Journal of Hydrology, 70 (1984) 251--263 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands 251 [5] FREE-SURFACE FLOW IN POROUS MEDIA AND PERIODIC SOLUTION OF THE SHALLOW-FLOW APPROXIMATION J.-Y. PARLANGE 1 , F. STAGNITTI 1 , J.L. STARR 2 and R.D. BRADDOCK 1 1School of Australian Environmental Studies, Griffith University, Brisbane, Qld. 4111 (Australia) 2Soil Nitrogen and Environmental Chemical Laboratory, U.S. Department of Agriculture (U.S.D.A.), Beltsville, MD 20705 (U.S.A.) (Received October 25, 1982; accepted for publication April 14, 1983) ABSTRACT Parlange, J.-Y., Stagnitti, F., Starr, J.L. and Braddock, R.D., 1984. Free-surface flow in porous media and periodic solution of the shallow-flow approximation. J. Hydrol., 70: 251--263. The equations describing the flow of liquid in a porous medium with a free surface are expanded when the shallow-flow assumption holds. Second-order theory is used to de- scribe the propagation of steady periodic motion in the medium, driven by the oscillating level of a reservoir in contact with it. A linearized solution of the second-order theory is compared with a numerical solution and is found adequate even when the amplitude of the motion is comparable to the mean depth of the liquid. The predictions of the analysis are found in good agreement with two laboratory experiments. INTRODUCTION We consider flow in a porous medium, with the liquid bounded by a well-defined free surface. The thickness of the capillary fringe is negligible compared to any other length entering the problem. Such problems are often treated by applying the standard Dupuit- Forchheimer assumption leading to the well-known Boussinesq equation ~(e.g., see Ch. 8 in Bear, 1972). The Boussinesq equation is an approximation which sometimes leads to paradoxical results (Kirkham, 1967; Bear, 1972). In a fundamental paper, Dagan (1967) showed that this Boussinesq equation is the first of a series of equations which can be derived from gen- eral flow equations by an expansion based on a shallow-flow approximation. With a slight correction of Dagan's second-order equation, the approach followed here is essentially the same. The hydraulic head is defined as qJ = p + z where p is the pressure head and z is the vertical distance above a horizontal impermeable layer located below the porous medium. We seek a steady periodic solution to the two-