A Two-Level Optimization Method for Geostatistical Integration of Production Data on Non-Uniform Grids

Fine scale heterogeneities can have significant effects on flow performances in subsurface formations. Highly detailed geostatistical realizations are needed to capture the quantitative effects of these heterogeneities on flow simulation results based on high resolution well-log and medium resolution seismic data. These realizations in many cases require number of grid cells on the order of 10⁶-10⁷, hence making it too demanding CPU wise to directly history match geostatistical models. Non-uniformly gridding these models for emphasizing the heterogeneities does not necessarily present a solution since the practice of geostatistics on non-uniform grids is inefficient. In this paper we provide an entirely new approach to these problems by proposing a solution which is geostatistics based yet having the flexibility to work on non-uniformly gridded models. Hence coupling geostatistics with non-uniformly upscaled models results in an effective history matching technique incorporating all information at their scales. With the proposed approach we mainly target three advantages. Well and seismic data will still be effectively incorporated in the realizations. Flow simulations will be performed on non-uniformly upscaled models hence will be relatively fast. Finally the resulting field which is history matched will include the geology and will be on a non-uniformly gridded model for emphasizing heterogeneities or desired specifications, hence future predictions can be made fast, no posterior upscaling is required. The latter will avoid upscaling errors. This new approach is based on making a fast non-uniform upscaling technique a part of the history matching process instead of performing upscaling first and then downscaling again. Two levels of optimization are introduced in this technique; one level for history matching (outer loop) and the other for optimizing the non-uniform grids based on the results of a fast but realistic flow response (inner loop). In this paper we show how we benefit from a pre-optimized solution in conditioning reservoir models to production data.