SPARSE PROBABILISTIC COLLOCATION FOR UNCERTAINTY QUANTIFICATION IN RESERVOIR ENGINEERING

Mendes, J. H.; Willmersdorf, R. B.;

This paper presents an application of uncertainty quantification to numerical reservoir simulation using the Sparse Probabilistic Collocation Method (SPCM). Reservoir simulation is used in several phases of the development and exploitation of a field, from the initial planning of the production strategy to sophisticated automated control schemes that schedule the operation of well controls on a daily basis in smart fields. It basically consists of solving numerically the complex nonlinear partial differential equations (PDEs) that models the fluid flow in porous media. The petrophysical properties of the rock matrix determine the coefficients in the PDEs and associated algebraic system of equations. Due to technological and economic constraints, the available data to determine these properties is scarce and subject to human interpretation. This problem becomes even more important for offshore fields, where wells are kilometers apart, the reservoirs several kilometers underground, there are very few wells and there is very little or no production history. The petrophysical properties are therefore very uncertain, and can be described only in a probabilistic manner. The simulation can no longer be considered deterministic, since uncertain inputs leads to uncertain results. Uncertainty propagation techniques become necessary tools for the robust and reliable application of numerical reservoir simulation. In the probabilistic collocation method, statistics of the uncertain output are computed directly through numerical integration, based on efficient quadrature rules like Gauss and Clenshaw-Curtis. However, this method is not suitable for dealing with high-dimensional models, because it suffers from the “curse of dimensionality”. Sparse grid integration techniques can be used with the probabilistic collocation method to alleviate this problem, creating the sparse probabilistic collocation method. We present our implementation of the SPCM and apply it to estimate the statistics of uncertain variables such as the cumulative oil production and water breakthrough date, using a simple but realistic reservoir model. Comparisons of the efficiency of this technique against classical methods such as Monte Carlo are shown, as well as a discussion on the necessary computational resources for this kind of analysis and it’s practical use.

Palavras-chave: Polynomial chaos, Stochastic collocation, Sparse grids, Reservoir engineering,

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DOI: 10.5151/meceng-wccm2012-19032

Referências bibliográficas
• [1] Brian M. Adams, Keith R. Dalbey, Michael S. Eldred, David M. Gay, Laura P. Swiler, William J. Bohnhoff, John P. Eddy, Karen Haskell, and Patricia D. Hough. DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis. Sandia National Laboratories, 5.0+ edition.
• [2] Nitin Agarwal and N. R. Aluru. A domain adaptive stochastic collocation approach for analysis of mems under uncertaintes. Journal of Computational Physics, 2009.
• [3] John Burkardt. Advanced numerical methods for computing statistical quantities of interest from solutions of stochastic partial differential equations. Technical report, Interdisciplinary Center for Applied Mathematics/Information Technology Department - Virginia Tech, August 2010.
• [4] CMG. IMEX Advanced Oil/Gas Reservoir Simulator. Computer Modelling Group Ltd., Office #150, 3553 - 31 Street N. W. Calgary, Alberta Canada T2L 2K7, 2008 edition, 2008.
• [5] M. S. Eldred, C. G. Webster, and P. G. Constantine. Evaluation of non-intrusive approaches for wiener-askey generalized polynomial chaos. American Institute of Aeronautics and Astronautics, 2008.
• [6] Jasmine Foo, Xiaoliang Wan, and George Em Karniadakis. The multi-element probabilistic collocation method (me-pcm): Error analysis and applications. Journal of Computational Physics, 2008.
• [7] Benjamin Ganis, Hector Klie, Mary F. Wheeler, Tim Wildey, Ivan Yotov, and Dongxiao Zhang. Stochastic collocation and mixed finite elements for flow in porous media. Computer methods in applied mechanics and engineering, 2008.
• [8] Thomas Gerstner and Michael Griebel. Numerical integration using sparse grids.
• [9] John Jakeman, Michael Eldred, and Dongbin Xiu. Numerical approach for quantification of epistemic uncertainty. Journal of Computational Physics, 2010.
• [10] G.J.A. Loeven, J.A.S. Witteveen, and H. Bijl. Probabilistic collocation: An efficient non-intrusive approach for arbitrarily distributed parametric uncertainties. Aerospace Sciences Meeting, 2007.
• [11] Sergey Smolyak. Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk SSSR, 1963.
• [12] Dongbin Xiu and George Em Karniadakis. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Computer methods in applied mechanics and engineering, 2002.
• [13] Dongbin Xiu and George Em Karniadakis. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 2003.
Como citar:

Mendes, J. H.; Willmersdorf, R. B.; "SPARSE PROBABILISTIC COLLOCATION FOR UNCERTAINTY QUANTIFICATION IN RESERVOIR ENGINEERING", p. 2802-2816 . In: In Proceedings of the 10th World Congress on Computational Mechanics [= Blucher Mechanical Engineering Proceedings, v. 1, n. 1]. São Paulo: Blucher, 2014.
ISSN 2358-0828, DOI 10.5151/meceng-wccm2012-19032

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