Efficient sampling techniques for uncertainty quantification in history matching using nonlinear error models and ensemble level upscaling techniques

The Markov chain Monte Carlo (MCMC) is a rigorous sampling method to quantify uncertainty in subsurface characterization. However, the MCMC usually requires many flow and transport simulations in evaluating the posterior distribution and can be computationally expensive for fine-scale geological models. We propose a methodology that combines coarse-and fine-scale information to improve the efficiency of MCMC methods. The proposed method employs off-line computations for modeling the relation between coarse-and fine-scale error responses. This relation is modeled using nonlinear functions with prescribed error precisions which are used in efficient sampling within the MCMC framework. We propose a two-stage MCMC where inexpensive coarse-scale simulations are performed to determine whether or not to run the fine-scale (resolved) simulations. The latter is determined on the basis of a statistical model developed off line. The proposed method is an extension of the approaches considered earlier where linear relations are used for modeling the response between coarse-scale and fine-scale models. The approach considered here does not rely on the proximity of approximate and resolved models and can employ much coarser and more inexpensive models to guide the fine-scale simulations. Numerical results for three-phase flow and transport demonstrate the advantages, efficiency, and utility of the method for uncertainty assessment in the history matching.