Performance of ensemble Kalman filter and Markov chain Monte Carlo under uncertainty in forecast model

Abstract

Abstract Ensemble Kalman filter (EnKF) and Markov chain Monte Carlo (MCMC) are popular methods to obtain the posterior distribution of unknown parameters in the reservoir model. However, millions of simulation runs may be required in MCMC for accurate sampling of posterior as subsurface flow problems are highly nonlinear and non-Gaussian. Similarly, EnKF formulated on the basis of linear and Gaussian assumptions may also require a large number of realizations to correctly map the solution space of the unknown model parameters, ultimately resulting in the high computational cost. Data-driven meta/surrogate/proxy models provide an alternative solution to alleviate the issue of high computational cost. Since these models are not as accurate as numerical solutions of partial differential equations (PDE), their implementation may add an uncertainty in the forecast model. In literature, the effect of uncertainty in forecast model on data assimilation is not well studied, especially with field-scale reservoir models. In this work, we take the opportunity to evaluate and compare the performance of EnKF and MCMC using polynomial chaos expansion (PCE) based forecast model. Proposed forecast model relies on reducing parameter space using Karhunen–Loeve (KL) expansion which preserves the two-point statistics of the field. Random variables from KL expansion and orthogonal polynomials corresponding to the prior probability density function (pdf) form the set of input parameters in PCE. Further, non-intrusive probabilistic collocation method (PCM) is used to compute PCE coefficients. PCE forecast model is then used in EnKF and MCMC to calculate the likelihood of the samples in place of high fidelity full physics simulation runs. A case study is performed using a 3D field scale model of a reservoir located near Fort McMurray in northern Alberta, Canada. Performance of EnKF and MCMC are assessed under forecast model uncertainty using rigorous qualitative and quantitative analysis and posterior distribution characterization. Results clearly depict that, although EnKF provided reliable mean and variance estimates of model parameters, MCMC outperformed the former even under the uncertainty associated with PCE metamodel. Inaccurate initial assumptions of model parameters were successfully handled by MCMC, although, with a longer burn-in period. Furthermore, characterization of posterior demonstrated reduced uncertainty in the estimation of model parameters using MCMC as compared to EnKF. Practical implications of the proposed approach and performance assessment under forecast model uncertainty will be consequential in designing accurate and computationally efficient reservoir characterization workflows and hence, improved decision-making in reservoir management.