Dept. of Soil & Water Sciences,
P.O. Box 34, Al-Khod 123,
Sultan Qaboos University,
Sultanate of Oman
E-mail: anvar@squ.edu.om

Dept. of Civil Engineering,
University of Kentucky,
Lexington, KY 40509, USA
E-mail: sergio@engr.uky.edu

Dynamics of sharp interfaces in one-, two-phase flows in porous media:
asymmetry in the Boussinesq and Charny equations

The phreatic surface dynamics in unconfined aquifers and soil massifs (e.g. dams or embankments) is mathematically described by the Boussinesq nonlinear diffusion equation (hydraulic model) or by the Laplace equation with a nonlinear condition on the free surface (hydrodynamic model). In two-phase flows, the sharp interface between two phases obeys the Charny PDE, which belongs to the same class as the Boussinesq equation. In this paper, we employ the well-known results (e.g. one derived by the Ukrainian mathematician Yu.D. Sokolov [6]) and derive new ones based on similarity solutions, the Adomian decomposition method and complex variables (the hodograph and Polubarinova-Kochina techniques). First, we prove that the inversion of the Barenblatt-Sokolov finite support parabolic (instantaneous source) solution can be interpreted (in a steady-state regime) as a solution to a nonlinear ODE with a finite reservoir depth but exponentially varying evaporation along the phreatic surface (a distributed sink term). Next, we study cyclostationary excitation of the reservoir water level and illustrate the phenomenon of "superelevation" discovered by J.R. Philip in 1973 for tide-affected coastal aquifers. Third, we arrive at an explicit rigorous solution for a periodic water-drive regime with a heavier fluid sweeping a lighter one from a porous formation that is characterized by the phenomena of "superpropagation" and "counterslumping". Fourth, we employ a travelling wave solution for a phreatic surface advancing in a tilted layer. Applications to groundwater hydrology and petroleum engineering are discussed [1-5].


  1. Kacimov, A.R. 2002. Analytical solutions in a hydraulic model of seepage with sharp interfaces. J. Hydrology. 258, 179-186.
  2. Kacimov, A.R. and Serrano, S. 2003. Comment on "On an Exact Analytical Solution of the Boussinesq Equation" by J.-Y. Parlange, W.L. Hogarth, R.S. Govindaraju, M.B. Parlange, and D. Lockington. Transport in Porous Media (in press).
  3. Kacimov, A.R. and Yakimov, N.D. 2001. Moving phreatic surface in a porous slab: an analytical solution. J. Engineering Mathematics. 40, 399-411.
  4. Serrano,S.E. and Workman,S.R. 1998. Modeling transient stream/aquifer interaction with the non-linear Boussinesq equation and its analytical solution. J. of Hydrology. 206. 245-255.
  5. Serrano,S.E. and Workman,S.R. 2000. Reply to comment on "Modeling transient stream/aquifer interaction with the non-linear Boussinesq equation and its analytical solution" by Serrano,S.E., Workman, S.R. J. of Hydrology. 206. 245-255. J.Hydrology, 235, 293-296.
  6. Sokolov, Yu.D. 1956. On some particular solutions of the Boussinesq equation, Ukrainian Mathematical J. 8, N1, 48-54 (in Russian).