Undrained joints, undrained matrix, short time

There are several different, but equally valid, choices of time scale on which to define Skempton-like coefficients for the matrix/fracture system under consideration. Elsworth and Bai [1992] use a definition based on the idea that for very short time both fluid systems will independently act undrained after the addition of a sudden change of confining pressure. This idea implies that $\delta\zeta^{(1)} = 0 = \delta\zeta^{(2)}$ which, when substituted into (generalstrainstress), gives

-e = a_11p_c + a_12p_f^(1) + a_13p_f^(2) 0 = a_12p_c + a_22p_f^(1) + a_23p_f^(2) 0 = a_13p_c + a_23p_f^(1) + a_33p_f^(2).   Defining

B_EB^(1) . p_f^(1)p_c|_ ^(1) = ^(2) = 0   and   B_EB^(2) . p_f^(2)p_c|_ ^(1) = ^(2) = 0   we can solve (EB1) for the two Skempton's coefficients and find the results

B_EB^(1) = a_23a_13-a_12a_33a_22a_33-a_23^2   and

B_EB^(2) = a_23a_12-a_13a_22a_22a_33-a_23^2.   The effective undrained modulus is found to be given by

1K_u^EB - ep_c|_^(1)= ^(2)=0 = a_11 + a_12B_EB^(1) + a_13B_EB^(2).  

These definitions will be compared to others.