Novel Multiple Resolutions Design of Experiment/Response Surface Methodology for Uncertainty Analysis of Reservoir Simulation Forecasts

Response surfaces (RS) are useful and simple proxies to simulators, which relate in a closed form output variables (oil recovery factor, oil rate, etc.) of a simulator to input parameters over the entire parameter space. These proxies can then be used to compute production forecast uncertainty profiles with Monte Carlo calculation, instead of relying on time consuming reservoir simulation. Typically, design of experiment (DOE) and response surface methodologies (RSM) are used to both identify statistically significant factors and to generate RS. RS are usually low order polynomials, which are generated by using the regression method. These methodologies work well if response surfaces are smooth with weak non-linearity. However, when these polynomials fail to meet reasonable accuracy criteria, adding more points to the design will only marginally improve RS accuracy. Response surface methodology can be adapted for such cases by using interpolation methods (Kriging, Spline, etc.) instead of regression method. But, most interpolation methods tend to smooth out non-linearity. These interpolated proxies can become substantially inaccurate, if: 1. Non-linear effects are very strong; 2. The non-linearity is not uniformly distributed throughout the parameter space. In this paper, we introduce a novel RSM and its supporting DOE methodology to construct response surfaces as proxies of a simulator when input factors of the simulator such as faults cause strong non-linear effects. These methodologies are used to generate RS of arbitrary shapes by iteratively interpolating on a multilevel grid in the experimental space, which allows the local subdivision of the parameter domain. New partitions and interpolation points are added adaptively in the selected parameters regions, if local errors of the constructed response surface exceed a pre-selected threshold. The art of these methodologies consists in: 1. Splitting the whole domain into sub-domains of multiple scale levels, where the components of a RS can be accurately modeled with ‘thin plate’ spline interpolants. The resultant RS is obtained by combining its global and local components. 2. Achieving adequate RS accuracy with the minimum number of simulation runs. We modify the SPE9 Benchmark problem to investigate the method’s validity and efficiency. We compare it with traditional DOE and RSM methodologies. Finally, we perform Monte Carlo simulation using the RS proxies to assess uncertainty of the production forecasts. Results show that the presented methodologies work accurately and efficiently for problems with strongly non-linear effects, especially when the non-linear effects are non-uniformly distributed.