Ali Alkhatib

Robust quantification of parametric uncertainty for surfactant–polymer flooding

Uncertainty in surfactant–polymer flooding is an important challenge to the wide-scale implementation of this process. Any successful design of this enhanced oil recovery process will necessitate a good understanding of uncertainty. Thus, it is essential to have the ability to quantify this uncertainty in an efficient manner. Monte Carlo simulation is the traditional uncertainty quantification approach that is used for quantifying parametric uncertainty. However, the convergence of Monte Carlo simulation is relatively low, requiring a large number of realizations to converge. This study proposes the use of the probabilistic collocation method in parametric uncertainty quantification for surfactant–polymer flooding using four synthetic reservoir models. Four sources of uncertainty were considered: the chemical flood residual oil saturation, surfactant and polymer adsorption, and the polymer viscosity multiplier. The output parameter approximated is the recovery factor. The output metrics were the input–output model response relationship, the probability density function, and the first two moments. These were compared with the results obtained from Monte Carlo simulation over a large number of realizations. Two methods for solving for the coefficients of the output parameter polynomial chaos expansion are compared: Gaussian quadrature and linear regression. The linear regression approach used two types of sampling: full-tensor product nodes and Chebyshev-derived nodes. In general, the probabilistic collocation method was applied successfully to quantify the uncertainty in the recovery factor. Applying the method using the Gaussian quadrature produced more accurate results compared with using the linear regression with full-tensor product nodes. Applying the method using the linear regression with Chebyshev derived sampling also performed relatively well. Possible enhancements to improve the performance of the probabilistic collocation method were discussed. These enhancements include improved sparse sampling, approximation order-independent sampling, and using arbitrary random input distribution that could be more representative of reality.

Robust optimization of subsurface flow using polynomial chaos and response surface surrogates

This study employs an inclusive framework for surrogate model-based optimization in the presence of parametric and spatial uncertainties. The framework is applied to optimize water injection rate for optimal hydrocarbon recovery from a synthetic subsurface model with uncertainty in the geological and fluid relative permeability properties. In one model of parametric uncertainty, geological properties such as the channel’s absolute permeability and the fault transmissibility multiplier and the fluid relative permeability parameters such as the residual oil saturation to water and the water relative permeability at residual oil are assumed to be non-informative. In another model, the channels positions are assumed uncertain and various realizations of the channelized permeability are parameterized and the spatial uncertainty is accounted for in the optimization. The uncertainty is quantified in each evaluation of the objective function via polynomial chaos expansions. The coefficients of polynomial chaos expansion are solved by probabilistic collocation method. The objective function is assigned with a risk-averse net present value computed from a distribution of values obtained from the probabilistic proxies. The proxies are updated for each round of objective function evaluation. Monte-Carlo simulations are also conducted to verify accuracy and to demonstrate the computational efficiency of the probabilistic collocation approach. The optimization is conducted in various random input cases (depending on the number of uncertain parameters) and for each case net present value is successfully maximized and optimal solutions of the water injection rates are determined.

Decision Making Under Uncertainty: Applying the Least-Squares Monte Carlo Method in Surfactant-Flooding Implementation

This study introduces a decision-making evaluation method for flexibility in surfactant flooding. The method aims to capture the effects of uncertainty in the time series for both technical and economic parameters and produce a near-optimal policy with respect to these uncertainties as they vary with time. The evaluation method used was the least-squares Monte Carlo (LSM) method, which is best-suited for evaluating flexibility in project implementation. The decision analyzed was that of finding the best time to start surfactant flooding during the lifetime of a field under uncertainty. The study was conducted on two reservoir models: a 3D homogeneous model and a 2D heterogeneous model. The technical uncertainties considered were the residual oil saturation (ROS) to the surfactant flood, surfactant adsorption, and reservoir heterogeneity, and the main economic uncertain parameters considered were oil price, surfactant cost, and water-injection and -production costs. The results show that the LSM method provides a decision-making tool that was able to capture the value of flexibility in surfactant-flooding implementation and provides some insight into the effect of uncertainty on decision making, which can help mitigate adverse circumstances should they arise or capture the upside potential if circumstances prove beneficial. The results found that the optimal policy obtained was reliable and that heterogeneity and different well-placement patterns affect the value of flexibility and optimal policy for different reservoir models. Furthermore, possible extensions to enhance the LSM method were discussed.

Robust optimization of well location to enhance hysteretical trapping of CO2: Assessment of various uncertainty quantification methods and utilization of mixed response surface surrogates

The paper aims to solve a robust optimization problem (optimization in presence of uncertainty) for finding the optimal locations of a number of CO2 injection wells for geological sequestration of carbon dioxide in a saline aquifer. The parametric uncertainties are the interfacial tension between CO2 and aquifer brine, the Lands trapping coefficient and the boundary aquifers absolute. The spatial uncertainties are due to the channelized permeability field which exhibits a binary channel-non-channel system. The objective function of the optimization is the amount of residually trapped CO2 due to the hysteresis of the relative permeability curves. A risk-averse value derived from the cumulative density function of the distribution of the amount of trapped gas is chosen as the objective function value. In order to ensure that the uncertainties are effectively taken into account, Monte Carlo simulation and Polynomial Chaos Expansion (PCE)-based methods are used and compared with each other. For different cases of parametric and spatial uncertainties, the most accurate uncertainty quantification (UQ) method is chosen to be integrated within the optimization algorithm. While for parametric uncertainty cases of up to two uncertain variables, PCE-based methods computationally outperform Monte Carlo simulations, it is shown that for the multimodal distributions of the function of trapped gas occurring for the spatial uncertainty case, Monte Carlo simulations are more reliable than PCE-based UQ methods. For the discrete (integer) optimization problem, various mixed response surface surrogate models are tested and the robust optimization resulted in optimal CO2 injection well locations. This article is protected by copyright. All rights reserved.

An approximate dynamic programming approachto decision making in the presence of uncertainty for surfactant-polymer flooding

The least squares Monte Carlo method is a decision evaluation method that can capture the effect of uncertainty and the value of flexibility of a process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on the Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations. The objective of this study was to enhance the performance of the least squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in obtaining the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus, we introduce the least squares probabilistic collocation method. The decision evaluation considered a number of technical and economic uncertainties. Three reservoir case studies were used: a simple homogeneous model, the PUNQ-S3 model, and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method. Different possible enhancements were discussed in order to practically adapt the least squares probabilistic collocation method to more realistic and complex reservoir models. Furthermore, it is desired to perform the method to evaluate high-dimensional decision scenarios for different chemical enhanced oil recovery processes using real reservoir data.

Robust Quantification of Uncertainty in Heterogeneity for Chemical EOR Processes: Applying the Multi-Level Monte Carlo Method

tial uncertainty for CEOR processes while greatly reducing the computational requirement — up to two orders of magnitude when compared to traditional MC for both the Gaussian and the non-Gaussian reservoir models. The method can be easily extended to other EOR processes to quantify different kinds of spatial uncertainty, such as carbon dioxide (CO 2) EOR. Other possible extensions of this method are also discussed. Reservoir heterogeneity can be detrimental to the success of chemical enhanced oil recovery (CEOR) processes. Therefore, it is important to evaluate the effect of uncertainty in reservoir heterogeneity on the performance of CEOR. Usually, a Monte Carlo (MC) sampling approach is used, where 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. Monte Carlo simulation (MCS), however, has a slow convergence rate and requires a large number of samples to produce accurate results. This can be computationally expensive when using large reservoir models. This study used a multiscale approach to improve the efficiency of uncertainty quantification regarding reservoir heterogeneity. This multiscale approach is known as the multilevel Monte Carlo (MLMC) method. This method is based on performing a small number of ex - pensive simulations on the fine scale model and a large number of less expensive simulations on coarser upscaled models, then combining the results to produce the quantities of interest. The purpose of this method is to reduce computational cost while still maintaining the accuracy of the fine scale model. The re - sults of this approach have been compared with a reference MCS that assumes a large number of simulations on the fine scale model. Other advantages of the MLMC method are its nonintrusiveness and its scalability, which allows it to incorpo - rate an increasing number of uncertainties. This study used MLMC to efficiently quantify the effect of uncertainty in heterogeneity on the recovery factor of CEOR processes. The permeability field was assumed to be the random input. This method was first demonstrated using a Gaussian 3D reservoir model. Different coarsening algorithms, such as the renormalization and pressure solver methods, were used, and the results were compared. The results were next com - pared with running the MC approach for the fine scale model while equating the computational cost for the MLMC method. Both of these results were then compared with the reference case, which is assumed to use a large number of runs of the fine scale model. Finally, the method was extended to a channelized non-Gaussian generated 3D reservoir model. The results show that it is possible to robustly quantify spa -

Enhanced Decision Making for Chemical EOR Processes under Uncertainty - Applying the LSPC Method

The Least Squares Monte Carlo method is a decision evaluation method that can capture the value of flexibility of a process. This method was shown to provide us with some insight into the effect of uncertainty on decision making and to help us capture the upside potential or mitigate the downside effects for a chemical EOR process. The method is a stochastic approximate dynamic programming approach to decision making. It is based on a forward simulation coupled with a recursive algorithm which produces the near-optimal policy. It relies on Monte Carlo simulation to produce convergent results. This incurs a significant computational requirement when using this method to evaluate decisions for reservoir engineering problems because this requires running many reservoir simulations. The objective of this study was to enhance the performance of the Least Squares Monte Carlo method by improving the sampling method used to generate the technical uncertainties used in producing the production profiles. The probabilistic collocation method has been proven to be a robust and efficient uncertainty quantification method. It approximates the random input distributions using polynomial chaos expansions and produces a proxy polynomial for the output parameter requiring a limited number of model responses that is conditional on the number of random inputs and the order of the approximation desired. The resulting proxy can then be used to generate the different statistical moments with negligible computational requirement. By using the sampling methods of the probabilistic collocation method to approximate the sampling of the technical uncertainties, it is possible to significantly reduce the computational requirement of running the decision evaluation method. Thus we introduce the least square probabilistic collocation method. Both methods are then applied to chemical EOR problems using a number of stylized reservoir models. The technical uncertainties considered include the residual oil saturation to chemical flooding, surfactant and polymer adsorption and the viscosity multiplier of the polymer. The economic uncertainties considered were the oil price and the surfactant and polymer price. Both methods were applied using three reservoir case studies: a simple homogeneous model, the PUNQ-S3 model and a modified portion of the SPE10 model. The results show that using the sampling techniques of the probabilistic collocation method produced relatively accurate responses compared with the original method.

Applying the Probabilistic Collocation Method to Surfactant-polymer Flooding

Enhanced oil recovery has achieved great attention during the past few years. However, broad scale implementation requires greater understanding of the relevant uncertainties and their effect on performance. Quantifying this uncertainty is very important for designing these processes, yet traditional methods which are usually based on Monte Carlo simulations require a large number of realizations to produce convergent results. We propose the use of a non-intrusive approach known as the Probabilistic Collocation Method (PCM) to quantify parametric uncertainty for surfactant-polymer flooding. The quantification of uncertainty was performed for surfactant/polymer related state variables such as adsorption rates and residual saturations. The PCM is performed on two reservoir models: a modified section of the SPE10 model and the PUNQ-S3 model. The random input variables PDFs are first approximated using polynomial chaos expansions and then probabilistic collocation is used to produce approximations of the reservoir model using the collocation points obtained via Gaussian quadrature and Chebyshev extrema. These approximations can then be used to produce PDFs for output variables such as the recovery factor. Results show that PCM produces similar results to those obtained via Monte Carlo simulation, which requires a large number of simulations, while requiring significantly lower number of simulation runs.