Evaluation of Non-intrusive Generalized Polynomial Chaos Expansion in the Context of Reservoir Simulation

Uncertainty quantification workflows which map input uncertainties to output uncertainties are an important part of many reservoir simulation applications. Examples are estimation of prediction uncertainties including history data, reserves estimation, or optimization under uncertainty. Over the last few years, proxy modeling techniques have turned out to be essential for improving the efficiency of the employed methods, with one example being stochastic sampling techniques like Markov Chain Monte Carlo. An established proxy model family is the generalized Polynomial Chaos Expansion (gPCE). Some applications to reservoir simulation exist - also in the context of history matching. In this work we focus on a non-intrusive, efficient way to construct such gPCE representations for multi-dimensional input uncertainties: sparse-grid spectral projection – also known as Smolyak grid. In standard approaches, which derive their multi-dimensional cubature rules from full tensor-products of one-dimensional rules, the number of required points grows exponentially with the input dimension. This causes a significant computational effort since each point represents a forward simulation. On the other hand, sparse-grid techniques used in this work show polynomial scaling behavior by “thinning out” the full tensor-product. However, in order to avoid possible numerical instabilities of this technique a design of simulation cases needs to be applied. Practical consequences and guidelines for real applications will be described in detail. The numerical framework for constructing gPCEs is applied to uncertainty quantification workflows in reservoir simulation. We investigate performance criteria for representing key performance indicators using gPCEs, e.g., cumulative properties like oil production total or transient properties such as bottom hole pressure. Practical application designs are derived and described.