Abstract
Reservoir history matching is a well-known inverse problem for production prediction where enormous uncertain reservoir parameters of a reservoir numerical model are optimized by minimizing the misfit between the simulated and history production data. Gaussian Process (GP) has shown promising performance for assisted history matching due to the efficient nonparametric and nonlinear model with few model parameters to be tuned automatically. Recently introduced Gaussian Processes proxy models and Variogram Analysis of Response Surface-based sensitivity analysis (GP-VARS) uses forward and inverse Gaussian Processes (GP) based proxy models with the VARS-based sensitivity analysis to optimize the high-dimensional reservoir parameters. However, the inverse GP solution (GPIS) in GP-VARS are unsatisfactory especially for enormous reservoir parameters where the mapping from low-dimensional misfits to high-dimensional uncertain reservoir parameters could be poorly modeled by GP. To improve the performance of GP-VARS, in this paper we propose the Gaussian Processes proxy models with Latent Variable Models and VARS-based sensitivity analysis (GPLVM-VARS) where Gaussian Processes Latent Variable Model (GPLVM)-based inverse solution (GPLVMIS) instead of GP-based GPIS is provided with the inputs and outputs of GPIS reversed. The experimental results demonstrate the effectiveness of the proposed GPLVM-VARS in terms of accuracy and complexity. The source code of the proposed GPLVM-VARS is available at https://github.com/XinweiJiang/GPLVM-VARS.
Highlights
As a well-known inverse problem in reservoir simulation, History Matching is significant for reservoir development, management and predictions, which tries to estimate the uncertain parameters of a reservoir numerical model based on observed historical production data [1,2,3]
Except GPLVM-based inverse solutions (GPLVMIS), other modules are similar to Gaussian Process (GP)-Variogram Analysis of Response Surface (VARS) with an iterative optimization process where some initial random set of reservoir parameters are initially generated by Latin hypercube sampling technique and the temporary solutions in each iteration are estimated to minimize the misfits between the GPFS-based proxy model response and historical production data [29]
The inverse mapping from low-dimensional LVMs to high-dimensional uncertain reservoir parameters is modeled by Gaussian Processes Latent Variable Model (GPLVM) rather than Gaussian Process Regression (GPR), and the reservoir parameters with high dimensionality can be efficiently learnt by GPLVMIS which could improve the performance of GPIS as discussed in GP-VARS
Summary
As a well-known inverse problem in reservoir simulation, History Matching is significant for reservoir development, management and predictions, which tries to estimate the uncertain parameters (such as porosity and permeability) of a reservoir numerical model based on observed historical production data (such as well rates and pressure) [1,2,3]. To address the issue, Assisted History Matching (AHM) techniques have been proposed to replace labor-intensive and costly manual history matching [1,4,5,6]. These methods for assisted history matching can be divided into two categories [7]: the data assimilation approaches (such as Ensemble Kalman Filter and Ensemble Smoother) and the optimization approaches (such as gradient, evolutionary or data-driven-based algorithms). EnKF is a sequential Monte Carlo approximation of the Kalman filter where the correlation between reservoir parameters and observed production data can be estimated from the ensemble with the uncertainty of estimation [9]. EnKF can efficiently assimilate various types of data to optimize numerous reservoir parameters, but it could fail if there are multimodal nonlinear data or discrete reservoir parameters
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