HYDROLOGICAL PROCESSES, VOL. 9, 1-18 (1995)

ACCURACY OF KINEMATIC WAVE AND DIFFUSION WAVE

INFILTRATING SURFACES APPROXIMATIONS FOR SPACE-INDEPENDENT FLOWS ON

V. P. SINGH Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, USA

ABSTRACT Error equations for the kinematic wave and diffusion wave approximations were derived under simplified conditions for space-independent flows occurring on infiltrating planes or channels. These equations specify error as a function of time in the flow hydrograph. The kinematic wave, diffusion wave and dynamic wave solutions were parameterized 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 and channel roughness when the initial condition is non-vanishing; it reflects the effect of bed slope, channel roughness, lateral inflow and infiltration when the initial condition is vanishing. The error equations were found to be the Riccati equation.

KEY WORDS Kinematic wave approximation Diffusion wave approximation Space-independent flows Infiltrating surfaces

INTRODUCTION

A wide variety of problems, including overland and channel flows, subsurface flow, soil moisture movement, sediment transport, solute transport and movement of glaciers, to name but a few, occurs on surfaces that are subject to lateral outflows such as infiltration, evaporation, plant root extraction, or similar. These problems can be adequately modelled by the shallow water wave (SWW) theory or its simpli- fied 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. The DYW representation includes all of the forces and embraces the KW and DW approximations as its special cases. The KW approximation includes the gravitational and frictional forces, and the DW approxi- mation includes the gravitational, frictional and pressure forces. Thus the DW approximation can be con- sidered to be a higher order approximation than the KW approximation.

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). Even though most overland and channel flows occur on infiltrating surfaces, infiltration has not been explicitly incorporated in evalu- ating the accuracy of these approximations. Singh, 1994 has presented a historical perspective of the criteria proposed to evaluate the adequacy of these approximations. These criteria take on fixed point values and do not describe either in time or space the errors resulting from either the KW or the DW approximation. He derived, under simplified conditions, error equations for the KW and DW approximations for space- independent flows for the rising part of the flow hydrograph, which specified errors as a function of time. Such flows are a function of time and do not depend on space. In that study, infiltration was not included in the governing equations. The objective of this study is to investigate the effect of infiltration

CCC 0885-6087/95/010OO 1 - 18 0 1995 by John Wiley & Sons, Ltd.

Received 12 June I993 Accepted I0 February 1994

2 V. P. SINGH

on the accuracy of the KW and DW approximations. Quantification of the effect of including infiltration on the accuracy of the KW or DW approximation has probably not yet been undertaken, even though infil- tration is a major factor affecting the flow dynamics. It is found that the structure of the error equations is very different when infiltration is accounted for from that when it is not.

SHALLOW WATER WAVE THEORY FOR SPACE-INDEPENDENT FLOWS

The SWW theory can be described by some form of the St Venant (SV) equations. The assumption of space independence implies omission of convective-inertial and pressure forces. These forces must be small by comparison with gravity, friction and local inertial forces. The pressure forces on a wave front are not generally small and are the difference between the KW and DW models. Thus, on a wave front, the assump- tion of space-independence will not be applicable. For space-independent flows over an infiltrating plane subject to uniform rainfall, these equations can be written in one-dimensional form on a unit width basis as

Continuity equation,

dh dt - = q o - h

Momentum equation,

- du = g(S0 - Sf) - h qou dt

where h is the depth of flow (L), u is the local mean velocity (L/T), qo is the lateral inflow or rainfall intensity (L/T),fo is the uniform infiltration rate (L/T), g is the acceleration due to gravity (L/T2), t is time (T), So is the bed slope and Sf is the friction slope. Note Q = uh is discharge [L3/(TL)] per unit width.

The term Sf can be approximated as

U' Sf = p- h (3)

where p is some resistance parameter (T2/L). If the Chezy relation is used for representing the friction, then ,d = g / C 2 , where C is Chezy's resistance parameter (L0.5/T).

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

qou h

g(S0 - Sf) - - = 0 (4)

Equation (4) accounts for only the gravity and frictional forces, and the momentum exchange between rainfall and planar (or channel) flow. Thus the KW approximation uses Equations (1) and (4). 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, frictional and inertial forces, and the momentum exchange 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 equations 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 kinematic wave 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 occur- ring 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

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 3

theory. Similar behaviour is observed in a variety of geophysical processes, including erosion and sediment transport in upland watersheds subject to rainfall, the movement of nutrients and fertilizers in agricultural areas, sedimentation in a tank and snowmelt runoff, to name but a few. Furthermore, the assumption of spatial independence 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) expressed as

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

(2) h(0) = 0 , ~ ( 0 ) = O (6)

The first is a non-vanishing condition and the other is a vanishing one. 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 diffusion wave solution Equation (1 j, subject to Equation (9, has the solution

h = ho + (40 - f o P (7) Equation (7) states that if (qo -fo) > 0, the depth of flow increases linearly with time. The 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 in terms of dimensionless depth or time as

1 h T = - [ h o + ( q o - f o ) t ] =-, T 3 1 , ho#O

h0 h0

Clearly, the initial depth acts as a normalizing depth. Therefore, in terms of T, the flow depth becomes

h(T) = h 0 ~ (9.1) The dimensionless depth of flow can be expressed as

With the introduction of Equations (3) and (S), Equation (4) reduces to

gP 2 40 -24 + - u - g s o = o h0T h0T

Its solution is

The dimensionless flow velocity becomes

where U is the normalizing velocity defined as

4 V. P. SINGH

and 7 is a dimensionless parameter defined as

This parameter reflects the effect of the initial depth of flow, bed slope, bed roughness, acceleration due to gravity and lateral inflow, but is independent of infiltration because the momentum of infiltration was neglected in Equation (2). The velocity increases with 1/2 power of T for a fixed y. For a fixed T, it increases with lj2 power of y.

In terms of discharge Q, the solution of Equation (4) is

The dimensionless discharge Q* = Q/Qo becomes 7

Q* = <[-1 + (1 + Y T ) " ~ ] L

where Qo is the normalizing discharge expressed as

The discharge increases with 312 power of T for a fixed y, and with 112 power of y for a fixed 7

7 1 .' I I

6

0 10 20 J O 4 0 5 0 6 0

DIMENSIONLESS TIME

y = 0.5 - - --- ys2.5

--- 7 = 3 . Q y = 1.0 - - -

y = 3.5 --- - _ _ _ _ _ _ _ y=1 .5

y = 2.0 Y = 4.0 - ----- - - --- - - - - -.-- Figure 1. Dimensionless kinematic wave velocity as a function of dimensionless time for the space-independent case with 9 = qo,f =fo,

h(0) = ho and u(0) = uo

(00 J

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 5

Figure 2. Dimensionless kinematic wave discharge as a function of dimensionless time for the space-independent case with q = qo, f =fo, h(0) = ho and u(0) = ~0

The KW or DW solution is given in dimensionless form by Equations (9.2) and (12) or (16). The dimensionless velocity and dimensionless discharge were plotted as a function of r for various values of y in Figures 1 and 2, respectively. The KW or DW solution depends not only on T but also on the initial conditions and channel characteristics reflected by a dimensionless parameter 7 defined by Equation (14).

Dynamic wave solution

and (8), can be expressed as Equation (1) has the solution given by Equation (7). Equation (2), with the introduction of Equations (3)

In terms of dimensionless v, Equation (1 8) takes the form

6 V. P. SINGH

7'

6 -

k 5 - u w s

4 - cn cn w

3 - P 2 ' W g 2 -

I -

Equation (18) can be written in terms of discharge as

When expressed in terms of dimensionless discharge, Equation (20) becomes

Equations (18) to (21) are the Riccati equations and have to be solved numerically. The occurrence of the Riccati equations can be ascribed to the presence of inertial forces.

The DYW solution is given by Equation (7) and the solution of Equation (19) or (21). The solution reflects an explicit dependence on infiltration. Equation (19) was solved by using the fourth order Runge-Kutta method. To initiate the Runge-Kutta solution, two initial conditions are needed that were obtained as follows.

Initially, at T = 1, the velocity from Equation (12) is

w = 0*5[-1 + ( 1 + Y ) ~ ' ~ ] and the derivative of the velocity from Equation (19) becomes

dv(l) = o dT

0 I ' ~ " ' ' ' " I " ' " " ' ' I ' ' " " ' " I ' " " " ' ' I ' ' ' " " ' ' i .

0 10 2 0 30 10 50 6 0

DIMENSIONLESS TIME

Figure 3. Dimensionless dynamic wave velocity as a function of dimensionless time for the space-independent case with 9 = 90,f =fo, 9r = 1.0, h(0) = ho and u(0) = uo

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 7

With these initial values, it was simple to advance the numerical solution for the next time step, and the procedure was continued. The value of time step AT was taken 0.05. For various values of y and qr, the dimensionless velocity and dimensionless discharge were computed and are plotted against T in Figures 3 and 4 for a sample case qr = 1. The velocity increased with increasing T for a fixed y. If T was fixed, the velocity increased with increasing y. The location of maximum velocity was obtained from Equation (19), which could be expressed as a quadratic relation in 21, which was the same as the KW velocity by Equation (12).

DEFINITION OF ERROR

The relative error E is defined as

SK - sD E = SD

where SK is the solution (either dimensional or dimensionless) from the KW or 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 = (h, u, Q).

400

Figure 4. Dimensionless dynamic wave discharge as a function of dimensionless time for the space-independent case with q = qo, f =fo, qr = 1.0, h(0) = ho and u(0) = uo

8 V. P. SINGH

If the solution is expressed by u, then the error differential equation is

dE ( E + 1) duK ( E + 1 ) 2 duD _- --- - - dT U K dT U K dT

ERROR IN KW AND DW APPROXIMATIONS

For the error Equation (25), duK/dT is obtained from Equation (1 1) and

and duD/dT from Equation (18) as

(26.1)

(26.2)

With the substitution of Equations (1 l), (26.1) and (26.2) in Equation (25), and after some algebraic simpli- fication, the following error equation is obtained:

where

Equation (27) is a Riccati equation and has to be solved numerically. This equation also holds for the error in discharge.

Equation (27) was solved numerically by using the fourth order Runge-Kutta method. To that end, the initial conditions were obtained as follows. At T = 1, E = 0, and

= f [ 1 + + 1

With the use of these initial conditions, the solution was advanced for the next time step, and the procedure was continued.

The error in the KW or DW solution as a function of T for various values of y (0.5 < 7 < 4) is shown in Figure 5 for the sample case of qr = 1. The maximum error and its time of occurrence are given in Table I. The magnitude of error depended on qr but its pattern of error variability was independent of qr. In each instance the graph of error was highly skewed, with a sharp rise to a peak, and an extended recession assymptotically approaching a constant value. For a fixed T , the error increased with decreasing 7. To further investigate the temporal variability of error, the error derivative was computed and is shown for various values of y in Figure 6 for qr = 1. For T 2 10, the error derivative was independent of 7 and was almost constant near zero. It decreased sharply over 1 < T < 5 . Coefficients Co, C1 and C2 were computed and are plotted for various values of y. Co is shown in Figure 7, C1 in Figure 8 and C2 in Figure 9. Co was always positive and was only slightly sensitive to y for 1 < r < 20, and was independent of 7 for T 2 20. Co was independent of qr. Both C1 and C2 were negative, had the lowest values in the

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 9

0 . 5 3

0 . 4 8

0.43

0 . 3 8

0.33 & s 0 . 2 8 3

0 . 2 5

0 . 1 8

0 . 1 1

O . O E

0.03

0 1 0 2 0 30 10 5 0 6 0

DIMENSIONLESS TIME

y = 0.5 ----- yz2 .5

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

_ _ _ - __----- -1 = 1.5

__---_- - y=2.0

*{ = 3.5 ----- ------ y z 4 . 0

Figure 5. Error in the kinematic or diffusion wave approximation as a function of dimensionless time for the space-independent case with q = q O , f =fo, qr = 1.0, h(0) = ha and u(0) = ua

Table I. Maximum error and its time of occurrence for the non-vanishing initial condition

7 Qr = 1 Qr = 2 Qr = 3

7 Maximum Error

7

0.5 3.50 1.0 3.00 1.5 2.75 2.0 2.75 2.5 2.50 3.0 2.25 3.5 2.25 4.0 2.25

0.497 0.384 0.324 0.286 0.259 0.237 0.22 1 0208

2.50 2.25 2.0 2.0 2.0 1.75 1.75 1-75

Maximum Error

0.331 0.245 0.202 0.174 0.155 0.141 0.130 0.121

7

2.25 1.75 1.75 1.75 1.75 1.50 1.50 1-50

Maximum Error

0.244 0.173 0.140 0.118 0.103 0.093 0.085 0.078

10 V. P. SlNGH

Figure 6. Error derivative as a function of dimensionless time for the space-independent case with q = qo . f =fo, qr = 1.0, h(0) = ho and u(0) = uo

beginning, and increased to almost constant values when r was large. For a fixed r, they both increased with increasing 7 . The maxima of Co and C, occurred at T = 1 but the minimum of C2 occurred at that point.

ERROR EQUATIONS: USE OF ZERO INITIAL CONDITIONS

Kinematic wave and diflusion wave solution Equation (1) has, subject to Equation (6), the solution

h = (40 - . a t (32)

Equation (32) shows that the flow depth increases linearly with time and is equal to the amount of effective rainfall up to that time. It is convenient to define a dimensionless depth or time parameter, T , as

Note that this definition of T is different from that given by Equation (8). In terms of T , Equation (32) becomes

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS

0.8-

0 . 1 -

11

1

0 10 20 30 4 0 50 6 0 DIMENSIONLESS TIME

y = 0.5 - ---- ~ ~ 2 . 5

7 = 3.0 --- - - - - _ _ _ _ _ _ _ _ - y z 1.0

- - - - ___- - -- y = 1.5

---_--- y = 2 . 0 ---- ----_-- Figure 7. Coefficient C o as a function of dimensionless time for the space-independent case with q = q0. f =A, qr = 1.0, h(0) = he and

u(0) = ug

y = 3.5

'{ = 4.0

The dimensionless depth of flow becomes

where ho is the normalizing depth defined as 2

4 t ho = - g

It should be noted that the definition of ho here is different from that defined in the preceding section. Equation (4), with the introduction of Equations (3) and (34), can be rewritten in terms of T as

2 40 s o 4 * T = o 2 u + - u - -

gP Pg

This is a quadratic equation and has the solution

(37)

40 = - [-1+ (1 + 7T)0'5 ] 2gP

12

- 0 . 2 -

- 0 . 4 -

- 0 . 6 -

Li - 0 . 8 - k [ - 1 . 0 -

- 1 . 2 -

- 1 . 4 -

V. P. SINGH

J

0 . 0 1

- z . o \ , , , , , , , , , I , , . , , , , , ( , ' , , , , , , , , I , , , , , , , , , I , , , , , , . , , , , , , , , , , . I

1 0 20 30 40 5 0 60 0

DIMENSIONLESS TIME

y = 0.5

- -__---__-------- ---- :/= 1.0

- - - - - y z 2 . 5

- - - r = 3.0 - -- - __---- - y = 1.5 7 = 3.5

_ _ - - - . -- y=2 .0 ----- -----_ '1Z4.0

Figure 8. Coefficient C, as a function of dimensionless time for the space-independent case with 9 = 90.f =fo, 9r = 1.0, h(0) = ho and u(0) = ug

Equation (38) has the same form as Equation (1 1) but with a different definition of y. Here the dimension- less parameter y is defined as

4 s O g ~ d 4%

Y'

and can be recast in the same form as Equation (14) but with a different definition of ho

the parameter y explicitly depends on infiltration. The dimensionless velocity can be expressed as

which has the same form as Equation (12), with the normalizing velocity U defined as

(39-1)

(39.2)

(41.1)

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS

- 0 . 2 -

- 0 . 4 -

- 0 . 6 -

- 0 . 8 -

- 1 . 0 -

u" E 2 k - 1 . 2 -

- 1 . 4 -

13

0 . 0 1

0 10 20 30 40 50 6 0

DIMENSIONLESS TIME

----- y=2.5

- - - '! = 3.0 .( = 0.5

- - ----------- --------. y = 1.0

- -- - ------- -/= 1.5

y = 2.0

7 3.5

------- ---- * /=4 .0 - - - --.-- Figure 9. Coefficient C2 as a function of dimensionless time for the space-independent case with 9 = q 0 , f =fo. qr = 1.0, h(0) = ho and

u(0) = uo

which has the same form as Equation (13). The solution of Equation (4) in terms of discharge takes the form

Q(T) =-~[- l hoqo +(1 + Y T ) " ~ ] 2gP

(41.2)

which has the same form as Equation (1 5) . Equation (41) can be written in terms of dimensionless discharge as

Q* =-=-[-1+(1+7T)0'5] Q T

Qo 2 which has the same form as Equation (16), with the normalizing discharge Qo defined as

Qo=x hoqo (43)

which has the same form as Equation (17). Thus, the KW and DW solution is given by Equations (34) and (38) and its dimensionless form does not explicitly depend on infiltration. The dimensionless velocity and dimensionless discharge as functions of T for various values of y have the same appearance as shown in Figures 1 and 2. The velocity increased, for a fixed y, with one-half power of T. For a hied T, the velocity increased with one-half power of y. It was also independent of qr = qo/(qo -fo).

14 V. P. SINGH

Dynamic wave solution Equation (1) has the solution given by Equation (32). Equation (2) can be expressed in terms of r as

du 40 U-- gPu2 = soq* - - - dr q*T q*r (44)

Equation (44) can be obtained from Equation (18) by setting q* = (qo -&), q: = gho. Equation (44) can be written in terms of u as

& dv = qr ti -; U - ;] Equation (45) is a Riccati equation, and can be expressed in terms of discharge Q as

(45)

Equation (46) can also be obtained from Equation (20) with use of definitions of q*. In terms of dimension- less Q*, Equation (46) takes the form

which has the same form as Equation (21). Equation (47) is also a Riccati equation. The DYW solution is given by Equation (32) and the solution of Equation (45). Clearly, the solution

depends explicitly on infiltration. The fourth-order Runge-Kutta method was used to solve Equation (45) numerically. To initiate the solution, initial conditions were obtained as follows, At r = 0, = 0, but dv/dr has singularity. However, the derivative at T = 0 was obtained by the use of forward differ- encing as follows:

T~ = 0 and ul = 0. Therefore

Let y = v 2 / r 2 . Then

--y + 1 + - y - - = o r2 2 ( 6) :

This equation is quadratic in y and has the following solution:

Assuming r2 = 0.25

With these initial values, the solution was advanced for the next time step and so on. The dimensionless velocity and dimensionless discharge as a function of r for various values of y and qr were found to have the same appearance as shown in Figures 3 and 4. For a fixed r, the velocity increased with increasing y, and for a fixed y, it increased with r.

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 15

0.8

0.7

0.6

0 . 5

g 0 g 0.4 cz kl

0.3

0 . 2

0. I

0.0

n

0 10 20 30 40 50

DIMENSIONLESS TIME

y = 0.5 - - - - - -{= 2.5

- _ - - _ - _ _ _ _ _ _ _ _ _ _ _ y = 1.0 --- ~ ~ 3 . 0

_- - - -_ - - - -- y = 1 . 5 7 = 3.5

y = 4.0 _I---. -- y = 2 . 0 ---- --___-- Figure 10. Error in kinematic or diffusion wave approximation as a function of dimensionless time for the space-independent case with

q = q O , f =fo, qr = 1.0, h(0) = 0 and u(0) = 0

Table 11. Maximum error and its time of occurrence for the vanishing initial condition

Y q r = 1 91 = 2 41 = 3

T Maximum T Maximum T Maximum Error Error Error

0.5 1.00 0.719 0.75 0.389 0.25 0.278 1.0 1.00 0.612 0.50 0.341 0.25 0.253 1.5 0.75 0.545 0.50 0.306 0.25 0,228 2.0 0.75 0.495 0.50 0.278 0.25 0.197 2.5 0.75 0.454 0.50 0.256 0.25 0.156 3.0 0.75 0.421 0.25 0.240 0.75 0.127 3.5 0.75 0.393 0.25 0.226 1 *oo 0.108 4.0 0.75 0.370 0.25 0.210 1 .o 0.094

16 V. P. SINGH

ERROR IN KW AND DW APPROXIMATIONS

Equation (25) was used to define the error in the KW or DW solution, where U K is given by Equation (38) and uD by Equation (44). The derivative duK/dr is obtained from Equation (38) as

dUK - 740 - - - (1 + y r)-0'5 d7- 4Pg

From Equation (44)

(53)

Substitution of Equations (53), (54) and (37) in Equation (25) and a little algebraic simplification yields

dE d r = C0(y1 r ) + C1(y, r ) E + C2(y, r ) E 2 , E(0) == 0, 7 2 0 ( 5 5 ) -

Equation (55) is a Riccati equation and also holds for error in discharge. At T = 0, dE/dr has a discon- tinuity. Equation (55) was solved using the fourth order Runge-Kutta method. The value of dE/dr at r = 0 was obtained as discussed in the Appendix. Thereafter, the solution can be advanced for the next time step and the procedure is continued.

The error in the KW or DW solution was computed and is plotted as a function of T for various values of y and qr. Figure 10 shows the error for a sample case 4r = 1. The maximum error and its time of occurrence are given in Table 11. The error graph was highly skewed, with a sharp rise to the peak and then an extended decline, asymptotically approaching a constant value. The shape of the graph was independent of r. For a fixed T , the error was higher for lower values of y. To further examine the error variation, dE/dr was com- puted against T for various values of y and 4r. The error derivative was almost independent of y, decreased sharply and then assumed an almost constant value. Coefficient Co was computed for various values of y and was independent of qr, and virtually independent of y. It decreased sharply for the highest value in the beginning to almost a constant value at T > 15 and was always positive. C1 was computed for qr = 1,2 and 3 for various values of y. It was always negative and quite fairly sensitive to y. C2 was computed for q7 = 1,2 and 3 for various values of y. Both C1 and C2 were negative. C2 was much less sensitive to y than C1. They both increased with decreasing y. Their variation was further investigated by use of Equations (56) to (58). Clearly, Co and C1 were maximum at T = 0 and Cz was at a minimum at that point.

CONCLUSIONS

For space-independent flows, the accuracy of KW and DW approximations was significantly influenced by the relative magnitudes of infiltration rate and rainfall rate, expressed as qr, and the dimensionless para- meter y. The parameter y reflects the effect of initial flow depth, bed slope, lateral inflow, infiltration and channel roughness. The error of these approximations decreased exponentially for dimensionless time exceeding five. The non-linear differential equations describing the error of these approximations were found to be the Riccati equations. The occurrence of these equations was traced to the neglect of inertial force in the momentum equation. For non-vanishing initial conditions, the KW and DW

KINEMATIC AND DIFFUSION WAVE APPROXIMATIONS 17

approximations were accurate (error < 15%) for y B 5 and qr > 1.5. for vanishing initial conditions, the accuracy of these approximations was higher.

REFERENCES

Beven, K. 1981. 'Kinematic subsurface flow', Wac. Resour. Res., 17, 1419-1424. Daluz Viera, J. H. 1983. 'Conditions governing the use of approximations for the Saint Venant equations for shallow water flow', J.

Ferrick, M. G. 1985. 'Analysis of river wave types', Wat. 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. B. 1955. 'On kinematic waves: 1. Flood movement in long rivers', Proc. R . SOC. London Ser. A , 229,

Morris, E. M., and Woolhiser, D. A. 1980. 'Unsteady, one dimensional flow over a plane: partial equilibrium and recession hydro-

Ponce, V. M., and Simons, D. B. 1977. 'Shallow wave propagation in open channel flow', J. Hydr. Div. A X E , 103, 1461-1475. Singh, V. P. 1976. 'Studies on rainfall-runoff modeling: 2. A distributed kinematic wave model of watershed surface runoff, WRRZ

Rep. 065, New Mexico State University, New Mexico Water Resources Research Institute, Las Cruces, New Mexico, 154 pp. Singh, V. P. 1988. Hydrologic Systems. Vol. 1. Rainfall-Runoff Modeling. Prentice Hall, Englewood Cliffs, NJ. Singh, V. P. 1989. Hydrological Systems. Vol. 2. Watershed Modeling. Prentice Hall, Englewood Cliffs, NJ. Singh, V. P. 1994. 'Accuracy of kinematic wave and diffusion wave approximations for space-independent flows', Hydrol. Process., 8,

Woolhiser, D. A., and Liggett, J. A. 1967. 'Unsteady one-dimensional flow over a plane - the rising hydrograph', Wat. Resour. Res.,

Hydrol., 60,43-58.

phys. Res., 69, 1531-1540.

281-31 6.

graphs', Wac. Resour. Res., 16, 355-360.

45-62.

3, 753-111.

APPENDIX: NUMERICAL SOLUTION OF RICCATI EQUATION (55)

Coefficients Co, C1 and C2 contain the terms (1 + ~ 7 ) " ~ . Let

z1 = (1 + y7)-O3

2 2 = (1 + y7)@5

(Al)

and

(A21

By expanding z1 and z 2

(A31

(A41

Y 7 2

77 2

(1 + y T ) - O ' ~ = 1 - - + HOT

(1 t y ~ ) ' ' ~ = 1 +-+HOT

where HOT = higher order terms. Assume HOT are negligible. By substituting Equation (A3) and (A4) in Equations (56) to (57)

27

1 Y 1 - [T 41 l + l + - Y 7 - - q r -+-

2

Substitution of Equations (A5) to (A7) in Equation (55) yields

18 V. P. SINGH

If dE/dT is bounded, then dE dT -=Ly+f(T)

near 0 where f(T) t 0 as T -t 0. On integration E = a + Q! T + F(T)

where a is constant of integration, f(T) is some function of T, and F(T) is the integral of f(7). Near T = 0, F(T) = 0 and E(0) = 0.

Therefore, Equation (A10) becomes

E=O+(YT (All) Substitution of Equation (A1 1) in Equation (A8) leads to

2 2 7 -74rc-u 7

For dE/dr to be bounded, Equation (A10) is to be satisfied. Therefore, near T = 0

dE 7

d7 4 - = Ly = - - + (Y(1 - qr)

Hence

This gives the value of the error derivative at T = 0.