Accurate Surrogate Modelling for Timeseries Using Functional Principal Component Analysis: Application to History-Matching and Uncertainty Quantification

Abstract

Summary Accurate surrogate models are essential for application of computational methods such as Markov chain Monte Carlo (McMC) using numerical reservoir simulation. Previous studies have often focused on building surrogates to represent the misfit (or likelihood) function. However, building an accurate non-negative surrogate for the likelihood is difficult for higher dimensions unless an overly large number of samples is simulated first. Fortunately, functional data analysis can provide a set of ensemble-based statistical tools which can be utilized to emulate the full simulation output rather than the misfit function itself. Consequently, the misfit can be easily calculated by the simulated timeseries. In this study, functional principal component analysis (fPCA) is utilized to reduce the dimensionality of the timeseries. In other words, the simulation output (e.g., oil rate) is represented in terms of a few optimal functional principal component scores (fPCS), which can be readily inverted to reconstruct the original timeseries. fPCA is a favorable tool for developing a new and efficient Bayesian history matching workflow in that it can be used to iteratively update the surrogate and search for the extrema of the likelihood function. In this proposed process, a few initial random samples are generated, and the corresponding timeseries are processed by fPCA. The resulting fPCSs for each simulation output are modelled individually using kriging or Random Forests, which are then used to estimate the likelihood and optimized to suggest the next best samples until a convergence criterion is met. This workflow is applied to a set of data obtained from a near critical gas condensate well from a Canadian shale reservoir. The proposed history matching workflow results in a 10-times faster convergence rate compared to an adaptive differential evolution algorithm. The history matching samples are combined with additional quasi-Monte Carlo samples to enhance the exploration aspect and predictability of the surrogate models. The results demonstrate that clustering of the timeseries and applying fPCA to each cluster separately can significantly enhance the accuracy of the surrogates. Finally, the surrogates are utilized to obtain accurate posteriors quickly through an McMC algorithm. This study introduces an adaptive sampling method for the first time that can be used to generate a highly accurate surrogate for timeseries and conduct the optimization efficiently. This surrogate modelling workflow can also be used for other reservoir engineering problems such as numerically assisted rate-transient analysis where new timeseries should be predicted from a limited number of numerical simulations.