Handling Multimodality with Ensemble MCMC for Uncertainty Quantification of Relative Permeability and Capillary Pressure

Abstract The uncertainty of relative permeability and capillary pressure can be naturally modeled with Bayesian statistics and corresponding probability distributions. These so-called posterior distributions are defined with Bayes’ theorem and infer the uncertainty on the curves’ parameters from an ansatz about the uncertainty associated with data measured in coreflooding or centrifuge experiments. In order to compute the resulting density function, a numerical solver is used to simulate the experiment together with a numerical integrator in the form of a Monte Carlo Markov Chain (MCMC) method. One of the challenges in obtaining faithful solutions for the posterior probability distribution is to cope well with the presence of multiple modes, i.e., separate regions with nonzero probability masses. In this work, we combine an in-house 1D simulator for two-phase flow in porous media, an ensemble MCMC method, implemented by the emcee library, and a clustered kernel density estimation for the Metropolis-Hastings transition distribution, with the objective to efficiently sample posterior distributions that may be anisotropic and multimodal. Results are shown quantifying the uncertainty for relative permeability and capillary pressure curves. We consider a synthetic USS experiment, which allows us to validate the obtained curves against their true values, as well as real USS and centrifuge experiments, for which the true curves are unknown. We show that our adapted sampler is able to map the posterior modes faithfully, enabling the user to deduce how to improve experimental data to mitigate uncertainty. It is also exemplified how the sampler ensures that the ensemble walkers escape easily out of negligible local maxima of the posterior landscape, a feature that is not present when employing the default stretch move of the emcee library.