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Comparisons of FV-MHMM with Other Finite Volume Multiscale Methods

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Abstract

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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Acknowledgements

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.

Author information

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
figure11

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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Keywords

  • Upscaling
  • Multiscale method
  • Finite volume
  • Heterogeneity