SIMULATING EPHEMERAL GULLY EROSION IN ANNAGNPS L. M. Gordon, S. J. Bennett, R. L. Bingner, F. D. Theurer, C. V. Alonso ABSTRACT. Ephemeral gully erosion can cause severe soil degradation and contribute significantly to total soil losses in agricultural areas. Physically based prediction technology is necessary to assess the magnitude of these phenomena so that appropriate conservation measures can be implemented, but such technology currently does not exist. To address this issue, a conceptual and numerical framework is presented in which ephemeral gully development, growth, and associated soil losses are simulated within the Annualized Agricultural Non-Point Source (AnnAGNPS) model. This approach incorporates analytic formulations for plunge pool erosion and headcut retreat within single or multiple storm events in unsteady, spatially varied flow at the sub-cell scale, and addresses five soil particle-size classes to predict gully evolution, transport-capacity and transport-limited flows, gully widening, and gully reactivation. Single-event and continuous simulations demonstrate the model’s utility for predicting both the initial development of an ephemeral gully and its evolution over multiple runoff events. The model is shown to recreate reasonably well the dimensions of observed ephemeral gullies in Mississippi. The inclusion of ephemeral gully erosion within AnnAGNPS will greatly enhance the model’s predictive capabilities and further assist practitioners in the management of agricultural watersheds. Keywords. AnnAGNPS, Cell scale, Concentrated flow, Ephemeral gully, Erodibility, Headcut, Modeling, Sediment transport, Soil erosion.
E
phemeral gullies are small erosional channels formed by concentrated flow within topographic swales on agricultural fields (Foster, 1986; Poesen et al., 2003; Foster, 2005). These features typically are decimeters wide and deep and may be tens of meters or more in length. They often form within individual storms, and they can easily be refilled by tillage operations (Laflen et al., 1986; Vandekerckhove et al., 1998; Bennett et al., 2000a; De Santisteban et al., 2005, 2006). Ephemeral gullies occur, often every year, in areas of concentrated flow that are invariably positioned on the landscape between hillslopes. It is this relationship to landscape position that distinguishes ephemeral gullies from rills, which form due to overland flow and soil erosion on hillslopes (Foster, 2005). As such, analysis of high-resolution digital elevation models suggests that simple topographic indices based on slope and upstream drainage area, as a surrogate for discharge, can be used to predict the location of ephemeral gullies (Zevenbergen and Thorne, 1987; Moore et al., 1988; Desmet and Govers, 1996; Vandaele et al., 1996; Vandekerckhove et al., 1998; Casalí et
Submitted for review in April 2006 as manuscript number SW 6427; approved for publication by the Soil & Water Division of ASABE in January 2007. The authors are Lee M. Gordon, ASABE Student Member, Graduate Research Assistant, and Sean J. Bennett, Associate Professor, Department of Geography, University at Buffalo, Buffalo, New York; Ronald L. Bingner, ASABE Member Engineer, Research Agricultural Engineer, USDA-ARS National Sedimentation Laboratory, Oxford, Mississippi; Fred D. Theurer, ASABE Member Engineer, Agricultural Engineer, USDA-NRCS National Water and Climate Center, Gaithersburg, Maryland; and Carlos V. Alonso, Supervisory Research Hydraulic Engineer, USDA-ARS National Sedimentation Laboratory, Oxford, Mississippi. Corresponding author: Lee M. Gordon, Department of Geography, University at Buffalo, 105 Wilkeson Quad, Buffalo, NY 14261; phone: 716-725-5727; fax: 716-645-2329; e-mail:
[email protected].
al., 1999; De Santisteban et al., 2005). Classical gullies also form on hillslopes in areas of concentrated flow, but these features are relatively larger in size as compared to ephemeral gullies, they tend to occur at the edge of fields rather than on fields, and they cannot be obliterated by common tillage operations (Foster, 2005). Ephemeral gully erosion can cause severe soil degradation and contribute significantly to total soil losses in agricultural areas. Poesen et al. (2003) noted that ephemeral gully erosion is an important soil degradation process affecting many environments worldwide. Estimates for the percentage of total soil loss from agricultural watersheds due to ephemeral gullies can range from 20% to 100% (Bennett et al., 2000a; De Santisteban et al., 2005, 2006), but values typically are in the range of 30% to 60% (Poesen et al., 2003). Because of this significant contribution and their widespread occurrence, the USDA-NRCS recently stressed the need to include ephemeral gully erosion processes within soil loss prediction technology (USDA-NRCS, 1996; Garen et al., 1999), with the added constraint that such models not require large amounts of user-defined data (Bosch et al., 1998). The only tool currently available to field practitioners to address ephemeral gully erosion on agricultural fields is the Ephemeral Gully Erosion Model (EGEM; Merkel et al., 1988; USDA-SCS, 1992; Woodward, 1999). EGEM simulates a single, non-bifurcating ephemeral gully on a planar surface whose hydrologic components are based on USDANRCS methods (USDA-SCS, 1986) to estimate peak discharge and runoff volume, and whose erosion components are based on those developed for CREAMS (USDA-ARS, 1980) to estimate gully width and soil loss. The application of EGEM to field data, however, has been problematic. Nachtergaele et al. (2001a, 2001b) and Capra et al. (2005) used EGEM to simulate gullies observed in southeast Spain, southeast Portugal, northern Belgium, and Sicily. In general, EGEM performed poorly in predicting the areal dimensions
Transactions of the ASABE Vol. 50(3): 857−866
2007 American Society of Agricultural and Biological Engineers ISSN 0001−2351
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of gullies and total soil losses. Ephemeral gully length was identified as the key input parameter that resulted in such aberrant simulations. As noted by Nachtergaele et al. (2001a, 2001b), a strong correlation exists between ephemeral gully length and volume of soil eroded. While topographic indices may be used to identify locations of potential gully erosion, there is no method currently available to predict gully length. The USDA Annualized Agricultural Non-Point Source model (AnnAGNPS) has been developed to facilitate assessment of watershed and landscape processes affecting agricultural areas (Bingner and Theurer, 2002; Bingner et al., 2003). The primary strength of AnnAGNPS lies in its ability to simulate runoff, sediment yields, and pollutant transport on hillslopes as affected by agricultural activities and best management practices through the use of well-established numerical methods and techniques. Within AnnAGNPS, the watershed is divided into cells that have uniform slope, soil type, land use, and land management, and the model uses the soil erosion routines of the Revised Universal Soil Loss Equation (RUSLE) to predict soil loss for each cell. At present, AnnAGNPS cannot predict ephemeral gully erosion. As recently noted by Bingner et al. (2006), ephemeral gullies are the primary source of sediment and sediment yields (73% of the total) within the Maumee River basin in Ohio, and AnnAGNPS consistently underpredicted total soil losses when applied to this region because RUSLE accounts only for sheet and rill erosion. The goal of the research program is to improve the physical basis and predictive capabilities of AnnAGNPS to explicitly address the processes of ephemeral gully erosion. The objectives of this article are: (1) to develop a conceptual and numerical framework for the prediction of gully dimensions, lengths, and soil losses within spatially varied, unsteady runoff events within the AnnAGNPS model, and (2) to document the time-evolution of ephemeral gullies and soil losses within individual and multiple runoff events.
CONCEPTUAL AND NUMERICAL FRAMEWORK
The simulation of ephemeral gully erosion processes will build upon those technologies described in EGEM (see references above) but adapted and modified for use within individual cells of AnnAGNPS. As such, gully erosion will only occur in those cells specified by the user, and all processes will ensue at the sub-cell scale. That is, each cell will have uniform topography, soil characteristics, and management practices, and these parameters will be time and space invariant during each runoff event. The erosion process can be conceptually presented as follows and fully discussed in subsequent sections. For a given runoff event, a hydrograph can be constructed at the mouth or outlet of the cell, and flow rate at a given location within the cell will be proportional to the upstream drainage area, depending on the length of the gully; thus, flow is unsteady and spatially varied. Once the flow rate at the mouth of the cell exceeds the erosion threshold of the soil (Foster et al., 1982), incision is initiated in the form of a headcut (Casalí et al., 1999; Bennett et al., 2000b). This headcut will first incise down to the tillage depth, an erosion-resistant layer, before it migrates upstream at a rate proportional to the flow
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rate (Alonso et al., 2002). The width of the gully downstream of the headcut and sediment transport, whether limited by sediment supply or flow capacity, also will be proportional to flow rate (Nachtergaele et al., 2002; Torri et al., 2006). Since flow is unsteady and spatially varied, the headcut migration rate, the gully width, and the rates of sediment entrainment, transport, and deposition all will vary accordingly in time and space. Erosion processes will cease at any given location once the flow rate at that location drops below this same soil erosion potential. Following the runoff event, the cell may be re-tilled, thus obliterating the developed gully, or it will be carried forward in time until another runoff event occurs, which may or may not modify the existing gully. Therefore, four fundamental improvements have been identified to overcome major limitations of current technology. These include: (1) allowing for unsteady, spatially varied runoff or storm events; (2) defining gully length through extending the gully by the upstream migration of a headcut; (3) determining gully width from discharge, thus allowing channel dimensions to be explicitly predicted at any point in time and space; and (4) routing five distinct particle-class sizes (clay, silt, sand, and small and large aggregates) through the gully and addressing the downstream sorting of these sediments, including erosion and deposition. The specific components of the model are presented in the following sections, and model input requirements are listed in table 1. HYDROLOGY The calculation of ephemeral gully erosion requires the peak discharge and total runoff volume for each simulated storm so that an event hydrograph can be constructed. These parameters may be supplied by the user or calculated based on TR-55 methods (USDA-SCS, 1986) using drainage area, rainfall, curve number, and storm type. Given the event peak discharge at the mouth of the gully (Qp , m3 s−1) and runoff volume (Vb , m3), a triangular hydrograph is constructed using an even number of timesteps t with the subscript i representing each timestep. At the gully mouth, this hydrograph has a time to base (tb ,s) of: Notation Re CN ST Qp Vr SCx τc kd S n dt Ad Rclay Rsilt Rsand Rsagg Rlagg δs [a] [b]
Table 1. Model input requirements. Description Event rainfall SCS curve number SCS storm type Event peak discharge[a] Event runoff volume[a] Soil condition Critical shear stress[a] Soil erodibility coefficient[a] Average thalweg slope Manning’s roughness Tillage depth, or depth to non-erodible layer Drainage area to gully mouth Clay ratio in surface soil Silt ratio in surface soil Sand ratio in surface soil Small aggregate ratio in surface soil Large aggregate ratio in surface soil Soil bulk density
Units mm −− −− m3 s−1 m3 −−[b] N m−2 cm3 N−1 s−1 m m−1 −− m ha decimal decimal decimal decimal decimal Mg m−3
Default to internal calculations if not user-defined. SC1 = no-till, SC2 = freshly cultivated, and SC3 = established crop.
TRANSACTIONS OF THE ASABE
tb =
2Vb Qp
(1)
tb 2
(2)
and time to peak tp (s) of: tp =
Discharge at the mouth of the gully (Qmi ) during each timestep i can be determined from: t Qmi = i Q p tp
(3)
wij = 2.51Qij0.412
when 0 < ti < tp on the rising limb, and: Qmi
=
(tb − ti )
(tb − t p )
Q p
(4)
when tp < ti < tb on the falling limb, so that flow is conserved: Qm =
t
∑ Qmi
i =1
V = b tb
(5)
As the headcut migrates upstream (see below), the contributing drainage area decreases, so the discharge at the head of the gully also decreases. A maximum ephemeral gully length (Lmax , m) for a given drainage area (Ad , ha) can be defined based on Leopold et al. (1964): Lmax = 80.3 Ad0.6
(6)
The proportion of the drainage area (Aj ) contributing to discharge at the gully head is defined as: L j A j = 1 − Lmax
5/3
(7)
where j represents the upstream location of the migrating gully. Thus, for any given time during the hydrograph, the flow discharge at the gully head (Qij ) is defined as: Qij = Qmi A j = wij d ij vij
GULLY INCISION AND HEADCUT MIGRATION Hillslope erosion processes within AnnAGNPS operate on fields that are internally homogeneous in terms of topography, soil profile characteristics, and management operations. At this sub-cell scale, the ephemeral gully is located initially at the mouth or outlet of the planar cell that has been identified. Flow is concentrated at the outlet of the cell. At this location, ephemeral gully width wij is determined using an empirical relationship developed by Nachtergaele et al. (2002) and Torri et al. (2006) and based on discharge Qij (m3 s−1) and Aj = 1:
(8)
where wij is flow width (m), dij is mean flow depth (m), and vij is mean flow velocity (m s−1). These hydraulic geometry parameters are defined below.
(9)
Mean flow depth (dij , m) and velocity (vij , m s−1) are determined using Manning’s equation, defined as: vij = R 2 3
S1 2 n
(10)
where n is Manning’s roughness coefficient, required as input, and R is hydraulic radius. Since both dij and velocity vij are unknowns, equation 10 is solved iteratively and constrained by flow continuity (eq. 8; see also EGEM). Mean boundary shear stress (ij , N m−2) is defined as: τij = ρ w gd ij S
(11)
where ρw is density of water (kg m−3), g is gravitational acceleration (m s−2), and S is bed slope assuming uniform flow. Soil erosion rate is typically assumed to be proportional to the excess shear stress (Foster et al., 1982). Thus, once boundary shear stress (ij) exceeds the critical shear stress for the soil (c ), a relatively small length of the ephemeral gully channel is incised until the tillage depth is reached, thus creating a scour hole (fig. 1) at the gully mouth. The detachment capacity (DCij, g m−2 s−1) of the flow is defined as: DCij = k d (τij − τ c )
(12)
where kd (g N−1 s−1) is the soil’s erodibility coefficient. The value of kd is calculated as a paired value with critical shear stress (see discussion below) The duration of each timestep ti (s) and soil bulk density s (Mg m−3) allow the depth of erosion during each timestep DEij (m) to be calculated as: DEij = ti
DCij δs
(13)
Figure 1. Definition sketch of an actively migrating headcut digitized directly from video recording of a laboratory experiment. Headcut related parameters are identified.
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Because these incision algorithms are responsible only for evacuating a very short length of channel in order to form a headcut, all soil material detached here is assumed to be evacuated (transport capacity is not addressed), and this small volume of soil does not contribute to the total ephemeral gully erosion calculated (initial plunge pool erosion is negligible). Once channel incision reaches the tillage depth, a headcut (fig. 1) forms at the step-change between the original soil level and the tillage depth, and an overfall now exists at the brinkpoint position. Equations to describe the brinkpoint conditions and headcut erosion developed by Alonso et al. (2002) are employed here during each timestep to determine a rate of headcut migration, and thus the length of ephemeral gully Lj . Headcut migration rate (Mij , m s−1) is determined by: M ij = Vij
µij qij
(14)
S D − hij
θij 1 µij = ρ w k d sin 2 2 2
dij
=
Vij =
(15) (16)
Frij2
(17)
Frij2 + 0.4 qij
(18)
Bij cos θij
hij θij = tan 1 + ελ Bij −1
+ ln (Te )1 + 1 + (Te )− 2
(19)
5
∑ Px
(21)
x =1
where x represents a particular size fraction, and Px is the proportion by mass of that size fraction in the soil. There are three possible sources of sediment available for transport within a gully section during a given timestep: (1) incoming sediment from upstream sections, (2) internal sediment due to headcut migration and/or channel widening within the gully section, and (3) previously deposited sediment that resides on the bed within the gully section. If C represents sediment flux (Mg), then accounting for volume, density, and the availability of each particle-size class, the mass conservation of each particle-class size leaving a gully section j during any timestep i is given as: Cijx = Ci( j +1) x + ∆wij ∆L j S D δ s Px
1/ 2
ΦH
Φ H = (Te )2 1 + (Te )− 2
(20)
where Vij is the jet entry velocity (m s−1), qij is unit discharge (m2 s−1), SD is the scour depth, which is assumed equal to the tillage depth (m), hij is vertical distance from the brink to the pool surface (m), ij is jet entry angle (radians), Bij is the flow depth at the headcut brinkpoint (m), Fr is Froude number upstream of the brinkpoint (Frij = uij/[gdij]0.5), Te is the arc tangent of the jet entry angle for gravitational ventilated jets, and and are calibrating pressure-gradient and suction-head coefficients (Alonso et al., 2002), herein taken as 2 and 0.3, respectively. Headcut dimensions and bulk flow parameters are assumed to adjust instantaneously to changes in flow rate and contributing drainage area. Flow depth (dij ) and channel width (wij ), as explicitly defined immediately downstream of the headcut, are used to address brinkpoint hydraulics immediately upstream of the headcut. Since discharge is reduced in upstream sections of an ephemeral gully, gullies are widest at their mouths and narrowest at the locations of the headcut, as observed by Smith (1992). Should a gully reach its maximum length through headcut migration, which
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SEDIMENT EROSION, TRANSPORT, AND DEPOSITION The adjustments in flow discharge over time and space are monitored with each successive runoff event. Flow associated with a specific event must be routed through the entire length of the ephemeral gully. Sediment transport capacity along the entire gully must be considered, since there often will be deposition in downstream gully sections formed during a previous event (e.g., Bennett et al., 2000a), and pre-existing gully sections may be further widened if a larger channel-forming discharge occurs (e.g., Nachtergaele et al., 2002). The original soil material is composed of up to five particle-size classes (table 1) and may be expressed as: 1.0 =
hij = S D − dij Bij
can occur in long, continuous simulations (months), headcut migration ceases but hydrologic and sediment transport processes are still addressed.
± w j L j Dd j δd
Px nm
(22)
where the terms on the right side of equation 22 are sediment flux from upstream, sediment erosion within the channel section (where w and L refer to the change in gully width and length during the timestep), and the change in sediment storage on the bed, where Dd (m) and d (Mg m−3) are the thickness and bulk density of the sediment deposit in a gully section, respectively, and nm is the number of timesteps during which headcut migration is occurring. The algorithms used in AnnAGNPS to calculate sediment transport capacity for each particle-class size have been adapted here (Bingner and Theurer, 2002). The equation for unit-width sediment transport capacity (tc, Mg m−1 s−1) for each size fraction x within each gully section j for each timestep i is defined as: tcijx =
ηijx k x τij uij FVx
(23)
where is the non-dimensional effective transport factor, k is the non-dimensional transport capacity factor, and FV is particle fall velocity (m s−1). The transport capacity (TC, Mg) of each particle-class size x over the duration of each timestep ti within each gully section j is: TCijx = tcijx ti wij
(24)
TRANSACTIONS OF THE ASABE
20
Table 2. Default hydrologic values calculated by Extended TR-55 for five event rainfall amounts on a 5.0 ha drainage area. Rainfall Runoff Volume Peak Discharge (mm) (m3) (m3 s−1) 5.9 106.5 305.5 907.0 1688.5
5
0.00025 0.01092 0.06888 0.27410 0.54870
0 20
Table 3. Default values for field-scale input and the range of values tested. Input Parameter Default Value Range Tested Slope (S) Manning’s roughness (n) Tillage depth (dt , m) Drainage area (Ad , ha)
0.02 0.04 0.20 5.00
0
0.005 to 0.100 0.03 to 0.06 0.1 to 0.3 0.5 to 20.0
30 28 73 38
30 17 14 40
5 0 0 0
5 0 0 0
1.5 1.8 1.6 1.5
1.09 3.12 0.54 0.78
0.096 0.057 0.137 0.113
SOIL ERODIBILITY The initial gully incision algorithms have been adapted from EGEM, in which soil erodibility is reflected by two key parameters: c and kd (table 1). The EGEM user manual (USDA-SCS, 1992) lists 36 pairs of values for c and kd , representing changes in erodibility for different soils and/or tillage operations. These relationships are based on the premise that the critical shear stress of a freshly cultivated soil (c ) is a function of its clay content (Pc, %): (25)
If the soil has an established crop, then the values of c are assumed to be twice that of the freshly tilled condition. If no-till conditions are prevalent, then values of c are assumed to be linearly related to clay content and calculated based on the following regression of the data contained in the EGEM documentation (USDA-SCS, 1992): τc = 0.179 Pc + 7.07 A regression of these 36 paired values of c (N kd (g N−1 s−1) yields the following relation:
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(26) m−2)
0
5
10 τ (N m−2) c
15
20
(d)
and
Re 38.1 mm 50.8 mm 76.2 mm 101.6 mm
0.002 0.000 0.010
(e)
0.005 0.000
Ad 0.5 ha 5.0 ha 20.0 ha
0.004
If the amount of available sediment is less than the sediment transport capacity for a given timestep, then all available sediment will be moved to the next downstream section, where it is again compared to that section’s transport capacity. This process is repeated until the gully mouth is reached. Should transport capacity be exceeded for a particular particle-class size, the excess amount is deposited in a uniformly thick layer on the channel bed within that particular gully section, and possibly re-entrained during subsequent timesteps. If in a given timestep the available sediment is less than transport capacity, then previously deposited sediment will be entrained and eroded until transport capacity is satisfied.
τc = 0.311 × 10 0.0182 Pc
(c)
0.006
Ma (m s−1)
30 55 13 22
S 0.25% 1.0% 5.0% 10.0%
10 5
Table 4. Default values for input soil properties. δs τc kd Soil Type Rclay Rsilt Rsand Rsagg Rlagg (g cm−3) (N m−2) (cm3 N−1 s−1) Default Clay loam Silt loam Loam
(b)
15
25 20 15 10 5 0
Re 38.1 mm 50.8 mm 76.2 mm 101.6 mm
10
ET (Mg)
25.4 38.1 50.8 76.2 101.6
(a)
15
dt 0.10 m 0.20 m 0.30 m
0
5
10 τ (N m−2) c
15
20
Figure 2. Effect of critical shear stress (tc ) on ephemeral gully erosion (ET ) and average headcut migration rate (Ma ) under (a and d) different rainfall amounts (Re ), (b) slopes (S), (c) drainage areas (Ad ), and (e) tillage depths (dt ).
k d = 29e −0.224 τ c
(27)
When calibrating their headcut migration model, Alonso et al. (2002) used representative values of kd obtained from the following relationship developed by Hanson and Simon (2001): k d = 0.1τc −0.5
(28)
where units for kd are cm3 N−1 s−1. After correcting units, the values of kd predicted by equation 27 are almost exactly two orders of magnitude larger than those predicted by equation 28, and no explanation could be found to rectify this difference. In order to maintain consistent values and units for kd in equations 12 and 15 as originally published, equation 28 is used to predict kd from c , and then multiplied by 100 when used in equation 12. Recently, Knapen et al. (2007) provided an exhaustive review on soil erodibility by upland, concentrated flows, focusing primarily on kd and c . In addition to a number of observations, they found the following: (1) kd and c can vary among soils by several orders of magnitude, (2) no statistically significant relationship exists between kd and c for all available data, and (3) a multitude of soil and environmental properties are responsible for the large range and temporal
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and spatial variation in kd and c . It is clear that user-defined or quantified values of kd and c would be far superior to any calculated values, given these discussions.
of sand, silt, and clay and smaller but equal parts as aggregates. Values for c and kd for the various soil types (table 4) are coupled parameters calculated according to methods described above.
RESULTS AND DISCUSSION
END-OF-EVENT SIMULATIONS Single hydrograph simulations were conducted to examine model output at the gully mouth following a runoff event. The ephemeral gully was assigned a length of zero prior to each simulation. Total event erosion (ET, Mg), cumulative erosion (CT, Mg), total event deposition (Md , Mg), gully length (LT, m), gully width (w, m), and headcut migration rate (M, m s−1) are presented for various simulations. For all event rainfalls, slopes, drainage areas, and tillage depths examined, an increase in c and a concomitant decrease in kd reduce the amount of simulated ephemeral gully erosion (fig. 2). Higher slopes (fig. 2b) result in higher bed shear stresses, causing greater rates of erosion. A higher critical shear stress not only limits the time for soil detachment (the number of timesteps for which c is exceeded), it also reduces the rate of headcut migration (figs. 2d and 2e)
Sensitivity analyses were performed in which model input parameters were varied within expected ranges. The model output presented and discussed below may be categorized as: (1) end-of-event simulations, which summarize erosion parameters at the gully mouth; (2) within-event simulations, which summarize erosion parameters along the gully length; and (3) end-of-event continuous simulations, which summarize erosion parameters at the gully mouth after multiple runoff events. Tables 2, 3, and 4 summarize default values for hydrology, field characteristics, and soil parameters used during the sensitivity analyses. Here, a 5.0 ha field on a 2% slope, freshly tilled to 0.2 m with a Manning’s roughness coefficient of 0.04, was subjected to rainfalls varying from 38.1 to 101.6 mm (1.5 to 4.0 in). The default soil contains equal parts Re= 38.1 mm
Re = 76.2 mm
(a)
ET (Mg)
1.2 0.8 0.4
(b)
SD (m)
0.0 0.3
dt (m)
0.2
dt (m)
0.1
(c)
M (m s−1)
0.0 0.006 0.004 0.002 0.000 80 60
LT (m)
(d)
40 20 0
0
50
100
150
200
250
(e)
w (m)
2.0
20 300 0 Event Time (min)
40
100
1.0 t = 1.0 N m−2 c t = 4.0 N m−2 c t = 8.0 N m−2
0.0 0.20
Md (Mg)
80
1.5 0.5
(f)
60
0.15
c
t = 10.0 N m −2 c t = 12.0 N m −2 c
0.10 0.05 0.00
0
20
40
60
80
0
5
10
15
20
25
30
35
Distance from gully mouth (m)
Figure 3. Gully erosion within runoff events showing (a) erosion (ET ), (b) depth of scour (incision; SD ), (c) headcut migration rate (M), and (d) ephemeral gully length (LT ) as a function of event time. Also shown are (e) ephemeral gully section widths (w) and (f) mass of deposition within gully sections (Md ) as a function of distance upstream from the gully mouth.
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TRANSACTIONS OF THE ASABE
END-OF-EVENT CONTINUOUS SIMULATIONS In this case, the 5.0 ha field is subjected to repeated rainfall events of the same magnitude to assess the cumulative effect that these events have on erosion processes and gully characteristics. For all rainfall intensities, erosion magnitudes (fig. 4a) increase from event 1 to event 2. During event 1, a small portion of the hydrograph is devoted to incision at the gully mouth. At the beginning of event 2, incision processes are complete and headcut migration begins as soon as critical shear stress is exceeded. In later events, as the gully approaches its maximum length (fig. 4c), reduced discharges at the location of the headcut due to smaller drainage areas result in less erosion and narrower channels. While even a few events of a small magnitude, e.g., 1.5 in. (38.1 mm) rainfall events (fig. 4c), can create relatively long gullies (due to long hydrograph base times), the magnitude of erosion associated with these gullies is relatively small (fig. 4b). This is because smaller rainfall events create relatively narrower ephemeral gully channels, while larger rainfalls create relatively wider channels.
Vol. 50(3): 857−866
ET (Mg)
(a)
10 5
(b)
CT (Mg)
0 100 80 60 40 20 0 200 (c)
LT (m)
WITHIN-EVENT SIMULATIONS Figure 3 displays model output at the gully mouth and along the gully length over the duration of a simulated hydrograph. The erosion rate (fig. 3a) remains zero until the incision at the gully mouth is completed (tillage depth is reached and headcut is formed, fig. 3b), and this incision occurs more quickly under larger rainfalls and lower critical shear stresses. Once the tillage depth is reached, and if the critical shear stress is exceeded, the headcut progresses upstream at a relatively constant rate, regardless of the unsteady, spatially varied flow (fig. 3c). As noted above, headcut migration rate is more sensitive to changes in soil resistance parameters than to changes in flow discharge. Hydrograph base-time effects become apparent when examining the event times in figure 3d. The 38.1 mm (1.5 in.) rainfall event has a much longer duration than that of the 76.2 mm (3.0 in.) rainfall event; hence, the longer event creates a longer, albeit narrower, gully. This is an artifact of the ratio of runoff volume to event peak discharge generated by the simplified triangular hydrograph used within AnnAGNPS. Implementing any other assumption for storm shape or actual time-varying storm data would result in the same logical pattern of increasing gully length with increasing storm duration. Since flow discharge determines ephemeral gully width, gully width is independent of critical shear stress (fig. 3e). Gullies are widest at their mouth and narrowest at the location of the headcut because of the spatially varied discharge, and higher flow discharges associated with larger rainfall events create wider gullies.
Clay Loam
15
Event Number
Lmax
100 Re = 38.1 mm
0
Re = 50.8 mm Re = 76.2 mm
1.0 (d)
Md (Mg)
by decreasing the erodibility coefficient. In fact, the average headcut migration rate is more dependent on the erodibility of the soil than the magnitude of the runoff, as noted by Alonso et al. (2002). While headcut migration rate remains relatively constant regardless of increased rainfall rate, ephemeral gully erosion is markedly higher because of the larger channel widths associated with higher discharges. As the depth to a non-erodible layer (tillage layer) increases (fig. 2e), headcut migration rate increases because the angle of the impinging jet below the overfall is increased. Finally, the erosion rate (gully volume) increases with increased tillage (gully) depth and migration rate (gully length).
Re = 101.6 mm
0.5 0.0
2
4 6 8 Number of runoff events
10
Figure 4. Gully erosion on a clay loam soil for continuous simulations showing (a) changes in total event erosion (ET ), (b) cumulative erosion (CT ), (c) ephemeral gully length (LT ), and (d) mass of deposited material (Md ) for repeated rainfall events (Re ) of varied magnitude.
Deposition is generally observed to peak after two or three runoff events, and diminish in later events (fig. 4d). As a gully grows longer and as discharge at upstream gully head is reduced, the width of the gully channel at these locations is also reduced. This causes the downstream stations to experience lower incoming sediment loads from upstream. Hence, the amount of sediment being transported falls below transport capacity within these gully sections, and the deposited material then becomes a source of sediment. While little deposition is simulated for larger rainfall events on the clay loam soil (fig. 4d), this is caused in part by the relatively high critical shear stress of this soil (3.11 N m−2). This trend would not hold for soils with minimal clay contents or higher amounts of sand. COMPARISON TO FIELD DATA There are currently no comprehensive datasets for ephemeral gullies containing measured hydrology, basin characteristics, and management operations as well as differentiating between deposited material and actual erosion (scour) depth. By using an existing less comprehensive dataset containing ephemeral gully dimensions and by estimating model hydrology, it is shown here that ephemeral gullies observed in the field can be simulated using the above formulations. The U.S. Army Corps of Engineers conducted a detailed study of ephemeral gullies at four agricultural field sites in central Mississippi (Smith, 1992). Gullies were generally measured only after they had reached their seasonal extent and before cultivation occurred. Gully length, width, and
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Table 5. Input values for field comparison simulations of gullies in Mississippi. Qp (m3 s−1) Vr (m3) A
Parameters Unique to Each Gully
Gully A Gully B Gully C Gully D Global Parameters for Soil Characteristics [a] [b]
24.3 9.8 17.1 7.9
(ha) 0.37 0.15 0.26 0.12
S 0.042 0.065 0.070 0.060
dt (m) 0.053 0.041 0.083 0.034
Rsilt 86
Rday 2
BDs (Mg m−3) 1.43
Manning’s n[b] 0.04
Event 1
Event 2
Event 1
Event 2
0.0574 0.0237 0.0404 0.0186
0.0225 0.0091 0.0158 0.0073
175.7 71.2 123.5 57.0 Rsand 12
τc 1 (N m−2) 0.514
kd [a] (cm3 N−1 s−1) 0.139
d
Calculated. Estimated.
depth were measured at multiple cross-sections along each gully, and a contour map was digitized so that the drainage area and slope of each gully could be identified. The NASIS soil database provided the particle size distribution and bulk density of the Memphis soil described at the field site. Two precipitation events, 84.6 mm (3.33 in.) and 29.7 mm (1.17 in.), were observed after tillage operations, and SCS methods (see references above) were used to determine runoff volumes for Type II storms based on a curve number of 85. Using equal rainfall rates of ~50 mm h−1 for both events (to eliminate possible artifacts of hydrograph shape), peak discharges were calculated, and hydrograph base times for the 84.6 and 29.7 mm rainfall events were 1.7 and 0.6 h, respectively. The average measured cross-sectional depth was assumed to be the tillage depth for each gully. Table 5 summarizes the model input for these simulations. Measured and simulated dimensions for ephemeral gullies A through D are shown in table 6. The root-mean-square relative error (rms), given as a percent, can be defined as: rms =
1 m (O − P )2 × 100 ∑ m i =1 O
(29)
where O and P are the observed and predicted values, respectively, and m is the number of observations. Simulated ephemeral gully lengths compare reasonably well to measured field data with rms = 31% (table 6). Because the gullies did not approach their maximum calculated length (Lmax ), the predicted lengths directly reflect gully extension through headcut migration. Simulated ephemeral gully channel widths compare less favorably to measured field data, with rms = 52% (table 6). This underprediction reflects the fact that gully widths are determined entirely by discharge; thus, the simulated hydrology is the only controlling factor. Here, peak discharges calculated for each gully’s drainage area were not sufficient to widen the simulated gully to its measured width. Because the gullies were measured at the end of the growing season, runoff events occurring after the prescribed events may have contributed to gully widening. It should be noted that no input values were adjusted to fit model output to the measured data. That is, predictions would have been significantly improved by adjusting parameters such as soil erodibility and peak discharge to fit the data.
More importantly, had soil erosion rates at these locations been predicted using technology such as RUSLE (Renard et al., 1997), the contribution that these gullies make to total losses would not have been considered at all. LIMITATIONS AND FUTURE RESEARCH Watershed-scale erosion models such as AnnAGNPS parameterize hillslope erosion processes at the sub-cell scale, and this leads to several necessary assumptions. First, it is assumed that an ephemeral gully begins at the cell (field) outlet and its occurrence is identified by the user. Digital elevation models at higher spatial resolution would facilitate the use of topographic indices to predict where ephemeral gullies initiate on a landscape (see references above). Second, the gullies simulated here are single, non-bifurcating channels. Accounting for such variations in local topography or tillage-induced flow might permit modeling of branching gully systems. Third, it is assumed that the tillage depth is non-erodible and that predictive equations developed for migrating headcuts are applicable in such layered soils. Fourth, ephemeral gully width is dependent only on discharge, which ignores bank erosion processes associated with lateral channel adjustment. Fifth, input values for soil erodibility parameters are difficult to assess and their temporal and spatial variations are not addressed (see Govers et al., 1990; Hanson and Robinson, 1993; Ghebreiyessus et al., 1994; Poesen et al., 1999; Knapen et al., 2007). Finally, it is assumed that the flow conditions above the headcut are the same as below the headcut, and additional research is needed to determine these relationships. However, the formulation presented herein is the necessary first step in addressing ephemeral gully erosion employing commonly used technology with easily accessible data.
CONCLUSIONS
Ephemeral gully erosion is now recognized as an important and sometimes dominant cause of soil loss and soil degradation in upland areas and on agricultural fields. In spite of this, current soil erosion prediction technology either does not address ephemeral gully erosion explicitly, or if it does, its application has proven to be problematic. A conceptual
Table 6. Observed and predicted dimensions of gullies in Mississippi. Gully A Gully B Gully C Length (m) Average width (m)
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Gully D
Observed
Predicted
Observed
Predicted
Observed
Predicted
Observed
Predicted
51.01 0.86
20.78 0.55
18.90 0.87
17.48 0.36
28.03 0.80
31.93 0.41
17.16 0.83
15.38 0.33
TRANSACTIONS OF THE ASABE
and numerical framework has been constructed to simulate the development and upstream extension of an ephemeral gully within AnnAGNPS based on a migrating headcut and discharge-dependent gully dimensions in unsteady, spatially varied runoff events. This model addresses transport-limited and capacity-limited flows, the routing of different particlesize classes, and the potential for deposition and re-entrainment of sediment within the evolving or reactivated gully. Model simulations illustrate the initial formation and the temporal and spatial evolution of ephemeral gullies in response to a range of runoff events, tillage conditions, and indices of soil erodibility. Comparison to a representative field dataset demonstrates the model’s utility in predicting soil losses due to ephemeral gully development. The inclusion of this technology will greatly enhance the current predictive capabilities of AnnAGNPS and its continued use in assessing and managing landscape processes in agricultural watersheds. ACKNOWLEDGEMENTS Funding for this research was provided by the USDA Agricultural Research Service. The comments of three anonymous referees provided many helpful suggestions for ways to improve the manuscript.
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