Effect of Sampling Strategies on Prediction Uncertainty Estimation

Abstract

Generating multiple history-matched reservoir models by stochastic sampling to quantify the uncertainty in oil recovery predictions has recently aroused interest in the industry. Coupling a stochastic sampling algorithm with a Bayesian analysis potentially allows incorporation of all sources of uncertainties including data, simulation and interpolation errors. However, the accuracy of the uncertainty estimations strongly depends on the sampling performance. In order to improve the robustness of the coupled Bayesian methodology, the factors that affect the accuracy of the estimations must be examined. This paper investigates how different sampling strategies affect the estimation of uncertainty in prediction of reservoir production. The sampling strategy involves the choice of algorithm and selection of algorithm parameters in sampling the high-dimensional parameter space. We present examples of using both the Neighbourhood Algorithm (NA) and a Genetic Algorithm (GA) to generate history-matched reservoir models for a real field case from the North Sea. Introduction Reservoir engineers often face the task of predicting the production performance of a subsurface hydrocarbon reservoir for decision making in field development. A computer model of the reservoir is constructed based on the geological information available, and the mathematical equations of the fluid flow in the reservoir are solved to simulate the reservoir response. The reservoir model is then calibrated by using the data observed in the field. Usually hydrocarbon production or reservoir pressure measurements from the wells are compared with the simulated results and the reservoir model is modified until a reasonable match between the simulated response and the observed data is achieved. This calibration process is called history-matching. The history-matched model is then simulated over the prediction period to obtain the recovery prediction. The data used to construct and calibrate the reservoir model are limited to the small-scale samples taken from sparse points (i.e. wells) in the field. Therefore, the model is always subject to interpolations at various scales, and so the recovery prediction based on the model is uncertain. Moreover, the errors in data measurements and the mathematics of reservoir modeling bring additional uncertainties. As a result of these uncertainties, the solution of the history-matching problem is not unique, i.e. multiple reservoir models can possibly match the history data equally well. Hence, recovery predictions are uncertain, although this uncertainty is not captured by a single history-matched model. To improve operational and investment decisions in reservoir management, the inherent uncertainty in predictions must be quantified, usually using the language of probability theory. The uncertainty is quantified by generating models that match observed measurements and are consistent with the known details of reservoir geology. The forecasts from these models then provide the probability distribution of the reservoir model response. In order to generate the models, an optimization (or sampling) technique is often employed, in which the discrepancy between the model response and the observed data is minimized by modifying the model parameters. The main challenges in the sampling process are the high number of unknown parameters, the existence of sharp local minima in the search space, and the high computational cost of reservoir simulation. These difficulties can have a detrimental effect on the accuracy of the sampling. This paper is concerned with the effect of any inaccuracies in the sampling process on the prediction uncertainty estimations. History Matching in a Bayesian Framework The Bayesian framework1 for statistical inference provides a formal and systematic procedure for updating current knowledge of a system on the basis of available data2. The basic formulation of Bayes theorem is given in Eq. (1).

Assisted History Matching Using Pattern Recognition Technology

“…In order to address the shortcomings of gradient-based optimization methods, global optimization approaches such as Simulated Annealing, Evolutionary Algorithms, and Evolution Strategy were proposed. Some of the successful methods are such as the ensemble Kalman filter (Van Leeuwen, 1999;Evensen, 2003;Haugen, et al, 2006;Aanonsen, et al, 2009;Hanea, et al, 2010;Szklarz, et al, 2011), Neighborhood Algorithm (Christie, et al, 2002;Stephen, et al, 2006;Rotondi, et al, 2006;Subbey, et al, 2003), Genetic Algorithms (Erbas, et al, 2007) (Castellini, 2005), Scatter search (Sousa, 2007), Tabu Search (Yang, et al, 2007), Hamiltonian Monte Carlo (HMC) (Mohamed, et al, 2009), Particle Swarm Optimization (PSO) (Eberhart, et al, 2001;Mohamed, et al, 2009;2010;Rwechungura, et al, 2011;Kathrada, 2009) Ant Colony Optimization (ACO) algorithm (Razavi, et al, 2008;2010), Markov chain Monte Carlo (Maucec, 2007), and Chaotic Optimization (Mantica, 2002).…”

Estimation of Distribution Algorithms Applied to History Matching

“…Population-based algorithms have been popular in recent years and use interactions between individual solutions to improve the quality of the search. Examples of stochastic algorithms that have been deployed in history matching include simulated annealing (Sultan et al 1994), Tabu search (Yang et al 2007), genetic algorithms (Romero et al 2000;Erbas and Christie 2007), neighborhood algorithm (Subbey et al 2003), particle-swarm optimization (Mohamed et al 2010), ant-colony optimization (Hajizadeh 2010), and differential evolution .…”

Bayesian Optimization Algorithm Applied to Uncertainty Quantification

“…These methods generally provide a flexible framework in which exploration of the search space for a diverse set of solutions followed by local search in previously found regions (exploitation) results in an efficient search. Simulated annealing (Sultan et al 1994), scatter search (Sousa et al 2006), tabu search (Yang et al 2007), genetic algorithm (Romero et al 2000;Erbas and Christie 2007), neighborhood algorithm (Subbey et al 2003), evolutionary strategy (Schulze-Riegert et al 2001), particle-swarm optimization (Mohamed et al 2010), differential evolution , and ant-colony optimization (Hajizadeh 2010) are among the stochastic optimization algorithms that have been applied to the uncertainty-quantification problem.…”