Fluid Substitution for an HTI Medium
Fluid substitution for rocks has been widely performed to understand their fluid- depend seismic response. Among many of the different methods, Gassmann’s equations was one of the most commonly used. However, these equations are only valid for isotropic cases while many rocks are anisotropic. Here we study the case of an HTI medium.
In Gassmann’s 1951 paper, he gave the equations for anisotropic fluid substitution. Based on his work, we derived the equations for an HTI medium using the linear slip model. The new equations have the similar form as the isotropic one but in terms of stiffness tensor other than bulk modulus. The linear slip model decompose an HTI rock into two parts: isotropic background rocks and vertical fractures. The isotropic background rock is characterized by Lamé’s parameter (λ) and shear rigidity (μ). The vertical fractures are characterized by two new terms as normal and tangential fracture weakness, δN and δT. We also proposed a method to obtain these four parameters from well logs. To test our theory, we made a 3D printed model as a synthetic rock. With the new technique of 3D printing, we can possibly print any models we want. These printed models are printed layer by layer and for the simplest case which has no structures inside the model, a symmetry of HTI (or VTI) was observed. We measured velocities from different angles before and after saturation and calculated the corresponding stiffness tensors. Then we applied our new equations to the dry model and predicted the velocity of the saturated models. Our predictions matched the measured velocities very well.
The new equations and innovative printed models are the highlights of our work. For further extension of our work, we will study orthorhombic case with more complicated printed models.
AAPG Search and Discovery Article #90182©2013 AAPG/SEG Student Expo, Houston, Texas, September 16-17, 2013