modelling of hillslope runoff processes

Environmental Geology 35 (2–3) Auguat 1998 7 Q Springer-Verlag 115

Received: 5 October 1996 7 Accepted: 25 June 1997

N. M. ShakyaCivil Engineering Department, Institute of Engineering,Laliptur, Nepal

S. Chander (Y)Civil Engineering Department, Indian Institute of Technology,Delhi, India

Modelling of hillslope runoffprocessesN. M. Shakya 7 S. Chander

Abstract The present study is aimed at modellinghillslope flows with emphasis on subsurface storm-flows that involve macropores. The physical pro-cesses connected with the runoff process on a hill-slope are identified. The components which areconsidered in modelling the hillslope flow are thenature of flows in the macropore and microporedomains, the spatial and temporal characteristics ofthe macropore network, the interaction between thedomains, and the initiation of flow in the macro-pores. Both Horton and Dunne’s variable sourcearea generation mechanisms are explicitly incorpo-rated in the model. The dominant physical pro-cesses governing hillslope runoff are conceptualizedin terms of parameters which are derived from thephysical properties of the soil, the nature of macro-pores, and hillslope geometry. The conceptualiza-tion of the model is then used to examine infiltra-tion and runoff production. This helps to computethe development of the groundwater table, runoffhydrograph, and soil moisture profile.

Key words Hydrology 7 Slope 7 Infiltration 7Runoff 7 Macropore

Introduction

The multiplicity of field catchment studies on densely in-strumented small areas, led to a new interest in de-scribing and modelling hydrological processes on slopes.This has resulted in the development of hillslope hydrolo-gy, which is concerned with the transformation of preci-

pitation as it passes through the vegetation and soillayers on a slope. In hilly areas, hillslopes are responsiblefor generating 95% of the water in the streams. The waterflows over or through the soil before reaching the chan-nel network. Therefore, modelling of hillslope hydrologyamounts to modelling of infiltration, percolation anddownslope flow. Earlier models describing rainfall runoffprocesses on a catchment generally relied on Horton’s(1933) infiltration excess phenomenon for runoff genera-tion. However, persistent failure to observe this phenom-enon in a wide variety of forested hillslopes has castdoubt on its validity. In order to explain these observa-tions Hewlett (1961) introduced the variable source areaconcept to explain the formation of runoff as a result ofsaturation of the toe of the slope. This concept has beenused by Dunne (1983) to develop a conceptual model forstreamflow generation in a watershed.In humid and hilly areas, surface flow may dominate thevolume, peak, and timing of the hydrograph. Under suchconditions water movement occurs in two domains:through the micropores (i.e., the soil matrix); andthrough the interconnected macropores of the soil sys-tem. Whipkey (1965), Weyman (1970), Sloan and others(1983), and Moore and others (1986) attribute rapid sub-surface flow response in forested hillslopes to macroporesin both saturated and unsaturated conditions. Macro-pores represent only a small percentage of the total porevolume, but account for the bulk of water movement.Approaches based on the theory of potential flow overes-timate either the volume of flow or the response time atthe hillslope scale when their results are compared withthe observations on forest soils. Macropores are thoughtto be the cause behind these discrepancies, indicating theneed for questioning the Darcian assumption for subsur-face flow. Despite the widespread observations of thesediscrepancies and a general agreement on the predomi-nance of non-Hortonian flow in humid watersheds, near-ly all existing hydrological models are based on the Hor-tonian flow concept. Ormsbee and Khan (1989) proposeda model which was developed to address this problem.The model, however, needs observed rainfall-runoff datato iteratively determine the various model parameters,i.e., it lacks the ability to estimate model parametersfrom the physical properties of the soil and the geometryof the slope. In this study we propose a model in whichthe parameters can be estimated using physical proper-ties of the soil an the geometry of the slope.

116 Environmental Geology 35 (2–3) August 1998 7 Q Springer-Verlag

Fig. 1Saturated and unsaturated zones on ahillslope

Literature review

The runoff processes on a hillslope, i.e., infiltration, per-colation, and downslope flow, depend on physical charac-teristics of the soil. The rate of flow through the unsatu-rated zone can be very different for soils with mainlyDarcian flow (in micropores) as compared to soils with asignificant macropore network (Mosely 1982; Ormsbeeand Khan 1989). This indicates the need for the introduc-tion of a two-domain concept to model combined micro-pore/macropore systems. The micropores form one do-main and the macropores the other. This is done by div-iding total soil porosity into two components. The firstcomponent contains mobile water and the second com-ponent contains immobile or slowly moving water.Many authors (Sloan and others 1983; Steenhuis and oth-ers 1988) have simplified Richard’s equation for flowthrough a single domain, porous media for applying it tohillslopes. These simplifications transformed Richard’sequation into different forms of a kinematic wave equa-tion. The form depends on the approximations, such aslayering of hillslope depth, consideration of vertical flowand downslope flow, or a combination of these. The in-corporation of a macropore system into such modelsposed a great challenge due to the complex geometry ofthe macropores. Three approaches were followed to de-velop two-domain models, namely: 1) analytical methods,by solving Richard’s equation with simplified assump-tions in which the micropores-macropore dichotomy ischaracterized by parameters obtained through in situmeasurement of hydraulic properties of soil; 2) incorpo-ration of macropore flow into existing subsurface flow

models; or 3) conceptual flow allocation using water bal-ance accounting procedures (Ormsbee and Khan 1989).In the first method, the diffusion assumption inherent inRichard’s equation would be violated. As a result, suchmodels neglect the nature of flow in the macropore sys-tem, initiation of flow and the multitude of flow pathstaken by macropore flows. In the second method, the useof subsurface flow models for a shallow sloping soil layeroverlaying an impermeable bed (Steenhuis and others1988) is not valid for hillslope with apparent depth. Inthe third method, the model lacks the ability to estimatethe parameters from the physical properties of the soiland the geometry of the slope. In addition, the model isnot capable of computing moisture profiles, which is veryimportant if the slope is not represented as a thin layer.The proposed model was developed emphasizing subsur-face stormflows that involve macropores. The model spe-cifies the nature of flows in micropore and macroporedomains, the spatial and temporal characteristics of themacropore network, the interaction between the two do-mains, and the initiation of flow in the macropores.

The model

The modal represents the hillslope as a rectangular stor-age element of length L and depth D with an impermea-ble bed making an angle a with the horizontal (Fig. 1).The hillslope is assumed to be comprised of m number ofparallel elementary layers, of depth Dz, of nearly thesame characteristics. Depending upon the extent of thewater table each layer is comprised of unsaturated and

Environmental Geology 35 (2–3) August 1998 7 Q Springer-Verlag 117

Fig. 2a Flow allocation in two zones of hillslopes; b flow allocation inunsaturated zones of different layers in hillslopes

saturated zones of length li and LPli (ip1, 2,. . . m), re-spectively (Figs. 1 and 2). The unsaturated zone, of vol-ume V, is assumed to be made up of two domains repre-sented by moisture stores, namely micropore store andmacropore store. The size of micropore store and macro-pore store is determined by the microporosity and ma-croporosity of the soil respectively. The micropore do-main is defined as soil whose pore size is less than theminimum macropore size (rmin). The hydraulic conduc-tivity in the micropore domain is function of its moisturecontent. The macropore domain comprises large voidsproducing greater flux than is produced by the saturatedmicropores. The hydraulic conductivity in such a domainis a function of macroporosity representing large-sizedvoids. Water which cannot be stored in these domainswill begin to be stored at the soil surface as surface store.Four situations can arise on initiation of rainfall event ofvolume Q1in.

Case 1: overland flowIn this case the input to the unsaturated portion of thehillslope is greater than the capacity of the uppermostlayer to the slope, i.e.,

Qluin1 (Qmaxm,incFScF) (1)

where,

QluinpQl

inPQsatout

Overland flow, Qintovr, occurs due to the infiltration excess

mechanism under QintovrpQl

uinP(Qmaxmac,incFscF).

In this case micropore store is full to its capacity (Fig. 3).The terms which appear for the first time are: Qsat

out, theoverland flow volume due to the saturation excess mech-anism; Quin, the component of total rainfall diverted intothe unsaturated zone; Qmax

min , the maximum input whichcan be received by the macropore domain of the con-cerned layer; Fs, the portion of infiltration in microporestore excluding the portion which occurs even after sur-face saturation; and F, the cumulative infiltration at satu-ration rate. Qmax

hp is the transaction which occurs betweenmacropore and micropore stores.

Case 2: Macropore-dominated flowIt occurs when the input to the unsaturated zone is lessthan the capacity of uppermost layer, as well as, the sumof the conductivity of micropores plus the capacity ofmacropores, i.e.,

Khl1 DtcQmaxmac,in 1Qluin~(Qmax

mincFscF) (2)


Fig. 3Case 1: flow allocation in the overland flow situation

where Kh is the hydraulic conductivity of soil in micro-pore domain; and I1 is the length of the unsaturated por-tion of the topmost layer. The allocations of water to var-ious stores is shown in Fig. 4.

Case 3: Transaction-dominated flowWhen the input to the unsaturated zone is greater thanthe conductivity of the micropores and is less than thesum of the conductivity in the micropores plus the ca-pacity of macropores, i.e.,

Khl1 Dt~Qluin~(Khl1 DtcQmax

min ) (3)

During this period small-sized macropores get saturatedwhile most of the large macropores remain partially satu-rated. The allocation of water to various stores is shownin Fig. 5.

Case 4: Empty macropores flowIn this case the input to the unsaturated zone does notexceed the conductivity of the micropores, i.e.,

Qluin~orpKhl1Dt (4)

Macropores are completely empty and the flow situationis similar to the flow in a single domain matrix system(Fig. 6). The hydraulic conductivity of matrix, Kh, is de-fined by the Brooks and Corey (1964) function as:

KhpKsh1uu1Pui

urPus2

1/n

(4a)

where ur and us are residual and saturated moisture con-tents, respectively; ui is the initial moisture content; uu1 isthe moisture content in the unsaturated micropore

Fig. 4Case 2: flow allocation in the macropore-dominated situation

stores; n is a constant depending upon soil water charac-teristics; and Ksh is the saturated hydraulic conductivityof micropores.

KhpKspKsm(p0) (4b)

where Ks is the observed saturated hydraulic conductivityof soil with macroporosity, p0; Ksm(p0) is the saturatedmacropores hydraulic conductivity.

Flow through macroporesMacropores are considered to have a minimum size ofradius rmin and maximum size of radius rmax with macro-porosity p0. Let Ksm(p0) be the saturated macropore hy-draulic conductivity, l the parameter which characterizesvertical water movement, and Am the macropore area.Assuming: the macropore flow as the average value offlow through the maximum and minimum size of macro-pores; laminar flow in small size macropores; and Man-ning’s flow in maximum size macropores, Shakya (1995)found that the vertical flow can be computed as:

QplAm (5)

where,

lp12 3

gr2 min8v

c1m 1rmax

2 22/34

Shakya (1995) computed the effective circumferencelength, L0, in the saturated macropore domain of area Am

as:

L0paAm (6)

where


Fig. 5Case 3: flow allocation in the transaction-dominated situation

Fig. 6Case 4: flow allocation in the empty macropore situation

a was found to be equal to 1 1rmin

c1

rmax2. (7)

Using a one-dimensional flow equation combined withDarcy’s equation, Shakya (1995) computed the value of Ih

as:

Ihp1Sr

2t21/2

(8)

where Sr is the sorptivity, which was computed using:

Sr(u)p6.3(usPui)1/2 K1/4sh (9)

Knowing these parameters, the lateral flow from macro-pores to micropore is given by

Qmaxhp pihL0DzDt (10)

Outflow from macroporesFor the different cases of flow described earlier, total out-flow Qm,out under free drainage from domain in the layerunder consideration is estimated using the continuityequation:

(qm,inPqm,out) AtDtPQhppd(pv)

dtDt (11)

where qm,in is the input rate per unit area in the macro-pore domain; and qm,out is the flow rate per unit area un-der free drainage from the macropore domain.Equation 11 can be solved to obtain qm,out as:

qm,outpqm,in

C 3DtPbDzlC 11Pexp1PDtC

bDt 224 (12)

where Cp1cihbDz

l; and b is a constant determined

during the calculation of mass balance.

Outflow from the micropore domainThe outflow from the micropore domain, Qh,out, duringtime interval Dt in the layer under consideration, is esti-mated with the help of the response equation:

C1dqout

dt

cqh,outpC2PC3 exp(C4Dt) (13)

where

C1pDtw

;

C2pqh,incihabDzqm,in

lC;

C3pihabDz

l

Qm,in

C;

C4plC

bDz; and

qh,in and qh,out are input and output rate per unit area inthe micropore domain.Qh,out can then be estimated as:

Qh,outpC2DtPOn(1Pexp(PDt/C1))

cC3

C4(1cC1C4)(1Pexp(C4 Dt)) (14)

where On is the percolation rate.


Water balance of different layersThe water balance for each layer can be written as:

Qinpp1v1Pp2v2cuu1v1Puu2v2cQsoutcQout (15)

where Qin is the total input to the layer under considera-tion; p1 and p2 are average effective macroporosity of thelayer under consideration at the beginning and end ofeach time step, respectively; Qsout is the total outflow div-erted into the saturated zone of the layer under consider-ation; and Qout is the total outflow, which becomes theinput to the subsequent layer.Flow allocation in different zones of the hillslope are rep-resented in Fig. 2. Using Eq. 15 the new moisture con-tent, uu2, at the end of time step Dt can be obtained,since all other terms are known.

The last layerFor the slope under consideration, which lies above animpermeable bed, there will be no vertical outflow fromthe last layer. The moisture moves downhill in this layerand contributes to the development of the water table. Inthe model this mechanism is simulated by dividing thelast layer into a number of segments along the slope. Thecharacteristic velocity of water in the layer decides thelength of the segment. The change in moisture content inany segment is due to the output from the upper layer,as well as the input from the previous uphill segment ofthe same layer. This makes the moisture content of eachsegment greater than the previous uphill segment, lead-ing to the saturation of the segment with time and thebuild up of a water table.Let the depth of water table at the begining be h1, and atthe end of the time step h2, then

h2

lsmfu2d

DtcKsh

lsmf sin aPh2 cosa

lsmf

2p

ph1

lsmiu1d

DtcKsh

lsmi sin aPh1 cosa

lsmi c

m

Ajpw

Qjsout

Dtc

c

2porDzm

(lsmfPlsmi)

Dt(16)

were lsmi and lsmf are the saturated length on the bed ofthe hillslope at the beginning and end of the time step,respectively; por is the microporosity of soil; and u1

d andu2

d are the drainable porosity of soil at the beginning andend of each time step, respectively.Equation 16 can be used to determine water-table depth,h2, at the end of time interval Dt, knowing the initial wa-ter-table depth, h1.

Model application

The model was applied to Beven and Germann (1981)simulation data as a result of different combination ofsoil-water characteristics in the micropore and macroporedomains (Table 1). The model simulation was also car-ried out for Weyman’s (1970) experimental hillslope ob-servations, and using drainage discharge data from a soiltrough measured by Hewlett and Hibbert (1963) at theCoweeta Hydrological Laboratory. The Coweeta study isof practical interest because it provides data that can beused to evaluate the ability of a subsurface model to sim-ulate flow in shallow soils overlying a steeply sloping im-permeable bed.

Results and discussionThe results of the model simulation for one- and two-do-main soil systems are compared with that of Beven andGermann (1981) in terms of moisture profile and cumu-lative infiltration. The comparison of moisture profile fora one-domain soil system shows that the timing and dis-tribution of moisture over depth are in good agreementwith that produced by the Beven and Germann (1981)model. The saturated hydraulic conductivity of soil typeD in Table 1 is only 3.74% more than that of D’ as a re-sult of an additional 0.3% macroporosity. In this case themacropores remain empty and the additional macropor-osity has little effect in changing the moisture profile(Fig. 7). For soils with lower hydraulic conductivity, likethe Guelph loam (Table 1), the extra 0.3% macroporosityincreases the saturated hydraulic conductivity of the soilsystem by 125%. In such soils macropores allow addition-al infiltration into the micropores depending upon therate of loss into the surrounding micropores. As a result,more moisture is received by the micropore domain, thuschanging the moisture profile (Fig. 8).The infiltration capacity of soil of type A, because of thelarger size macropores, is larger than that of soil of typeC (Table 1). Consequently, significant moisture transac-tion from the macropore to the micropore domain takesplace, keeping a larger number of macropores empty.This results in a complete adjustment of moisture intosuch soil within the topmost 50 cm. However, in soil oftype C more macropores get saturated than could be pos-sible in soil of type A, Therefore, an appreciable quantityof moisture percolates down, resulting in the moistureprofile shown in Fig. 9. The simulation runs suggest thatthe macroporosity and distribution of macropore sizehave direct effect on the flow situation in the macroporedomain. This influences the moisture distribution in themicropore domain of the soil system, even though the to-tal saturated hydraulic conductivity (Ks) remains con-stant. For soil of type B (Table 1), exhibiting a macropor-osity of 0.003, the relative change in moisture contentduring the same time interval remains nearly constant.This is due to the negligible interaction of macroporeswith highly permeable micropores (Fig. 10).


Table 1Characteristic parameters of different soils

Soil type Microporesaturatedhydraulicconductivity(Ks)[cm sP1]

Microporosity[vol/vol]

Macroporosity[vol/vol]

Totalsaturatedhydraulicconductivity(Ksh)[cm sP1]

Maxmacroporesize[m]

Minmacroporesize[m]

A 1.23!10P3 0.495 0.003 2.23!10P2 0.5!10P2 0.5!10P5

B 1.23!10P2 0.495 0.003 3.34!10P2 0.5!10P2 0.5!10P5

Bb 1.23!10P2 0.495 Nil 1.23!10P2 Nil NilC 1.23!10P3 0.495 0.003 1.69!10P3 0.5!10P4 0.5!10P5

Cb 1.23!10P3 0.495 Nil 1.23!10P3 Nil NilD 1.23!10P2 0.495 0.003 1.28!10P2 0.5!10P4 0.5!10P5

Db 1.23!10P2 0.495 Nil 1.23!10P3 Nil NilGuelph Loam 3.67!10P4 0.523 Nil 3.67!10P4 Nil Nil

Fig. 7Moisture profiles of highly permeable soil with and withoutmacropores

Fig. 8Moisture profiles for Guelph loam with and without additionalmacroporosity after 10 min

The proposed model was used to compute the runoff hy-drograph for four bursts of rain using the data of theEast Twin-Brook catchment. The model was run with amacroporosity of 0% and 0.1%, The hydrographs ob-tained were compared with the observed hydrographs ofWeyman (1970) (Fig. 11). The timing of the hydrographsin terms of both the start of the rise and occurrence ofthe peak are remarkably well reproduced. The compari-son of hydrographs obtained without considering macro-pores shows early generation of runoff compared to theobserved one, suggesting the presence of large poreswithin the soil. These large pores allow the distributionof infiltrated water as a result of transaction of moisturefrom the macropore wall to the micropore domain withinthe layer nearer to the slope surface. This also results ina delay in the start of the hydrograph if compared to thatpredicted by the model which considers the slope as a

one-domain soil system. The result of a simulation con-sidering a 0.1% macroporosity throughout the depth ofthe slope and preserving the same total hydraulic con-ductivity (Ksh) supports the above explanation.Lastly, the model was used to simulate subsurface flowoccurring within and through the soil micropores inhillslopes. The cumulative drainage was computed usingthe model and compared with the model obtained usingSloan and Moore’s (1986), the kinematic storage model,Hewlett and Hibbert’s (1963) observed value, Nieber andWalter’s (1981) 2-D finite element model, and the simpli-fied drainage model (Fig. 12). The results obtained are ingood agreement with Nieber’s two-dimensional numericalsolution of Richard’s equation.


Fig. 9Moisture profiles for soils with the same KSh value andmacroporosity but different maximum macropore sizes after1186 s

Fig. 10Moisture profiles after different time periods for highlypermeable soil with the same KSh values but different porosity

Fig. 11Comparison of observed and predicted hydrographs forweyman’s slope

Fig. 12Comparison between observed and predicted cumulative runoffcurves for the present model and other existing models

Conclusions

The following conclusions can be drawn from this study:1. Both Horton and Dunne’s variable source area genera-

tion mechanisms can be explicitly used to model therunoff processes on hillslopes.

2. A one-domain model is inappropriate in modellinghillslope runoff process in the following situations: a)when rainfall is higher than the infiltration capacity ofsoil micropores; b) where the total saturated hydraulicconductivity of the soil system and the saturated hy-

draulic conductivity of the soil macropores are differ-ent; c) where slopes are comprised of layered soilswith decreasing macroporosity with depth; and d) ifsoil moisture content in the slope is maintained closeto the saturation level, as it is the case in humid areas.

3. Macropores allow the distribution of infiltrated waternearer to the surface of the slope, resulting in a delayin the start of the hydrograph, as observed on hill-slopes.

4. Highly permeable homogeneous soils exhibit similarmoisture movement as permeable homogeneous soils


with an appropriate additional macroporosity. Thissuggests that highly porous soils have to be classifiedin terms of macroporosity along with other character-istics, so that the saturated hydraulic conductivity ofthe matrix can be estimated with the help of observedsaturated hydraulic conductivity of such soils.

References

Beven K, Germann P (1981) Waterflow in soil macropores, 2,a combined flow model. J Soil Sci 32 : 15–29

Brooks RH, Corey AT (1964) Hydraulic properties of porousmedia. Hydrology Paper no. 3. Colorado State Univ., FortCollins, Cdo.

Dunne T (1983) Relation of field studies and modelling in theprediction of storm runoff. J Hydrol 65 : 25–48

Hewlett JD (1961) Soil moisture as a source, of base flowfrom steep mountain watersheds. USDA Forest Service PaperSE 132. pp 1–10

Hewlett JD, Hibbert AR (1963) Moisture and energy condi-tions within a sloping soil mass during drainage. J GeophysRes 68 : 1081–1987

Horton RE (1933) The role of infiltration in the hydrologicalcycle. Trans Am Geophys Union 14 : 446–460

Moore ID, Bruch GJ, Wallbrink PJ (1986) Preferential flowand hydraulic conductivity of forest soils. Soil Sec Soc Am J50 : 876–881

Mosely MP (1982) Subsurface flow velocities selected forestsoils, South Island. N Z J Hydrol 55 : 65–92

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Ormsbee LE, Khan AQ (1989) A parametric model for steeplysloping forested watersheds. Water Resour Res 20 : 1815–1822

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Sloan PG, Moore ID (1986) Modelling subsurface stormflowon steeply sloping forested watersheds. Water Resour Res20 : 631–634

Sloan PG, Moore ID, Colthrap GB, Eigel JD (1983) Modell-ing surface and subsurface stormflow on steeply sloping for-ested watershed. Report 142, Water Resources Institute

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(1988) A simple model for flow on hillslopes. Agric WaterManage 14 : 153–168

Weyman DR (1970) Throughflow on hillslopes and its relationto the stream hydrograph. Bull Int Assoc Sci Hydrol 15 : 25–33

Whipkey RZ (1965) Subsurface stormflow from forested slopes.Int Assoc Sci Hydrol Bull 10 : 74–84

modelling of hillslope runoff processes

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