This paper presents a comparison study in which several partners have applied methods to quantify uncertainty on production forecasts for reservoir models conditioned to both static and dynamic well data. A synthetic case study was set up, based on a real field case. All partners received well porosity/permeability data and ‘historic’ production data. Noise was added to both data types. A geological description was given to guide the parameterization of the reservoir model. Partners were asked to condition their reservoir models to these data and estimate the probability distribution of total field production at the end of the forecast period. The various approaches taken by the partners were categorized. Results showed that for a significant number of approaches the truth case was outside the predicted range. The choice of parameterization and initial reservoir models gave the largest influence on the prediction range, whereas the choice of reservoir simulator introduced a bias in the predicted range.

Summary A synthetic reservoir model, known as the PUNQ-S3 case, is used to compare various techniques for quantification of uncertainty in future oil production when historical production data is available. Some results for this case have already been presented in an earlier paper.1 In this paper, we present some additional results for this problem, and also argue an interpretation of the results that is somewhat different from that presented in the earlier paper. The additional results are obtained with the following methods: importance sampling, history matching of multiple models using a pilot-point approach, and Markov Chain Monte Carlo (MCMC). Introduction It is widely recognized that the future production performance of oil and gas reservoirs cannot be predicted exactly. There will always be some uncertainty. Nowadays, more and more effort is being made to quantify this uncertainty. The aim of the work described in this paper is to compare a number of different methods for quantifying uncertainty in future reservoir performance. In particular, it considers reservoirs where some production data (beyond well testing) is available. Such data is particularly difficult to incorporate in an uncertainty analysis because of the time-consuming nature of the computations necessary to simulate fluid flow in the reservoir. The work was carried out as part of the PUNQ-2 project,2 partly funded by the European Union. PUNQ is an acronym for Production forecasting with UNcertainty Quantification. The project involved 10 European universities, research institutes, and oil companies. As part of the project, one of the participating organizations created a synthetic reservoir model known as PUNQ-S3. Eight years of production were simulated using a commercial reservoir simulator. The simulated production data (with noise added) were revealed to the other participants, together with some other information about the model (see below for details). The participants were asked to predict the cumulative recovery after 16.5 years of production for a given development scheme. They were also asked to quantify the uncertainty associated with their forecast. The various participants used different techniques to answer these questions and came up with a wide range of answers. The results obtained by several of the participants were presented in an earlier paper.1 In this paper, we will present some additional results for this problem, and we also will argue an interpretation of the results that is somewhat different from that presented in the earlier paper. Uncertainty Quantification Methods In the PUNQ-2 project, a Bayesian approach to uncertainty quantification is taken. The uncertainty in any quantity to be predicted is quantified by a probability density function (pdf). Bayes theorem relates the posterior pdf to a prior pdf and a likelihood function. In the present context, the prior pdf characterizes the uncertainty before the production data is taken into account (i.e., it is the expression of the knowledge of the reservoir derived from logs, cores, seismic, and general geological knowledge). The prior model can be used to construct initial reservoir models before any history matching. The posterior pdf characterizes the uncertainty after the production data is taken into account, and is the pdf that the project seeks to define. The likelihood function measures the probability that the actual production data would have been observed for any given model of the reservoir. We assume that the production data are statistically independent, unbiased measurements with normally distributed measurement errors. The likelihood function, L, is then given byEquation 1 where c is a normalization constant and Q is given by:Equation 2 Here, si=the standard deviation of the measurement error. Note that each evaluation of the likelihood function requires a reservoir simulation to be run over the history period. Thus, characterization of the posterior pdf in this context is difficult and time-consuming. Indeed, prior to Floris et al.,1 previous published work3–7 had focused on limited aspects of the problem or had been restricted to matching of pressure data only. Other methods for uncertainty estimation have been proposed that are easier and quicker to implement, but that are inevitably less precise. Linear uncertainty analysis,8,9 perturbation methods, and the Scenario Test Method (STM)10 all consider perturbations around a single "most likely" reservoir model. Some of these have been reviewed by Lépine et al.9 They fail to account for the possible existence of other, quite different models that would also respect the available data. They are thus likely to underestimate the true uncertainty. Generation of "downside" and "upside" models as well as a "most likely" model is a fairly standard industry practice, though it is not always done when production data are available because of the time it would take to history-match three models rather than just one. The models are often referred to as "P10," "P50," and "P90," because they represent an attempt to sample the corresponding quantiles of the posterior pdf. However, the models are usually derived in a heuristic way, and there is no guarantee that they do in fact correspond to these levels of probability, especially after a history match.

TX 75083-3836, U.S.A., fax 01-972-952-9435.Abstract A synthetic reservoir model, known as the PUNQ-S3 case, is used to compare various techniques for quantification of uncertainty in future oil production when historical production data is available. Some results for this case have already been presented in an earlier paper 1 .In this paper, we present some additional results for this problem, and also argue an interpretation of the results that is somewhat different from that presented in the earlier paper. The additional results are obtained with the following methods: (i) rejection sampling, (ii) history matching of multiple models using a pilot-point approach, and (iii) Markov Chain Monte Carlo (MCMC).

The method of Ding for scaling-up in the near-well region is evaluated on a variety of two-and three-dimensional problems, including cases with partially penetrating wells and some with non-vertical wells. The method is found to work well for all cases tested although accuracy is in general lower for 3D cases and possibly for wells producing at constant bottom hole pressure. The computational effort involved in the scaling-up can be minimized by use of a reduced computational domain with only a slight degradation of the results. Both well index and modified horizontal transmissibility are required for satisfactory results, but use of modified vertical transmissibility appears to be unnecessary.

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