Field Properties in Surface Irrigation Management and Design

Field Properties in Surface Irrigation Managementand Design

Theodor S. Strelkoff, M.ASCE1; Albert J. Clemmens, M.ASCE2; and Eduardo Bautista, M.ASCE3

Abstract: Field properties—topography, hydraulic resistance, and infiltration—play an important role in the performance of surfaceirrigation systems, and appropriate characterizations of these are required as data input to simulation or design software. The EWRI/ASCETask Committee on Soil and Crop Hydraulic Properties has been charged with preparing a guide for practitioners faced with such dataentry. The result is this special section of the Journal of Irrigation and Drainage Engineering in which this paper is the first in the seriespresented. It describes the characteristics of these field properties and notes a series of caveats to be considered when dealing with themin the course of analyses or designs of surface irrigation systems.

DOI: 10.1061/�ASCE�IR.1943-4774.0000119

CE Database subject headings: Channel beds; Hydraulic roughness; Infiltration; Surface irrigation.

Introduction

Field properties such as topography, soil infiltration, and hydrau-lic resistance �of soil grains and clods, mulch, and plant parts�have a profound effect on the performance of surface irrigationsystems, and quantification of these parameters is crucial to goodmanagement and design. Surface irrigation systems are oftencharacterized as exhibiting poor performance, with nonuniformapplication of water and excessive deep percolation and runoff.With the goal of improved design and operation, software hasbeen developed for predicting and optimizing performance withtrial values of the controlling variables affecting operation �SIR-MOD, Walker �2004�; WinSRFR, USDA Agricultural ResearchService �USDA-ARS� �2009�; Bautista et al. 2009��. Based onlaws of physics like conservation of mass and momentum, thesoftware predicts the outcome of a particular set of control vari-ables, such as inflow rate and cutoff time as these interact with theextant set of field variables. It is not immediately evident how tocharacterize those variables, like topography, system geometry,surface roughness, plant drag, and infiltration, nor how to evalu-ate them in a given field. To sort through the many availablemethods of estimating field characteristics, the Environmental andWater Resources Institute of ASCE authorized formation of aTask Committee on Soil and Crop Hydraulic Properties, a child ofthe parent technical committee for On-Farm Irrigation Systems, a

1Research Hydraulic Engineer, U.S. Arid Land Agricultural ResearchCenter, USDA-ARS, 21881 N. Cardon Ln., Maricopa, AZ 85238 �corre-sponding author�. E-mail: [email protected]

2Research Hydraulic Engineer, Center Director, U.S. Arid Land Agri-cultural Research Center, USDA-ARS, 21881 N. Cardon Ln., Maricopa,AZ 85238. E-mail: [email protected]

3Research Hydraulic Engineer, U.S. Arid Land Agricultural ResearchCenter, USDA-ARS, 21881 N. Cardon Ln., Maricopa, AZ 85238. E-mail:[email protected]

Note. This manuscript was submitted on July 31, 2008; approved onMay 12, 2009; published online on May 15, 2009. Discussion periodopen until March 1, 2010; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Irrigation andDrainage Engineering, Vol. 135, No. 5, October 1, 2009. ©ASCE, ISSN
0733-9437/2009/5-525–536/$25.00.
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member of the Irrigation and Drainage Council. The output of thecommittee is this special issue of the Journal of Irrigation andDrainage Engineering. In this, the first paper of the group, thevarious functional forms used to describe surface roughness andvegetative drag and infiltration are reviewed along with some ofthe features of these field characteristics. Methods for quantifyingthe parameters of those formulas and sensitivity of simulation anddesign software to the results comprise the subject of companionpapers.

Soil-Surface Geometry

Field-surface configuration is measured in a level survey. It issubject to change over time, during an irrigation �say, as a freshlytilled soil settles� and over a season or between irrigations. Thetopography of an irrigated field affects performance at severaldifferent scales, in part because departures of the field surfacefrom a theoretical plane occur at various scales, from grain size tofield dimensions. Field-scale-average slopes have long been rec-ognized as a major influence in surface irrigation performance.Spatial variations at a scale on the order of field dimensions, forexample a gradual decrease in slope with distance along a borderstrip or furrow, or simply a break in slope at some point in thelength of run can be input to computer-simulation models toevaluate the effect of such departures from a plane surface. Fig. 1shows simulated post-irrigation distributions of infiltrated waterunder different grading treatments in an Egyptian basin �Clem-mens et al. 1999a�. The markedly greater total infiltration withtraditional leveling stems from the Egyptian practice of cuttingoff inflow when the stream arrives at the downstream end—theirregularity in the soil surface slows stream advance. The contri-bution of cross slope to nonuniformity of infiltration can be in-vestigated with two-dimensional simulation models �Playan et al.1996; Strelkoff et al. 2003�. Small-scale variations, microtopog-raphy, typically random departures from a smooth surface on aspatial scale of, say, a few meters, tend to decrease advance ratesand so affect distribution uniformity, but their major influence isthrough low spots with longer than average recession times, es-
pecially if the average field surface is level, or inflow rates are
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ASCE license or copyright; see http://pubs.asce.org/copyright

small �Clemmens et al. 1999b�. Practical distribution uniformitydepends as well on the plant-element scale �Burt et al. 1997�. Forexample, random variations in elevation in the area under anorchard-tree canopy are of little concern. On the other hand, sys-tematic variations in soil-surface elevations can become importantin their relationship to plant placement.

The scale of nonuniformities can be too small for a completemeasured description of surface elevations, but, for entry intosimulation models, kriging �Fonteh and Podmore 1994� can makethe most of a limited number of measurements. Statistically ran-dom generated elevations with standard deviation and trends cor-responding to field conditions have been used in simulationstudies showing the relation between distribution uniformity andnonuniformity of surface elevations �Fangmeier et al. 1999�. Itshould be noted that elevation measurements in level surveys withstandard techniques should be assumed subject to some 3–5 mmerrors. Still smaller scales—clods, aggregates—truly microtopog-raphy, can be classified as characterizing surface roughness, andwill be discussed in the next section.

Resistance to Flow

Measurement of hydraulic resistance and, in fact, appreciation forthe influence of soil-surface roughness or plant drag, are rela-tively new. It is recognized now that resistance has an effect onstream advance comparable to that of infiltration. However, therequired accuracy in estimating resistance is not as great as thatfor infiltration, especially in border irrigation; there, errors in re-sistance parameters affect computed advance and recession in acompensatory way �Fangmeier and Strelkoff 1979�. Still, there isno consensus even on the appropriate formulas for describingresistance as a function of flow depth or plant drag as a functionof vegetative density.

The friction slope Sf is the resistive drag force of the soilsurface and inundated vegetation on the irrigation-stream flow perunit length of channel, divided by the weight of the stream, alsoper unit length. In surface irrigation flows, with their low Froudenumbers and virtually zero inertia, it represents the slope of thewater surface. It is related to flow velocity V and hydraulic radiusR �cross-sectional area divided by wetted perimeter� through the

Fig. 1. Simulated post-irrigation distribution of infiltrated water asinfluenced by levelness of field surface �after Clemmens et al.�1999a��

Chézy C, defined in

526 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE


Sf =V�V�C2R

�1�

in which the absolute value provides the correct sign for Sf if Vturns negative in a flow reversal.

While calculating cross section geometry is for the most partstraightforward, Strelkoff and Clemmens �2000� note that thecommon assumption, that area and wetted perimeter �or hydraulicsection� both follow empirical power laws of depth, is inherentlycontradictory. They show it can be a source of error on the orderof 5–20% over the range of depths encountered in an irrigation.Alternate formulations are suggested for controlled accuracy.

Boundary Drag

At the high Reynolds numbers and relative roughness encoun-tered in surface irrigation, the drag is nearly independent of Rey-nolds number. Soil-surface drag then depends primarily onrelative roughness. The widely used Manning formula

C = cu

R1/6

n�2�

assumes that n is a measure of absolute roughness-element size�measured in m1/6�, with the formula itself accounting for relativeroughness. The units coefficient, cu, is 1.0 m1/2 /s in the metricsystem and 1.486 ft1/3 m1/6 /s in the English system, the mixedunits allowing n to have the same numerical value in either sys-tem. Devised without the benefit of any theory of resistance toturbulent flow, the formula, in order to correctly predict C whenflow depths vary over a significant range on a very rough surface,requires n to vary with R, e.g., a power-law variation with depth�Bassett and Fitzsimmons 1976�. More theoretical studies �Sayreand Albertson 1961� in very rough open channels, followingColebrook and White’s pipe-flow analyses with equivalent sand-grain size k, led to the logarithmic formula

C = 6.06�g log10R

��3�

in which �=absolute roughness, measured in meters or feet andg=ratio of weight to mass. Fig. 2 shows a comparison of theManning and Sayre and Albertson formulas for a case of highroughness �U.S. Army Corps of Engineers 1961�. The small varia-

Fig. 2. Variation of Manning n and Sayre-Albertson � with depth ina rectangular very rough open channel with 19-mm �3/4-in.� alumi-num angles mounted across the flow, 152 mm �6 in.� apart on flumesides and bottom—computed from measured discharge and depth.Width, B=1.22 m, bottom slope, S0=0.005 �physical data from U.S.Army Corps of Engineers �1961��.

tion of the Sayre and Albertson �, when back calculated from

/ SEPTEMBER/OCTOBER 2009


measured C, compared to the large variation in n as the relativeroughness changes, testifies to the advantages of the Sayre andAlbertson formula in this case. The impact on irrigation-performance prediction of the error in characterizing boundaryresistance is not known, and both the Manning and Sayre-Albertson formulas are in WinSRFR, for future study of this ques-tion.

To allow estimation of � from direct, in-the-dry measurementof absolute-roughness, extensive laboratory and field comparisonsin the mid 1960’s �Kruse et al. 1965; Heermann et al. 1969;Heermann and Evans, “Estimate of resistance from roughnessmeasurements,” unpublished report, 1969� of measured surfacemicrogeometry and hydraulically determined values of � led tothe approximate correlation between � and the standard deviationof surface elevations � in the microgeometry

�F = 12.9�F1.66; �M = 28.3�M

1.66 �4�

the subscripts F and M referring to feet and meters, respectively.Relative roughness can be expected to vary with distance down afurrow as the cross sectional dimensions vary, but there are indi-cations that absolute roughness varies also �Walker and Kasilin-gam 2004�. Absolute roughness can be expected to vary with timeas well, as soil clods are melted by the flowing water. This canlead to flow depths behind the advancing water front considerablylarger than at the upstream end of a furrow �Clemmens et al.2001�

Vegetative Drag

In grass waterways, the grass typically bends into increasinglyprone positions with increasing flow velocities. In surface irriga-tion, plant parts tend to be stiffer and flow velocities lower suchthat they act more like rigid bodies. Thus earlier, empirical studieson grass waterways are supplanted by more fundamental, physi-cally based approaches �Kouwen et al. 1981; Rahmeyer et al.1999�.

Form drag on vegetation penetrating the flow is distributedover the entire depth of flow and is the sum of the drag forces onthe individual plant parts. For a plant element of area �Ap pro-jected on a plane perpendicular to the flow velocity �cross sec-tion�, the drag force is �FD=CD�Ap ·�V2 /2, in which the dragcoefficient CD is close to 1, and � is water density. With all veg-etation elements within a unit surface area �unit length times unitwidth� projected onto a flow cross section, the volumetric densityof projected area, ap, is defined such that the projected area perunit length of stream dAp /dx is the area integral of ap over thestream cross section Ay. Then, with the vegetative drag per unitlength �gAySfVeg, the friction slope chargeable to the vegetation is

SfVeg = CD

V2

2g

1

Ay�

0

Ay

apdA

With uniform density

SfVeg = CDap

V2

2g�5�

The preceding pertains primarily to plant spacings independent ofdirection �see Two-dimensional considerations, next�. In the espe-cially simple laboratory studies with vertical wires of diameter D,placed in the flow at a density of M “stems” per unit plan area,ap=MD. If, instead, it is assumed only that plant density is uni-form over the planted area and that each plant is identical, then ap

in the first of Eqs. �5�, independent of transverse and longitudinal



coordinates, would be a function only of height � above the soilsurface, e.g., in a border strip.

Evaluation of vegetative density ap can proceed either geo-metrically �e.g., optically�, or hydraulically, i.e., calculating back-ward from measurements of friction slope �or bottom slope inuniform flow�, average velocity, and flow depth. The hydraulicmeasurements necessarily include the soil-surface drag, a signifi-cant component if the vegetation is sparse, and which must beestimated, to isolate ap. Characterizations of ap�� can be foundin Dudley et al. �1996� and Dunn et al. �1996�.

The drag coefficient CD is influenced by the screening effectsof plant parts in tandem, as well as by modification of the flowpattern from lateral crowding of stems and leaves. The deforma-tion of plants by the flow can affect CD as well as ap. Theseeffects are typically not currently included in theoretical determi-nations of drag of submerged vegetation.

A dense, submerged growth makes the velocity distribution inthe vertical more uniform, and hence, at the boundary, leads to asteeper gradient and greater shear. Furthermore, the nonunifor-mity in velocity distribution induced by wall shear affects thedrag on the plant parts as a function of depth. But in a firstapproximation, the drag of the soil surface and that of the plantparts can be considered independent. Then the two components ofdrag can be individually calculated and simply added.

Neither the soil-surface �, nor the vegetative ap are in commonuse today for surface irrigation simulation or design. While bothcan be established hydraulically in any given flow, there is nobody of experience suggesting appropriate values a priori. Thecommon measure of resistance is, currently, the Manning n in allcases. The Natural Resources Conservation Service �NRCS� in itsdesign manuals has suggested values for different border-stripconditions.

While some of these theoretical considerations have been pro-grammed into simulation models, their significance to the accu-racy of performance prediction is not generally known. Foradditional information, the reader is referred to the extensive bib-liographic review of Maheshwari �1992�.

Two-Dimensional Considerations: Sf as a Vector

In flat-planted basins and other systems in which the flow is notconfined by furrows, the flow direction is not known a priori andcan assume any orientation. If the resistance is independent ofdirection, the drag-force vector is always opposite the flow-velocity vector. Then the Chézy C is a scalar and has no direc-tional attributes. But if drag is anisotropic, say in the presence ofcorrugations or if drilled grain is grown, with greater resistance toflow across the rows than along them, then the Chézy C must beviewed as a tensor, and Sf and the velocity vector are no longernecessarily collinear. Fig. 3 illustrates the relationships, with thelocal unit flow rate vector q �velocity vector times depth� orientedat some angle �1 to the coordinate system. The concomitant headgradient �h, drag FD, and friction slope S f are collinear with qonly in the isotropic case �Strelkoff et al. 2003�.

Infiltration

Probably the most important field factor in the surface irrigationprocess and also the one most difficult to quantify is the volumeof infiltration per unit stream length, Az. In principle, it is possibleto solve the Richards equations of unsaturated flow in the soil
medium with given boundary geometry and initial and boundary


conditions, but even this theoretical approach requires empiricalsoil data relating moisture content to piezometric head and hy-draulic conductivity. Moreover, in natural soils, surfaceconditions—cracking, sealing, consolidating when wetted, etc.—often time dependent, can be a major factor, as can other inho-mogeneity and anisotropy. Furthermore, the task of coupling thesubsurface and surface flows for simultaneous solution in a singlecomputer model remains formidable �Abbasi et al. 2003; Zerihunet al. 2002; Skonard 2002; Tabuado et al. 1995�. All of thesesuggest a more empirical approach.

Infiltration has traditionally been measured at discrete pointsin a field—with ring infiltrometers, or short sections of a furrow.Accuracy of measurement is a concern, especially as infiltrationrates change rapidly after water is first introduced to the soil.Spurious infiltration into the dry soil surrounding the measure-ment site has led to recommendation, for example, that rings bebuffered by irrigation water. But at best, a measurement applies tothat point alone. Spatial variability can be of such magnitude thatappropriate field-averaged values are not easily deduced. Tech-niques have arisen that use instrumented test irrigations to deducelikely values of the extant field parameters. The tests can be ascrude as pressing a metal rod into the soil after an irrigation todetermine the depth of penetration, or as complicated as measur-ing water-surface profiles during the irrigation along with the in-flow and outflow hydrographs �e.g., Strelkoff et al. 1999�. Indirectdeterminations �“backing into” the field parameters�, from com-parisons of simulation results and measurements of advanceand/or recession and runoff, are popular because the data are rela-tively easy to collect and the results represent field averages insome sense.

The major issues are the infiltrated depth �volume per unitinfiltrating area� as a function of time and the inclusion of furrowwetted perimeter in calculations of the infiltrated volume per unitlength of furrow. The effect of successive periods of dewateringand rewetting are important in surge flow and, potentially, withsome other inflow-hydrograph patterns, such as cutback flow orcablegation. Again, the issues are appropriate functional form,estimation of the parameters and their spatial and temporal vari-ability, and the effect of error on performance prediction, and

Fig. 3. Relationship among friction slope, discharge vector, and headgradient under �a� isotropic; �b� anisotropic roughness conditions �x1,x2: orthogonal coordinate directions in horizontal plane; �1: angle ofvelocity vector with x1 axis; q: unit discharge vector; S f: frictionslope vector; FD: drag-force vector; �h: gradient of water surface�

hence, design.



Cumulative Infiltration as a Function of OpportunityTime

In border strips, infiltration is essentially vertical and almost in-dependent of the hydrostatic pressure variations caused by vari-able depth and, so, if one discounts soil spatial variability, it isessentially a function of infiltration time alone. The observed fea-tures of this function are a very large rate of infiltration uponinitial wetting, followed by a decrease in rate, typically approach-ing a constant at large infiltration times. The three-parameter Hor-ton formula for infiltration rate, I, developed in rainfall/runoffhydrology and not fitting most agricultural soils very well �Philip1957,1969�, postulates an initial infiltration rate f1, a final rate f2,and an exponential transition from one to the other with inunda-tion time �, I= f2+ �f1-f2�e−r�, in which r is a third empirical con-stant governing the rate at which the transition between the twoinfiltration rates takes place. Theoretical studies based on the Ri-chards equation for unsaturated flow �e.g., Philip 1957,1969�,supported by laboratory experiments with homogeneous isotropicporous media, led to a two-parameter formula for cumulative in-filtration

z = S�1/2 + A� �6�

exhibiting initially an infinite rate, a feature missing in the Hortonformulation �Philip cautions that the term A in Eq. �6� is not thelong-term final infiltration rate—saturated hydraulicconductivity—but merely the second term of a truncated seriesintended for small to moderate inundation times�.

In certain circumstances �typically, with cracked vertisols� asignificant volume is infiltrated as soon as the water is madeavailable at the surface. A general formula exhibiting observedfeatures is a modification of the Kostiakov power law forcumulative-infiltration depth

z = k�a + b� + c �7�

in which k, a, b, and c are empirical constants. Convenient unitsfor k are mm /ha, since k then represents the cumulative infiltra-tion in the first hour.

Theoretical justification for this empirical formulation �with-out the c term�, as well as quantification of the effects of initialwater content and surface-water depth, was provided by Furmanet al. �2006�, who solved the Richards equation for vertical infil-tration numerically �Šimůnek et al. 1998� for a range of soil types,initial water content, and surface depths. The numerical experi-ments yield the pertinent Kostiakov parameters by fitting. Theabsence of a c term is unnoticed, as the theoretical soils exhibit nocracking.

The formula, Eq. �7�, with the constant c deleted, is sometimesmisused, for cracking, swelling clay soils, by specifying a verysmall value of exponent a, well under 0.1; a better representationof physical reality would be a nonzero value of c. Furthermore,numerical simulations of irrigation-stream flow with a highlynonlinear infiltration function are troublesome, as the computa-tions attempt to track the sharp “dog leg” in the infiltration pro-file. Similarly, an observed relatively large infiltration rate at largetimes is often described by a large value of a in the basic formula.A better physical representation is achieved by incorporating anonzero value of b and a moderate value of a, closer to Philip’s0.5 theoretical value.

The convenient and practical Kostiakov-Clemmens branchfunction �Kostiakov �1932� and Clemmens �1981�, respectively�,avoids the slow, very gradual approach to the basic infiltration
rate of Eq. �7�, branching instead, from the power-law monomial


k�a to the constant final rate b, at that inundation time tB at whichthe monomial form yields the same rate. With the addition of theconstant term c, this is

z = c + k�a, � � �B; z = cB + b�, � �B �8�

in which

�B = �ak

b�1/�1−a�

cB = c + k�Ba − b�B

The modified Kostiakov equation and the branch function can bemade to exhibit similar infiltration rates and cumulative infiltra-tion, as shown in Figs. 4 and 5, respectively, for the Flowellnonwheel furrow data from Walker and Humpherys �1983�, inwhich c=0. The parameters selected for Branch function 2 pro-vide a good fit to the modified Kostiakov equation during the first24 h, but the two increasingly diverge at greater times. If one usesthe same parameters for the branch function as were used in themodified Kostiakov function, the results are quite different�Branch function 1�. Although both have the same basic infiltra-tion rate, this is achieved very much later in the modified Kostia-kov formulation. Figs. 4 and 5 show times beyond the practicalrange to underscore the theoretical trends.

With some soils, and, especially, cracking soils, the entire firstbranch can be effectively assumed to take place in zero time�relative to the total time of simulation�, leading to an exception-ally simple one-branch infiltration formula consisting of an initialdepth upon wetting, followed by a constant rate

z = c + b� �9�

�Collis-George 1974; Mailhol and Gonzalez 1993; Mailhol 2003�.In 1974, the NRCS �USDA Soil Conservation Service 1974�

proposed a characterization of infiltration in borders and basins bymembership in a family. The concept implies that the curve ofcumulative infiltration versus time for any soil at any initial mois-ture content will fit within one of the regions, or families, intowhich the z-� plane is divided. Each family is characterized by afunction of the form z=k�a+c, passing through the middle of theregion, and is named for the long-time final basic infiltration ratefor the soil �in inches per hour�, although the form itself does notsupport a final rate. The c-value in the functional form �a constant

Fig. 4. Comparison of infiltration rates as functions of time withmodified Kostiakov and branch functions applied to data from Walkerand Humpherys, 1983 �dashed line: modified Kostiakov formula,Branch function 1: coefficients same as in Modified Kostiakov for-mula; Branch function 2: best fit to Modified Kostiakov formula�

for all families� does not represent anything physical, but was



introduced to better fit empirical data. The families are repre-sented in Fig. 6, which covers a greater range of infiltration timesand cumulative infiltration than typically encountered in the fieldin order to show the theoretical trends. The nonzero c value isresponsible for the curvature in the lower portions of the curveson logarithmic scales. While many soils fail to fit any of thefamilies and cross a number of them, some are successfully in-

Fig. 5. Comparison of modified Kostiakov and branch function cu-mulative infiltration versus time. Branch function 1: coefficients iden-tical to those in modified Kostiakov formula; Branch function 2: bestfit to modified Kostiakov formula. �a� arithmetic scales; �b� logarith-mic scales.

Fig. 6. NRCS �SCS� infiltration families �z: cumulative-infiltrationdepth—volume per unit infiltrating area� �from NRCS/USDA �SCS��1984��



corporated within the Soil Conservation Service �SCS� groupings.If experience justifies describing a soil in this way, the familiesallow a simple characterization of infiltration, with prescribed val-ues of k, a, and c.

Merriam and Clemmens �1985�, arguing that the time requiredto infiltrate a desired depth of water is the key infiltration param-eter for surface irrigation management, built upon the NRCS con-cept of infiltration families—characterized by a generalcorrelation between the Kostiakov k and a—and developed thetime-rated intake families. They examined cumulative-infiltrationdata for many soils and, excluding the cracking clays, empiricallyfound a correlation between the exponent in the Kostiakov for-mula and the time �100 �in hours� to infiltrate a characteristicdepth of, specifically, 100 mm, namely

a = 0.675 – 0.2125 log10��100 �10�

Thus, for noncracking soils, a single measurement, the time toinfiltrate 100 mm, estimates both k and a in the Kostiakov powerlaw. Fig. 7 shows the time-rated families, each family named for�100.

Fig. 8 contrasts the interdependence between k and a in thetwo sets of families. Figs. 6–8, by comparing the two, show that

Fig. 7. Merriam and Clemmens �1985� time-rated intake families.The time to infiltrate 100 mm is shown on each curve �hours� �z:cumulative-infiltration depth—volume per unit infiltrating area�.

Fig. 8. Relationship between Kostiakov k and a in the NRCS/USDA�SCS� �1984� intake families and the Merriam and Clemmens �1985�time-rated families. The squares identify family members, while thecurves that fit them are identified by specific functions, given in thecited references.



both sets of families cover roughly the same region of the �, zplane, but with the NRCS families including lower k values. Thetime-rated families exhibit a greater variation of a, dropping tolower values in the tighter soils than the NRCS families. Thecurious crossing of the time-rated families pictured at the lowerend of Fig. 7 stems from extending an empirical expression be-yond its intended range. Current research is aimed at relatingthese purely empirical families to soil texture and other physicalcharacteristics.

As with other field properties, the key issues in the empiricalestimation of infiltrated depth are the selection of a suitable for-mula, estimation of its parameters and their spatial and temporalvariation, the likely error, and the effect of that error on simulatedperformance. Different formulations and parameter value combi-nations will give similar results in simulation, if the infiltratedvolume at the average opportunity time is similar.

Influence of Wetted Perimeter

With all infiltration passing through the irrigation-stream/soil-surface boundary, wetted perimeter plays a role in determiningthe volume infiltrated per unit length of run. In border strips andbasins, infiltration into the berms is typically negligible comparedto that into the border-strip surface. Wetted perimeter is then es-sentially independent of flow depth and is known; volume infil-trated per unit length is directly proportional to the width. Infurrows, on the other hand, wetted perimeter varies significantlywith flow depth. During typical stream advance, runoff, and re-cession, the depth and wetted perimeter at any section of a furrowrise rapidly when the stream arrives, hold more or less steady, andeventually, after cutoff, return to zero. During runoff, dischargeand flow depths at the tail end of a long furrow can be signifi-cantly smaller than at the head end. Under cutback or cablegationinflows, additional changes in wetted perimeter occur because ofchanges in flow rate. In the early stages of wetting of a furrowsection, after the stream arrives, infiltration through the furrowwall is entirely dependent on the wetted perimeter and the soilproperties—which may themselves change upon contact with theirrigation water and exhibit some surface sealing, or crusting.Infiltration is not, at first, dependent on coincident infiltrationfrom neighboring furrows, but later, as the wetted soil regionsaround each furrow expand and approach each other, lateral mois-ture flow is inhibited, and furrow spacing becomes an importantfactor. In the event of extensively cracked soils, the transversecracks quickly convey infiltrated water �water that has crossed thenominal furrow boundary� laterally to meet with water spreadingfrom the neighboring furrow. In this case, the effective wettedperimeter for infiltration is the furrow spacing from the start.

In contrast to the previous theoretical considerations on therole of wetted perimeter in furrow infiltration, a recent review ofextensive field data �Walker and Kasilingam 2004� indicated thatmeasured wetted perimeter failed to show any consistent variationwith either time or distance during the course of the irrigations.Expected decreases in wetted perimeter with distance down thefurrow in many cases failed to materialize, possibly as the resultof significantly greater measured roughness in the lower reachesof a furrow and erosive changes in the furrow-boundary geom-etry. These circumstances lead to doubts that incorporating theo-retical changes based upon the assumption of constant crosssections �in the dry� in the course of a simulated irrigation pro-vides any enhancement of accuracy.

Nonetheless, despite the incompleteness of theoretical descrip-
tions of furrow infiltration, something can be learned from solu-


tion of the Richards unsaturated-soil water flow equations inhomogeneous isotropic soils. Infiltration from furrows is, strictlyspeaking, a three-dimensional phenomenon, as some longitudinalsoil-moisture movement occurs during an irrigation. However,much of the phenomenon is captured by viewing it in two dimen-sions, in a plane normal to the furrow length.

The two-dimensional version of the HYDRUS software�Šimůnek et al. 1999�, with coordinate directions vertical andlateral, transverse to the furrow length, provides numerical solu-tions to the Richards equation for variably saturated water flow

��m

�t=

�

�xiKij� �hm

�xj+ �iy�� 11�

in which �m=volumetric water content �L3L−3; hm=pressurehead �L� in the soil matrix; xi �i=1,2�=spatial coordinates �L�;t=time �T�; Kij =components of the unsaturated hydraulic con-ductivity anisotropy tensor �LT−1 �a simple scalar in the casesdescribed here�, and �iy =Kronecker delta with y the vertical di-rection �equal to one if i corresponds to the vertical and zerootherwise�. Unlike the sudden step change at the wetting frontassumed for development of the Green-Ampt equation, the soilwater content changes gradually, with soil suction and hydraulicconductivity both a function of water content. The Richards equa-tion assumes that the porous medium is rigid and that the airphase plays an insignificant role in the liquid flow process.

In a special version of HYDRUS-2D, the assigned boundaryconditions in the furrow were made to change dynamically de-pending on the position of the water level in the furrow. Positivehydrostatic equilibrium pressures were assumed below the waterlevel and a potential seepage face was established on the furrowwall above water level. In this variant, water is made available tothe soil matrix at all points of the furrow wall below the freesurface. A seepage face is a special type of a system-dependentboundary condition through which water can flow out of the soilfrom the saturated part of the flow domain �such as during adrawdown of the water level in the furrow�. Without these specialconsiderations, a negative pressure head would be specified onthe boundaries in the furrow above the water level. This type ofboundary condition does not allow controlling the flux throughthe boundary and both inflow and outflow into or out of the soilprofile could occur depending on pressure heads in the soil. Thismodification has subsequently been made a standard feature ofthe HYDRUS-2D upgrade called HYDRUS �2D/3D� and is de-scribed in detail in its technical manual and online help �Šimůneket al. 2006�.

Fig. 9 illustrates the case of steady furrow flow, at a constantdepth of 100 mm, in a trapezoidal furrow formed in a silty loamsoil. The corresponding unsteady flow of soil moisture as calcu-lated by HYDRUS-2D is shown by the distribution of moistureafter 600 and 1,800 min. The rise of moisture above the waterlevel in the furrow as well as lateral penetration into the soil isclearly evident. The character of the solution changes over time asthe initial unfettered penetration of water into the soil in all di-rections is increasingly constrained by the lateral advance ofwater from adjacent furrows �modeled in HYDRUS-2D as animpenetrable boundary condition�. This theoretical solution sup-ports the observation by the NRCS/USDA �SCS� �1984�, quanti-fied in a subsequent section of this paper, that in furrows, theinfiltration per unit area of infiltrating surface is greater than in aborder strip.

This phenomenon has led to two schools of thought emerging
around the empirical assessment of furrow infiltration. Some


models of the surface irrigation process �e.g., SIRMOD, Walker�2004�� expect as data input the parameters of empirical infiltra-tion formulas directly expressing cumulative volume per unitlength of furrow, AZ, as a function of time �in the literature, this isoften given the symbol, Z, with the corresponding formula param-eters then, K, a, B, and C; the use of AZ here underscores the factthat the units of volume per unit length are area units, i.e., lengthsquared�. This implies that at some point in time, inflow andoutflow measurements were taken in the furrow to yield a timerate of growth of infiltrated volume, thus incorporating the rami-fications of whatever wetted perimeter was present when inflowand outflow were being measured. Infiltration-formula parametersare often obtained by measuring the increasing infiltrated volumein a section of furrow containing water at some constant depthand then fitting the per-unit-length data with the formula. Thus,the measured infiltration is viewed as a consequence of both soilproperties �including initial moisture content� and the details ofthe irrigation system �furrow size, shape, spacing, flow depth,etc.� But how these data should be adjusted to apply to water at,say, a different depth—a different wetted perimeter—even a con-stant one, is not obvious �however, see Alvarez �2003� and Langatet al. �2007� for a suggested approach�.

In models from the other school of thought, such as SRFR�Strelkoff et al. 1998�, or in the furrow design procedures of theNRCS/USDA �SCS� �1984�, the infiltration-data input is consid-ered a function of soil properties alone, with the contribution ofthe irrigation system then calculated by the model. The data input,then, consists of formula parameters for volume infiltrated perunit infiltrating area �z, cumulative depth of infiltration�, basically,one-dimensional �1D� infiltration, as in a basin. The volume in-filtrated per unit length of furrow, AZ, is then calculated for thevarying flow depths and wetted perimeters of the simulated irri-gation stream—in the case of SRFR, through a convenient butheuristic algorithm �A. J. Clemmens, personal communication,March 2, 1990� described in the following paragraph.

Accounting for Wetted Perimeter

It is assumed that the infiltration rate pertinent to the entire cur-rent wetted perimeter P of some cross section in the flow is given

Fig. 9. Cross section of furrow and surrounding silt loam soil. Con-tours of moisture content computed by HYDRUS-2D �Šimůnek et al.1999� stemming from a constant 100-mm depth of irrigation water inthe furrow. Initial moisture content: 0.24; saturated: 0.45. �a� time=600 min; �b� time=1,800 min.

by an empirical function of the time of wetting of the furrow



bottom, �. This is simply current time minus the stream-advancetime to the given location. The rate of growth of AZ reflecting thisinfiltration rate, i, is then

dAZ

d�= i · P �12�

This expression does not account for the possibility of a constantterm c in the formula for cumulative depth z reflecting essentiallyinstantaneous accumulation of water in cracks as the irrigationstream arrives. Clemmens �personal communication, 1990� addsan additional term to the right hand side of Eq. �12�, as a reason-able description of such a cracking soil

dAZ

d�= i · P + c

dP

d�for

dP

d� 0 �13�

For example, with an extensively cracked tight clay soil, whichexhibits virtually no additional infiltration once the cracks havebeen filled, Eq. �12� would predict no infiltration at all as thedepth and wetted perimeter at a furrow section increase with time,while Eq. �13� would respond reasonably. However, it must benoted that Eq. �13� can be valid only for increasing depth andwetted perimeter with time—AZ does not decrease at a location asthe irrigation-stream depth there reduces to zero in the recessionprocess. Eq. �13� is the basis for the algorithm incorporated inSRFR, and in WinSRFR when the local wetted-perimeter optionis selected, for dealing with cracking soils.

To simplify the exposition, no further consideration is givenherein to the contribution of a possible constant term in thecumulative-infiltration formula, which is straightforward. Eq. �12�leads to the following expression for the growth of AZ over time

AZ =�0

�

i · P�t�d� �14�

For implementation in a simulation that proceeds over a series oftime steps, the increase in AZ over a time step is

�AZ = AZ2 − AZ1 =��1

�2 dz

d�P�t�d� =�

z1

z2

P�t�dz �15�

in which the subscripts 1 and 2 refer to the beginning and end ofthe time step, respectively. Following the rules for integration byparts, we obtain

�AZ = P2z2 − P1z1 −�1

2

zdP

dtdt �16�

With �P= P2− P1, and assumed minor linear variation of z overthe time interval such that

�1

2

zdP � z̄�P =z1 + z2

2�P �17�

it follows that �AZ is given by the increase in depth �z multipliedby the average wetted perimeter over the time step, i.e., combin-ing Eqs. �16� and �17�

�AZ = �zP1 + P2

2�18�

Thus, the SRFR algorithm operates on the data describing 1Dinfiltration to allow it to be used to approximate the results of anessentially two-dimensional process. In the following, this ap-
proach is compared with theoretical solution of the two-


dimensional Richards equation applied to a single cross section toyield cumulative volume infiltrated per unit length of furrow as afunction of time �Perea et al. 2003�.

First, an initial HYDRUS-2D run was performed to define the1D infiltration into the soil. In this case infiltration time � issimply equal to total time t. Depth of infiltration z was sought asa function of time t—to be fitted by the modified Kostiakov for-mula

z = kta + bt �19�

For this scenario, in a border strip 1,000 mm wide, the soil wasassumed a silt loam �from a HYDRUS-2D soils database� with auniform initial moisture content �m=0.24. Water depth on theborder strip was assumed constant at h=100 mm. HYDRUS-2Dthen calculated the moisture changes in the soil for a period of 30h. Cumulative flux at the soil surface �in mm2� output byHYDRUS-2D was converted to depth �in millimeters� throughdivision by the 1,000-mm-width of the border strip. Formula pa-rameters, k, a, and b in Eq. �19� were selected interactively. Fig.10�a� illustrates the fit that proved possible for the given condi-tions, while Fig. 10�b� shows the difference between the twofunctions, in millimeters and as a percent. In this case, during thetime period of interest, a fit within 1/2 mm proved possible ev-erywhere, with the values k=18.71 mm /ha, a=0.426, b=3.77 mm /h in Eq. �19� �see also Furman et al. �2006��.

Fig. 11 shows the results of the SRFR algorithm in a trapezoi-dal furrow with base width 100 mm and side slopes 1/1 H:Vcompared to HYDRUS at two different spacings, 1,000 and 500mm. Initially, the SRFR algorithm fits the HYDRUS results, butat large infiltration times it inexorably underpredicts the infiltra-

Fig. 10. Modified Kostiakov formula fitted to solution of Richardsequation �HYDRUS-2D: Šimůnek et al. 1999� in a border strip �1Dflow�. Silt loam soil at initial water content of 0.24. �a� fit; �b� error.

tion, the effect being greater at the wider spacing.



A modification is suggested by the SCS �NRCS/USDA �SCS�1984�, which has long recognized increased infiltration per unitwetted perimeter of furrows as compared to borders. In theirmanual on furrow irrigation, they propose an “empirical” wettedperimeter, Pe, for calculating volume infiltrated per unit length offurrow from the infiltration families, which define volume infil-trated per unit area of infiltrating surface. This nominal wettedperimeter consists of two parts: an expression, PGe, representingan approximate fit to geometrical wetted perimeters at normaldepth for the inflowing discharge over a range of trapezoidal crosssections, plus a constant P=0.7 ft �0.213 m� to account for“horizontal intake” from the furrow. The original SCS expressionin feet and gallons per minute is

Pe = PGe + P = 0.2686�Qn

S00.5�0.4247

+ 0.0462 + 0.7 �20�

Fig. 11. Trapezoidal furrow, base=100 mm, side slopes=1 /1 H /V,depth constant at 100 mm. Silt loam soil, initial moisture content of0.24. HYDRUS-2D �Šimůnek et al. 1999� solutions with furrowspacing=1,000 mm �W�; 500 mm �N�. Calculated by SRFR algo-rithm �A—dashed�.

Fig. 12. SCS �NRCS/USDA �SCS� 1984� empirical wetted perim-eter, Pe, for furrow infiltration contrasted with geometrical wettedperimeters, PG, for base widths of 0, 100, 200, and 300 mm and sideslopes of 0, 1, 2 �H/W�. PGe �shown as dashed� is Pe minus theadditional 0.7 ft �213 mm� for horizontal intake. The two abscissascales are equivalent. Q=flow rate, n=Manning n, cu=units coeffi-cient, S0=bottom slope; and Ay =cross-sectional flow area, R=hydraulic radius, both at normal depth.



in which Q=inflow rate �gpm� and S0=bottom slope. This expres-sion is plotted in Fig. 12 in terms of the equivalent AyR

2/3. As canbe seen in Fig. 12, the SCS single formula for PGe does a fair jobof approximating geometrical wetted perimeters over a range ofbase widths and side slopes �contrary to an assertion in Strelkoff�1992�, that the error can exceed 100%—the result of an error incalculation�.

The SRFR algorithm, with its true geometrical wetted perim-eter, PG, was modified, following the NRCS concept, by addingan increment P to PG. The P in the modified SRFR algorithmwas then adjusted to fit HYDRUS-2D results for cumulative in-filtration. The aim was to select such a value for P as wouldyield nearly identical results for AZ�t� in the practical range ofinfiltration times. The results, with two values of furrow spacingand two values of P in contrast to those in Fig. 11 for whichP=0, are shown in Fig. 13. A little numerical experimentationwith P showed that the trend of the algorithm-generated AZ�t� iswrong and that changing P does not correct that. Clearly, theselection of an optimum P in this approach depends on whichportion of the HYDRUS-2D curve one wants to fit. Furthermore,values of P considerably smaller than the NRCS value of 213mm suffice for any part of the time range up to the 30 h investi-gated. For the 1-m furrow spacing, P=157 mm provides a fit at30 h, with an error then of but 1 mm; however, that incurs anerror of 4 mm at around 7 h. A smaller value, P=85 mm, pro-vides a good fit at 30 h for a 1 /2-m furrow spacing, with errorsunder 1 mm everywhere in the 30-h period. But at the 1-m spac-ing, past 11 h, the errors exceed this value, and at 30 h comprise15 mm.

Additional examples �including time-variable flow depthssimilar to those in an actual surface irrigation hydrograph� and thesuggestion that it may be more profitable to change k for furrowsrather than P can be found in Perea et al. �2003�. What is ab-solutely clear is that the selection of numerical values of k, b, andc for entry into simulation or design software depends on thewetted-perimeter assumption made in the software. In particular,for a given soil, the k, b, and c values will be less if furrowspacing is taken as the nominal wetted perimeter than if the latteris calculated locally or approximated by wetted perimeter at theupstream furrow end or by wetted perimeter at normal depth

Fig. 13. Trapezoidal furrow, base=100 mm, side slopes=1 /1 H /V, at constant depth=100 mm. Silt loam soil; initial mois-ture content of 0.24. Furrow spacing=1,000 mm �W�; 500 mm �N�.Infiltration calculated by modified SRFR algorithm with P=157 mm �A1�; P=85 mm �A2—shown dashed�.



there. On the other hand, if the simulation software accepts K, B,and C as input instead, the empirical values entered should begleaned from data gathered at wetted perimeters close to those tobe simulated. As an approximation, the modified Kostiakov pa-rameters except for a should be selected inversely proportional tothe wetted perimeter in effect.

Influence of Rewetting on Infiltration

Following recession of water from a furrow, upon rewetting, theinfiltration rate is typically smaller than it would have been hadthe furrow been wet continuously. Various theoretical reasonshave been suggested for this behavior, including redistribution ofsoil moisture after the initial wetting, but a likely significant causeis the tightening of the soil structure near the surface under theinfluence of the negative soil pore pressures as water disappearsfrom the surface �Stringham 1988�. The issue of rewetting arisesalso with surge irrigation and in the case of cablegation or cut-back furrow streams if the reduced flow rate is incapable of main-taining infiltration over the entire length of furrow previouslywetted at the full flow rate. For then the stream contracts, thefront receding back toward the source, until the infiltration ratehas decreased enough to allow the stream to advance once again.

From the standpoint of simulation, some albeit approximatetheoretical estimate of infiltration changes upon rewetting is de-sirable, in lieu of totally new measured values for the constants inthe infiltration equations after a surge, or renewed advance �as inWalker and Humpherys 1983�. Various models have been pro-posed �e.g., Blair et al. 1984; Izuno and Podmore 1985; Purkeyand Wallender 1989�. A simulation that calculates AZ by incre-ments in each time step �infiltration rate times extant wetted pe-rimeter times time step� allows easy introduction of the Blair etal. approach to calculate infiltration for successive surges, and, infact, periods of successive inundations and dewatering from anycause. Accordingly, infiltration rate is based on time from initialinundation, while infiltration depth accumulates only during wet-ted periods. More recent investigations �Palomo et al. 1996� sug-gest an alternative, simpler, and more accurate approach for soilsdescribed by the branch function of Eq. �8�. The Kostiakov for-mulation of the first branch, in effect for the first surge, is re-placed for all subsequent surges by the basic infiltration rate, thesecond branch.

Approximate Physically Based Approaches

The shortcoming shared by all of the empirical formulations ofinfiltration is the need for field measurements for each individualcase, even for simple 1D vertical infiltration. Formula parametersfor one level of antecedent soil moisture cannot be rationallyadjusted for use at a different level. The influence of soil texturescannot be assessed a priori, etc. And with the difficulties of incor-porating numerical solutions of the Richards equation into surfaceirrigation simulation models, simplified solutions are in demand.

Midway in complexity between solutions of the Richardsequation and the pure empiricism of the Kostiakov formulationsis the Green and Ampt �1911� equation

z = Ks� + �� ln�1 +z

�� 21�

with Ks=saturated hydraulic conductivity of the given soil; �=difference in volumetric moisture content between the initialmoisture level and saturation; and �=suction at the wetting
front. This physically based implicit equation for cumulative in-


filtration represents an approximate solution to the Richards equa-tion, under the assumption that behind the wetting front, the soilis saturated and that the soil-moisture content exhibits a disconti-nuity across this front. The parameters can be determined fromthe same kind of measurements as required for a general solutionof the Richards equation, but it is also possible to fit the equationto measured infiltration data. As in the Richards equations, noformal recognition is given to possibly significant surface soilconditions �e.g., cracking, consolidation on wetting, surface seal-ing, etc.�

Following Philip’s algebraic simplification of solutions to theRichards equation �Philip 1957,1969�, Warrick and Lazarovitch�2007� and Warrick et al. �2007� developed a theoretical algebraicsolution for infiltration from a saturated strip at the soil surface�based on Philip’s sorptivity�, and thence to an approximation ofinfiltration from a furrow. Cracking soils with their stepwise ini-tiation of infiltration were treated by Mailhol and Gonzalez�1993� and Mailhol �2003� on the basis of a simplification of theHorton equation. Additional information on the method can befound in the work by Clemmens and Bautista �2009�.

Irrigation-Induced Erosion

In some areas where surface irrigation is practiced, irrigation-induced soil erosion is significant. Methods for modeling it are intheir infancy, and even the physical processes involved are inquestion. But it seems clear that soil-particle and aggregate sizesand their distribution in the soil mix at the soil/water interfaceplay a major role �Harmon et al. 1989�. So does a related, butseparate, field characteristic, soil erodibility, the rate of entrain-ment of soil particles on a given slope, subject to a clear waterflow of some depth and velocity. This depends on many physicaland chemical soil characteristics, and current approaches to simu-lation of sediment entrainment/transport/deposition call for site-specific empirical determinations �Strelkoff 2009�.

Conclusions

There are four fundamental concerns in the empirical character-ization of field properties• Selecting a suitable mathematical formulation.• Estimating the parameters of the chosen formula and their

spatial/temporal variation.• Likely error of the estimate.• Resulting error in predicted performance, design, and manage-

ment recommendations.The first is the subject of the present paper. While not exhaus-

tive, it has presented many theoretical aspects of field properties.Sorting the available scientific information for those salient fea-tures that have the greatest practical bearing on surface irrigationperformance is relegated to the remaining papers in this specialissue.

In addition, once the four fundamental issues have been ad-dressed, the practical use of the collected data would be encour-aged by the gradual development, over the course of a growingseason or several seasons, of local databases of field infiltrationand roughness. These could be used subsequently for circum-stances similar to those from which the data stemmed—stage ofplant growth, irrigation number, wheel row/nonwheel row,etc.—as input into simulation or design programs for improving
performance.


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