A fully implicit mimetic finite difference scheme for general purpose subsurface reservoir simulation with full tensor permeability.(English)Zbl 1453.76120
Summary: In the previous article [A. S. Abushaikha et al., J. Comput. Phys. 346, 514–538 (2017;Zbl 1378.76042)], we presented a fully-implicit mixed hybrid finite element (MHFE) method for general-purpose compositional reservoir simulation. The present work extends the implementation for mimetic finite difference (MFD) discretization method. The new approach admits fully implicit solution on general polyhedral grids. The scheme couples the momentum and mass balance equations to assure conservation and applies a cubic equation-of-state for the fluid system. The flux conservativity is strongly imposed for the fully implicit approach and the Newton-Raphson method is used to linearize the system. We test the method through extensive numerical examples to demonstrate the convergence and accuracy on various shapes of polyhedral. We also compare the method to other discretization schemes for unstructured meshes and tensor permeability. Finally, we apply the method through applied computational cases to illustrate its robustness for full tensor anisotropic, highly heterogeneous and faulted reservoirs using unstructured grids.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76T30 | Three or more component flows |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
Keywords:
mixed formulation;mimetic finite difference;reservoir simulation;unstructured grids;tensor permeability;fully implicitCitations:
Zbl 1378.76042Software:
MRSTCite
Full Text:DOI
References:
| [1] | Abushaikha, A. S.; Voskov, D. V.; Tchelepi, H. A., Fully implicit mixed-hybrid finite-element discretization for general purpose subsurface reservoir simulation, J. Comput. Phys., 346, 514-538 (2017), URL ·Zbl 1378.76042 |
| [2] | Aziz, K.; Settari, A., Petroleum Reservoir Simulation (1979), Chapman & Hall |
| [3] | Gunasekera, D.; Cox, J.; Lindsey, P., The generation and application of k-orthogonal grid systems, (SPE Reservoir Simulation Symposium (1997), Society of Petroleum Engineers) |
| [4] | Settari, A.; Aziz, K., Use of irregular grid in reservoir simulation, Soc. Pet. Eng. J., 12, 02, 103-114 (1972) |
| [5] | Aavatsmark, I., An introduction to multipoint flux approximations for quadrilateral grids, Comput. Geosci., 6, 3, 405-432 (2002) ·Zbl 1094.76550 |
| [6] | Nordbotten, J. M.; Eigestad, G. T., Discretization on quadrilateral grids with improved monotonicity properties, J. Comput. Phys., 203, 2, 744-760 (2005) ·Zbl 1143.76540 |
| [7] | Wheeler, M. F.; Yotov, I., A multipoint flux mixed finite element method, SIAM J. Numer. Anal., 44, 5, 2082-2106 (2006) ·Zbl 1121.76040 |
| [8] | Cao, Y.; Helmig, R.; Wohlmuth, B., Geometrical interpretation of the multi-point flux approximation l-method, Int. J. Numer. Methods Fluids, 60, 11, 1173-1199 (2009) ·Zbl 1166.76042 |
| [9] | Edwards, M. G., Higher-resolution hyperbolic-coupled-elliptic flux-continuous cvd schemes on structured and unstructured grids in 2-d, Int. J. Numer. Methods Fluids, 51, 1059-1077 (2006) ·Zbl 1158.76363 |
| [10] | Eigestad, G. T.; Klausen, R. A., On the convergence of the multi-point flux approximation o-method: numerical experiments for discontinuous permeability, Numer. Methods Partial Differ. Equ., 21, 6, 1079-1098 (2005) ·Zbl 1089.76037 |
| [11] | Aavatsmark, I.; Eigestad, G.; Klausen, R.; Wheeler, M.; Yotov, I., Convergence of a symmetric MPFA method on quadrilateral grids, Comput. Geosci., 11, 4, 333-345 (2007) ·Zbl 1128.65093 |
| [12] | Chen, Q.-Y.; Wan, J.; Yang, Y.; Mifflin, R. T., Enriched multi-point flux approximation for general grids, J. Comput. Phys., 227, 3, 1701-1721 (2008), URL ·Zbl 1221.76123 |
| [13] | Pal, M.; Edwards, M. G., q-families of CVD(MPFA) schemes on general elements: numerical convergence and the maximum principle, Arch. Comput. Methods Eng., 17, 2, 137-189 (2010), URL ·Zbl 1269.76075 |
| [14] | Terekhov, K. M.; Mallison, B. T.; Tchelepi, H. A., Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem, J. Comput. Phys., 330, 245-267 (2017), URL ·Zbl 1380.65335 |
| [15] | Le Potier, C., Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles non structurés, C. R. Math., 341, 12, 787-792 (2005) ·Zbl 1081.65086 |
| [16] | Nikitin, K.; Terekhov, K.; Vassilevski, Y., A monotone nonlinear finite volume method for diffusion equations and multiphase flows, Comput. Geosci., 18, 3, 311-324 (2014) ·Zbl 1378.76076 |
| [17] | Schneider, M.; Flemisch, B.; Helmig, R.; Terekhov, K.; Tchelepi, H., Monotone nonlinear finite-volume method for challenging grids, Comput. Geosci., 22, 2, 565-586 (2018) ·Zbl 1405.65145 |
| [18] | Le Potier, C., A nonlinear finite volume scheme satisfying maximum and minimum principles for diffusion operators, Int. J. Finite Vol., 1-20 (2009) ·Zbl 1490.65242 |
| [19] | Jackson, M.; Percival, J.; Mostaghimi, P.; Tollit, B.; Pavlidis, D.; Pain, C.; Gomes, J.; Elsheikh, A. H.; Salinas, P.; Muggeridge, A., Reservoir modeling for flow simulation by use of surfaces, adaptive unstructured meshes, and an overlapping-control-volume finite-element method, SPE Reserv. Eval. Eng., 18, 02, 115-132 (2015) |
| [20] | Abushaikha, A. S.; Blunt, M. J.; Gosselin, O. R.; Pain, C. C.; Jackson, M. D., Interface control volume finite element method for modelling multi-phase fluid flow in highly heterogeneous and fractured reservoirs, J. Comput. Phys., 298, 41-61 (2015) ·Zbl 1349.76163 |
| [21] | Nick, H.; Matthai, S., A hybrid finite-element finite-volume method with embedded discontinuities for solute transport in heterogenous media, Vadose Zone J., 10, 299-312 (2011) |
| [22] | Abushaikha, A., Numerical methods for modelling fluid flow in highly heterogeneous and fractured reservoirs (2013), Imperial College London: Imperial College London London: United Kingdom, Ph.D. thesis |
| [23] | Salinas, P.; Pavlidis, D.; Xie, Z.; Osman, H.; Pain, C. C.; Jackson, M. D., A discontinuous control volume finite element method for multi-phase flow in heterogeneous porous media, J. Comput. Phys., 352, 602-614 (2018) |
| [24] | Chavent, G.; Roberts, J., A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in water flow problems, Adv. Water Resour., 14, 329-348 (1991) |
| [25] | Lipnikov, K.; Manzini, G.; Shashkov, M., Mimetic finite difference method, J. Comput. Phys., 257, 1163-1227 (2014) ·Zbl 1352.65420 |
| [26] | Shashkov, M.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys., 129, 2, 383-405 (1996) ·Zbl 0874.65062 |
| [27] | Hyman, J. M.; Shashkov, M., Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Appl. Numer. Math., 25, 4, 413-442 (1997) ·Zbl 1005.65024 |
| [28] | Droniou, J.; Eymard, R.; Gallouët, T.; Herbin, R., A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci., 20, 02, 265-295 (2010) ·Zbl 1191.65142 |
| [29] | A.K. Pergament, Y.A. Poveshenko, Support operator method on irregular grids in computer technology for prognosis of oil and gas recovery. Preprints of Keldysh Institute of Applied Mathematics RAS (1997) 23-1 (in Russian). |
| [30] | Alpak, F. O., A mimetic finite volume discretization method for reservoir simulation, SPE J., 15, 02, 436-453 (2010) |
| [31] | Lie, K.-A.; Krogstad, S.; Ligaarden, I. S.; Natvig, J. R.; Nilsen, H. M.; Skaflestad, B., Open-source Matlab implementation of consistent discretisations on complex grids, Comput. Geosci., 16, 2, 297-322 (2012) ·Zbl 1348.86002 |
| [32] | Nilsen, H. M.; Natvig, J. R.; Lie, K.-A., Accurate modeling of faults by multipoint, mimetic, and mixed methods, SPE J., 17, 02, 568-579 (2012) |
| [33] | Zhang, N.; Abushaikha, A. S., An efficient mimetic finite difference method for multiphase flow in fractured reservoirs, (SPE Europec Featured at 81st EAGE Conference and Exhibition (2019), Society of Petroleum Engineers) |
| [34] | Zhang, N.; Abushaikha, A. S., Fully implicit reservoir simulation using mimetic finite difference method in fractured carbonate reservoirs, (SPE Reservoir Characterisation and Simulation Conference and Exhibition (2019), Society of Petroleum Engineers) |
| [35] | Alpak, F. O.; Pal, M.; Lie, K.-A., A multiscale adaptive local-global method for modeling flow in stratigraphically complex reservoirs, SPE J., 17, 04, 1-056 (2012) |
| [36] | Zhou, Y.; Tchelepi, H.; Mallison, B., Automatic differentiation framework for compositional simulation on unstructured grids with multi-point discretization schemes, (Proceedings of SPE Reservoir Simulation Symposium. Proceedings of SPE Reservoir Simulation Symposium, 21-23 February, The Woodlands, Texas, USA (2011)) |
| [37] | Voskov, D.; Tchelepi, H., Comparison of nonlinear formulations for two-phase multi-component EoS based simulation, J. Pet. Sci. Eng., 82-83, 101-111 (2012) |
| [38] | Garipov, T.; Karimi-Fard, M.; Tchelepi, H., Discrete fracture model for coupled flow and geomechanics, Comput. Geosci., 20, 1, 149-160 (2016) ·Zbl 1392.76079 |
| [39] | Brooks, R.; Corey, A., Hydraulic Properties of Porous Media, Hydrology Papers, vol. 3 (1964), Colorado State University: Colorado State University Fort, Collins |
| [40] | Michelsen, M. L., The isothermal flash problem. Part I. Stability, Fluid Phase Equilib., 9, 1, 1-19 (1982) |
| [41] | Iranshahr, A.; Voskov, D.; Tchelepi, H., A negative-flash tie-simplex approach for multiphase reservoir simulation, Soc. Pet. Eng. J., 18, 6, 1140-1149 (2013), URL |
| [42] | Michelsen, M. L., The isothermal flash problem. Part II. Phase-split calculation, Fluid Phase Equilib., 9, 1, 21-40 (1982) |
| [43] | Voskov, D. V.; Tchelepi, H. A., Compositional space parameterization: theory and application for immiscible displacements, Soc. Pet. Eng. J., 14, 431-440 (2009) |
| [44] | Iranshahr, A.; Voskov, D.; Tchelepi, H., Tie-simplex based compositional space parameterization: continuity and generalization to multiphase systems, AIChE J., 59, 5, 1684-1701 (2013) |
| [45] | Coats, K. H., An equation of state compositional model, SPE J., 20, 5 (1980), URL ·Zbl 0455.76090 |
| [46] | Voskov, D., An extended natural variable formulation for compositional simulation based on tie-line parameterization, Transp. Porous Media, 92, 3, 541-557 (2012) |
| [47] | Zaydullin, R.; Voskov, D. V.; James, S. C.; Lucia, A., Fully compositional and thermal reservoir simulation, Comput. Chem. Eng., 63, 51-65 (2014) |
| [48] | da Veiga, L. B.; Lipnikov, K.; Manzini, G., The Mimetic Finite Difference Method for Elliptic Problems, vol. 11 (2014), Springer ·Zbl 1286.65141 |
| [49] | Younis, R., Modern advances in software and solution algorithms for reservoir simulation (2011), Stanford University, PhD Thesis |
| [50] | Zhou, Y., Parallel General-Purpose Reservoir Simulation with Coupled Reservoir Models and Multi-Segment Wells (2012), Stanford University, PhD Thesis |
| [51] | Kuzmin, A.; Luisier, M.; Schenk, O., Fast methods for computing selected elements of the green’s function in massively parallel nanoelectronic device simulations, (Euro-Par 2013 Parallel Processing (2013), Springer), 533-544 |
| [52] | Brezzi, F.; Lipnikov, K.; Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15, 10, 1533-1551 (2005) ·Zbl 1083.65099 |
| [53] | Christie, M.; Blunt, M., Tenth SPE comparative solution project: a comparison of upscaling techniques, (SPE Reservoir Simulation Symposium (2001), Society of Petroleum Engineers) |
| [54] | Cao, H., Development of Techniques for General Purpose Simulators (2002), Stanford University, PhD Thesis |
| [55] | Jiang, Y., Techniques for modeling complex reservoirs and advanced wells (2007), Stanford University, Ph.D. thesis |
| [56] | Worden, R.; Smalley, P., H2s-producing reactions in deep carbonate gas reservoirs: Khuff formation, abu dhabi, Chem. Geol., 133, 1, 157-171 (1996), URL |
| [57] | Ferronato, M.; Franceschini, A.; Janna, C.; Castelletto, N.; Tchelepi, H. A., A general preconditioning framework for coupled multiphysics problems with application to contact- and poro-mechanics, J. Comput. Phys., 398, Article 108887 pp. (2019), URL ·Zbl 1453.65065 |
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.
