Hierarchical simultaneous upscaling of porosity and permeability features using the bandwidth of kernel function and wavelet transformation in two dimensions: Application to the SPE-10 model

In this paper, two methods of kernel bandwidth and wavelet transform are used for simultaneous upscaling of two features of hydrocarbon reservoir. In the bandwidth method, the criterion for upscaling is the cell variability, and by calculating the optimal bandwidth and determining the distance matrix, the upscaling process is performed in a completely non-uniform and unregularly manner. In areas with extreme variability, the bandwidth is considered small enough to maintain the fine scale characteristics of model. Conversely in homogenous areas, with the choice of large bandwidth, the maximum rate of upscaling will occur. The bandwidth upscaling algorithm is an iterative and hierarchical algorithm. The bandwidth method, unlike conventional scale-up methods, focuses on how to upgrid cells and, by determining the optimal averaging window, we will have the least loss information for the fine scale model. Upscaling is a pre-processing to building a simulator model with lower cell number, and thus, reducing volume and computational cost, while maintaining and retaining the basic information of the fine model. Due to the various variability of the reservoir features, the attribute upscaling pattern differs, and in order to show the variability of two features in the upscaling model simultaneously, it is suggested in this paper to upscale two features simultaneously. For simultaneous upscaling, we applied two different approaches; minimum and maximum bandwidth. Moreover, wavelet transformation is applied to upscaling the model. Then, as a result, the variance of the scale-up models based on wavelet is about one-third of the variance of the bandwidth method. Simulation results show that the bandwidth method is a good approach for upscaling the heterogeneous reservoirs.