A fast markov chain Monte Carlo method for conditioning reservoir models to dynamic data

The integration of dynamic data, such as well-test or production data, has become an increasingly important task for accurate reservoir characterization. Integration of dynamic data is mathematically treated as an ill-posed inverse problem. To solve such problem one faces two major challenges: the forward model, i.e. the flow simulator is computationally demanding; the resulting multiple reservoir models generated should honor a prior geological vision. In geostatistics such prior information is usually provided through a variogram and the assumption of Multi-Gaussianity for the resulting reservoir models. In this paper, we propose an entirely new approach to solving this problem within the framework of Markov chain Monte Carlo simulation. First, we develop a Markov chain method for sampling from the prior distribution. We assume that a Markov property holds in the sense that the local conditional distribution of permeability at any location depends only on the closest neighboring permeability values. We show how these local conditional distributions can be calculated from a given variogram. Given a limited set of simulated realizations from this Gauss-Markov prior distribution, we run the flow-simulator on these models: and observe the pressure and recovery responses for the actual well-field configuration. Given these responses, we train neural networks to approximate the non-linear multivariate function between permeability field and field-response. Using a realistic dataset, we show that the neural'network approximation is.good to excellent; The neural network can then be used to extend the Markov chain formulation to generate realizations that are constrained to the dynamic data and honor a given variogram.