Seismic modeling using the fractional, diffusive-propagatory wave equation for the study of anelastic media: application to oil traps and VSP data

Obtaining subsurface images with quality spatial resolution is essential for seismic exploration in the search for hydrocarbons. However, the images of the structures located under areas with gas saturation are generally of low quality since the seismic waves are attenuated (lose energy) when propagated by these media. This paper proposes a seismic modeling method based on fractional differential equations: The diffusion-wave equation, which interpolates two physical phenomena, diffusion, and propagation. This equation is studied both in the time domain and frequency domain to observe its amplitude and phase behaviour when the wave propagates in different anelastic materials with different quality factor Q (inverse factor to the attenuation) values. The equation with the time derivative of fractional order is solved numerically using a finite difference scheme, where the mathematical expression of the stability and convergence criteria of the method was established. Wave propagation was modelled in structures with hydrocarbon traps with gas saturation. In addition, a real VSP (Vertical seismic profile) Zero-offset acquisition in which the source is located on the surface and the receivers inside the well was compared with the data obtained from the simulation.