Effect of fluid patch clustering on the P-wave velocity-saturation relation: a critical saturation model

Quantitative analysis of the relationship between seismic wave velocities and fluid saturation in porous media is of great significance for any fluid injection and extraction operation in subsurface rock formations. However, seismic velocities are not only dependent on the amount of saturation, but also on the distribution of fluid patches and their size. The patch size variation during changes in saturation is oftentimes ignored in modeling studies, even though it is natural to assume that with increasing saturation, fluid patches will form larger and, at some critical saturation, percolating clusters. To capture the evolution of patch size with saturation implied in the velocity-saturation relations, we are inspired by percolation theory. By incorporating the connectivity of water-filled patches in the continuous random medium model, we develop a critical saturation model. We apply this critical saturation model to examine recently reported experimental measurements, specifically analyzing the patch size changes. For measurements of drainage or imbibition processes in four sandstone samples, we indeed find a clear indication of growing patch size with water saturation. The predictions of the critical saturation model are in reasonable agreement with observations. Our approach improves the accuracy of the interpretation of the velocity-saturation relations in partially saturated rocks and forms a basis for exploring its underlying mechanisms.