Pressure/Saturation System for Immiscible Two-Phase Flow: Uniqueness Revisited

Author(s)
Koffi B. Fadimba

ABSTRACT

We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct "mobilities" (or mobility-like functions) that satisfy the new condition (when the parameter is 2). For the constructed "mobilities", we do graphical experiments that show, empirically, that this condition could be satisfied for other values of . These empirical experiments indicate that the usual smoothness condition on the fractional flow function (and on the total mobility), for uniqueness and convergence, might not be necessary. This condition is also sufficient for the convergence of a family of perturbed problems to the original pressure/saturation problem.

We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct "mobilities" (or mobility-like functions) that satisfy the new condition (when the parameter is 2). For the constructed "mobilities", we do graphical experiments that show, empirically, that this condition could be satisfied for other values of . These empirical experiments indicate that the usual smoothness condition on the fractional flow function (and on the total mobility), for uniqueness and convergence, might not be necessary. This condition is also sufficient for the convergence of a family of perturbed problems to the original pressure/saturation problem.

KEYWORDS

Porous Medium, Uniqueness of a Solution, Degenerate Equation, Immiscible Two-Phase Flow, Regularization, Phase Mobility.

Porous Medium, Uniqueness of a Solution, Degenerate Equation, Immiscible Two-Phase Flow, Regularization, Phase Mobility.

Cite this paper

nullK. Fadimba, "Pressure/Saturation System for Immiscible Two-Phase Flow: Uniqueness Revisited,"*Applied Mathematics*, Vol. 2 No. 5, 2011, pp. 541-550. doi: 10.4236/am.2011.25071.

nullK. Fadimba, "Pressure/Saturation System for Immiscible Two-Phase Flow: Uniqueness Revisited,"

References

[1] G. Chavent and J. Jaffre, “Mathematical Models and Finite Element for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows through Porous Media,” North-Holland, New York, 1986.

[2] R. E. Ewing, “Problems Arising in the Modeling of Processes for Hydrocarbon Recovery,” In R. E. Ewing, Ed., The Mathematics of Reservoir Simulation, S.I.A.M., Philadelphia, 1983, pp. 3-34.

[3] K. B. Fadimba, “On Existence and Uniqueness for a Coupled System Modeling Immiscible flow through a Porous Medium,” Journal of Mathematical Analysis and Applications, Vol. 328, No. 2, 2007, pp. 1034-1056. doi:10.1016/j.jmaa.2006.06.012

[4] Z. J. Chen and N. L. Khlopina, “Degenerate Two-Phase Flow Problems: Error Estimates,” Communication in Applied Analysis, Vol. 5, 2001, pp. 503-521.

[5] K. B. Fadimba, “Regularization and Numerical Methods for a Class of Porous Medium Equations,” PhD Thesis, University of South Carolina, Columbia, 1993.

[6] K. B. Fadimba and R. C. Sharpley, “A Priori Estimates and Regularization for a Class of Porous Medium Equations,” Nonlinear World, Vol. 2, 1995, pp. 13-41.

[7] Z. Chen and R. E. Ewing, “Degenerate Two-Phase Incompressible Flow-III: Sharp Error Estimates,” Numerische Mathematik, Vol. 90, No. 2, 2001, pp. 215-240. doi:10.1007/s002110100291

[8] K. B. Fadimba and R. C. Sharpley, “Galerkin Finite Element Method for a Class of Porous Medium Equations,” Nonlinear Analysis: Real World Applications, Vol. 5, No. 2, 2004, pp. 355-387. doi:10.1016/j.nonrwa.2003.07.001

[9] M. E. Rose, “Numerical Methods for Flows through Porous Media-I,” Mathematics of Computations, Vol. 40, No. 162, 1983, pp. 437-467.

[10] J. Bear and A. Verruijt, “Modeling Groundwater Flow and Pollution,” D. Reidel Publication Company, Dodreich, 1987.

[11] R. E. Ewing and H. Wang, “A Summary of Numerical Methods for Time-Dependent Advection Dominated Partial Differential Equations,” Journal of Computational and Applied Mathematics, Vol. 128, No. 1-2, 2001, pp. 423-445. doi:10.1016/S0377-0427(00)00522-7

[12] Z. Chen, G. Huan, and Y Ma, “Computational Methods for Multiphase Flows in Porous Media,” SIAM Computational Science and Engineering, Philadelphia, 2006.

[13] D. L. Smylie, “A near Optimal Order Approximation to a Class of Two-Sided Nonlinear Parabolic Partial differential Equations,” PhD Thesis, University of Wyoming, Laramie, 1989.

[14] Z. Chen, “Degenerate Two-Phase Incompressible Flow-I: Existence, Uniqueness, and Regularity of a Weak Solution,” Journal of Differential Equations, Vol. 171, 2001, pp. 203-232. doi:10.1006/jdeq.2000.3848

[1] G. Chavent and J. Jaffre, “Mathematical Models and Finite Element for Reservoir Simulation: Single Phase, Multiphase and Multicomponent Flows through Porous Media,” North-Holland, New York, 1986.

[2] R. E. Ewing, “Problems Arising in the Modeling of Processes for Hydrocarbon Recovery,” In R. E. Ewing, Ed., The Mathematics of Reservoir Simulation, S.I.A.M., Philadelphia, 1983, pp. 3-34.

[3] K. B. Fadimba, “On Existence and Uniqueness for a Coupled System Modeling Immiscible flow through a Porous Medium,” Journal of Mathematical Analysis and Applications, Vol. 328, No. 2, 2007, pp. 1034-1056. doi:10.1016/j.jmaa.2006.06.012

[4] Z. J. Chen and N. L. Khlopina, “Degenerate Two-Phase Flow Problems: Error Estimates,” Communication in Applied Analysis, Vol. 5, 2001, pp. 503-521.

[5] K. B. Fadimba, “Regularization and Numerical Methods for a Class of Porous Medium Equations,” PhD Thesis, University of South Carolina, Columbia, 1993.

[6] K. B. Fadimba and R. C. Sharpley, “A Priori Estimates and Regularization for a Class of Porous Medium Equations,” Nonlinear World, Vol. 2, 1995, pp. 13-41.

[7] Z. Chen and R. E. Ewing, “Degenerate Two-Phase Incompressible Flow-III: Sharp Error Estimates,” Numerische Mathematik, Vol. 90, No. 2, 2001, pp. 215-240. doi:10.1007/s002110100291

[8] K. B. Fadimba and R. C. Sharpley, “Galerkin Finite Element Method for a Class of Porous Medium Equations,” Nonlinear Analysis: Real World Applications, Vol. 5, No. 2, 2004, pp. 355-387. doi:10.1016/j.nonrwa.2003.07.001

[9] M. E. Rose, “Numerical Methods for Flows through Porous Media-I,” Mathematics of Computations, Vol. 40, No. 162, 1983, pp. 437-467.

[10] J. Bear and A. Verruijt, “Modeling Groundwater Flow and Pollution,” D. Reidel Publication Company, Dodreich, 1987.

[11] R. E. Ewing and H. Wang, “A Summary of Numerical Methods for Time-Dependent Advection Dominated Partial Differential Equations,” Journal of Computational and Applied Mathematics, Vol. 128, No. 1-2, 2001, pp. 423-445. doi:10.1016/S0377-0427(00)00522-7

[12] Z. Chen, G. Huan, and Y Ma, “Computational Methods for Multiphase Flows in Porous Media,” SIAM Computational Science and Engineering, Philadelphia, 2006.

[13] D. L. Smylie, “A near Optimal Order Approximation to a Class of Two-Sided Nonlinear Parabolic Partial differential Equations,” PhD Thesis, University of Wyoming, Laramie, 1989.

[14] Z. Chen, “Degenerate Two-Phase Incompressible Flow-I: Existence, Uniqueness, and Regularity of a Weak Solution,” Journal of Differential Equations, Vol. 171, 2001, pp. 203-232. doi:10.1006/jdeq.2000.3848