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This paper presents an application of uncertainty quantification to numerical reser- voir simulation using the Sparse Probabilistic Collocation Method (SPCM). Reservoir simu- lation 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 coeffi- cients 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 there- fore 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 appli- cation 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 re- sources for this kind of analysis and it's practical use.

SPARSE PROBABILISTIC COLLOCATION FOR UNCERTAINTY QUANTIFICATION IN RESERVOIR ENGINEERING