We present a probabilistic approach for integrating multiple data into subsurface flow models. Our approach is based on a Bayesian framework whereby we exhaustively sample the multi-dimensional posterior distribution to define a Pareto front which represents the trade-off between multiple objectives during history matching. These objectives can be matching of water-cut, GOR, BHP and time-lapse seismic data. For field applications, these objectives do not necessarily move in tandem because of measurement errors and also interpretative nature of the seismic data.
Our proposed method is built on a Differential Evolution Markov Chain Monte Carlo (DEMC) algorithm in which multiple Markov Chains are run in parallel. First, a dominance relationship is established amongst multiple models. This is followed by construction of the posterior distribution based on a hypervolume measure. A unique aspect of our method is the proposal selection which is based on a random walk on two arbitrarily selected chains. This promotes effective mixing of the chains resulting in improved convergence.
We illustrate the algorithm using a nine-spot waterflood model whereby we use water-cut and bottomhole flowing pressure data to calibrate the permeability field. The permeability field is reparameterized using a previously proposed Grid Connectivity Transform (GCT) which is defined based on a decomposition of the grid Laplacian. The compression power of GCT allows us to reconstruct the permeability field with few parameters, thus significantly improving the computational efficiency of the MCMC approach. Next, we applied the method to the Brugge benchmark case involving 10 water injectors and 20 producers. For both cases, the algorithm provides an ensemble of models all constrained to the history data and defines a probabilistic Pareto front in the objective space. Several experimental runs were conducted to compare the effectiveness of the algorithm with NonDominated Sorting Genetic Algorithms (NSGA). Higher hypervolume was constantly measured using our algorithm which indicates that more optimal solutions were sampled. Our method provides a novel approach for subsurface model calibration and uncertainty quantification using MCMC in which the communication between parallel Markov chains enhances adequate mixing. This significantly improves the convergence without loss in sampling quality.
We demonstrate that the proposed approach can efficiently integrate seemingly conflicting data into reservoir models while quantifying uncertainties. Besides reservoir model calibration, we expect that the algorithm will find application in probabilistic production optimization problems as well.