## Recharge and Downslope Flow in a Saturated Zone

In fully saturated soil the propagation of the effects of changes in the boundary conditions, such as those due to recharge, is much more rapid than in the unsaturated zone. In shallow subsurface systems, such as where a shallow soil overlays an impermeable rock bed, most of the downslope flow toward stream channels will take place in the saturated zone. Because of the more rapid dissipation of local pressure differences in the saturated zone, a description of flow processes based on Darcy's law is generally more acceptable, even if preferential flow pathways are still contributing to the flow, since those pathways will be subject to similar pressure gradient conditions to the saturated matrix (with the reservation that in large pipe systems the flow may be turbulent and transitional rather than laminar and Darcy's law will not be valid).

Again, in relatively shallow soil systems a kinematic description is a useful analogy for the saturated zone, if we can assume that the local hydraulic gradient is approximately equal to the local slope angle (or even better the local bedslope angle). In this case, the equation of flow is the kinematic wave equation

where h is the depth of saturation above the bed, x is distance measured along the slope, Wx, is the width of the slope at point x, a is the local slope angle, r is the recharge rate at point x and time t, and Th is the integral of the saturated soil hydraulic conductivity function Kh over the depth of saturation h and is called the transmissivity; Th and Kh will be a function of h that may be nonlinear for many soil profiles. The local downslope Darcian velocity of flow (volume flux per unit cross-sectional area) is then given by

The mean pore water velocity is then given by

where 4> is the porosity of the soil. This is the mean velocity of the water itself. The kinematic wave velocity is given by

Vc = Kh sina/^>c where (j)c is an effective storage coefficient for the soil, or effectively in this case, the difference between the soil water content just above the water table and saturation. This last velocity is the rate at which disturbances to the flow are propagated in the direction of flow. The effective storage coefficient will generally be much less than the porosity of the soil, particularly in soils that are near saturation above the water table. Thus the wave speed may be very much faster than the mean pore water velocity, which will be faster than the Darcian velocity (since the porosity must itself be less than 1). The implications of this are that the effects of a recharge to the water table will move downslope much faster than the speed at which the water itself is moving. This will, in general, cause a rise in the subsurface outflow into the downslope stream channel more quickly than the inputs can flow toward that channel. This is one reason why in humid environments subsurface flow processes can make more rapid contributions to storm runoff than has been generally accepted in the past.

Again, Eq. (2), the kinematic wave equation, is an approximation. A fuller description will allow for fully three-dimensional flows in both soil and bedrock, perhaps including flow through bedrock fractures, with time-variable hydraulic gradients (see, e.g., Rasmussen et al, 2000). The same behavior of pressure wave transmission being faster than pore water velocities being in turn faster than Darcian velocities will hold. This is important in understanding the results of tracer experiments.

Equation (2) has been written in a way that allows for the width of the hillslope to vary downslope. It has been known for some time that an important control on the production of subsurface runoff, and of the occurrence of saturated dynamic contributing areas, is the form of hillslopes, in terms of both convergence or divergence in plan, and convexity or concavity in section. Soil water contents will tend to be higher and the saturated zone nearer to the surface in areas that are both convergent and concave. Such areas tend to be found particularly in zero-order headwater basins close to the heads of channels or the appearance of springs. These are areas where the soil is most likely to be close to saturation and consequently will show the greatest potential for acting as runoff source areas or dynamic contributing areas by either surface or subsurface flow processes.

Hillslope topography is, however, not the only cause of variability in flow rates. There is an increasing appreciation of the role of the geological structure of a catchment in controlling the subsurface flow pathways, even in catchments where there is no deep aquifer. The tracing experiments of Genereux et al. (1993), for example, revealed strong variability in the channel inputs in the Walker Branch catchment, Tennessee, that appeared to result from the bedding structure of the underlying rocks. Fracture systems in the near surface geology can also lead to the concentration of flow in certain locations. The occurrence of local perennial or seasonal springs is an indication that such effects may be important. Deeper fracture systems and flows along fault lines may also have an effect on subsurface flow pathways but are very difficult to study. Usually it is necessary to infer the presence of such flow pathways from the geochemical characteristics of baseflows.

For areas of relatively homogeneous shallow soils, it has been demonstrated that one way of predicting the location of such source areas is by use of the pattern of the topographic index a/s (e.g., Kirkby, 1978; see also O'Loughlin, 1981), which is the ratio of the area draining from upslope through unit contour length at any point in the catchment, to the slope angle at that point. The upslope area a represents the propensity for water to collect at a point; while the slope 5 represents the ease with which that point will drain. Approximate steady-state theory suggests that the index can be used as an index of hydrological similarity in that, other things being equal, points with similar values of the index should respond in a hydrologically similar way (Fig. 2). The topographic index has been incorporated into the rainfall-runoff model TOPMODEL and land surface parameterization TOPLATS, which aim to predict the dynamics of the surface and subsurface contributing areas and spatial patterns of latent heat flux in a simple way (Beven et al, 1995; Famiglietti et al, 1992). For recent critiques of the success of using the topographic index to represent contributing area dynamics see Beven (1997).

Use of the topographic index assumes that there is a consistent downslope flow of water on the hillslopes, but this is not always the case, particularly in catchment areas that are subject to extended drying periods. In such catchments, the effective subsur-

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Figure 2 Pattern of the topographic index a/s in comparison with measured areas of surface saturation in basin WC-4 Sleepers River, Vermont (after Kirkby, 1978).

face contributing areas to the channel may not normally extend to the catchment divides and the development and connectivity of local saturation zones may be important. As noted earlier, in some soils, perched water tables may develop over a permeability break in the profile, resulting in increased downslope flow velocities without the profile being saturated to its base. This will tend to occur first in areas at the base of slopes and in hillslope hollows where the soil is normally wetter prior to an event (e.g., Weyman, 1970).

A final subsurface flow process that can lead to rapid subsurface responses is flow in natural soil pipes or along percolines of higher permeability (see, e.g., Beven and

Germann, 1982; McDonnell, 1990). Artificial drainage can have a similar effect, at least under wet conditions (under dry conditions artificial drainage can enhance the storage deficit of the soil prior to a storm and thereby lead to a lower runoff coefficient for an event.

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