Efficient Optimization and Robust Value Quantification of Enhanced Oil Recovery Strategies

With an increasing demand for hydrocarbon reservoir produces such as oil, etc., and difficulties in finding green oil fields, the use of Enhanced Oil Recovery (EOR) methods such as polymer, Smart water, and solvent flooding for further development of existing fields can not be overemphasized. For reservoir profitability and reduced environmental impact, it is crucial to consider appropriate well control settings of EOR methods for given reservoir characterization. Moreover, finding appropriate well settings requires solving a constrained optimization problem with suitable numerical solution methods. Conventionally, the solution method requires many iterations involving several computationally demanding function evaluations before convergence to the appropriate near optimum. The major subject of this thesis is to develop an efficient and accurate solution method for constrained optimization problems associated with EOR methods for their value quantifications and ranking in the face of reservoir uncertainties. The first contribution of the thesis develops a solution method based on the inexact line search method (with Ensemble Based Optimization (EnOpt) for approximate gradient computation) for robust constrained optimization problems associated with polymer, Smart water, and solvent flooding. Here, the objective function is the expectation of the Net Present Value (NPV) function over given geological realizations. For a given set of well settings, the NPV function is defined based on the EOR simulation model, which follows from an appropriate extension of the black-oil model. The developed solution method is used to find the economic benefits and also the ranking of EOR methods for different oil reservoirs developed to mimic North Sea reservoirs. Performing the entire optimization routine in a transformed domain along with truncations has been a common practice for handling simple linear constraints in reservoir optimization. Aside from the fact that this method has a negative impact on the quality of gradient computation, it is complicated to use for non-linear constraints. The second contribution of this thesis proposes a technique based on the exterior penalty method for handling general linear and non-linear constraints in reservoir optimization problems to improve gradient computation quality by the EnOpt method for efficient and improved optimization algorithm. Because of the computationally expensive NPV function due to the costly reservoir simulation of EOR methods, the solution method for the underlying EOR optimization problem becomes inefficient, especially for large reservoir problems. To speedup the overall computation of the solution method, this thesis introduces a novel full order model (FOM)-based certified adaptive machine learning optimization procedures to locally approximate the expensive NPV function. A supervised feedforward deep neural network (DNN) algorithm is employed to locally create surrogate model. In the FOM-based optimization algorithm of this study, several FOM NPV function evaluations are required by the EnOpt method to approximate the gradient function at each (outer) iteration until convergence. To limit the number FOM-based evaluations, we consider building surrogate models locally to replace the FOM based NPV function at each outer iteration and proceed with an inner optimization routine until convergence. We adapt the surrogate model using some FOM-based criterion where necessary until convergence. The demonstration of methodology for polymer optimization problem on a benchmark model results in an improved optimum and found to be more efficient compared to using the full order model optimization procedures

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.publishedVersio

Efficient Optimization and Robust Value Quantification of Enhanced Oil Recovery Strategies

With an increasing demand for hydrocarbon reservoir produces such as oil, etc., and difficulties in finding green oil fields, the use of Enhanced Oil Recovery (EOR) methods such as polymer, Smart water, and solvent flooding for further development of existing fields can not be overemphasized. For reservoir profitability and reduced environmental impact, it is crucial to consider appropriate well control settings of EOR methods for given reservoir characterization. Moreover, finding appropriate well settings requires solving a constrained optimization problem with suitable numerical solution methods. Conventionally, the solution method requires many iterations involving several computationally demanding function evaluations before convergence to the appropriate near optimum. The major subject of this thesis is to develop an efficient and accurate solution method for constrained optimization problems associated with EOR methods for their value quantifications and ranking in the face of reservoir uncertainties. The first contribution of the thesis develops a solution method based on the inexact line search method (with Ensemble Based Optimization (EnOpt) for approximate gradient computation) for robust constrained optimization problems associated with polymer, Smart water, and solvent flooding. Here, the objective function is the expectation of the Net Present Value (NPV) function over given geological realizations. For a given set of well settings, the NPV function is defined based on the EOR simulation model, which follows from an appropriate extension of the black-oil model. The developed solution method is used to find the economic benefits and also the ranking of EOR methods for different oil reservoirs developed to mimic North Sea reservoirs. Performing the entire optimization routine in a transformed domain along with truncations has been a common practice for handling simple linear constraints in reservoir optimization. Aside from the fact that this method has a negative impact on the quality of gradient computation, it is complicated to use for non-linear constraints. The second contribution of this thesis proposes a technique based on the exterior penalty method for handling general linear and non-linear constraints in reservoir optimization problems to improve gradient computation quality by the EnOpt method for efficient and improved optimization algorithm. Because of the computationally expensive NPV function due to the costly reservoir simulation of EOR methods, the solution method for the underlying EOR optimization problem becomes inefficient, especially for large reservoir problems. To speedup the overall computation of the solution method, this thesis introduces a novel full order model (FOM)-based certified adaptive machine learning optimization procedures to locally approximate the expensive NPV function. A supervised feedforward deep neural network (DNN) algorithm is employed to locally create surrogate model. In the FOM-based optimization algorithm of this study, several FOM NPV function evaluations are required by the EnOpt method to approximate the gradient function at each (outer) iteration until convergence. To limit the number FOM-based evaluations, we consider building surrogate models locally to replace the FOM based NPV function at each outer iteration and proceed with an inner optimization routine until convergence. We adapt the surrogate model using some FOM-based criterion where necessary until convergence. The demonstration of methodology for polymer optimization problem on a benchmark model results in an improved optimum and found to be more efficient compared to using the full order model optimization procedures

Ensemble-based constrained optimization using an exterior penalty method

In science and engineering, non-linear constrained optimization has been a useful mathematical technique for many practical applications. Of interest to us is its applicability in the modeling and prediction of hydrocarbon reservoir production. In this paper, a new efficient, robust, and accurate optimal solution strategy based on the exterior penalty function (EPF) method and the adaptive ensemble-based optimization (EnOpt) approach (with backtracking line-search technique) for non-linear constrained optimization problems is presented. The purpose of this work is to provide a better user-friendly strategy which mitigates the problem often faced with the current constraints handling technique utilized when using the EnOpt method to solve constrained problems of water or EOR flooding. This study notes that the problem contributes to uncertainties in the gradient computation of the objective function and hence leads to the poor convergence rate of the standard EnOpt method. In this work, we used the EPF method to transform a given constrained optimization problem to a sequence of unconstrained subproblems and then sequentially solve the subproblems by unconstrained EnOpt procedure until convergence to the solution of the original problem. To demonstrate the advantage of the proposed methodology, we used it to solve analytical 2D bound constrained Rosenbrock’s problem and a practical high dimensional bound constrained water flooding optimization problem associated with a 2D 5Spot field and a 3D Reek reservoir field. The numerical results are compared with EnOpt using classical Lagrangian approach, as well as the traditional EnOpt. Our findings showed that the proposed solution method has a fast convergence rate and is more accurate and robust.publishedVersio

Ensemble-based constrained optimization using an exterior penalty method

In science and engineering, non-linear constrained optimization has been a useful mathematical technique for many practical applications. Of interest to us is its applicability in the modeling and prediction of hydrocarbon reservoir production. In this paper, a new efficient, robust, and accurate optimal solution strategy based on the exterior penalty function (EPF) method and the adaptive ensemble-based optimization (EnOpt) approach (with backtracking line-search technique) for non-linear constrained optimization problems is presented. The purpose of this work is to provide a better user-friendly strategy which mitigates the problem often faced with the current constraints handling technique utilized when using the EnOpt method to solve constrained problems of water or EOR flooding. This study notes that the problem contributes to uncertainties in the gradient computation of the objective function and hence leads to the poor convergence rate of the standard EnOpt method. In this work, we used the EPF method to transform a given constrained optimization problem to a sequence of unconstrained subproblems and then sequentially solve the subproblems by unconstrained EnOpt procedure until convergence to the solution of the original problem. To demonstrate the advantage of the proposed methodology, we used it to solve analytical 2D bound constrained Rosenbrock’s problem and a practical high dimensional bound constrained water flooding optimization problem associated with a 2D 5Spot field and a 3D Reek reservoir field. The numerical results are compared with EnOpt using classical Lagrangian approach, as well as the traditional EnOpt. Our findings showed that the proposed solution method has a fast convergence rate and is more accurate and robust