Sonic log prediction in carbonates

This work is conducted to study the complications associated with the sonic log prediction in carbonate logs and to investigate the possible solutions to accurately predict the sonic logs in Traverse Limestone. Well logs from fifty different wells were analyzed to define the mineralogy of the Traverse Limestone by using conventional 4-mineral and 3-mineral identification approaches. We modified the conventional 3-mineral identification approach (that completely neglects the gamma ray response) to correct the shale effects on the basis of gamma ray log before employing the 3-mineral identification. This modification helped to get the meaningful insight of the data when a plot was made between DGA (dry grain density) and UMA (Photoelectric Volumetric Cross-section) with the characteristic ternary diagram of the quartz, calcite and dolomite. The results were then compared with the 4-mineral identification approach. Contour maps of the average mineral fractions present in the Traverse Limestone were prepared to see the basin wide mineralogy of Traverse Limestone. In the second part, sonic response of Traverse Limestone was predicted in fifty randomly distributed wells. We used the modified time average equation that accounts for the shale effects on the basis of gamma ray log, and used it to predict the sonic behavior from density porosity and average porosity. To account for the secondary porosity of dolomite, we subtracted the dolomitic fraction of clean porosity from the total porosity. The pseudo-sonic logs were then compared with the measured sonic logs on the root mean square (RMS) basis. Addition of dolomite correction in modified time average equation improved the results of sonic prediction from neutron porosity and average porosity. The results demonstrated that sonic logs could be predicted in carbonate rocks with a root mean square error of about 4μsec/ft. We also attempted the use of individual mineral components for sonic log prediction but the ambiguities in mineral fractions and in the sonic properties of the minerals limited the accuracy of the results.

Fading pdf of free-space optical communication system with pointing error

A free-space optical (FSO) laser communication system with perfect fast-tracking experiences random power fading due to atmospheric turbulence. For a FSO communication system without fast-tracking or with imperfect fast-tracking, the fading probability density function (pdf) is also affected by the pointing error. In this thesis, the overall fading pdfs of FSO communication system with pointing errors are calculated using an analytical method based on the fast-tracked on-axis and off-axis fading pdfs and the fast-tracked beam profile of a turbulence channel. The overall fading pdf is firstly studied for the FSO communication system with collimated laser beam. Large-scale numerical wave-optics simulations are performed to verify the analytically calculated fading pdf with collimated beam under various turbulence channels and pointing errors. The calculated overall fading pdfs are almost identical to the directly simulated fading pdfs. The calculated overall fading pdfs are also compared with the gamma-gamma (GG) and the log-normal (LN) fading pdf models. They fit better than both the GG and LN fading pdf models under different receiver aperture sizes in all the studied cases. Further, the analytical method is expanded to the FSO communication system with beam diverging angle case. It is shown that the gamma pdf model is still valid for the fast-tracked on-axis and off-axis fading pdfs with point-like receiver aperture when the laser beam is propagated with beam diverging angle. Large-scale numerical wave-optics simulations prove that the analytically calculated fading pdfs perfectly fit the overall fading pdfs for both focused and diverged beam cases. The influence of the fast-tracked on-axis and off-axis fading pdfs, the fast-tracked beam profile, and the pointing error on the overall fading pdf is also discussed. At last, the analytical method is compared with the previous heuristic fading pdf models proposed since 1970s. Although some of previously proposed fading pdf models provide close fit to the experiment and simulation data, these close fits only exist under particular conditions. Only analytical method shows accurate fit to the directly simulated fading pdfs under different turbulence strength, propagation distances, receiver aperture sizes and pointing errors.

Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.