Reduced-Order Modeling for Compositional Simulation by Use of Trajectory Piecewise Linearization
- Jincong He (Stanford University) | Louis J. Durlofsky (Stanford University)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- October 2014
- Document Type
- Journal Paper
- 858 - 872
- 2014.Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 6.1.5 Human Resources, Competence and Training
- Production optimization, Reduced-order modeling, Trajectory piecewise linearization, Proper orthogonal decomposition, Compositional simulation
- 1 in the last 30 days
- 361 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 10.00|
|SPE Non-Member Price:||USD 30.00|
Compositional simulation can be very demanding computationally as a result of the potentially large number of system unknowns and the intrinsic nonlinearity of typical problems. In this work, we develop a reduced-order modeling procedure for compositional simulation. The technique combines trajectory piecewise linearization (TPWL) and proper orthogonal decomposition (POD) to provide a highly efficient surrogate model. The compositional POD-TPWL method expresses new solutions in terms of linearizations around states generated (and saved) during previously simulated "training" runs. High-dimensional states are projected (optimally) into a low-dimensional subspace by use of POD. The compositional POD-TPWL model is based on a molar formulation that uses pressure and overall component mole fractions as the primary unknowns. Several new POD-TPWL treatments, including the use of a Petrov-Galerkin projection to reduce the number of equations (rather than the Galerkin projection, which was applied previously), and a new procedure for determining which saved state to use for linearization are incorporated into the method. Results are presented for heterogeneous 3D reservoir models containing oil and gas phases with up to six hydrocarbon components. Reasonably close agreement between full-order reference solutions and compositional POD-TPWL simulations is demonstrated for the cases considered. Construction of the POD-TPWL model requires preprocessing overhead computations equivalent to approximately three or four full-order runs. Runtime speedups by use of POD-TPWL are, however, very significant—up to a factor of 800 for the cases considered. The POD-TPWL model is thus well suited for use in computational optimization, in which many simulations must be performed, and we present an example demonstrating its application for such a problem.
|File Size||2 MB||Number of Pages||15|
Antoulas, A.C. 2005. Approximation of Large-Scale Dynamical Systems. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics.
Aziz, K. and Settari, A. 1986. Fundamentals of Reservoir Simulation. New York City, New York: Elsevier Applied Science.
Bechtold, T., Striebel, M., Mohaghegh, K., et al. 2008. Nonlinear Model Order Reduction in Nanoelectronics: Combination of POD and TPWL. Proc. App. Math. Mech. 8 (1): 10057–10060. http://dx.doi.org/10.1002/pamm.200810057/.
Berkooz, G. and Titi, E.Z. 1993. Galerkin Projections and the Proper Orthogonal Decomposition for Equivariant Equations. Phys. Lett. A 174 (1–2): 94–102. http://dx.doi.org/10.1016/0375-9601(93)90549-F.
Bui-Thanh, T., Damodaran, M. and Willcox, K. 2004. Aerodynamic Data Reconstruction and Inverse Design Using Proper Orthogonal Decomposition. AIAA J. 42 (8): 1505–1516. http://dx.doi.org/10.2514/1.2159.
Cai, L. and White, R.E. 2009. Reduction of Model Order Based on Proper Orthogonal Decomposition for Lithium-Ion Battery Simulations. J. Electrochem. Soc. 156 (3): A154–A161. http://dx.doi.org/10.1149/1.3049347.
Cao, H. 2002. Development of Techniques for General Purpose Simulators. PhD dissertation, Stanford University, Stanford, California (2002).
Cardoso, M.A. and Durlofsky, L.J. 2010a. Linearized Reduced-Order Models for Subsurface Flow Ssimulation. J. Comput. Phys. 229 (3): 681–700. http://dx.doi.org/10.1016/j.jcp.2009.10.004.
Cardoso, M.A. and Durlofsky, L.J. 2010b. Use of Reduced-Order Modeling Procedures for Production Optimization. SPE J. 15 (2): 426–435. http://dx.doi.org/10.2118/119057-MS.
Cardoso, M.A., Durlofsky, L.J. and Sarma, P. 2009. Development and Application of Reduced-Order Modeling Procedures for Subsurface Flow Simulation. Int. J. Numer. Meth. Eng. 77 (9): 1322–1350. http://dx.doi.org/10.1002/nme.2453.
Carlberg, K., Bou-Mosleh, C. and Farhat, C. 2011. Efficient Non-Linear Model Reduction via a Least-Squares Petrov–Galerkin Projection and Compressive Tensor Approximations. Int. J. Numer. Meth. Eng. 86 (2): 155–181. http://dx.doi.org/10.1002/nme.3050.
Castro, S.A. 2007. A Probabilistic Approach to Jointly Integrate 3D/4D Seismic, Production Data and Geological Information for Building Reservoir Models. PhD dissertation, Stanford University, Stanford, California (2007).
Chaturantabut, S. and Sorensen, D.C. 2010. Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM J. Sci. Comput. 32 (5): 2737–2764. http://dx.doi.org/10.1137/090766498.
Chaturantabut, S. and Sorensen, D.C. 2011. Application of POD and DEIM to Dimension Reduction of Nonlinear Miscible Viscous Fingering in Porous Media. Math. Comp. Model. Dyn. 17 (4): 337–353. http://dx.doi.org/10.1080/13873954.2011.547660.
Chen, Y. and White, J. 2000. A Quadratic Method for Nonlinear Model Order Reduction. Proc., the 2000 International Conference on Modeling and Simulation of Microsystems, 477–480.
Chien, M.C.H., Lee, S.T. and Chen, W.H. 1985. A New Fully Implicit Compositional Simulator. Paper SPE 13385 presented at the SPE Reservoir Simulation Symposium, Dallas, Texas, 10–13 February. http://dx.doi.org/10.2118/13385-MS.
Christie, M.A. and Blunt, M.J. 2001. Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques. SPE Res Eval & Eng 4 (4): 308–317. http://dx.doi.org/10.2118/72469-PA.
Coats, K.H. 1980. An Equation of State Compositional Model. SPE J. 20 (5): 363–376. http://dx.doi.org/10.2118/8284-PA.
Echeverría Ciaurri, D., Isebor, O.J. and Durlofsky, L.J. 2011. Application of Derivative-Free Methodologies to Generally Constrained Oil Production Optimisation Problems. IJMMNO 2 (2): 134–161. http://dx.doi.org/10.1504/IJMMNO.2011.039425.
Fan, Y., Durlofsky, L.J. and Tchelepi, H.A. 2012. A Fully-Coupled Flow-Reactive-Transport Formulation based on Element Conservation, with Application to CO2 Storage Simulation. Adv. Water Resour. 42 (June): 47–61. http://dx.doi.org/10.1016/j.advwatres.2012.03.012.
Gildin, E., Ghasemi, M., Romanovskay, A., et al. 2013. Nonlinear Complexity Reduction for Fast Simulation of Flow in Heterogeneous Porous Media. Paper SPE 163618 presented at the 2013 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. http://dx.doi.org/10.2118/163618-MS.
Gildin, E., Klie, H., Rodrigues, A., et al. 2006. Development of Low-Order Controllers for High-Order Reservoir Models and Smart Wells. Paper SPE 102214 presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 24–27 September. http://dx.doi.org/10.2118/102214-MS.
Haukas, J., Aavatsmark, I., Espedal, M., et al. 2007. A Comparison of Two Different IMPSAT Models in Compositional Simulation. SPE J. 12 (1): 145–151. http://dx.doi.org/10.2118/101700-PA.
He, J. 2013. Reduced-Order Modeling for Oil-Water and Compositional Systems, with Application to Data Assimilation and Production Optimization. PhD dissertation, Stanford University, Stanford, California (2013).
He, J. and Durlofsky, L.J. 2013. Reduced-Order Modeling for Compositional Simulation Using Trajectory Piecewise Linearization. Paper SPE 163634 presented at the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 18–20 February. http://dx.doi.org/10.2118/163634-MS.
He, J., Sætrom, J. and Durlofsky, L.J. 2011. Enhanced Linearized Reduced-Order Models for Subsurface Flow Simulation. J. Comput. Phys. 230 (23): 8313–8341. http://dx.doi.org/10.1016/j.jcp.2011.06.007.
He, J., Sarma, P. and Durlofsky, L.J. 2013. Reduced-Order Flow Modeling and Geological Parameterization for Ensemble-Based Data Assimilation. Comput. Geosci. 55 (June): 54–69. http://dx.doi.org/10.1016/j.cageo.2012.03.027.
Hooke, R. and Jeeves, T.A. 1961. “Direct Search” Solution of Numerical and Statistical Problems. J. ACM 8 (2): 212–229. http://dx.doi.org/10.1145/321062.321069.
Liberge, E. and Hamdouni, A. 2010. Reduced Order Modelling Method via Proper Orthogonal Decomposition (POD) for Flow Around an Oscillating Cylinder. J. Fluid. Struct. 26 (2): 292–311. http://dx.doi.org/10.1016/j.jfluidstructs.2009.10.006.
Malik, S., Chugh, S. and Mckishnie, R.A. 2006. Field-Scale Compositional Simulation of a CO2 Flood in the Fractured Midale Field. J. Cdn. Pet. Tech. 45 (2): 41–50. http://dx.doi.org/10.2118/06-02-03.
Onwunalu, J. and Durlofsky, L.J. 2010. Application of a Particle Swarm Optimization Algorithm for Determining Optimum Well Location and Type. Comput. Geosci. 14 (1): 183–198. http://dx.doi.org/10.1007/s10596-009-9142-1.
Peng, D.Y. and Robinson, D.B. 1976. A New Two-Constant Equation of State. Ind. Eng. Chem. Fund. 15 (1): 59–64. http://dx.doi.org/10.1021/i160057a011.
Rewienski, M. and White, J. 2003. A Trajectory Piecewise-Linear Approach to Model Order Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices. IEEE Tans. Comput. Aid. D. 22 (2): 155–170. http://dx.doi.org/10.1109/TCAD.2002.806601.
Rousset, M.A.H., Huang, C.K., Klie, H., et al. 2012. Reduced-Order Modeling for Thermal Recovery Processes. Oral presentation given at ECMOR XIII – 13th European Conference on the Mathematics of Oil Recovery, Biarritz, France, 10–13 September.
van Doren, J.F.M., Markovinovic, R. and Jansen, J.D. 2006. Reduced-Order Optimal Control of Water Flooding Using Proper Orthogonal Decomposition. Comput. Geosci. 10 (1): 137–158. http://dx.doi.org/10.1007/s10596-005-9014-2.
Vasilyev, D., Rewienski, M. and White, J. 2006. Macromodel Generation for BioMEMS Components Using a Stabilized Balanced Truncation Plus Trajectory Piecewise-Linear Approach. IEEE Tans. Comput. Aid. D. 25 (2): 285–293. http://dx.doi.org/10.1109/TCAD.2005.857389.
Vermeulen, P.T.M., Heemink, A.W. and Stroet, C.B.M.T. 2004. Reduced models for linear groundwater flow models using empirical orthogonal functions. Adv. Water Resour. 27 (1): 57–69. http://dx.doi.org/10.1016/j.advwatres.2003.09.008.
Voskov, D.V. and Tchelepi, H.A. 2012. Comparison of Nonlinear Formulations for Two-Phase Multi-Component EoS Based Simulation. J. Pet. Sci. Eng. 82–83 (February–March): 101–111. http://dx.doi.org/10.1016/j.petrol.2011.10.012.
Wang, P., Balay, S., and Sepehrnoori, K. 1999. A Fully Implicit Parallel EOS Compositional Simulator for Large Scale Reservoir Simulation. Paper SPE 51885 presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 14–17 February. http://dx.doi.org/10.2118/51885-MS.
Yang, Y.J. and Shen, K.Y. 2005. Nonlinear Heat-Transfer Macromodeling for MEMS Thermal Devices. J. Micromech. Microeng. 15 (2): 408–418. http://dx.doi.org/10.1088/0960-1317/15/2/022.
Young, L.C. and Stephenson, R.E. 1983. A Generalized Compositional Approach for Reservoir Simulation. SPE J. 23 (5): 727–742. http://dx.doi.org/10.2118/10516-PA.
Zhou, Y. 2012. Parallel General-Purpose Reservoir Simulation with Coupled Reservoir Models and Multi-Segment Wells. PhD dissertation, Stanford University, Stanford, California (2012).