Green’s Function of the Wave Equation for a Fractured Dissipative HTI Medium Taking the Viscoelasticity of the System into Account

In this paper we derive the Green’s function of the wave equation for a fractured dissipative HTI medium. Inside the fractures there is a viscous fluid which adds to the attenuation of the wave. Previous works have been done for the elastic medium where the stiffness tensor have all real components. In this scenario the host rock and the fluid inside the fractures both have viscoelastic properties. Thus, complex terms in the stiffness tensor has been introduced to account for this viscoelasticity. Finally, we arrive to a Green Christoffel type of equation with additional complex terms due to the introduction of viscoelasticity. We then perform a Fourier Transform to solve for the Green’s function and finally an Inverse Fourier Transform to obtain the Green’s function in (x,t) space. This Green’s function can be used to determine how a wave passing through a viscoelastic layer (e.g. hydrocarbon layer) is changed after passing through it. Thus, in turn it can be used to detect hydrocarbon layers.