Generalized flux law, with an application

Generalized ¯ux law, with an application

V. P. Singh* and M. PrasanaDepartment of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA 70803-6405, U.S.A.

Abstract:A generalized ¯ux law, which can specialize into power- and di�usive-type ¯ux laws, is proposed. When coupledwith the continuity equation, a generalized ¯ow equation is the result. The generalized equation specializes intoseveral cases, one of which is the kinematic wave equation. By applying the generalized ¯ow equation and its

variants to ¯ow routing in two rivers, the usefulness of the generalized ¯ux law is evaluated. Copyright # 1999John Wiley & Sons, Ltd.

KEY WORDS generalized ¯ux law; kinematic wave equation; power-type ¯ux law; di�usive-type ¯ux law; ¯ow

routing; advective±dispersive equation; mixing cell method

INTRODUCTION

For physically based modelling of a wide range of surface and subsurface ¯ow processes (such as overland¯ow, channel ¯ow, glacial motion, snowmelt runo�, subsurface storm ¯ow, unsaturated ¯ow, ¯ow throughsnowpacks, sedimentation in a tank, etc.) the law of conservation of mass (yielding the continuity equation)and a ¯ux law are required.

A ¯ux law is a relationship between ¯ux and concentration, or some characteristic thereof. Most of the ¯uxlaws commonly used in hydrology can be distinguished as one of two types: (i) power and (ii) di�usive orgradient types. Denoting the ¯ux by u and the concentration by h, one of the most popular power-type ¯uxlaws is the kinematic wave relationship.

u � ahm �1�Where a andm are parameters and their physical meaning depends on the type of ¯ow. In surface ¯ow, a is

the resistance parameter and m is the ¯ow parameter indicating whether the ¯ow is turbulent or laminar. Inthe case of overland or channel ¯ow, u is the cross-sectionally averaged ¯ow velocity, and h is the ¯ow depth.The parameters a andm are generally determined from either Chezy's or Manning's equation. The kinematicwave relationship has been extensively applied in surface-water hydrology (Singh, 1996), subsurfacehydrology (Charbeneau, 1984; Singh and Joseph, 1994) and agricultural irrigation (Singh and Ram, 1983).

In vandose-zone hydrology, Equation (1) reduces to a ¯ux law where u denotes the moisture ¯ux and hdenotes the moisture concentration Ð usually the e�ective saturation. Parameter m indicates the grain-sizedistribution index, and parameter a is related to the hydraulic conductivity.

CCC 0885±6087/99/010073±15$17�50 Received 21 November 1996Copyright # 1999 John Wiley & Sons, Ltd. Accepted 15 May 1997

HYDROLOGICAL PROCESSESHydrol. Process. 13, 73±87 (1999)

* Correspondence to: V. P. Singh, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge,LA 70803-6405, USA.

A gradient- or di�usive-type ¯ux can be expressed as

u � ÿg @h@x

�2�

where g is a parameter. This form is similar to Darcy's law in groundwater hydrology (where g is thehydraulic conductivity) to Fourier's law in heat ¯ow (where g is the thermal conductivity) to Ohm's law inelectricity (where g is the resistivity) and to Fick's law in solute ¯ow (where g is the di�usion coe�cient).

Each of the above two types of ¯ux law accounts for certain ¯ow mechanisms. For example, Equation (1),in the case of surface ¯ow, arises from the balance between frictional and gravitational forces, whereasEquation (2), in case of subsurface ¯ow, includes gravitational and pressure forces. However, in nature,many ¯ow phenomena are governed, or at least signi®cantly in¯uenced by, all of these forces and not just bysome of them. This means that neither of the above-mentioned ¯ux laws will be adequate for those ¯ows anda di�erent, or new, ¯ux law will have to be sought. The objective of this paper is to propose such a law andvalidate it for the particular case of ¯ow routing.

GENERALIZED FLUX LAW

A generalized ¯ux law is hypothesized by combining Equations (1) and (2) as

u � ahm ÿ g@h

@x�3�

Equation (3) has an advantage in that it includes Equations (1) and (2) as special cases. For surface ¯ows itretains the convective as well as the di�usive characteristics of ¯ow, through incorporation of gravitationaland frictional forces as well as pressure. In the case of subsurface ¯ow it accounts for the convectiveand di�usive characteristics of ¯ow, through inclusion of gravitational and capillary forces. For m� 1,Equation (3) results in a ¯ux law ®rst employed by Burgers (1948). Thus, Equation (3) can be thought of as ageneralization of Burgers' ¯ux law.

GOVERNING EQUATION

When the generalized ¯ux law is coupled with the continuity equation, the resulting equation governs a rangeof ¯ow phenomena. In the case of surface ¯ows, a one-dimensional form of the continuity equation (withoutsources and sinks or lateral ¯ow) for a wide rectangular geometry on a unit width basis can be written as

@h

@h� @Q@x� 0 �4�

where h is water ¯ow depth (L), Q is discharge per unit width (L2/T), t is time (T) and x is distance along the¯ow (L). Equation (4) has two dependent variables, h and Q. Its solution therefore requires an additionalexpression relating Q to h, which is given by the generalized ¯ux law of Equation (3).

The volumetric discharge, Q, can be expressed as Q� uA, where A is the cross-sectional area. For arectangular geometry, A� bh, where b is the width �1, and h is the depth of ¯ow. In terms of discharge,Equation (3) takes the form

Q � ahn ÿ b@h

@x�5�

where n�m � 1 and b� gh. Equation (5) is a generalized ¯ow depth±discharge relationship. To coupleEquations (4) and (5), we ®rst di�erentiate Equation (5) with respect to x

@Q

@x� anhnÿ1

@h

@xÿ b

@2h

@x2ÿ g

@h

@x�6�

Copyright # 1999 John Wiley & Sons, Ltd. HYDROLOGICAL PROCESSES, VOL. 13, 73±87 (1999)

74 V. P. SINGH AND M. PRASANA

Substituting Equation (6) for @Q/@x in the continuity Equation (4), we have

@h

@t� �anhnÿ1 ÿ g� @h

@x� b

@2h

@x2�7�

which is the generalized governing ¯ow equation in terms of ¯ow depth. This equation specializes into severalparticular cases.

An important case of Equation (7) arises when b� constant

@h

@t� C

@h

@x� b

@2h

@x2�8a�

C � nahnÿ1@h

@x�8b�

where C is the kinematic wave celerity.Equation (8a) is a non-linear advective±dispersive equation. Another special case of Equation (7) is the

Burgers equation (Burgers, 1948; Whitham, 1974) which is obtained when n� 2 or m� 1 and b� constant

@h

@t� ah

@h

@x� b

@2h

@x2�9a�

A full account of Burgers' equation is given by Fletcher (1982). If n� 1 and b� constant, Equation (7)reduces to a linear parabolic equation (i.e. advection±dispersion equation) (Dooge, 1973)

@h

@t� a

@h

@x� b

@2h

@x2�9b�

Burgers equation is a useful mathematical model for such diverse physical problems as shock ¯ows, tra�c¯ow, acoustic transmission in fogs, etc., and other non-linear wave propagation problems in which eithershocks or viscous dissipation are signi®cant in part of the ¯ow region (Whitham, 1974; Warrick and Parkin,1995). In the case of ¯uid ¯ow, the dissipation may arise from viscosity, heat conduction, chemical reaction,mass di�usion, thermal radiation, etc. A popular special case results when g� 0 (and consequently b� 0)

@h

@t� nahnÿ1

@h

@x� 0 �10�

which is the kinematic wave equation.

SOLUTION METHODOLOGY

Equation (7) is a non-linear partial di�erential equation and its analytical solution in general is not tractable.Thus the numerical solution is the only resort. One of the simpler methods to solve Equation (7) is the mixingcell method, which is employed here and is compared with a simple ®nite di�erence method.

Mixing cell method

The mixing cell method employs the backward ®nite di�erence to approximate the ®rst-order spatialderivative, and the forward ®nite di�erence to approximate the time derivative. Such approximationsproduce numerical dispersion that can be forced to equate with the physical dispersion involved in thegoverning equation. A complete discussion of the method is presented by Noye (1982), Van Ommen (1985),Appelo and Williamson (1987) and Bajracharya and Barry (1992), among others. Applying Taylor series


GENERALIZED FLUX LAW, WITH AN APPLICATION 75

expansion to the ®rst-order terms of Equation (7) at i, k, where i is the discrete length step and k is thediscrete time step, and neglecting the terms of third and higher order we get

@h

@t� hk�1i ÿ hki

Dtÿ Dt

2

@2h

@t2�11�

@h

@x� hki ÿ hkiÿ1

Dt� Dx

2

@2h

@x2�12�

The second term on the right-hand side of Equation (11) can be expressed using the continuityEquation (4) as

@h

@t� ÿ @Q

@x�13�

Di�erentiation of Equation (13) with respect to t yields

@2h

@t2� @

@t

@h

@t

� �� @

@tÿ @Q@x

� �� @

@t�ÿanhnÿ1 � g� @h

@x� gh

@2h

@x2

� ��14�

or

@2h

@t2� @


@x

� �� @

@tgh@2h

@x2

� ��15�

Equation (15) can be written as

@2h

@t2� Q1 � Q2 �16�

where

Q1 � @


@x

� �� 17�

and

Q2 � @

@tgh@2h

@x2

� ��18�

Equation (17) can be expressed as

Q1 � @

@t�ÿanhnÿ1� @h

@x

� �� @

@t�g� @h

@x

� �� 19�

or

Q1 � @

@x�ÿan�n ÿ 1�hnÿ2� @h

@t

� �� ÿ �ÿanhnÿ1� @h

@x

� �� g

@

@t

@h

@x

� ��20�

Substituting for

@h

@t� ÿ @Q

@x� �ÿanhnÿ1 � g� @h

@x� gh

@2h

@x2



from Equations (4) and (5) in Equation (20), and simplifying, one gets

Q1 � 2AB@h

@x

� �2

�Bgh @h@x

@2h

@x2

� �� @2h

@x2

� ��A ÿ 2Ag ÿ g2� � @h

@x

@2h

@x2

� �� h

@3h

@x3

� �� g2 ÿ Ag� �21�

where

A � anhnÿ1 �22�B � an�n ÿ 1�hnÿ2 �23�

Similarly, Q2 can be expressed as

Q2 � @

@tgh@2h

@x2

� �� g

@2h

@x2@h

@t�A � g� ÿ Cg

@h

@t

@h

@t

� �3

ÿ3Bgh @h@t

@2h

@x2

ÿ h@3h

@x3�Ag � g2� � g2h2

@4h

@x4� 2g2h

@h

@x

@3h

@x3� 2g2h

@2h

@x2

� �2�24�

where

C � an�n ÿ 1��n ÿ 2�hnÿ3 �25�

Substituting Q1 and Q2 in Equation (16), and rearranging, we obtain

@2h

@t2� 2AB

@h

@x

� �2

ÿ2Agh @3h

@x3ÿ 2Bgh

@h

@x

@2h

@x2

� �ÿ Cgh

@h

@x

� �3

� 2g2@2h

@x2

� �@h

@x

� �� 2g2h

@h

@x

@3h

@x3� 2g2h

@2h

@x2

� �2

�g2h2 @4h

@x4

�26�

Substituting for (@2h)/(@t2) in Equation (11) and then substituting Equations (11) and (12) in the generalizedequation, and simplifying, we get

Dx � DtX � 2ghX

�27�

where

X � �ÿanhnÿ1 � g� �28�

In the ®nite di�erence form the generalized equation can be expressed as

hk�1i � h

ki ÿ

XDtDx�hki ÿ h

kiÿ1� � AB�Dt�2 @h

@x

� �2

ÿ�Dt�2Aghki@3h

@x3

� �ÿ �Dt�2Bghki

@h

@x

@2h

@x2

� �

ÿ Cghki�Dt�22

@h

@x

� �3

�g2h2 @4h2

@x4� g2h

@h

@x

@3h

@x3� g2h

@2h

@x2

� �2

�29�



where the partial derivatives in the ®nite di�erence form are expressed as

@h

@x�hki ÿ hki�1

Dx�30�

@2h

@x2� hki ÿ 2hkiÿ1 � hkiÿ2

�Dx�2 �31�

@3h

@x3� hki ÿ 3hkiÿ1 � 3hkiÿ2 ÿ hkiÿ3

�Dx�3 �32�

@4h

@x4� hki ÿ 4hkiÿ1 � 6hkiÿ2 ÿ 4hkiÿ3 � h4iÿ4

�Dx�4 �33�

Equation (29) is the generalized numerical analogue of Equation (7) owing to the mixing cell method. Itis, however, possible to truncate the general representation by neglecting higher powers of di�erentialexpressions. To that end, the following cases are considered:

. a general or a full form that considers the complete expression, which is Equation (29);

. a less general form with b� constant; and

. a simpli®ed or a short form that considers the truncated version of Equation (29) and is given by

hk�1i � h

ki ÿ

ADtDx�hki ÿ h

kiÿ1� �34�

Each of the above cases will have two scenarios: one with all three parameters optimized and the otherwith n constant at 1.667 and the other two parameters optimized.

Finite di�erence method

The equation to be solved using a simple ®nite di�erence method is Equation (7), where the ®rst- andsecond-order derivatives can be expressed in ®nite di�erence form as

@h

@t� hk�1i ÿ hki

Dt�35�

@h

@x� hki�1 ÿ hkiÿ1

2Dx�36�

@2h

@x2� hki�1 ÿ 2hki � hkiÿ1

�Dx�2 �37�

Substituting Equations (35), (36) and (37) into Equation (7), we have

hk�1i � h

ki ÿ

ADt2Dx�hki�1 ÿ h

kiÿ1� � ghki Dt

hki�1 ÿ 2hki � hkiÿ1�Dx�2

!�38�

A � anhnÿ1 ÿ g �39�

Equation (38) is the ®nite di�erence solution of Equation (7).



APPLICATION

The generalized ¯ow equation was applied to route ¯ow in two di�erent river reaches in the People'sRepublic of China. The ®rst is Yuanling±Wangjiahe River reach, which is 112 km long with an average widthof 445 m and a mean bed slope, S0 of 0

.0004. In August 1968 a ¯ood event was measured at two stationslocated at the upstream and downstream ends of the reach. Initially, h(x, 0)� h(x) should be known for thereach. For want of additional information this value was considered uniform throughout the reach. At theupstream end h(0, t)� h(t) is known (which is the upstream hydrograph). With these initial and boundaryconditions, ¯ow was routed from the upstream station to the downstream station using the mixing cellmethod and the ®nite di�erence method.

For ¯ow routing using Equation (29), or its simpli®cations, the parameter values should be known.Equation (29) has three parameters: n, a and g. The value of n can be taken as 1.667 but it is not clear if a and gcan be prescribed a priori. Therefore, two cases, resulting from Equation (7), were considered: (i) n� 1.666and a and g optimized; and (ii) n, a and g optimized. For the less general case, the cases dealt with were:(i) n� 1.667 and a and g optimized; and (ii) n, a and g optimized. These cases result from Equation (8a) andtheir numerical solution will, accordingly, be a simpli®ed version of Equation (29). Two further simpli®-cations result from use of Equation (34) with optimization of n, a and g; and with optimization of a and g only,and keeping n� 1.667.

Thus, the numerical solution was considered for six cases.

1. The full form solution owing to the mixing cell method was used keeping n constant at 1.667, andoptimizing a and g.

2. The full form solution owing to the mixing cell method was used optimizing a, g and n.

3. The less generalized solution owing to the mixing cell method, keeping n constant at 1.667 andoptimizing a and b.

4. The less generalized solution owing to the mixing cell method, optimizing a, b and n.

5. The short form solution owing to the mixing cell method, keeping n constant at 1.667, and optimizinga and b.

6. The short form solution owing to the mixing cell method was used, optimizing a, b and n.

The values of ¯ow depth were determined using the mixing cell method and the ®nite di�erence method,with the optimal values of a, b and n obtained by using the least-squares method. These were then comparedwith observed ¯ow depths for each case. For the ®rst case of the full form solution, with a and b optimizedand keeping n constant at 1.666, Figure 1 shows a graphical representation of the observed and computeddepth hydrographs. It is clear that the observed and computed hydrographs are in close agreement for boththe ®nite di�erence method and the mixing cell method. In the second case, parameters a, b and n werevaried to ®nd an optimal solution for the mixing cell method. Figure 2 graphs the observed and computedhydrographs. The agreement between observed and computed hydrographs is close but this case does noto�er any advantage over the previous case with only a and g optimized. In this case the ®nite di�erencemethod did not perform well. Figures 3 and 4 sketch the computed and observed hydrographs for cases 3and 4, representing two simpler versions. Clearly the simpler versions are quite good and their computedhydrographs were in close agreement with observed hydrographs. However, the ®nite di�erence methoddid not perform well in these two cases. For two further simpli®ed cases, 5 and 6, Figures 5 and 6 graph thecomputed and observed depth hydrographs. The agreement between computed and observed hydrographs isquite good for both the mixing cell and ®nite di�erence methods.

The other reach is the Longjie±Qiaojia River reach located upstream of the Yangtze River in China. Thelength of the reach is 247 km, with an average width of 279 m and a mean bed slope of 0.00112. A ¯oodevent was measured on 18±20 June, 1967, at two stations located at the upstream and the downstream ends



Figure 1. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjiahe River reach (full form)(optimized for a and b) a� 0.564, g� 6.62, n� 1.666

Figure 2. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjiahe River reach (full form)(optimized for a, b and n) a� 0.484, g� 4.32, n� 1.006



Figure 3. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjihe River reach(less generalized form) (optimized for a and b keeping n constant at 1.666) a� 0.752, b� 5960, n� 1.666

Figure 4. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjiahe River reach(less generalized form) (optimized for a, b and n) a� 0.662, b� 8310, n� 1.73



Figure 5. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjiahe River reach (short form)(optimized for a and b keeping n at 1.666) a� 0.704, b� 7960, n� 1.666

Figure 6. Comparison of calculated depth hydrograph with the observed hydrograph for Yuanling±Wangjiahe River reach (short form)(optimized for a, b and n) a� 0.65, b� 8466, n� 1.7



of the reach. Figures 7±11 graph observed and calculated depth hydrographs for the six aforementionedcases.

As shown in Figure 7 for case 1, the agreement between the observed hydrographs and the hydrographscomputed using the ®nite di�erence method method was satisfactory. The hydrograph peak by the mixingcell method method exceeded the observed peak. When only a and b were optimized in case 2, theagreement between observed and computed hydrographs remained the same for both methods, as shown inFigure 7. When the simpler version (a less generalized form of the full form solution), with a, and boptimized and n constant at 1.667 was used, the agreement between the observed and computed hydro-graphs improved for the mixing cell method but worsened for the ®nite di�erence method, as shown inFigure 8. When all three parameters were optimized in case 4, the agreement between observed andcomputed hydrographs was better than in the preceding case, as shown in Figure 9. For case 5, with a andb optimized, the agreement between observed and computed hydrographs was quite good but the ®nitedi�erence method did not fare as well, as shown in Figure 10. When all three parameters were optimized inthe simplest case 6, the agreement improved only slightly for the ®nite di�erence method, as shown inFigure 11.

The high value of b indicates a high dispersion rate, the values of a and n relate to the slope, friction andthe cross-section of the channel in which the ¯ow occurs. A comparison of observed and computedhydrograph peaks and times to peak for di�erent cases is shown in Table I, and parameters for these casesare summarized in Table II. It is seen that all six cases produced peak ¯ow and time to peak quite accurately.Indeed a more complete formulation did not have any particular advantage over simpler versions, whichmay partly be due to the versatility of the generalized ¯ux law.

Figure 7. Comparison of calculated depth hydrograph with the observed hydrograph for Longjie±Qiaojia River reach (full form)(optimized for a and b) a� 0.0001, g� 4.80, n� 1.666



Figure 8. Comparison of calculated depth hydrograph with the observed hydrograph for Longjie±Qiaojia River reach (less generalizedform) (optimized for a and b keeping n constant) a� 0.6, b� 400, n� 1.666

Figure 9. Comparison of calculated depth hydrograph with the observed hydrograph for Longjie±Qiaojia River reach (less generalizedform) (optimized for a and b and n) a� 4.336, b� 589, n� 1.05



Figure 10. Comparison of calculated depth hydrograph with the observed hydrograph for Longjie±Qiaojia River reach (short form)(optimized for a and b keeping n constant) a� 0.556, b� 30 000, n� 1.666

Figure 11. Comparison of calculated depth hydrograph with the observed hydrograph for Longjie±Qiaojia River reach (short form)(optimized for a and b and n) a� 1.979, b� 27 062, n� 1.26



CONCLUSION

A generalized ¯ux law was hypothesized and the resulting ¯ow equation applied to ¯ood routing. The ¯owrouting equation was then solved using the mixing cell and ®nite di�erence methods. Higher powers ofdi�erential expressions were found to be relatively small. Two numerical examples with and without theincorporation of these di�erential terms were analysed and the calculations show that these terms arenegligibly small and do not contribute signi®cantly to the ®nal ¯ow depth. A comparison of ¯ow depths ofthe numerical methods for two ¯ood events shows that the values of the calculated peak depth arecomparable with those of the observed depth. The time at which the peak ¯ow depth is reached for each¯ood event using the two numerical methods compared well with the observed peak out¯ow time. Acomparison of six cases shows that although all six cases produced reasonably accurate hydrographs, thesimpler versions performed better. Indeed the two simpler versions with a, b, and n varying produced thebest result.

REFERENCES

Appelo, C. A. J. and Willemsen, A. 1987. `Geochemical calculations and observations on salt water intrusions, 1. A combinedgeochemical/mixing cell method', J. Hydrol., 94, 313±330.

Bajracharya, K. and Barry, D. A. 1992. `Mixing cell methods for nonlinear nonequilibrium single species adsorption and transport',Wat. Resour. Res., 29, 1405±1413.

Burgers, J. M. 1948. `A mathematical model illustrating the theory of turbulence', Adv. Appl. Mech., 1, 171±199.Charbeneau, R. J. 1984. `Kinematic models for soil moisture and solute transport', Wat. Resour. Res., 20, 699±706.Dooge, J. C. I. 1973. `Linear theory of hydrologic systems', Technical Bulletin No. 1468. US Dept. Agricultural Research Services,

Washington D.C.Fletcher, C. A. J. 1982. Burger's Equations: A Model for all Reasons. North-Holland Publishing Company, Amsterdam, The

Netherlands.Noye, J. 1982. Finite di�erence methods for partial di�erential equations, North-Holland Publishing Company, Amsterdam, The

Netherlands.

Table I. Comparison of observed peak and time to peak with those calculated using di�erent solutions

River reach Event date Observed peak Computed peak

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Yuanling±Wangjiahe August 1968 6.5 6.69 6.57 6.58 6.6 6.47 6.47Longjie±Qiaojia June 1967 13.12 14.56 14.56 12.94 13.32 13.55 13.69

Time to PeakYuanling±Wangjiahe August 1968 36 36 36 36 36 36 36Longjie±Qiaojia June 1967 24 24 24 27 27 24 24

Table II. Summary of parameter values for di�erent cases

River reach Parameters

Case 1 Case 2 Case 3 Case 4 Case 5 Case 6

Yuanling±Wangjiahe

n� 1.666 n� 1.006 n� 1.666 n� 1.73 n� 1.666 n� 1.7alpha� 0.564 alpha� 0.484 alpha� 0.752 alpha� 0.662 alpha� 0.704 alpha� 0.65

gamma� 6.62 gamma� 4.32 beta� 5960 beta� 8310 beta� 7960 beta� 8466

Longjie±Qiaojia

n� 1.666 n� 1.666 n� 1.666 n� 1.05 n� 1.666 n� 1.26alpha� 0.0001 alpha� 0.0001 alpha� 0.7 alpha� 4.336 alpha� 0.556 alpha� 1.979

gamma� 4.80 gamma� 4.80 beta� 400 beta� 589 beta� 30 000 beta� 27 062



Singh, V. P., Wang, G. and Adrian, D. 1997. `Flood routing based on di�usion wave equation using the mixing cell method',Hydrol. Process., 11, 1181±1894.

Singh, V. P. and Joseph, E. S. 1994. `Kinematic wave model for soil moisture movement with plant-root extraction', Irrig. Sci., 14,189±198.

Singh, V. P. and Ram, R. S. 1983. `A kinematic model for surface irrigation: veri®cation by experimental data', Wat. Resour. Res., 19,1599±1612.

Singh, V. P. 1996. Kinematic Wave Modeling in Water Resources: Surface-Water Hydrology, John Wiley, New York. 1399 pp.Van Ommen, H. C., 1985. `The mixing-cell concept applied to transport of non reactive and reactive components in soils and

groundwater', J. Hydrol., 78, 201±213.Warrick, A. W. and Parkin, Y. W., 1995. Analytical solution for one-dimensional drainage: Burgers' and simpli®ed forms, Wat. Res.

Res., 31, 2891±2894.Whitham, G. B., 1974. Linear and Nonlinear Waves, Wiley-Interscience, New York.



Generalized flux law, with an application

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