In the paper there are discussed certain issues concerning ill-posed problems that frequently appear in inverse problems, modal analysis, acoustics and other methods making use of matrix algebra. There is presented the mathematical definition, application of the singular value decomposition method to ill-posed problems detecting as well as a new method of improving such problems conditioning by the use of the Tikhonov regularization method. In the paper are presented some results of solution estimation of the Fredholm integral equation of the first kind that is the classical ill-posed problem. Analysis was carried out in the Matlab environment by means of the least squares and Tikhonov regularization methods for both the noiseless and noisy cases.
New ways of calculating narrow microparticle size distributions based on using the Tikhonov and the modified Twomey methods for the laser diffraction technique are presented. These allow to have reduced the broadening (over-smoothing) of the result occurring in these methods for narrow distributions both singular and their sum. The calculated singular distributions and their distribution sum were then approximated by a Gaussian function and a bimodal Gaussian function, respectively, using the Levenberg- Marquardt method. The angular distribution of scattering power was measured for polystyrene particles with radii of 0.676 μm and 1.573 μm, and for their sum. The tests were carried out for linearly polarized He-Ne laser light scattered by a dilute aqueous suspension of these particles. The results obtained were compared with those obtained with the nanoDS instrument (CILAS). It turned out that using the way based on the Twomey method, the parameters of the narrow distribution sought could be determined quite well.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
A dozen of inversion methods are applied and tested to estimate the permeability of the area where gas-fngering event has taken place in an Iranian carbonate reservoir located southwest of Iran. In a previous work, the gas-fingering event was detected by inverting the 3D seismic data and in this study the permeability model in that area is estimated. Because the lateral area of the gas-fingering event is narrow, the whole system conducting the injected gas can be considered as one rock unit system and therefore the assumption of horizontal linear steady-state fow can be applied. Inversion methods are exploited to determine the permeability in the interval of interest. The interval of interest is located at the crest and involves four wells among which one is the gas-injection well. To investigate the feasibility of such an approach and select the best possible inversion method, frst a controlled experiment for the system is designed and studied. The porosity values of the system are known from seismic data inversion and the permeability values are the desired parameters. The permeability values at well locations are known via well-test data and are used as constraints in the inversion procedure. The interval of interest is discretized and a simulator is used to simulate the fuid fow in the controlled system in order to apply and validate the inversion methods. All calculations are performed in the MATLAB environment. According to the results from the controlled experiment, the Maximum Entropy and Total Variation methods were found to be the best two inversion methods which were successful in retrieving the true permeability model. Similar comparative study using diferent inversion methods is performed for the real case for which the results retrieved by the Total Variation method is most reliable as it suggests the best recovery of the permeability value for the check-well. An estimation of the fracture permeabilities for the area under study also indicated that the inverted permeability values are most representing the fracture permeabilities rather than the matrix. The results of this study will be used to tune the feld simulation model in terms of rock and fuid properties, consider the inverted permeability model as further constraints for the reservoir history-matching of the oil feld, reconsider the factors involving the gas injection plan for the oil feld, and obtain insights for further feld development plans in other nearby oil felds.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.