Stein Variational Gradient Descent for Reservoir History Matching Problems

Abstract

AbstractReservoir history matching problem estimates the system (i.e., reservoir morel) parameters based on noisy observed data. Examples can be estimating the permeability and porosity fields from time series of oil, water, and gas production rates. The estimation of parameters is formulated in the form of estimating their probability distributions; it is a required step for reservoir management operation and planning under subsurface uncertainty. The Bayesian framework is commonly used to estimate the posterior distribution of parameters, which may contain multiple modes that correspond to distinct reservoir scenarios. Here, we study the application of Stein Variational Gradient Descent (SVGD) method, originally proposed by Liu & Wang (2016), in reservoir history matching problems. The rationale and mechanics of SVGD method is discussed and the adaptation of this method to the reservoir characterization application is presented. More specifically, we propose to formulate the gradient-based SVGD method using stochastic gradients for reservoir history matching applications. To the best of our knowledge, this paper presents the first application of SVGD method for reservoir characterization problem. The utilization of stochastic approximation of gradients within a gradient-based SVGD is another novelty aspect of this work. The formulated algorithm is benchmarked using synthetic test problems with multimodal known posterior distributions. Also, the application of the proposed algorithm is investigated to solve synthetic and real history matching problems including the IC Fault model and an unconventional well simulation model. The reservoir test problems are further investigated to evaluate the method's performance in comparison with application of implementations of a Gauss Newton optimization and an iterative Ensemble Smoother method for sampling the posterior distribution. We show that the proposed implementation of SVGD can capture the posterior distribution and complicated geometry. For reservoir IC Fault test problem, the method effectively samples multiple modes. For the unconventional test problem, the samples are compared with the ones obtained using a Gauss Newton or iterative Ensemble Smoother methods.