Analyses of Slope and Runoff Factors Based on the WEPPErosion Model

Chi-hua Huang* and Joe M. Bradford

ABSTRACTUnder process-based erosion model development, the source of sed-

iment transported off a field is separated into that from interrill andrill areas, and separate detachment equations are developed. Limi-tation of this spatial separation is that, in many conditions, rill andinterrill areas are not clearly defined a priori. We interpreted theerosion and deposition equations used in the current Water ErosionPrediction Project (WEPP) in an alternative fashion such that spatialseparation of rill and interrill areas is no longer required. Analyticsolutions were derived for the WEPP erosion and deposition equationsunder general conditions. Simplified solutions for a specific case, uni-form rain on uniform slope, were examined closely for slope andrunoff effects on sediment delivery. Under both erosion and depositionconditions, analytic solutions show a linear relationship between sed-iment yield, q, and slope, 5. The dependency of q, on runoff, gw, iseither linear or quadratic depending on whether the system is domi-nated by an erosion or deposition regime. These analytic findingsexplain results obtained from laboratory studies in which sedimentyield was collected under variable slope and rain intensities.

PROCESS-BASED erosion prediction models, such asthe USDA's WEPP (Nearing et al., 1989), sep-

arate erosion processes into rill and interrill subpro-cesses and use different parameters and equations toquantify these subprocesses. Interrill erosion processesinclude detachment by raindrop impact and transportby shallow sheet flow in the interrill areas, while therill erosion model quantifies shear detachment byflowing water and transport by concentrated flow. Sinceseparate processes are used in different areas, the modelcan be applied as long as there are spatially distin-guishable areas that are defined as rill or interrill areas.An example of a well-defined system is the ridge-furrow system where the side slopes of the ridges arethe interrill areas and the furrows are rills. Problemsarise when the surfaces do not have such well-definedridges and furrows and the rill network gradually evolvesto some branching and interconnecting pattern. If aprior determined rill vs. interrill partition is not avail-able, it will be difficult to apply this model.

Parallel to erosion model development, there havebeen many experiments designed to examine slopeand rain intensity (or runoff) effects on either totalerosion or interrill erosion only. Results of these stud-ies were analyzed by regression procedures to produceempirical relationships for slope and intensity effects(Evett and Dutt, 1985; Hahn et al., 1985; McCool etal., 1987; Mclsaac et al., 1987; Meyer, 1981; Meyeret al., 1975a; Watson and Laflen, 1986). Since thedominant erosion process will change as either theslope or rain-runoff rate is changed and if one uses aset of process-based equations to model erosion, then

C. Huang, USDA-ARS National Soil Erosion Research Lab., 1196SOIL Bldg., Purdue Univ., W. Lafayette, IN 47907; and J.M.Bradford, USDA-ARS, 2413 E. Hwy. 83, Weslaco, TX 78596.Received 25 Sept. 1992. * Corresponding author.

Published in Soil Sci. Soc. Am. J. 57:1176-1183 (1993).

solutions of these equations should automatically con-tain those empirical relationships derived for slope andintensity effects.

We proposed an alternative interpretation of theWEPP erosion and deposition equations that does notrequire the spatial partition of rill and interrill areas.This makes it simpler to apply these equations to gen-eral field conditions. We will present analytic solu-tions to both equations and examine slope and runoff(or intensity) effects on sediment yield based on de-tachment and transport-limiting conditions. Analyticresults will be compared with experimental data andempirical relationships reported in the literature.

THEORYBasic Erosion and Deposition Equations in WEPPThe WEPP steady-state sediment routing equation uses the

mass balance formulation:

^ = Dr + D, [1]

where qs is sediment rate per unit width of rill channel, M L-1

T-1; x is the length scale in the direction of the rill flow, L;and D, and D( are rill and interrill net detachment or depositionrates, M L~2 T-1. Foster and Meyer (1972) proposed to usethe first-order rate process to model rill detachment:

D, = [2]

where Tc is the transport capacity per unit width of the rill, ML-1 T-', and a is analogous to the reaction rate constant, I/L. Since I/a has a length unit, we considered it as the char-acteristic erosional length scale. The transport capacity, as itsname implies, is the maximum amount of sediment that a flowcan cany.

The first-order rate process concept was adopted from in-tuitive thinking and observations that clear water had the great-est amount of sediment detachment potential and, as the sedimentconcentration reaches its saturation level, or the transport ca-pacity, the detachment potential would gradually diminish (Fosterand Meyer, 1972). Once the sediment concentration reachesthe transport capacity, deposition takes place. Thus, the max-imum rill detachment rate, aTc, is called detachment capacity,Dc. Equation [1] can be rewritten as

dx

=4-1) [3]

Equation [3] is the sediment routing equation embedded withthe first-order rate process concept.

In the WEPP model, the detachment terms in Eq. [3] arefurther expanded to incorporate the critical shear concept forrill detachment and intensity dependency for interrill process

1176

HUANG & BRADFORD: SLOPE AND RUNOFF FACTOR ANALYSIS 1177

and

Dc = KT(r - rc),

A = KtSfI2

[4]

[5]where K, and K{ are rill and interrill erodibilities, r and TC areshear and critical shear stresses, Sf is the slope adjustmentfactor, and / is the rain intensity. Note that the interrill deliveryfunction is formulated by multiplication of factors, which issimilar to the model structure of the Universal Soil Loss Equa-tion (USLE).

The full erosion equation now becomes

I -I- [6]

Equation [6] is basically the same as Eq. [3] except that thereare two additional detachment concepts being incorporated. Ifone is interested in solving the erosion equation, it seems log-ical to work with Eq. [3] instead of [6] due to its simpler formand fewer parameters. On the other hand, if the first-order rateconcept in Eq. [3] was found invalid, Eq. [6] automaticallybecomes invalid. Much effort was placed into acquiring thecredibility parameters, K{, K,, and re; contrarily, little wasinvested in the validation of the rate process concept.

The deposition equation has a slightly different form fromEq. [3], and is given as

mwhere /? is a depositional rate parameter, M L,-2 T-1, and q^is the runoff discharge rate per unit width of rill, M L~' T-1.According to Foster and Meyer (1975), j8 is related to the fallvelocity of the sediment.

Limitations of the Rill and Interrill FormulationSince the WEPP governing equations are written for unit

width of rill, the actual implementation of the model requiresknowledge of rill width and spacing between rills. The inputrequirement of rill geometry is an inherent limitation of themodel. In general, one expects an erosion prediction model toproduce results in terms of rill development on the hillslope.Thus rill geometry is probably more appropriate as part of theoutput of the prediction model instead being an input require-ment.

The use of a linear threshold concept, K, (T - rc), for rilldetachment implies that these formulations are designed for apreset scale. This poses further limitations if we consider thatthe development of a rill network in the hillslope involves acascade of scales and processes.

Experimental evidence shows that severe rilling can occurin 0.5- to 1-m-scale interrill plots (see Fig. 1 of Huang andBradford, 1993). These microrills are developed from flowconcentrations caused by surface microtoppgraphy. Surface flowsin the network of microrills carry a major portion of the sed-iment. Under conditions of high erosion potential, such as witherodible soil, high slope gradient, or high rainfall-runoff rates,these microrills will gradually deepen, widen, and eventuallybecome a significant feature of the surface in the same fash-ion as rilling in the field hillslope. The shear stress on thesmall-scale interrill plot is relatively small compared withthe shear stress from a concentrated flow channel 10 to 100m long in the field. Since rilling is an indication of localizedshear excess due to flow, the question is how to model themicrorill detachment in a "typical" interrill setting? We

believed that there should not be a scale limitation on ap-plication of the first-order rate process concept for sedimentdetachment-deposition and transport if the concept is indeeda valid one.

Alternative Interpretation of the Erosion EquationHere we offer an alternative interpretation of the sediment

routing Eq. [3] and [7]. We put aside the concept of separatingthe surface into rill and interrill areas, and applied the massbalance equation to a unit width of the surface instead of aunit width of a rill. Since we no longer have interrill and rillareas, we replaced D{ with a new symbol, E, in Eq. [3] and[7]. The E term is the sediment supply rate per unit area dueto raindrop impact or other non-flow-related processes such asdispersion or slaking. Note that qs, Tc, and <?w are now alsoexpressed per unit width of the area. The detachment and dep-osition terms in these equations are still governed by the rate-process concept based on saturation deficit or excess relativeto the transport capacity, except that their interpretations arebroadened to the width of the plot, not just limited to the rillarea. With the new interpretation, we are able to separate thesource of sediment based on physical processes. The sourcesof sediment transported off a field area are now separated intothose that become available for transport from nonflow sources,such as raindrop impact, slaking, and dispersion, and thoseadditional amounts that are due to flow detachment, if Tc ishigher than <?s. When Tc < qs, deposition becomes the domi-nant process and the transported sediment is mainly from theE term.

Advantages of this alternative interpretation are that physicalprocesses of detachment and transport are maintained in themodel formulation, and spatial and scale limitations of sepa-ration into rill and interrill areas are avoided completely. Theerosion and deposition equations are basically mass balanceequations for detachment and transport, except that they canbe applied readily anywhere on the surface. This modificationalso simplifies the interpretation of collected sediment data,normally directly related to qs, from field plots. Since the con-cepts of rill and interrill are no longer used, the need to definerill and interrill erodibilities becomes redundant.

Note that, in this derivation, we have used a generalizedapproach to interpret parameters in Eq. [3] and [7] as beinggrossly averaged across the spatial scale (i.e., width of thesurface) and particle size classes. A more elaborate formulationis to expand these equations and define parameters such as Tc,qs, E, a, and 0 in terms of individual size classes.

Analytic Solution of the Erosionand Deposition Equations

To solve Eq. [3] and [7], we need to introduce a parametricrelationship for Tc. Finkner et al. (1989) showed that

Tc « T1-5, and T « (q^Sf3

where 5 is slope. Therefore,

Tc « qwS, or Tc = [8]

Equation [8] was derived from a simplified Yalin sedimenttransport equation and the Chezy form of the uniform flowequation. A detailed derivation for Eq. [8] can be found inFoster and Meyer (1972).

We let water discharge vary as a linear function of down-stream distance:

<?w = <?w [9]

where #wo is run-on water at the top of the slope segment and

1178 SOIL SCI. SOC. AM. J., VOL. 57, SEPTEMBER-OCTOBER 1993

a is the discharge rate parameter. Slope, S, is also expressedas a linear function:

5 = Sn + fee [10]

where S0 is the slope at the top of the plot and b is the slopegradient parameter. Positive, zero or negative values of b rep-resent convex, uniform, or concave slope forms, respectively.Combining Eq. [8], [9], and [10], Tc is now a quadratic func-tion of*:

Tc = abcx2 cqwoS0. [11]

The boundary conditions (jc = 0) are: qs = q^, qw = qwo,S = S0, and Tc = cq^0 S0, where q^ is the sediment rateassociated with the run-on water. Solution to the erosion equa-tion, Eq. [3], is:

- 2— }x + abcx2

(2abc _ qwjbc + Sjic\ a2 a

[12]

Solution to the deposition equation, Eq. [7], under the sameboundary conditions is:

+ ax)

= abcx2 + c(qvl(b + S<fl)xE — 2abcx — q^Jbc — S0ac

j8 + a

2abc

/\

+ 2a)

E -

(qwo + ax)2 [13]

+ a

2abca)(p + 2a)< + ax

Note that the analytic solution of the deposition equation isslightly different from the one presented in the WEPP model,which used different boundary conditions (Foster et al., 1989;Flanagan and Nearing, 1990). •

Solutions [12] and [13] were derived based on the assump-tion that parameters a, j3, and E are constant and do not varyas functions of downstream distance, x. It is foreseeable toapply Eq. [12] and [13] in small segments where the param-eters are relatively constant and to vary their values betweensegments if their downstream dependencies are known. At thistime, making a, /3, and E into spatially dependent functionsis an unnecessary complication because we still are not ableto prove the validity of the first rate process concept.

Since boundary conditions for both erosion and depositionequations are set up in a general fashion, these solutions canbe applied contiguously on slope segments where the dominantregime changes from erosion to deposition and vice versa dueto changes in slope shape or surface conditions. These solu-tions can also simulate erosion under furrow irrigation wheredischarge decreases with increasing distance from the inlet.

Despite the wide range of applications of these general solu-tions, we refrain from engaging a full-scale simulation beforevalidity of the first-order rate process concept and assumptionsused in the boundary conditions can be checked.

Simplified Solution for Uniform Slopewith No Run-on Water

We now present simplified solutions for boundary condi-tions that are applicable to many laboratory and field situa-tions. We assume that the slope is uniform, 6=0, and thereis no water or sediment being supplied at the top of the slopesegment: qwo = 0, and q-K = 0. Solution to the erosion equa-tion has the simple form

-(Or

Similarly, the solution to the deposition equation is

„ ' E -qs =

These two simplified solutions reveal some interesting slopeand runoff effects on the sediment delivery, #s. We first ex-amine, conditions where slope is the sole variable. Under eithererosion or deposition conditions, qs is always a linear functionof slope with positive intercept. When .runoff, <7W, becomesthe only variable, there are different effects depending on whetherthe system, is dominated by erosion or deposition. Under ero-sion, conditions, qs is a linear function of qw. Under depositionconditions, qs dependency on qw varies from quadratic to lineardepending on the ratio of /3 to a. When /3 » a, a/(j8 + a) «a and E/(f3 + a) « E. Under an oscillating nozzle type of rainsimulator, the rain intensity is controlled by the frequency ofthe sweep. The rainfall energy per unit volume of rainwater,therefore, is constant. Under this specific condition, we canassume that the amount of soil detached by raindrop impactand made available for transport, E, is proportional to the rainintensity of approximate runoff, qw. Thus, when /3 » a, qswill be a quadratic polynomial of qw. When /3 = a or /3 < a,the relationship approaches linearity.

MATERIALS AND METHODSSurface soil materials from Russell silt loam (fine-loamy,

mixed, mesic Typic Halpudalf, near Lafayette, IN) and Say-brook silt loam (fine-silty, mixed, mesic Typic Argiudoll, nearLexington, IL) were collected from fields and air dried in thelaboratory. Both soils were from conventional-tillage fields.The study of Saybrook soil was part of a large-scale projectdesigned to quantify tillage effects on soil erodibility.

After drying, the Russell soil was sieved through a 20-mm-opening sieve before placing into the erosion pan. The Say-brook soil was placed directly into the soil pan without sievingexcept for the removal of surface crop residues and breakingup large (>30 mm) clods by hand. The soil pan used is 1.2by 1.2 m in area and 0.12 m deep. The pan has a 50-mmbottom layer of sand, and the overlying soil was packed to adepth of 70 mm. A trough was placed at the lower edge of thepan to collect runoff and sediment samples. The pan was placed

HUANG & BRADFORD: SLOPE AND RUNOFF FACTOR ANALYSIS 1179

BU

80

70

<60

E 50

1 4O

20

10

n

. 1 1 , . . . . . . , .

-.

•

•

.

-

, •

. • 1

»

•-

-

m

•

Rain 70 50 100 70 50 100 70 50 100 70 50 100Slopa 5 X || 9 X || 20 X || 20 X Crust

IU

§ «i •^ 7

Sedim

ent l

am R

otev>

-no

-» M

u *

en a

« A

•

• - ._ " •

rrm mr, 1 \ rrm Iml

• -

1

' •

.

iln 7O 50 10O 70 50 1OO 7O 50 100 70 SO 100pa 5 X || 9 X || 20 X || 20 X Crust

Fig. 1. Runoff and sediment delivery rate from the soil pan study on Russell soil.

beneath a rainfall simulator using oscillating nozzles (Veejet80150, Spraying Systems Co., Wheaton, IL) suspended 3.05m above the pan. Deionized water with electrical conductivity<0.8 mS/m was used as rainwater. The rain simulator wasprogrammed to apply rainfall intensities of 50, 70 and 100mm/h.

The soil pan was rained for 15 min at 70 mm/h at its levelposition. This initial rain was used to consolidate loose aggre-gates on the surface and bring soil surface moisture content toa near-saturation level. The pan was then set to a 5% slope(Russell soil) and rained under the intensity sequence of 70,50, and 100 mm/h for 20, 20, and 12 min each. For Saybrooksoil, the first slope was set to 4%, similar to the field plotslope. During the rain, the shift in intensity was instantaneousby turning a knob on the rain controller. The rain sequencewas repeated for 9 and 20% slopes. Adjustment of the pan toa new slope took approximately 5 min.

After the first set of three slope and three intensity combi-nations, the soil pan was dried under a fan for 3 to 5 d untilthe surface appeared dry and developed cracks. The soil panwas then subjected to the same sequence of three intensityevents at 20% slope. The soil pan was replicated at least twice.

Four equally timed runoff samples, either 5 or 3 min, werecollected in 8-L buckets for each intensity-slope combination.The sediment samples were later transferred to autoclavablebottles for oven drying and sediment weight determination.

RESULTS AND DISCUSSIONExperimental Results

A runoff hydrograph and its corresponding sedimentdelivery rate from one of the two replicate runs for Rus-

sell soil are plotted in Fig. 1. This plot showed that thefirst sample from each event reflects residual effects fromthe previous event. For example, runoff from the firstsample of the 50 mm/h event after a 70 mm/h rain tendedto be higher then the other three samples. Therefore,only the last three samples were used to obtain averagevalues of runoff and sediment yield. Differences betweenrainfall and runoff values are due to infiltration as wellas water splashed off the pan. Inability to perform awater balance on the soil pans is one of the inherentdeficiencies of this type of setup. On the other hand, ifsediment delivery is interpreted in terms of runoff insteadof rainfall rate, then water loss due to infiltration andsplash off the pan will not affect the interpretation.

Soil erosion data from the Russell soil are presentedin two graphs. Figure 2 is a plot of average sedimentdelivery rate (qs) as a function of slope steepness (5)under three different runoff rates (qw). Figure 3 showsqs as functions of qw for different 5. Sediment data col-lected from runs on dried crusts are not shown in Fig. 2because these runs were carried out for the 20% slopeonly. Best-fit curves from regression analysis are drawnon the figures, and statistical parameters are presentedin Tables 1 and 2.

These plots show some interesting relationships. Fig-ure 2 shows that, for all three runoff rates, qs is a linearfunction of 50, and the slope of the line increases asrunoff rate is increased. Figure 3 contains two types ofrelationships. At low slopes the sediment delivery, qs, is

1180 SOIL SCI. SOC. AM. J., VOL. 57, SEPTEMBER-OCTOBER 1993

tiinM ,o 3-

® «E2:

Russell SoilRunoff, mm/hO D O O O 75

5134

20 250 5 10 15Slope, o/o

Fig. 2. Sediment delivery rate of Russell soil as a function ofslope for three levels of runoff with least square regressionlines.

10

9-3

> 8^

O» 7-

•' 6:e

5-

I *1^

2-(ft

1-

Russell SoilSlope

opa<u> 20 o/o (Dried Crust)o°S>S>S 20 o/oAjtAA* 9 0/0

5 o/o

<* .'

.' **

XX

20 30 40 50 60 70Runoff, mm/h

so

Fig. 3. Sediment delivery rate of Russell soil as a functions ofrunoff for three levels of slope steepness. Also shown is erosionfrom a dried crust at 20% slope. Best-fit curves are plottedon the graph.

a curvilinear relationship to runoff, <?w. When plot slope,S0, is increased to 20%, qs becomes linearly dependenton q^. Effects of slope steepness on sediment yield willbe discussed first.

Erosion data from the Saybrook conventionally tilledsoil also have similar slope and runoff dependency ex-cept at a lower magnitude (Fig. 4). Since our objectivewas to provide an analytic explanation of experimentaldata, only data from Russell soil will be used as exam-ples in the following discussion.

Table 1. Results of regression analysis on sediment yield (qs)of Russell soil as a function of slope (5) at different runoffrates.

Runoffmm/h

345175

Model

q, = aS + b

q, = aS + b

Model coefficientsa b

7.173 -0.057917.845 -0.205828.207 0.3915

i2

0.99900.99870.9954

Effects of Slope Steepness on Sediment YieldEffects of slope steepness on erosion are generally

reported under two contexts: total soil loss or the interrillerosion only. For rill erosion, the slope effects are im-plicitly contained in transport capacity, Tc, and shearstress, T. Recent publications on the slope adjustmentfactors on total erosion were centered on the modifica-tion of the USLE slope factor, which relates soil loss atany slope to soil loss at 9% slope by

SUSLE = 65.41(sin 0)2 + 4.56 sin 6 + 0.065

where 6 is slope angle (Wischmeier and Smith, 1978).At low slopes, sin0 is approximately equal to slope gra-dient, or tan0. McCool et al. (1987) reanalyzed historicdata used to derive SUSLE and incorporated an extensivedatabase to derive a revised USLE slope factor that hastwo linear segments with a break point at 9%. Parallelto the effort of McCool et al. (1987), Mclsaac et al.(1987) also reevaluated the USLE slope factor on erosionfrom disturbed lands and confirmed linear relationshipsas those reported by McCool et al. (1987) except oneslight modification on the linearity coefficient. Under theUSLE, erosion from plots with a slope length shorterthan 4 m is to be considered as interrill erosion. Afteranalyzing data from Lattanzi et al. (1974) and Meyer etal. (1975b), Foster (1982) proposed the following inter-rill adjustment factor:

SUSLE-I = 2.96(sin 0)°-79 + 0.56.

Mclsaac et al. (1987) stated that, for slopes <4 m long,the slope factor was equally well described by either alinear function of percent slope, sin0, or sin0°-8. Exper-imental data reported in the literature showed either alinear trend (Murphree and Mutchler, 1981; Hahn et al.,1985; Evett and Dutt, 1985; Schroeder, 1987) or an ap-proximately linear dependency at low slopes and a grad-ual taper at high slopes (Lattanzi et al., 1974; Meyer etal., 1975b; Singer and Blackard, 1982; Watson and La-flen, 1986; Meyer and Harmon, 1989). Meyer et al.(1975b) attributed the gradual diminishing of slope ef-fects at high slopes on interrill plots to the shift from atransport-limiting regime at low slopes to detachmentlimiting at high slopes.

The analytic solutions Eq. [14] and [15] readily ex-plain those data sets, including data from this study,which displayed linear dependency on slope steepness.These solutions also suggested that the slope of the straightline depends on gw and the detachment, a, or deposition,/3, parameters. Figure 5 is a plot of the linearity param-eter from Table 1 against qw and it shows a general trend

HUANG & BRADFORD: SLOPE AND RUNOFF FACTOR ANALYSIS 1181

Table 2. Results of regression analysis on sediment yield (qs) of Russell soil as a function of runoff (<?„) at different slope steepnesses.

Slopem/m0.05

0.09

0.200.20

(Dried crust)

Model

4> = a4i4s = aq* + ft<jw + c4, = ag£q, = agj, + fti/w + c4s = a4« + fr4. = a<7» + ft

a

0.0001480.0005230.0003510.0005900.1145440.148094

Model coefficientsb

2.1708-0.020914

2.1084-0.002230-2.43808-3.00735

C I2

0.9952f0.425269 0.9974

0.9995-0.014625 0.9990

0.99450.9957

t The i2 for the q, - atf, model is calculated from the log-transformed model: In q, = In n + ft In <jw.

of increasing value of a with #w. The break of the lineartrend may be due to the shift from an erosion-dominatingregime of the high-intensity rainfall to the depositionregime of low intensities.

The analytic solutions also show a positive intercept,indicating that at zero slope erosion is occurring. Theintercept, or qs at 5 = 0, is a function of E, a, /3, and#w. Under some conditions, e.g., small values of E or<7W, the intercept may approach zero. To validate thepositive intercept concept, only data collected at lowslopes, 0 to 2%, can be used. Data reported by Lattanziet al. (1974), Meyer et al. (1975b), Murphree and Mutchler(1981), and Schroeder (1987) supported the positive in-tercept concept. In addition, Lattanzi et al. (1974) alsoshowed an increased zero-slope intercept as erosion rateis increased, a characteristic implicitly contained in theanalytic solution.

Effects of Runoff on Sediment YieldSimilar to the development of slope factors for ero-

sion, the rain erosivity factor was either treated as alumped parameter, EI30, in the USLE approach or sep-arated into rill and interrill processes in the WEPP pro-

cedure. For interrill erosion, a power-law-type intensityfactor was proposed and the exponent was found ap-proximately equal to 2 (Meyer, 1981; Watson and La-flen, 1986; Meyer and Harmon, 1989). For rill erosion,the erosivity is contained in the transport capacity, Tc,and flow shear, T (Foster and Meyer, 1975).

Although rainfall energy and flow shear are drivingforces for soil detachment in interrill and rill areas, it isthe transport process that actually moves the sedimentoff the field. One misconception is the direct translationof sediment transported off a field or an erosion plot todetachment rate. This exercise essentially ignores theseparation between detachment and transport processes.Conceptually, the detachment process makes sedimentavailable for transport and the transport process movesthe sediment. Therefore, sediment collected at the outletis what has been transported and it is not necessarilyequal to the amount that has been detached. The amountof transported sediment, qs, can be related back to de-tachment rate if the relationship between detachment andtransport is known. If processes of erosion are to bemodeled anew, it seems logical to start with the transportequation as the first step. This is the reason why weexamined only the transport equation and intentionally

10

9t

8-

£ 6oa:

3-

*

Saybrook Soil, Conv. TillSlope

O D D a o 20 o/o (Dried Crust)02<L°J> 20 0/0».»».»» 9 o/o* • « « » 4 o/o

70

35

BOi i I i i i

20 30 40 50 60Runoff, mm/h

Fig. 4. Sediment delivery rate of Saybrook soil as a functionof runoff for three levels of slope steepness. Also shown iserosion from a dried crust at 20% slope. Best-fit curves areplotted on the graph.

30-

825

c20o>•oo>Q.15Q>QOg-10

V)

5-

20 30 40 50 60 70Runoff, mm/h

so 90

Fig. 5. Slope dependency coefficients obtained from linearregression as a function of runoff rate for Russell soil, showingthe increasing trend.

1182 SOIL SCI. SOC. AM. J., VOL. 57, SEPTEMBER-OCTOBER 1993

did not include any detachment parameters in this ex-ercise.

Since sediment is carried off the field by runoff water,sediment yield, qs, should be directly related to runoff,qw. The analytic solutions given by Eq. [14] and [15]indeed have explicit relations between qs and qw.

Experimental results from this study (Fig. 3 and Table2) showed that, at low slopes, qs is nonlinearly relatedto gw. Regression analyses showed that this nonlinearrelationship can be described either by a power law modelor a quadratic polynomial. As the slope is increased to20%, <?s becomes linearly related to qw. After drying,the surface became more erodible and the sediment yieldwas increased by roughly 30% for all three intensities.Effects of surface conditions, such as drying, and tillageeffects on soil credibility will not be discussed here. Theshift from a nonlinear to linear runoff dependency asslope is increased indicates the difficulty in deriving ageneralized empirical relationship through statistical pro-cedures without taking into account the processes in-volved. Both quadratic polynomial and linear relationshipscan be explained by solutions of the deposition and ero-sion equations, as discussed above.

At 20% slope, severe rilling occurred during the rain.Thus, we can use the solution of the erosion equation(Eq. [14]) for this case. One discrepancy between Eq.[14] and the regression results is the intercept term. Thenegative intercept for the two linear relationships at 20%slope suggested a threshold runoff value for sedimentyield. This is directly in agreement with the runoff-basedmodel for rill erosion discussed by Meyer et al. (1975a).As an interesting note, both linear relationships give avery similar threshold runoff value of 20 mm/h.

At low slopes, the surface may be dominated by thedeposition regime. Solution to the deposition equation,Eq. [15], can yield a quadratic polynomial dependencyon runoff. Regression analyses showed both quadraticpolynomial and power law relationships described equallywell for sediment yield at low slopes. If the interrillcondition is typified by the lack of transport capacity tocarry the detached sediment, then the deposition equa-tion would describe the interrill erosion as well. Thereason we make this linkage is due to the extensive useof a power-law-type relationship with an exponent of 2to describe the interrill erosion. Our analysis apparentlyprovides an analytic basis to the empirically derived re-lationship.

CONCLUSIONSErosion and deposition equations used in the WEPP

model are interpreted as lumped processes in the spatialscale, thus the prior knowledge of rill and interrill areasare no longer needed. The analytic solutions, based ona specific transport capacity model, were derived. Spe-cific solutions for a general case, uniform rainfall on auniform slope, give an explicit relationship between sed-iment yield, qs, slope S, and runoff discharge, <?w. Ifslope or discharge is considered as the sole variable, theanalytic solutions appear capable of explaining the ex-perimental data and other published empirical relation-ships. Other empirical relationships that deviate from theanalytic expressions, such as the nonlinear slope de-pendency, suggest that some of the constant-valued pa-

rameters used in the current derivation may be functionsof runoff, slope, and downstream distance, x.

Although some of the experimental data can be ex-plained by the analytic solutions, this by no means con-stitutes a proof of the validity of the rate process conceptproposed by Foster and Meyer (1972) and now incor-porated in the WEPP model. The structure of Eq. [12]and [13], or the simplified case of Eq. [14] and [15],indicates that experimental procedures to validate theconcepts and assumptions used in this analysis are muchmore involved. There are more unknowns in these equa-tions than can be measured under the current experi-mental setup. Data already suggested that the reactionrate constants, a and /3 parameters, may not be constantat all. Therefore, the validity of the rate process conceptused to couple the difference between Tc and qs andbetween flow detachment and deposition rate is yet tobe proved.

TARIQ & DURNFORD: ANALYTICAL VOLUME CHANGE MODEL FOR SWELLING SOILS 1183