## 1. Introduction

[2] The representation of groundwater flow processes in hydrologic models is usually referred to as the horizontal or lateral redistribution of soil water [*Wood*, 1991]. Even though at small spatial scales these processes cannot automatically be neglected, in many models, applied at large (continental) scales, these processes are not represented. An example of a widely used model that does not account for lateral redistribution is the Common Land Model (CLM) [*Dai et al.*, 2003]. On the other hand, the Richards equation [*Richards*, 1931] can be solved numerically in a full three-dimensional manner, as is demonstrated by, for example, *Paniconi and Wood* [1993] and *Camporese et al.* [2009]. This methodology has the advantage that the horizontal redistribution of soil water is modeled in a physically based way. The drawbacks are that a large number of spatially distributed parameter values are required, that the computational demands can render this approach infeasible for larger modeling domains, and that effective parameter values cannot be determined in the presence of heterogeneity. If one assumes the aquifers to be homogeneous and isotropic (a reasonable assumption if one wants a relatively simple model, taking into account all other uncertainties), the need for large parameters sets is strongly reduced, but the computational demands can in many cases still be excessive. A middle of the road approach is to model the saturated groundwater flow processes in a lumped manner, meaning that the lateral redistribution of soil water is modeled using simple methods that are obtained using a variety of assumptions, strongly reducing the required computer time. Examples of models that apply this approach are the Variable Infiltration Capacity (VIC) model [*Liang et al.*, 1994] and the TOPMODEL-based Land Atmosphere Transfer Scheme [*Famiglietti and Wood*, 1994].

[3] For the latter approach, aquifer hydraulic parameters, such as the saturated hydraulic conductivity, are required at the catchment scale. Although in theory these parameters can be measured in laboratories using soil samples, the difference between the scale of the observations and the scale at which the processes occur renders these measurements inadequate for application in hydrologic models [*Bear*, 1972]. For this reason, methods have been developed, in which these parameters are estimated through a hydrograph recession analysis. More specifically, using theoretically derived equations, the rate of the recessions can be related to the aquifer hydraulic parameters. In order to eliminate the time variable in these relationships, the discharge recessions are analyzed as a function of the discharge [*Brutsaert and Nieber*, 1977; *Troch et al.*, 1993; *Rupp and Selker*, 2005; *Huyck et al.*, 2005]. Usually, approximations for short and large times are derived. Based on these expressions, the aquifer parameters can be related to observed recession dynamics. *Rupp and Selker* [2006a] provide an overview of such relationships, including the assumptions used in their derivation. These assumptions apply to the slope of the aquifer, the linearization of the governing equation, the incorporation of variable geometry, the uniformity of the aquifer parameters, and the omission of certain terms in the governing equation.

[4] In the application of the short-time equations, a number of inconsistencies are usually neglected. For example, although conditions of probable full saturation have been documented for an isolated hydrograph caused by extreme rainfall for a small catchment [*Lyon et al.*, 2008], full saturation combined with zero recharge at the onset of a recession and a sudden drawdown at the bottom of the aquifer is a condition that arguably occurs very rarely in reality. Further, it is known that hydraulic conductivity values are not uniform over the soil profile [*Beven and Kirkby*, 1979; *Rupp and Selker*, 2005, 2006a]. It can be expected that the inconsistency between the conditions under which the equations have been derived and the conditions under which they are applied will lead to errors in the parameter estimates.

[5] The objective of this paper is to develop a methodology to estimate catchment-scale hydraulic conductivity values of the lower layers of the underlying aquifer, consistent with the conditions under which the required equations are derived. Such estimates are crucial to quantify catchment behavior during drought periods when aquifer storage is near minimum. To develop such a method, the theory related to the governing equation (the extended Boussinesq equation) is first summarized. A general equation for the short-time behavior of the base flow, valid for realistic initial conditions and recharge rates, is derived using the linearized extended Boussinesq equation. This expression is further simplified for the case of the rising limb of a base flow hydrograph succeeding a long rainless period. It should be noted that, recently, advances have been made in the estimation of catchment averaged net rainfall rates using fluctuations in streamflow time series [*Kirchner*, 2009]. However, the methodology developed in this paper requires time series of base flow instead of streamflow. Instead of estimating catchment averaged net rainfall rates, in this paper it is demonstrated that the rising limbs of base flow hydrographs can be related to the hydraulic properties on an aquifer. Synthetic experiments are used to demonstrate the accuracy of this new equation, and to assess the impact of the unrealistic assumptions in the traditionally used expressions for the short-time behavior of base flow. The results of a traditional base flow recession analysis are compared to the results of the newly developed methodology. Further, a dimensional analysis is performed, for the purpose of assessing whether the parameter sensitivity of the newly derived short-time expression is consistent with the parameter sensitivity of the linearized extended Boussinesq equation. A suggestion is then made in order to apply the new equation for the estimation of aquifer hydraulic conductivities, in case the nonlinear extended Boussinesq equation is used. The paper finally concludes with a discussion on the strengths and weaknesses of the newly developed methodology, based on the use of the rising limbs of hydrographs, versus a traditional recession analysis.