Markov-Chain Monte Carlo with Locally Varying Mean Kriging for Improved Reservoir Model Calibration and Uncertainty Assessment
Authors: Tarrahi Mohammadali , Afra Sardar 
Published: April 2016

Abstract
Calibrating complex subsurface geological models against dynamic well observations yields to a challenging inverse problem which is known as history matching in oil and gas literature. The highly nonlinear nature of interactions and relationships between reservoir model parameters and well responses demand automated, robust and geologically consistent inversion techniques. The ensemble of calibrated and history matched models quality determines the reliability of production uncertainty assessment. Reliable production forecasting and uncertainty assessment are essential steps toward reservoir management and field development. The Bayesian framework is a widely accepted approach to incorporate dynamic production data to the prior probability distribution of reservoir models and obtain the posterior distribution of reservoir parameters. Uncertainly assessment is performed by sampling the posterior probability distribution which is a computationally challenging task.
Markov-Chain Monte Carlo (MCMC) algorithm has shown successful application in reservoir model calibration and uncertainty quantification is recent years. MCMC can efficiently sample the high-dimensional and complex posterior probability distribution of reservoir parameters and generate history matched reservoir models that consequently can be used for production forecasting uncertainty assessment. MCMC method is a gradient-free approach which makes is favorable when gradient information is not available through reservoir simulation. In MCMC method normally to march to next iteration the new sample is independent of the previous sample and the proposal distribution is rather random. To improve the sampling procedure and make MCMC process more efficient we propose an approach based on locally varying mean (LVM) Kriging to base the new sample generation on the previous iteration sample. In this method, the previous sample is used as the varying mean map in the geostatistical simulation approach to generate the new proposal for the next iteration.
Using LVM Kriging to relate the new sample to previous iteration sample, make the chain of samples in MCMC more correlated and geologically consistent. Also this new proposal distribution makes the sampling procedure more efficient and avoids random and arbitrary movements is the parameter space. We applied MCMC with LVM Kriging to a suite of 2D and 3D reservoir models and obtained the calibrated model. We observed that the application of the new proposal distribution based on LVM Kriging along with MCMC improved the quality of the samples and resulted in promising uncertainty quantification. We also observed meaningful improvement in calibrated reservoir models quality and uncertainty interval while utilizing LVM comparing to random proposal or transition distribution in MCMC.
MCMC with LVM Kriging as proposal distribution results in improved uncertainty assessment through enhancing the quality of the generated samples from posterior probability distribution of reservoir model parameters. Traditional random or independent proposal distribution does not represent the dependency of the samples through MCMC chain and iterations while this challenge is addressed by combining MCMC with LVM.