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.

[1]  George E. Karniadakis,et al.  The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications , 2008, J. Comput. Phys..

[2]  Y. P. Cheong,et al.  Experimental Design and Analysis Methods for Assessing Volumetric Uncertainties , 2005 .

[3]  Reidar Brumer Bratvold,et al.  Valuing Oil and Gas Options by Least-Squares Monte Carlo Simulation , 2009 .

[4]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[5]  Kamy Sepehrnoori,et al.  An Efficient Reservoir-Simulation Approach To Design and Optimize Improved Oil-Recovery-Processes With Distributed Computing , 2005 .

[6]  Wisup Bae,et al.  Development of Isotherm Polymer/Surfactant Adsorption Models in Chemical Flooding , 2011 .

[7]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[8]  Mircea Grigoriu,et al.  Convergence properties of polynomial chaos approximations for L2 random variables. , 2007 .

[9]  Heng Li,et al.  Efficient and Accurate Quantification of Uncertainty for Multiphase Flow With the Probabilistic Collocation Method , 2009 .

[10]  Jeroen A. S. Witteveen,et al.  Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties , 2007 .

[11]  H. Bijl,et al.  Probabilistic collocation used in a Two-Step approached for efficient uncertainty quantification in computational fluid dynamics , 2009 .

[12]  Manuel Moreno,et al.  On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives , 2007 .

[13]  Andrea Gamba,et al.  An Extension of Least Squares Monte Carlo Simulation for Multi-options Problems 1 , 2002 .

[14]  Menner A. Tatang,et al.  An efficient method for parametric uncertainty analysis of numerical geophysical models , 1997 .

[15]  W. W. Weiss,et al.  Planning and Implementing a Large-Scale Polymer Flood , 1985 .

[16]  W. Gerbacia,et al.  The Evaluation Of Surfactant Systems For Oil Recovery Using Statistical Design Principles And Analysis , 1978 .

[17]  Michael Andrew Christie,et al.  Tenth SPE Comparative Solution Project: a comparison of upscaling techniques , 2001 .

[18]  Reidar Brumer Bratvold,et al.  Taking Real Options Into the Real World: Asset Valuation Through Option Simulation , 2009 .

[19]  Heng Li,et al.  A Comparative Study of the Probabilistic-Collocation and Experimental-Design Methods for Petroleum-Reservoir Uncertainty Quantification , 2011 .

[20]  Martin J. Blunt,et al.  The Design and Optimization of Polymer Flooding under Uncertainty , 2011 .

[21]  Gonzalo Cortazar,et al.  The valuation of multidimensional American real options using the LSM simulation method , 2008, Comput. Oper. Res..

[22]  Monte Carlo Simulation of Stochastic Processes , 2006 .

[23]  L. Lake,et al.  Enhanced Oil Recovery , 2017 .

[24]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[25]  Ben Wang,et al.  SENSITIVITY STUDY OF MICELLAR/POLYMER FLOODING. , 1979 .

[26]  L. Mathelin,et al.  A Stochastic Collocation Algorithm for Uncertainty Analysis , 2003 .

[27]  Warrren B Powell.  Approximating Value Functions , 2007 .

[28]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[29]  C. A. Kossack,et al.  The Sensitivity of Micellar Flooding to Reservoir Heterogeneities , 1976 .

[30]  Mohammad Saber Karambeigi,et al.  Neuro-simulation modeling of chemical flooding , 2011 .

[31]  Guang Lin,et al.  An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media , 2009 .

[32]  Eduardo S. Schwartz,et al.  Investment Under Uncertainty. , 1994 .

[33]  H. Bijl,et al.  COMPARISON OF INTRUSIVE AND NON-INTRUSIVE POLYNOMIAL CHAOS METHODS FOR CFD APPLICATIONS IN AERONAUTICS , 2010 .

[34]  Peter R. King,et al.  Decision Making Under Uncertainty: Applying the Least-Squares Monte Carlo Method in Surfactant-Flooding Implementation , 2013 .

[35]  G. Uhlenbeck,et al.  On the Theory of the Brownian Motion , 1930 .

[36]  Dongxiao Zhang,et al.  Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods , 2007 .

[37]  Julian Richard Barnes,et al.  Surfactant Systems for EOR in High-Temperature, High-salinity Environments , 2012 .

[38]  Sunil Kumar,et al.  Optimizations in financial engineering: The Least-Squares Monte Carlo method of Longstaff and Schwartz , 2008, 2008 IEEE International Symposium on Parallel and Distributed Processing.

[39]  WALTER GAUTSCHI.  Algorithm 726: ORTHPOL–a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules , 1994, TOMS.

[40]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[41]  P. King,et al.  Uncertainty Quantification of a Chemically Enhanced Oil Recovery Process: Applying the Probabilistic Collocation Method to a Surfactant-Polymer Flood , 2013 .

[42]  Sara Thomas,et al.  Chemical EOR: The Past - Does It Have a Future? (Russian) , 2006 .

[43]  Martin J. Blunt,et al.  The Design and Optimization of Polymer Flooding Under Uncertainty , 2011 .

[44]  Gary A. Pope,et al.  Optimization of chemical flooding in a mixed-wet dolomite reservoir , 2006 .

[45]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[46]  Gary A. Pope,et al.  Measurement and analysis of surfactant retention , 2012 .

[47]  Clarence A. Miller,et al.  Recent Advances in Surfactant EOR , 2011 .

[48]  R. S. Bryant,et al.  Effects of Polymer Adsorption and Flow Behavior on Two-Phase Flow in Porous Media , 1998 .

[49]  Jiang Xie,et al.  Efficient and Robust Uncertainty Quantification in Reservoir Simulation with Polynomial Chaos Expansions and Non-intrusive Spectral Projection , 2011, ANSS 2011.

[50]  Harry Surkalo,et al.  Economics of Field Proven Chemical Flooding Technologies , 2008 .

[51]  Nicholas P. Hankins,et al.  Case studies for the feasibility of sweep improvement in surfactant-assisted waterflooding , 1997 .

[52]  G. Michael Shook,et al.  Interwell Tracer Tests To Optimize Operating Conditions for a Surfactant Field Trial: Design, Evaluation, and Implications , 2012 .

[53]  Warren B. Powell,et al.  “Approximate dynamic programming: Solving the curses of dimensionality” by Warren B. Powell , 2007, Wiley Series in Probability and Statistics.

[54]  Ingebret Fjelde,et al.  Surfactant Flooding in Heterogeneous Formations , 2012 .

[55]  Reidar Brumer Bratvold,et al.  Making Good Decisions , 2010 .

[56]  N. Wiener.  The Homogeneous Chaos , 1938 .

[57]  C. Brown,et al.  The Evaluation of Uncertainty in Surfactant EOR Performance Prediction , 1984 .