Dispersion and attenuation of P wave in a periodic layered-model with Patchy saturation for three-phase

Based on the White periodic layered-model, this work extended the traditional two fluid (gas, water) model to a three-fluid (oil, gas, water) periodic layered-model, and further extended it to high frequency under the Biot theory framework, and studied P-wave's propagation law in this three-fluid model with Patchy saturation at the mesoscale of the seismic frequency band. Using the poroelastic equation uncoupled method and combining with equivalent boundary conditions, we derived the analytical solution of P-wave velocity dispersion and attenuation under the framework of the Biot1956 theory. Employing finite difference method in the frequency domain, we also derived the numerical solution of P-wave velocity dispersion and attenuation under the framework of the Biot1956 and Biot1941 theory. Combining the Biot-Gassmann-Hill and Biot-Gassmann-Wood equations, the upper and lower limits of calculated P-wave velocity were obtained, proving the correctness of analytical solution from the three-fluid model. Furthermore, the effect of three fluid's order on the P-wave velocity and attenuation was analyzed. From the force point of view, oil can offset part of force produced by water, but water can promote oil flowing into the gas layer, forming a superimposed force, so changing the fluids' order can lead to different forces to the gas layer, resulting in different P-wave dispersion and attenuation. Finally, the effect of gas saturation on the P-wave velocity dispersion and attenuation was studied.