Wave Attenuation in Porous Media Under Multiple Energy Excitations and a Statistical Periodicity Ratio Method with its Application in Nonlinear System Behavior Diagnosis
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To comprehend the mechanism of seismic vibration technique used as an external excitation method in the oil production industry, a three dimensional wave model was developed based on Biot’s theory (1956a, b) for describing the wave field in a medium excited by multiple point energy sources under spherical sources. The displacements governing equations of compressional and shear wave propagation in non-viscous fluid saturated elastic porous media are derived separately. The relative displacement between the fluid and solid is investigated because it is critical to understanding the dynamic behavior of the whole domain. The superposition characteristic is considered for spherical waves under the multiple energy source models. In viscous fluid-saturated elastic porous media, the spherical shear wave displacement equations are developed based on a modified wave frame proposed by Sahay (2008). Analytical forms of the solutions are introduced by considering the wave magnitude dispersion due to the viscosity of Newtonian fluids. The attenuation of relative displacement between the fluid and solid parts of the medium is then quantified. The attenuation of compressional wave and shear wave are studied separately. Many advantages of multiple sources over a single source have been demonstrated. By the implementation of this model, a desired controllable wave field can be obtained by selecting proper frequencies, amplitudes, and locations of the wave sources. By altering the energy frequencies, the magnitude of the relative displacement can be adjusted. The direction of wave propagation and the vibration can be controlled by either changing the frequencies or the amplitudes of the wave energy sources. As such, the desired relative displacement at a certain location over a time range can be achieved under multiple energy sources. Because the vibration of the particles in the reservoir is crucial for a good understanding of the wave propagation characteristics, the vibration of a single particle can be taken as a dynamic system that can be reflected by differential equations and whose vibration behavior can be very complex due to the physical properties mostly related to the nonlinear terms. The nonlinear dynamic systems governed by differential equations with damping and regular or periodic external excitations may demonstrate complex behaviors with different parameters and under different initial conditions. Thus, highly accurate and effective novel methods are always in need for the systematic study of the nonlinear dynamic system behavior diagnosis and predication.