Availability and performance of sediment detachment and transport functions for overland flow conditions

Abstract

Soil erosion is a global environmental problem. To quantify water erosion rates at the field, hillslope or catchment scale, several spatially-distributed soil erosion models have been developed. The accuracy of those models depends largely on the sediment detachment and transport functions used, many of which were developed from empirical research. In this paper, the physical basis of the available sediment detachment and transport functions is reviewed, and their application boundaries determined. Well-known and widely-used sediment detachment and transport functions are discussed on the basis of composite force predictors, i.e. shear stress, stream power, unit stream power and effective stream power, and their suitability is elucidated based on information in the literature. It was found that only a few sediment detachment functions are available, and those have been poorly tested. Most erosion models ignore direct calculation of sediment detachment, but use the sediment transport capacity deficit approach to estimate detachment rate. Many more sediment transport functions are available that also tested better for overland flow conditions. However, our tests did not result in a single function that appeared to perform best under a range of experimental conditions. The unit stream power-based functions developed by Govers seem to be the most promising ones for water erosion modelling. It is therefore recommended to evaluate the performance of existing sediment transport functions with more detailed field and laboratory datasets.

Editor Z.W. Kundzewicz

Résumé

L’érosion des sols est un problème environnemental mondial. Pour quantifier les taux d’érosion de l’eau à l’échelle terrain, versant ou bassin versant, plusieurs modèles d’érosion des sols spatialement distribués ont été développés. La précision de ces modèles dépend en grande partie sur les fonctions de détachement de sédiments et de transport utilisés, dont beaucoup ont été développés à partir de recherches empiriques. Dans cet article, la base physique des fonctions de détachement de sédiments et de transport disponibles est examiné, et leurs limites d’application déterminée. Bien connus et fonctions largement utilisés détachement de sédiments et de transport sont examinés sur la base de prédicteurs de force composites, soit contrainte de cisaillement, la puissance de flux, la puissance du flux de l’unité et de la puissance de flux efficaces, et leur adéquation élucidé basées sur des informations dans la littérature. Il a été constaté que seules quelques fonctions de détachement de sédiments sont disponibles, et ceux qui ont été mal testé. La plupart des modèles d’érosion ignorent calcul direct de détachement de sédiments, mais utilisent l’approche du déficit de capacité de transport des sédiments pour estimer le taux de détachement. Beaucoup de fonctions de transport plus de sédiments sont disponibles qui a également testé mieux pour des conditions d’écoulement terrestres. Toutefois, nos tests ne ont pas abouti à une fonction unique qui semble fonctionner le mieux dans une gamme de conditions expérimentales. Les fonctions à base de puissance-courant unitaire développés par Govers semblent être les plus prometteuses pour la modélisation de l’érosion de l’eau. Il est donc recommandé d’évaluer la performance des fonctions de transport des sédiments existant avec des ensembles de données de terrain et de laboratoire plus détaillées.

Key words:

• literature review
• sediment transport function
• sediment detachment function
• performance of sediment functions

Mots clefs:

• revue de la littérature
• fonction de transport des sédiments
• fonction de détachement de sédiments
• exercice des fonctions de sédiments

1 INTRODUCTION

Soil erosion is a common global problem that adversely affects the productivity of agriculture (Lal and Stewart 1990, Pimentel et al. 1995, Yang et al. 2003). Severe erosion may occur when unprotected soil is exposed to rain or wind energy (Barrow 1991). According to Barrow (1991), globally 75 billion tons of soil are eroded from agricultural lands and around 20 million hectares of land are lost due to erosion each year. Soil erosion rates are high in Asia, Africa and South America, averaging 30–40 t ha^-1  year^-1 (Barrow 1991). Estimates for Asia are higher than the averages given by Barrow (1991), and are in the order of 138 t ha^-1 year^-1 (Sfeir-Younis 1986). Erosion causes land degradation and reduces crop production potential, while the eroded sediment contaminates surface waters and reduces the storage capacity of reservoirs that directly affects irrigated agriculture and hydro-electricity generation. Soil erosion is also one of the main causes of global warming because it emits CO₂ and CH₄ gases from soil to the atmosphere (Lal 2004).

To assess water erosion problems in catchments, scientists have developed several spatially distributed soil erosion models with various degrees of sophistication. Examples are CREAMS (Knisel 1980), KYERMO (Hirschi and Barfield 1988a, 1988b), PRORILL (Lewis et al. 1994a, 1994b), KINEROS2 (Smith et al. 1995), LISEM (De Roo et al. 1996), RUSLE (Renard et al. 1997), EUROSEM (Morgan et al. 1998a, 1998b), EGEM (Woodward 1999), GLEAMS (Knisel and Davis 2000), WEPP (Flanagan et al. 2001), MUSLT (Sadeghi et al. 2013), and DREAM (Ramsankaran et al. 2013). Some of those models are fully empirical (e.g. RUSLE), while others are physically-based (e.g. KINEROS2), approaching the erosion problem from physical laws (Beven 2001). Those empirical and physically-based models have been applied in catchment-scale erosion studies with varying degrees of success (e.g. Finney et al. 1993, Summer and Walling 2002, Kim et al. 2007, Larsen and MacDonald 2007, Grismer 2012, Ramsankaran et al. 2013, Wieprecht et al. 2013).

Sediment detachment and transport are important sub-processes in water erosion. These two components of soil erosion are critically interlinked (Foster and Meyer 1972). Accurate prediction of sediment detachment and transport rates is very important in the development of a water erosion model. Correct calculations of the amounts of sediment detachment and transport play a vital role in the accuracy of the outcomes of a given spatially-distributed soil erosion model. For both sub-processes in water erosion there is a variety of predictive functions available.

The detachment functions used in most models are of empirical nature and were derived from experimental data (e.g. Foster 1982, Elliot and Laflen 1993). The empiricism may cause problems when those detachment functions are used outside the experimental domain for which they were derived. Most of the existing sediment transport functions were originally derived for channel flow. These functions are commonly used in several physically-based soil erosion models to estimate sediment transport in shallow overland flows (Smith et al. 1995, Flanagan et al. 2001). But, the applicability of stream flow functions has become questionable under overland flow, because the water layer depths and discharges are usually much smaller in overland flow. Moreover, hillslope surfaces are usually rougher than streams. This is due to obstacles at the surface, such as stones, plant stems, leaves, etc. Such higher values of roughness substantially reduce the transport capacity of the flow (Govers and Rauws 1986, Abrahams and Parsons 1994, Abrahams et al. 2000). Also raindrops can disturb the thin overland flow layers, which is not the case for channel flow, where the water depth is sufficient. It is therefore questionable if those functions give reliable results for water erosion predictions under overland flow conditions.

Given the importance of sediment detachment and transport functions for accurate water erosion modelling, it is important to know what the physical basis of the main functions is and how well these functions perform under different experimental conditions. The main aim of this paper was to review and summarize the available information about sediment detachment and transport functions that are often used in spatially-distributed soil erosion models. The review encompasses (a) processes engaged in soil erosion; (b) the availability and performance of soil detachment functions for overland flow conditions; and (c) the availability and performance of sediment transport functions for overland flow conditions.

2 PROCESSES OF SEDIMENT DETACHMENT AND TRANSPORT BY OVERLAND FLOW

The soil erosion process by water is usually described in two steps, i.e. detachment and transport (Ellison 1947). Soil detachment is the dislodgement of soil particles from the soil mass, while transport is the movement of soil particles from one location to another (Foster and Meyer 1972). The dislodgement of soil particles is mainly caused by the forces applied by raindrops and by overland flow (Meyer and Wischmeier 1969). Detachment of soil particles by raindrops depends on several variables, such as rain drop size, fall velocity, rainfall intensity and soil erodibility (Owoputi and Stolte 1995). The impact of raindrops on sediment transport in the absence of overland flow, i.e. splash erosion, has been comprehensively studied (Poesen and Savat 1981, Savat 1981, Moss and Green 1983), and is not considered in this paper. Here we deal with detachment and transport of sediment by layers of overland flow only.

Detachment by overland flow is caused by the forces of the layer of flowing water affecting the soil surface. Theoretically, a given layer of overland flow on a certain slope can detach a maximum amount of sediment, indicated by the detachment capacity (D_c). The actual rate of detachment (D_r) will normally be lower than the detachment capacity, because the maximum detachment can only occur with clean water that contains no sediments (Foster and Meyer 1972).

Sediment transport is also an important component of the soil erosion process. Under overland flow conditions, sediment can be transported in the form of bedload and suspended load (Allen 1994). The sediment transport rate (T_r) mainly depends upon the transport capacity (T_c) of overland flow, defined as the maximum amount of sediment that can be transported at a particular discharge on a certain slope (Merten et al. 2001). Several studies have shown that the T_c is dependent on bed slope, discharge, flow velocity, flow depth and sediment particle size (Nearing et al. 1991, Zhang et al. 2003, Aksoy et al. 2013).

Foster and Meyer (1972) developed a first-order detachment–transport coupling approach for overland flow. This approach assumes that the available flow energy is preferentially used for sediment transport and any excess energy will be utilized for soil particle detachment. The ratio of detachment rate D_r (kg m^-2 s^-1) and detachment capacity D_c (kg m^-2 s^-1), plus the ratio of sediment transport T_r (kg m^-1 s^-1) and transport capacity T_c (kg m^-1 s^-1) is equal to a constant value of 1:

(1)

The resulting soil erosion can be described by three different processes: inter-rill, rill and gully erosion. Inter-rill or sheet erosion can be defined as the removal of thin soil layers from the soil surface (Foster and Meyer 1972). Raindrops and overland flow are both responsible for sediment detachment and transport in inter-rill areas (Zhang et al. 2003). Detachment by raindrop impact is the dominant process under inter-rill erosion, while the impact of overland flow to detach soil particles is often considered negligible. Overland flow is only considered as a transporting agent. The term ‘rain splash’ is also commonly used under inter-rill erosion, and is defined as the capability of raindrops to dislodge and transport soil particles (Owoputi and Stolte 1995).

Rill erosion is the removal of soil by concentrated flow running through small channels that can be easily obliterated under normal tillage practice (Loch et al. 1989). Foster et al. (1982) also defined rill erosion in a similar way and used a maximum channel depth of 300 mm to define a rill. In rill erosion, sediment is mainly detached and transported by overland flow (Owoputi and Stolte 1995). Several studies have specified that rill erosion contributes significantly to sediment yield (e.g. Young and Wiersma 1973, Fullen and Reed 1987).

Gullies are larger than rills and cannot be obliterated by ordinary tillage operations (Toy et al. 2002). Poesen et al. (2003) estimated that gully erosion can make up 10–94% of the total sediment production within a catchment. The term ‘ephemeral gully’ is commonly used in soil erosion studies, and is defined as a small incised channel formed on agricultural lands by concentrated flows that is normally refilled by regular farming operations during the next cropping season.

Here, only those sediment detachment and transport functions are reviewed that are used under inter-rill and rill erosion conditions. Hence, only thin layers of overland flow are considered, thus excluding gully formation as the layers of overland flow become much thicker then.

3 DETACHMENT AND TRANSPORT PREDICTORS

Most detachment and transport functions were derived from laboratory (flume) experimental data, in which the transport of non-cohesive homogeneous sediment was studied under controlled hydraulic conditions. The rate of sediment detachment and transport under overland flow depends on many hydraulic and sediment parameters, such as flow discharge, bed slope gradient, flow depth, flow velocity, particle size, bed roughness elements and rainfall impact. Several researchers studied the impact of slope gradient, flow velocity and flow depth on sediment transport capacity (e.g. Govers 1992, Abrahams et al. 1996, Liu et al. 2000, Ali et al. 2012). These hydraulic variables are well parameterized in existing detachment and sediment transport capacity functions to predict detachment and transport capacity of overland flow as a function of a driving force. The most widely used composite force predictors are shear stress, stream power, unit stream power and effective stream power.

3.1 Shear stress

Shear stress (τ) is the force per unit area (N m^-2) applied by water flowing on the soil surface. It is defined as (Beven 2001):

(2)

where ρ (kg m^-3) is the water density; g (N kg^-1) is the acceleration due to gravity; R is the hydraulic radius, which is assumed equal to flow depth, h (m) under overland flow; and S (m m^-1) is the slope gradient.

3.2 Stream power

Bagnold (1966) introduced the concept of stream power (ω) by assuming that the sediment transport rate should be related to the time rate of potential energy expenditure per unit bed area. His study clearly demonstrated that the ability of a stream to transport sediment depends on its available power, and not on its available energy (Yang 1973). Stream power (J m^-2 s^-1) is defined as the product of shear stress and the average flow velocity:

(3)

where u (m s^-1) is the mean flow velocity.

3.3 Unit stream power

Yang (1972) assumed that the sediment transport rate should be related to the rate of potential energy dissipation per unit weight of water. He defined unit steam power, ω_u (m s^-1) as the product of average flow velocity and slope gradient:

(4)

The basic difference between stream power and unit stream power is that the stream power deals with the power per unit bed area and unit stream power deals with the power per unit weight of water.

3.4 Effective stream power

Effective stream power (ω_eff) is an empirically derived relationship and is fundamentally based on the concept of shear stress (Bagnold 1980). This concept was primarily used by Govers (1990) and Everaert (1991) for the development of empirical relationships to predict the sediment transport rate. Effective stream power (N^1.5 s^-1.5 m^-2.17) is defined as:

(5)

4 SEDIMENT DETACHMENT FUNCTIONS

The dislodgement of soil particles is mainly caused by the forces applied by overland flow and raindrop impacts. A consensus has yet to develop on the mathematical formulation of soil detachment parameters in the soil erosion process. Because of the complexities in developing physical relationships between detachment parameters of overland flow, most of the existing functions were developed empirically through regression analyses of experimental data. The derived functions use a composite predictor for prediction of the detachment capacity by the given characteristics of the overland flow. The actual rate of detachment will subsequently depend on the amount of sediment in transport, similar to the approach of Foster and Meyer (1972). Several researchers adopted the shear stress and stream power concepts (equations (2) and (3)) for the estimation of soil detachment rate and capacity under overland flow conditions (e.g. Foster 1982, Hairsine and Rose 1992). In addition, an alternative approach using the sediment transport capacity deficit for estimating the detachment rate was adopted in some erosion models (e.g. Blau et al. 1988, Morgan et al. 1998a, 1998b). No functions using unit stream power and effective stream power for detachment rate or capacity were found in the literature.

4.1 Shear stress-based functions

In the functions that use shear stress as the driving force, the detachment of soil particles occurs when the shear stress applied by overland flow is high enough to pull soil particles away from the parent material. The shear stress applied by overland flow is primarily a function of flow depth and slope gradient (Giménez et al. 2007). Because the flow depth is usually significant and more easily measured than the mean flow velocity, most researchers considered shear stress the best predictor for soil detachment by overland flow in rills (Foster 1982, Wicks and Bathurst 1996, Zhu et al. 2001).

4.1.1 Foster (1982)

derived a function for prediction of rill detachment capacity D_c (kg m^-2 s^-1) from the DuBoys bedload transport function (DuBoys 1879). It uses the concept that detachment capacity depends upon the shear stress value of the flow exceeding a critical shear stress (τ_cr) value for detachment (Foster et al. 1981). If the shear stress is below the critical value, no detachment will occur. When shear stress exceeds the critical value, the amount of detachment depends upon the soil erodibility. The detachment capacity function of Foster (1982) is defined as:

(6)

where K is the soil erodibility factor (unit depends on value of a and becomes s m^-1 for a = 1); τ_cr is the critical shear stress (N m^-2), and a is an empirical constant. Many water erosion models, e.g. CREAMS (Knisel 1980), PRORIL (Lewis et al. 1994a, 1994b) and WEPP (Flanagan et al. 2001), use detachment capacity functions that are based on the same concept as equation (6).

4.1.2 Wicks and Bathhurst (1996)

derived a function for the SHESED model to predict detachment capacity, D_c (kg m^-2 s^-1). This function is somewhat different from equation (6), because both rill and inter-rill erosion processes were lumped:

(7)

where K′ is the soil erodibility factor (kg m^-2 s^-1).

4.1.3 Zhu et al. (2001)

The concept of a critical condition has often been criticized, because the detachment of soil particles may occur for shear stresses even below a certain defined critical value, which means this term is subjective and rather vague (Prosser and Rustomji 2000). Therefore, a simplified version of equation (6) was used by Zhu et al. (2001), neglecting the critical shear stress term:

(8)

where K″ is the soil erodibility factor (unit depends on value of b and becomes s m^-1 for b = 1) and b is an empirical constant. However, the disadvantage of this function is that it always predicts sediment detachment, even for very low shear stress values. A similar function was developed by Onstad (1984).

4.2 Stream power-based functions

A few studies have shown that shear stress is not necessarily always the best hydraulic predictor for detachment of all soil types. Elliot and Laflen (1993) found that stream power was a better predictor for detachment rate when compared with shear stress. The stream power approach was also adopted by Zhang et al. (2003) to predict the detachment capacity, and by Hairsine and Rose (1992) to predict detachment rate.

4.2.1 Hairsine and Rose (1992)

proposed a complex function using stream power to estimate rill detachment rate, D_r (kg m^-2 s^-1). In this function, the detachment rate of a certain flow depends on the difference between the stream power and its critical value (ω_cr), which implies that, if the value of stream power is below its critical value, then there will be no detachment. This is similar to the approach of Foster (1982) for shear stress (equation (6)). Hairsine and Rose (1992) argued that available flow energy is consumed by four sub-processess: (i) overcoming resistance of cohesive soil against detachment, (ii) entrainment of soil particles from rill bed, (iii) entrainment of previously detached sediments, and (iv) dissipation of energy:

(9)

where H is the fractional protection of the underlying soil provided by the deposited layer; w is the rill width (m); h is the flow depth(m); F is the effective fraction of flow energy causing detachment; ω_cr (J m^-2 s^-1) is the critical stream power, N is the number of sediment settling velocity classes, and E (J kg^-1) is the energy required for detachment per unit mass of soil matrix. A similar approach was incorporated in the GUEST model (Misra and Rose 1996).

4.2.2 Elliot and Laflen (1993)

developed an empirical function from their experimental data for head cutting in rill erosion based on the stream power concept. They also used the difference between the stream power and its critical value (ω_cr) to quantify the detachment capacity (D_c):

(10)

where K_c is the soil erodibility factor (s² m^-2). This approach is similar to equation (6), but now stream power is used instead of shear stress.

4.2.3 Zhang et al. (2003)

carried out flume experiments to quantify the detachment capacity of natural undisturbed soils. They correlated detachment capacity directly to stream power with a power function by neglecting the fuzzy term of critical stream power, similar to equation (8):

(11)

where K_c′ is the soil erodibility factor (unit depends on value of c and s² m^-2 is only valid for c = 1); and c is an empirical constant. Zhang et al. (2003) determined the values of K_c′ and c by regression analysis that were equal to 0.0088 and 1.07, respectively, for natural, undisturbed, mixed mesic typical undorthent soil.

4.3 Other functions

An alternative approach to estimate soil detachment in rills is based on the transport capacity deficit concept (Knapen et al. 2007). According to this concept, the detachment rate D_r (kg m^-2 s^-1) is linearly dependent on the difference between the sediment transport capacity T_c (kg m^-1 s^-1), and the current sediment transport rate T_r (kg m^-1 s^-1). Hence, to apply such a function, it is required that T_c and T_r are known, or can be calculated (see Section 5).

4.3.1 Blau et al. (1988)

developed a function based on the transport capacity deficit approach, which was later incorporated in the KINEROS2 model (Smith et al. 1995) to predict soil detachment rate in rill flows. The function is defined as:

(12)

where A is the cross-sectional area (m²) of the rill flow; and c_g is the transport rate coefficient (m^-3), which accounts for the resistance of a soil particle against erosion.

4.3.2 Morgan et al. (1998a, 1998b)

The EUROSEM model (Morgan et al. 1998a, 1998b) also calculates detachment rate in rills by using the transport capacity deficit approach in combination with settling velocity (v_s) of soil particles. Moreover, the soil erosion resistance was also incorporated as a flow detachment efficiency coefficient B (s m^-3). The Morgan et al. (1998a, 1998b) function is defined as:

(13)

where v_s is the settling velocity (m s^-1) of sediment particles. This approach is also used in the LISEM (De Roo et al. 1996) and DREAM (Ramsankaran et al. 2013) soil erosion models.

4.4 Suitability of soil detachment functions

Among the available detachment rate and detachment capacity functions, only the performance of existing shear stress-based detachment functions has been evaluated by Zhu et al. (2001). They checked the suitability of linear and power detachment capacity functions (equations (6) and (8)) with and without critical shear stress by using laboratory (Ghebreiyessus 1990, Zhu et al. 1996) and field (Elliot et al. 1989) datasets. To obtain a linear function, they considered the value for the exponent of shear stress in equation (6) equal to unity (a = 1). Five midwestern USA soils with textures ranging from sandy loam to silty clay loam, i.e. Barnes loam (fine loamy), Forman clay loam (fine loamy), Sverdrup sandy loam (sandy), Mexico silt loam (fine) and Sharpsburg silty clay loam (fine silty), were used to conduct the experiments. Zhu et al. (2001) found that the shear stress range in the laboratory experiments was 2–5 times smaller than the range of shear stresses obtained in field experiments. In addition to this, the procedures adopted to conduct laboratory experiments were also different from the field experiments. Zhu et al. (2001) concluded that the linear function is simple to use and its parameters can easily be estimated. But, the linear function did not fit well to the data collected under low and high shear stresses. It underestimated detachment capacity by 25% at high shear stress and overestimated detachment capacity by 30% at low shear stress. Moreover, the values of soil erodibility parameter (K) for the linear function varied by a factor of 3. More stable erodibility parameters were obtained with the power function (a ≠ 1), because this function fitted better to the observations.

No study was found that checked the suitability of stream power-based functions and the functions using the transport capacity deficit approach. Hence, there is a need to evaluate the performance of the functions based on the stream power and transport capacity deficit approaches under a variety of laboratory and field conditions.

5 SEDIMENT TRANSPORT FUNCTIONS

Most of the existing sediment transport capacity functions were derived for channel flow, while only a few functions were derived under overland flow conditions by using small-scale laboratory experimental data (Govers 1990, Everaert 1991, Abrahams et al. 2001). Therefore, one should be careful when applying these functions outside the range of conditions for which they were originally developed. The available sediment transport capacity functions are using shear stress, stream power, unit stream power or effective stream power as driving force. The described sediment transport capacity functions considered in this review are well-known and widely used for soil erosion studies.

5.1 Shear stress-based functions

The concept of shear stress was first utilised by DuBoys (1879) for the derivation of a bedload sediment transport function. Afterwards, many other scientists (Meyer-Peter and Muller 1948, Laursen 1958, Yalin 1963, Smart 1984, Low 1989, Lu et al. 1989, Abrahams et al. 2001) introduced modifications to DuBoys’ original function to estimate bedload.

5.1.1 DuBoys (1879)

proposed a bedload function for channel flow. The concept was based on the theory that sediment moves in layers along the stream bed. For the initiation of sediment motion, the shear stress (τ) applied by the flow on the stream bed must exceed the critical shear stress (τ_cr). If the applied shear stress is below its critical value, than there will be no sediment transport:

(14)

where ρ_s is the density (kg m^-3) of the sediment particles; and D (m⁴ s³ kg^-2) is a coefficient that depends upon sediment particle diameter. Similar to DuBoys function, Shields (1936) derived a bedload function for channel flow by incorporating the effect of mean flow velocity with excess shear stress theory.

5.1.2 Meyer-Peter and Muller (1948)

derived an empirical bedload function for channel flow at slopes ranging between 0.04 and 2.0% on the basis of DuBoys’ excess shear stress theory. For regression analysis, they used experimental results of uniform and non-uniform sediments of specific gravities ranging from 1.3 to 4.0:

(15)

where s (-) is the sediment specific gravity (ρ_s ρ^-1).

Many transport capacity functions have been derived that use the excess shear stress approach and are similar to equations (14) and (15). The most important examples are the functions of Bagnold (1956), who introduced D₅₀ in the equation; Yalin (1963); Smart (1984), who introduced slope and mean velocity in the equation; Low (1989); Lu et al. (1989) and Govers (1992).

5.1.3 Laursen (1958)

developed a total sediment load function for channel flow, in which the sediment concentration was related with relative roughness and excess shear stress. This relationship is corrected by an empirical function of, which accounts the effectiveness of turbulence in maintaining the bed material in suspension. The function was derived using wide and narrowly graded grain sizes ranging from 0.011 to 4.08 mm, and is defined as:

(16)

where q is the unit flow rate (m² s^-1), u_* is the shear velocity (m s^-1), defined as , and is an empirical function that was derived from flume experiments.

5.1.4 Abrahams et al. (2001)

A total-load sediment transport function was developed by Abrahams et al. (2001) using dimensional analysis for inter-rill flows, both with and without rainfall by using the excess shear stress theory. The derived function was based on the dataset obtained from flume experiments which were conducted by using non-cohesive sediments. Moreover, they also parameterized the effect of bed roughness in terms of roughness concentration and roughness diameter in the proposed function:

(17)

(18)

(19)

where C_r is the roughness concentration (%) and H_r is the mean roughness height (m). This function can also be used for plane beds by taking the values for regression coefficient x and dimensionless mean flow velocity exponent y equal to 1.

5.2 Stream power based functions

Bagnold (1966) introduced the concept of stream power (ω) and used this to develop a total load sediment transport function. Later, the stream power concept was used by Engelund and Hansen (1967) and Bagnold (1980) in their well-known transport capacity functions.

5.2.1 Bagnold (1966)

proposed a total load function using stream power (ω), in which total load was calculated by summing bedload and suspended load. For the derivation of this function, the immersed weight of bedload is related to the momentum transfer due to grain collision, while the suspended load immersed weight is related to an upward momentum transfer due to fluid turbulence. The available flow power is the only supply of energy to both transport mechanisms. The function is defined as:

(20)

where e_b (-) is the bedload efficiency factor.

5.2.2 Engelund and Hansen (1967)

The Engelund and Hansen (1967) function was derived from flume data of four median sand particle diameters (0.19, 0.27, 0.45 and 0.93 mm) and flow velocities ranging between 0.20 and 1.90 m s^-1. They proposed a total load function using the stream power concept (ω). In order to derive this function, the work done by the tractive forces was equated to the potential energy gained by the grains during entrainment:

(21)

5.2.3 Bagnold (1980)

derived a stream power based bed load function using flume and river datasets. Similar to the DuBoys (1879) approach for shear stress, the value of stream power (ω) must be greater than that of the critical stream power (ω_cr) for the initiation of sediment motion. If ω < ω_cr, there will be no sediment transport:

(22)

5.3 Unit stream power based functions

Yang (1972) introduced the concept of unit stream power (ω_u) for the development of sediment transport functions. Later on Yang (1984) evaluated the performance of unit stream power in predicting sediment concentrations by comparing its results with shear stress and stream power based predictions. Yang (1972) found that both shear stress and stream power would not be ideal for the estimation of bed load concentration, while unit stream power gave better results. On the basis of Yang’s (1972, 1973, 1984) studies, Govers (1990) derived a sediment transport function using unit stream power, while Smith et al. (1995) modified the Engelund and Hansen (1967) function (equation (21)) by using unit stream power theory for overland flow conditions.

5.3.1 Yang (1973)

A frequently-used total load function for channel flows was developed by Yang (1973), based on the concept of dimensionless effective unit stream power. He defined effective unit stream power as the difference between the unit stream power (ω_u) and its critical value (ω_ucr), which is actually the available energy used for the transport of sediment. According to this terminology, the value of unit stream power must exceed its critical value for sediment initiation, otherwise there will be no transport of sediment. For this function, ω_u and ω_ucr were used in dimensionless form by dividing both with settling velocity (v_s) of sediment particles:

(23)

where ω_ucr is the critical unit stream power (m s^-1); and I and J are coefficients. Yang (1973) found that the value of dimensionless unit stream power (ω_ucr/v_s) depends on the shear velocity Reynolds number (), with ν (m² s^-1) being the kinematic viscosity. For instance, if the calculated value of is equal to or greater than 70, then ω_ucr/v_s is considered equal to 2.05, otherwise it can be calculated from:

(24)

Moreover, regression analysis was used to determine the coefficients I and J of equation (23), which can be calculated as:

(25)

(26)

5.3.2 Govers (1990)

conducted flume experiments using non-cohesive materials that ranged from coarse silt (D₅₀ = 0.058 mm) to coarse sand (D₅₀ = 1.098 mm). The experiments were carried out with slopes ranging from 1.7 to 21% and unit discharges ranging from 0.2 to 10.0 × 10^-3 m² s^-1. By using the concept of Yang (1973), Govers (1990) derived an empirical relationship by regression analysis between sediment concentration and effective unit stream power (ω_u – ω_ucr):

(27)

Govers (1990) assumed an absolute value for critical unit stream power i.e. 0.004 m s^-1. At present, this function is being used in the EUROSEM model (Morgan et al. 1998a, 1998b) for rill erosion and in the LISEM (De Roo et al. 1996) and DREAM (Ramsankaran et al. 2013) models for rill and inter-rill erosion.

5.3.3 Govers (1992)

derived another empirical relationship between sediment concentration and unit stream power by regression analysis using his flume experimental results for particle sizes ranging between 0.058 and 0.218 mm. The function is defined as:

(28)

5.3.4 Modified Engelund and Hansen function

The original Engelund and Hansen (1967) sediment transport function (equation (21)) has certain limitations for its application. The channel bed material should have a minimum particle diameter of 0.15 mm and not have a wide variation about the median particle diameter (D₅₀). Therefore, the original function of Engelund and Hansen (1967) was modified by Smith et al. (1995) with the results of Govers (1990) to be more generally applicable:

(29)

This modified Engelund and Hansen function is incorporated in the KINEROS2 model (Smith et al. 1995).

5.4 Effective stream power functions

5.4.1 Everaert (1991)

conducted flume experiments with sediment size ranging from 0.033 to 0.390 mm and slope gradient varied between 3.5 and 17.6% to measure the sediment transport capacity of inter-rill flows with or without rainfall. For the runs with rain, a rainfall intensity of 60 mm h^-1 was applied. Everaert (1991) performed a stepwise multiple regression analysis to relate sediment transport capacity with effective stream power:

(30)

The derived relationship showed significant variation with grain size and only a minor effect of rainfall on the transport capacity. The function is used in the EUROSEM (Morgan et al. 1998a, 1998b) model for the estimation of sediment transport capacity for inter-rill flows.

5.4.2 Govers (1992)

Apart from equation (28), Govers (1992) also developed an empirical relationship between sediment transport rate and effective stream power by regression analysis using his flume experimental results for particle sizes ranging between 0.127 and 0.414 mm. The derived function is defined as:

(31)

where ω_{eff cr} (N^1.5 s^-1.5 m^-2.17) is the critical effective stream power.

5.5 Other functions

5.5.1 Schoklitsch (1962)

developed an empirical bed load function for sand bed streams, in which discharge and slope gradient are the main variables:

(32)

where q_cr (m² s¹) is the critical unit discharge at initiation of sediment motion. Although the function is based on a limited range of experimental data and lacks a theoretical basis, it provides reasonable estimates for bed load discharge over a broad range of conditions (Graf 1971).

5.5.2 Rickenmann (1991)

conducted flume experiments with fine sediments to derive a bed load function for channel flow. During experimentation, the various concentrations of clay suspension at steep slopes (5–20%) were re-circulated. Rickenmann (1991) studied the effect of an increasing fluid density and viscosity on the flow behavior and on bed-load transport capacity. The function is defined as:

(33)

(34)

where D₃₀ and D₉₀ (m) are the grain sizes at which 30 and 90%, respectively, of material is finer by weight.

5.6 Suitability of sediment transport functions

Several studies have been carried out to evaluate the performance of existing sediment transport capacity functions under overland flow conditions with laboratory and field data. In each study different datasets were used to evaluate the sediment transport function and, therefore, results about the suitability of transport capacity functions were also different. The most important studies that were found in the scientific literature are listed below.

Alonso et al. (1981) evaluated the suitability of transport capacity functions (Meyer-Peter and Muller 1948, Bagnold 1956, Laursen 1958, Yalin 1963, Engelund and Hansen 1967, Yang 1973) using five datasets, which were obtained from laboratory flume and erosion plot experiments. The median grain size of the bed materials used to conduct experiments ranged from 0.156 to 0.342 mm and slopes from 0.0008 to 0.08 m m^-1. They used the discrepancy ratio (the ratio of predicted sediment transport rate and measured sediment transport rate) for the evaluation of the selected sediment transport functions. The mean values of the discrepancy ratio obtained ranged from 1.11 to 4.91 for the Meyer-Peter and Muller (1948) function, from 0.24 to 0.63 for the Bagnold (1956) function, from 5.16 to 17.42 for the Laursen (1958) function, from 0.63 to 1.28 for the Yalin (1963) function, from 0.17 to 0.48 for the Engelund and Hansen (1967) function, and from 0.01 to 2.99 for the Yang (1973) function. The results clearly showed large difference between predicted and measured sediment transport rates for the selected functions, except for the Yalin (1963) function. Therefore, Alonso et al. (1981) recommended the Yalin (1963) function to estimate sediment transport rate in overland flows.

Low (1989) carried out 187 flume experiments with light weight plastic particles of median grain diameter equal to 3.5 mm and sediment specific gravity of between 1.0 and 2.5. The simulated flow rates ranged from 0.75 to 5.5 × 10^-3 m³ s^-1 and slopes varied between 0.0046 and 0.0149 m m^-1. The experimental results were used to evaluate the performance of several bedload transport functions (Shields 1936, Meyer-Peter and Muller 1948, Bagnold 1956, Yalin 1963, Smart 1984) by comparing measured and predicted sediment transport rates. Low (1989) found that the Shields (1936) and Meyer-Peter and Muller (1948) functions over-predicted transport rates, while the Bagnold (1956), Yalin (1963) and Smart (1984) functions under-predicted the results. On the basis of his own data, Low (1989) modified the coefficient of the Smart (1984) function by introducing sediment specific gravity, since the original Smart (1984) function only estimated transport rates reasonably for sediments having a specific gravity of 2.5. As a consequence, the sediment transport rates estimated by using the Low (1989) function were comparable with the measured transport rates, since the equation was based on the same data.

Guy et al. (1992) tested the applicability of six fluvial transport functions (DuBoys 1879, Laursen 1958, Schoklitsch 1962, Yalin 1963, Bagnold 1966, Yang 1973) for shallow uniform flow with or without rainfall impact. They carried out 207 laboratory experiments in a rectangular flume by using four narrowly graded materials with median grain diameters equal to 0.151, 0.271, 0.328 and 0.381 mm, and respective particle densities of 1496, 2638, 2647 and 2650 kg m^-3. The flume was inclined at angles of between 0.01 and 0.12 m m^-1 and various flow rates (applied range was not given) were used to attain different sediment transport rates. During selected experimental runs, rainfall was also applied at intensities of 33, 108 and 140 mm h^-1 in combination with overland flow. Guy et al. (1992) evaluated the performance of the selected functions by calculating the discrepancy ratio. The derived mean values of the discrepancy ratio were equal to 0.36 for the Yang (1973) function, 0.54 for the DuBoys (1879) function, 1.49 for the Bagnold (1966) function, 6.12 for the Laursen (1958) function, 0.19 for the Yalin (1963) function, and 0.89 for the Schoklitsch (1962) function for overland flows only. For the condition with rainfall impacting the overland flow the discrepancy ratios were 0.06 for the Yang (1973) function, 0.75 for the DuBoys (1879) function, 0.99 for the Bagnold (1966) function, 1.89 for the Laursen (1958) function, 0.06 for the Yalin (1963) function, and 0.53 for the Schoklitsch (1962) function. Hence, the results showed that the Schoklitsch (1962) function predicted transport rates best for overland flow only, while the Bagnold (1966) function gave the best results for rain-impacted overland flow conditions.

Govers (1992) evaluated the performance of five existing channel transport functions (Meyer-Peter and Muller 1948, Yalin 1963, Yang 1973, Low 1989, Lu et al. 1989) under overland flow conditions. Flume data collected by Govers (1990) were used for the evaluation of selected functions. Govers (1990) carried out 465 flume experiments at slopes varying between 0.017 and 0.21 m m^-1, and unit discharges ranging from 0.2 to 10.0 × 10^-3 m² s^-1. Five well sorted quartz materials with median grain diameters equal to 0.058, 0.127, 0.218, 0.414, 1.098 mm and with a sediment density of 2650 kg m^-3 were used. Performance of the selected functions was evaluated using logarithmic graphs of observed against predicted sediment transport rates. None of the selected functions performed well over the range of conditions tested. The best performing function was the one of Low (1989). On the basis of the dataset used, Govers (1992) developed new empirical functions (equations (28) and (31)) and checked their performance on datasets collected by Kramer and Meyer (1969), Meyer et al. (1983), Rauws (1984), Riley and Gorey (1988), and Aziz and Scott (1989), Govers (1992) found that the proposed functions exhibited better agreement of results with other datasets. He also recommended that the new developed functions can be used in erosion models, especially for rill erosion.

Hessel and Jetten (2007) checked the suitability of eight sediment transport functions (Schoklitsch 1962, Yalin 1963, Yang 1973, Bagnold 1980, Low 1989, Govers 1990, Rickenmann 1991, Abrahams et al. 2001) by incorporating them in the LISEM model (De Roo et al. 1996). The model was applied to the Danangou catchment (3.5 km²) of the Chinese loess plateau, where the soils are mainly erodible silt loams with a median grain diameter equal to 0.035 mm. Discharges were measured for the period 1998–2000 at a weir that was constructed at the catchment outlet. In this study, slopes were very steep and ranged from 0.5 to 2.5 m m^-1. The performance of the selected sediment transport functions was evaluated by comparing the predicted and measured sediment yield at the catchment outlet and also by comparing the predicted and measured total soil loss. Graphical comparison of predicted and measured sediment yield revealed that most of the selected functions, such as Schoklitsch (1962), Yalin (1963), Bagnold (1980), Low (1989), Rickenmann (1991) and Abrahams et al. (2001), over-predicted the transport rates at steep slopes and under-predicted the transport rates at gentle slopes. Hence, these functions were found to be sensitive to slope angle. The Yang (1973) function over-predicted sediment yield for both mild and steep slopes, because it appeared to be sensitive to sediment particle size. Only the Govers (1990) function performed well, because this function showed lower dependency on both slope and grain size. For one particular storm, the Govers (1990) function over-predicted the total soil loss by only 21%, while all other functions resulted in larger over- or under-predictions. Therefore, Hessel and Jetten (2007) recommended the use of the Govers (1990) function (equation (27)) in erosion models, because it gave the best agreement with measured erosion data.

Nord and Esteves (2007) applied the PSEM_2D numerical model on five different texture soils by incorporating four sediment transport functions for rill flows: Yalin (1963), Low (1989), Govers (1992) Unit Stream Power, and Govers (1992) Effective Stream Power. The selected soils for the experiments included three cohesive sediments (the Pierre, the Collamer, and the Barnes_ND soils, with D₅₀ equal to 0.004, 0.014 and 0.028 mm, respectively) and two non-cohesive sediments (the Amarillo and the Bonifay soils, with D₅₀ equal to 0.23 and 0.31 mm, respectively). The slopes ranged from 0.04 to 0.09 m m^-1. A pair of identical rills was considered for each soil and the initial shape of a rill was 9.0 m long, 0.5 m wide, and 0.05 m deep, with a uniform trapezoidal cross-section. Applied flow discharge in a single rill was 0.66 × 10^-3 m³ s^-1. The predicted sediment loads were compared with the observed sediment loads to evaluate the performance of selected functions. Nord and Esteves (2007) found that the Govers (1992) Unit Stream Power function (equation (28)) exhibited the best results for cohesive soils, but all of the selected functions showed poor performance for non-cohesive soils.

6 SUMMARY AND CONCLUSIONS

Given the large number of existing sediment detachment and transport capacity functions, the main questions regarding their application are: which function will give promising results when and where? Selection of appropriate sediment detachment and sediment transport functions for a particular condition requires information about their data requirements, complexities involved, assumptions made during derivation, and boundary conditions for which they were derived. Therefore, as a step towards this understanding, soil detachment and transport capacity functions considered widely in soil erosion modelling were reviewed herein.

Most of the existing detachment functions were empirically derived by directly relating the detachment capacity with a composite force predictor. In the literature only detachment functions were found that are based either on shear stress or on stream power. In these functions, the soil’s resistance against erosion is represented by a soil erodibility factor, which principally depends on soil characteristics. Among the existing functions, only the performance of shear stress-based detachment capacity functions (equations (6) and (8)) was evaluated with a limited set of laboratory and field data (Zhu et al. 2001). The best results were obtained when using a power function of shear stress (equation (6)). The use of shear stress based functions has been criticized by several scientists (Elliot and Laflen 1993, Owoputi and Stolte 1995, Zhang et al. 2003, Knapen et al. 2007), but still this concept is used in several spatially-distributed soil erosion models, such as CREAMS (Knisel 1980), PRORIL (Lewis et al. 1994a, 1994b), and WEPP (Flanagan et al. 2001).

During soil erosion modelling, the detachment capacity functions are normally coupled with the Foster and Meyer (1972) detachment–transport coupling approach (equation (1)) to calculate the actual detachment rate. Furthermore, on the basis of the stream power concept, a complex function was developed by Hairsine and Rose (1992) to directly determine the detachment rate using physical principles. Due to the complexities and shortcomings of the shear stress and stream power functions, several erosion models use a more simplified approach to determine detachment rate. In this approach the energy utilized for soil detachment is computed by taking the difference between sediment transport capacity and actual sediment transport rate (transport capacity deficit concept), and is used in the models KINEROS2 (Smith et al. 1995), LISEM (De Roo et al. 1996), EUROSEM (Morgan et al. 1998a, 1998b), and DREAM (Ramsankaran et al. 2013).

Many sediment transport capacity functions exist. Most of those functions were derived using limited datasets, which implies that their applicability becomes questionable when applied outside the domain for which they were developed. Contrary to the available detachment functions, a lot more studies have been done to evaluate the performance of several sediment transport functions (Alonso et al. 1981, Low 1989, Govers 1992, Guy et al. 1992, Hessel and Jetten 2007, Nord and Esteves 2007). However, each study came to different conclusions on which function is best for overland flow conditions. Hence, a single sediment transport capacity function cannot be recommended globally, because hydraulic and sediment variables such as grain size, bed roughness, flow depth, flow velocity, flow discharge and bed slope gradient are variable from place to place. Based on this review, a general guidance can be prepared for the selection of a suitable sediment transport capacity function:

1. For fine to medium-range sands and at mild slopes (<0.08 m m^-1), the Yalin (1963) function may be applicable (Alonso et al. 1981).

2. Low’s (1989) function could be used to predict transport capacity for mild to steep slopes (0.0046–0.21 m m^-1), when the grain size of bed material ranges from medium to coarse sands (Low 1989, Govers 1992).

3. Preference may be given to the unit stream power-based Govers (1990, 1992) functions for slopes ranging from 0.017 to 0.21 m m^-1 and bed materials from clay to coarse sands (Govers 1992, Hessel and Jetten 2007, Nord and Esteves 2007).

Furthermore, bedload stream flow functions always showed their capability to predict transport capacity for shallow flows when their ability was compared with total load stream flow functions (Alonso et al. 1981, Low 1989, Guy et al. 1992). These results unveiled the fact that rolling, sliding and saltation are the major modes of motion of sediment particles under overland flow conditions, which is in agreement with the findings of Lu et al. (1989). When the performance of both overland flow and stream flow functions was evaluated for shallow flows, researchers always recommended the use of an overland flow function (Govers 1992, Hessel and Jetten 2007, Nord and Esteves 2007). Overall, it is recommended to evaluate the performance of unit stream power theory-based functions (i.e. Govers 1990, 1992) with more detailed field and laboratory datasets.

Among the available sediment detachment and transport capacity functions, none of the available functions could simulate the impact of rock fragment and rainfall on sediment detachment and transport capacity, except the Everaert (1991) and Abrahams et al. (2001) transport capacity functions. The Abrahams et al. (2001) function was found un-suitable for the Danangou catchment of the Chinese loess plateau (Hessel and Jetten 2007), while the main drawback of Everaert’s function is that its suitability under laboratory and field conditions has not been tested yet. Owing to their application limitations, it is expected that these functions may produce errors when applied to natural hillslopes. Hence, in the light of results of the previous studies, it is recommended that the impact of rock fragments and rainfall should be incorporated by selecting appropriate variables in a suitable transport capacity function.

Disclosure statement

No potential conflict of interest was reported by the author(s).