CONSISTENCY WITH EXACT RESULTS

One particularly powerful means of checking the validity of any estimation scheme is to compare the results with those of various exact results that may be known for special cases. In the present problem, a result of Berryman and Milton (1991) provides a convenient check on all the formulas derived so far. This result states that for an arbitrary two-component mixture of Gassmann materials the Biot-Willis parameter must satisfy the conditions

^* - ^(1)K^* - K^(1)= ^* - ^(2)K^* - K^(2)= ^(2) - ^(1)K^(2) - K^(1).   It is not hard to show that all the formulas presented satisfy these constraints as long as the side condition that has been mentioned previously, i.e., $\sum v_i (1-P^{*i})=0$ for CPA or the corresponding side condition $P^{h*} = \sum v_i P^{hi}$for Kuster-Toksöz. For the other two methods, the formulas were actually designed to guarantee satisfaction of (BMresult) directly. For CPA and Kuster-Toksöz, this satisfaction is especially easy to check in the case of spherical inclusions, but is not limited to that case. Thus, the theories presented here all satisfy this important additional condition that any ``good'' theory should satisfy, when these constraints are satisfied. Part of the art of constructing good approximations then is contained in our choices of models that satisfy these constraints at least approximately.