This paper focuses on the numerical simulation of geothermal systems in complex geological settings. The physical model is based on two-phase Darcy flows coupling the mass conservation of the water component with the energy conservation and the liquid-vapor thermodynamical equilibrium. The discretization exploits the flexibility of unstructured meshes to model complex geology including conductive faults as well as complex wells. The polytopal and essentially nodal Vertex Approximate Gradient scheme is used for the approximation of the Darcy and Fourier fluxes combined with a Control Volume approach for the transport of mass and energy. Particular attention is paid to the faults which are modelled as two-dimensional interfaces defined as a collection of faces of the mesh and to the flow inside deviated or multi-branch wells defined as a collection of edges of the mesh with rooted tree data structure. By using an explicit pressure drop calculation, the well model reduces to a single equation based on complementarity constraints and only one well unknown, the bottom hole pressure, implicitly coupled to the reservoir unknowns. The coupled systems are solved at each time step using efficient nonlinear and linear solvers on parallel distributed architectures. The convergence of the discrete model is investigated numerically on a simple test case with a Cartesian geometry and a single vertical producer well. Then, the ability of our approach to deal efficiently with realistic test cases is assessed on a high energy faulted geothermal reservoir operated using a doublet of two deviated wells.
Keywords: Geothermal systems, thermal well, two-phase Darcy flow, mixed-dimensional model, faults, finite volume scheme, parallel algorithm.
CC-BY 4.0
@article{SMAI-JCM_2023__9__121_0,
author = {Antoine Armandine Les Landes and Daniel Castanon Quiroz and Laurent Jeannin and Simon Lopez and Roland Masson},
title = {Two-phase geothermal model with fracture network and multi-branch wells},
journal = {The SMAI Journal of computational mathematics},
pages = {121--149},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {9},
year = {2023},
doi = {10.5802/smai-jcm.97},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.97/}
}
TY - JOUR AU - Antoine Armandine Les Landes AU - Daniel Castanon Quiroz AU - Laurent Jeannin AU - Simon Lopez AU - Roland Masson TI - Two-phase geothermal model with fracture network and multi-branch wells JO - The SMAI Journal of computational mathematics PY - 2023 SP - 121 EP - 149 VL - 9 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.97/ DO - 10.5802/smai-jcm.97 LA - en ID - SMAI-JCM_2023__9__121_0 ER -
%0 Journal Article %A Antoine Armandine Les Landes %A Daniel Castanon Quiroz %A Laurent Jeannin %A Simon Lopez %A Roland Masson %T Two-phase geothermal model with fracture network and multi-branch wells %J The SMAI Journal of computational mathematics %D 2023 %P 121-149 %V 9 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.97/ %R 10.5802/smai-jcm.97 %G en %F SMAI-JCM_2023__9__121_0
Antoine Armandine Les Landes; Daniel Castanon Quiroz; Laurent Jeannin; Simon Lopez; Roland Masson. Two-phase geothermal model with fracture network and multi-branch wells. The SMAI Journal of computational mathematics, Volume 9 (2023), pp. 121-149. doi : 10.5802/smai-jcm.97. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.97/
[1] Well Index in Reservoir Simulation for Slanted and Slightly Curved Wells in 3D Grids, SPE Journal, Volume 8 (2003) no. 01, pp. 41-48 | arXiv | DOI
[2] A hybrid-dimensional compositional two-phase flow model in fractured porous media with phase transitions and Fickian diffusion, J. Comput. Phys., Volume 441 (2021), p. 110452 | DOI | MR | Zbl
[3] Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model, J. Comput. Phys., Volume 284 (2015), pp. 462-489 | DOI | MR
[4] Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model, J. Comput. Phys., Volume 303 (2015), pp. 470-497 | DOI | MR | Zbl
[5] Modeling fractures as interfaces for flow and transport in porous media, Fluid flow and transport in porous media: mathematical and numerical treatment (Contemporary Mathematics), Volume 295 (2002), pp. 13-24 | DOI | MR | Zbl
[6] Asymptotic and numerical modelling of flows in fractured porous media, ESAIM, Math. Model. Numer. Anal., Volume 43 (2009) no. 2, pp. 239-275 | DOI | Numdam | MR | Zbl
[7] Mimetic finite difference approximation of flows in fractured porous media, ESAIM M2AN, Volume 50 (2016), pp. 809-832 | DOI | Numdam | MR | Zbl
[8] Wellbore Models GWELL, GWNACL, and HOLA, user’s guide (1991) no. LBL-31428 http://www.osti.gov/scitech/servlets/purl/5785189 (Technical report)
[9] Petroleum Reservoir Simulation, Applied Science Publishers, 1979
[10] Parallel Geothermal Numerical Model with Fractures and Multi-Branch Wells, ESAIM: ProcS, Volume 63 (2018), pp. 109-134 | DOI | MR | Zbl
[11] Non-isothermal compositional liquid gas Darcy flow: formulation, soil-atmosphere boundary condition and application to high-energy geothermal simulations, Comput. Geosci., Volume 23 (2019) no. 3, pp. 443-470 | DOI | MR | Zbl
[12] Two-phase flow through fractured porous media, Phys. Rev. E, Volume 68 (2003) no. 2 | DOI | MR
[13] Gradient discretization of hybrid-dimensional Darcy flows in fractured porous media, Numer. Math., Volume 134 (2016) no. 3, pp. 569-609 | DOI | MR | Zbl
[14] Vertex Approximate Gradient scheme for hybrid-dimensional two-phase Darcy flows in fractured porous media, ESAIM, Math. Model. Numer. Anal., Volume 2 (2015) no. 49, pp. 303-330 | DOI | Numdam | MR
[15] Immiscible two-phase Darcy flow model accounting for vanishing and discontinuous capillary pressures: application to the flow in fractured porous media, Comput. Geosci., Volume 21 (2017) no. 5, pp. 1075-1094 | DOI | MR | Zbl
[16] Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix-fracture interfaces, IMA J. Numer. Anal. (2016) | DOI
[17] Nodal Discretization of Two-Phase Discrete Fracture Matrix Models, Springer (2021), pp. 73-118 | DOI
[18] Hybrid-dimensional modelling of two-phase flow through fractured porous media with enhanced matrix fracture transmission conditions, J. Comput. Phys., Volume 357 (2018), pp. 100-124 https://www.sciencedirect.com/science/article/pii/s0021999117308781 | DOI | MR | Zbl
[19] A Hybrid High-Order Method for Darcy Flows in Fractured Porous Media, SIAM J. Sci. Comput., Volume 40 (2018) no. 2, p. A1063-A1094 | DOI | MR | Zbl
[20] A Hybrid High-Order method for passive transport in fractured porous media, GEM - International Journal on Geomathematics, Volume 10 (2019) no. 1, p. 12 | DOI | MR | Zbl
[21] Well flow models for various numerical methods, J. Numer. Anal. Model., Volume 6 (2009), pp. 375-388 | MR | Zbl
[22] Small-stencil 3D schemes for diffusive flows in porous media, ESAIM, Math. Model. Numer. Anal., Volume 46 (2012) no. 2, pp. 265-290 | DOI | Numdam | MR | Zbl
[23] Model reduction and discretization using hybrid finite volumes of flow in porous media containing faults, Comput. Geosci., Volume 20 (2016), pp. 317-339 | DOI | MR | Zbl
[24] Domain decomposition for an asymptotic geological fault modeling, C. R. Méc. Acad. Sci. Paris, Volume 331 (2003) no. 12, pp. 849-855 | Zbl
[25] A singularity removal method for coupled 1D–3D flow models, Comput. Geosci., Volume 24 (2020) no. 2, pp. 443-457 | DOI | MR | Zbl
[26] A two-phase flow simulation of a fractured reservoir using a new fissure element method, Journal of Petroleum Science and Engineering, Volume 32 (2001) no. 1, pp. 35-52 http://www.sciencedirect.com/science/article/pii/s0920410501001462 | DOI
[27] Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture-matrix system, Adv. Water Resources, Volume 32 (2009), pp. 1740-1755 | DOI
[28] BoomerAMG: A parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., Volume 41 (2002) no. 1, pp. 155-177 | DOI | MR | Zbl
[29] An efficient numerical model for incompressible two-phase flow in fractured media, Adv. Water Resources, Volume 31 (2008) no. 6, pp. 891-905 | DOI
[30] An efficient discrete-fracture model applicable for general-purpose reservoir simulators, SPE Journal, Volume 9 (2004) no. 02, pp. 227-236 | DOI
[31] The semi-smooth Newton method for multicomponent reactive transport with minerals, Adv. Water Resources, Volume 34 (2011), pp. 137-151 | DOI
[32] Decoupling preconditioners in the implicit parallel accurate reservoir simulator (IPARS), Numer. Linear Algebra Appl., Volume 8 (2001) no. 8, pp. 537-549 | DOI | MR | Zbl
[33] A fully-coupled thermal multiphase wellbore flow model for use in reservoir simulation, Journal of Petroleum Science and Engineering, Volume 71 (2010) no. 3, pp. 138-146 http://www.sciencedirect.com/science/article/pii/s0920410509002563 (Fourth International Symposium on Hydrocarbons and Chemistry) | DOI
[34] Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., Volume 26 (2005) no. 5, pp. 1667-1691 | DOI | MR | Zbl
[35] Finite element - node-centered finite-volume two-phase-flow experiments with fractured rock represented by unstructured hybrid-element meshes, SPE Reservoir Evaluation & Engineering, Volume 10 (2007) no. 06, pp. 740-756 | DOI
[36] Control-volume model for simulation of water injection in fractured media: incorporating matrix heterogeneity and reservoir wettability effects, SPE Journal, Volume 12 (2007) no. 03, pp. 355-366 | DOI
[37] Unified approach to discretization of flow in fractured porous media, Comput. Geosci., Volume 23 (2019), pp. 225-237 | DOI | MR | Zbl
[38] Interpretation of Well-Block Pressures in Numerical, Reservoir Simulation Symposium Journal SEPJ (1978), pp. 183-194
[39] Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Permeability, Reservoir Simulation Symposium Journal SEPJ (1983), pp. 531-543
[40] TOUGH2 user’s guide, version 2 (1999) http://esd.lbl.gov/files/research/projects/tough/documentation/tough2_v2_users_guide.pdf (Backup Publisher: Earth Sciences Division, Lawrence Berkeley National Laboratory, University of California Issue: LBNL-43134 Volume: LBNL-43134) (Technical report) | DOI
[41] A mixed-dimensional finite volume method for two-phase flow in fractured porous media, Adv. Water Resources, Volume 29 (2006) no. 7, pp. 1020-1036 | DOI
[42] An efficient multi-point flux approximation method for Discrete Fracture-Matrix simulations, J. Comput. Phys., Volume 231 (2012) no. 9, pp. 3784-3800 | DOI | MR | Zbl
[43] Decoupling and block preconditioning for sedimentary basin simulations, Comput. Geosci., Volume 7 (2003) no. 4, pp. 295-318 | DOI | MR | Zbl
[44] Properties of water and steam in S.I. units, Springer, 1969
[45] Drift-Flux Modeling of Two-Phase Flow in Wellbores, SPE Journal, Volume 10 (2005) no. 01, pp. 24-33 | DOI
[46] A model for conductive faults with non-matching grids, Comput. Geosci., Volume 16 (2012) no. 2, pp. 277-296 | DOI | Zbl
[47] Calculation of Well Index for Nonconventional Wells on Arbitrary Grids, Comput. Geosci., Volume 7 (2003) no. 1, pp. 61-82 | DOI | Zbl
[48] Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media, J. Comput. Phys., Volume 345 (2017), pp. 637-664 | DOI | MR | Zbl
[49] Parallel Vertex Approximate Gradient discretization of hybrid-dimensional Darcy flow and transport in discrete fracture networks, Comput. Geosci. (2016) | Zbl
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