Polynomial Chaos Based Solution to Inverse Problems in Petroleum Reservoir Engineering

Abstract

In reservoir simulations, model parameters such as porosity and permeability are often uncertain and therefore better estimates of these parameters are obtained by matching the simulation predictions with the production history. Bayesian inference provides a convenient way of estimating parameters of a mathematical model, starting from a probable range of parameter values and knowing the production history. Bayesian inference techniques for history matching require computationally expensive Monte Carlo simulations, which limit their use in petroleum reservoir engineering. To overcome this limitation, we perform accelerated Bayesian inference based history matching by employing polynomial chaos (PC) expansions to represent random variables and stochastic processes. As a substitute to computationally expensive Monte Carlo simulations, we use a stochastic technique based on PC expansions for propagation of uncertainty from model parameters to model predictions. The PC expansions of the stochastic variables are obtained using relatively few deterministic simulations, which are then used to calculate the probability density of the model predictions. These results are used along with the measured data to obtain a better estimate (posterior distribution) of the model parameters using the Bayes rule. We demonstrate this method for history matching using an example case of SPE1CASE2 problem of SPEs Comparative Solution Projects. We estimate the porosity and permeability of the reservoir from limited and noisy production data.