Relating Time Series in Data to Spatial Variation in the Reservoir Using Wavelets


Abeeb Adebowale Awotunde







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An important issue in reservoir parameter estimation is to develop methods that are computationally efficient and reliable. Furthermore, in history matching, one of the challenges in the use of gradient-based Newton algorithms such as Gauss-Newton and Levenberg-Marquardt in solving the inverse problem is the huge cost associated with the computation of sensitivity matrix. Although the Newton type of algorithm gives faster convergence than most other gradient-based inverse solution algorithms, its use is limited to small and medium-scale problems in which the sensitivity coefficients are easily and quickly computed. Modelers often use less efficient algorithms such as conjugate-gradient and quasi-Newton to model large-scale problems because these algorithms avoid the direct computation of sensitivity coefficients. To find a direction of descent, such algorithms often use less precise curvature information that would be contained in the gradient of the objective function. Using a sensitivity matrix gives more detailed information about the curvature of the function; however this comes with a significant computational cost for large-scale problems.

We present a multiresolution wavelet approach to estimate the spatial distribution of reservoir parameters, by performing the nonlinear least squares regression in the wavelet domains of both time and space. Wavelet transforms have the ability to reveal important events in time signals or spatial images. Thus we transformed both the model space and the time series pressure data into spatial wavelet and time wavelet domains and used a thresholding to select a subset of wavelet coefficients from each of the transformed domains. These subsets were used subsequently in nonlinear regression to estimate the appropriate description of reservoir parameters. The appropriate subset is not only smaller; the problem is also reduced to the consideration of only the important components of the measured data and only the part of the reservoir description that depends on them. Furthermore, the approach developed in this research improves parameter estimation by reducing the dimension of the inverse problem without significant loss in accuracy.

We transformed the inverse problem into reduced wavelet spaces and subsequently coupled the transformation with adjoint sensitivity formulation to reduce the cost associated with computing sensitivity coefficients. The improved adjoint sensitivity formulation presented in this work exploits the reduction in observation (data) space to enforce a reduction in the adjoint linear system of equations thus leading to a reduction in the memory and time required to compute sensitivities. We were able to achieve up to 20 times reduction in computational requirements in some example cases.

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Copyright 2010, Abeeb Adebowale Awotunde: Please note that the reports and theses are copyright to their original authors. Authors have given written permission for their work to be made available here. Readers who download reports from this site should honor the copyright of the original authors and may not copy or distribute the work further without the permission of the author, Abeeb Adebowale Awotunde.

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