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Comparisons of FV-MHMM with Other Finite Volume Multiscale Methods

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Upscaling and multiscale methods in reservoir engineering remain a complicated task especially when dealing with heterogeneities. In this study, we focus on flow field problem with a Darcy’s equation considered at the fine scale. The main difficulty is then to obtain an accurate description of the flow behavior by using multiscale methods available in petroleum engineering. We cross-compare three of the main finite volume formulations: multiscale finite volume method (MsFv), multiscale restriction smoothed (MsRSB) and a new finite volume method, FV-MHMM. Comparisons are done in terms of accuracy to reproduce the fine scale behavior.

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  1. Aarnes, J.E.: On the use of a mixed multiscale finite element method for greaterflexibility and increased speed or improved accuracy in reservoir simulation. Multiscale Model. Simul. 2(3), 421–439 (2004)

  2. Arbogast, T.: Implementation of a locally conservative numerical subgrid upscaling scheme for two-phase Darcy flow. Comput. Geosci. 6(3–4), 453–481 (2002)

  3. Arbogast, T., Cowsar, L.C., Wheeler, M.F., Yotov, I.: Mixed finite element methods on nonmatching multiblock grids. SIAM J. Numer. Anal. 37(4), 1295–1315 (2000)

  4. Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model. Simul. 6(1), 319–346 (2007)

  5. Bosma, S., Hajibeygi, H., Tene, M., Tchelepi, H.A.: Multiscale finite volume method for discrete fracture modeling on unstructured grids (ms-dfm). J. Comput. Phys. 351, 145–164 (2017)

  6. Castelletto, N., Hajibeygi, H., Tchelepi, H.A.: Multiscale finite-element method for linear elastic geomechanics. J. Comput. Phys. 331, 337–356 (2017)

  7. Christie, M., Blunt, M., et al.: Tenth spe comparative solution project: a comparison of upscaling techniques. In: SPE Reservoir Simulation Symposium, Society of Petroleum Engineers (2001)

  8. Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (gmsfem). J. Comput. Phys. 251, 116–135 (2013)

  9. Franc, J., Jeannin, L., Debenest, G., Masson, R.: Fv-mhmm method for reservoir modeling. Comput. Geosci. 21(5–6), 895–908 (2017)

  10. Hajibeygi, H., Jenny, P.: A general multiscale finite-volume method for compressible multiphase flow in porous media. In: ECMOR XI-11th European Conference on the Mathematics of Oil Recovery (2008)

  11. Hajibeygi, H., Jenny, P.: Adaptive iterative multiscale finite volume method. J. Comput. Phys. 230(3), 628–643 (2011)

  12. Hajibeygi, H., Bonfigli, G., Hesse, M.A., Jenny, P.: Iterative multiscale finite-volume method. J. Comput. Phys. 227(19), 8604–8621 (2008)

  13. Hajibeygi, H., Lunati, I., Lee, S.H.: Error estimate and control in the msfv method for multiphase flow in porous media. In: Proceedings of XVIII International Conference on Computational Methods in Water Resources (CMWR XVIII), Barcelona, Spain (2010)

  14. Harder, C., Paredes, D., Valentin, F.: A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys. 245, 107–130 (2013)

  15. Harder, C., Paredes, D., Valentin, F.: On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogeneous coefficients. Multiscale Model. Simul. 13(2), 491–518 (2015)

  16. Hilden, S.T., Møyner, O., Lie, K.A., Bao, K.: Multiscale simulation of polymer flooding with shear effects. Transp. Porous Media 113(1), 111–135 (2016)

  17. Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

  18. Jenny, P., Lunati, I.: Modeling complex wells with the multi-scale finite-volume method. J. Comput. Phys. 228(3), 687–702 (2009)

  19. Jenny, P., Lee, S., Tchelepi, H.A.: Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. J. Comput. Phys. 187(1), 47–67 (2003)

  20. Lee, S.H., Zhou, H., Tchelepi, H.A.: Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations. J. Comput. Phys. 228(24), 9036–9058 (2009)

  21. Lie, K.A.: An introduction to reservoir simulation using matlab: user guide for the matlab reservoir simulation toolbox (mrst). sintef ict (2014)

  22. Lunati, I., Jenny, P.: The multiscale finite volume method. In: 10th European Conference on the Mathematics of Oil Recovery (ECMOR X) (2006a)

  23. Lunati, I., Jenny, P.: Multiscale finite-volume method for compressible multiphase flow in porous media. J. Comput. Phys. 216(2), 616–636 (2006b)

  24. Lunati, I., Jenny, P.: Multiscale finite-volume method for density-driven flow in porous media. Comput. Geosci. 12(3), 337–350 (2008)

  25. Lunati, I., Lee, S.H.: An operator formulation of the multiscale finite-volume method with correction function. Multiscale Model. Simul. 8(1), 96–109 (2009)

  26. Lunati, I., Tyagi, M., Lee, S.: Multiscale finite volume method for reservoir simulation, Patent (2009)

  27. Lunati, I., Tyagi, M., Lee, S.H.: An iterative multiscale finite volume algorithm converging to the exact solution. J. Comput. Phys. 230(5), 1849–1864 (2011)

  28. Manea, A.M., Sewall, J., Tchelepi, H.A., et al.: Parallel multiscale linear solver for highly detailed reservoir models. SPE J. 21(06), 2–62 (2016)

  29. Møyner, O.: Construction of multiscale preconditioners on stratigraphic grids. In: ECMOR XIV-14th European Conference on the Mathematics of Oil Recovery (2014)

  30. Møyner, O., Lie, K.A.: A multiscale two-point flux-approximation method. J. Comput. Phys. 275, 273–293 (2014)

  31. Møyner, O., Lie, K.A.: A multiscale restriction-smoothed basis method for high contrast porous media represented on unstructured grids. J. Comput. Phys. 304, 46–71 (2016)

  32. Moyner, O., Lie, K., Natvig, J.: Adaptive multiscale multi-fidelity reservoir simulation, Patent (2015)

  33. Møyner, O., Lie, K.A., et al.: A multiscale restriction-smoothed basis method for compressible black-oil models. SPE J. 21(06), 2–079 (2016)

  34. Møyner, O., Tchelepi, H., et al: A multiscale restriction-smoothed basis method for compositional models. In: SPE Reservoir Simulation Conference, Society of Petroleum Engineers (2017)

  35. Raviart, P.-A., Thomas, J.M.: Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput. 31(138), 391–413 (1977a). http://www.ams.org/publications/journals/journalsframework/mcom

  36. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods, pp. 292–315. Springer (1977b)

  37. Shah, S., Møyner, O., Tene, M., Lie, K.A., Hajibeygi, H.: The multiscale restriction smoothed basis method for fractured porous media (f-msrsb). J. Comput. Phys. 318, 36–57 (2016)

  38. Ţene, M., Wang, Y., Hajibeygi, H.: Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media. J. Comput. Phys. 300, 679–694 (2015)

  39. Ţene, M., Al Kobaisi, M.S., Hajibeygi, H.: Algebraic multiscale method for flow in heterogeneous porous media with embedded discrete fractures (f-ams). J. Comput. Phys. 321, 819–845 (2016)

  40. Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56(3), 179–196 (1996)

  41. Wang, Y., Hajibeygi, H., Tchelepi, H.A.: Algebraic multiscale solver for flow in heterogeneous porous media. J. Comput. Phys. 259, 284–303 (2014)

  42. Zhou, H., Tchelepi, H.A., et al.: Two-stage algebraic multiscale linear solver for highly heterogeneous reservoir models. SPE J. 17(02), 523–539 (2012)

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The authors would like to thank ENGIE and STORENGY for the financial support of J. Franc. INP Toulouse is also acknowledged for the travel grant, BQR SMI, of J. Franc. We also thank Prof. Margot Gerritsen, ICME-Stanford, for all the advices and discussions on the multiscale finite volume methods. The authors also want to thank the editors for their understanding during the submission process and the final publication.

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Correspondence to Gerald Debenest.

A Complexity Comparison

A Complexity Comparison

Following the steps which are written in Jenny et al. (2003) for complexity analysis of MsFv multiscale algorithm, we remind here the results for FV-MHMM (Franc et al. 2017) and proposed the same kind of analysis for MsRSB. The notations introduced are presented in Table 8.

Table 8 Parameters

Assuming that \(t_{1}(n)\sim ct_{\mathrm{m}}n^{\alpha }\) where \(t_{\mathrm{m}}\) is the time spent for one multiplication, c and \(\alpha \) are constants depending on the solvers. This complexity analysis neglects time spent in the reconstruction of the pressure and the fluxes. For MsFv, reproducing the analysis of Jenny et al. (2003), it leads to:

$$\begin{aligned} t_{\mathrm{MsFv1}}&\approx N_{\mathrm{n}}a_{\mathrm{n}}ct_{\mathrm{m}}\gamma ^{\alpha },\\ t_{\mathrm{MsFv2}}&\approx N_{\mathrm{v}}(a_{v}+1)ct_{\mathrm{m}}\gamma ^{\alpha },\\ t_{\mathrm{MsFv3}}&\approx ct_{\mathrm{m}}N_{\mathrm{v}}^{\alpha } \end{aligned}$$

with the step MsFv1, the time spent in constructing the basis function on the dual grid; MsFv2, the time spent building coarse scale equivalent transmissivities on the coarse grid and MsFv3, the time needed to invert the coarse system.

In the same manner, FV-MHMM can be cast into three main steps:

$$\begin{aligned} t_{\mathrm{LLP}}&\approx 2d(l+1)N_{\mathrm{v}}ct_{\mathrm{m}}\gamma ^{\alpha },\\ t_{\mathrm{LSP}}&\approx N_{\mathrm{v}}ct_{\mathrm{m}}\gamma ^{\alpha },\\ t_{\mathrm{GP}}&\approx ct_{\mathrm{m}}[N_{\mathrm{v}}+(l+1)N_{\mathrm{f}}]^{\alpha } \end{aligned}$$

where d is the dimension number, l the order of the polynomial space used to approximate \(\Lambda _{H}\) and considering an average number of faces as \(N_{\mathrm{f}}\approx d(N_{\mathrm{v}}^{1/d}+1){\mathop {\prod }\limits _{i=1}^{d-1}}N_{\mathrm{v}}^{1/d}\). The LLP step denotes the time spent in the construction of Lagrange multipliers-related basis functions; LSP, the time spent in the construction of source term-related basis functions and GP the time spent inverting the global problem.

Table 9 Counting for MsRSB overlap

For the MsRSB, it has to be noted that its basis functions computations do not require inverting any operator. Indeed, in the Jacobi smoother formulation, only diagonal preconditioner \(D^{-1}\) is required, which is trivially inverted. It left us with:

$$\begin{aligned} t_{\mathrm{MsRSB1}}&\approx n_{\mathrm{it}}(\gamma _{\mathrm{overlap}}+\gamma )N_{\mathrm{v}}t_{m}\\ t_{\mathrm{MsRSB2}}&\approx n_{\mathrm{it}}\gamma _{\mathrm{overlap}}N_{\mathrm{v}}t_{\mathrm{m}}\\ t_{\mathrm{MsRSB3}}&\approx ct_{m}N_{\mathrm{v}}^{\alpha } \end{aligned}$$

with the step MsRSB1 stands for the time spent in smoothing the basis functions on the overlapping supports; MsRSB2 represents the time spent restricting the spreading of the basis functions to ensure mass conservation and, as in MsFv, MsRSB3, the time needed to invert the coarse system. Estimating the overlap size is done counting nodes \(nu_{\mathrm{n}}\), edges \(nu_{\mathrm{e}}\) and faces \(nu_{\mathrm{f}}\) of a coarse cell depending on d (see Table 9). To define \(\gamma _{\mathrm{overlap}}\), let us introduce \(\overline{nc}\) the mean number of fine cells per dimension embedded in a coarse cell, that is do say \(\overline{nc}\approx \gamma ^{1/d}\).Then, the overlapping area is counted as:

$$\begin{aligned} \gamma _{\mathrm{overlap}}&=nu_{\mathrm{f}} \overline{nc}^{(d-1)}\left( \frac{\overline{nc}}{2}\right) \\&\quad +nu_{\mathrm{e}} \overline{nc}^{(d-2)}\left( \frac{\overline{nc}}{2}\right) ^{2}\\&\quad +nu_{\mathrm{n}} \left( \frac{\overline{nc}}{2}\right) ^{d} \end{aligned}$$

Moreover, the step MsFv1 and step MsFv2 are reported to be parallelizable for the MsFv method (Jenny et al. 2003). As for the FV-MHMM, all the local problems lambda (LLP) and local problems source (LSP) are independent from each others and therefore can be treated in parallel. Moreover, the TPFA matrix for a selected coarse cell has to be generated only one time and can be used for solving both LLP and LSP associated with this coarse cell (see Fig. 11).

Fig. 11

Complexity plot for Table 8 parameters. From left to right, the first region stands for MsFv, the second for FV-MHMM, the third for MsRSB. The right most region represents the total amount of computational time spent for each method

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Franc, J., Debenest, G., Jeannin, L. et al. Comparisons of FV-MHMM with Other Finite Volume Multiscale Methods. Transp Porous Med 125, 151–171 (2018). https://doi.org/10.1007/s11242-018-1111-5

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  • Upscaling
  • Multiscale method
  • Finite volume
  • Heterogeneity