Aplicación del Cálculo Fraccional a la Pérdida de Energía en la Propagación de Ondas Sísmicas

Abstract

When there is an earthquake in a seismic active zone the energy that gets free spreads through the earth is shape of mechanical waves, this type of waves are called seismic waves. A similar phenomenon happens when an artificial disturbance is done to get subsoil seismic images in the oilfieldŠs exploration. To describe the propagation of these waves certain hypothesis are used in the waveŠs nature and also in the characteristic of the place where they propagate. One of the phenomenonŠs that should be taked into account in the study of the seismic waveŠs propagation is the attenuation. A way to diminish the energy propagation of a seismic wave is called anelastic. The attenuation is produced by the internal grain friction that composes the rock or for the fractures the rock has. The attenuation brings as consequence the change in the initial pulse and the scattering phenomenon, what means that the components of different frequency are propagation in a different velocity. The attenuation process depends of the characteristics of the medium such as: porosity, density of the fractures, the kind of fluid fills the pores between others. Also, the attenuation of the seismic waves generally is represented to be inversely related with the quality factor which is denoted by Q. Arbitrary order integration and differentiation or fractional calculus is a mathematic part that extends over the differential and integral calculus which is used to provide a better description of the material properties; it has been showed that fractional order models are more appropriate than integer order models to describe some materials properties.
This project will try to explain the energy lost of the wave propagation process in an anelastic medium. Using the interpolation between the heat or diffusion equation (the temporary derivative is 1) and the wave equation (the temporary derivative is 2) through the arbitrary order integration and differentiation (Fractional calculus), what means to take orders of the temporary derivative between 1 and 2 to provide a mixed behavior of the diffusive process to the propagative.