Similarity solution of the Boussinesq equation D.A. Lockington a, * , J.-Y. Parlange b , M.B. Parlange c , J. Selker d a Department of Civil Engineering, University of Queensland, Brisbane 4072, Australia b Department of Agricultural and Biological Engineering, Cornell University, Ithaca, NY 14853, USA c Department of Geography and Environmental Engineering, Johns Hopkins University, Baltimore, MD 21218, USA d Department of Bioresource Engineering, Oregon State University, Corvallis, OR 13906, USA Received 31 August 1999; received in revised form 18 December 1999; accepted 19 January 2000 Abstract Similarity transforms of the Boussinesq equation in a semi-in®nite medium are available when the boundary conditions are a power of time. The Boussinesq equation is reduced from a partial dierential equation to a boundary-value problem. Chen et al. [Trans Porous Media 1995;18:15±36] use a hodograph method to derive an integral equation formulation of the new dierential equation which they solve by numerical iteration. In the present paper, the convergence of their scheme is improved such that nu- merical iteration can be avoided for all practical purposes. However, a simpler analytical approach is also presented which is based on ShampineÕs transformation of the boundary value problem to an initial value problem. This analytical approximation is remarkably simple and yet more accurate than the analytical hodograph approximations. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Similarity solution; Boussinesq equation; Aquifer recharge 1. Introduction Groundwater ¯ow in an uncon®ned aquifer may be approximately modeled by the nonlinear Boussinesq equation, assuming DupuitÕs hypothesis of horizontal ¯ow applies [3]. Solutions of the Boussinesq equation are applied in catchment hydrology and base¯ow studies (e.g., [12,14]) as well as agricultural drainage problems [10] and constructed, subsurface wetlands. Consider the usual Boussinesq equation for one-di- mensional ¯ow in an initially dry, uncon®ned aquifer: h s oh ot K s o ox h oh ox 1 with h0; t h 0 t rt a ; 2 hx; 0 0 3 and hx !1; t 0; 4 where h s is the porosity, K s the saturated hydraulic conductivity, h the height of the water table above a horizontal impermeable layer and t and x are the time and distance, respectively (see Fig. 1). The parameter a is constant. The objective of this paper is to determine accurate, analytical approximations of Eqs. (1)±(4). Several approximate analytical solutions already exist for the simple case when a 0 such that h 0 is constant [4,13]. Eq. (2) is more general, but more importantly, admits a similarity solution that facilitates accurate solution. Although useful in their own right, such sol- utions are also valuable in that they provide `benchmarks' to validate more general numerical solutions. While the physical interpretation of the solutions is straightfor- ward for a > 0 (boundary water supply head increases with time), the implications of a < 0 are not so clear. It will be seen later that a )1/3 corresponds to the re- distribution of a ®nite quantity of water introduced at x 0. For values of a less than this, the solution is nonmonotonic and therefore physically unacceptable in the present situation. A similarity solution is sought in the form h rt a H n; 5 where n x 2h s a 1 rK s t a1 s 6 so that Eqs. (1)±(4) become www.elsevier.com/locate/advwatres Advances in Water Resources 23 (2000) 725±729 * Corresponding author. 0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 0 4 - X