Transport in Porous Media

, Volume 18, Issue 1, pp 15–36 | Cite as

An integral equation formulation for the unconfined flow of groundwater with variable inlet conditions

  • Z. -X. Chen
  • G. S. Bodvarsson
  • P. A. Witherspoon
  • Y. C. Yortsos


We combine an integral equation formulation with a hodograph transformation to solve self-similar problems describing the unconfined flow of groundwater with variable inlet conditions. A class of new semi-analytical solutions is obtained for both rectilinear and radial flow geometries. The solutions are in general agreement with those derived by Barenblatt, although there are some discrepancies for the case of radial flow. The formulation presented provides additional analytical insight, and for computational purposes is simpler than Barenblatt's. In addition, the method proposed can be successfully used for the solution of a host of other nonlinear problems that admit self-similarity.

Key words

Analytical solution unconflned flow groundwater 



parameter defined in (3) [L/T]


relative total discharge function for the case of constant head


gravitational constant [L/T2]


hydraulic head [L]


dimensionless hydraulic head


permeability [L2]


hydraulic conductivity [L/T]


total discharge per unit width [L2/T]


relative total discharge function


radial distance [L]


time [T]


dimensionless total discharge


cumulative volume [L3]


total discharge per unit head


distance [L]


exponent of time variation of inlet head


dimensionless constant defined in (9)


viscosity [M/TL]


similarity variable


density [M/L3]




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  1. Aronson, G., 1985, The porous media equation, some problems in nonlinear diffusion (A. Fasano and M. Primicerio, eds.),Lecture Notes in Mathematics (CIME Foundation Series), Springer-Verlag, No. 1224.Google Scholar
  2. Barenblatt, G. I., 1952a, Some transient flow of fluid and gas in a porous medium (in Russian),Prikladnaia Matematika i Mekhanika 16(1), 67–78.Google Scholar
  3. Barenblatt, G. I., 1952b, Self-similar movements of compressible fluid in a porous medium (in Russian),Prikladnaia Matematika i Mekhanika 16(6), 679–698.Google Scholar
  4. Barenblatt, G. I., 1954, Some problems of transient flow of fluids through porous media (in Russian),Izv. Akad. Nauk, U.S.S.R., Otdel Tech. Nauk, No. 6, 97–110.Google Scholar
  5. Barenblatt, G. I. and Vishek, M. I., 1956, On the finiteness of velocity of propagation in the problems of transient flow of liquid and gas in a porous medium (in Russian),Prikladnaia Matematika i Mekhanika 20(3), 411–417.Google Scholar
  6. Barenblatt, G. L, Entov, V. M. and Ryzhik, V. M., 1972,Theory of Unsteady-State Flow of Liquid and Gas through Porous Media (in Russian), 288pp., Nedra Moscow, pp. 69, 72.Google Scholar
  7. Barenblatt, G. I., Entov, V. M. and Ryzhik, V. M., 1990,Theory of Fluid Flows through Natural Rocks 395pp., Kluwer, Dordrecht.Google Scholar
  8. Bear, J., 1972,Dynamics of Fluids in Porous Media, American Elsevier, New York, NY, 763 pp.Google Scholar
  9. Boussinesq, J., 1904, Recherches théoretiques sur l'écoulement des nappes d'eau infiltrées dans le sol et sur le debit des sources,Jour, de Math. Pures et Appl. 10, 5–78.Google Scholar
  10. Clarkson, P. A., Fokas, A. S. and Ablowitz, M. J., 1989, Hodograph transformations of linearizable partial differential equations,SIAM J. Appl. Math. 49(4), 1188–1209.Google Scholar
  11. Chen, Z.-X., Bodvarsson, G. S. and Witherspoon, P. A., 1990, An integral equation formulation for two-phase flow and other nonlinear flow problems through porous media, paper SPE 20517 presented at the 1990 SPE Annual Fall Meeting, New Orleans, LA, Sept. 23–26.Google Scholar
  12. Chen, Z.-X., Bodvarsson, G. S. and Witherspoon, P. A., 1991a, One-dimensional horizontal infiltration in an unsaturated porous media including air viscosity and applied pressure, preprint.Google Scholar
  13. Chen, Z.-X., Bodvarsson, G. S. and Witherspoon, P. A., 1991b, Exact semi-analytical solution for unconfined Dupuit flow of groundwater, preprint.Google Scholar
  14. van Duijn, C. J. and Peletier, L. A., 1992, A boundary-layer problem in fresh-salt groundwater flow,Q.Jl. Mech. appl. Math. 45, 1–23.Google Scholar
  15. Fokas, A. S. and Yortsos, Y. C., 1982, On the exactly solvable equationS t=[(ΒS+λ)−2 S X]X+α (ΒS+λ)−2 S X occurring in two-phase flow in porous media,SIAM J. Appl. Math. 42(2) 318.Google Scholar
  16. Lake, L. W., 1989,Enhanced Oil Recovery, Prentice Hall, New York.Google Scholar
  17. Leibenson, L. S., 1929, Gas-flow in a porous medium (in Russian),Neftianoe Khoziaistvo 10, 497–519.Google Scholar
  18. Lenormand, R., 1990, Liquids in Porous Media,J. Phys.: Condens. Matter 3, SA79–89.Google Scholar
  19. McWhorter, D. B., 1990, Unsteady radial flow of gas in the vadose zone,Journal of Contaminant Hydrology 5(3), 297–314.Google Scholar
  20. McWhorter, D. B. and Sunada, D. K., 1990, Exactintegral solutions for two-phase flow,Water Resour. Res. 26(3), 399–413.Google Scholar
  21. Polubarinova-Kochina, P. Ya, 1948, A nonlinear partial differential equation encountered in theory of fluid flow through porous media (in Russian),Dokl. Akad. Nauk, U.S.S.R. 63(6), 623–626.Google Scholar
  22. Polubarinova-Kochina, P. Ya, 1949, Unsteady movements of groundwater when it flows from a reservoir (in Russian),Prikladnaia Matematika i Mekhanika 13(2), 187–206.Google Scholar
  23. Shankar, K. and Yortsos, Y. C., 1983, Asymptotic analysis of single pore gas-solid reactions,Chem. Engng. Sci. 38(8), 1159–1165.Google Scholar
  24. Yortsos, Y. C., 1991, A theoretical analysis of Vertical Flow Equilibrium, paper SPE 22612 presented at the 66th SPE Annual Fall Meeting, Dallas, TX, Oct. 6–9. Also,Transport in Porous Media, in press (1995).Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Z. -X. Chen
    • 1
  • G. S. Bodvarsson
    • 1
  • P. A. Witherspoon
    • 1
  • Y. C. Yortsos
    • 2
  1. 1.Lawrence Berkeley LaboratoryUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Chemical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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