Efficient Markov Chain Monte Carlo sampling using Polynomial Chaos Expansion

Markov Chain Monte Carlo (MCMC) methods are often used to probe the posterior probability distribution in inverse problems. This allows for computation of estimates of uncertain system responses conditioned on given observation data by means of approximate integration. However, these methods suffer from the computational complexities in reservoir simulation. It is hence of great interest to propose a sparse approximation technique to represent the posterior probability density function and conditional expectations given the data.

Polynomial Chaos Expansion (PCE) is a powerful tool to quantify uncertainty in dynamical systems when there is probabilistic uncertainty in the system parameters. In reservoir simulation it has been shown to be more accurate and efficient compared with traditional experimental design (ED). PCEs have a significant advantage over other response surfaces as the convergence to the true probability distribution when the order of the PCE is increased can be proved. PCEs as applied with Non-Intrusive Spectral Projection (NISP) to compute the coefficients, have successfully and efficiently approximated the distribution of output random variables of interest such as cumulative oil production. Accordingly polynomial chaos proxy can be used as the pseudo-simulator to represent the probability density function of the uncertain variables.

We propose the application of polynomial chaos proxy as applied with the MCMC method to efficiently sample from the posterior probability density function of the system parameters. We present a two dimensional example of fluvial channels to demonstrate that with a few hundred trial runs of the actual reservoir simulator, it is feasible to construct a polynomial chaos proxy which accurately approximates the posterior distribution of the high permeability zones, in an analytical form. We show that precision improves as the order of PCE and the number of trial runs used to calculate the PCE coefficients is increased. Then we use Markov Chain Monte Carlo methods to sample from this sparse analytical representation of the posterior distribution.