Constraints on the a_ij coming from the long-time limit

Before passing on to the specific models for the various coefficients, we state here several general constraints (independent of any modeling assumptions) on the geomechanical constants a_ij. Note that in order to measure the a_ij's in the laboratory, we need only consider an isolated sample immersed in a ``reservoir'' characterized by three control parameters: p_c, p_f⁽¹⁾, and p_f⁽²⁾; i.e., gradients in these quantities and the subsequent flow induced by those gradients do not enter the definition of the a_ij's.

The constraints are obtained from the limiting case in which the rate at which p_c, p_f⁽¹⁾, and p_f⁽²⁾ are all changing is much slower than the rate at which internal fluid equilibration can take place. In this ``long-time limit'', we are always in the quasi-static state where

p_f⁽¹⁾ = p_f⁽²⁾.

Left to itself, any system having finite permeability will achieve this state as $t \to \infty$.