Elsevier

Journal of Hydrology

Volume 295, Issues 1–4, 10 August 2004, Pages 140-148
Journal of Hydrology

Soil erosion due to rainfall impact with inflow: an analytical solution with spatial and temporal effects

https://doi.org/10.1016/j.jhydrol.2004.03.007Get rights and content

Abstract

An approximate analytical solution is obtained for a physically based model of soil erosion on a gentle slope driven by rainfall impact in which there is inflow of clear or nearly clear water to the top of the soil bed. Comparison of the approximate analytical and numerical solutions shows very good agreement, except for the first few minutes of an erosion event. The approximate analytic solution is applied using data from an illustrative experiment to explore its physical features. The importance of adequately defining the soil's settling velocity characteristic through the use of a sufficient number of sediment size classes, especially for prediction at short times, is illustrated. The temporal variation in sediment concentration, except at short times, is shown to be more significant than the spatial variation down the eroding surface. Solution of the equations allows visualization of the rate of convective transport of sediment down the eroding surface, this rate decreasing as sediment size increased due to more frequent return of such particles to the soil surface in deposition.

Introduction

The effects on water quality and river health of non-point source delivery of sediment and nutrients has given stimulus to research into the dynamic processes of soil erosion. Of particular interest is the size distribution of transported sediment since finer particles have much lower settling velocity, thus tending to move further and more rapidly in overland flow, so being more likely to reach waterways. These finer particles are also more effective than larger particles in absorbing nutrients (Rose and Dalal, 1988). Thus, the loss of nutrients from agricultural land is very dependent on both the size distribution of sediment and its overall concentration and delivered mass (Palis et al., 1990).

Hairsine and Rose (1991) developed a physically based theory for rainfall-driven soil erosion which treated soil as particles or aggregates of particles with a range of sizes. Size distribution is of particular importance in considering the rate of settling and deposition of sediment in the water layer above the soil surface. The model also considers the formation of a deposited layer of soil formed on the soil surface by deposition from the water layer. This deposited layer acts as a shield to protect the original soil from erosion. Clearly, there is a dynamic interaction between the sediment in the water and deposited layers which is reflected in the mass conservation equations for sediment in each layer.

A series of approaches have been used to solve the rainfall-driven erosion theory of Hairsine and Rose (1991). These authors provided an equilibrium solution, whereas Sander et al. (1996) presented a time-dependent solution at the end of an eroding slope both for sediment concentration and size distribution. In this approximate analytical solution, at any time sediment concentration was assumed to be spatially constant along the eroding soil surface. A fully analytical solution, making the same approximation of Sander et al. (1996), was given by Parlange et al. (1999) by exploiting differing short and long-term behaviour in the solution. Hairsine et al. (1999) obtained an event based solution by integrating the equations over both the length of the eroding land element and the duration of the erosion event. Hogarth et al. (2004) provided a numerical solution of the dynamic model for low slopes with no entrainment with the driving erosive process being rainfall impact. This confirmed that the assumption made in the previous work of Sander et al., 1996, Parlange et al., 1999 would provide good accuracy. It also enabled an exploration of the dynamic changes in settling velocity characteristics (SVC) of soil particles. The solution given by Hogarth et al. (2004) assumed no inflow of water at the top of the eroding slope. It is sometimes also assumed that changes in the dynamic situation can be adequately described as a continuous series of steady states. Examples of that type are the WEPP series of programs (Lane and Nearing, 1989, Nearing et al., 1989), and EUROSEM (Morgan, 1994).

In this paper, we consider the consequences for solution of the theory if there is inflow of clear water at the top of the eroding slope. This allows for the modelling of the erosion below a water source, which may contain at most very fine particles with very little settling. For instance, this is the case when water comes out of an area not susceptible to erosion such as a grass field, an overflowing irrigation ditch, or a forest onto a bare field. An approximate analytical solution is developed for this situation, and compared with a numerical solution obtained using the method of Hogarth et al. (2004) modified to account for inflow. We make similar assumptions to Hogarth et al. (2004), for example, the problem is one-dimensional, the water flow is steady flow and there is no breakdown of soil aggregate during the erosion event. The solution is also compared with experimental data taken from Profitt et al. (1991) to provide an illustration of the solution method.

Section snippets

The mathematical model

Hairsine et al. (1999) presented a set of equations to describe the dynamics of soil erosion by rainfall impact. The water layer was described using(qci)x+(ciD)t=aP(1−H)I+βPmdi−νiciwhile the deposited layer was given bymditici−βPmdiwithH=∑mdimdtandβ=admdtwhere mdt is the mass of the sediment in the deposited layer required to completely shield the original soil from rainfall impact, q is the water flux per unit width of slope, ci is the sediment concentration in size class i, D is

Appropriate analytical solution

Using Eq. (2) in Eq. (1), we can write(Dci)t+mdit+(qci)x=aP(1−H)I

Eq. (11) can be reformulated asDcit+mdit+PmcilnD=aP(1−H)I−Pci

Eqs. , are linear with coefficients independent of time. Thus, using Laplace transforms we can eliminate the transform of mdi. This procedure was followed by Lisle et al. (1998) and the transform of ci could be inverted exactly as there was only one size class. Unfortunately, this is not possible here because of H, which involves all size classes. Instead,

Experimental data

To illustrate the approximate analytical solution, a comparison was undertaken with the experimental data of Profitt et al. (1991). This data was obtained from a tilting flume with an impermeable base with adjustable slope of length 5.8 and 1 m wide described by Misra and Rose (1995). The experiment used here for comparison was of low slope (0.1%) with erosion driven by rainfall impact. The constant rainfall rate was 9.33×10−4 m/min delivered by sprinklers 9 m above the flume and with a ponding

Results and discussion

Analytical solutions have an advantage over numerical solutions for problems involving partial differential equations in that they are computationally more efficient, and for particular values of the independent variables the solution is easily obtained without the requirement of starting from the initial condition each time. Also an analytical solution generally enables the physical processes involved in a problem to be explored more readily. Before using the analytical solution for such

Conclusion

An approximate analytical solution of a model of rainfall-driven soil erosion with inflow at the top of the eroding soil surface has been developed. Except at very early times the analytical solutions are found to be in excellent agreement with the numerical solutions of the same model. The approximate analytical solution is used to explore physical processes during soil erosion.

The definition of a soil's SVC is improved as the number of size classes adopted is increased. Due to the nature of

References (15)

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