×

Global/local model order reduction in coupled flow and linear thermal-poroelasticity. (English) Zbl 1434.86013

Summary: Coupled flow and geomechanics computations are very complex and require solving large nonlinear systems. Such simulations are intense from both runtime and memory standpoint, which strongly hints at employing model order reduction (MOR) techniques to speed them up. Different types of Reduced-Order Models (ROM) have been proposed to alleviate this computational burden. MOR approaches rely on projection operators to decrease the dimensionality of the problem. We first execute a computationally expensive “offline” stage, during which we carefully study the full order model (FOM). Upon creating a ROM basis, we then perform the cheap “online” stage. Our reduction strategy estimates a ROM using proper orthogonal decomposition (POD). We determine a family of solutions to the problem, for a suitable sample of input conditions, where every single realization is so-called a “snapshot.” We then ensemble all snapshots to determine a compressed subspace that spans the solution. Usually, POD employs a fixed reduced subspace of global basis vectors. The usage of a global basis is not convenient to tackle problems characterized by different physical regimes, parameter changes, or high-frequency features. Having many snapshots to capture all these variations is unfeasible, which suggests seeking adaptive approaches based on the closest regional basis. We thus develop such a strategy based on local POD basis to reduce one-way coupled flow and geomechanics computations. We partition the time window to adequately capture regimes such as depletion/build-up and decreasing the number of snapshots per basis. We focus on linear elasticity and consider factors such as the role of the heterogeneity. We also assess how to tackle different degrees of freedom, such as the displacements (intercalated and coupled), pressure, and temperature, with MOR. Preliminary 2- and 3-D results show significant compression ratios up to 99.9% for the mechanics part. We formally compare FOM and ROM and provide time data to demonstrate the speedup of the procedure. Examples focus on linear and nonlinear poroelasticity. We employ continuous Galerkin finite elements for all of the discretizations.

MSC:

86-10 Mathematical modeling or simulation for problems pertaining to geophysics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage

Software:

MRST; Armadillo
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abousleiman, Y.; Cheng, AD; Cui, L.; Detournay, E.; Roegiers, JC, Mandel’s problem revisited, Geotechnique, 46, 187-195 (1996) · doi:10.1680/geot.1996.46.2.187
[2] Amsallem, D.; Zahr, MJ; Farhat, C., Nonlinear model order reduction based on local reduced-order bases, Int. J. Numer. Meth. Eng., 92, 891-916 (2012) · Zbl 1352.65212 · doi:10.1002/nme.4371
[3] Argáez, M., Ceberio, M., Florez, H., Mendez, O.: A Model Order Reduction Method for Solving High-Dimensional Problems. Proceedings of NAFIPS. IEEE, El Paso (2016)
[4] Aziz, K., Settari, A.: Petroleum reservoir simulation. Elsevier Applied Science Publishers (1986)
[5] Batselier, K.; Yu, W.; Daniel, L.; Wong, N., Computing low-rank approximations of large-scale matrices with the Tensor Network randomized SVD, SIAM J. Matrix Anal. Appl., 39, 3, 1221-1244 (2018) · Zbl 1416.65109 · doi:10.1137/17M1140480
[6] Becker, E., Carey, G., Oden, J.: Finite Elements: An Introduction, The Texas Finite Element Series. Prentice-Hall Inc., Englewood Cliffs, vol. I (1981) · Zbl 0459.65070
[7] Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Meth. Eng., 86, 155-181 (2011) · Zbl 1235.74351 · doi:10.1002/nme.3050
[8] Carlberg, K.; Farhat, C.; Cortial, J.; Amsallem, D., The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys., 242, 623-647 (2013) · Zbl 1299.76180 · doi:10.1016/j.jcp.2013.02.028
[9] Chaturantabut, S.; Sorensen, D., Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media, Math. Comput. Model. Dyn. Syst., 17, 4, 337-353 (2011) · Zbl 1302.76127 · doi:10.1080/13873954.2011.547660
[10] Chaturantabut, S.; Sorensen, DC, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 5, 2737-2764 (2010) · Zbl 1217.65169 · doi:10.1137/090766498
[11] Corigliano, A.; Dossi, M.; Mariani, S., Model order reduction and domain decomposition strategies for the solution of the dynamic elastic-plastic structural problem, Comput. Methods Appl. Mech. Eng., 290, 127-155 (2015) · Zbl 1423.74874 · doi:10.1016/j.cma.2015.02.021
[12] Coussy, O., Poromechanics (2004), New York: Wiley, New York
[13] Dean, R., Gai, X., Stone, C., Minkoff, S.: A comparison of techniques for coupling porous flow and geomechanics. No. 79709 in SPE Reservoir Simulation Symposium. SPE, Houston (2003)
[14] Enriquez-Tenorio, O., Knorr, A., Zhu, D., Hill, D.: Relationships Between Mechanical Properties and Fracturing Conductivity for the Eagle Ford Shale. no. 181858 in asia pacific hydraulic fracturing conference SPE (2016)
[15] Everson, R.; Sirovich, L., Karhunen-Loeve procedure for gappy data, JOSA A, 12, 8, 1657-1664 (1995) · doi:10.1364/JOSAA.12.001657
[16] Florez, H.: Domain Decomposition Methods for Geomechanics. Ph.D. thesis, The University of Texas at Austin (2012)
[17] Florez, H.: Applications of Model-Order Reduction to Thermo-Poroelasticity. In: 51St US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association (2017)
[18] Florez, H.: Linear Thermo-Poroelasticity and Geomechanics, chap. 10, pp. 223-242. in Finite Element Method - Simulation, Numerical Analysis and Solution Techniques, editor R. Pacurar. InTech Open. 10.5772/intechopen.71873. ISBN 978-953-51-3849-5 (2018)
[19] Florez, H., A Novel Mesh Generation Algorithm Based on the Elasticity Operator, To Appear J. Comput. Phys., 1, 1-20 (2019)
[20] Florez, H.; Argáez, M., A model-order reduction method based on wavelets and POD to solve nonlinear transient and steady-state continuation problems, Appl. Math. Model., 53, 12-31 (2018) · Zbl 1480.65310 · doi:10.1016/j.apm.2017.08.012
[21] Florez, H., Argáez, M.: A Reduced Order Gauss-Newton Method for Nonlinear Problems Based on Compressed Sensing for PDE Applications, chap. 1, pp. 1-20. in Nonlinear Systems - Volume 1, editor M. Reyhanoglu. InTech Open. https://www.intechopen.com. ISBN 978-953-51-6134-9 (2018)
[22] Florez, H., Ceberio, M.: A Novel Mesh Generation Algorithm for Field-Level Coupled Flow and Geomechanics Simulations. In: 50Th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association, Houston (2016)
[23] Florez, H., Ceberio, M., Bravo, L., et al.: Uncertainty Quantification in Dynamic Systems with Applications to Combustion-related Problems: Analysis, Approaches, and Challenges. In: Joint Propulsion Conference. AIAA Propulsion and Energy Forum, Cincinnati. 10.2514/6.2018-4920 (2018)
[24] Florez, H., Gildin, E.: Model-order reduction applied to coupled flow and geomechanics. In: Proceedings of the ECMOR XVI - 16th European Conference on the Mathematics of Oil Recovery. Barcelona (2018) · Zbl 1434.86013
[25] Florez, H., Gildin, E.: Model-Order Reduction of Coupled Flow and Geomechanics in Ultra-Low Permeability (ULP) Reservoirs. No. 193911 in SPE Reservoir Simulation Conference, Galveston, Texas (2019)
[26] Florez, H.; Manzanilla-Morillo, R.; Florez, J.; Wheeler, MF, Spline-based reservoir’s geometry reconstruction and mesh generation for coupled flow and mechanics simulation, Comput. Geosci., 18, 6, 949-967 (2014) · Zbl 1392.86008 · doi:10.1007/s10596-014-9438-7
[27] Florez, H.; Wheeler, M., A mortar method based on NURBS for curved interfaces, Comput. Methods Appl. Mech. Engrg., 310, 535-566 (2016) · Zbl 1439.74411 · doi:10.1016/j.cma.2016.07.030
[28] Florez, H., Wheeler, M., Rodriguez, A.: A Mortar Method Based on NURBS for Curved Interfaces Proceedings of the 13Th European Conference on the Mathematics of Oil Recovery (ECMOR XIII), Biarritz, France (2012)
[29] Florez, H., Wheeler, M., Rodriguez, A., Monteagudo, J.: Domain Decomposition Methods Applied to Coupled Flow-Geomechanics Reservoir Simulation. No. 141596 in SPE Reservoir Simulation Symposium. The Woodlands, Texas (2011)
[30] Freifeld, B.; Zakim, S.; Pan, L.; Cutright, B.; Sheu, M.; Doughty, C.; Held, T., Geothermal energy production coupled with CCS: a field demonstration at the SECARB Cranfield site, Cranfield, Mississippi, USA, Energy Procedia, 37, 6595-6603 (2013) · doi:10.1016/j.egypro.2013.06.592
[31] Gai, X.: A Coupled Geomechanics and Reservoir Flow Model on Parallel Computers. Ph.D. Thesis, The University of Texas at Austin (2004)
[32] Ghasemi, M.; Gildin, E., Localized model order reduction in porous media flow simulation, J. Pet. Sci. Eng., 145, 689-703 (2016) · doi:10.1016/j.petrol.2016.06.030
[33] Ghommem, M., Gildin, E., Ghasemi, M.: Complexity reduction of multiphase flows in heterogeneous porous media. SPE Journal (2015)
[34] Gunawan, F.E.: Levenberg Marquardt Iterative Regularization for the Pulse-Type Impact-Force Reconstruction, vol. 331. 10.1016/j.jsv.2012.07.025. http://www.sciencedirect.com/science/article/pii/S0022460X12005512 (2012)
[35] He, J.; Durlofsky, LJ, Reduced-order modeling for compositional simulation by use of trajectory piecewise linearization, SPE J., 19, 5, 858-872 (2014) · doi:10.2118/163634-PA
[36] Hernández, J.; Oliver, J.; Huespe, AE; Caicedo, M.; Cante, J., High-performance model reduction techniques in computational multiscale homogenization, Comput. Methods Appl. Mech. Eng., 276, 149-189 (2014) · Zbl 1423.74785 · doi:10.1016/j.cma.2014.03.011
[37] Kerfriden, P.; Gosselet, P.; Adhikari, S.; Bordas, S., Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: An adaptive model order reduction for highly nonlinear mechanical problems, Comput. Methods Appl. Mech. Eng., 200, 850-866 (2011) · Zbl 1225.74092 · doi:10.1016/j.cma.2010.10.009
[38] Kerfriden, P.; Passieux, JC; Bordas, SPA, Local/global model order reduction strategy for the simulation of quasi-brittle fracture, Int. J. Numer. Methods Eng., 89, 2, 154-179 (2012) · Zbl 1242.74130 · doi:10.1002/nme.3234
[39] Killough, J., et al.: Ninth Spe Comparative Solution Project: a Reexamination of Black-Oil Simulation. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (1995)
[40] Kim, J., Tchelepi, H., Juanes, R.: Stability; Accuracy and Efficiency of Sequential Methods for Coupled Flow and Geomechanics. No 119084 In 2009 SPE Reservoir Simulation Symposium. SPE, The Woodlands, Texas, USA (2009) · Zbl 1228.74106
[41] Kim, J.; Tchelepi, HA; Juanes, R., Rigorous coupling of geomechanics and multiphase flow with strong capillarity, SPE J., 18, 6, 1-123 (2013) · doi:10.2118/141268-PA
[42] Kováčik, J., Correlation between Young’s modulus and porosity in porous materials, J. Mater. Sci. Lett., 18, 13, 1007-1010 (1999) · doi:10.1023/A:1006669914946
[43] Lewis, R.; Schrefler, B., The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (1998), New York: Wiley, New York · Zbl 0935.74004
[44] Lie, KA; Krogstad, S.; Ligaarden, IS; Natvig, JR; Nilsen, HM; Skaflestad, B., Open-source matlab implementation of consistent discretisations on complex grids, Comput. Geosci., 16, 2, 297-322 (2012) · Zbl 1348.86002 · doi:10.1007/s10596-011-9244-4
[45] Longuemare, P., Geomechanics in reservoir simulation: Overview of coupling methods and field case study, Oil Gas Sci. Technol. Rev. IFP, 57, 471-483 (2002) · doi:10.2516/ogst:2002031
[46] Lu, S.; Ren, T.; Gong, Y.; Horton, R., An improved model for predicting soil thermal conductivity from water content at room temperature, Soil Sci. Soc. Am. J., 71, 1, 8-14 (2007) · doi:10.2136/sssaj2006.0041
[47] Mandel, J., Consolidation des sols (etude mathhatique), Geotechnique, 3, 287-299 (1953) · doi:10.1680/geot.1953.3.7.287
[48] Marquardt, DW, An Algorithm for Least-Squares Estimation of Nonlinear Parameters, J. Soc. Ind. Appl. Math., 11, 2, 431-441 (1963) · Zbl 0112.10505 · doi:10.1137/0111030
[49] Minkoff, S.; Stone, C.; Bryant, S.; Peszynska, M.; Wheeler, M., Coupled fluid flow and geomechanical deformation modeling, J. Pet. Sci. Eng., 38, 37-56 (2003) · doi:10.1016/S0920-4105(03)00021-4
[50] Mokhtari, M., Honarpour, M, Tutuncu, A, Boitnott, G: Acoustical and Geomechanical Characterization of eagle ford shale-anisotropy, Heterogeneity and Measurement Scale. No. 170707 in Annual Technical Conference and Exhibition. SPE (2014)
[51] Niroomandi, S.; Alfaro, I.; Cueto, E.; Chinesta, F., Model order reduction for hyperelastic materials, Int. J. Numer. Methods Eng., 81, 9, 1180-1206 (2010) · Zbl 1183.74365
[52] Niroomandi, S.; Alfaro, I.; González, D.; Cueto, E.; Chinesta, F., Model order reduction in hyperelasticity: a proper generalized decomposition approach, Int. J. Numer. Methods Eng., 96, 3, 129-149 (2013) · Zbl 1352.74417
[53] Pao, W.; Lewis, R.; Masters, I., A fully coupled hydro-thermo-poro-mechanical model for black oil reservoir simulation, Int. J. Numer. Anal. Meth. Geomech., 25, 1229-1256 (2001) · Zbl 1016.74023 · doi:10.1002/nag.174
[54] Phillips, P.: Finite element methods in linear poroelasticity: Theoretical and computational results. Ph.D. thesis, The University of Texas at Austin (2005)
[55] Roussel, N., Florez, H., Rodriguez, A.A.: Hydraulic Fracture Propagation from Infill Horizontal Wells. In: SPE Annual Technical Conference and Exhibition held in New Orleans, Louisiana. Society of Petroleum Engineers. 10.2118/166503-MS (2013)
[56] Sanderson, C.; Curtin, R., Armadillo: a template-based c++ library for linear algebra, J. Open Sour. Softw., 1, 2, 26-32 (2016) · doi:10.21105/joss.00026
[57] Tan, X.; Gildin, E.; Florez, H.; Trehan, S.; Yang, Y.; Hoda, N., Trajectory-based DEIM (TDEIM) model reduction applied to reservoir simulation, Comput. Geosci., 23, 1, 35-53 (2019) · Zbl 1411.86003 · doi:10.1007/s10596-018-9782-0
[58] Tan, X., Gildin, E., Trehan, S., Yang, Y., Hoda, N., et al.: Trajectory-Based DEIM TDEIM Model Reduction Applied to Reservoir Simulation. In: SPE Reservoir Simulation Conference. Society of Petroleum Engineers (2017) · Zbl 1411.86003
[59] Vosteen, HD; Schellschmidt, R., Influence of temperature on thermal conductivity, thermal capacity and thermal diffusivity for different types of rock, Phys. Chem. Earth, Parts A/B/C, 28, 9-11, 499-509 (2003) · doi:10.1016/S1474-7065(03)00069-X
[60] Walton, S.; Hassan, O.; Morgan, K., Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions, Appl. Math. Model., 37, 20-21, 8930-8945 (2013) · Zbl 1426.76576 · doi:10.1016/j.apm.2013.04.025
[61] Winget, JM; Hughes, TJ, Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies, Comput. Methods Appl. Mech. Eng., 52, 1-3, 711-815 (1985) · Zbl 0579.73119 · doi:10.1016/0045-7825(85)90015-5
[62] Yin, S.; Dusseault, MB; Rothenburg, L., Thermal reservoir modeling in petroleum geomechanics, Int. J. Numer. Anal. Meth. Geomech., 33, 449-485 (2009) · Zbl 1273.74092 · doi:10.1002/nag.723
[63] Yoon, H., Kim, J., et al.: Rigorous Modeling of Coupled Flow and Geomechanics in Largely Deformable Anisotropic Geological Systems. In: 50Th US Rock Mechanics/Geomechanics Symposium. American Rock Mechanics Association (2016)
[64] Yoon, S.; Alghareeb, ZM; Williams, JR, Hyper-Reduced-Order Models for Subsurface Flow Simulation, SPE J., 21, 6, 2-128 (2016) · doi:10.2118/181740-PA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.