Abstract
We consider prediction and uncertainty analysis for the ‘history matching’ problem in petroleum reservoir evaluation. Unknown reservoir properties are represented on a fine three dimensional lattice. A ‘reservoir simulator’ takes the reservoir properties as input and gives production properties as output. The history matching problem is to infer the reservoir properties from the observed production history. To run the reservoir simulator on the lattice size of interest is computer intensive, and this severely limits the number of runs possible.
We formulate the problem in a Bayesian setting and, following suggestions in the statistical literature, consider the reservoir simulator as an unknown function. To obtain a realistic prior distribution for this function, we propose to combine a coarse lattice (faster) version of the simulator with parameters correcting for bias introduced by the coarser lattice. We simulate from the resulting posterior by Markov chain Monte Carlo (MCMC). We construct an artificial reference reservoir, generate corresponding flow observations, and use our procedure to evaluate the reservoir properties in the resulting posterior distribution. Convergence and mixing are acceptable. The case study demonstrates how the observed production history provides information about both the reservoir properties and the bias correcting parameters included in the prior specification.