HYDROLOGICAL PROCESSES. VOL. 9. 783-796 (1995)

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE

INFILTRATING SURFACES WITH LATERAL INFLOW NEGLECTED IN THE MOMENTUM EQUATION

APPROXIMATIONS FOR SPACE-INDEPENDENT FLOWS ON

V. P. SINGH Department of Civil Engineering, Louisiana State University, Baton Rouge, LA 70803-640.5, USA

ABSTRACT Error equations for the kinematic wave and diffusion wave approximations with lateral inflow neglected in the momen- tum equation are derived under simplified conditions for space-independent flows. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions are parame- terized through a dimensionless parameter y which is dependent on the initial conditions. This parameter reflects the effect of initial flow depth, channel-bed slope, lateral inflow, infiltration and channel roughness when the initial con- dition is non-vanishing; it reflects the effect of bed slope, channel roughness and acceleration due to gravity when the initial condition is vanishing. The error equations are found to be the Riccati equation. The structure of the error equations in the case when the momentum equation neglects lateral inflow is different from that when the lateral inflow is included.

KEY WORDS: kinematic wave models; diffusion wave models; errors; space-independent flows

INTRODUCTION

A wide variety of problems, including overland and channel flows, subsurface flow, the soil moisture move- ment, sediment transport, solute transport and the movement of glaciers, to name but a few, can be adequately modelled by using the shallow water wave (SWW) theory or its simplified representations. The three most commonly used representations of the SWW theory are the kinematic wave (KW) approximation (Lighthill and Whitham, 1955), the diffusion wave (DW) approximation and the dynamic wave (DYW) representation. These representations result from the mechanics of force balancing. For example, the KW approximation assumes that the gravitational and frictional forces are the dominant forces and the inertial and pressure forces are negligible. The DW approximation includes the gravita- tional, frictional and pressure forces, but neglects the inertial force. Thus, the DW approximation can be considered to be a higher order approximation than the KW approximation. The DYW representation includes all of the forces and embraces the KW and DW approximations as its special cases.

The accuracy of the KW and DW approximations compared with the DYW representation for modelling overland and channel flows has been a subject of much discussion (Woolhiser and Liggett, 1967; Ponce and Simons, 1977; Morris and Woolhiser, 1980; Daluz Viera, 1983; Ferrick, 1985). Singh (1994) presented an historical perspective on the criteria proposed to evaluate the adequacy of these approximations. It was stated that all of these criteria take on fixed point values and do not describe either in time or space the errors resulting from these approximations. He derived, under simplified conditions, error equations for the KW and DW approximations for space-independent flows for the rising part of the flow hydro- graph, which specified errors as a function of time. In that study, lateral inflow or rainfall was included in the momentum equation. The objective of this study is to investigate the effect of neglecting lateral

CCC 0885-6087/95/070783- 14 0 1995 by John Wiley & Sons, Ltd.

Received 1 March 1994 Accepted 20 May I994

784 V. P. SlNGH

inflow in the momentum equation on the accuracy of the KW and DW approximations. This neglect is based on the assumption that the momentum exchange between lateral inflow and longitudinal flow is negligible. Quantification of the effect of this assumption on the accuracy of the KW or DW approxi- mation has probably not yet been undertaken. What is clear, however, is that the structure of the error equations is different under this assumption from that when this assumption is revoked.

SHALLOW WATER WAVE THEORY FOR SPACE-INDEPENDENT FLOWS

The SWW theory can be described by some form of the St Venant (SV) equations. For space-independent flows over an infiltrating plane subject to uniform rainfall, these equations with the momentum-exchange term deleted can be written in one-dimensional form on a unit width basis as

Continuity equation,

dh - = 4 0 - h dt

Momentum equation,

du dt - = g(S0 - Sf)

where h is the depth of flow (L), u is the local mean velocity (L/T), qo is lateral inflow or rainfall intensity (LIT)& is the infiltration rate (L/T), g is acceleration due to gravity (L/T2), t is time (T), So is bed slope and Sf is friction slope. Equation (2) neglects the effect of lateral inflow and, as a result, is simpler than when lateral inflow is included. Note Q = uh is discharge (L3/[TL]) per unit width.

The assumption of space independence implies omission of convective-inertial and pressure forces. These forces must be small compared with gravity, friction and local inertial forces. The term Sf can be approxi- mated as

2 U

Sf = Ph (3)

where /3 is some resistance parameter (T2/L). If the Chezy relation is used for representing the friction, then /3 = g/C2, where C is Chezy’s resistance parameter (Lo.’/T).

The KW approximation is based on Equation (1) and Equation (2) with the left-hand side (or local inertia) deleted

s, = Sf (4)

Thus Equation (3) is the KW approximation of the momentum equation. Equation (4) accounts for only the gravity and frictional forces. For space-independent flows, the DW approximation is the same as the KW approximation. The DYW representation is based on Equations (1) and (2), where the gravity, fric- tional and inertial forces are all included.

Although the assumption of spatial independence in surface flows is restrictive, its use does allow a useful first approximation and gives a lot of physical insight. When the SV equation or its variants are solved in the x-t plane, surface flow exhibits spatial independence during part of its rising limb or for part of the solution domain. For example, when the effective rainfall occurs at a constant rate on a plane, then accord- ing to the KW theory, the flow (e.g. depth) observed over the plane is spatially independent during the duration bounded by the time of concentration (Singh, 1976). This duration is the period of the rising limb of the hydrograph. As another example, consider the case of subsurface stormflow occurring in a thin zone of saturation overlying a steeply sloping bedrock on a hillslope and subject to a constant rate of recharge (Henderson and Wooding, 1964; Beven, 1981). The subsurface flow is observed to be spatially independent during part of the rising hydrograph of subsurface flow, according to the KW theory. Similar behaviour is observed in a variety of geophysical processes, including erosion and sediment transport in

KINEMATIC A N D DlFFUSlON WAVE 785

upland watersheds subject to rainfall, movement of nutrients and fertilizers in agricultural areas, sedimen- tation in a tank and snowmelt runoff, to name but a few. Furthermore, the assumption of spatial indepen- dence can be likened to the concept underlying the systems approach commonly used in surface water hydrology (Singh, 1988. 1989). In this approach, the hydrological system is treated as lumped, without any consideration of spatial variability of flow.

INITIAL CONDITIONS

The KW and DYW solutions were obtained for two types of initial conditions (t = 0). The first is a non- vanishing condition and the other is vanishing, and these can be expressed as

( 1 ) h(0) = ho, u(0) = uo ( 5 )

(2) h(0) = o , u(0) = o ( 6 ) Equation (5) implies that the channel is wet and has uniform flow at the beginning, whereas Equation (6) considers the channel to be dry.

ERROR EQUATIONS: USE OF NON-ZERO INITIAL CONDITIONS

Kinematic wave and diflusion wave solution Equation (1) subject to Equation ( 5 ) has the solution

h = ho + (40 -m (7) Equation (7) states that the depth of flow increases linearly with time for (qo -so) > 0. The flow depth at any time is equal to the sum of the initial depth and the difference between rainfall and infiltration accumulated (or effective rainfall depth) up to that time. It may be convenient to define a dimensionless parameter T as

Therefore, in terms of T , the flow depth becomes

h ( T ) = h0T

and the dimensionless depth, h,, is

h h =- * ho

where the initial depth serves as a normalizing depth, ho. From the KW approximation

In terms of T , Equation (1 1) becomes

(9)

The dimensionless flow velocity becomes

786 V. P. SlNGH

where U is the normalizing flow velocity defined as

The flow discharge Q can be expressed as

Equation (15) can

The dimensionless

be expressed in terms of T as

discharge, Q* = Q/Qo, can be expressed as

Q*(T) = 7'''

where Qo is the normalizing discharge defined by 0.5

Q~ = ($) The KW and DW solution is given by Equations (10) and (13) or (17). The dimensionless velocity and dimensionless discharge are sketched in Figures 1 and 2, respectively. The flow velocity increases with 0.5 power of T , and flow discharge with one and one-half power of r.

Dynamic wave solution

takes the form Equation (1) has the solution given by Equation (7). With the introduction of Equation (3), Equation (2)

du UZ - = sso - g p h dt

Figure 1. Dimensionless kinematic wave velocity as a function of dimensionless time for a spdcc-indcpcndcnt case with lateral inflow neglected in the momentum equation, and q = q O , f = j & h(0) = /I , , and r t ( 0 ) = It,,

KINEMATIC AND DIFFUSION WAVE 787

DIMENSIONLESS Time

Figure 2. Dimensionless kinematic wave discharge as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momentum equation, and q = q0 , f =fo, h(0) = ho and u(0) = uo

In terms of 7, Equation (19) becomes

which can be written in dimensionless form as

Equation (21) can be simply written as

where 7 is a dimensionless parameter defined as

Equation (22) is a special case of the Riccati equation with the linear term disappearing, and has to be solved numerically.

Equation (21) can be expressed in terms of Q as

In terms of T, Equation (24) becomes

788 V. P. SINGH

I . " . ' . ' - " " " ' 10 40 50 10 0 I 0 20

DIMENSIONLESS TIME

Figure 3. Dimensionless dynamic wave velocity as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momentum equation, and 9 = qo,.f =fo. h(0) = ho and u(0) = uo

In terms of dimensionless discharge Q* = Q/Qo, Equation (25) becomes

Equation (26) is a Riccati equation. The dynamic wave solution is given by Equation (10) and the solution of Equation (22) or (26). In dimensionless domain, the solution is independent of infiltration.

Equation (22) was solved by using the fourth-order Runge-Kutta method. The Runge-Kutta solution requires two initial conditions that were obtained as follows: at 7 = 1, dv/dr = 0, and u = 1. The dimen- sionless velocity and dimensionless discharge as a function of T are sketched for various values of y in Figures 3 and 4. The velocity increased, for a fixed y, with increasing T. For a fixed T, it also increased with increasing y. For y 2 1.5, it was almost insensitive to y.

DEFINITION O F ERROR

The relative error E can be defined as

where SK is the solution (either dimensional or dimensionless) from the Dr DW approximation, and SD is the solution from the DYW representation. The subscripts K and D correspond to the KW and DYW solutions, respectively. The solution can be either in terms of depth, velocity or discharge, i.e. S = (11 , u, Q}. If the solution is expressed by dimensionless velocity, v, then the dimensionless error equation is

KINEMATIC AND DIFFUSION WAVE 789

0 I0 10 10 40

DIMENSIONLESS TIME

Figure 4. Dimensionless dynamic wave discharge as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momentum equation, and q z q o . f =fo. h(0) = ho and u(0) = uo

ERROR IN KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS

With the substitution of Equations ( 1 3) and (22) in Equation (28), and after some algebraic manipulation, the error equation is found to be

where 1

27 Co(7) = -

l-

Equation (29) is a Riccati equation, also holds for the error in discharge, and has to be solved numerically. Equation (29) was solved by using the fourth-order Runge-Kutta method. For this method, two initial

conditions are needed that were defined as: at T = 1, E( 1 ) = 0, and dE( l ) / d ~ = 0.5. The error in the KW or DW solution is plotted in Figure 5 for various values of y. The graph is highly skewed with a steep rise to a peak and an extended decrease. Table I shows maximum error and its time of occurrence. For a fixed T , the error increased with decreasing y. The error derivative is plotted in Figure 6. For 7230, it is almost independent of y and T. Even for rG30, it is only mildly sensitive to y. Coefficients Co, C1 and C2 were computed and are plotted, as shown in Figures 7-9. Co was positive, independent of y and inversely pro- portional to 7. C , was negative, and increased with the square root of T. For a fixed T, it increased with

790 V. P. SlNGH

0 . 3 8 .

0.11-

0 . 2 8 -

g 0 . 2 3 - 0

0 . 1 8 -

0 . 1 3 .

0.01.

Figure 5. Error in the kinematic or diffusion wave approximation as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momentum equation, and q = qo,f =fo, h(0) = ho and u(0) = uo

decreasing y. C2 was negative, inversely proportional to the square root of T , and was indirectly propor- tional to y. C, and C1 were maximum at T = 1, but C2 was minimum at that point.

ERROR EQUATIONS: USE OF ZERO INITIAL CONDITIONS

Kinematic wave and difusion wave solution Equation (l), subject to Equation (6), has the solution

h = (40 -m

Table I . Maximum error and its time of occurrence

i- Maximum Y error

4.5 3.75 3.25 3.00 2.75 2.75 2.50 2.50

0.373 0.285 0.243 0.2 I7 0. I98 0.184 0. I72 0. I63

0.5 1 .O 1.5 2.0 2.5 3.0 3.5 4.0

(33)

KINEMATIC AND DIFFUSION WAVE 79 1

0 I0 10 10 40 50 I 0

DIMENSIONLESS TIME

Figure 6 . Error derivative as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momen- tum equation, and 4 = 40,.f' =.fo. h(0) = ho and u(0 ) = u0

I

0 . 0 0 I . . ~ ' " ' . " ~ . . ' . " . ~ " " . . . ~ . ' I ~ ' ~ ~ ' ' ~ ~ ~ l ' ~ ' ' ~ ' ~ ~ ~ l

0 I0 10 a0 40 5 0 10

DIMENSIONLESS TIME

Figure 7. Coefficient C, as a function of dimensionless time for a space-indcpcndcnt ciisc with Iiitcral inflow neglected in thc momcn- turn equation, and 4 = 4",.f' =.f;,. h(0) = /I,) and ~ ( 0 ) = r / ( ,

792 V. P. SlNGH

Figure 8. Coefficient C , as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momen- tum equation, and 9 = qo. f =.&, h(0) = h0 and u(0) = uo

Equation (33) shows that the flow depth increases linearly with time for (qo -fo) > 0, and is equal to the amount of effective rainfall up to that time. It is convenient to define a dimensionless parameter 7 as

(34)

Note that this definition of 7 is different from the one given by Equation (8). In terms of 7 , Equation (33) can be written as

4: h ( 7 ) = -7 = hor, q* = qo - f o g

The dimensionless flow depth becomes

where ho is the normalizing depth defined as 2

4* g

ho = -

(35)

(36)

(37)

This definition of ho is different from that defined in the preceding section. From the KW approximation

which can be written in terms of 7 as

KINEMATIC A N D DIFFUSION WAVE

0 . 0 1

- 0 . I -

- 0 . 1 -

-0.1-

’ -0.k- !2 $ - 0 . 5 -

5 8 - o . c -

- 0 . 1 -

-0.1-

- 0 . 9 -

793

Figure 9. Coefficient Cz as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momen- tum equation, and q = q O , f = f o , h(0) = ho and u(0) = uo

The dimensionless flow velocity becomes

where U is

The flow

the normalizing velocity defined as

u = q* ($ discharge Q can be written as

In terms of T, Equation (42) can be expressed as

The dimensionless discharge Q* = Q/Qo becomes

Q* ( T ) = TI.’

where Qo is the normalizing discharge defined as

(43)

(44)

794 V. P. SlNGH

The KW and DW solution is given by Equations (36) and (40) or (44). In dimensionless domain, the solu- tion is independent of infiltration. The dimensionless velocity and dimensionless discharge as a function of r exhibited similar variability as in Figures 1 and 2. The velocity increased with the square root of r, and discharge with one and one-half power of r.

Dynamic wave solution Equation (1) has the solution given by Equation (33). In terms of r , Equation (2) can be written as

du g u2 - = soq* - p-- d r 4* 7

which, in dimensionless form, becomes

or simply

where

(49) 0.5 r = 2 ( ~ , g p ) ~ . ~ , = 4gsop, r =

Equation (48) is a special form of the Riccati equation with the linear term vanishing and has to be solved numerically. The dynamic Equation (2) can also be expressed in terms of discharge Q as

9 Q gP Q2 = - + gs,q*t - -- dt t 4: t2

In terms of T , Equation (50) can be written as

which, in terms of dimensionless discharge, Q* becomes

Equation (52) is a Riccati equation and has to be solved numerically. The DYW solution is given by Equation (36) and the solution of Equation (48) or (52). The dimensionless solution does not explicitly depend on infiltration. Equation (48) was solved by using the fourth-order Runge-Kutta method. At r = 0, u = 0, and dv/dr had a discontinuity which was circumvented by the use of forward differ- encing. The patterns of variability of the dimensionless velocity and dimensionless discharge for various values of y were similar to those exhibited in Figures 3 and 4. The velocity was independent of q3% and increased with increasing r. For a fixed r, it increased with y, but became nearly independent of y for Y = 1.5.

ERRORS IN KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS

Equations (40) and (48) are of the same form as Equations (13) and (22), respectively. Therefore, by analogy, the error equation for this case can be written as

d E - = C C ~ ( T ) + C , ( Y , T ) E + C * ( Y , T ) E ~ , d r E(0) = O , ~ 2 0 (53)

KINEMATIC AND DIFFUSION WAVE 795

0 10 10 30 40

DIMENSIONLESS TIME

y = 0.5 ----- ys2 .5

- y = 3.5

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ---- y=1.0 --- y=3.0

_ _ _ _ _ _ _ _ _ _ _ y=1.5

Figure 10. Error in the kinematic or diffusion wave approximation as a function of dimensionless time for a space-independent case with lateral inflow neglected in the momentum equation, and 9 = q 0 . f =fo. h(0) = 0 and u(0) = 0

where

Equation ( 5 3 ) is a Riccati equation, and also describes errors in discharge. It has a discontinuity at T = 0. The discontinuity can be circumvented by the use of forward differencing. Equation (53) was solved by using the fourth-order Runge-Kutta equation.

The variation of error in the KW or DW solution is sketched in Figure 10 for various values of y. The maximum error and its time of occurrence are given in Table 11. The error graph is highly skewed with a

Table 11. Maximum error and its time of occurrence

i- Maximum Y error

2.00 1 .oo 0.75 0.50 0.50 0.50 0.50 0.50

0.857 0.522 0.386 0-307 0.258 0.2 19 0.187 0.160

0.5 1 .o 1 .5 2.0 2.5 3.0 3.5 4.0

796 V. P. SlNGH

steep rise to a peak and then extended decrease approaching a constant value for large values of T . The error increased with decreasing y for a fixed T. To further investigate the error variability, the error derivative was computed. The derivative decreased steeply over a short range of T , reached a near-constant value for 7.220, and became independent of y. Coefficients Co, CI and C2 were also computed. Co was positive, independent of T, and inversely proportional to T. CI could take on negative or positive values. For a fixed T, CI increased with decreasing y. C2 was negative, inversely proportional to the square root of T , and increased with decreasing y. Co and CI were maximum at T = 0, but C2 was minimum at that point.

CONCLUSIONS

For space-independent flows, the KW and DW approximations are accurate (error < 10%) when the dimensionless parameter y 2 10, and are acceptable (error < 15%) for y> 5. This parameter reflects the effect of initial flow depth, bed slope, lateral inflow, acceleration due to gravity and channel roughness. The error of these approximations decreases exponentially when the dimensionless time exceeds five. The non-linear differential equations describing the error of these approximations are found to be the Riccati equation and their structure is significantly influenced by the neglect of lateral inflow in the momentum equation. The error differential equations arise due to the neglect of inertial force in the momentum equation. The effect of inertial forces and the lateral inflow decreases with increasing y as well as with increasing T .

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Ferrick, M. G . 1985. ‘Analysis of river wave types’, W a f . Resour. Res., 21, 209-220. Henderson, F. M. and Wooding, R. A. 1964. ‘Overland flow and groundwater flow from a steady rainfall of finite duration’, J . Geo-

Lighthill, M. J. and Whitham, G. 8. 1955. ‘On kinematic waves: 1. flood movement in long rivers’, Proc. Roy. Soc. London. Ser. A , 229,

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