Linear dynamics of double-porosity dual-permeability materials. II. Fluid transport equations

For the purpose of understanding the acoustic attenuation of double-porosity composites, the key macroscopic equations are those controlling the fluid transport. Two types of fluid transport are present in double-porosity dual-permeability materials: (1) a scalar transport that occurs entirely within each averaging volume and that accounts for the rate at which fluid is exchanged between porous phase 1 and porous phase 2 when there is a difference in the average fluid pressure between the two phases and (2) a vector transport that accounts for fluid flux across an averaging region when there are macroscopic fluid-pressure gradients present. The scalar transport that occurs between the two phases can produce large amounts of wave-induced attenuation. The scalar transport equation is derived using volume-averaging arguments and the frequency dependence of the transport coefficient is obtained. The dual-permeability vector Darcy law that is obtained allows for fluid flux across each phase individually and is shown to have a symmetric permeability matrix. The nature of the cross coupling between the flow in each phase is also discussed.