 

 Any value plotting to the right of the origin indicates a borrow material 

 finer than the native material (Mj^^j > Ma^) „ Any point plotting above the 

 horizontal axis indicates a borrow material more poorly sorted than the 

 native material (.^^b ^ ^An)* Hence the four categories discussed above 

 are separated into the four quadrants on this plot. 



The curves in Figure 5-3 indicate equal-value contours of the criti- 

 cal ratio. Contours are dashed lines in quadrant 2, because here the 

 critical ratio is assumed to be an upper bound to the true loss ratio. 

 In quadrant 1, the critical ratio is assumed to be a conservative esti- 

 mate of the true loss ratio and the curves are solid lines. No curves 

 are plotted in quadrants 3 and 4, because the computed value of critical 

 ratio has no meaning when the borrow material is well sorted in compari- 

 son to the native material. 



This plot shows the general behavior of the critical ratio as func- 

 tions of the differences in textural characteristics between borrow and 

 native materials. The following relationships are noteworthy: 



(a) For any fixed ratio between the sorting of borrow and native 

 material, the critical ratio changes only slowly with small differences 

 in phi means, then more rapidly as this difference becomes larger. 



(b) For larger ratios of the sorting parameter o±, the stability 

 of the computed critical ratio is greater, i„e., if the ratio of borrow 

 to native sorting is large the computed critical ratio is nearly insensi- 

 tive to the difference in phi means. 



(c) For any fixed finite difference in phi means, there will be 

 some ratio of borrow to native sorting for which the critical ratio will 

 be a minimum. For sorting ratios less than this value, the critical ratio 

 rises rapidly and approaches infinity as the sorting ratio approaches 

 unity. For sorting ratios larger than this optimal value, the critical 

 ratio increases slowly. 



These relations indicate that the computed value for critical ratio 

 is generally more sensitive to the phi sorting ratio than to differences 

 in phi means. If the borrow material is poorly sorted in comparison to 

 the native material, errors in determination of the difference in phi means 

 will not cause significant errors in the computation of the critical ratio. 



5-15 



