DISCUSSION Steady state groundwater seepage in sloping unconfined aquifers RP Chapuis, Bull Eng Geol Environ 70:89–99. doi:10.1007/s10064-010-0282-2 Antonis D. Koussis • Evangelos Akylas Received: 30 May 2011 / Accepted: 20 November 2011 Ó Springer-Verlag 2011 This Commentary has three aims: (1) to state the complete extended Boussinesq equation, from which its abridged and commonly used form derives, and the condition under which the latter represents the complete one with sufficient accuracy; (2) to draw attention: (a) to an analytical steady- state solution of the nonlinear extended equation of Boussinesq derived by Henderson and Wooding (1964) and reworked by Basha and Maalouf (2005), and (b) to McEnroe’s (1993) solution of Eq. 21 in Chapuis (2011); and (3) to discuss steady-state solutions of two linearised forms of the extended equation of Boussinesq, giving cri- teria under which the linear solutions approximate the nonlinear solutions well. The complete form of the extended equation of Boussinesq Subsurface flow on a sloping base (also called subsurface stormflow or hillslope flow) has been studied extensively. Henderson and Wooding (1964) and Wooding and Chap- man (1966) laid its mathematical foundations based on the Dupuit-Forchheimer theory of unconfined flow (hydrostatic pressure and thus constant potential over the depth, H, measured normal to the bed) deriving the extended equation of Boussinsq that accounts for a sloping base. Wooding (1966) examined the accuracy of that hydraulic equation via application of conformal mapping. In the notation of Chapuis (2011), the soil has saturated hydraulic conductivity k sat and specific yield f, rests on an impervious bed inclined against the horizontal at an angle a, and is recharged at a constant rate per unit horizontal area N. Then, the discharge per unit width (planar flow), at time t and location x 0 , measured from the top of hill along the inclined base of length L, is given by (Henderson and Wooding 1964; Wooding and Chapman 1966; Childs 1971) q ðx; tÞ¼ Hk sat sin a  o H o x 0 cos a   ; ð1Þ where flow is properly positive in the ?x 0 direction. The storage balance equation is f o H o t þ o q o x 0 ¼ N cos a þ N o H o x 0 sin a: ð2Þ The term N(qH/qx 0 )sina on the right-hand side of Eq. 2 derives from the scalar product of the recharge vector and the unit normal of a free surface element, but is rarely included in the volume balance. Combining Eq. 1 with Eq. 2 yields the complete extended equation of Boussinesq for unconfined flow over an inclined base (Akylas et al. 2006): f o H o t þ k sat sin a 1  N k sat   o H o x 0  k sat o o x 0 H o H o x 0   cos a ¼ N cos a: ð3Þ Henderson and Wooding (1964) neglected the last term in Eq. 2, obtaining Eq. 3 with (1 - N/k sat ) replaced by 1. That equation has been adopted widely (e.g., Beven 1981; A. D. Koussis (&) Institute for Environmental Research and Sustainable Development, National Observatory of Athens, I. Metaxa & Vas. Pavlou, Palea Penteli, 15452 Athens, Greece e-mail: akoussis@noa.gr E. Akylas Department of Civil Engineering and Geomatics, Technical University of Cyprus, Limassol, Cyprus 123 Bull Eng Geol Environ DOI 10.1007/s10064-011-0413-4