Analysis of the nonlinear storage–discharge relation for hillslopes through 2D numerical modelling Melkamu Ali,* Aldo Fiori and Giorgio Bellotti Dipartimento di Scienze dell’Ingegneria Civile, Universita di Roma Tre, Via Vito Volterra 62 00146, Rome, Italy Abstract: Storage–discharge curves are widely used in several hydrological applications concerning flow and solute transport in small catchments. This article analyzes the relation Q(S) (where Q is the discharge and S is the saturated storage in the hillslope), as a function of some simple structural parameters. The relation Q(S) is evaluated through two-dimensional numerical simulations and makes use of dimensionless quantities. The method lies in between simple analytical approaches, like those based on the Boussinesq formulation, and more complex distributed models. After the numerical solution of the dimensionless Richards equation, simple analytical relations for Q(S) are determined in dimensionless form, as a function of a few relevant physical parameters. It was found that the storage–discharge curve can be well approximated by a power law function Q/(LK s )= a(S/ (L 2 (f  θ r ))) b , where L is the length of the hillslope, K s the saturated conductivity, f  θ r the effective porosity, and a, b two coefficients which mainly depend on the slope. The results confirm the validity of the widely used power law assumption for Q(S). Similar relations can be obtained by performing a standard recession curve analysis. Although simplified, the results obtained in the present work may serve as a preliminary tool for assessing the storage–discharge relation in hillslopes. Copyright © 2012 John Wiley & Sons, Ltd. KEY WORDS hillslope; nonlinear reservoir; flow recession; storage–discharge relation Received 10 January 2012; Accepted 3 May 2012 INTRODUCTION The storage–discharge relation is a very important component of catchment hydrology and it is widely used for several engineering applications, such as estimating design floods (Rahman and Goonetilleke, 2001), forecasting of low flows for water resource management (Vogel and Kroll, 1992), estimating groundwater poten- tial of basins (Wittenberg and Sivapalan, 1999) and rainfall runoff models (Sriwongsitanon et al., 1998). The matter has been investigated in the framework of base flow recession, hydrograph separation and other related areas of hillslope hydrology (Brooks et al., 2004; Weiler and McDonnell, 2004; Fiori and Russo, 2007; Graham and McDonnell, 2010; McGuire and McDonnell, 2010). Based on different principles and approaches, the recession of subsurface flow has been studied by Brutsaert and Nieber (1977), Wittenberg (1994, 1999), Fenicia et al. (2006), Aksoy and Wittenberg (2011), Wang (2011) and Moore (1997). Tallaksen (1995) has discussed various methods and approaches widely used to determine the storage–discharge relationship by the recession curve analysis. One of the widely used storage–discharge relations is the linear reservoir model, originally defined by Maillet (1905), which implies that the aquifer behaves like a single reservoir with storage S, linearly proportional to outflow Q, namely Q = aS. In this case, the plot of logQ t against S yields a straight line (Barnes, 1939). Moreover, Fenicia et al. (2006) have also derived an actual S–Q relation in which the percolated water from the unsaturated reservoir as well as the preferential recharge as inflow to the saturated reservoir are taken into account, for which the S–Q relation was treated as a linear and second-order polynomial. However, linear reservoirs (Q = aS) can very well describe the ground- water behaviour for most of the catchments they have studied. In most actual cases, however, semilogarithmic plots of flow recessions are still concave (Aksoy and Wittenberg, 2011), indicating nonlinear storage– discharge relationships. Recent studies agreed that the outflow of a lumped storage model can be characterised by a general power law function, Q = aS b , where a and b are constants. The constant b varies between 0 and 2 or higher for some cases (Chapman, 1997; Wittenberg, 1999; Wittenberg and Sivapalan, 1999; Harman and Sivapalan, 2009) and it has also been proved by physical experiments (Wittenberg, 1994; Chapman, 1999). The power law formulation is only occasionally chosen in recession analysis (Wittenberg, 1994), and the recession process is commonly formulated in terms of the reservoir inflow and outflow, which can be calculated using the continuity equation. On the other hand, Brutsaert and Nieber (1977) have proposed to determine the outflow rate from a low-flow hydrograph derived from direct measurements of Q(t). They have developed their models based on the nonlinear *Correspondence to: Melkamu Ali, Dipartimento di Scienze dell’Ingegneria Civile, Universita di Roma Tre, Via Vito Volterra 62, 00146 Rome, Italy. E-mail: melkamualebachew.ali@uniroma3.it HYDROLOGICAL PROCESSES Hydrol. Process. (2012) Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/hyp.9397 Copyright © 2012 John Wiley & Sons, Ltd.