Adjusting geological properties of reservoirs to match production data obtained from dynamic well observations is a nontrivial task and thus results in a challenging inverse problem which is known as history matching in oil and gas literature. Of special interest in reservoir engineering is to develop and establish automated, robust and geologically consistent inversion approaches such that the highly nonlinear and underdetermined nature of interactions and relationships between reservoir model parameters and well responses can accurately be modeled. Furthermore, a reliable production data uncertainty assessment can be gained by the quality of the ensemble of calibrated and history matched models. Reliable production forecasting and uncertainty assessment are fundamental 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.
A very common and well-stablished technique towards reservoir model calibration and uncertainty quantification is Markov-Chain Monte Carlo (MCMC) algorithm that has recently attracted many due to its powerful and successful performance. The high-dimensional and complex posterior probability distribution of reservoir parameters can efficiently be sampled and generate history matched reservoir models by employing MCMC and thus can be utilized for production forecasting uncertainty assessment. Once gradient information is not available, e.g., in many reservoir simulation problems, MCMC approach is the method of choice due to its gradient-free procedure. 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.