Chapter 6: Dealing With Decision-Making Under Uncertainty

certainty

One of the essential decisions from reservoir simulation studies is field development by optimally and robustly placing an infill well(s) to maximise recovery from the field and minimise operational expenditure. Robust optimal decisions are the ones that would hold in the presence of geological uncertainty, such as when a realisation of reservoir setting deviates from the base case. Would the decision optimised for the base case remain op- timal for the updated reservoir condition? To answer this question, the optimisation across multiple realisations is required. Even though attempts on the optimisation across multiple realisations have been conducted in the literature, the computational cost involved in performing the task and resulting in the proper estimation of uncertainty remains a great challenge.

In Chapter 6, we introduce a new workflow for robust and reliable well placement opti- misation. It accounts for geological uncertainty in choosing the optimal well location. The proposed workflow combines multi-objective history matching, Bayesian posterior inference based on PPD approximation of Pareto history-matched models and well placement optimisation across multiple geological models. The application and validation of the workflow are demonstrated on an industry-standard reservoir model case study.

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The Impact of Model Parameterisation in

Reservoir History Matching and

Forecasting

3.1

Introduction

A general workflow in the reservoir history matching and uncertainty quantification consists of three steps: (i) model parameterisation; (ii) history-matching process; and (iii) forecasting with uncertainty quantification, as shown in Figure 3.1. In the model parameterisation, we can assume that a limited number of model parameters will contain a combination of parameter values that accurately represents the reservoir. The choice of model parameterisation itself, either less or more detailed description, can be based on geological prior information, engineering knowledge, or a combination of both to represent uncertainty in the reservoir description. For instance, the model can be parameterised to different zones based on the geological interpretation of channel distri- bution in the reservoir, or it can be parameterised on the well drainage area and relative permeability curve based on the engineering knowledge.

In the history-matching process, powered by a stochastic optimisation algorithm, we can choose to use either single or multi-objective optimisation approach. Either way, the

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ensemble of history-matched models is used for forecasting with uncertainty quantifica- tion. The uncertainty forecasting is done based on the posterior probability of each matched model that approximates the posterior probability distribution (PPD).

Figure 3.1—General workflow in the reservoir history matching and uncertainty quantification in forecasting.

Many techniques of model parameterisation have been developed in the literature to capture uncertainty in the geometry of the reservoir, the spatial distribution of rock properties, and reservoir fluid. In general, the aim of these techniques is to produce geologically realistic reservoir models that preserve spatial variability of reservoir properties inferred from the available static data for meaningful property estimates. These techniques include a gradual deformation method (GDM) [222], pilot point method (PPM) [223], gradual pilot point method (GPPM) [224], facies proportion calibration technique [225], coordinate-free approach called stochastic elliptic partial differential equations [226], and machine learning techniques including principal component analysis (PCA) [227], support vector machines (SVM) [228,229], and multiple kernel learning (MKL) [230,231]. A comprehensive description can be found in [232–234].

A common approach of model parameterisation in the petroleum engineering is the zo- nation or regionalisation technique [235] that is adopted from hydrology [236]. It has an extensive history of its application since the early history-matching studies of Jacquard and Jain in 1965 [237] and Jahns in 1966 [64]. The zonation technique involves dividing the reservoir model into a small number of zones or regions, in each of that the properties are treated as uniform. A modelling error is thus introduced through the assumption of uniform properties within each zone and the more or less arbitrary assignment of the zone boundaries. The modelling error increases as the number of zones are decreased [62].

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Gavalas et al. [62] and Shah et al. [63] studied the use of zonation technique for history matching and concluded that zonation is preferable when the location of the zone boundaries is aligned with geological information. Both studies in [62] and [63] demonstrated that the zonation approach reaches the minimum modelling error at some intermediate level of parameterisation, i.e. at a particular number of zones, that can be regarded as the optimum level. Both studies [62,63] also suggested using prior geological information in the form of a prior probability density of the reservoir proper- ties formulated in Bayesian estimation. The results of Bayesian estimation were found to be more accurate than those of zonation in a simulated case of a one-dimensional reservoir. The Bayesian formulation also resulted in improved convergence of the itera- tive minimisation algorithms.

The combination of zonation technique and prior geological information in the model parameterisation have been favoured by many researchers in the history-matching studies [7–9,92]. It offers a straightforward approach to reducing a large number of un- known parameters in the reservoir, and the problem becomes statistically better deter- mined by using a prior statistical information on the unknown parameters. This tech- nique has also managed to produce history-matched models with meaningful results. However, the choice of the model parameterisation based on this technique is also one of the sources of uncertainty in reservoir model simulation. More or less detailed zona- tion in the model parameterisation can be formed based on geological information and engineering knowledge that can vary subject to interpretation and data analysis.

When we come to model the parameterisation uncertainty in history matching, we encounter the situation where we can have different model parameterisations with different number of model parameters describing the reservoir system. These could be a simple “black box” models that are based on some simple physical theory or our best attempt at a “full” physics evaluation. Within each choice of model parameterisation, we could have a range of unknown parameters that we are trying to calibrate. Pickup et

al. [238] examined two techniques to handle this problem. The first technique is based

on Bayesian model comparison to choose the optimum number of parameters for a model. This technique includes the use of minimum description length (MDL) [239] and

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combines two different model parameterisations with different level of parameterisation’s complexity to improve the predictive capability compared to the use of individual models, which is also studied in [79].

In this chapter, contrast to the previous studies to handle the uncertainty on model parameterisation, we investigate the impact of different geological model parameterisa- tions on history matching and uncertainty quantification in the forecasting. The parameterisations are based on zonation or regionalisation with prior geological infor- mation. We briefly discuss two different approaches in history-matching process powered by stochastic optimisation algorithm: single and multi-objective optimisation methods. The strategy for the comparison of the different approaches is then described. This is followed by a case study of PUNQ-S3 reservoir model to demonstrate the ad- vantage of multi-objective approach over single-objective history matching in respect to forecasting with different model parameterisations. We describe two different model parameterisations and compare the results in history matching and uncertainty quantifi- cation in the forecasting between single and multi-objective optimisation approaches within the Bayesian framework. Finally, we discuss the verification of the possible im- provement that the choice of history-matching approach can bring to the history match- ing and forecasting under model parameterisation uncertainty.