Models for three different P-wave attenuation mechanisms were presented that differ only in the values of the aij constants and in the nature of the mesoscopic transport coefficient . These three models correspond to (1) mesoscopic-scale heterogeneity in the frame moduli (``double porosity''), (2) mesoscopic-scale heterogeneity in the fluid type (``patchy-saturation''), and (3) grain-scale heterogeneity due to microcracks in the grains (``squirt''). In all three models, the amount of attenuation is controlled principally by the contrast of elastic compressibility among the constituents. In the double-porosity model, it is the contrast between the frame bulk-modulus of the two porous phases that is key, while in the patchy-saturation model it is the contrast in the fluid bulk modulus (immiscible patches of different fluids that have nearly identical bulk moduli would not produce much attenuation), and in the squirt model, it is the contrast between the drained modulus of an isolated cracked grain and that of the entire packing of grains.
Putting in small pockets of unconsolidated sand grains into an otherwise consolidated sandstone can produce attenuation in the seismic band that is comparable to what is measured in the field even when the pockets represent only a small amount of the total volume (<1% volume fractions). Since mesoscopic-scale heterogeneity is rather ubiquitous throughout the earth's crust, it seems reasonable to suppose that this mechanism may be responsible for most of the attenuation observed in seismograms. The squirt mechanism produces a great deal of attenuation at the ultrasonic frequencies used in laboratory measurements, but has trouble explaining attenuation in the seismic band. This result is good news for some important applications of the theory because the mesoscopic-scale flow is affected by the permeability of the material, while squirt flow is not. This leaves open the possibility of extracting permeability information from the frequency dependence of seismically measured Q.