Seismic and ultrasonic velocities in permafrost

We calculate the compressional- and shear-wave velocities of permafrost as a function of unfrozen water content and temperature. Unlike previous theories based on simple slowness and/or moduli averaging or two-phase models, we use a Biot-type three-phase theory that considers the existence of two solids (solid and ice matrices) and a liquid (unfrozen water). The compressional velocity for unconsolidated sediments obtained with this theory is close to the velocity computed with Wood's model, since Blot's theory involves a Wood averaging of the moduli of the single constituents. Moreover, the model gives lower velocities than the well-known slowness averaging theory (Wyllie's equation). For consolidated Berea sandstone, the theory underestimates the value of the compressional velocity below 0 degrees C. Computing the average built moduli by slowness averaging the ice and solid phases and Wood averaging the intermediate moduli with the liquid phase yields a fairly good fit of the experimental data. The proportion of unfrozen water and temperature are closely related. Fitting the wave velocity at a given temperature allows the prediction of the velocity at the whole range of temperatures, provided that the average pore radius and its standard deviation are known.