Abstract
Upscaling of flow from pore to Darcy scale is a long-standing research field within flow in porous media. It is well known that non-linearities can occur in near-well regions and high-porosity or fractured media. At the same time, the upscaled non-linear effects associated with high flow rates are hard to quantify a priori with single-scale models. Advances in pore-scale imaging combined with increased computational have made flow simulations in small pore-scale domain feasible, but computations on domains larger than at most a few centimeters are still elusive. In this work, we present a multiscale simulation framework that automatically adapts to non-linear effects as they arise. We formulate a control volume heterogeneous multiscale method (CVHMM) by coupling of a Darcy-scale control volume method with a constitutive relation that is captured based on the fine-scale physics. While the CVHMM formulation works with arbitrary upscaled laws, we emphasize its ability to be applied in fully discrete multiscale context, in particular when a finite element solver is used for solving Navier-Stokes equations on the fine-scale pore geometry. Previous versions of CVHMM are consistent only when the coarse grid is aligned with the upscaled permeability. Herein, we generalize CVHMM by introducing a new coarse solver, thus significantly improving the applicability of the method. The presented method is applied to study flow in near-well regions, as well as media with fractures and irregular grain shapes. The examples show that the method successfully copes with general grids and pore geometries and handles flows with varying degree of non-linearities even outside the domain of applicability of classical upscaled models. In terms of computational efficiency, the method seamlessly localizes computations to regions where non-linear effects are important.
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Aavatsmark, I.: Interpretation of a two-point flux stencil for skew parallelogram grids. Comput. Geosci. 11(3), 199–206 (2007). https://doi.org/10.1007/s10596-007-9042-1
Aavatsmark, I., Barkve, T., Bøe, O., Mannseth, T.: Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods. SIAM J. Sci. Comput. 19(5), 1700–1716 (1998). https://doi.org/10.1137/S1064827595293582
Aavatsmark, I., Eigestad, G.T., Mallison, B.T., Nordbotten, J.M.: A compact multipoint flux approximation method with improved robustness. Numer. Methods Partial Differ. Equ. 24, 1329–1360 (2008). https://doi.org/10.1002/num.20320. http://onlinelibrary.wiley.com/doi/10.1002/num.20320/abstract
Abdulle, A.: The finite element heterogeneous multiscale method: a computational strategy for multiscale PDEs. Mult. Scales Probl. Biomath. Mech. Phys. Numer. 31, 133–181 (2009)
Abdulle, A.: A priori and a posteriori error analysis for numerical homogenization: a unified framework. In: Damlamian, A., Miara, B., Li, T. (eds.) Multiscale Problems, vol. 16, pp. 280–305. World Scientific (2011). https://doi.org/10.1142/8267. http://anmc.epfl.ch/Pdf/abdulle_cam_final_pr.pdf. http://www.worldscientific.com/doi/abs/10.1142/9789814366892_0009
Abdulle, A., Budác, O.: An adaptive finite element heterogeneous multiscale method for stokes flow in porous media. Multiscale Model. Simul. 13(1), 256–290 (2015). https://doi.org/10.1137/130950136
Abdulle, A., Budáč, O.: A reduced basis finite element heterogeneous multiscale method for stokes flow in porous media. Comput. Methods Appl. Mech. Eng. 307, 1–31 (2016). https://doi.org/10.1016/j.cma.2016.03.016
Alyaev, S., Keilegavlen, E., Nordbotten, J.: Multiscale simulation of non-Darcy flows. In: CMWR, pp. 1–8. http://cmwr2012.cee.illinois.edu/Papers/SpecialSessions/HybridMultiscaleModelsinSubsurfaceFlowandTransport/Alyaev.Seregy.pdf (2012)
Alyaev, S., Keilegavlen, E., Nordbotten, J.M.J.: Analysis of control volume heterogeneous multiscale methods for single phase flow in porous media. MMS 12(1), 335–363 (2014). https://doi.org/10.1137/120885541
Andrȧ, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E.H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rock physics benchmarks-part II: Computing effective properties. Comput. Geosci. 50, 33–43 (2013). https://doi.org/10.1016/j.cageo.2012.09.008
Andrȧ, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E. H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A., Zhan, X.: Digital rock physics benchmarks-Part I: Imaging and segmentation. Comput. Geosci. 50, 25–32 (2013). https://doi.org/10.1016/j.cageo.2012.09.005
Aulisa, E., Ibragimov, A., Valko, P., Walton, J.: Mathematical framework of the well productivity index for fast Forchheimer (non-Darcy) flow in porous media. Math. Models Methods Appl. Sci. 19(8), 1241–1275 (2009). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.182.5461&rep=rep1&type=pdf
Balhoff, M., Mikeliċ, A., Wheeler, M.F.: Polynomial filtration laws for low Reynolds number flows through porous media. Transp. Porous Media 81(1), 35–60 (2010). https://doi.org/10.1007/s11242-009-9388-z
Balhoff, M., Wheeler, M.: A predictive pore-scale model for non-Darcy flow in porous media. SPE J. 14, 9 (2009). https://doi.org/10.2118/110838-PA. http://www.onepetro.org/mslib/servlet/onepetro preview?id=SPE-110838-PA
Balhoff, M.T., Thomas, S.G., Wheeler, M.F.: Mortar coupling and upscaling of pore-scale models. Comput. Geosci. 12(1), 15–27 (2008). https://doi.org/10.1007/s10596-007-9058-6
Barree, R.D., Conway, M.W.: Beyond beta factors: a complete model for Darcy, Forchheimer, and Trans-Forchheimer flow in porous media. In: Proceedings of SPE Annual Technical Conference and Exhibition, p. 8. https://doi.org/10.2523/89325-MS (2004)
Berkowitz, B.: Characterizing flow and transport in fractured geological media: a review. Adv. Water Resour. 25(8), 861–884 (2002). https://doi.org/10.1016/S0309-1708(02)00042-8
Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modelling. Adv. Water Resour. 51, 197–216 (2013). https://doi.org/10.1016/j.advwatres.2012.03.003
Chan, S., Elsheikh, A.H.: A machine learning approach for efficient uncertainty quantification using multiscale methods. J. Comput. Phys. 354, 493–511 (2018)
Chen, Z., Lyons, S.L., Qin, G.: Derivation of the Forchheimer Law via Homogenization, pp. 325–335 (2001)
Chu, J., Engquist, B., Prodanovic, M., Tsai, R.: A multiscale method coupling network and continuum models in porous media I: steady-state single phase flow. Multiscale Model. Simul. 10(2), 515–549 (2012). https://doi.org/10.1137/110836201
Chu, J., Engquist, B., Prodanovic, M., Tsai, R.: A multiscale method coupling network and continuum models in porous media Ii: single-and two-phase flows. In: Advances in Applied Mathematics, Modeling, and Computational Science, pp. 161–185. Springer (2013)
Weinan, E., Engquist, B.: The heterogeneous multiscale methods. Comm. Math. Sci. 1(1), 87–132 (2003)
Ewing, R.E., Lazarov, R.D., Lyons, S.L., Papavassiliou, D.V., Pasciak, J., Qin, G.: Numerical well model for non-Darcy flow through isotropic porous media. Comput. Geosci. 3(3-4), 185–204 (1999). https://doi.org/10.1023/A:1011543412675. papers2://publication/uuid/94680680-1A1A-4E3C-BDDE-D9182D7F93EC
Faigle, B., Helmig, R., Aavatsmark, I., Flemisch, B.: Efficient multiphysics modelling with adaptive grid refinement using a MPFA method. Comput. Geosci. 26, 625–636 (2014). https://doi.org/10.1007/s10596-014-9407-1
Hornung, U.: Homogenization and Porous Media, vol. 6. Springer, Berlin (1997)
Iliev, O., Kirsch, R., Lakdawala, Z., Rief, S., Steiner, K.: Currents in Industrial Mathematics. Springer, Berlin (2015). https://doi.org/10.1007/978-3-662-48258-2
Lao, H.W., Neeman, H.J., Papavassiliou, D.V.: A pore network model for the calculation of non-Darcy flow coefficients in fluid flow through porous media. Chem. Eng. Commun. 191(10), 1285–1322 (2004). https://doi.org/10.1080/00986440490464200
Logg, A., Mardal, K., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method Lecture Notes in Computational Science and Engineering, vol. 84. Springer, Berlin (2012). https://doi.org/10.1007/978-3-642-23099-8
Macini, P., Mesini, E., Viola, R.: Laboratory measurements of non-Darcy flow coefficients in natural and artificial unconsolidated porous media. J. Pet. Sci. Eng. 77(3-4), 365–374 (2011). https://doi.org/10.1016/j.petrol.2011.04.016
Mattila, K., Puurtinen, T., Hyvȧluoma, J., Surmas, R., Myllys, M., Turpeinen, T., Robertsėn, F., Westerholm, J., Timonen, J.: A prospect for computing in porous materials research: Very large fluid flow simulations. J. Comput. Sci. 12, 62–76 (2016). https://doi.org/10.1016/j.jocs.2015.11.013
Mavis, F., Wilsey, E.: A study of the permeability of sand. University of Iowa Studies in Engineering (Bulletin 7), pp. 1–29. http://ir.uiowa.edu/cgi/viewcontent.cgi?article=1007&context=uisie (1936)
McClure, J.E., Gray, W.G., Miller, C.T.: beyond anisotropy: examining Non-Darcy flow in asymmetric porous media. Transp. Porous Media 84(2), 535–548 (2010). https://doi.org/10.1007/s11242-009-9518-7
Mehmani, Y., Balhoff, M.T.: Bridging from pore to continuum: a hybrid mortar domain decomposition framework for subsurface flow and transport. Multiscale Model. Simul. 12 (2), 667–693 (2014). https://doi.org/10.1137/13092424X
Narsilio, G.A., Buzzi, O., Fityus, S., Yun, T.S., Smith, D.W.: Upscaling of Navier-Stokes equations in porous media: theoretical, numerical and experimental approach. Comput. Geotech. 36(7), 1200–1206 (2009). https://doi.org/10.1016/j.compgeo.2009.05.006
Ovaysi, S., Piri, M.: Direct pore-level modeling of incompressible fluid flow in porous media. J. Comput. Phys. 229(19), 7456–7476 (2010). https://doi.org/10.1016/j.jcp.2010.06.028
Oyewole, E., Garcia, A.P., Heidari, Z.: A new method for assessment of directional permeability and conducting pore network using electric conductance in porous media. In: SPWLA 57th Annual Logging Symposium (2016)
Peszynska, M., Trykozko, A.: Pore-to-core simulations of flow with large velocities using continuum models and imaging data. Comput. Geosci. 17(4), 623–645 (2013). https://doi.org/10.1007/s10596-013-9344-4
Peszynska, M., Trykozko, A., Sobieski, W.: Forchheimer law in computational and experimental studies of flow through porous media at porescale and mesoscale. Math. Sci. Appl. 32, 463–482 (2010)
Ramstad, T., Idowu, N., Nardi, C., Øren, P.E.: Relative permeability calculations from two-phase flow simulations directly on digital images of porous rocks. Transp. Porous Media 94(2), 487–504 (2012). https://doi.org/10.1007/s11242-011-9877-8
Santillȧn, D., Mosquera, J.C., Cueto-felgueroso, L.: Fluid-driven fracture propagation in heterogeneous media: Probability distributions of fracture trajectories. Phys. Rev. E. 96(5), 1–10 (2017). https://doi.org/10.1103/PhysRevE.96.053002
Scheibe, T.D., Perkins, W.A., Richmond, M.C., McKinley, M.I., Romero-Gomez, P.D.J., Oostrom, M., Wietsma, T.W., Serkowski, J.A., Zachara, J.M.: Pore-scale and multiscale numerical simulation of flow and transport in a laboratory-scale column. Water Resour. Res. 51(2), 1023–1035 (2015). https://doi.org/10.1002/2014WR015959
Sheng, Q., Thompson, K.: Dynamic coupling of pore-scale and reservoir-scale models for multiphase flow. Water Resour. Res. 49(9), 5973–5988 (2013). https://doi.org/10.1002/wrcr.20430
Skjetne, E., Auriault, J.L.: New insights on steady, non-linear flow in porous media. Eur. J. Mech. - B/Fluids 18(1), 131–145 (1999). https://doi.org/10.1016/S0997-7546(99)80010-7
Skjetne, E., Kløv, T., Gudmundsson, J.: Experiments and modeling of high-velocity pressure loss in sandstone fractures. SPE Journal (March):61–70. https://doi.org/10.2118/69676-PA (2001)
Sun, T., Mehmani, Y., Balhoff, M.T.: Hybrid multiscale modeling through direct substitution of pore-scale models into near-well reservoir simulators. Energy Fuels 18(5), 5828–5836 (2012). https://doi.org/10.1021/ef301003b
Swift, G.W., Kiel, O.G.: The prediction of gas-well performance including the effect of non-Darcy flow. J. Petrol. Tech. 14(07), 791–798 (1962)
Tomin, P., Lunati, I.: Hybrid multiscale finite volume method for two-phase flow in porous media. J. Comput. Phys. 250, 293–307 (2013). https://doi.org/10.1016/j.jcp.2013.05.019. http://linkinghub.elsevier.com/retrieve/pii/S0021999113003513
Tomin, P., Lunati, I.: Spatiotemporal adaptive multiphysics simulations of drainage-imbibition cycles. Comput. Geosci. 20(3), 541–554 (2016). https://doi.org/10.1007/s10596-015-9521-8
Wang, Y., Li, X., Zheng, B., Zhang, Y.X., Li, G.F., Wu, Y.F.: Experimental study on the non-Darcy flow characteristics of soil-rock mixture. Environ. Earth Sci. 75(9), 756 (2016). https://doi.org/10.1007/s12665-015-5218-5
Whitaker, S.: Flow in porous media {II}: the governing equations for immiscible, two-phase flow. Transp. Porous Med. 1, 105–125 (1986)
Whitaker, S.: The forchheimer equation: a theoretical development. Transp. Porous Media 25, 27–61 (1996)
Wildenschild, D., Sheppard, A.P.: X-ray imaging and analysis techniques for quantifying pore-scale structure and processes in subsurface porous medium systems. Adv. Water Resour. 51, 217–246 (2013). https://doi.org/10.1016/j.advwatres.2012.07.018
Zaretskiy, Y., Geiger, S., Sorbie, K., Fȯrster, M.: Efficient flow and transport simulations in reconstructed 3D pore geometries. Adv. Water Resour. 33(12), 1508–1516 (2010). https://doi.org/10.1016/j.advwatres.2010.08.008
Zeng, Z., Grigg, R.: A criterion for non-darcy flow in porous media. Transp. Porous Media 63(1), 57–69 (2006). https://doi.org/10.1111/j.1439-0523.2006.01169.x
Acknowledgments
SA acknowledges the Research Council of Norway and the industry partners; ConocoPhillips Skandinavia AS, BP Norge AS, Det Norske Oljeselskap AS, Eni Norge AS, Maersk Oil Norway AS, DONG Energy A/S, Denmark, Statoil Petroleum AS, ENGIE E&P NORGE AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS of The National IOR Centre of Norway for support. SA also thanks Anna Kvashchuk, Trine Solberg Mykkeltvedt and Kundan Kumar for helpful discussions during the preparation of the manuscript. The authors thank the reviewers for constructive criticism that helped to improve and better frame the paper.
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Appendix: Numerical validation of the fully-discrete method
Appendix: Numerical validation of the fully-discrete method
In this appendix, we validate newly formed MPFA-based fully discrete CVHMM and highlight its superiority compared to two-point multiscale methods. The method falls within the HMM family and thus assumes scale separation. The method is not designed to approximate the full fine-scale solution, but to use the fine-scale information for estimation of the missing coarse-scale relations consistently. Thus, the numerical examples study the error propagation to the coarse scale.
We consider tests on two different pore geometries, which allow us to highlight different properties of the method. The first test considers a pore structure from [9] which created challenges for two-point method due to orientation of its upscaled permeability tensor. In the second test, we focus on two-scale convergence under refinement of both the coarse and the fine grid.
1.1 Consistency on arbitrary porous media
It is well known [1] that TPFA is convergent only if the principal axis of the permeability coincide with the principal directions of the grid. As shown in [9], this inconsistency carries over to HMM-approaches based on the two-point flux. The MPFA methods were developed to overcome the consistency errors of TPFA, and the purpose of this test is to verify that a similar improvement can be seen for the corresponding multiscale methods (see schematics in Fig. 2). To that end, we revisit the experiment presented in [9]. We consider a randomly generated pore structure (see Fig. 1) which is extended periodically to form the fine scale of the whole computational domain. Based on the homogenization theory [26], the coarse linear problem can be defined in the closed form as
We benchmark the numerical method by comparing a coarse-scale numerical solution on the unit square to a chosen function of the form:
To compute the source term f corresponding to the analytical solution (21), we first estimate A by solving the cell problem from Fig. 1 numerically on a highly refined fine-scale grid (3 uniform refinements), and then substitute A and Eqs. 20 and 21.
The initial coarse-scale grid has a resolution of H = 1/4 and fine grid starts with triangulation shown in Fig. 1b; we then consider uniform refinements on the coarse and the fine scale. Due to linearity of the system, the period ε does not play an explicit role in the error, but the size of fine-scale discretization should be understood relative to the period.
The comparison of the experimental orders of convergence with respect to coarse-grid resolutions for both two-point flux method and the multi-point flux CVHMM are presented in Table 2. While the two-point method suffers from inconsistency errors and fails to converge the multi-point method achieves the order of convergence of around 1.9 after only two fine-scale refinements. Table 2 gives a clear indication of the consistency errors of TPFA-based method, that can be seen in rows 3–5. Regardless of the fine-scale refinement, the final error stays almost the same and method stops converging.
On contrary, for the case of MPFA-based method, two h-refinements is sufficient (the last two columns in Table 2) to reach the optimal order of convergence with respect to considered coarse scale grids.
The observed order of convergence with respect to the coarse scale discretization is close to the second order observed for single-scale MPFA methods with a discontinuous right hand side [3]. The L2 norm of the error has the same qualitative behavior and is omitted for brevity.
We note that for single-scale problems, the consistency errors in the two-point flux to some extent can be overcome by orienting the grid along the principal axes of the permeability (see, e.g., [1]). Such techniques are however less viable in two-scale simulations, where the equivalent permeability is not known a priori.
1.2 Two-scale convergence
The pore structure of Fig. 1 requires a highly resolved grid to conform to the geometry. As a result, the error from the solution of cell problems gives only marginal contribution to the overall error after the first h-refinement.
Here, we consider an example from Fig. 2 where a coarse-scale grid is aligned with the xy-axis of the fine-scale unit cell, which has simple diagonal grain arrangement
that is purposely misaligned with the coarse geometry. Moreover, the fine-scale geometry can be meaningfully meshed by a relatively coarse grid. This leaves room for multiple grid refinements on the fine scale and allows us to study the interplay between coarse- and fine-scale convergence.
Again, we consider the linear coarse-scale (20). The numerical errors are computed as the difference of fully,discrete numerical solutions and the chosen analytical expression on the unit square:
The right hand side in Eq. 20 in this example is computed as follows. First the tensor A is estimated by numerical upscaling: we solve fine-scale cell problem numerically on a very fine grid (after four refinements). Then, the semi-analytical source term f is computed knowing P and A, that can be written as:
We note that the analytical solution is smooth, and we therefore expect high rates of convergence.
To study the interaction between the errors, we refine both the coarse- and fine-scale grid and compare the approximated pressure solutions with the analytical solution (22) in the H1 semi-norm (measured in terms of the coarse grid). Note again, that choosing L2 norm yields the same qualitative behavior. Table 3 shows the experimental order of convergence of the H1-seminorm (convergence of the pressure gradients) as both the coarse- and the fine-scale grid are refined. As can be seen from the table, both the fine- and coarse discretization errors contributes to the overall accuracy, i.e., the expected order of convergence with respect to each scale is only achieved when the other scale is sufficiently refined.
These results are in perfect agreement with the error analysis for the two-point flux method in [9].
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Alyaev, S., Keilegavlen, E. & Nordbotten, J.M. A heterogeneous multiscale MPFA method for single-phase flows in porous media with inertial effects. Comput Geosci 23, 107–126 (2019). https://doi.org/10.1007/s10596-018-9787-8
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DOI: https://doi.org/10.1007/s10596-018-9787-8