Dispersion of waves in porous cylinders with patchy saturation: formulation and torsional waves

Laboratory experiments on wave propagation through saturated and partially saturated porous media have often been conducted on porous cylinders that were initially fully saturated and then allowed to dry while continuing to acquire data on the wave behavior. Since it is known that drying typically progresses from outside to inside, a sensible physical model of this process is concentric cylinders having different saturation levels-the simplest example being a fully dry outer cylindrical shell together with a fully wet inner cylinder. We use this model to formulate the equations for wave dispersion in porous cylinders for patchy saturation (i.e., drainage) conditions. In addition to multiple modes of propagation obtained numerically from these dispersion relations, we find two distinct analytical expressions for torsional wave modes. We solve the resulting torsional wave dispersion relation for two examples: Massillon sandstone and Sierra White granite. One essential fact that comes to light during the analysis is that the effective shear moduli of the gas- and liquid-saturated regions must differ, otherwise it is impossible to account for the laboratory torsional wave data. Furthermore, the drainage analysis appears to give improved qualitative and quantitative agreement with the data for both of the materials considered.