Abstract
A technique for locally rescaling (upscaling) the functions of the relative phase permeability (RPP) has been developed, which minimizes the error in the approximating the phase filtration rates for the superelement modeling of waterflooding a layered heterogeneous oil reservoir. The RPP is locally upscaled for each superelement based on the solution of two-dimensional two-phase filtration problems on a refined computational grid. Modified RPP functions (MFRPPs) are represented in the parametric form; i.e., the values of the parameters are sought when solving the problem of minimizing the deviations of the averaged and approximated phase velocities at the sites corresponding to the faces of the superelement. The efficiency of applying MFRPP to superelement modeling is illustrated by an example of a model reservoir region where oil is extracted using injection and production wells and by an example of waterflooding at a real oilfield.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Motion of Liquids and Gases in Natural Strata (Nedra, Moscow, 1984) [in Russian].
A. B. Mazo and D. V. Bulygin, “Super elements: new approach towards oil field development modeling,” Neft. Gaz. Novatsii, No. 11, 6–8 (2011).
A. B. Mazo, K. A. Potashev, E. I. Kalinin, and D. V. Bulygin, “Oil reservoir simulation with the superelement method,” Mat. Model. 25 (8), 51–64 (2013).
A. Mazo, K. Potashev, and E. Kalinin, “Petroleum reservoir simulation using Super Element Method,” Proc. Earth Planet. Sci. 15, 482–487 (2015).
L. G. Strakhovskaya and R. P. Fedorenko, “A version of the finite element method,” USSR Comput. Math. Math. Phys. 19, 162–173 (1979). 90164-2 doi 10.1016/ 0041-5553(79)
Z. I. Burman, O. M. Aksenov, V. I. Lukashenko, and M. T. Timofeev, Superelement Calculation of Stiffened Shells (Mashinostroenie, Moscow, 1982) [in Russian].
The Method of Superelements in Calculations of Engineering Structures, Ed. by V. A. Postnov (Sudostroenie, Leningrad, 1979) [in Russian].
A. V. Akhmetzyanov and V. N. Kulibanov, “Nontraditional mathematical models of fluid filtration on porous media,” Autom. Remote Control 65, 1177–1188 (2004).
Y. Efendiev, V. Ginting, T. Hou, and R. Ewing, “Accurate multiscale finite element methods for two-phase flow simulations,” J. Comput. Phys. 220, 155–174 (2006).
A. Kh. Pergament, V. A. Semiletov, and P. Yu. Tomin, “On some multiscale algorithms for sector modeling in multiphase flow in porous media,” Math. Models Comput. Simul. 3, 365–374 (2010).
X. H. Wu, Y. Efendiev, and T. Y. Hou, “Analysis of upscaling absolute permeability,” Discrete Continuous Dyn. Syst., Ser. B 2, 185–204 (2002).
L. J. Durlofsky, “Coarse scale models of two phase flow in heterogeneous reservoirs: volume averaged equations and their relationship to existing upscaling techniques,” Comput. Geosci. 2, 73–92 (1998).
H. Dykstra and R. Parsons, “The prediction of oil recovery by waterflooding,” in Secondary Oil Recovery of Oil in the United States, 2nd ed. (API, 1950), pp. 160–174.
D. V. Bulygin and V. Ya. Bulygin, Geology and Development Imitation of Oil Reservoirs (Nedra, Moscow, 1996) [in Russian].
A. K. Kurbanov, “On some generalization of equations of two-phase liquid filtering,” Nauch.-Tekh. Sb. VNII 15, 32–38 (1961).
K. H. Coats, J. R. Dempsey, and J. H. Henderson, “The use of vertical equilibrium in two-dimensional simulation of three-dimensional reservoir performance,” Soc. Pet. Eng. J. 11, 63–71 (1971).
C. L. Hearn, “Simulation of stratified waterflooding by pseudo relative permeability curves,” J. Pet. Technol. 23, 805–813 (1971).
G. E. Pickup, P. S. Ringrose, and A. Sharif, “Steady-state upscaling: from lamina-scale to full-field model,” Soc. Pet. Eng. J. 5, 208–217 (2000).
S. Ekrann and J. O. Aasen, “Steady-state upscaling,” Transport Porous Media 41, 245–262 (2000).
S. E. Gasda and M. A. Celia, “Upscaling relative permeabilities in a structured porous medium,” Adv. Water Res. 28, 493–506 (2005).
H. H. Jacks, O. J. E. Smith, and C. C. Mattax, “The modelling of a three-dimensional reservoir with a twodimensional reservoir simulator- the use of dynamic pseudo functions,” Soc. Pet. Eng. J. 13, 175–185 (1973).
J. R. Kyte and D. W. Berry, “New pseudofunctions to control numerical dispersion,” Soc. Pet. Eng. J. 15, 269–276 (1975).
J. W. Barker and F. J. Fayers, “Transport coefficients for compositional simulation with coarse grids in heterogeneous media,” Soc. Pet. Eng. Adv. Technol. Ser. 2, 103–112 (1994).
S. P. Rodionov and L. N. Orekhova, “Determination of modified relative permeabilities at transformation of the geological model to the hydrodynamic one. Part 1,” Neft Gaz, No. 6, 12–17 (2008).
R. Eymard, T. Gallouet, and R. Herbin, in Finite Volume Methods, Handbook of Numerical Analysis, Ed. by Ph. Ciarlet and J. L. Lions (North Holland, Amsterdam, 2000), pp. 713–1020.
T. A. Hewett and T. Yamada, “Theory for the semi-analytical calculation of oil recovery and effective relative permeabilities using streamtubes,” Adv. Water Res. 20, 279–292 (1997).
X. H. Wen, L. J. Durlofsky, and M. G. Edwards, “Upscaling of channel systems in two dimensions using flowbased grids,” Transport Porous Media 51, 343–366 (2003).
A. H. Zarifi, E. Sahraei, and H. Parvizi, “Pseudo relative permeability compensation for numerical dispersion,” Pet. Sci. Technol. 30, 1529–1538 (2012).
K. A. Potashev, “Upscaling of relative phase permeabilities in an isolated stratified reservoir,” Uch. Zap. Kazan. Univ., Fiz.-Mat. Nauki 156, 120–134 (2014).
J. Kozeny, Über kapillare Leitung des Wassers im Boden: Aufstieg, Versickerung und Anwendung auf die Bewaesserung (Sitzungsber. Akad. Wissensch., Wien, 1927), Vol. 136, pp. 271–306.
H. Daigle and B. Dugan, “Extending NMR data for permeability estimation in fine-grained sediments,” Marine Pet. Geol. 26, 1419–1427 (2009).
Y. Yang and A. C. Aplin, “Permeability and petrophysical properties of 30 natural mudstones,” J. Geophys. Res. B: Solid Earth 112 (3), 1–14 (2007).
M. A. Christie and P. J. Clifford, “A fast procedure for upscaling in compositional simulation,” Soc. Pet. Eng. J. 3, 272–278 (1998).
P. Dupouy, J. W. Barker, and J.-P. Valois, “Grouping pseudo relative permeability curves,” In Situ 22, 1–33 (1998).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.B. Mazo, K.A. Potashev, 2017, published in Matematicheskoe Modelirovanie, 2017, Vol. 29, No. 3, pp. 81–94.
Rights and permissions
About this article
Cite this article
Mazo, A.B., Potashev, K.A. Upscaling relative phase permeability for superelement modeling of petroleum reservoir engineering. Math Models Comput Simul 9, 570–579 (2017). https://doi.org/10.1134/S207004821705009X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S207004821705009X