To obtain the transport law , the mesoscopic flow is analyzed in the limits of low and high frequencies. These limits are then connected using a frequency function that respects causality constraints. The linear fluid response inside the patchy composite due to a seismic wave can always be resolved into two portions: (1) a vectorial response due to macroscopic fluid-pressure gradients across an averaging volume that generate a macroscopic Darcy flux across each phase and that corresponds to the macroscopic conditions and ; and (2) a scalar response associated with internal fluid transfer and that corresponds to the macroscopic conditions and . The macroscopic isotropy of the composite guarantees that there is no cross-coupling between the vectorial transport and the scalar transport within each sample.
The mesoscopic flow problem that defines is the internal equilibration of fluid pressure between the patches when a confining pressure has been applied to a sealed sample of the composite. Having the external surface sealed is equivalent to the required macroscopic constraint that . Upon taking the divergence of (2) and using equation (3), the diffusion problem controling the mesoscopic flow becomes