×

Comparison of the response to geometrical complexity of methods for unstationary simulations in discrete fracture networks with conforming, polygonal, and non-matching grids. (English) Zbl 1453.86005

Summary: The aim of this study is to compare numerical methods for the simulation of single-phase flow and transport in fractured media, described here by means of the discrete fracture network (DFN) model. A Darcy problem is solved to compute the advective field, then used in a subsequent time-dependent transport-diffusion-reaction problem. The numerical schemes are benchmarked in terms of flexibility in handling geometrical complexity, mass conservation, and stability issues for advection-dominated flow regimes. To this end, two benchmark cases, along with an additional one from a previous work, have been specifically designed and are here proposed and investigated, representing some of the most critical issues encountered in DFN simulations.

MSC:

86-08 Computational methods for problems pertaining to geophysics
76S05 Flows in porous media; filtration; seepage
86A05 Hydrology, hydrography, oceanography
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aavatsmark, I., Interpretation of a two-point flux stencil for skew parallelogram grids, Comput. Geosci., 11, 3, 199-206 (2007) · Zbl 1124.65101 · doi:10.1007/s10596-007-9042-1
[2] Antonietti, PF; Facciolà, C.; Russo, A.; Verani, M., Discontinuous galerkin approximation of flows in fractured porous media on polytopic grids, SIAM J. Sci. Comput., 41, 1, A109-A138 (2019) · Zbl 1407.65255 · doi:10.1137/17M1138194
[3] Antonietti, PF; Formaggia, L.; Scotti, A.; Verani, M.; Verzotti, N., Mimetic finite difference approximation of flows in fractured porous media, ESAIM:, M2AN, 50, 3, 809-832 (2016) · Zbl 1381.76231 · doi:10.1051/m2an/2015087
[4] Bear, J., Dynamics of Fluids in Porous Media (1972), New York: American Elsevier, New York · Zbl 1191.76001
[5] Benedetto, M.; Berrone, S.; Borio, A.; Pieraccini, S.; Scialò, S., Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems, Comput. Methods Appl. Mech. Eng., 311, 18-40 (2016) · Zbl 1439.76051 · doi:10.1016/j.cma.2016.07.043
[6] Benedetto, MF; Berrone, S.; Borio, A.; Pieraccini, S.; Scialò, S., A hybrid mortar virtual element method for discrete fracture network simulations, J. Comput. Phys., 306, 148-166 (2016) · Zbl 1351.76048 · doi:10.1016/j.jcp.2015.11.034
[7] Benedetto, MF; Berrone, S.; Pieraccini, S.; Scialò, S., The virtual element method for discrete fracture network simulations, Comput. Methods Appl. Mech. Eng., 280, 135-156 (2014) · Zbl 1423.74863 · doi:10.1016/j.cma.2014.07.016
[8] Benedetto, MF; Berrone, S.; Scialò, S., A globally conforming method for solving flow in discrete fracture networks using the virtual element method, Finite Elem. Anal. Des., 109, 23-36 (2016) · doi:10.1016/j.finel.2015.10.003
[9] Benedetto, MF; Borio, A.; Scialò, S., Mixed virtual elements for discrete fracture network simulations, Finite Elem. Anal. Des., 134, 55-67 (2017) · doi:10.1016/j.finel.2017.05.011
[10] Berrone, S.; Borio, A., Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method, Finite Elem. Anal. Des., 129, 14-31 (2017) · doi:10.1016/j.finel.2017.01.006
[11] Berrone, S.; Borio, A.; Scialò, S., A posteriori error estimate for a PDE-constrained optimization formulation for the flow in DFNs, SIAM J. Numer. Anal., 54, 1, 242-261 (2016) · Zbl 1382.76159 · doi:10.1137/15M1014760
[12] Berrone, S.; Borio, A.; Vicini, F., Reliable a posteriori mesh adaptivity in discrete fracture network flow simulations, Comput. Meth. Appl. Mech. Eng., 354, 904-931 (2019) · Zbl 1441.76053 · doi:10.1016/j.cma.2019.06.007
[13] Berrone, S.; D’Auria, A.; Vicini, F., Fast and robust flow simulations in discrete fracture networks with GPGPUs, GEM - Int. J. Geomath., 10, 1, 8 (2019) · Zbl 1422.86001 · doi:10.1007/s13137-019-0121-y
[14] Berrone, S.; Pieraccini, S.; Scialò, S., A PDE-constrained optimization formulation for discrete fracture network flows, SIAM J. Sci. Comput., 35, 2, B487-B510 (2013) · Zbl 1266.65188 · doi:10.1137/120865884
[15] Berrone, S.; Pieraccini, S.; Scialò, S., On simulations of discrete fracture network flows with an optimization-based extended finite element method, SIAM J. Sci. Comput., 35, 2, 908-935 (2013) · Zbl 1266.65187 · doi:10.1137/120882883
[16] Berrone, S.; Pieraccini, S.; Scialò, S., An optimization approach for large scale simulations of discrete fracture network flows, J. Comput. Phys., 256, 838-853 (2014) · Zbl 1349.76806 · doi:10.1016/j.jcp.2013.09.028
[17] Berrone, S.; Pieraccini, S.; Scialò, S., Towards effective flow simulations in realistic discrete fracture networks, J. Comput. Phys., 310, 181-201 (2016) · Zbl 1349.76179 · doi:10.1016/j.jcp.2016.01.009
[18] Berrone, S.; Pieraccini, S.; Scialò, S., Flow simulations in porous media with immersed intersecting fractures, J. Comput. Phys., 345, 768-791 (2017) · Zbl 1378.76107 · doi:10.1016/j.jcp.2017.05.049
[19] Berrone, S.; Scialò, S.; Vicini, F., Parallel meshing, discretization and computation of flow in massive Discrete Fracture Networks, SIAM J. Sci. Comput., 41, 4, C317-C338 (2019) · Zbl 1433.65277 · doi:10.1137/18M1228736
[20] Brezzi, F.; Falk, RS; Marini, DL, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48, 4, 1227-1240 (2014) · Zbl 1299.76130 · doi:10.1051/m2an/2013138
[21] Burman, E.; Hansbo, P.; Larson, MG; Larsson, K., Cut finite elements for convection in fractured domains, Comput. Fluids, 179, 726-734 (2019) · Zbl 1411.76055 · doi:10.1016/j.compfluid.2018.07.022
[22] Chave, F.; Di Pietro, DA; Formaggia, L., A hybrid high-order method for darcy flows in fractured porous media, SIAM J. Sci. Comput., 40, 2, A1063-A1094 (2018) · Zbl 1448.65196 · doi:10.1137/17M1119500
[23] Chave, F.; Di Pietro, DA; Formaggia, L., A hybrid high-order method for passive transport in fractured porous media, GEM - Int J Geomath, 10, 1, 12 (2019) · Zbl 1419.76606 · doi:10.1007/s13137-019-0114-x
[24] D’Angelo, C.; Scotti, A., A mixed finite element method for Darcy flow in fractured porous media with non-matching grids, Math. Model. Numer. Anal., 46, 2, 465-489 (2012) · Zbl 1271.76322 · doi:10.1051/m2an/2011148
[25] Dowd, PA; Martin, JA; Xu, C.; Fowell, RJ; Mardia, KV, A three-dimensional fracture network data set for a block of granite, Int. J. Rock Mech. Min. Sci., 46, 5, 811-818 (2009) · doi:10.1016/j.ijrmms.2009.02.001
[26] de Dreuzy, J.R., Méheust, Y., Pichot, G.: Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (dfn). J. Geophys. Res. Solid Earth 117(B11). doi:10.1029/2012JB009461. doi:10.1029/2012JB009461 (2012)
[27] de Dreuzy, JR; Pichot, G.; Poirriez, B.; Erhel, J., Synthetic benchmark for modeling flow in 3d fractured media, Comput Geosci, 50, 59-71 (2013) · doi:10.1016/j.cageo.2012.07.025
[28] Erhel, J.; de Dreuzy, JR; Poirriez, B., Flow simulation in three-dimensional discrete fracture networks, SIAM J. Sci. Comput., 31, 4, 2688-2705 (2009) · Zbl 1387.65124 · doi:10.1137/080729244
[29] Facciolà, C.; Antonietti, PF; Verani, M., Mixed-primal discontinuous galerkin approximation of flows in fractured porous media on polygonal and polyhedral grids, PAMM, 19, 1, e201900117 (2019) · doi:10.1002/pamm.201900117
[30] Flemisch, B.; Berre, I.; Boon, W.; Fumagalli, A.; Schwenck, N.; Scotti, A.; Stefansson, I.; Tatomir, A., Benchmarks for single-phase flow in fractured porous media, Adv. Water Resour., 111, 239-258 (2018) · doi:10.1016/j.advwatres.2017.10.036
[31] Formaggia, L.; Fumagalli, A.; Scotti, A.; Ruffo, P., A reduced model for Darcy’s problem in networks of fractures, Esaim Math. Model. Numer. Anal., 48, 1089-1116 (2014) · Zbl 1299.76254 · doi:10.1051/m2an/2013132
[32] Franca, LP; Frey, SL; Hughes, T., Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Eng., 95, 2, 253-276 (1992) · Zbl 0759.76040 · doi:10.1016/0045-7825(92)90143-8
[33] Fumagalli, A., Dual virtual element method in presence of an inclusion, Appl. Math. Lett., 86, 22-29 (2018) · Zbl 1412.65209 · doi:10.1016/j.aml.2018.06.004
[34] Fumagalli, A.; Keilegavlen, E., Dual virtual element method for discrete fractures networks, SIAM J. Sci. Comput., 40, 1, B228-B258 (2018) · Zbl 1380.76140 · doi:10.1137/16M1098231
[35] Fumagalli, A.; Keilegavlen, E., Dual virtual element methods for discrete fracture matrix models, Oil Gas Sci. Technol., 74, 41, 1-17 (2019) · doi:10.2516/ogst/2019008
[36] Fumagalli, A.; Keilegavlen, E.; Scialò, S., Input and benchmarking data for flow simulations in discrete fracture networks, Data Brief, 21, 1135-1139 (2018) · doi:10.1016/j.dib.2018.10.088
[37] Fumagalli, A.; Keilegavlen, E.; Scialò, S., Conforming, non-conforming and non-matching discretization couplings in discrete fracture network simulations, J. Comput. Phys., 376, 694-712 (2019) · Zbl 1416.76306 · doi:10.1016/j.jcp.2018.09.048
[38] Fumagalli, A.; Scotti, A., A numerical method for two-phase flow in fractured porous media with non-matching grids, Adv. Water Resour., 62 Part C, 454-464 (2013) · Zbl 1273.76398 · doi:10.1016/j.advwatres.2013.04.001
[39] Fumagalli, A.; Scotti, A., A Reduced Model for Flow and Transport in Fractured Porous Media with Non-Matching Grids, 499-507 (2013), Berlin: Springer, Berlin · Zbl 1273.76398
[40] Geuzaine, C.; Remacle, JF, Gmsh: a 3-d finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., 79, 11, 1309-1331 (2009) · Zbl 1176.74181 · doi:10.1002/nme.2579
[41] Hardebol, NJ; Maier, C.; Nick, H.; Geiger, S.; Bertotti, G.; Boro, H., Multiscale fracture network characterization and impact on flow: a case study on the latemar carbonate platform, J. Geophys. Res. Solid Earth, 120, 12, 8197-8222 (2015) · doi:10.1002/2015JB011879
[42] Hyman, JD; Gable, CW; Painter, SL; Makedonska, N., Conforming delaunay triangulation of stochastically generated three dimensional discrete fracture networks: a feature rejection algorithm for meshing strategy, SIAM J. Sci. Comput., 36, 4, A1871-A1894 (2014) · Zbl 1305.74082 · doi:10.1137/130942541
[43] Hyman, JD; Karra, S.; Makedonska, N.; Gable, CW; Painter, SL; Viswanathan, HS, Dfnworks: a discrete fracture network framework for modeling subsurface flow and transport, Comput. Geosci., 84, 10-19 (2015) · doi:10.1016/j.cageo.2015.08.001
[44] Keilegavlen, E., Berge, R., Fumagalli, A., Starnoni, M., Stefansson, I., Varela, J., Berre, I.: Porepy: an open-source software for simulation of multiphysics processes in fractured porous media. Tech. rep., arXiv:1908.09869 [math.NA] (2019)
[45] Keilegavlen, E., Fumagalli, A., Berge, R., Stefansson, I.: Implementation of mixed-dimensional models for flow in fractured porous media. In: Radu, F.A., Kumar, K., Berre, I., Nordbotten, J.M., Pop, I.S. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2017, vol. 126, pp. 573-580. Springer International Publishing (2019). doi:10.1007/978-3-319-96415-7_52 · Zbl 1425.76141
[46] Lee, S.; Lee, YJ; Wheeler, MF, A locally conservative enriched galerkin approximation and efficient solver for elliptic and parabolic problems, SIAM J. Sci. Comput., 38, 3, A1404-A1429 (2016) · Zbl 1337.65128 · doi:10.1137/15M1041109
[47] Martin, V.; Jaffré, J.; Roberts, JE, Modeling fractures and barriers as interfaces for flow in porous media, SIAM J. Sci. Comput., 26, 5, 1667-1691 (2005) · Zbl 1083.76058 · doi:10.1137/S1064827503429363
[48] McClure, M., Babazadeh, M., Shiozawa, S., Huang, J.: Fully coupled hydromechanical simulation of hydraulic fracturing in three-dimensional discrete fracture networks. In: SPE Hydraulic Fracturing Technology Conference. Society of Petroleum Engineers. doi:10.2118/173354-MS (2015), doi:10.2118/173354-MS
[49] Mustapha, H.; Mustapha, K., A new approach to simulating flow in discrete fracture networks with an optimized mesh, SIAM J. Sci. Comput., 29, 4, 1439-1459 (2007) · Zbl 1251.76056 · doi:10.1137/060653482
[50] Ngo, TD; Fourno, A.; Noetinger, B., Modeling of transport processes through large-scale discrete fracture networks using conforming meshes and open-source software, J. Hydrol., 554, 66-79 (2017) · doi:10.1016/j.jhydrol.2017.08.052
[51] Nordbotten, J.M., Boon, W., Fumagalli, A., Keilegavlen, E.: Unified approach to discretization of flow in fractured porous media. Comput Geosci. https://link.springer.com/article/10.1007/s10596-018-9778-9 (2018) · Zbl 1414.76064
[52] Pichot, G.; Erhel, J.; de Dreuzy, JR, A generalized mixed hybrid mortar method for solving flow in stochastic discrete fracture networks, SIAM J. Sci. Comput., 34, 1, B86-B105 (2012) · Zbl 1387.65121 · doi:10.1137/100804383
[53] Pichot, G., Poirriez, B., Erhel, J., de Dreuzy, J.R.: A mortar Bdd method for solving flow in stochastic discrete fracture networks. In: Erhel, J., Gander, M.J., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds.) Domain Decomposition Methods in Science and Engineering XXI, pp. 99-112. Springer International Publishing, Cham (2014) · Zbl 1380.74112
[54] Raviart, PA; Thomas, JM, A mixed finite element method for second order elliptic problems, Lecture Notes in Mathematics, 606, 292-315 (1977) · Zbl 0362.65089 · doi:10.1007/BFb0064470
[55] Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. In: Handbook of Numerical Analysis, Vol.II, Handbook of Numerical Analysis, II, pp. 523-639. North-Holland, Amsterdam (1991), 10.1016/S1570-8659(05)80041-9 · Zbl 0875.65090
[56] Schön, JH, Physical Properties of Rocks, Handbook of Petroleum Exploration and Production, vol. 8 (2011), Amsterdam: Elsevier, Amsterdam
[57] Shewchuk, J.R.: Triangle: engineering a 2d quality mesh generator and delaunay triangulator. In: Lin, M.C., Manocha, D. (eds.) Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science. From the First ACM Workshop on Applied Computational Geometry, vol. 1148, pp. 203-222. Springer-Verlag (1996)
[58] Kadeethum, T., Nick, H.M., Lee, S., Ballarin, F.: Flow in porous media with low dimensional fractures by employing enriched galerkin method. Adv. Water Resour. (2020)
[59] Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, LD; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 1, 199-214 (2013) · Zbl 1416.65433 · doi:10.1142/S0218202512500492
[60] Beirão da Veiga, L.; Brezzi, F.; Marini, LD; Russo, A., H(div) and H(curl)-conforming VEM, Numer. Math., 133, 2, 303-332 (2014) · Zbl 1343.65133 · doi:10.1007/s00211-015-0746-1
[61] Beirão da Veiga, L.; Brezzi, F.; Marini, LD; Russo, A., The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 8, 1541-1573 (2014) · Zbl 1291.65336 · doi:10.1142/S021820251440003X
[62] Beirão da Veiga, L.; Brezzi, F.; Marini, LD; Russo, A., Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model Numer. Anal., 50, 3, 727-747 (2016) · Zbl 1343.65134 · doi:10.1051/m2an/2015067
[63] Wangen, M., Physical Principles of Sedimentary Basin Analysis (2010), Cambridge: Cambridge University Press, Cambridge · doi:10.1017/CBO9780511711824
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.