AnalyticalNumerical Methods for
Simulating Full Waveform Sonic Logs Principal Investigators Inclusive Dates: 02/01/06 – 05/01/07 Background  Borehole acoustic (sonic) logging has become an indispensable tool for petroleum reservoir exploration, reserve estimation, well completion, and hydrocarbon production. Unlike surface seismic, borehole sonic allows reservoir characterization to be conducted at the scale of about one foot. To make use of the rich information provided by sonic logs, a solid understanding of the acousticseismic wave field in formations with depth dependent properties and boreholes with irregular caliper is crucial. This project developed a solution to simulate complex borehole geometries. However, the difficult numerical problems presented by this work could not have been resolved with the time and resources we had to accomplish our final goals. Therefore the scope of work was changed to produce a borehole modeling methodology for the oil and gas industry. This new technology will address a critical industry need — exploration in the deep ocean environment. Major oil companies are drilling deviated wells from platforms in different directions into ocean floor sediments to either explore for new fields or to delineate existing reservoirs. The ocean floor geology is formed by sequences of sands and shales. The shale units are anisotropic, and their characteristics preclude the use of current technology for estimating formation slowness from well logs acquired in deviated boreholes. Understanding the signatures recorded in deviated wells is important when developing processing techniques to estimate vertical and horizontal velocities of propagation from sonic logs for interpreting 3D seismic data. Modeling deviated boreholes in anisotropic formations has been addressed using a finitedifference timedomain approach. This work shows that cross dipole data is very sensitive to borehole deviation. Other approaches for predicting sonic and dipole logs of deviated boreholes in anisotropic formations include the KirchhoffHelmholtz theory, the perturbation method, and the variational method. Another is an analytical method to derive Stoneley wave speed for a deviated borehole penetrating an anisotropic formation. All these methods have limitations in terms of radiation conditions, attenuation, degree of deviation, or degree of anisotropy. SwRI previously developed the transform boundary integral equation (TBIE) method for a borehole in uniform transversely isotropic media with a horizontal axis of symmetry or with vertical fractures. This method is being extended to the Deviated Borehole Transversely Isotropic (DBTI) model to overcome the limitations of some of the above approaches. TBIE is a rigorous, fullwaveform approach with a number of advantages. The boundary and radiation conditions at infinity are satisfied automatically. There are no limitations regarding frequency, deviated angle, or amount of anisotropy. The model makes it straight¬forward to include attenuation and frequency dependent parameters. The TBIE method allows accurate and efficient synthetic full waveform sonic logs (monopole, dipole, quadrupole, etc.) and normal modes. This method is particularly efficient when many receivers and azimuthal angles are involved. Approach  In the TBIE method, an integral transform with kernel exp(ik_{z}z) is applied to the unknown wave components in the original BIEs on the borehole wall. The integral transform replaces the zdependence by the k_{z} dependence. The transformed boundary is the cross section of the borehole surface at z=0, i.e., a circle. Thus, instead of discretizing the boundary by boundary elements in both z and θ, the mesh is reduced to one dimension (in θ only) for each k_{z}. As a tradeoff, an inverse integral transform is needed to replace k_{z} by z and obtain the solution of the original BIEs. The integral transform and its inverse are a direct integration of given functions in z and k_{z}. The numerical integration can be very efficient using the MCC integration method. Therefore, the benefit from the reduction in degree of freedom (DOF) significantly surpasses the drawback due to the additional two folded (z and k_{z}) integration. A TBIEbased computer program was implemented, debugged, and tested for some basic capacity, namely a dipole in the plane perpendicular to the borehole axis, which in general is tilted, and a transversely isotropic formation whose axis of symmetry is in general tilted. The program works well when the angle between these two axes is between 55 and 90 degrees. When this angle is between 0 and 20 degrees, the program produces waveforms that look reasonably good, but there are minor oscillations before the first arrival. When this angle is between 20 and 55 degrees, there are signs of numerical problems. Time limitations prevented completion of this work. For the examples given, boreholes at various deviated angles in viscoelastic TI formations were considered. A dipole source was assumed in the direction of x_{1} perpendicular to the borehole axis. Its time function is the Ricker wavelet with a center frequency of 3 kHz. There are 12 inline dipole receivers. Their offsets from the source are 2.70, 2.85, 3.00, 3.15, 3.30, 3.45, 3.60, 3.75, 3.90, 4.05, 4.20, and 4.35 meters, respectively. Accomplishments  The TBIE method has been successfully extended to the synthetic study of full waveform multipole sonic logs for deviated boreholes in anisotropic formations. The arrival time, wave shape and amplitude of the TBIE results have been verified with results produced with other methods. Numerical examples of synthetic dipole sonic logs confirm the significant role of deviated angle in the presence of formation anisotropy. The preliminary results demonstrate interesting patterns of the spectra, waveforms, timefrequency spectrogram and velocityfrequency plots as a result of the interaction between the anisotropy and deviation angle, which are important to an understanding of sonic logs in deviated wells. The project team presented a paper describing this method at the Society of Exploration Geophysicists (SEG) International Meeting in 2007. The presentations attracted members of Chevron and Marathon oil companies, who showed interest in our work. To fully offer this product to the oil and gas industry, processing capabilities to extract reservoir parameters from full waveform sonic data need to be developed. SwRI’s modeling program can be used to generate seismograms with which to develop processing techniques to extract reservoir parameters from cross dipole data that may be associated with stressinduced anisotropy or fractureinduced anisotropy.
