Sampling from the posterior in reservoir simulation

In recent years, there has been significant effort within reservoir engineering to move towards reservoir simulation studies that rigorously accounts for uncertainties in the model parameters when developing production forecasts. Probabilistic history matching approaches play an increasingly integral role in the calibration of subsurface reservoir models to dynamic data. In a Bayesian fashion, the calibration process employs prior distributions for the uncertain parameters and the likelihood function to generate posterior distributions. Due to the impossibility of having a closed form for the reservoir simulation evaluations, uncertainty quantification is done via sampling methodologies. Markov Chain Monte Carlo (MCMC) is the widelyaccepted uncertainty quantification technique used for this purpose. MCMC applications often require several millions of evaluations of a forward model. In reservoir engineering, each forward evaluation (reservoir simulation) can take hours, direct application of MCMC is too computationally expensive in most real-world applications. In reservoir simulation applications, sampling is usually performed indirectly via proxies, which replace each forward evaluation with an inexpensive one. The accuracy of the results relies strongly on the ability of the proxy to robustly represent the high-dimensional relationship between the model parameters and the simulator output. The industry standard, polynomial response surface regression, can be quite inaccurate when assessed using blind testing. In addition, the number of simulation runs required to construct an adequate proxy model can itself be prohibitively large. We present a novel application of Bayesian inference within the reservoir simulation domain. A parallelized nested sampling workflow is integrated with ultra-fast reservoir simulation technology running on Graphics Processing Units (GPUs). MultiNest, a variant of nested sampling, has been shown to require fewer number of runs than conventional MCMC algorithms such as Metropolis-Hasting. In large models, GPUs significantly accelerate each forward evaluation compared to models running solely on Central Processing Units (CPUs). The combination of nested sampling and accelerated simulations results in a rigorous methodology that can be used to quantify uncertainty in reservoir simulation workflows. The methodology has been applied to synthetic models and field models. It is affordable in terms of cost and elapsed time. Examples are shown to illustrate the capabilities of the workflow.