A similarity framework to assess controls on shallow subsurface flow dynamics in hillslopes


  • Ciaran Harman,

    1. Department of Geography and Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
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  • Murugesu Sivapalan

    1. Department of Geography and Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
    2. Also at Department of Water Management, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, Netherlands.
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[1] An understanding of the dominant controls on water fluxes through landscapes at scales smaller than the resolution of a hydrologic model is useful for choosing appropriate parameterizations, and for developing a classification system for landscape hydrology. The links between these controls and the parameters in hillslope-scale models of subsurface lateral saturated flow have been treated implicitly in both physical (e.g., Characteristic Response Function) and empirical formulations, such as power law storage-discharge relations. In particular, the controls exerted by the boundary condition at the toe of the hilslope and the temporal variability of the recharge have not been integrated with the understanding of the controls of topography and soil properties. In this work, we develop a dimensionless similarity framework for assessing these controls on the subsurface flow dynamics, based on the Boussinesq equation in an idealized hillslope system. Using this framework we demonstrate the relationship between (1) the exponents of the storage-discharge relations, (2) the boundary conditions and (3) the characteristic storage thickness used in the Characteristic Response Function parameterization. We use the concept of hydrologic regimes to incorporate information about the population of recharge events that drive the system and set up the initial condition of saturated storage for each storm event. The analysis demonstrates that for planar hillslopes with uniform soil properties, exponents range between 0 (when the flow through the hillslope is dominated by the topographic gradients, and is in the early stages of draining) and 2 (when the flow is driven by water table gradients, and the thickness of the aquifer at the seepage face is small compared to the average thickness). The similarity framework can be used to make first-order predictions of these exponents, and classify hillslopes using a simple typology of subsurface flows based on two components of the behavior: the relative dominance of the topographic and water-table gradients, and the responsiveness of the hillslope discharge to temporal variability of the recharge inputs.

1. Introduction

[2] To model and predict the flow of water through catchments we must choose a scale at which flows are resolved explicitly, and below which their variability is parameterized in some way [Reggiani et al., 1998, 1999; Beven, 2006]. These parameterizations of the flow must capture in some parsimonious way how the landscape and climate properties interact to control the flow of water at scales below the resolution of the model. The shallow subsurface lateral flow that occurs in hillslopes when infiltrating water meets a low-permeability layer, such as the bedrock, is one such flow requiring parameterization. It is an important hydrologic flux both as a component of stormflow runoff generation in steep hillslopes with transmissive soils, as well as in lower gradient landscapes where saturated flow through an unconfined aquifer is the primary cause of variable source areas and saturation overland flow [Dunne and Black, 1970], and may provide baseflow to a stream, integrating the inputs over many storms.

[3] Parameterizations of this flux have been developed in the past both from physical arguments by upscaling Richards equation to the hillslope scale and approximating the solution in terms of the “characteristic response function” (CRF) [e.g. Brutsaert, 1994; Berne et al., 2005; Akylas and Koussis, 2007; Verhoest and Troch, 2000; Troch et al., 2003, 2004]. The CRF in this context is the discharge response normalized by the total discharge corresponding to the free drainage of a hillslope aquifer (with given initial and boundary conditions) [Berne et al., 2005]. Other parameterizations of this flux have used simpler, empirical approaches that assume a power law relationship between storage and discharge in the hillslope [e.g., Chapman, 1999].

[4] While effective in many circumstances, both these methods have their drawbacks. In particular, neither provides an account of the way the climatic variability, boundary conditions and hillslope properties interact to control the subsurface runoff behavior. Climatic variability affects the initial storage in the hillslope, which is known to control the form of the CRF [Brutsaert, 1994]. The boundary condition at the base of a hillslope represents the connection of the hillslope to the remainder of the catchment. The role of the boundary condition is generally not explored in the CRF approaches, which usually adopt the mathematically convenient assumption that the saturated thickness at the foot of the slope is fixed at zero depth [Brutsaert, 1994; Ram and Chauhan, 1987; Koussis and Lien, 1982; Troch et al., 2004] (an exception is Akylas and Koussis [2007]). In fact the thickness of a free seepage face is known to vary with the configuration of the hillslope [Chapman, 2005]. The empirical approach avoids all such considerations by subsuming the controls on hillslope behavior into the parameters and calibrating their values from data [e.g., Wittenberg, 1999]. A variety of exponents for these relationships has been observed from less than 1 to 9.1, with typical values around 2.5 [Chapman, 1999; Wittenberg, 1994; Wittenberg and Sivapalan, 1999].

[5] A full accounting of the physical mechanisms that control the variety of coefficients and exponents has not been made. Nor has the link between this and the CRF approach been fully articulated. These relationships are important for developing the closure relations necessary for distributed catchment modeling [Reggiani et al., 1998, 1999; Beven, 2006]. In this work we will examine the dynamics underlying subsurface hillslope flow using an idealized model based on the Boussinesq equation and use the concept of dimensionless similarity and regimes [Robinson and Sivapalan, 1997] to resolve some of these issues.

2. Regimes of Subsurface Lateral Flow

[6] The storage of water in the saturated zone of a hillslope is controlled by the balance between the recharge that is filtered through the unsaturated zone from storms, and the discharge from the hillslope. The relationship between the variability of recharge and discharge for the case of a linear reservoir was laid out in the work of Robinson and Sivapalan [1997], who showed that the peak discharge (per unit area) from a catchment qpeak was a function of the storm duration and time between storms tr and tb and the catchment timescale tc. A linear reservoir is a simple idealization of catchment behavior in which discharge is linearly proportional to storage and inversely proportional to the timescale tc. By writing storage as the product of the storage thickness h and a drainable porosity ϕ, we can obtain a simple expression for the peak storage in a linear reservoir from Robinson and Sivapalan [1997]equation (3):

equation image

Robinson and Sivapalan [1997] described a range of regimes of variability from ‘very fast’ to ‘very slow’, which we will adapt here for our purposes. When tr/tc and tb/tc are small, the hillslope may build up a store of water that carries over between events, and is more sensitive to seasonal variations in storminess than to the variability within the storms themselves. At the other extreme, where the ratios tr/tc and tb/tc are large, the hillslope responds quickly to the rainfall, creating a ‘flashy’ response that is sensitive to variations in recharge intensity.

[7] This provides a simple, first-order prediction of the relationship between storage thickness and the characteristics of the recharge regime. To make use of the regime concept, three new issues need to be addressed. Firstly, the hillslope properties encapsulated in the characteristic timescale tc must be related to measurable quantities of the hillslope. Second, we must determine the circumstances under which hillslope behavior does resemble a linear store, and those where it does not. Where it does not, a new relationship for hmax must be found. Finally, the effects of the boundary condition must be accounted for.

3. Subsurface Flow Dynamics and the Similarity Framework

[8] To address these issues, we will analyze the hillslope flow dynamics using a simple model based on the well-known Boussinesq equation for saturated unconfined groundwater flow over a sloping impermeable base [Brutsaert, 2005; Boussinesq, 1904; Childs, 1971]. This approach has been shown to be accurate where flow lines are approximately parallel to the base, and vertical fluxes are negligible, and where the effects of capillary rise are small [Brutsaert, 2005; Paniconi et al., 2003].

[9] We only consider flow in one dimension (downslope). The hillslope domain is assumed to have a length L with a constant slope tan Θ and a coordinate x [L] with an origin at the bottom of the hillslope. We will assume that the soil is homogeneous and isotropic, with a constant drainable porosity ϕ and hydraulic conductivity K [L/T]. The thickness of the saturated layer measured perpendicular to the bedrock is h(x, t) [L], which we will assume never exceeds the soil thickness.

[10] Based on these assumptions, the flow (per unit width) qx, and Boussinesq equation are given by the following equations [Childs, 1971]:

equation image
equation image

where r(t) [L/T] is the (spatially uniform) recharge rate perpendicular to the bedrock.

3.1. Dimensionless Similarity Framework

[11] We can collapse together the mathematical representation all the hillslopes that are essentially stretched versions of each other in either (or both) the vertical or horizontal dimension. To do this, we rescale the lateral axes by the length of the hillslope equation image = x/L, and the vertical (or almost vertical) dimensions by the height of the hillslope l* = (Ltan Θ). We therefore define the dimensionless thickness η = h/(Ltan Θ). Since this is simply a re-scaling of dimensions, no error or approximation is introduced by doing so, but it should be noted that since h is measured perpendicular to the bedrock (rather than vertically), the similarity will not be based purely on simple stretching distortions where Θ is large.

[12] Substituting these dimensionless variables into the above equations, we can obtain the following:

equation image
equation image

where the time coordinate τ = t/t* has been rescaled by the timescale t*:

equation image

and the flux and recharge rate are expressed in dimensionless form as:

equation image
equation image

[13] The division by the porosity ϕ ensures that the recharge rate is expressed in terms of its effective volume in the soil matrix. Areally averaged flux out of the hillslope in dimensionless terms is simply equation image(t) = equation imagex(0, t).

[14] Although it has been developed independently, this nondimensional form of the Boussinesq equation is similar to that of Koussis [1992] but differs from others (such as Brutsaert [1994]). Despite any differences, the forms are equivalent.

[15] There is some field evidence that the timescale t* obtained in this similarity framework is indeed a controlling factor in the hillslope response. McGuire et al. [2005] used isotope data to estimate the age of water leaving seven catchments in the Cascades in Oregon. They found that the residence time scaled with the ratio of travel path length L and slope Θ. This ratio also appears in the expression for t*. While it should be acknowledged that the residence time and hydrologic response time characterized by t* are not the same thing, there ought to be a relationship between them, as the results of McGuire et al. [2005] suggest.

3.2. Advective and Diffusive Dynamics

[16] The similarity framework immediately throws light on the way the saturated thickness controls the hillslope dynamics. The gradient driving the saturated flow through the hillslope has two terms. Taking the cosine out of the parentheses in equation (2) the first represents flow driven by the head gradient equation image. The second term represents flow driven by the topographic gradient tan Θ. Following from Berne et al. [2005] the former is the ‘diffusive’ component, and the latter is the ‘advective’ component. Noting that equation image scales with the areal mean of the thickness h, denoted equation image, and inversely with the hillslope length L, it is clear that to a first order, the relative importance of the two terms is given by the ratio equation image/(Ltan Θ), which is simply equation image, the areal mean value of η.

[17] We can also define timescales that scale with (1) the time taken for flow to advect through the hillslope domain (advective timescale) and (2) the time taken for a perturbation to the water table to diffuse through the domain. Following Berne et al. [2005] (for the case of a straight hillslope) and others [Beven, 1981; Brutsaert, 1994], these are given by: Advective timescale

equation image

Diffusive timescale

equation image

[18] Note that the ratio of these two timescales is again η.

3.3. Boundary Conditions

[19] A fixed-depth boundary condition is specified by a fixed value h(0) = hb, which may be zero or positive. In dimensionless terms, the fixed depth boundary is simply η(0) = ηb, for some fixed constant ηb. For ηb > 0 and zero recharge or discharge, the water table is everywhere horizontal, which implies that η = max[ηbequation image, 0].

[20] An alternative is to assume that the value of ∂h/∂x is fixed at the base at a constant hb. The thickness at the base is then permitted to vary to accommodate the flow. A special case is where the gradient of h is specified as zero hb = 0, which implies that the water table gradient is parallel to the bedrock gradient at the base. This is called the ‘kinematic’ boundary condition, since the underlying assumption is identical to that of kinematic wave theory. A fixed gradient boundary in dimensionless form is:

equation image

with ηb = 0 being the special case of a kinematic boundary condition. It is common to assume that the upper boundary of the hillslope is a drainage divide and thus qx(L) = 0.

4. Controls on the Nonlinearity of the Hydrologic Response

[21] Placed in the dimensionless framework presented above, storage-discharge relations can be expressed as:

equation image

where equation image is a dimensionless constant. When n is constant and equals 1, the hillslope behaves as a linear store, and the arguments leading to equation (1) are valid. With the similarity framework in place, we are now in a position to examine the controls of the storage thickness and boundary condition on the linearity of the storage-discharge relationship.

[22] To do so, we will consider how the storage and discharge dynamics behave for the asymptotic cases of large storage thickness equation image ≫ 1 and equation image ≪ 1. We will use the terms ‘thick’ and ‘thin’ as shorthand for these cases, but it should be remembered that they refer to dimensionless quantities. This is done separately for kinematic and zero-depth boundary conditions. This will lead directly to a set of approximate scaling relationships describing the storage-discharge relations at (1) steady state under constant intensity recharge, (2) for the early part of subsequent drainage, (3) for the early part of drainage following an impulse and (4) for late drainage. The results of this analysis are summarized in Table 1.

Table 1. Summary of Exponents n for the Storage-Discharge Relation (Equation (12)) Obtained From the Analysis of the Boussinesq Equation in the Similarity Frameworka
 equation image ≪ 1equation image ≫ 1
ηb = 0ηb = 0ηb = 0ηb = 0
  • a

    The exponents depend on the dimensionless storage thickness equation image, the boundary condition (ηb = 0, ηb= 0) and the history of recharge.

Steady state1112
Early draining following steady stateequation imageequation image12
Early draining following impulse0inf,01inf,2
Late drainingequation imageequation image12

4.1. Kinematic Boundary Condition

[23] Consider the flow equation (4) evaluated at the hillslope base:

equation image

[24] We can apply the boundary conditions to this equation directly. For a kinematic boundary condition the gradient term at the boundary is zero. Then the flow thickness at the base is numerically equal to the discharge rate in our dimensionless framework.

equation image

The gradient term at the upper boundary is ∂η/∂equation image = −1. When equation image ≫ 1 the diffusion timescale is very short (tKtU), meaning that gradients in η will decay in times smaller that t*. Since the water table gradients are thus guaranteed to be small (of order 1 or less) compared to the thickness of the hillslope in general (≫ 1), the storage thickness at the boundary will be generally similar to the storage thickness in the hillslope as a whole, either at steady state, during subsequent drainage or following an impulse, and we can write:

equation image

At steady state the storage thickness will therefore vary linearly with the recharge intensity: equation image = equation image. During draining we have:

equation image
equation image

Discharge from thick aquifer (equation image ≫ 1) hillslopes with kinematic boundary conditions will therefore have exponential recession curves with characteristic timescales of t*.

[25] When equation image ≪ 1 (thin aquifer hillslopes), the diffusive term is small compared to the advective term, and the flow out of the hillslope can be approximated by a kinematic wave. In this circumstance the governing equation (5) reduces to [Beven, 1981; Troch et al., 2002]:

equation image

Thus the change in η at the base (and thus the flow) depends primarily on the shape of the water table upslope. The diffusive timescale is large compared to the advective (tKtU), meaning that variations in the water table thickness do not decay in the time taken for them to reach the hillslope base. Consequently the thickness at the base is not a good approximation of the average storage thickness. Although equation (14) still holds, equation (15) does not.

[26] Following an impulse of recharge that produces a uniform initial thickness ηi, equation (18) suggests that initially the discharge at the base remains constant even as the hillslope drains, and the average storage thickness declines linearly in time. If we were to insist on writing the discharge as a function of storage, we would have to write:

equation image

This produces a characteristic ‘sill’ of constant discharge over time in the discharge response to an impulse (see Figure 1). The no-flow boundary condition creates a discontinuity in η at the upslope boundary, which advects downslope and diffuses. For very small η this diffusion is minimal, and the discharge will drop rapidly when this discontinuity reaches the base around time t*. For larger values of η the diffusion will be more pronounced, producing a more gradual decline in discharge.

Figure 1.

Drainage of a homogeneous hillslope with a kinematic boundary condition subject to (left) instantaneous impulses of various sizes (arrows indicate ηi) and (right) steady state recharge of various intensities (arrows indicate intensity equation image). Discharge is normalized by impulse size and recharge intensity, respectively.

[27] Following constant recharge to almost steady state in thin hillslopes, the spatial variation in thickness will reflect the accumulation of discharge downslope [Henderson and Wooding, 1964; Beven, 1981], which is linear. This can be demonstrated by setting the time derivative to zero in equation (18), and noting that equation image = equation image = η(0) at steady state. This yields η(equation image) = equation image(equation image − 1). Thus the saturated aquifer at steady state is approximately a wedge, which produces an average storage thickness of equation image = equation image. Following the cessation of recharge, this drains out of the hillslope as a kinematic wave. The discharge declines linearly as equation image(1 − τ)equation image, and the average storage declines as equation image(1 − τ)2equation image. Combining these and eliminating τ, we obtain:

equation image

That is, the discharge decreases with the square-root of storage as the hillslope drains. This relation will hold following low intensity recharge, but will cease to be accurate for higher intensities because the hillslope will be thicker and the shape of the water table will be influenced by the diffusive dynamics.

[28] Thus we find that the exponents relating storage and discharge for a kinematic boundary condition can vary from 0 to 1, depending on the storage thickness and the nature of the recharge. Thick hillslopes behave as linear reservoirs, while thin hillslopes do not.

4.2. Fixed-Depth Boundary Condition

[29] When there is a fixed-depth boundary condition, the gradient term /dequation image at the base of the hillslope must vary to deliver the outflow. Immediately following an impulse of recharge this gradient is infinitely large: the added water creates a step-change in storage at the hillslope base.

[30] When equation image ≪ 1 the control exerted by the boundary condition is necessarily limited. Because of the disparity in timescales involved, the gradient in η in the vicinity of the boundary cannot diffuse upslope at a rate faster than it is advected downslope. The storage-discharge behavior derived above for a kinematic boundary condition will therefore also apply for the fixed boundary condition case. However, when equation image ≫ 1 the diffusive timescale is much faster, and the behavior of the whole hillslope may be dictated by the gradients generated by the boundary condition.

[31] The gradient of η at the boundary may be positive or negative. Positive gradients imply that flow is contracting towards the outlet, and negative implies that it is expanding. The value of the gradient under steady state recharge can be determined from equation describing the boundary flux (equation (13)). Substituting the steady state condition equation image, and rearranging we obtain

equation image

When the right hand side of this equation is equal to zero the water table gradient at the boundary is equal to the bed slope. At that point the kinematic and fixed-depth boundary conditions are equivalent. Positive values of the boundary gradient (contracting flow) occur when the dimensionless fixed boundary depth ηb is smaller than the dimensionless recharge rate equation image, and negative (expanding flow) when it is larger. These conditions represent a hillslope equivalent of the positive and negative backwater curves in open channel flow, which depend on whether a downstream control produces flow above or below ‘normal’ depth [see Chow, 1959]. ‘Normal’ depth in this case is that produced by the kinematic boundary condition.

[32] When the flux across the boundary is zero but the depth is fixed and nonzero, equation (21) requires that the dimensionless gradient at the boundary is equal to −1. In other words, the gradient across the boundary exactly cancels the topographic gradient. When there is no flow within the hillslope at all, the water table slope must always exactly cancel the topographic gradient, producing a wedge extending back from the hillslope base. Let us call the dimensionless average thickness of the storage under this condition of zero flux as equation image0. When ηb < 1 the wedge intersects the base of the hillslope (since this imples that the elevation at the base h0 is less than the elevation of the bedrock at the ridge L tan Θ). The value of equation image0 can be calculated from the area of this wedge, and is equation imageηb2. When ηb > 1 the water table extends above the top of ridge to a dimensionless thickness (hbL tan Θ)/L tan Θ = ηb − 1. The wedge has a dimensionless mean thickness of 1/2, and so the overall mean thickness is equation image0 = ηb − 1 + equation image = ηbequation image.

[33] We can obtain a storage-discharge for the case of a fixed boundary condition from the following scaling arguments. When the storage is greater than equation image0, the gradient driving flow through the hillslope (∂η/∂equation image + 1 in equation (7)) at a point equation image will scale with the difference between the elevation of the water table ηequation image and the elevation at the hillslope base ηb, and inversely with equation image. Thus as a crude approximation we have:

equation image

Assuming that the flow rate equation image is of a similar order of magnitude everywhere in the hillslope (including the base) and approximating η (equation image) with equation image and equation image = 1/2 (the midpoint of the hillslope) gives the approximate storage-discharge relation:

equation image

[34] For the case where equation image ≫ 1 ≫ equation imageb, this reduces to a simple power law storage discharge relationship with an exponent of 2.

equation image

[35] Thus in thick hillslopes with small, fixed-depth boundary conditions, the exponents of the storage-discharge relations vary between 0 when equation image ≪ 1 and 2 when equation image ≫ 1.

5. Comparison With Numerical Simulation Results

[36] The asymptotic behavior derived above can be compared to numerical solutions of the Boussinesq equation to determine how valid they are. Solutions are obtained for single events, covering the asymptotic regimes examined above: impulse-response, steady state recharge, and draining after steady state.

[37] The numerical model used a boundary integral version of the Boussinesq equation (equation (3)) to obtain a finite area solution. The domain was divided into 32 × 32 cells. The differential equation was integrated using an ordinary differential equation solver in MATLAB [Mathworks, 2007]. Tests indicated that the model conserved mass and correctly predicted the evolution of the water table with acceptable accuracy.

5.1. Discharge Response to Impulses and Constant Recharge

[38] Figures 1, 2, and 3 each represent a different boundary condition imposed on the base of the hillslope. The first is the kinematic (fixed gradient) boundary condition. The second is the fixed depth of ηb = 0, and the third is a fixed depth of ηb = 0.5. This third is included to demonstrate the effect of a ‘backwater’ boundary condition, such as where the hillslope adjoins a riparian aquifer. The plots show the outflow response of the hillslopes following instantaneous inputs (impulses) of recharge to various thicknesses (left plots) and the recession following constant recharge of various intensities to almost steady state (right plots). The thickness equation image is small in the figures when the impulse size is small (ηi < 1) or the recharge intensity is small (equation image < 1).

Figure 2.

Same as Figure 1, but with a fixed depth boundary condition ηb = 0.

Figure 3.

Same as Figure 1, but with a fixed depth boundary condition ηb = equation image.

[39] The results agree with the suggestion that the discharge response varies with the storage thickness and the boundary condition, and that the effect and importance of the boundary condition differs depending on the thickness. For small equation image cases the difference between the kinematic or zero depth boundary condition is small (compare ηi = 0.01 and equation image = 0.01 in Figures 1, 2, and 3). The linear recession following constant, low intensity recharge is seen in the right-hand plot of each figure. The ‘sill’ in the impulse response predicted by equation (19) is evident for very small thicknesses (ηi = 0.01) with the kinematic and fixed zero-depth boundary conditions (Figures 1 and 2).

[40] The behavior of the hillslope with a ‘backwater’ boundary condition is slightly different under the small impulse (ηi = 0.01, Figure 3, left). It exhibits a secondary peak. This is a result of the kinematic wave being generated in the upslope portion of the hillslope then decelerating as it reaches the aquifer at the base and undergoing diffusive deformation.

[41] In the thick hillslope, there are much greater differences between the responses depending on the boundary condition. Under a kinematic boundary condition, the hillslope has an exponential recession, as predicted in equation (15). The match with the exponential recession is apparent in Figure 1.

[42] The impulse responses for the fixed zero depth mirror the characteristic response functions used in the Brutsaert [1994] linearization. The high initial flow due to the boundary gradient is evident.

5.2. Storage-Discharge Relationships

[43] The storage-discharge relationships for homogeneous hillslopes with kinematic and zero fixed-depth boundary conditions are shown in Figures 4 and 5. The predicted storage-discharge relationships are plotted in each figure as dotted and dashed lines, and agree well with the observed behavior. The squares in the figure are the predicted steady state conditions, all of which agree very well with the observed steady state conditions, shown as circles. As predicted, during low intensity recharge (equation image ≪ 1) the discharge is initially equal to storage thickness, but then increases to 2 times equation image at steady state, regardless of the boundary condition. The initial discharge is also equal to equation image in the high intensity cases (equation image ≫ 1) with a kinematic boundary. In the fixed zero-depth case under high intensity recharge the initial discharge is infinitely large (this may not be apparent in the figures due to integration tolerances). At steady state the discharge is proportional to the storage raised to the power of n = 2.

Figure 4.

Storage discharge relations for a homogeneous hillslope with a kinematic boundary condition subject to (left) impulses of recharge and (right) constant recharge to steady state (thin lines), and then drainage (thick lines). Circles indicate the relationship immediately after an impulse or at steady state. The squares are the relationships predicted by the scaling relationships. The dotted and dashed lines are the predicted storage-discharge relationships.

Figure 5.

Storage discharge relations for a homogeneous hillslope with a fixed depth of ηb = 0 at the base. The left plot is for impulses of recharge and the right is for constant recharge to steady state (thin lines), and then drainage (thick lines). Steady state is indicated by the circles. The squares are the steady state conditions predicted by the scaling relationships. The dotted and dashed lines are scaling relationships.

[44] The modeled storage-discharge relations for the draining phase are plotted as thicker lines in the above figures. The period of n = 0 is present for thin (equation image ≪ 1) hillslopes following impulse recharge, as predicted in equation (19). Following low intensity recharge, the n exponent is close to 1/2, as predicted in equation (20). Moreover, this relationship also seems to hold even when the hillslope is draining from a thick state to the point when the thickness at the upper boundary of the hillslope reaches zero, and the water table intersects the bed. However, the aquifer is not wedge shaped at that point and so the relationship is slightly changed. The coefficient in this limiting case appears to be 1/2, giving:

equation image

[45] This implies that this relationship is applicable when equation image > equation image, and equation (20) is appropriate at other times. This relationship is plotted in the figures as a dash-dot line.

[46] In hillslopes with equation image ≫ 1 with fixed zero depth boundary conditions, the discharge is proportional to equation image to the power of 2, as equation (24) predicts. In the kinematic boundary case, the discharge for these hillslopes is linearly proportional to the storage (n = 1). These results suggest that the storage-discharge scaling relations derived in the previous section are a good approximation to the hillslope behavior in a range of circumstances.

6. Regimes of Subsurface Lateral Flow: Linear and Nonlinear Storage-Discharge Relationships

[47] Now that we have an expression for the characteristic timescale t* (equation (6)) and have determined the origins of linear and nonlinear scaling relationships between storage and discharge, we can return to the question of the role of the recharge variability in the hillslope behavior. Equation (1) gives an expression for the peak storage in the case of a linear reservoir. We will refer to this as the ‘regime thickness’ ηR for convenience.

[48] The hillslope does indeed sometimes behave as a linear reservoir for the case of a kinematic boundary condition. Figure 6 compares the values of ηR with the result obtained by running the numerical solution of the Boussinesq equation with kinematic boundary condition with periodic storms until periodic steady state is reached. The approximation is very good, particularly for high intensity storms, where the storage thickness is always ≫ 1 and the system behaves as a linear store according to the scaling relation in equation (15).

Figure 6.

Comparison of the relationship between the event duration τr, interstorm period τb and recharge intensity equation image for a range of values and three boundary conditions. The grey lines represent the values predicted by the similarity framework. The solid lines are those predicted by the hillslope model. Dotted lines refer to the the condition given by equation (28).

[49] For low intensity recharge, the regime thickness tends to overestimate the peak thickness. It does so for three reasons. For fast regimes when the recharge approaches steady state, the thickness of flow is not uniform in thin hillslopes; it will vary from zero at the top of the hillslope to a value equal to equation image at the base. Similarly for slow regimes, the recharge rate converges to steady state recharge at a reduced rate equal to the total recharge averaged over the event and interevent period, and the same internal distribution effect occurs. For intermediate regimes with a short interstorm duration, the difference in the shape of the response between the hillslope model and the linear store becomes important: the truncated tail of the kinematic wave has a shorter ‘memory’ than the exponential recession, and so storage is not carried over between storms.

[50] Overall however, the fit is very good, and we can reasonably assert that for hillslopes with kinematic boundary conditions, the peak storage thickness can be predicted as:

equation image

[51] Figure 6 shows the thickness of the hillslopes at periodic steady state with zero depth (ηb = 0) and backwater (ηb = 0.5) boundary conditions for a variety of recharge characteristics. The behavior differs from that of a kinematic bounded hillslope in two ways: the maximum thickness for high-intensity recharge is depressed in both cases, and there is a minimum storage level for the case with the backwater boundary condition. We can account for these effects using the scaling relationships developed previously.

[52] Consider first the case of the zero depth boundary. Given the scaling relationship expressed in equation (24), we may expect that in a thick hillslope that is approaching steady state, the thickness will scale with the intensity as:

equation image

[53] This is in contrast to the linear scaling observed previously. We may expect that in fast regimes (where the hillslope drains after each event) this relationship will apply when the duration and intensity of events are sufficiently large to supply water to that thickness in each event. That is when τrequation image is similar to or greater than the steady state storage. Substituting this condition into the above equation to eliminate the intensity, we obtain the condition that:

equation image

[54] Thus we may assume that when the regime thickness is greater than 1 and meets the condition stated above, the ‘true’ thickness will be given by equation (27). This area lies above and to the left of the dashed line in each plot in Figure 6.

[55] Thick hillslopes that do not reach steady state in each storm will also be affected by the boundary condition. The combination of interaction between storms and the rapid recessions of these diffusive hillslopes makes finding an appropriate scaling relationship more difficult. However we can approximate their thickness by assuming that the scaling with the intensity at steady state carries through as scaling with the regime thickness, though without the factor of a half, so that for slow regimes:

equation image

[56] As noted previously, where the boundary condition prescribes a thickness at the base ηb greater than 0, there is a minimum storage equation imageb that cannot drain out of the hillslope. This minimum storage is equal to equation imageηb2 for ηb < 1, and ηbequation image for ηb ≥ 1. When the storage in the hillslope approaches this value, the gradient driving the flow out of the hillslope approaches zero. Thus the gradient term in the flow equation (13) scales not with equation image, but with equation imageequation imageb. This correction alters the relations in equations (27)(29), so that we can write:

equation image

[57] For the zero depth boundary this reduces to:

equation image

[58] These relationships are plotted in grey in Figure 6. The overestimate of storage for low intensity is still evident, but the only area where the prediction is not as good as it was in the kinematic hillslope is in the case of slow regimes with diffusive dynamics, where scaling is difficult for the reasons already mentioned.

7. Discussion

[59] The novelty of the similarity framework developed here is the integration of the internal (soil properties, slope) and external (boundary condition, recharge) controls on the dynamics of subsurface flow in a hillslope. This framework allows us to predict (to a first order) how these controls affect the dynamics, through the prediction of the characteristic dimensionless storage thickness equation image (equations (29)(31)).

[60] This framework enriches the understanding of controls on this process provided by the dimensionless parameters of the CRF approach: the Peclet Pe and Hillslope Hi numbers of Berne et al. [2005] and Brutsaert [2005].These parameters are functionally identical to the characteristic storage thickness equation image. For straight hillslopes, assuming that p = 1/3 (following Brutsaert [1994]) and equating D with the maximum (in time) of the mean thickness (in space) of the aquifer equation imagemax, it is trivial to show that

equation image

Indeed, the framework developed here could be used to make predictions of appropriate values of Pe where observations or calibration data are inadequate.

[61] This framework also unifies this understanding with that provided by Beven [1981], when he showed that the hillslope drainage under steady state recharge could be approximated by a kinematic wave if the λ-index of Henderson and Wooding [1964] was less than about 0.75. The λ-index is identical to the dimensionless recharge intensity as it has been formulated here, multiplied by a factor of 4:

equation image

As the regime concept suggests, under steady state recharge, the aquifer thickness depends on the recharge intensity, and so the dependence on λ noted by Beven [1981] is itself a dependence on equation image. Under the assumption of linear storage-discharge scaling, a λ of 0.75 implies that at steady state equation image ≈ 0.18, and so is well into the ‘advective’ mode of hillslope behavior, as would be expected.

7.1. A Simple Typology of Subsurface Flow

[62] We can use the similarity framework to develop a typology of subsurface hillslope flow. The gradient driving subsurface saturated flow can be advective (dominated by the bedrock gradient), or diffusive (dominated by the water table gradient. They can also be fast or slow. Thus four broad types of hillslope dynamics possible, as illustrated in Figure 7. The advective and diffusive dynamics are not independent of the recharge regime, as we have just shown. However, if we allow the dimensionless recharge intensity to vary freely, we can see that for any combination of dimensionless storm and interstorm period (and thus fast or slow regime) it is at least theoretically possible to have any value of the dimensionless storage thickness, and therefore advective or diffusive dynamics. Distinctions between these categories are not sharp, but involve transitions from one type of behavior to another. These might be called ‘medium’ regimes and ‘transitional’ dynamics respectively. Medium regimes occur when storm durations and interevent periods are around unity, say 10−1 < τr < 101 and 10−1 < τb < 101. Note that the nonlinearity of the response when there is a fixed-depth lower boundary shifts some slow diffusive hillslopes into the fast-diffusive regime. Transitional dynamics might be defined (again somewhat loosely) as being when 10−1 < equation imagemax < 101.

Figure 7.

Subsurface flow in hillslopes classified according to whether their hydrologic regimes are fast or slow, and whether their dynamics are dominated by advective or diffusive dynamics, in terms of the values of equation image and τr. Divisions between these types are not as sharp as indicated. The division between fast and slow diffusive types depends on the boundary condition at the base of the hillslope.

7.2. Implications for Storage-Discharge Relations

[63] As well as a simple typology, the similarity analysis and regime concept provide a basis for predicting the storage-discharge relations. The exponent n was shown to be related to the dimensionless storage thickness equation image and to the boundary condition. The boundary condition was shown to be particularly important for thick aquifer (equation image ≫ 1) cases. The strength of the gradient of the saturated thickness that drives the flow in these cases is directly related to the difference between the average storage and the depth at the boundary. In the fixed-depth case examined here, this gradient is responsible for increasing the exponent of the storage-discharge relationship to 2. In the kinematic boundary condition, the boundary thickness adjusts to keep a constant gradient, so the variation in the discharge is controlled only by the thickness of the flow.

[64] In reality, the boundary condition of a hillslope will be controlled by the ability of the receiving landscape element (be it a wide riparian zone, seepage face, or something else) to carry the water away. Thus the boundary condition is not a fixed quantity, but rather emerges out of the relationship between parts of the landscape. This relationship may change over time, as the water levels in the receiving element fluctuate (due to seasonal variations or human extractions). Given the importance of the boundary condition to the value of n, this would suggest that the storage-discharge relations will also change in these cases.

[65] This work also suggests that it may be possible to observe variations in the exponent n as the characteristics of the recharge regime change, due to the link demonstrated between the regime, the storage thickness and the value of n. These variations may be apparent at seasonal timescales in slow hillslopes, or even between storms in very fast cases. They will likely be most apparent for transitional hillslopes (equation image ≈ 1), where the behavior of n is most sensitive to equation image. Single calibrated values of n fixed over the duration of a simulation will not capture these effects.

[66] The similarity analysis suggests that a different set of issues need to be considered when applying storage discharge relations to thin aquifer (equation image ≪ 1) cases. In these cases, the relationship between storage and discharge is not constant over a hydrograph. This is because the discharge in these cases is provided by the lateral transport of water through the hillslope, rather than by pressure gradients set up at the scale of the hillslope aquifer as a whole. Consequently, n varies from 0 following an impulse to 1 at steady state. This implies that hysteresis will be apparent when examining storage discharge relations from a population of storms in these cases. Results presented by Norbiato and Borga [2008] suggest that this is indeed the case. Hysteresis has not been examined here, but will be explored in future work.

[67] The similarity framework also sheds light on the controls on the coefficient in these relationships. Placing the dimensionless storage-discharge relationship in equation (12) in terms of the total discharge per width Q [L2/T] and total storage per unit width S [L2] we obtain:

equation image

The numerical simulations presented above suggest that in the thick hillslope aquifer cases where this relationship is most applicable, equation image is of the order of unity. When n = 1, the coefficient scales with the conductivity and slope, and inversely with the porosity and the length. However, as n increases, the coefficient begins to scale nonlinearly with porosity and length, and the dependence on slope decreases (since the ratio of sin Θ and tan Θ is ≈ 1 for small Θ). When n = 2 the coefficient scales inversely with the cube of the hillslope length L. This suggests that the discharge rates from landscapes where n is likely to be high (according to the classification system described above) will be very sensitive to the degree of dissection, which controls the length of hillslopes discharging into rivers.

8. Conclusion

[68] This work has provided a framework for assessing the controls of the hillslope properties, boundary condition and regime of temporally variable recharge on subsurface lateral hillslope flow. The framework was used to develop tools for making first-order predictions of the dimesionless storage thickness parameter used in CRF approaches.

[69] It also provides a dynamical explanation for some of the variation in the exponents of storage-discharge relations found in the literature. However a full accounting of the variations in n observed in nature has not been achieved in this work. In the homogeneous, planar hillslopes, examined here, n never exceeded 2. Values observed in the field range between as low as less than 1 and as high as 9.1 [Wittenberg, 1994; Chapman, 1999]; values around 2.5 are typical. It has been suggested that these higher exponents may be attributable to the horizontal and vertical convergence of the flow in source areas [Chapman, 1999]. However, this has not been conclusively demonstrated to account for the observed values as far as we know. Alternative explanations may be found by relaxing some of the assumptions made here, such as the uniformity of the porous medium and the impermeability of the bed.

[70] This work has highlighted the importance of the interaction between landscapes and climatic inputs in determining the hillslope behavior. The typology presented integrates information on the soil properties, topography, climate and receiving landscape element, and provides insight into the driver of flow through the hillslope (and therefore what parameterizations of this flux are likely to be successful), and the impact of long and short-term variability of the recharge on the hillslope behavior. Similar typologies that account for more factors, and developed for more processes, could be developed into a classification system for landscape hydrology, as envisioned by McDonnell et al. [2007]. This is left for future work.


[71] Support for this project was provided by a Beatty Fellowship to the first author from the Department of Geography, University of Illinois at Urbana-Champaign, and by a Teragrid Development Allocation through the National Center for Supercomputing Applications. Partial support was also provided by NSF grants ATM 06-28687 (PI: Praveen Kumar) and EAR 06-35752 (PI: Thorsten Wagener). The second author is grateful to Delft University of Technology for providing facilities and partial financial support during a stay as visiting professor, during which time this work was completed. Thanks also to Peter Troch and an anonymous reviewer for their detailed and helpful comments.