A Distance-based Representation of Reservoir Uncertainty: the Metric EnKF

Uncertainty quantification is probably one of the most important and challenging aspects of reservoir modeling, engineering and management. Traditional approaches have failed to provide manageable solution models. In most cases, uncertainty is represented through either a multi-variate distribution model (posterior) in a Bayesian modeling context or through a set of multiple realizations in a more frequentist view. The traditional Monte-Carlo simulation approach, whereby multiple alternative high-resolution models are generated and used as input to reservoir simulation or optimization codes, has not been proven to be practical mainly due to CPU limitation, nor is it effective or flexible enough in terms of addressing a varying degree of uncertainty assessment issues. In this paper, we propose a reformulation of reservoir uncertainty in terms of distances between any two reservoir model realizations. Instead of considering a reservoir model realization as some point in a high-dimensional space on which a complex posterior distribution is defined, we consider a distance matrix which contains the distances between any two model realizations. The latter matrix is of size NR×NR, NR being the number of realizations, much less for example that an N×N covariance matrix, N being the number of grid-blocks. Note that, unlike a co-variance matrix or probability function, a distance can be tailored to the specific problem at hand, for example water-breakthrough, or OOIP, even though the realizations remain the same. Next, the classical Karhunen-Loeve expansion of a Gaussian random field, based on the eigenvalue decomposition of an N×N covariance table, can now be formulated as function of the NR×NR distance matrix. To achieve this, we construct an NR×NR kernel matrix using the classical radial basis function, which is function of the distance and perform eigenvalue decomposition of this kernel matrix. In kernel space, new realizations can be generated or adjusted to new data. As application we discuss a new technique, termed metric EnKF yo simulataneously update multiple non-Gaussian realizations with production data. Using this simple dual framework to probabilistic-based uncertainty, I show how many types of reservoir uncertainty (spatial, non-spatial, scenario-based etc..) can be modeled in this fashion. I show various applications of this framework to the problem of history matching, model updating and optimization.