Using Distances and Kernels to Parameterize Spatial Uncertainty for Flow Applications

Modeling spatial uncertainty is challenging due to the large dimensionality of the problem. In Petroleum Geostatistics, this problem is compounded by the CPU demand of flow simulations, making traditional Monte Carlo approaches not always practical. Traditional ranking techniques for selection of P10, P50 and P90 flow responses are highly dependent on the ranking property used (e.g. OOIP). In this paper, we propose to parameterize the spatial uncertainty represented by a large set of geostatistical models through a distance function measuring “dissimilarity” between any two geostatistical realizations. The distance function allows, through multi-dimensional scaling, mapping the space of uncertainty into any dimension to visualize uncertainty. The distance function should be tailored to the particular problem, in this case, flow responses. This nD space can be modeled using kernel techniques, such as kernel principal component analysis (KPCA). KPCA allows for the selection of a subset of representative realizations containing similar properties to the larger set. Without losing accuracy, production decisions can then be performed using flow simulation on this subset of realizations. A case study is presented, where spatial uncertainty of channel facies is modeled through multiple realizations generated using a multi-point geostatisitcal algorithm and several training images.