Conjunctive surface–subsurface modeling ofoverland flow

Vivekanand Singh* & S. Murty BhallamudiDepartment of Civil Engineering, Indian Institute of Technology, Kanpur, India

(Received 10 June 1996; revised 10 January 1997; accepted 28 May 1997)

In this paper, details of a conjunctive surface–subsurface numerical model for thesimulation of overland flow are presented. In this model, the complete one-dimensional Saint-Venant equations for the surface flow are solved by a simple,explicit, essentially non-oscillating (ENO) scheme. The two-dimensional Richardsequation in the mixed form for the subsurface flow is solved using an efficient stronglyimplicit finite-difference scheme. The explicit scheme for the surface flow componentresults in a simple method for connecting the surface and subsurface components. Themodel is verified using the experimental data and previous numerical results availablein the literature. The proposed model is used to study the two-dimensionality effectsdue to non-homogeneous subsurface characteristics. Applicability of the model tohandle complex subsurface conditions is demonstrated.q 1998 Elsevier ScienceLimited. All rights reserved

Key words:Overland flow, Saint-Venant equations, Richards equation, Conjunctivemodel.

1 INTRODUCTION

Overland flow is a dynamic part of a watershed response tothe rainfall and has many engineering applications, such as(i) water budgeting, (ii) flood prediction, (iii) soil erosionstudies, (iv) investigations into infiltration process, (v) trans-port of pollutant to streams, and (vi) design of runways andparking lots. Mathematical modeling of overland flowinvolves the solution of the governing equations for boththe surface flow and the groundwater flow with seepage atthe ground surface acting as the connecting link. The levelof model sophistication depends on the assumptions madefor simplifying the governing equations before they aresolved. Many models consider only the surface flow systemalong with an empirical treatment of the infiltration term.On the other hand, some models consider both the surfaceand the subsurface flow systems, but use either a diffusionwave or a kinematic wave approximation for modelingoverland flow.

Akan 3,4, Sunada and Hong30, De Lima and Van derMolen 13, and Govindarajuet al. 17 developed overlandflow models based on the kinematic wave approximation.

It should be noted here that the kinematic wave approxima-tion fails for highly subcritical flows on flat slopes and whenthe downstream boundary condition is an important factor25. Diffusion wave models are applicable over a wider rangeof flow conditions and, therefore, may be used for suchcases. Hromadkaet al. 21 developed a two-dimensional dif-fusion wave model assuming a constant effective rainfallintensity. Govindarajuet al. 15 derived an approximateanalytical solution to overland flow under a specified netlateral inflow using the diffusion wave approximation.They also provide the complete numerical solutions forthe diffusion wave equation. Todini and Venutelli32 alsodeveloped a similar two-dimensional diffusion wave modelin which the governing equations were solved by the finite-difference as well as the finite-element methods.

Vieira 33 conducted a comprehensive study of variousapproximations usually applied to the Saint-Venant equa-tions. It was suggested that the complete Saint-Venant equa-tions should be used in cases where the kinematic wavenumber is less than 5. Although, the overland flow modelsbased on the complete Saint-Venant equations are compu-tationally very intensive, they are more versatile and canalso act as standards of comparison for simplified models.Among the earliest full dynamic models are those developed

Advances in Water ResourcesVol. 21, pp. 567–579, 1998q 1998 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0309-1708/98/$19.00 + 0.00PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 00 2 0 - 1

567

*Corresponding author.

by Woolhiser and Liggett34 and by Linet al. 24. Chow andBen-Zvi 12 and Kawahara and Yokoyama22 developed two-dimensional overland flow models for impervious surfacesbased on the Lax–Wendroff method23 and a finite-elementmethod, respectively. Zhang and Cundy37 developed a two-dimensional overland flow model for a temporally constant,but spatially varying infiltration. MacCormack finite-differ-ence method was used to obtain the numerical solution.Tayfur et al. 31 also developed a two-dimensional overlandflow model based on an implicit method8 for solving thegoverning equations and applied the model to experimentalhill slopes. It was reported that consideration of microtopo-graphy in fine detail could result in numerical instability andconvergence problems.

All the overland flow models discussed so far describethe infiltration process in an empirical way. Smith andWoolhiser 29 were probably the first to develop a con-junctive model for overland flow. They solved the one-dimensional Richards equation for unsaturated flow in thesubsurface together with the one-dimensional kinematicwave equation for the surface flow. The Richards equationwas solved by an implicit Crank–Nicolson finite-differenceformulation. Akan and Yen5 developed a sophisticated con-junctive model in which they solved the complete Saint-Venant equations and the two-dimensional Richardsequation. The Saint-Venant equations were solved by thefour-point implicit method, while Richards equation wassolved by the SLOR technique. In the very versatile, com-plete catchment model, SHE2, the overland flow componentis simulated using the diffusion wave approximation. Thesubsurface flow in the unsaturated zone is represented by theone-dimensional Richards equation. The surface flow equa-tions are solved by the explicit procedure developed byPreissmann and Zaoui28, while the Richards equation issolved by the implicit finite-difference scheme. Govindarajuand Kavvas16, in a detailed study of hillslope hydrologythrough a stream flow—overland flow—subsurface flowmodel, used the diffusion wave approximation for the sur-face flow component. The surface flow equation was solvedby the centred implicit scheme, while the two-dimensionalRichards equation was solved by the SLOR technique.

In this paper, an alternative conjunctive numerical modelis presented for simulating the overland flow. The numericalmodel is based on a conjunctive solution of the completeSaint-Venant equations for the surface flow and the two-dimensional Richards equation. Presently available fulldynamic models for the overland flow employ classicalsecond-order accurate finite-difference methods for solvingthe Saint-Venant equations. These methods often result inhigh frequency oscillations when there are sharp changes inthe flow parameters and require ad hoc procedures forobtaining smooth solutions6. In this study, a simple toimplement high-resolution essentially non-oscillating(ENO) scheme27 is applied to solve the surface flow com-ponent in the conjunctive model. The explicit ENO schemealso results in a simple method for connecting the surfaceand subsurface flow components. Presently available

conjunctive models for the overland flow use Richardsequation in pressure-head form, which may result in signi-ficant mass balance error for complex subsoil conditions.Many studies7,19,9 have shown that the ‘mixed form’ ofthe Richards equation results in better numerical behaviourthan the other forms. Therefore, in this study, the two-dimensional Richards equation in the mixed form is usedfor simulating the subsurface component. This equation issolved by a recently developed strongly implicit procedure20. This is computationally more efficient than the SLORtechnique adopted in the currently available conjunctivemodels. The proposed numerical model is verified usingthe experimental data available in the literature29. Robust-ness of the numerical model is demonstrated by consideringa test case in which a clay lens is present in the subsurface.One of the objectives of developing a sophisticated model isthat it can act as a standard of comparison for simplifiedmodels. Therefore, the proposed numerical model is used tostudy the errors that may result in the prediction of the out-flow hydrograph if a one-dimensional simulation of the sub-surface flow is used.

2 GOVERNING EQUATIONS

Mathematical modeling of overland flow involves the solu-tion of the governing equations for both the surface flow andthe groundwater flow with seepage at the ground surfaceacting as the connecting link. In the present study (Fig. 1),the surface runoff is represented by the one-dimensionalflow equations in thex-direction, while the groundwaterflow is represented by the two-dimensional Richards equa-tion in thex andz directions.

2.1 Surface flow equations

Surface flow is assumed to occur in a prismatic channel ofrectangular section. The one-dimensional shallow water

Fig. 1. Definition sketch for a conjunctive model.

568 V. Singh, S. M. Bhallamudi

flow equations in conservation form for such a case aregiven by:

]U]t

þ]F]x

¼ S (1)

in which,

U ¼h

q

( ), F ¼

q

q2

hþ

gh2

2

8><>:9>=>;, S¼

(R¹ I )

gh(S0 ¹ Sf )

( )

(2)

In eqns (1) and (2):h ¼ flow depth (m);q ¼ discharge perunit width (m2/s); g ¼ acceleration due to gravity (m/s2); R¼ volumetric rate of rainfall per unit surface area (m/s);I ¼

volumetric rate of infiltration per unit area (m/s);S0 ¼

bottom slope in the direction of flow;Sf ¼ friction slope;x ¼ distance along the flow direction (m); andt ¼ time (s).The assumptions and the derivation of the above equationshave been reported earlier36,1,11and are not repeated here.

The friction slope,Sf is computed using the Darcy–Weisbach formula taking into account the effect of rainfallon frictional resistance.

Sf ¼ fdq2

8gh3 (3)

in which, fd ¼ frictional resistance coefficient. Evaluationof fd depends on the instantaneous state of flow. The flow islaminar in all the studies reported here and, therefore,fd isgiven by the following equation5.

fd ¼CL

Re(4)

where Re¼ Reynolds number¼ q/v (v ¼ kinematicviscosity of water), andCL is a constant which dependson the rainfall intensity.

2.2 Subsurface flow equations

In the present study, subsurface flow is modeled as two-dimensional motion of a single-phase incompressible fluidin an incompressible porous medium. The two-dimensional,transient flow equation in an isotropic porous medium isderived by applying the principle of conservation of mass.This equation without sources and sinks within the flowdomain can be written as

]v

]tþ

]Vx

]xþ

]Vz

]z¼ 0 (5)

where,v ¼ volumetric moisture content;Vx andVz ¼ darcyflow velocity in thex and z direction, respectively, andxand z are distances along the two coordinate directions;z is taken positive downwards. It is assumed that theDarcy’s law is applicable for evaluating the velocitycomponents.

Vx ¼ ¹ K(w)]w

]x, Vz ¼ ¹ K(w)

]w

]z¹ 1:0

� �(6)

in which, w ¼ pressure head (m) andK(w) ¼ unsaturatedhydraulic conductivity (m/s) which depends on the pressurehead, w. Substitution of eqn (6) in eqn (5) yields theRichards equation14:

]v

]t¼

]

]xK(w)

]w

]x

� �þ

]

]zK(w)

]w

]z¹ 1:0

� �� �(7)

eqn (7) is said to be in ‘mixed form’ since it includes boththe dependent variablesv and w. Hydraulic relationshipsbetween the pressure head,w, and the hydraulic conductiv-ity, K (w–K relationship), and between the moisture con-tent, v, and the pressure headw (w–v relationship), areneeded while solving eqns (5) and (6) in the unsaturatedzone. In general,w–K and w–v relationships are notunique and soils exhibit different behaviour duringwetting and drying phases. This hysteresis in soil charac-teristics is not considered for the cases studied in thepresent work. However, the hysteresis can be includedby employing differentw–K andw–v relationships for wet-ting and drying processes. Although semi-analyticalequations are available for describingw–K and w–v rela-tionships, the soil characteristics derived from the experi-mental and field data are employed wherever available.These relationships are specific to the case studied andare described later.

3 NUMERICAL SOLUTION

3.1 Surface flow

Surface flow equations constitute a set of nonlinear hyper-bolic partial differential equations. A recently developedhigh-resolution essentially non-oscillating (ENO) schemefor solving the shallow water flow equations27 is employedin the present study to solve the surface flow equations. Thescheme, proposed by Nujic27, is modified suitably toaccount for the non-zero source term in the continuity equa-tion. This scheme, unlike many other classical second-orderaccurate schemes such as the MacCormack method, is non-oscillatory even when sharp gradients in the flow variablesare present. The main advantages of this scheme are itssimplicity, ease of implementation and ease of extensionto a two-dimensional case. It is also very attractive for thepresent application because it is possible to account for thevariable bottom topography in a convenient and accurateway. It is an explicit, two-step predictor–corrector schemewhich results in second-order accuracy in both space andtime. Only a brief description of the method as applied in thepresent study is outlined here.

3.1.1 Predictor partThe finite-difference analog of eqn (1) is written here for afinite-difference grid where the subscripti refers to the gridpoint in thex direction and the superscriptsn and * refer tothe values at the known time level and the predictor level,respectively. The finite-difference form of eqn (1) for the

Conjunctive surface–subsurface modeling of overland flow 569

explicit determination ofUi* is written as

Upi ¼ Un

i ¹DtDx

Fni þ 1=2 ¹ Fn

i ¹ 1=2

� �þ DtSn

i (8)

whereFiþ1/2n represents the numerical flux through the cellface between nodesi þ 1 andi. Dx is the grid spacing andDt is the computational time step. All the terms on theright-hand side of eqn (8) are evaluated at the knowntime leveln and, therefore,Ui

* can be computed explicitly.The numerical fluxFiþ1/2 is computed using the followingformula.

Fi þ 1

2¼

12[FR þ FL ¹a(UR ¹ UL)] (9)

in which, a ¼ a positive coefficient,FR ¼ f(UR) the fluxcomputed using the information from the right side of thecell face andFL ¼ f(UL) ¼ the flux computed using theinformation from the left side of the cell face.UL andUR

are obtained using a MUSCL (monotone upwind schemefor conservation laws) approach.

UL ¼ Ui þdUi

2(10)

UR ¼ Ui þ 1 ¹dUi þ 1

2(11)

There are several ways of determiningdUi and dUiþ1

using different slope limiter procedures6,35. The‘minmod’ limiter is followed in the present study. Accord-ing to this,

dUi ¼ minmod(Ui þ 1 ¹ Ui , Ui ¹ Ui ¹ 1) (12)

where the minmod function is defined as

minmod(a,b) ¼

a if lal , lbl andab. 0

b if lbl , lal andab. 0

0 if ab# 0

8>><>>: (13)

The positive coefficienta is determined using the maxi-mum value (for all the grid points) of the largest eigenvalue of the Jacobian of the system of equations. This isgiven as

a $ max

����� qi

hiþ

������ghi

p ����� !

i ¼ 1 to N (14)

where, N is the total number of grid points. The vectorequation (eqn (8)) gives the predicted values ofh and qat the unknown time level at any nodei.

3.1.2 Corrector partThe vectorU at the unknown time leveln þ 1 and at nodei,i.e. Ui

nþ1 is computed using the predicted values and thevalues at the time leveln.

Unþ 1i ¼ 0:5 Un

i þ Upi ¹

DtDx

Fpi þ 1=2 ¹ Fp

i ¹ 1=2

ÿ �þ DtSp

i

� �(15)

where

Fpi þ 1=2 ¼

12[Fp

R þ FpL ¹ a(Up

R ¹ UpL)] (16)

Following the recommendations of Alcrudoet al. 6, UR* and

UL* are determined fromUiþ1

* andUi* using the samedUi þ 1

anddUi as determined in the predictor step. This procedureresults in better numerical stability.

In eqn (15), only the source term in the momentum equa-tion (the second component of the vector equation) is eval-uated using the predicted values ofh andq. However, thesource term in the continuity equation i.e. (R ¹ I) i is eval-uated using the values at the known time level instead of thepredicted values. Strictly speaking, this procedure decou-ples the subsurface flow computations and the surfaceflow computations during the computational time stepDt.However, the response of the subsurface flow to the varia-tion in surface flow depth is much slower than the responseof surface flow to changes in the rate of infiltration5. There-fore, the above decoupling does not affect the results sig-nificantly. In fact, numerical experimentation showed thatdetermination of the infiltration rate,I, during the correctorstep by using the predicted flow depth did not alter theresults. On the other hand, the decoupling procedureresulted in significant savings of the computational time,since the subsurface flow is computed only once during atime step.

3.2 Initial and boundary conditions

Values of the flow depth, the discharge and the infiltrationrate are specified at all the nodes at timet ¼ 0 as the initialconditions. The initial infiltration rate is equal to the rainfallrate. Although the initial flow depth and the discharge areequal to zero (overland flow on an initially dry surface), avery thin water film of depthhini and corresponding uniformflow discharge,qini are assumed to exist at timet ¼ 0. Thisassumption is made to overcome the numerical singularityin a simple way. However, it should be noted that the out-flow hydrograph may be sensitive to thehini value and,therefore, it should be chosen as small as possible.

The explicit finite-difference scheme described earliercan be used to determineh and q at the unknown timelevel only at the nodesi ¼ 2 to N ¹ 1. The values of thevariables at the upstream and the downstream ends of thedomain are determined using the appropriate boundary con-ditions. The discharge at the upstream end is equal to zero.However, the discharge at the upstream end is specified asqini to be consistent with the initial conditions. The flowdepth at the upstream end can then be determined usingthe negative characteristic equation11. In the presentstudy, a simple extrapolation procedure is adopted to deter-mine the upstream flow depth from the depth at the interiornodes. Numerical experimentation in the initial stages of themodel development showed that the extrapolation pro-cedure gave satisfactory results. Similar extrapolation pro-cedure is adopted to determineh andq at the last nodeN.

570 V. Singh, S. M. Bhallamudi

3.3 Stability condition

The high-resolution Lax–Friedrichs scheme adopted in thepresent study is an explicit scheme. Therefore, the computa-tional time step,Dt, is chosen using the CFL condition.

Cn ¼DtDx

ghþ

�����gh

ph i# 1:0 (17)

in which Cn ¼ Courant number.Dt is chosen dynamicallyin the numerical model such that eqn (17) is satisfied at allthe nodesi ¼ 1,2…N.

3.4 Subsurface flow

In order to determine the infiltration rateI in the continuityequation for the surface flow, the subsurface flow equationshave to be solved along with an appropriate boundary con-dition at the ground surface. In the present study, a recentlydeveloped strongly implicit finite-difference scheme20 forthe mixed based formulation of the Richards equation isused to simulate the unsaturated subsurface flow conditions.This scheme ensures mass balance in its solution regardlessof time step size and nodal spacings, and has no limitationswhen applied to field problems. It is also easy to incorporatedifferent types of boundary conditions in this scheme. Inthe present study, two different models 1DS2DSS and1DS1DSS are developed for simulating the overland flow.In the model 1DS2DSS (one-dimensional surface flow withtwo-dimensional subsurface flow), the infiltration rate,I, isdetermined by solving the two-dimensional Richards equa-tion. In the model 1DS1DSS (one-dimensional surface flowwith one-dimensional subsurface flow), the subsurface flowis assumed to occur only in the vertical direction. Fig. 2shows the definition sketch for the model 1DS1DSS. Inthis model, the infiltration rate at any distancex is deter-mined using the one-dimensional Richards equation withthe surface flow depth at that point as the top boundarycondition. Model 1DS1DSS results in significant savingsof CPU time. Numerical solution of the Richards equationis described in the following section.

The subsurface flow domain is divided into a number ofrectangular blocks. The moisture content,v, and the

pressure head,w, are specified at the center of the block(the node), while the velocities are specified at the inter-block faces. The subscripti refers to the block number inthex direction and the subscriptj refers to the block numberin thez direction. The superscriptsn andn þ 1 refer to theknown and the unknown time levels, respectively. Thefinite-difference form of the eqn (5) is

vnþ 1i, j ¹ vn

i, j

Dtþ

V̄i þ 1=2, j ¹ V̄i ¹ 1=2, jÿ �

Dx

þV̄i, j þ 1=2 ¹ V̄i, j ¹ 1=2

ÿ �Dz

¼ 0 ð18Þ

where the bar is used to denote the time-averaged value ofthe velocity.Dx andDzare the nodal spacings in thex andzdirections, respectively. The time-averaged velocities aredetermined by

V̄ ¼ wVnþ 1 þ (1¹ w)Vn (19)

in which, w ¼ time weighting factor. The velocity at anyinterblock face is determined using the pressure heads atthe neighbouring cell centers. For example;

Vi, j þ 1=2 ¼ ¹ Ki, j þ 1=2[(wi, j þ 1 ¹ wi, j) ¹ Dz]=Dz (20)

in which, Ki,jþ1/2 is the unsaturated hydraulic conductivityevaluated at the interblock faces between the nodes (i,j þ 1)and (i,j). Substitution of eqns (19) and (20) in eqn (18)yields

wDt

Dx2

�¹ Knþ 1

i þ 1=2, j wnþ 1i þ 1, j ¹ wnþ 1

i, jÿ �

þ Knþ 1i ¹ 1=2, j wnþ 1

i, j ¹ wnþ 1i ¹ 1, j

ÿ ��þ

wDt

Dz2

�¹ Knþ 1

i, j þ 1=2 wnþ 1i, j þ 1 ¹ wnþ 1

i, j ¹ Dzÿ �

þ Knþ 1i, j ¹ 1=2 wnþ 1

i, j ¹ wnþ 1i, j ¹ 1 ¹ Dz

ÿ ��þ vnþ 1

i, j

¹

�vn

i, j ¹ (1¹ w)DtDx

Vni þ 1=2, j ¹ Vn

i ¹ 1=2, jÿ �

¹ (1¹ w)DtDz

Vni, j þ 1=2 ¹ Vn

i, j ¹ 1=2

ÿ ��¼ 0 ð21Þ

The unsaturated hydraulic conductivity at an interblockface is estimated using the pressure heads at the neighbour-ing cell centers. Cooley10, Haverkamp and Vauclin18 andNarasimhan and Witherspoon26 have suggested variousways for this purpose. Haverkamp and Vauclin18 statethat the geometric mean is the best choice for estimatingthe interblock conductivities. However, Honget al. 20

reported (also experienced during the development of thepresent model) that the iterative solution of eqn (21) fails toconverge if the above procedure is adopted for estimatingthe K value. This is especially true for infiltration intoinitially very dry soils. The geometric mean is stronglyweighted towards the lower value and, therefore, watercannot drain easily if the soil is initially dry. This resultsin a non-physical build up of pressure. In this study, theinterblock hydraulic conductivity is estimated by theFig. 2. Definition sketch for model 1DS1DSS.

Conjunctive surface–subsurface modeling of overland flow 571

weighted arithmetic mean. For example,

Ki, j þ 1=2 ¼gK(wi, j) þ (1¹ g)K(wi, j þ 1) (22)

in which, g is the weight coefficient. Honget al. 20 suggesta value of 0.5 forg. Eqn (21) is written for all the blocks inthe flow domain and this results in a set of simultaneousalgebraic equations in the unknownsw i,j

nþ1. These simulta-neous equations are highly non-linear sincevnþ1 andKnþ1

are non-linear functions ofwnþ1. In the present study, theyare solved by using the Newton–Raphson technique.

3.5 Boundary conditions

3.5.1 Flux-type boundary conditionsIn the adopted scheme, the grid is arranged in such a mannerthat the boundaries of the flow domain coincide with aninterblock. Therefore, flux or velocity-type boundary con-ditions can be incorporated in a natural way in eqn (18).

3.5.2 Pressure head-type boundary conditionsThe imposed pressure head at the boundary is used alongwith the values of pressure head at the neighbouring interiorpoints to determine the flux at the boundary. Second-orderone-sided finite-difference analog is used for this purpose

3.6 Surface and subsurface flow interaction

Surface and subsurface flow components are interrelated bya common pressure head and the infiltration at the groundsurface. The top boundary condition for the subsurface flowis determined by the surface flow depth. In turn, the infiltra-tion term in the surface flow equation is controlled by thesubsurface flow conditions. The following procedure isadopted for simulating the interaction between the surfaceand the subsurface flow components.

1. Subsurface flow solution at time leveln is used todetermine the infiltration rate at the ground surface.

2. Surface flow equations are now solved using the infil-tration rate from step 1 to determineq and h at theunknown time leveln þ 1.

3. The surface flow depth at the time leveln þ 1 is usedas the top boundary condition and the subsurface flowequations are solved. This gives thev andw distribu-tion in the subsurface at time leveln þ 1.

4. Steps 1–3 are repeated up to the required time level.

As mentioned earlier, the above is a decoupled approachwhich reduces the CPU time by half without significantlyaffecting the accuracy of the results.

3.7 Boundary conditions

For subsurface flow resulting from rainfall infiltration, thetop boundary condition changes with time. During the initialstages of the rainfall, there is no ponding and the infiltrationrate is equal to the rainfall rate. The top boundary conditionfor such a situation is the specification of the flux equal to

the rainfall rate. As time progresses, the upper layers of thesubsoil become saturated and then infiltration rate startsdecreasing. Before starting the solution of the Richardsequation for any time step, the flux at the top boundary isestimated by takingwb ¼ 0. If this velocity is greater thanthe rainfall rate, then the flux type of boundary condition isapplied. Otherwise, a head type of boundary condition(wb(x) ¼ h(x) water flow depth at that point) is applied.The time to ponding comes out as a part of the solution.

Specified head or specified flux conditions can beimposed at the lateral and bottom boundaries. For thephysical situations considered in the present study, ano-flux boundary condition is imposed at the right and leftboundaries. Thew values at the bottom boundary areobtained using a simple extrapolation from the interiorpoints. This approximation does not introduce errorsbecause the bottom boundary is taken fairly deep and themoisture front does not reach that far for the computationaltimes considered.

4 VERIFICATION OF THE MODEL

The model presented in the previous section is tested withthe experimental data of Smith and Woolhiser29 for a con-junctive surface–subsurface flow. The experiments29 wereconducted in a soil flume of 12.2 m length, 51 mm wide and1.22 m deep. The slope of the channel was 0.01. The fluidused was a light oil whose viscosityv ¼ 1.943 10¹4 m2/s.A rainfall simulator was used to produce a rainfall of inten-sity 250 mm/h over a duration of 15 min. The flow regime islaminar throughout and theCL value5 in eqn (4) is taken as92.0. The soil used was Poudre fine sand and it was placed inthree layers of differing bulk densities. Thicknesses of thethree layers, 1, 2 and 3, were 76.5, 229.5 and 761 mm,respectively. The hydraulic properties (v vs. w andK vs. wcurves) of the three layers were determined experimentally29. The relative hydraulic conductivity,K r and the effectivesaturation,Se are defined as

Kr ¼KKs

(23)

Se ¼(v ¹ vr)(vs ¹ vr)

(24)

in which,Ks andvs are the saturated hydraulic conductivityand saturated moisture content,v r is the residual moisturecontent. The curves forSe vs. pressure head andK r vs.pressure head (Figs. 4 and 5 of Smith and Woolhiser29)are represented in the present study using the followingequations.Layer 1:

Se ¼

wb

w

� �l

if w # ¹ 12:85 cm

0:8434¹ 0:0148w ¹ 0:0026w2 if ¹ 12:85# w # ¹ 2:8 cm

0:8642 if w . ¹ 2:8 cm

8>>>><>>>>:(25)

572 V. Singh, S. M. Bhallamudi

Kr

wb

w

� �h

if w # ¹ 13:729 cm

0:5228þ 0:0204w ¹ 0:0009w2 if ¹ 13:729# w # ¹ 1:0 cm

0:5015 if w . ¹ 1:0 cm

8>>>><>>>>:(26)

Layer 2:

Se ¼

wb

w

� �l

if w # ¹ 17:196 cm

0:9163þ 0:0011w¹ 0:0012w2 if ¹ 17:196# w # ¹ 2:0 cm

0:9093 if w . ¹ 2:0 cm

8>>>><>>>>:(27)

Kr

wb

w

� �h

if w # ¹ 16:784 cm

0:6229þ 0:0134w ¹ 0:0013w2 if ¹ 16:784# w # ¹ 1:0 cm

0:6082 if w . ¹ 1:0 cm

8>>>><>>>>:(28)

Layer 3:

Se ¼

wb

w

� �l

if w # ¹ 25:18 cm

0:9842þ 0:01w ¹ 0:000536w2 if ¹ 25:18# w # ¹ 2:0 cm

0:962 if w . ¹ 2:0 cm

8>>>><>>>>:(29)

Kr

wb

w

� �h

if w # ¹ 21:34 cm

0:7387þ 0:0063w ¹ 0:0011w2 if ¹ 21:34# w # ¹ 1:0 cm

0:7313 if w . ¹ 1:0 cm

8>>>><>>>>:(30)

The values ofKs, vs, v r, l, h andwb (hypothetical bubblingpressure potential) for the three soil layers are given in theTable 1.

The numerical model is run for a length of 15 m with2.8 m acting as the buffer length for application of thedownstream boundary condition. The finite-difference gridspacings in thex and z directions,Dx and Dz, are equalto 0.25 m and 12.7 mm, respectively, and Courant numberCn ¼ 0.65. A very small value of 0.2 mm is taken as theinitial flow depth, hini. The numerical parameters forthe subsurface flow computations are:w ¼ 1.0, g ¼ 0.5and e ¼ 10¹3. Although, only a high-resolution (ENO)scheme for the surface flow is presented in this paper, thesurface flow equations were also solved using the classical

MacCormack scheme11 as a double check. Both themethods gave essentially the same results for this run.

Fig. 3 shows the runoff hydrograph at the downstreamend (x ¼ 12.25 m) for the above experiment obtained (i)experimentally29, (ii) numerically by Akan and Yen5,(iii) numerically by the model 1DS1DSS and (iv) numeri-cally by the model 1DS2DSS. It is clear from this figure thatboth the models 1DSlDSS and 1DS2DSS give almost thesame results. Model 1DS2DSS shows only some marginaleffect of two-dimensional subsurface flow. It can be seenthat the numerical results obtained by the models 1DS1DSSand 1DS2DSS compare satisfactorily with the experimentalas well as with the earlier numerical results of Akan andYen5. The present numerical results are lagging as far as therising limb of the hydrograph is concerned. However, theyare better for the recession phase. The discrepancy betweenthe present numerical results and the earlier numericalresults of Akan and Yen5 could be due to minor differencesin processing the soil-moisture characteristic curves asgiven by Smith and Woolhiser29. It should be noted herethat the results for the outflow hydrograph are sensitive tothe hydraulic properties of the subsurface soil. A 2.8%decrease in theKs value for layer 1 and a corresponding2.8% increase in theKs value for layer 2 resulted in anoutflow hydrograph which matched very well with theexperimental data. Fig. 4 shows the volumetric saturationprofile under the ground along the depth at a distance of5.5 m from the upstream end at different times. Thisfigure shows the present numerical results and the previousnumerical results obtained by Smith and Woolhiser29. Noattempt is made to compare the numerical results with theexperimental data because the data points are very few innumber. It can be seen from Fig. 4 that the present andprevious numerical results of Smith and Woolhiser29

match satisfactorily at early times. However, as time pro-gresses, the wetting front in the present model moves fasterthan the wetting front in Smith and Woolhiser’s29 model

Table 1. Hydraulic properties of the soil

Layer Ks

(cm/day)vs v r l h wb

(cm)

1 566.9 0.460 0.020 2.03 8.09 ¹ 10.02 365.7 0.504 0.025 2.20 8.60 ¹ 13.03 268.2 0.543 0.025 2.31 8.93 ¹ 17.0

Fig. 3. Verification of models 1DS1DSS and 1DS2DSS: outflowhydrograph.

Conjunctive surface–subsurface modeling of overland flow 573

because the computed infiltration rate is larger. This is con-sistent with the results presented in Fig. 3, where the outflowhydrograph by the present model is lagging the previousresults. Fig. 4 also shows that the two-dimensional subsur-face flow effects are marginal for the particular case studiedhere.

Sensitivity of the numerical results to the numericalparametershini, and the weighting coefficient,g, is studiedusing the model 1DS1DSS. Fig. 5 shows the effect of ad hocspecification of the initial flow depth,hini. Initial flow depthin these runs varied from 0.2 to 0.5 mm. The numericalresults for the rising limb of the hydrograph are sensitiveto the specifiedhini value. However, there is no effect of the

hini on the hydrograph beyond a certain time (11 min).Although it appears that better results can be obtained bytakinghini ¼ 0.5 mm, this value should be chosen as small aspossible in order to faithfully represent the initial dry con-ditions. The effect of weighting coefficient,g (in the deter-mination of the interblock hydraulic conductivity), on theoutflow hydrograph is shown in Fig. 6. It is obvious fromFig. 6 that the numerical results are highly sensitive to theg

value. Honget al. 20 recommended a value of 0.5 forg,which results in equal weightage to higher and lowervalues of K on the two sides of the face. In an earlierstudy, Haverkamp and Vauclin18 concluded that the inter-block conductivity should be weighted more towards the

Fig. 4. Verification of models 1DS1DSS and 1DS2DSS: volumetric saturation in subsurface.

Fig. 5. Effect of hini on the overland flow hydrograph.

574 V. Singh, S. M. Bhallamudi

lower value ofK and, accordingly, they used the geometricmean. In the limit of grid convergence,g ¼ 0.35 gave betterresults thang ¼ 0.5 for the test case studied here. However,the above conclusion cannot be generalized because it isbased on a comparison of numerical results with experimen-tal data which may have uncertainties in them. This aspectneeds further study.

Often soil deposits are encountered in the field (Ex: Allu-vial deposits) which are more or less homogeneous exceptfor the presence of an isolated clay lens. This clay lens actsas a barrier to the vertical movement of the wetting front inthe ground, and can significantly affect overland flow. Insuch a case, sharp gradients may occur in the overland flowhydrograph, and models which use traditional numericalmethods result in numerical oscillations. This is demon-strated by considering a case (Fig. 7) in which a clay lensof length 4 m is present at a depth of 11.4 cm from thesurface. The subsoil characteristics for this run were sameas those for layer 1 in the experiments by Smith and Wool-hiser29. Other input parameters, such as the rainfall rate andlength of the domain, were also same. The outflow hydro-graphs for this case, obtained using (i) the model 1DS2DSSwith ENO scheme, and (ii) the model 1DS2DSS with atraditional finite-difference scheme, such as the Mac-Cormack method, are compared in Fig. 8. A Courantnumber of 0.8 was used for the ENO scheme while avalue of 0.75 was used for the MacCormack scheme. It isvery clear from Fig. 8 that the numerical results obtainedusing the ENO scheme are superior to those obtained usingthe MacCormack scheme. A Courant number value as lowas 0.2 was required to suppress the numerical oscillations inthe MacCormack scheme. This is also the order of Courantnumber used by Zhang and Cundy37 in some of their simu-lations. It should be noted here that low Courant numbervalues result not only in higher computational times, butalso in diffusive errors11. Numerical results for this testcase also demonstrate that the model 1DS2DSS can be

used to simulate the effect of complex subsurface conditionson an overland flow hydrograph. Immediately after the arri-val of the wetting front at the clay lens, the infiltration ratefor that part of the channel which is directly above the claylens becomes negligible, and there is a sharp rise in theoutflow hydrograph. Model 1DS2DSS was able to simulatethis effect.

5 SIMPLIFIED MODELING

One of the objectives of developing a sophisticated model isthat it can act as a standard of comparison for simplifiedmodels. Some of the currently available conjunctive modelsfor the overland flow2,29assume that the subsurface flow inthe unsaturated zone is predominantly one-dimensional.Therefore, a one-dimensional Richards equation is used intheir models. Govindaraju and Kavvas16 have demonstratedhow such an assumption is not valid if the groundwater tableis close to the surface. Herein, the model 1DS2DSS is usedto study the error in the prediction of the outflow hydrographusing a one-dimensional simulation of the subsurface flowfor the case of a deep groundwater table. Three differenthypothetical cases, as represented in Fig. 9, are consideredfor this purpose. In this figure, the soil characteristics oflayers 1, 2 and 3 are same as the soil characteristics forlayers 1, 2 and 3 in the experiments by Smith and Woolhiser29. The same surface flow conditions (length¼ 15 m, slope¼ 0.01,v¼ 1.943 104 m2/s,R¼ 250 mm/h,CL ¼ 92.0) areused in these simulation runs. The numerical simulations areperformed usingDx ¼ 0.25 m,hini ¼ 2 3 10¹3 m, a ¼ 1.0andg ¼ 0.5, in both models, 1DS1DSS and 1DS2DSS. InFig. 10, the outflow hydrograph for the case shown in

Fig. 6. Effect of g on the overland flow hydrograph.

Fig. 7. Schematic representation of case considered for studyingENO scheme vs. MacCormack scheme.

Conjunctive surface–subsurface modeling of overland flow 575

Fig.9(a) is compared with the outflow hydrograph for Smithand Woolhiser’s29 experiment. Both the hydrographs areobtained using the model 1DS1DSS. It can be observedfrom Fig. 10 that the ad hoc simplification of layered soilconditions in Smith and Woolhiser’s29 experiment by asingle layer, as given in Fig. 9(a), affects the outflow hydro-graph significantly. This is because the spatial variation insoil characteristics in the vertical direction affects the infil-tration rate. Appropriate homogenization procedures shouldbe adopted if simple analytical solutions for infiltration

based on homogeneous soil characteristics are to be usedwhile simulating overland flow on layered subsoil. In theabsence of appropriate homogenization, model 1DS1DSScan be used to study the effect of subsoil layering on theoutflow hydrograph. Application of the computationallymore intensive model 1DS2DSS is not required if there isonly vertical layering in the subsoil. Both models,1DS1DSS and 1DS2DSS, gave essentially the same results(not shown here) for the case shown in Fig. 9(a). For thiscase of uniform rainfall rate, the whole surface length isponded at the same time, and the wetting front moves essen-tially in the vertical direction. The two-dimensional subsur-face flow effects are only marginal and models 1DS1DSSand 1DS2DSS give similar results. Model 1DS1DSS is alsorun for Smith and Woolhiser’s29 experiment by assumingthat the infiltration rate is uniform throughout the length, asschematically represented in Fig. 11(a). Such an assumptionis valid if the surface flow depth does not have a significanteffect on the infiltration rate. Fig. 11(b) compares the out-flow hydrograph from this run with the previous outflowhydrograph obtained using the general 1DS1DSS model.It can be clearly seen that both the hydrographs are essen-tially the same. This indicates that the infiltration rate ismore or less uniform throughout the length as long asthere is no spatial variability of soil characteristics in thexdirection. The non-uniform surface flow depth does not havea significant effect on the infiltration rate. This assumptionof uniform infiltration rate greatly reduces the computa-tional time (as much as by 10 times) while running themodel 1DS1DSS and, therefore, is attractive from an engi-neering point of view.

The outflow hydrograph for the case shown in Fig. 9(b) isconsidered for study if the two-dimensionality effects in thesubsurface flow become significant due to the spatial varia-tion of the soil characteristics in thex direction. The outflowhydrographs for this case obtained using models 1DS1DSSand 1DS2DSS are shown in Fig. 12. It can be clearly seenfrom this figure that, even in this case, the two-dimensional

Fig. 8. Outflow hydrograph for case in Fig. 7: ENO scheme vs. MacCormack scheme.

Fig. 9. Schematic representation of subsurface conditionsconsidered in the numerical simulation studies.

576 V. Singh, S. M. Bhallamudi

effects are only marginal and the application of the simplemodel 1DS1DSS gives satisfactory results. The modelsdeveloped in the present work are also applied to simulatethe outflow hydrograph for the case shown in Fig. 9(c), inwhich the subsurface soil characteristics vary in a com-plicated manner. It should be noted here that even themodel 1DS1DSS considers the heterogeneity of the sub-surface and there is no homogenization. The onlydifference between models 1DS1DSS and 1DS2DSS isthat in the former case the flow is assumed to be only inthe vertical direction. The outflow hydrographs obtainedusing models 1DS1DSS and 1DS2DSS for the above caseare shown in Fig. 13. The effect of two-dimensionality in

the subsurface flow is very much evident in this case. Theone-dimensional model shows an early ponding and areduced peak. It can be concluded from the above simula-tion studies that, in many cases where the groundwatertable is very deep, the unsaturated flow in the groundmay be considered one-dimensional for the purpose ofsimulating the overland flow. This assumption is valid,especially when the spatial variation in the soil character-istics is only in the vertical direction. However, two-dimensionality comes into the picture if the spatial varia-tion of the soil characteristics is very complex and is inbothx andzdirections. Model 1DS2DSS should be used insuch cases.

Fig. 10. Effect of simplification of layered subsoil conditions using a single layer on outflow hydrograph.

Fig. 11. Effect of simplification of model 1DS1DSS using the assumption of uniform infiltration rate.

Conjunctive surface–subsurface modeling of overland flow 577

6 CONCLUSIONS

In this study, a conjunctive surface–subsurface numericalmodel is developed for the simulation of overland flow. Inthis model, the complete Saint-Venant equations are solvedusing a simple, explicit, high-resolution essentially non-oscillating (ENO) finite-difference scheme. The subsurfaceflow component in the proposed model is represented by thetwo-dimensional Richards equation in the mixed form. Thisequation is solved by a recently developed efficient stronglyimplicit finite-difference scheme. The proposed conjunctivemodel is verified using experimental data available in theliterature. The ability of the proposed model to handlecomplex subsurface conditions is demonstrated throughthe numerical simulation studies. It is shown that the pro-posed model does not result in high-frequency numerical

oscillations even for hydrographs with steep rising limbs.It is found from the numerical simulation studies that, inmany cases where the groundwater table is deep, theunsaturated flow in the ground may be considered one-dimensional. It is also possible to obtain further simplifica-tion if it is assumed that the infiltration rate is more or lessuniform. Such an assumption may be valid if the spatialvariation in the subsoil characteristics is only in the verticaldirection. However, two-dimensional effects do becomesignificant if the spatial variation of the soil characteristicsis complex and is in bothx andz directions.

ACKNOWLEDGEMENTS

The authors thank Prof. K. Subramanya of I.I.T. Kanpurfor the many interesting discussions. They also thank theanonymous reviewers whose comments helped in signifi-cantly improving the quality of this paper.

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