A pseudospectral approach to the McWhorter and Sunada equation for two-phase flow in porous media with capillary pressure
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Two well-known mathematical solutions for two-phase flow in porous media are the Buckley–Leverett equation and the McWhorter and Sunada equation (MSE). The former ignores capillary pressure and can be solved analytically. The latter has traditionally been formulated as an iterative integral solution, which suffers from convergence problems as the injection saturation approaches unity. Here, an alternative approach is presented that solves the MSE using a pseudospectral Chebyshev differentiation matrix. The resulting pseudospectral solution is compared to results obtained from the original integral implementation and the Buckley–Leverett limit, when the capillary pressure becomes negligible. A self-contained MATLAB code to implement the new solution is provided within the manuscript. The new approach offers a robust and accurate method for verification of numerical codes solving two-phase flow with capillary pressure.
KeywordsAnalytical solutions Two-phase flow Porous media Pseudospectral Differentiation matrix Chebyshev Capillary pressure
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- 1.Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions: with formulas, graphs, and mathematical tables. Courier Dover Publications (1964)Google Scholar
- 2.Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover (2001)Google Scholar
- 3.Buckley, S.E., Leverett, M.C.: Mechanism of fluid displacement in sands. Trans. AIME 146(1), 107–116 (1942)Google Scholar
- 7.Mathias, S.A., Gluyas, J., Gonzalez, G., Hosseini, S.A.: Role of partial miscibility on pressure buildup due to constant rate injection of CO2 into closed and open brine aquifers. Water Resour. Res. 47(12) (2011). doi: 10.1029/2011WR011051
- 9.Orr, F.M.J.: Theory of Gas Injection Processes. Tie-Line Publications (2007)Google Scholar
- 11.Piche, R., Kanniainen, J.: Matrix-based numerical modelling of financial differential equations. Int. J. Math. Model. Numer. Optim. 1(1/2), 88–100 (2009)Google Scholar
- 12.van Reeuwijk, M., Mathias, S.A., Simmons, C.T., Ward, J.D.: Insights from a pseudospectral approach to the elder problem. Water Resour. Res. 45 (2009)Google Scholar
- 13.Schmid, K.S., Geiger, S.: Universal scaling of spontaneous imbibition for water-wet systems. Water Resour. Res. 48 (2012)Google Scholar
- 14.Schmid, K.S., Geiger, S., Sorbie, K.S.: Semianalytical solutions for cocurrent and counter countercurrent imbibition and dispersion of solutes in immiscible two-phase flow. Water Resour. Res. W02550 (2011)Google Scholar
- 16.Trefethen, L.N.: Spectral Methods in MATLAB Society for Industrial and Applied Mathematics (SIAM). Philadelphia (2000)Google Scholar
- 17.Weideman, J.A.C., Reddy, S.C.: A matlab differentiation matrix suite. ACM Trans. Math. Softw. 24(4), 465–519 (2000). the codes are available at http://dip.sun.ac.za/~weideman/research/differ.html CrossRefGoogle Scholar