Extraction of density-layered fluid from a porous medium

Abstract

We consider axisymmetric flow towards a point sink from a stratified fluid in a vertically confined aquifer. We present two approaches to solve the equations of flow for the linear density gradient case. Firstly, a series method results in an eigenfunction expansion in Whittaker functions. The second method is a simple finite difference method. Comparison of the two methods verifies the finite difference method is accurate, so that more complicated nonlinear, density stratification can be considered. Such nonlinear profiles cannot be considered with the eigenfunction approach. Interesting results for the case where the density stratification changes from linear to almost two-layer are presented, showing that in the nonlinear case there are certain values of flow rate for which a steady solution does not occur.

References

• Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, 9th ed. National Bureau of Standards, Washington, 1972.
• Bear, J. and Dagan, G. Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69(8):1563–1572, 1964. doi:10.1029/JZ069i008p01563
• Bear, J. Dynamics of fluids in porous media. Elsevier, New York, 1972. https://store.doverpublications.com/0486656756.html
• COMSOL Multiphysics. COMSOL Multiphysics Programming Reference Manual, version 5.3. https://doc.comsol.com/5.3/doc/com.comsol.help.comsol/COMSOL_ProgrammingReferenceManual.pdf
• Farrow, D. E. and Hocking, G. C. A numerical model for withdrawal from a two layer fluid. J. Fluid Mech. 549:141–157, 2006. doi:10.1017/S0022112005007561
• Henderson, N., Flores, E., Sampaio, M., Freitas, L. and Platt, G. M. Supercritical fluid flow in porous media: modelling and simulation. Chem. Eng. Sci. 60:1797–1808, 2005. doi:10.1016/j.ces.2004.11.012
• Lucas, S. K., Blake, J. R. and Kucera, A. A boundary-integral method applied to water coning in oil reservoirs. ANZIAM J. 32(3):261–283, 1991. doi:10.1017/S0334270000006858
• Meyer, H. I. and Garder, A. O. Mechanics of two immiscible fluids in porous media. J. Appl. Phys., 25:1400–1406, 1954. doi:10.1063/1.1721576
• Muskat, M. and Wycokoff, R. D. An approximate theory of water coning in oil production. Trans. AIME 114:144–163, 1935. doi:10.2118/935144-G
• GNU Octave. https://www.gnu.org/software/octave/doc/v4.2.1/
• Yih, C. S. On steady stratified flows in porous media. Quart. J. Appl. Maths. 40(2):219–230, 1982. doi:10.1090/qam/666676
• Yu, D., Jackson, K. and Harmon, T. C. Disperson and diffusion in porous media under supercritical conditions. Chem. Eng. Sci. 54:357–367, 1999. doi:10.1016/S0009-2509(98)00271-1
• Zhang, H. and Hocking, G. C. Axisymmetric flow in an oil reservoir of finite depth caused by a point sink above an oil-water interface. J. Eng. Math. 32:365–376, 1997. doi:10.1023/A:1004227232732
• Zhang, H., Hocking, G. C. and Seymour, B. Critical and supercritical withdrawal from a two-layer fluid through a line sink in a bounded aquifer. Adv. Water Res. 32:1703–1710, 2009. doi:10.1016/j.advwatres.2009.09.002
• Zill, D. G. and Wright, W. S. Differential Equations with Boundary-value problems, 8th Edition. Brooks Cole, Boston USA, 2013.