G. C. Sander

Exact nonlinear solution for constant flux infiltration

Recently∗ an analytical nonlinear solution to the problem of two phase oil and water infiltration under a constant flux boundary condition was derived. We show that this solution also applies to the problem of constant infiltration of water by introducing a very simple change in the independent variables of space and time.

Unsteady soil erosion model, analytical solutions and comparison with experimental results

Hairsine and Rose developed a soil erosion model which described the erosion transport of the multiparticle sizes in sediment for rain-impacted flows in the absence of entrainment in overland flow. In this paper we extend their steady-state solutions to account for the time variation of suspended sediment concentration during an erosion event. A very simple approximate analytical solution is found which agrees extremely well with experimental data obtained from nine experiments. We are able to reproduce the rapid initial increase to a peak in the total sediment concentration, which occurs about 3–5 min after the commencement of rainfall, as well as the subsequent declining exponential tail towards steady-state conditions. We are also able to show that the fraction of shielding of the original soil bed resulting from depositing sediment reaches its equilibrium value on about the same time-scale as the total peak suspended sediment concentration. Interestingly, we find that the masses of the individual particles which form this deposited layer are far from equilibrium, and that there is a great deal of continuous reworking and sorting of this material during the erosion event. Finally, our solution shows that the initial peak in the total sediment concentration is due to the enrichment of this sediment by the finer size classes and that as the event continues their percentage contribution diminishes.

Stochastic sediment transport in soil erosion

A stochastic model governing the downslope transport of a sediment particle during an erosion event is described. Particles alternate between resting on the soil bed and being transported by the overland flow. Probability densities and moments are constructed for the distribution of a particles position at a given time, and also for time of passage to a given location. We show that a suitable averaging of the stochastic particle motions in our model gives rise to the deterministic erosion differential of Hairsine and Rose (Hairsine, P., Rose, C., 1991. Soil Sci. Soc. Am. J. 55 (2), 320–324). The model generalized H. Einsteins stochastic model (Der Geschiebebetrieb als Wahrscheinlichkeitsproblem, Verlag Rascher, Zurich, 1937) for bedload transport in streams.

Testing a mechanistic soil erosion model with a simple experiment

A simple experiment was used to test the development of a “shield” over the original soil and associated changes in sediment concentrations as described in the mechanistic Rose erosion model. The Rose model, developed for rain-induced erosion and sediment transport on hillslopes (J. Hydrol., 217 (1999) 149; Trends Hydrol., 1 (1994) 443), was applied to a simple experimental set-up, consisting of a small horizontal soil surface (7 £ 7c m 2 ) under constant shallow (5 mm) overland flow with raindrop impact. The soil consisted of two particle size classes, clay and sand, greatly simplifying the analytical solution of the Rose model by reducing the unknown system parameters to one, the soil detachability. Photographic documentation of shield formation corroborated the conceptual validity of the Rose model. Using a single, best-fit value for the soil detachability, quantitative agreement between modeled and experimental results is excellentOR 2 a 0:9U: This research provides lucidity to the primary processes enveloped in the Rose model and these mechanisms can be extrapolated to more complicated or realistic systems in which the individual processes may be more difficult to recognize. q 2001 Elsevier Science B.V. All rights reserved.

Unsteady soil erosion due to rainfall impact: a model of sediment sorting on the hillslope

A new method is presented for predicting sediment sorting associated with soil erosion by raindrop impact for non-equilibrium conditions. The form of soil erosion considered is that which results from raindrop impact in the presence of shallow overland flow itself where the flow is not capable of eroding sediment. The method specifically considers early time runoff and erosion when sediment leaving an eroding area is generally finer and thus may have a higher potential for transport of sorbed pollutants. The new mechanism described is the formation of a deposited layer on the soil surface, which is shown to lead to sediment sorting during an erosion event. The deposited layer is taken to have two roles in this process: to temporarily store sediment on the surface between successive trajectories, and to shield the underlying soil from erosive stresses. Equations describing the dynamics of the suspended sediment mixture and the deposited layer are developed. By integrating these equations over the length of eroding land element and over the duration of the erosion event, an event-based solution is proposed which predicts total sediment sorting over the event. This solution is shown to be consistent with experimentally observed trends in enrichment of fine sediment. Predictions using this approach are found to only partly explain measured enrichment for sets of experimental data for two quite different soils, but to be in poor agreement for an aridsol of dispersive character. It is concluded that the formation of the deposited layer is a significant mechanism in the enrichment of fine sediment and associated sorbed pollutants, but that processes in the dispersive soil are not as well described by the theory presented.

Kinematic flow approximation of runoff on a plane: An exact analytical solution

Abstract A general analytical solution to the kinematic flow approximation is presented for an excess rainfall which is any arbitrary function of time. The present solution generalizes an earlier solution which applies when the excess rainfall is constant for a finite time interval. Application of the solution is illustrated by considering the runoff resulting from a variable rate of excess rainfall, and also a problem in border irrigation.

Modeling the dynamics of soil erosion and size-selective sediment transport over nonuniform topography in flume-scale experiments

Soil erosion and the associated nutrient fluxes can lead to severe degradation of surface waters. Given that both sediment transport and nutrient sorption are size selective, it is important to predict the particle size distribution (PSD) as well as the total amount of sediment being eroded. In this paper, a finite volume implementation of the Hairsine-Rose soil erosion model is used to simulate flume-scale experiments with detailed observations of soil erosion and sediment transport dynamics. The numerical implementation allows us to account for the effects of soil surface microtopography (measured using close range photogrammetry) on soil erosion. An in-depth discussion of the model parameters and the constraints is presented. The model reproduces the dynamics of sediment concentration and PSD well, although some discrepancies can be observed. The calibrated parameters are also consistent with independent data in the literature and physical reason. Spatial variations in the suspended and deposited sediment and an analysis of model sensitivity highlight the value of collecting distributed data for a more robust validation of the model and to enhance parametric determinacy. The related issues of spatial resolution and scale in erosion prediction are briefly discussed.

Kinematic flow approximation to runoff on a plane: An approximate analytic solution

Abstract A new theory is given which relates excess rainfall to runoff for overland flow on a plane. The theory assumes that rate of change of water flux with distance down the plane is constant at any time, but its variation with time is such that runoff from the plane is exact. It follows that mass of water is conserved for the entire runoff event, but not exactly at all times within the event. The magnitude of the error in the solution due to this approximation is investigated by comparison with the more complex exact theory. If runoff from the plane is measured, the theory allows simple direct estimation of excess rainfall as a function of time. Alternatively, if rainfall and infiltration rates are known then runoff can be calculated. This approximate theory and the exact theory are also compared by using them to estimate infiltration rate into a small planar catchment from measured rates of rainfall and runoff. The accuracy and great simplicity of the approximate theory makes it useful in practice.

Limitation of the transport capacity approach in sediment transport modeling

In a recent paper by Polyakov and Nearing (2003) it was shown experimentally that the sediment transport capacity in a rill is not unique for a given soil type, slope, and flow rate. Indeed, they found that the transport capacity was dependent on whether sediment transport in the rill was occurring under net erosion or net deposition conditions. They concluded that this nonuniqueness in transport capacity is a discrepancy that needs addressing in soil erosion models. Here we postulate that this behavior occurs as a result of defining transport capacity as an model input to distinguish between net erosion and net deposition regimes, instead of determining it as an outcome between the separate but continuous rate processes of deposition and entrainment such as is the case for the multisize class erosion model of Hairsine and Rose (1992a, 1992b). This model is used to reinterpret and reproduce the results of Polyakov and Nearing (2003). The analysis shows that the transport capacity cannot be unique for a soil composed of a range of size classes and that uniqueness only occurs for the exceptional case of single size class soil. Consequently, when used as a model input, the transport capacity concept is deficient in modeling sediment transport of real soils across different flow conditions.

Addendum to unsteady soil erosion model

A simple analytical approximation is obtained for erosion on a hillslope involving sediment transport when rainfall is the only source of water. The solution extends an earlier study requiring some numerical evaluation. Owing to its simplicity, the solution gives a clear understanding on the influence of the physical processes involved. Fundamentally, there is a very short time behavior associated with rainfall impact and a longer time behavior controlled by convection. It is this time decoupling which permits the solution to be expressed in simple terms. The short time analytical results can be used to simplify greatly numerical procedures.