Undrained test, long time

The long-time undrained test for a double-porosity system should also produce the same physical results as a single-porosity system (assuming only that it makes sense at some appropriate larger scale to view the medium as homogeneous). The conditions for this test are that

$$\begin{eqnarray} \delta p_f^{(1)} = \delta p_f^{(2)} = \delta p_f, \nonumber \ \delta\zeta\equiv \delta\zeta^{(1)} + \delta\zeta^{(2)} = 0, \end{eqnarray}$$

from which follow

$$\begin{eqnarray} \delta e = - a_{11}\delta p_c - (a_{12}+a_{13})\delta p_f, \qqu... ... -(a_{21}+a_{31})\delta p_c - (a_{22}+2a_{23}+a_{33})\delta p_f. \end{eqnarray}$$

These require that the overall pore-pressure buildup coefficient be given by  

$$B \equiv \left. {{\partial p_f}\over{\partial p_c}}\right \v... ...a\zeta= 0} = - {{a_{21}+a_{31}}\over{a_{22}+2a_{23}+a_{33}}},$$ (20)

and that the undrained bulk modulus be given by  

$${{1}\over{K_u}} \equiv \left. {{\delta e}\over{\delta p_c}}\right \vert _{\delta\zeta= 0} = a_{11} + (a_{12}+a_{13})B.$$ (21)