Conditioning Stochastic Reservoir Models to Well-Test Data
Authors: He Nanqun , Oliver Dean S. , Reynolds Albert C. 
Published: December 2007

Abstract
Generating realizations of reservoir permeability and porosity fields that are conditional to static and dynamic data is difficult. The constraints imposed by dynamic data are typically non-linear and the relationship between the observed data and the petrophysical parameters is given by a flow simulator which is expensive to run. In addition, spatial organization of real rock properties is quite complex. Thus most attempts at conditioning reservoir properties to dynamic data have either approximated the relationship between data and parameters so that complex geologic models could be used, or have used simplified spatial models with actual production data.
In this paper, we describe a multistep procedure for efficiently generating realizations of reservoir properties that honor dynamic data from complex stochastic models. First, we generate a realization of the rock properties that is conditioned to static data. Second, we generate a realization of the production data (i.e. add random errors to the production data). Third, we find the property field that is as close as possible to the uncalibrated realization and also honors the realization of the production data. The ensemble of realizations generated by this procedure provides a good empirical approximation to the posteriori probability density function for reservoir models and can be used for Monte Carlo inference.
We apply the above procedure to the problem of conditioning a three-dimensional stochastic model to data from two well tests. The real-field example contains two facies. Permeabilities within each facies were generated using a "cloud transform" that honored the observed scatter in the crossplot of permeability and porosity. We cut a volume, containing both test wells, from the full-field model, then scaled it up to about 9,000 cells before calibrating to pressure data. Although the well-test data were of poor quality, the data provided information to modify the permeabilities within the regions of investigations and on the overall permeability average.

Introduction
Over the past several years, we have worked to develop methods for generating plausible reservoir models that are conditional to dynamic or production-type data. A large part of our effort has gone into ensuring that the ensemble of realizations that we generated would be representative of the uncertainty in the reservoir properties. In order to do this rigorously, we limited ourselves to Gaussian random fields and to fairly small synthetic models. We recently applied Markov chain Monte Carlo methods to generate ensemble of realizations because we believe they provide the best framework for ensuring that we obtain a representative set of realizations suitable for making economic decisions. Unfortunately, it is usually impractical to use MCMC methods for generating realizations that are conditional to production data. If realizations are proposed from a relatively simple probability density function (e.g. multivariate Gaussian), then most realizations are rejected and the method is inefficient. Alternatively, if realizations are proposed from a PDF that is complicated but close to the desired PDF, the Metropolis-Hastings criterion is difficult to evaluate.
In 1996, we proposed a methodology for incorporating production data that followed the second approach but ignored the Metropolis-Hastings criterion, instead accepting every realization. P. 129^