A novel method is presented for history matching and uncertainty quantification for channelized reservoirs using Reversible Jump Markov Chain Monte Carlo (RJMCMC) methods. Our objective is to efficiently sample realizations of channelized permeability fields conditioned to production data and permeability values at the wells.
In our approach, the channelized permeability field is reparameterized using the discrete cosine transform (DCT). The parameters representing the channel structure are the coefficients in truncated frequency domain. The parameter space is searched using a RJMCMC, where the dimension of the parameter space is assumed to be unknown. For each step of the RJMCMC, the dimension of the uncertainty space can be increased or decreased according to a prescribed prior distribution. This flexibility in the parameter dimension allows an efficient search of the uncertainty space. To speed up the computation and improve the acceptance rate of the RJMCMC algorithm, we employ two-stage methods whereby coarse-scale simulations are used to screen out the undesired proposals. After simulations, multi-dimensional scaling and cluster analysis are used to select realizations from the accepted models to adequately represent the diversity of the models.
We demonstrate the effectiveness of the RJMCMC algorithm using both 2D and 3D examples involving waterflood history matching. The 2-D example shows that the RJMCMC algorithm can successfully match the data and identify the orientation of the channels in the reference model. The 3-D results show that the proposed algorithm can determine the large-scale features of the reference channelized permeability field based on the production data. The MCMC algorithms naturally provide multiple realizations of the permeability field conditioned to well and production data and thus, allow for uncertainty quantification in the forecasting.