Quantifying Uncertainty in Production Forecasts: Another Look at the PUNQ-S3 Problem

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 by

Equation 1

where c is a normalization constant and Q is given by:

Equation 2

Here, σi=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.