The Study of Wave Propagation in a Borehole Using the Finite Difference Method

Abstract

Synthetic microseismograms of elastic wave propagation in a fluid-filled borehole were generated using both the finite difference technique and the discrete wavenumber summation technique. For the finite difference calculations, the solid-liquid borehole boundary was handled as a sharp boundary using a second order Taylor expansion of the displacements. and additionally, a rigid solid-liquid sharp interface is used to model the existence of the logging tool. A heterogeneous formulation was used to handle variations in formation properties. The finite difference grid has absorbing boundaries on two sides and axes of symmetry on the remaining two sides. A grid size no less than 10 points per wavelength was used. The results from the finite difference modeling were compared with the synthetic microseismograms generated by the discrete wavenumber summation method. A detailed comparison between the microseismograms generated by the two methods showed that the body waves (refracted P and S waves) are identical, while the guided waves showed a slight difference in both phase and amplitude. These differences are believed to be due to the dispersion generated by the finite difference method. We have studied the depth of investigation of the retracted body waves in an invaded or damaged borehole using the conventional ray theory approach and compared the results to those obtained by the finite difference method. The results show that the minimum source-receiver separation necessary to observe the unaltered formation depends on both the velocity gradient and the lowest and highest velocity of the damaged zone. Such an investigation shows us the importance of the length of the logging tool to be able to "see" past the damaged and invaded zone, and thus enables us to measure the true formation properties, as well as to estimate the depth of the damaged or invaded zone.