A Multistage Sampling Method for Rapid Quantification of Uncertainty During Subsurface Characterization

We discuss a novel sampling method for rapid quantification of uncertainty during subsurface characterization via inverse modeling of dynamic data, specifically multiphase production response and time-lapse (4D) seismic data. Uncertainty evaluation is generally carried out in a Bayesian framework whereby multiple subsurface models can be evaluated by sampling from a posterior distribution that incorporates the observed data and the prior parameter distribution. Rigorous sampling methods such as the Markov Chain Monte Carlo (MCMC) method provide accurate sampling but at a high cost because of their high rejection rates and the need to run a full flow and transport simulation for every proposed candidate. Approximate sampling methods like the randomized maximum likelihood (RML) are often used for computational efficiency but the assumptions are too restrictive, particularly for nonlinear problems in multiphase flow and transport. We discuss here a two-stage Markov Chain Monte Carlo (MCMC) method that utilizes a combination of a fast linearized approximation of the dynamic data and the MCMC algorithm. Our proposed sampling approach is rigorous, computationally efficient and has a significantly higher acceptance rate compared to traditional MCMC algorithms. In the first stage we utilize streamlines or trajectory-based analytic sensitivities to obtain an approximation in a small neighborhood of the previously computed dynamic data. These analytic sensitivities do not require any additional flow simulations. The approximation of the dynamic data is then used to modify the instrumental proposal distribution in the MCMC. Only those proposals that pass the acceptance criterion in the first stage are then assessed by running full flow simulations to assure rigorousness in sampling and are either accepted or rejected using the Metropolis-Hastings criterion. It is shown that the modified Markov chain converges to a stationary state corresponding to the posterior distribution. Both two dimensional synthetic examples and three dimensional field applications will be used to demonstrate the power and utility of the two-stage sampling method for dynamic data integration and uncertainty analysis.