Robust Quantification of Uncertainty in Heterogeneity for Chemical EOR Processes: Applying the Multi-Level Monte Carlo Method
Authors: Alkhatib A. M. 
Published: March 2015

Abstract
Reservoir heterogeneity can be detrimental to the success of chemical enhanced oil recovery (EOR) processes. Therefore, it is important to evaluate the effect of uncertainty in reservoir heterogeneity on the performance of chemical EOR. Usually, a Monte Carlo sampling approach is followed were a number of stochastic reservoir model realizations are generated and then numerical simulation is performed to obtain a certain objective function, such as the recovery factor; however, Monte Carlo simulation (MCS) has a slow convergence and requires a large number of samples to produce accurate results. This is computationally expensive when using reservoir simulators. This study applies an extension to MCS using a multi-scale approach. The applied method is known as the multilevel Monte Carlo (MLMC) method and has been only recently applied to problems of flow in porous media.
This method is based on running a small number of expensive simulations on the finer scale model and a large number of less expensive simulations on coarser scale models — these are upscaled models of the fine scale model — and then combining the results to produce the quantities of interest. The purpose of this method is to reduce computational cost while maintaining the accuracy of the finer scale model. The results of this approach are compared with reference MCS, assuming a large number of simulations on the fine scale model.
This study used MLMC to efficiently quantify the effect of uncertainty in heterogeneity on the recovery factor of different chemical EOR processes. The permeability field was assumed to be the random input. This approach was implemented by writing a MATLAB code to generate the stochastic realizations for the permeability field and also performing the coarsening processes. The code is then coupled with ECLIPSE, which was used as the numerical simulator for the chemical EOR processes to obtain the recovery factor. The code then combines the results obtained from the different scale models to produce the statistical moments for the recovery factor, such as the mean and variance.
This method was applied for two-dimensional (2D) and three-dimensional (3D) stylized reservoir models using Gaussian randomly generated permeability fields. Different coarsening algorithms were used and compared, such as the renormalization and pressure solver methods, and polymer and surfactant-polymer (SP) flooding processes where the chemical EOR processes were considered. The results were compared with running Monte Carlo for the fine scale model while equating the computational cost for the multilevel Monte Carlo method. Both of these results were then compared with the reference case, which uses a large number of runs of the fine scale model.
The results show that it is possible to robustly quantify spatial uncertainty for chemical EOR processes while greatly reducing the computational requirement, up to two orders of magnitude compared to traditional Monte Carlo. The method can be easily extendable to other EOR processes to quantify spatial uncertainty such as carbon dioxide (CO2) EOR.