## 1. Introduction

[2] Following the pioneering recession flow analysis work by *Brutsaert and Nieber* [1977], *Troch et al*. [1993], and *Szilagyi et al*. [1998], attempts have been made to use analytical solutions to (linearizations of) the nonlinear Boussinesq equation to infer hydraulic properties of sloping aquifers, e.g., while using the linearized Boussinesq equation [*Huyck et al*., 2005], focusing on the rising limb of a hydrograph [*Pauwels and Troch*, 2010], or testing a hillslope subsurface flow similarity number [*Lyon and Troch*, 2007]. These attempts make use of the method of hydrograph analysis, first introduced by *Brutsaert and Nieber* [1977], when considering recession flow from horizontal aquifers. *Brutsaert and Nieber* [1977] showed how under certain initial and boundary conditions, the recession discharge *Q*(*t*) from a horizontal Boussinesq aquifer can take the form

where *a* is a function of the aquifer properties and *b* is a constant. For instance, for flat-lying aquifers, characterized by spatially and vertically uniform saturated hydraulic conductivity *k* and drainable porosity (or specific yield) *f*, it can be shown analytically [*Brutsaert and Nieber*, 1977] that after initial effects are dissipated (“late time”)

with *A* catchment area and *L*_{d}/*A* drainage density.

[3] The direct use of equation (2a) to infer, for instance, *k* from recession data and known *A*, *L*_{d}, and *f*, requires that *b* = 3/2 and that the assumptions underlying the nonlinear Boussinesq equation hold. Any *b* ≠ 3/2 thus raises challenges for the use of equation (2a) to infer *k* for flat-lying aquifers. Indeed, in applications of equation (2a), *Brutsaert and Nieber* [1977] and *Troch et al*. [1993] showed that their discharge data suggested *b* = 3/2 during late time. *Eng and Brutsaert* [1999] and *Brutsaert and Lopez* [1998] found *b* = 1, and consequently replaced equation (2a) with a similar approach based on the linearized Boussinesq equation, that does correspond to *b* = 1. Using an alternative set of assumptions, especially a drawdown that is small relative to saturated thickness, *van de Giesen et al*. [2005] demonstrated that solutions to aquifer recession based on the Laplace equation leads to *b* = 1 during late time as well.

[4] It bears noting that the use of a similar approach to infer aquifer properties for sloping hillslopes or catchments hinges on the correspondence between observed values of *b*, and the valid value of *b* associated with the version of the Boussinesq equation and the underlying assumptions made.

[5] The nonlinear Boussinesq equation for a sloping aquifer with gradient α is derived by combining the Darcy equation

with the continuity equation

yielding [*Brutsaert*, 1994]

where *x*^{*} is a coordinate along the hillslope (*x*^{*} = 0 at the foot), *h*^{*} = *h*(*x*^{*}, *t*^{*}) is the thickness of the water layer perpendicular to the bedrock, and *N* is the recharge rate.

[6] A generalized variant of equation (5) is due to *Rupp and Selker* [2006] who relaxed the assumption of uniform conductivity, instead allowing conductivity *k* to vary with depth as a power-law:

where *k _{D}* is the saturated hydraulic conductivity at distance

*D*perpendicular above the bedrock. The constant

*n*is the exponent that describes the rate of change in saturated hydraulic conductivity

*k*with distance perpendicular to the aquifer base. When

*n*= 0 (i.e., vertical homogeneity in

*k*), equation (6) reduces to equation (5).

[7] Application of the method of *Brutsaert and Nieber* [1977], i.e., linking recession parameters *a* and *b* to aquifer properties, to sloping aquifers requires analytical solutions to equation (5) or (6). *Daly and Porporato* [2004] have provided the only known exact solutions, which are for special cases of flow in a homogeneous aquifer that is infinite in both the up and downslope directions. For more practical cases in finite aquifers, analytical solutions have mainly been arrived at by making the kinematic-wave approximation [e.g., *Henderson and Wooding*, 1964; *Beven*, 1981; *Harman and Sivapalan*, 2009a] or by linearizing the Boussinesq equation (see review in *Rupp and Selker* [2006]).

[8] *Huyck et al*. [2005] showed how an analytical solution to the linearized form of equation (5) leads to discharge of the form

after sufficient time has elapsed following the cessation of recharge to the aquifer, suggesting that *b* ≈ 1. *Brutsaert* [1994, p. 2762], however, had warned of the (potential) lack of validity of the late-time result arising from the linearization of equation (5), when applied to steep shallow aquifers. This is because the linearized equation, unlike the nonlinear equation, does not permit the lowering water table, or drying front, to arrive at the impermeable base at the upslope aquifer boundary and then progress downslope along the aquifer base [*Stagnitti et al*., 2004].

[9] Given the above limitations of linearization, *Rupp and Selker* [2006] made a closer examination of the late-time discharge behavior of equation (6) through numerical solutions of the nonlinear equation. They also arrived at the result given by equation (7) for *n* = 0, though their definition of the parameter *a* was different from *Huyck et al*. [2005]. Furthermore, *Rupp and Selker* [2006] concluded that the late-time discharge could be approximated as

for *n* ≥ 0, which also suggest that *b* = 1 for the case of a homogeneous aquifer (*n* = 0). This result of *b* = 1 is in contradiction to the drainage behavior of steep hillslopes if one assumes that kinematic-wave assumptions holds. For steep hillslopes, the term *∂h*/*∂x* in equation (3) becomes small with respect to sin *α*, such that *q* ≈ *kh* sin *α*. It has been shown that under these circumstances (the discharge predicted by) the kinematic-wave equation is a good approximation of (the discharge predicted by) the Boussinesq equation [*Henderson and Wooding*, 1964; *Beven*, 1981].

[10] For constant *k* (*n* = 0), the kinematic recession behavior of such a hillslope will depend on initial conditions, where two extreme cases can be distinguished [*Rupp and Selker*, 2006]. First, prolonged rainfall on initial dry soil leads to the formation of a wedge-shaped steady state geometry of the saturated zone. For each location along the hillslope of length *L*, *q* = *N*(*L*−*x*), such that *h* = *N*(*L*−*x*)/(*k* sin *α*). During recession, this wedge travels downhill, with constant velocity such that *h*(*t*) is linearly decreasing, and d*Q*/d*t* a constant, such that *b* = 0. Second, after a short pulse-like rainstorm an initial dry soil becomes saturated to some depth *h*. During the poststorm recession period, this rectangular “saturated slab” travels downhill. Because *h* is constant, *q* is constant, and therefore d*Q*/d*t* = 0. Solving equation (1) for *a* and *b* yields *a* = 0 with *b* left undetermined. Note that *Rupp and Selker* [2006] in their Figure 3 mistakenly list *b* = 0 instead of *b* = undefined for the uniform saturation case.

[11] A similar result (*q* decreasing linearly with time, the equivalent of *b* = 0) for the kinematic-wave approximation of the Boussinesq equation applied to the case of steep hillslopes with shallow soils was reached by [*Harman and Sivapalan*, 2009a], while investigating storage-discharge relationships derived from the Boussinesq equation for a range of forcing and boundary conditions. However, as shown above, the value of *b* that arises from the kinematic-wave approximation “long” after the cessation of recharge is dependent upon the profile of the water table at the onset of recession, which is in turn dependent on the history of recharge events. Whether the kinematic wave approximation is an accurate representation of the behavior of the nonlinear Boussinesq equation, in terms of d*Q*/d*t* versus *Q*, has not, to our knowledge, been demonstrated.

[12] The aim of this paper is to investigate the behavior of *b* in numerical solutions of the nonlinear Boussinesq equation, to evaluate to what extent the results of *b* = 1 in the literature are due to the physics of the system (in case of *b* derived from observations), or caused by numerical artifacts (in case of *b* derived from numerical simulations). These results are relevant for any interpretation of field observed *a* and *b* for steep hillslopes and catchments, analogous to the application of equation (2a) to flat-lying aquifers. These results are also relevant for our understanding of the numerical aspects of hydrological models [*Kavetski et al*., 2003; *Clark and Kavetski*, 2010; *Kavetski and Clark*, 2010].