Uncertainty quantification in history matching of channelized reservoirs using Markov Chain level set approaches

We present a method for history matching and uncertainty quantification for channelized reservoir models using Level Set Method and Markov Chain Monte Carlo. Our objective is to efficiently sample realizations of the channelized permeability fields conditioned to the production data and facies observation at the wells. In our approach, the channel field boundary is first described by a level set function, e.g., a signed distance function or any other indicator function. By solving the level set equation (motion in a prescribed direction), we are able to gradually move the channel boundaries and evolve the channelized reservoir properties. Our approach allows representing facies via a parameterization of the velocity field that deforms the interface. Thus facies can be parameterized in the space of smooth velocity fields. The dimension reduction can be achieved for covariance-based velocity fields by re-parameterizing with SVD techniques. After parameterization, Markov Chain Monte Carlo method is utilized to perturb the coefficients of principal components of velocity field to update channel reservoir model matching production history. One advantage of this approach is that it is easy to condition the channel model to the facies observations at well locations by constraining the random velocity field to zero at well locations. To speed up the computation and improve the acceptance rate of the MCMC algorithm, we employ two stage methods where coarse-scale simulations are used to screen out the undesired proposals. The MCMC algorithms naturally provide multiple realizations of the permeability field conditioned to well and production data and thus, allow for uncertainty assessment in the forecasting. We demonstrate the effectiveness of the level set MCMC algorithm using both 2D and 3D examples involving waterflood history matching.