Efficient sampling techniques for uncertainty

5 The Markov Chain Monte Carlo (MCMC) is a rigorous sampling method to quan6 tify uncertainty in subsurface characterization. However, the MCMC usually requires 7 many flow and transport simulations in evaluating the posterior distribution and can be 8 computationally expensive for fine-scale geological models. We propose a methodology 9 that combines coarseand fine-scale information to improve the efficiency of MCMC 10 methods. The proposed method employs offline computations for modeling the relation 11 between coarseand fine-scale error responses. This relation is modeled using nonlinear 12 statistical maps which are used in efficient sampling within MCMC framework. We 13 propose two-stage MCMC where inexpensive coarse-scale simulations are performed 14 to determine whether or not to run the fine-scale (resolved) simulations. The latter 15 is determined based on a statistical model developed offline. The proposed method 16 is an extension of the approaches considered earlier (e.g., [10]) where linear relations 17 are used for modeling the response between coarse-scale and fine-scale models. The 18 Department of Mathematics, Texas A&M University, College Station, TX 77843 Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843 Department of Petroleum Engineering, Texas A&M University, College Station, TX 77843 Department of Statistics, Texas A&M University, College Station, TX 77843 1 approach considered here does not rely on the proximity of approximate and resolved 19 models and can employ much coarser and inexpensive models to guide the fine-scale 20 simulations. Numerical results for three-phase flow and transport demonstrate the ad21 vantages, efficiency and utility of the method for uncertainty assessment in the history 22 matching. 23