Summary
Generating realizations of reservoir permeability and porosity fields that are conditional to static and dynamic data are difficult. The constraints imposed by dynamic data are typically nonlinear 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, but not to the pressure 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 often 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
The problem of generating plausible reservoir models that are conditional to dynamic or production-type data has been an active area of research for several years. Existing studies can be classified by the way in which they approach three key aspects of the problem:Complexity of the stochastic geologic or petrophysical model.Method of computing pressure response from a reservoir model.Attention to the problem of sampling realizations from the a posteriori probability density function.
Most researchers have worked with simple models (e.g., characterized by a variogram), an effective well-test permeability instead of a flow simulator, and largely ignored the problem of sampling. Other, more sophisticated examples include the use of a complex stochastic geologic model (channels), and simulated annealing to sample from the a posteriori probability distribution function (PDF), but an effective well-test permeability instead of pressure data (and a simulator) for conditioning.1 The works by Oliver2 and by Chu et al.3 provide other examples. In these cases, a flow simulator was used for conditioning but the geology was relatively simple and realizations were generated using a linearization approximation around the maximum a posteriori model. Landa4 treated the problem of conditioning two-dimensional channels, but chose a simple model that could be described by a few parameters.
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 have used the actual pressure data but have had to limit ourselves to Gaussian random fields and to fairly small synthetic models. We recently applied Markov chain Monte Carlo (MCMC) methods5 to generate an 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. The principal advantage of MCMC is that it provides a method for sampling realizations from complicated probability distributions such as the distributions of reservoirs conditional to production data. The method consists of a proposal of a new realization, and a decision as to whether to accept the proposed realization, or to again accept the current realization. The "chain" refers to the sequence of accepted realizations and "Monte Carlo" refers to the stochastic aspect in the proposal acceptance steps.
Unfortunately, it appears to be 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, which involves the ratio of the probability of proposing the proposed realization to the probability of proposing the current realization, is difficult to evaluate.
Oliver et al.6 proposed a methodology for incorporating production data that followed the second approach but ignored the Metropolis-Hastings criterion, instead accepting every realization. We showed that the method is rigorously valid for conditioning Gaussian random fields to linear data (i.e., weighted averages of model variables) and is easily adapted to more complex geostatistical models and types of data. Although the method is then not rigorously correct, we have shown that the distribution of realizations is good for simple, but highly nonlinear problems. The realizations generated using this methodology still honor all the data—the ensemble of realizations is, however, not a perfect representation of the true distribution even as the number of realizations becomes very large.
