Imperial College London Department of Earth Science and Engineering DECISION MAKING AND UNCERTAINTY QUANTIFICATION FOR SURFACTANT-POLYMER FLOODING A dissertation submitted to the Department of Earth Science and Engineering of Imperial College London in partial fulfilment of the requirements for the degree of Doctor of Philosophy Ali Alkhatib November 2013 Supervised by Prof. Peter R. King 1 Declaration I hereby declare that this thesis titled ‘DECISION MAKING AND UNCERTAINTY QUANTIFICATION FOR SURFACTANT-POLYMER FLOODING’ is entirely my own work and all else is appropriately referenced. This work has not been previously submitted in its entirety or in part to any other academic institute for a degree or qualification. Ali Alkhatib Department of Earth Science and Engineering Imperial College London © 2013 Ali M. Alkhatib The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Abstract The aim of this thesis is to develop a robust parametric uncertainty quantification method and a decision making method for a chemical EOR process. The main motivation is that uncertainty is detrimental to the wide scale implementation of chemical EOR. Poor scale-up performance is not in line with the success in laboratory applications. Furthermore, economic uncertainty is also an important factor as low oil prices can deter EOR investment. As an example of chemical EOR we used Surfactant- polymer flooding due to its high potential and complexity. The approach was based on using Value of Flexibility evaluation in order to optimize the surfactant-polymer flooding in the presence of economic and technical uncertainty. This method was inspired by real options theory which provides a framework to value flexibility and captures the effect of uncertainty as the process evolves through time. By doing so, it provides the means to capitalize on the upside opportunities that these uncertainties present or to help mitigate worsening circumstances. In addition, it fulfils a secondary objective to develop a decision making process that combines both technical and economic uncertainty. The Least Squares Monte Carlo (LSM) method was chosen to value flexibility in surfactant-polymer flooding. The algorithm depends on two main components; the stochastic simulation of the input state variables and the dynamic programming approach that produce the optimal policy. The produced optimal policy represents the influence of uncertainty in the time series of the relevant input parameters. Different chemical related parameters were modelled stochastically such as surfactant and polymer adsorption rates and residual oil saturation. Static uncertainty in heterogeneity was incorporated using Gaussian and multiple-point statistics generated grids and dynamic uncertainty in heterogeneity was modelled using upscaling techniques. Economic uncertainties such as the oil price and surfactant and polymer cost were incorporated into the model as well. The results obtained for the initial case studies showed that the method produced higher value compared with static policy scenarios. It showed that by designing flexibility into the implementation of the surfactant-polymer flood, it is possible to create value in the presence of uncertainty. An attempt to enhance the performance of the LSM algorithm was introduced by using the probabilistic collocation method (PCM) to sample the distributions of the technical state input parameters more efficiently, requiring significantly less computational time compared to Monte Carlo sampling. The combined approach was then applied to more complex decisions to demonstrate its scalability. It was found that the LSM algorithm could value flexibility for surfactant-polymer flooding and that it introduces a new approach to highly uncertain problems. However, there are some limitations to the extendibility of the algorithm to more complex higher dimensional problems. The main limitation was observed when using a finer discretization of the decision space because it requires a significant increase in the number of stochastic realization for the results to converge, thus increasing the computational requirement significantly. The contributions of this thesis can be summarized into the following: an attempt to use real options theory to value flexibility in SP flooding processes, the development of an approximate dynamic programming approach to produce optimal policies, the robust quantification of parametric uncertainty for SP flooding using PCM and an attempt to improve the efficiency of the LSM method by coupling it with the PCM code in order to extend its applicability to more complex problems. 3 List of Publications The following publications are a result of this work: Alkhatib, A. and King, P. 2011. Applying Real Options Theory in Determining Optimal Policies for a Surfactant Flood. Paper SPE 144869-MS presented at the SPE Enhanced Oil Recovery Conference, Kuala Lumpur, Malaysia, 19-21 July. Alkhatib, A., Babaei, M. and King, P. 2012. Decision Making under Uncertainty in EOR: Applying the Least Squares MonteCarlo (LSM) Method in Chemical EOR Implementation. Paper SPE 154467-MS presented at the SPE Europec/EAGE Annual Conference, Copenhagen, Denmark, 4-7 June. Alkhatib, A. and King, P. 2013. Uncertainty Quantification of a Chemically Enhanced Oil Recovery Process: Applying the Probabilistic Collocation Method to a Surfactant-Polymer Flood. Paper SPE 164244-MS presented at the 18th Middle East Oil & Gas Show and Conference (MEOS), Manama, Bahrain, 10-13 March. Alkhatib, A., Babaei, M. and King, P. 2013. Decision Making Under Uncertainty: Applying the Least-Squares Monte Carlo Method in Surfactant-Flooding Implementation. SPE J. (in press: published online 9 April 2013, SPE154467-PA). Alkhatib, A. and King, P. 2013. Applying the Probabilistic Collocation Method to Surfactant-Polymer Flooding. Paper EAGE 15560 presented at the 17th European Symposium on Improved Oil Recovery, St. Petersburg, Russia, 16-18 April. Alkhatib, A. and King, P. 2013. The Use of the Least Squares Probabilistic Collocation Method in Decision Making in the Presence of Uncertainty for Surfactant-Polymer Flooding. Paper SPE 164860-MS presented at the 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013, London, UK, 10-13 June. Alkhatib, A. and King, P. 2013. Robust Quantification of Parametric Uncertainty for Surfactant-Polymer Flooding. Submitted to Computational Geosciences. Alkhatib, A. and King, P. 2013. An Approximate Dynamic Programming Approach to Decision Making in the Presence of Uncertainty for Surfactant-Polymer Flooding. Submitted to Computational Geosciences. 4 Table of Contents Abstract __________________________________________________________________________ 3 List of Publications __________________________________________________________________ 4 Table of Contents ___________________________________________________________________ 5 List of Figures ______________________________________________________________________ 9 List of Tables ______________________________________________________________________ 13 1 Introduction ___________________________________________________________________ 14 2 Enhanced Oil Recovery: Surfactant-Polymer Flooding _________________________________ 17 2.1 Surfactants Chemistry and Structure ___________________________________________________18 2.2 IFT Reduction _____________________________________________________________________18 2.3 Phase Behaviour ___________________________________________________________________20 2.3.1 Effect of Salinity _______________________________________________________________________ 20 2.3.2 IFT and Phase Behavior _________________________________________________________________ 22 2.3.3 Variables that Affect IFT and Phase Behavior ________________________________________________ 22 2.4 Retention _________________________________________________________________________23 2.4.1 Adsorption ___________________________________________________________________________ 23 2.4.2 Ionic Exchange ________________________________________________________________________ 24 2.4.3 Precipitation _________________________________________________________________________ 24 2.4.4 Phase Entrapment _____________________________________________________________________ 24 2.5 Mobility Control ___________________________________________________________________25 2.5.1 Polymer _____________________________________________________________________________ 25 2.5.2 Retention ____________________________________________________________________________ 25 2.5.3 Viscosity _____________________________________________________________________________ 26 2.5.4 Mobility Control Design _________________________________________________________________ 26 2.5.5 Surfactant-Polymer Interaction and Compatibility ____________________________________________ 27 2.6 Design Considerations ______________________________________________________________28 2.6.1 Laboratory Results _____________________________________________________________________ 29 2.6.2 Field Implementation __________________________________________________________________ 30 2.6.3 Improvements and Variations ____________________________________________________________ 31 2.7 Performance Forecasting-Simulation ___________________________________________________32 2.7.1 Commonly Used Simulators _____________________________________________________________ 32 2.7.2 Reservoir Simulator used for this Study: ECLIPSE 100 __________________________________________ 33 2.7.3 Recent Developments and Other approaches________________________________________________ 37 2.8 Sources of Uncertainty ______________________________________________________________38 2.9 Concluding Remarks ________________________________________________________________39 3 Decision Making in E&P _________________________________________________________ 40 5 3.1 Decisions and Decision Analysis _______________________________________________________40 3.1.1 Definitions ___________________________________________________________________________ 40 3.1.2 Decisions in the Presence of Uncertainty ___________________________________________________ 41 3.2 Decision analysis in Exploration and Production __________________________________________41 3.2.1 Review of Decision Analysis Methods used in E&P ____________________________________________ 41 3.2.2 Review of Decision Analysis Methods for Chemical EOR Processes _______________________________ 43 3.3 Real Options Theory ________________________________________________________________46 3.3.1 Definition ____________________________________________________________________________ 46 3.3.2 Valuation Methods ____________________________________________________________________ 48 3.3.3 Examples in E&P ______________________________________________________________________ 51 3.4 Concluding Remarks ________________________________________________________________58 4 Theory and Methodology ________________________________________________________ 60 4.1 LSM Theory and Implementation ______________________________________________________60 4.1.1 The Least-Squares Monte Carlo Algorithm __________________________________________________ 62 4.1.2 Implementation _______________________________________________________________________ 64 4.2 Uncertainty in Heterogeneity ___________________________________________________________ 68 4.4 PCM Background and Theory _________________________________________________________69 4.4.1 Polynomial Chaos Expansion _____________________________________________________________ 70 4.4.2 Probabilistic Collocation ________________________________________________________________ 71 4.4.2.1 Gaussian Quadrature ________________________________________________________ 71 4.4.2.2 Linear Regression __________________________________________________________ 72 4.4.3 Selection of Collocation Nodes ___________________________________________________________ 73 4.4.4 Computation of Statistical Moments_______________________________________________________ 75 4.4.5 Implementation _______________________________________________________________________ 76 4.5 The LSPCM Algorithm _______________________________________________________________76 4.5.1 Coupling of LSM with PCM Algorithm ______________________________________________________ 77 4.5.2 Implementation of the LSPCM Algorithm ___________________________________________________ 78 5 The Least Squares Monte Carlo Method: Case Studies _________________________________ 79 5.1 Application: Case Study 1 ____________________________________________________________79 5.1.1 Problem Definition ____________________________________________________________________ 79 5.1.2 Monte Carlo Simulation _________________________________________________________________ 82 5.1.3 Code Validation _______________________________________________________________________ 89 5.1.4 Results for Case Study 1 ________________________________________________________________ 89 5.1.5 VoF Sensitivity Analysis _________________________________________________________________ 91 5.1.5.1 Varying the Static Policy Scenario _________________________________________________________ 91 5.1.5.2 Convergence as a Function of the Number of Realizations ______________________________________ 92 5.1.5.3 Decision Space Discretization ____________________________________________________________ 93 5.1.5.4 Mobility Control ______________________________________________________________________ 105 5.2 Basis Function Design and Regression ________________________________________________ 106 5.2.1 Simple Polynomials ___________________________________________________________________ 106 5.2.2 Orthogonal Polynomials _______________________________________________________________ 107 5.2.3 Clustered Linear Regression ____________________________________________________________ 110 5.3 Application: Case Study 2 __________________________________________________________ 113 5.3.1 Mobility Control ______________________________________________________________________ 116 5.3.2 Well Placement ______________________________________________________________________ 117 6 5.3.3 Permeability Field ____________________________________________________________________ 118 5.3.4 Summary ___________________________________________________________________________ 120 5.4 Uncertainty in Heterogeneity _______________________________________________________ 121 5.4.1 Moving Average Method _______________________________________________________________ 121 5.4.2 Multiple-Point Statistics _______________________________________________________________ 124 5.4.3 Summary ___________________________________________________________________________ 128 5.5 Concluding Remarks ______________________________________________________________ 128 6 The Probabilistic Collocation Method: Case Studies __________________________________ 130 6.1 Case Studies ____________________________________________________________________ 130 6.1.1 Example 1: Homogeneous three-dimensional reservoir model _________________________________ 132 6.1.2 Example 2: Heterogeneous two-dimensional reservoir model __________________________________ 136 6.1.3 Example 3: PUNQ-S3 three-dimensional reservoir model _____________________________________ 141 6.1.4 Example 4: Modified SPE10 three-dimensional reservoir model ________________________________ 144 6.2 Discussion ______________________________________________________________________ 145 6.3 Concluding Remarks ______________________________________________________________ 148 7 The Least Squares Probabilistic Collocation Method: Case Studies ______________________ 149 7.1 LSPCM Case Studies ______________________________________________________________ 149 7.1.1 Initial Application _____________________________________________________________________ 149 7.1.2 Incorporating Uncertainty in the Time Series _______________________________________________ 155 7.2 Possible Extensions _______________________________________________________________ 157 7.2.1 Mutually Exclusive Decisions ____________________________________________________________ 157 7.2.2 Application to Alkaline-Surfactant-Polymer Flooding _________________________________________ 160 7.3 Concluding Remarks ______________________________________________________________ 163 8 Conclusions and Recommendations _______________________________________________ 165 8.1 Conclusions _____________________________________________________________________ 165 8.2 Recommendations for Future Work__________________________________________________ 166 References ______________________________________________________________________ 169 Appendices ______________________________________________________________________ 186 Appendix A: Reservoir and Surfactant Properties for Homogeneous and SPE10 Layers 1, 5, 10, 15 and 20 Reservoir Models ______________________________________________________________________ 186 Appendix B: Basis Function Orthogonal Polynomials __________________________________________ 188 Appendix C: Reservoir and Fluid Data for Moving Average and FILTERSIM Reservoir Models __________ 189 Appendix D: Orthogonal Polynomials Used for PCM __________________________________________ 190 Appendix E: Golub-Welsch Algorithm for Gaussian Quadrature _________________________________ 191 Appendix F: Inverse Cumulative Distribution Function and Inverse Rosenblatt Transform ____________ 192 Appendix G: Surfactant-Polymer Properties and Simulation Constraints __________________________ 193 Appendix H: Alkaline Model in ECLIPSE _____________________________________________________ 194 Appendix I: Alkaline Properties ___________________________________________________________ 196 7 8 List of Figures Fig. 2.1─ The effect of salinity on IFT and solubiliztion of surfactant. C / C refers to solubilization parameter between the 23 33 mircoemulsion-oleic phases (for type II(-) and III phase behavior) and C /C refers to solubilization parameter between the 13 33 microemulsion-aqueous phases (for type II(+) and III phase behavior). This figure is reproduced from Lake (1989). _______ 19 Fig. 2.2─ The effect of the log of capillary number on final residual oil saturation. This is based on the correlation by Healy and Reed (1977). is the microemulsion-oil interfacial tension and is the microemulsion-water interfacial tension. Reproduced from Green and Willhite (1998). _____________________________________________________________ 20 Fig. 2.3─ The effect of salinity on a generalized phase diagrams. Reproduced from Green and Willhite (1998). _________ 21 Fig. 2.4─ Surfactant adsorption as a function of surfactant concentration. This is based on stylized data. ______________ 24 Fig. 2.5─ Effect of adding polymer to a surfactant flood on oil recovery. This is based on adding a 500ppm polymer slug to surfactant core flood using a Bentheim core ( Reproduced from Taugbol et al. 1995). _____________________________ 26 Fig. 2.6─ Field performance plot of the ratio of actual to predicted SP flooding recovery vs. incremental oil recovery (plot data from Hammershaimb et al. 1983). _________________________________________________________________ 31 Figure 3.1─ Discrepancy between DCF (Discounted Cash Flow) and ROV(Real Options Valuation) Approaches. Reproduced from Jafarizadeh and Bratvold (2009). __________________________________________________________________ 47 Fig. 3.2─ Compound option diagram. This diagram shows decision contingent on previous decisions over a two year period. Once a decision is made after the first time step (1 year), a new decision becomes available contingent on the previous decision. __________________________________________________________________________________________ 48 Fig. 3.3─ Binomial model lattice. (p) and (1-p) are the probabilities of the up (u) and down (d) moves. ________________ 49 Fig. 4.1─ The LSM method framework __________________________________________________________________ 61 Fig 4.2─ Algorithm flow chart illustrating the steps followed in the MATLAB code. ________________________________ 66 Fig 4.3─ Algorithm flow chart illustrating decision rule in the MATLAB code. ____________________________________ 67 Fig. 4.4─ Flow chart illustrating the steps in generating the stochastic realizations for the technical and economic parameters and heterogeneity fields which are used to obtain the production profiles and determine the objective function for each scenario. ___________________________________________________________________________________ 68 Fig. 4.5─ Two-dimensional quadrature grid, where moving from left plot to the right, are the second, third and fourth order PCM nodes. _______________________________________________________________________________________ 73 Fig. 4.6─ Two-dimensional mapped Fejer grid, where moving from left plot to the right, are the second, third and fourth order PCM nodes.___________________________________________________________________________________ 74 Fig. 4.7─ Two-dimensional boxed Fejer grid, where moving from left plot to the right, are the second, third and fourth order PCM nodes. _______________________________________________________________________________________ 74 Fig. 4.8 ─Number of nodes for a four-dimensional random input scenario, as a function of approximation order. _______ 75 Fig. 4.9─ PCM flowchart illustrating the implementation steps. _______________________________________________ 76 Fig. 4.9─ LSPCM code flowchart illustrating the implementation steps. _________________________________________ 78 Fig. 5.1─Decision (flexibility) flow chart. _________________________________________________________________ 80 Fig. 5.2─Decision state space representation where X(ω) represents the state variables per ω realization. _____________ 81 Fig. 5.3─ Surface plot for recovery factor and NPV as a function of D and S . Each surface color is illustrates a different s orc policy (surfactant flooding initiation). ___________________________________________________________________ 82 Fig. 5.4─ Serial test plots for six iterations for the pseudo-random number generator used in MATLAB. Each iteration consists of 1000 samples of a uniformly distributed variable with unit interval. _________________________________________ 83 Fig. 5.5─ Residual oil saturation to chemical flooding and surfactant adsorption variation with time for case study 1. ____ 85 Fig. 5.6─ Production profiles for case study 1. ____________________________________________________________ 85 Fig 5.7─ Ornstein-Uhlenbeck price model, showing the mean of 1000 paths and the corresponding standard deviation. __ 87 Fig 5.8─ Normalized NPV for the case study 1. ____________________________________________________________ 88 Fig 5.9─ Value of flexibility probability density function (PDF) plot and policy histogram for case study 1. ______________ 90 Fig 5.10─ VoF PDF plot and policy histogram for case study 1 using different static scenarios. _______________________ 91 Fig 5.11─ Histogram of optimal policy (adjusted to a basis of 103 realizations in order to compare with the original run) and plot of convergence of the averaged normalized option value for the homogeneous reservoir model with increasing number of realizations. The averaged normalized option value is taken as the difference between the averaged (over all realizations) normalized option and the averaged normalized static scenarios, the normalization is with respect to the corresponding 9 waterflood scenario NPV (averaged over all realizations). A value > 0 is considered to be favouring flexibility while a value <0 favours the static scenario. ___________________________________________________________________________ 93 Fig 5.12─ VoF PDF plot for case study 1 for different decision space discretizations. _______________________________ 94 Fig 5.13─ Policy histograms for case study 1 for different decision space discretization’s. __________________________ 94 Fig 5.14─ VoF PDF for one-dimensional homogeneous reservoir model assuming S as the technical uncertainty for orc different decision space discretization’s. _________________________________________________________________ 95 Fig 5.15─ Policy histograms for one-dimensional homogeneous reservoir model assuming S as the only technical orc uncertainty for different decision space discretization’s. ____________________________________________________ 96 Fig 5.16─ VoF PDF for three-dimensional homogeneous reservoir model assuming S as the technical uncertainty for orc different decision space discretization’s. _________________________________________________________________ 96 Fig 5.17─ Policy histograms for three-dimensional homogeneous reservoir model assuming S as the only technical orc uncertainty for different decision space discretization’s. ____________________________________________________ 97 Fig 5.18─ VoF PDF for one-dimensional homogeneous reservoir model assuming S and D as the technical uncertainties for orc s different decision space discretization’s. _________________________________________________________________ 97 Fig 5.19─ Policy histograms for one-dimensional homogeneous reservoir model assuming S and D as the technical orc s uncertainties for different decision space discretization’s. ___________________________________________________ 98 Fig 5.20─ VoF PDF for three-dimensional homogeneous reservoir model assuming S and D as the technical uncertainties orc s for different decision space discretization’s. ______________________________________________________________ 98 Fig 5.21─ Policy histograms for three-dimensional homogeneous reservoir model assuming S and D as the technical orc s uncertainties for different decision space discretization’s. ___________________________________________________ 99 Fig 5.22─ Plot of mean NPV and RF (recovery factor) over 1000 realizations as a function of injection policy. The legend in the upper right corner refer to the right axis (RF) and the legend in the upper left corner refers to the left axis (NPV). The plot on the left was determined for the base case three dimensional homogeneous reservoir model and the plot on the right was obtained using a one dimensional homogeneous reservoir model.____________________________________________ 100 Fig 5.23─ Figure showing the VoF PDF and the optimal policy histogram obtained for each decision space discretization case where the results for each water cost multiplier is plotted. These were obtained for the three-dimensional homogeneous reservoir model. ___________________________________________________________________________________ 101 Fig 5.24─ Plot of relative mean VoF as a function of the water cost multiplier. The mean VoF is normalized using the low water cost multiplier case (0.01) as the base case. These results were obtained using the three-dimensional homogeneous reservoir model base case. ___________________________________________________________________________ 102 Fig 5.27─ VoF PDF plot for case study 1 using mobility control. ______________________________________________ 105 Fig 5.28─ VoF PDF and optimal policy histogram using 1st order polynomial. ___________________________________ 106 Fig 5.29─ VoF PDF and optimal policy histogram using 3rd order polynomial. ___________________________________ 106 Fig 5.30─ VoF PDF and optimal policy histogram using 2nd order polynomial with cross-products. ___________________ 107 Fig 5.31─ VoF PDF and optimal policy histogram using 3rd order polynomial with cross-products. ___________________ 107 Fig 5.32─ VoF PDF and optimal policy histogram using 3rd order Legendre polynomials. __________________________ 108 Fig 5.33─ VoF PDF and optimal policy histogram using 3rd order Weighted Laguerre polynomials. __________________ 109 Fig 5.34─ VoF PDF and optimal policy histogram using 3rd order Laguerre polynomials. ___________________________ 109 Fig 5.35─ VoF PDF and optimal policy histogram using 3rd order Hermite polynomials. ___________________________ 109 Fig 5.36─ VoF PDF and optimal policy histogram using 3rd order Chebyshev polynomials. _________________________ 110 Fig 5.37─ Cross-plot of NPV for surfactant injection for different policies as a function of adsorption. The data is separated into four different clusters. __________________________________________________________________________ 111 Fig 5.38─ VoF PDF and optimal policy histogram for results determined by using dynamic clustered regression based on adsorption. _______________________________________________________________________________________ 111 Fig 5.39─ Cross-plot of NPV for surfactant injection for different policies as a function of residual oil saturation to chemical flooding. The data is separated into four different clusters. _________________________________________________ 112 Fig 5.40─ VoF PDF and optimal policy histogram for results determined by using dynamic clustered regression based on residual saturation. ________________________________________________________________________________ 112 Fig. 5.41─ Heterogeneity variation with time, with the far left showing the coarsest realization and the far right showing the finest realization of the permeability field. ______________________________________________________________ 114 Fig. 5.42─ Production profiles for case study 2. __________________________________________________________ 114 10