Most simulation models go through a series of iterations before being judged as giving an adequate representation of the physical system. This can be difficult because the input space to be searched may be high dimensional, the collection of outputs to be matched may be very large, and each single evaluation may take a long time. Because the uncertainty analysis is complex and time consuming, in this paper, a stochastic representation of the computer model, called an emulator, was constructed to quantify the reduction in the parameter input space.

## Introduction

Reservoir simulators are important and widely used in reservoir management. They are used in reservoir-performance prediction and for decision making. These simulators are computer implementations of high-dimensional mathematical models for reservoirs, where the model inputs are physical parameters and the outputs are observable characteristics such as well-pressure measurements and fluid production. Uncertainties are always present in the reservoir-characterization process; thus, input parameters are usually uncertain and so is the simulator output.

The procedure to calibrate the reservoir-simulation model is called history matching. On the basis of observed data, a set of possible input choices for the reservoir model is identified. Two different procedures can be used to perform the history matching: deterministic and probabilistic approaches.

The deterministic approach involves running the initial simulation model with different input values to obtain one simulation model between many probable matches to the field data.

In a probabilistic approach, in which several reservoir-model scenarios are considered, the uncertainty analysis procedure is used. Identifying the input parameters for which the simulation outputs match the observed data can be a difficult task because the input space to be searched may be high dimensional, the collection of outputs to be matched may be very large, and each single evaluation may take a long time.

To deal with the large number of iterations and high computational resources commonly encountered in the probabilistic approach, proxy models are used.

Because history matching and uncertainty-reduction quantification are complex and time consuming, this work shows the work flow used to quantify the reduction in the parameter input space from production data over different production periods. This work flow comprises the construction of a proxy model called an emulator. This technique was applied to a synthetic reservoir simulation model that was built to represent the region of an injector and related producers.

## Proposed Methodology

The work flow used to construct the emulator was designed to quantify the reservoir-simulation-model uncertainty reduction from production data. The objective was to identify the inputs of a reservoir-simulation model within a possible input-parameter space, whose outputs match the hypothetical historical production data. **Fig. 1 (above) ** shows the work flow. Each stage is described as follows.

**Input- and Output-Parameter Definition.** In reservoir simulation, uncertain inputs are physical parameters determined through an uncertainty analysis performed on the base model. The outputs of the model, for a given choice of inputs, are observable characteristics such as well bottomhole pressure, water rate at production wells, and water-saturation maps. The input-variable selection depends on the underlying problem and the knowledge of the engineer.

The physical state of the reservoir uncertainty varies because of the amount of information available and the production period. In this study, the analysis is being performed in the field-development phase and the uncertainty of the appropriate choices of the input parameters for the reservoir model is high.

**Input-Data-Set Sampling.** The input-data-set sampling is an important stage in creating an adequate emulator. Different sampling methods exist and have been applied in reservoir simulations. The Latin hypercube design (LHD) is an efficient design and was selected as a sampling method for this work. Scenarios were generated on the basis of the input-parameter space and sampled with the LHD. The selected scenarios were simulated with commercial simulation software to obtain the production outputs. The sampled input parameters and resulting simulation outputs were used to construct the emulator.

**Emulator Estimation.** The emulator is an approximation of the existing numerical reservoir model. It should be able to replicate the response of the simulation model. For a reservoir-simulation model, it is infeasible to evaluate the simulator at enough choices to search the input space exhaustively. Therefore, a representation of the output uncertainty at each input choice must be constructed. This representation is termed an emulator.

**Emulator Diagnostics.** Emulator diagnostics is the process of assessing an emulator’s prediction accuracy and quality. The response values predicted by the emulator must comprise the results of the full numerical simulation for the input data set.

**Implausibility Analysis.** The implausibility analysis is performed to obtain the input parameters whose outputs match the hypothetical historical data. The hypothetical historical data are derived from a hypothetical reality selected from all possible scenarios generated in the uncertainty analysis. Moreover, these inputs are obtained to improve the emulator reliability and to evaluate the uncertainty reduction at the end of the process.

The maximum-acceptable-implausibility-value cutoff determines whether an input-parameter vector is viewed as nonimplausible or not. This value can be defined on the basis of various considerations, but, often, the cutoff used is equal to the critical value of some appropriate distribution (for example, the standard normal distribution).

**Nonimplausible-Inputs Evaluation.** The nonimplausible input parameters obtained at the end of the process represent the input parameters of the reservoir simulation whose outputs match the hypothetical historical production data. These parameters are evaluated to identify how much the production data improved the reservoir-model understanding.

While carrying out this analysis considering different production periods, it is possible to evaluate the effect of the production period on the reservoir-uncertainty reduction.

## Conclusions

A work flow to determine the input parameters whose output values match historical data using emulation techniques was presented. The work flow was applied successfully to a five-spot synthetic case that was built to represent the region of an injector and related producers. The uncertainty reduction of a reservoir model because of new information acquisition for different production periods was quantified. The field production data used were obtained by considering a hypothetical reality among all possible scenarios because the analysis was performed at the development stage and used a synthetic model. Two periods of production were evaluated: at an early production stage (1,000 days) and at an intermediate production stage (3,500 days).

The results obtained showed the importance of using emulators in uncertainty-reduction quantification and history matching. The number of input parameters considered nonimplausible was a small set of the initial input space. At an early stage, it was possible to reduce the uncertainty by identifying the hypothetical real field permeability and identifying possible values for channel positioning. However, other important physical features were not identified, such as the channel permeability, width, and length. At an intermediate stage, the uncertainty reduction was higher. However, some important physical features that affect production prediction, such as channel permeability and width, still were not identified; therefore, further research will test the application of the emulation technique with 4D-seismic data to reduce uncertainty.

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 169405, “Use of Emulator Methodology for Uncertainty-Reduction Quantification,” by **C. Ferreira,** Universidade Estadual de Campinas; **I. Vernon,** Durham University;** D.J. Schiozer,** SPE, Universidade Estadual de Campinas; and** M. Goldstein,** Durham University, prepared for the 2014 SPE Latin American and Caribbean Petroleum Engineering Conference, Maracaibo, Venezuela, 21–23 May. The paper has not been peer reviewed.