Water Infiltration and Runoff under Rain Applications1

AHMED Y. HACHUM AND JOSE F. ALFAROZ

ABSTRACT intensity is not efficiently matched with the soil's intakeExperiments to verify the validity and to evaluate the performance properties valuable irrigation water and energy would be

of rain infiltration models based on extending the Green and Ampt lost bv surface runoff and not enough Water would Penetrateapproach are lacking in the literature. There are severe gaps between tne root zone for optimum crop production.theory, validity, and practicality of these models. In this study, a In an irrigation event, the application rate can be eitherphysically-based model for describing infiltration into a homogeneous constant or varied with time. In conventional sprinkler irri-deep stable soil profile with uniform initial water content distribution gation, the application rate is usually considered constantis presented. The derivation of the model is based on the concept of and equal to the average application rate. Center pivotmapping the actual wetted soil zone into an equivalent rectangular sat- sprinkler irrigation produces variable application rate pat-urated zone with constant effective hydraulic conductivity and capil- tems Because of the geometrical characteristics of this sys-lary pressure head at the abrupt wetting; front The model takes into and ide uniform d h f app,ication> the appli.account the intial water mobility in the sod profile and predicts the sys- . . , ,• ,- , .tern response under any rainfall pattern. The proposed model is fur- Catl0n rat" lncreaSC 3S *f. d!StanCC fr°m the .Plvot

ther simplified, for practical purposes, by grouping three basic input increases. Moreover, the application rate, at any point onparameters into a single parameter, the Surface Saturation Index. the soil surface, vanes continuously with time during appll- -

The validity of the model was tested and strongly supported by the cation. Addink et al. (1975) studied the problem of runoffresults of laboratory experiments conducted on samples of three soils, under center pivots by using a numerical solution of thetwo silt loams and a silty clay loam, under five different (variable and general flow equation (Richards' Equation) along with aconstant) water application rate patterns. It was found that the laboratory experiment. Tfe»y found that forward-skewedamount of runoff from a nearly symmetrical convex variable rainfall application rate patterns have potential for reducing runoffpattern was very close to that which resulted from its equivalent b n% QJ. mQK Jn ̂ ̂ ^ ̂ laborat patterns wereaverage constant application rate pattern, regardless of the significant stairstepped rather than smooth curveS as encountered in thedifferences between the times at which runoff begins and the charac- , jj . ,teristics of infiltration rates thereafter. actual field Center Plvot Pattems-

A review of literature revealed that there are many valu-Additional Index Words: incipient ponding, Surface Saturation able theoretical attempts to analyze the rain infiltration

index, wetting pressure head, wetting front, sprinkler irrigation, problem. Most attempts are either empirical (Kincaid et al.,physically-based models, modeling infiltration, rainfall pattern shape, 1969; Y. Z. El-Shafei, 19703; and M. Neyestani, 19684) orrainfall infiltration, initial water content, rainfall simulation. highly complicated (Rubin and Steinhardt, 1964; Hanks et

________________ al., 1969; Whisler and Klute, 1967). Also among these at-tempts are physically-based models (Mein and Larson,

PREDICTING THE RESPONSE of a soil profile under rain 1973; Farrell and Larson, 1972; Swartzendruber, 1974)applications is of great importance in irrigation and based on the "piston displacement" concept of Green and

hydrology practices. In irrigation, if the water application Ampt (1911).

'Contribution from the Utah Agric. Exp. Stn. (Project 591), Utah State 3Y. Z. El-Shafei. 1970. A study of flooded and rain infiltration relationsUniv., Logan, UT 84322. Received 13 Dec. 1976. Approved 25 March with surface ponding. Ph.D. Thesis. Utah State Univ., Logan.1977. <M. Neyestani. 1968. The effect of water application rate on infiltration

'Postdoctoral Fellow, and Associate Professor, respectively, Dep. and wetting front characteristics of unsaturated silt loam soil. Ph.D. The-Agric. and Irrig. Eng., Utah State Univ., Logan. sis. Utah State Univ., Logan.

HACHUM & ALFARO: WATER INFILTRATION & RUNOFF 961

CONSTANT INFLOWRATE

CONTAINER -I

Table 1—Summary of soil analysis and intial conditions.

CONTAINER-2

AIR VENTn

FLEXIBLE TUBING

RAINAPPLICATOR

\

-SWIVELCONNECTION

CONTROLPOINT

SOIL COLUMNCONTROL-

POINT

Fig. 1—Schematic diagram of the basic components of the rainfallsimulator used in the study.

Soil

Millville silt loam (MSL)Nibley silt loam (NSL)Nibley silty clay loam (NSCL)

Sand Silt Clay k(6s)-\ Ss 6;§ pb

hour30.0 53.0 17.0 4.2 0.47 0.23 1.385.0 70.0 25.0 12.0 0.52 0.26 1.259.0 43.0 48.0 22.0 0.55 0.31 1.20

t Average of three tests for each soil.§ k(6i) for all soils ̂ O.

hydraulic conductivity of the soil, and tyw = the capillary pressurehead (suction) at the equivalent wetting front.

From the continuity principle, z can be defined as:

where W(tJ = D(tg) ~ ts k(0t) = the net depth of rain water infil-trated and stored in the equivalent wetted soil zone of depth z,D(tJ = the cumulative depth of rain applied and infiltrated in thesoil until the moment of surface saturation, k(6i) = the hydraulicconductivity of the soil at 0,, and A0 = 0S - 0t, where 0S = the sat-urated water content of the soil.

By combining Eq. [1] and [2], the following equation thatdescribes the system behavior at the moment of surface saturationcan be obtained:

R(t.) 1]. [3]

Few models have shown a success when subjected to lab-oratory and/or field experimental tests. None of the physi-cally-based models have been subjected to field or labora-tory test. Experiments to verify the validity and evaluate theperformance of rain infiltration models are notably lackingin the literature. There are severe gaps between theory, va-lidity, and practicality in this field of study. The objectivesof this work are to develop a physically-based model and toexperimentally test its performance for different soils andrain conditions.

THEORETICALWater infiltration under a relatively high rain intensity, lasting

for a relatively long time, can be divided into two major stages:before and after surface saturation. In the "pre-surface saturation"stage, the infiltration is governed by the rainfall characteristics.Therefore, during this stage, the infiltration rate equals the instan-taneous rainfall rate. Once surface saturation is reached, a floodtype of infiltration takes place. After surface saturation time,runoff becomes potential because the infiltration rate drops belowthe application rate; thus, infiltration during the second stage or the"post-surface saturation" stage is controlled by the characteristicsof the soil profile.

Consider a deep, homogeneous, and stable soil profile with auniform initial water content, Ot, sprinkled by a variable flux rain-fall pattern in which the application rate R(t) is a function of time,t. It is assumed that the kinetic energy of the falling raindrops isnegligible and that one dimensional downward flow exists in thesystem. The "piston displacement" concept is based on mappingthe actual wetted soil zone into an equivalent rectangular zone ofwidth Os and depth z, with constant effective hydraulic conduc-tivity and capillary pressure head at the equivalent wetting front.Based on these assumptions, Darcy's equation can be used todescribe the system behavior at the instant of surface saturation:

R(ts) = [1]

where ts = the time of surface saturation, R(ts) = the instantaneousapplication rate at time of surface saturation, K(0S) = the saturated

For a given soil profile with certain, 6t, Eq. [3] can be expressedin general as:

W ( t s ) = f 1 [ R ( t s ) ] . [4]

For a given rainfall pattern, W(t) can be expressed at any time,t •& t., as follows:

W(t)= f R(t)dt-tk(6t)=f2[R(t)lJ o

[5]

The moment of surface saturation, ts, is defined as the intersec-tion of the two functions given in Eq. [4] and [5].

In this analysis, tyw is defined as given by Mein and Larson(1973) with a slight modification to take into consideration the ef-fect of 0f on Vw:

[6]

where kr = the relative hydraulic conductivity, and ty = capillarypressure head function of kr.

The infiltration rate / after surface saturation is given by:

[7]

in which D = cumulative depth of infiltration.In Eq. [7], / can be replaced by dD/dt. Then, the resulting dif-

ferential equation can be solved subject to the condition that at t =ts, D = DftJ, The final equation will be:

in which Dr = and

[9]

SOIL SCI. SOC. AM. J . , VOL. 41, 1977

O 8 16 24 32 4OTIME - minutes

Fig. 2—Rainfall intensity patterns studied.

Therefore, for a given set of conditions described by tyw, A0,k(6s), fc(0;), Dr and ;5, Eq. [8]can be used in conjunction with Eq.[7] to predict the infiltration rate at any time / > ts. The approachof Eq. [8] and [7] to describe/ at any t is undesirable due to its im-plicit nature.

By assuming a zero surface storage capacity, the runoff rate, Q,at any time t> ts is defined as:

G = [10]

If the computed value of I, in Eq. [7], at any / > ts is greaterthan or equal to R(t), then the infiltration rate, under the assump-tion of zero surface storage capacity, is equal to the instantaneousapplication rate. Consequently, Q, in Eq. [10], will be equal tozero.

EXPERIMENTALFor the experiment, a rainfall simulator was built using a new

technique for simulating smoothly varying, time-dependent, rain-fall intensity patterns. This technique utilizes the dynamics of asmoothly varying hydraulic head in a specially shaped container(A. Y. Hachum, 19765; Alfaro and Hachum, 1977). Figure 1shows a schematic illustration of the rainfall simulator used in theexperiment.

Cylindrical soil columns were prepared to simulate soil profileshaving uniform initial water contents and bulk densities. The soilcolumns measured 14.60 cm in diameter and 30 cm long. In orderto collect and measure water runoff, each column was providedwith a spout and a collecting graduate cylinder. Prior to waterapplication, the water content at saturation, 0S, the saturated hy-draulic conductivity, k(0s), and the water desorption character-istics were determined. The two parameters 6S and k(0s) weremeasured from the soil sample which was compacted in a per-meameter at a desired bulk density. Table 1 briefly describes thesoil samples utilized and the basic initial properties of the soil col-umns.

Figure 2 shows the five rainfall patterns used in the experiment.5A. Y. Hachum. 1976. Water infiltration and runoff under variable

application rate patterns. Ph.D. Thesis. Utah State Univ., Logan.

Fig. 3—A complete soil filled cylinder in position under the rainsimulator.

These patterns were selected to simulate the resulting water appli-cation patterns from a center pivot sprinkler irrigation system,making one revolution every 2 days. These patterns represent ap-proximately the same total depth of water application. Pattern IVis the average constant application rate of Pattern I, while PatternV represents the average constant application rate of Patterns IIand III.

Figure 3 shows a complete soil column in position under thewater applicator of the rain simulator. The soil column is slightlytilted to provide a surface slope of about 4% towards the runoffoutlet. Figure 4 shows a detail of the runoff collecting system andwater applicator. Small and highly sensitive tensiometers were, insome cases, installed in the soil column at various depths throughaccess holes in the cylinder wall. These holes acted as vents toreduce air entrapment in the soil.

For each experimental run, the time to surface saturation andcumulative runoff volume with time were measured and recorded,and in some cases, tensiometer readings were also obtained.Soilsurface was considered to be saturated when about 75% of the soilsurface area was covered with a thin layer of water. The runoff ratewas calculated by dividing the volume of runoff collected by thetime interval. The time interval was either 0.5, 1, or 2 minutes

Table 2—Values of *„, and We of soils.Soil *.t *« §

Millville silt loam (MSI,)Nibley silt loam (NSL)Nibley silty clay loam (NSCL)

25.0 7.521.012.0

19.75.3

0.300.940.44

t Computed by Eq. [6] for the desorption phase.§ Computed by Eq. [3] for one rainfall-induced runoff event.

HACHUM & ALFARO: WATER INFILTRATION & RUNOFF

NSL A8 = 0.26

NSCL A0 « 0.24SOLID LINES EQ. [41BROKEN LINES EQ. [5]

Fig. 4—Close-up of the soil column top under the rotating waterapplicator of the rain simulator.

depending on the runoff stream size. The maximum depth of soilwetted in any column did not exceed 25 cm.

The experiment was conducted in a laboratory where the tem-perature was 22.0 ± 0.5°C. There was no measurable temperaturevariation during any of the experimental runs.

More details on the experimental procedure can be found else-where (Hachum, 1976).

RESULTS AND DISCUSSIONEstimation of Ww

In the model under consideration, Ww is the most difficultparameter to estimate. The estimation of tyw, as given byEq. [6], requires the knowledge of the k - ^ relationshipfor the wetting (sorption) phase. In this study, <PW was es-timated by Eq. [3] using the known values of k(9s) and A0and a rate and depth of rain application at a given time ofsurface saturation. For each soil, the estimated value of ^wis then used as a soil property for any other application ratepattern.

The effect of 0( on tyw is shown implicitly in Eq. [6].Since kr(Qj) ~ 0 for Qi less than field capacity, "fyw can beconsidered constant for practical purposes.

If the k — W data for the drying (desorption) phase areused in Eq. [6], a value defined as tye will be obtained.Bouwer (1966) stated that y?w may range between 0.50^Peto tye, depending on the degree of air entrapment in the soil.Mein and Larson (1973) estimated Vw as 0.63^. Table 2shows the values of the paramenters ^Pe, as computed byEq. [6] from the desorption data, tyw as estimated by Eq.[3], and the ratio of tyj'fy,, for each soil. An inspection ofthe ratios of tyjtye indicates that they vary greatly betweensoils, with an average value of 0.56 for the three soils stud-ied. This value falls in the average range of 0.40 — 0.60found in other literature reported by Bouwer (1966).

Time of Surface SaturationIn Fig. 5, the solid lines represent the soil characteristics

expressed by Eq. [4] and the broken lines represent the rain-fall pattern characteristics expressed by Eq. [5]. The time tosurface saturation, ts, is defined by the intersection of these

24 32 40 48 56 64 72 80 88

RAIN RATE - mm/hrFig. 5—Example of determining W(ts) for Nibley Silt Loam (NSL) and

Nibley Silty Clay Loam (NSCL).

two equations. The/2[7?(fj] of Pattern I does not intersectwith the fi[R(ts)] of either soils. Therefore, surface satura-tion does not occur under this pattern. However, it doesoccur under Pattern III. In this procedure, ts is found indi-rectly by determining the cumulative depth W(tJ first, thenusing the rainfall pattern characteristics to determine ts.

There were 19 final runs in the experiment. Table 3presents a summary of the average measured and predictedtimes of surface saturation. Table 3 indicates that the pre-dicting ability of the model is, in general, promising. How-ever, it seems that the agreement between the predicted andmeasured t, improves as the water is applied faster. This canbe partially attributed to the assumption of the equivalentrectangular wetted region in the soil or the "piston displace-ment" concept used in developing the model. As the wateris applied faster, the wetting front becomes steeper (orsharper) and the wetted region above it reaches a higherdegree of saturation than that for slower application rates.

Infiltration After Surface SaturationAfter knowing ts and D(ts) or Dr, Eq. [8] and [7] can be

used to evaluate D and / at any t > ts. Figures 6 and 7 showthe observed and predicted behaviors of the system for Mill-ville silt loam (MSL) under Patterns II and III, respectively.The agreement between the measured and predicted infiltra-tion rates is close and reasonable.

Figures 8 and 9 show an example of the results for NSLand NSCL soils under Patterns II and HI, respectively, andindicate that there has been an accumulation of some sur-face storage water on the soil surface. As a result, the modeloverestimated the runoff; however, the agreement between

Table 3—Summary of average measured and predicted times of sur-face saturation, in minutes, for all soils studied.

Rainfall patternI

Soil Meas. Fred.

Millville silt loam 13.0 15.0Nibley silt loam § §Nibley silty clay loam g §

IIMeas.

3.51114.5#10.0

III IV

Pred.

3.5

9.5

Meas.

6.018.013.0

Pred. Meas. Pred.

7.0 12.0 14.017.5 § §14.0 § §

VMeas.

1.7#17.012.0

Pred.

t19.5t

Model calibration points in which predicted ts = measured ts.I Indicates no surface saturation.

Average of three replications.f Average of two replications.

964 SOIL SCI. SOC. AM. J . , VOL. 41, 1977

1 ' ' I ' I U4-'—j._L4_L4-i-4-i-v] ' I ' I ' I_

RAINFALL PATTERN II

PREDICTED INFILTRATION RATE

MEASURED INFILTRATION, NSL, REPLICATION 1MEASURED INFILTRATION, NSL, REPLICATION 2

MEASURED INFILTRATION, NSCL

8 10 12 14 16 18 20 22 24 12 14 16 18 20 22 24

Fig. 6—Predicted and measured infiltration rates for MSL soil underrainfall Pattern II.

80

== 70

I 60LU

of 50

20

' | ' | T [-. | I T, .

oMEASURED INFILTRATION RATE

RAINFALL PATTERN III

6 S 10 12 14 16TIME - minutes

20 22 24

Fig. 7—Predicted and measured infiltration rates for MSL soil underrainfall Pattern III.

the predicted and measured rates improves with time.Due to the well-aggregated structure of the two Nibley

soils, these soils behaved in the laboratory like a coarse soilin their hydraulic properties. The surface storage capacityfor the Nibley soils was relatively greater than that for theMillville soil due to the numerous "microdepressions" or"microbasins" which exist at the soil surface of the well-aggregated Nibley soils. Moreover, it is more difficult todefine ts experimentally when the surface is rough than if itwas smooth.

Effect of Rainfall Pattern ShapeTable 4 gives the depth of water applied, average mea-

sured total depth infiltrated, average measured total depth ofrunoff, and percentage of runoff for MSL soil under all rain-fall patterns presented in Fig. 2. For the soils and rainfallpatterns studied, lower peak rainfall patterns resulted in lessrunoff than their equivalent patterns with higher peak. In ir-rigation practice, this means that in an ideal center pivotsystem, the runoff in the innermost part of the lateral shouldalways be less than that in the outermost part of the system.In the field, however, the opposite may sometimes occur.This can be attributed to differences in depths of applica-tion, soils, and consequently, in 0t at time of irrigation.

Fig. 8—Predicted and measured infiltration rates for Nibley Silt Loam(NSL) and Nibley Silty Clay Loam (NSCL) under rainfall PatternII.

1"£20

oMEASURED INFILTRATION RATE, NSLoMEASURED INFILTRATION RATE, NSCL

0 2 4 6 8 10 12 14 16

TIME - minutes20 22 24

Fig. 9—Predicted and measured infiltration rates for Nibley Silt Loam(NSL) and Nibley Silty Clay Loam (NSCL) under rainfall Patternin.Comparisons between a variable rainfall pattern with its

respective average constant application rate pattern, such asPattern I with Pattern IV, and Patterns II and III with Pat-tern V revealed that the total amounts of runoff are almostthe same for both. This is true regardless of the differencebetween the times to surface saturation and/or the character-istics of infiltration for the two patterns. However, thisstatement holds better if the variable rainfall pattern is con-vex or symmetrical as indicated by comparing the agree-ment between Patterns II and V versus the agreement be-tween Patterns III and V.

Pattern III is an inefficient water application rate patternbecause it does not take advantage of the high initial infiltra-tion capacity of the soil. Therefore, it resulted in a 63%runoff, the highest amount of runoff observed in the experi-ment. Similar results for the effect of the rainfall patternshape on the response of the system are also obtained for theNSL and NSCL soils.

MODEL SIMPLIFICATIONThe theoretical and experimental analyses presented in

the previous sections have helped clarify many aspects of

HACHUM & ALFARO: WATER INFILTRATION & RUNOFF 965

Table 4—Summary of water budgets for Millville silt loam under dif- ior under different initial water content conditions without_____________ferent rainfall patterns._____________ the n£ed to conduct additional tests.

Total Average measured Average measureddepth total depth total depth Amount SniUMAWV AND PONPT ITSIONSPattern applied of infiltration______of runoff of runoff SUMMAKX ATN1J CUr>l,LU»lUfNa

————————— mm ——————— % Models based on extending the Green and Ampt ap-i 20.85 13.68 7.17 34 proach to analyze the infiltration under rain conditions have

j{| joiss 7.75 is!io 63 gained a great deal of attention recently. However, experi-iv 21.00 14.59 6.4i si mental work for testing these models is notably lacking.v 20'83_____1!!_______u'98______58 In this paper, a physically-based model for describing the

infiltration under any water application rate pattern is pre-TableS—T2 values for Millville silt loam, sented. The model involves three basic input parameters:

~^^ ~ — — — — — ~ — — — — p ~ ~ — — k(9s), A0, and Vw which are related to soil properties and——————————-——————-^——————-—————^—— initial conditions of the system. From the theoretical stand-

mm mm/hour mm mm /hour .T r • •, ,point, ww is constant tor a given soil as long as 0, remainsI 13.0 22.99 3.31 76.1 f , c

w,, , , „ , e „ & 'ii 3.5 46.52 1.75 81.4 below field capacity, /c(0j) = 0.III 6-° 32-98 2-02 66-6 The structural condition of the soil has a considerable ef-IV 12.0 21.00 4.20 88.2 f t ,T. , . ,„ . ™ ,. ,. ,v 1.7 50.00 1.46 73.0 reel °n TW and k(6s). These parameters are highly depen-

Average value of r2 - 77.1 ^ent on me hydraulic properties of the soil near or at satura-————————————————————————————————— tion. At small capillary pressure heads, it is difficult to

secure accurate measurements. Moreover, obtaining a reli-the model in terms of its operation and input parameters. able k — ty relation is an exacting and expensive task.When the application rates are relatively much higher than Therefore, it is recommended to estimate the value of ^?wk(0s), which is unavoidable in certain irrigation systems and by using Eq. [3] with known data for one rainfall-inducedpossible under natural rainfall, a simple index for estimating runoff event. More than one event can be used if greater ac-ts can be derived from Eq. [3]. Based on the assumption that curacy is desired.0j is below field capacity (i.e., fc(04) = 0) at time of irriga- The proposed model is further simplified for practicaltion, Eq. [3] can be written in the following form: applications by grouping the three basic input parameters

into a single parameter, the Surface Saturation Index.R(ts) _ ^u,A0 r , The validity of the model was tested and supported by thek(0s) ~ D (ts) *• •• results of laboratory experiments conducted on samples of

three soils, two silt loams and a silty clay loam, under fiveWhenfl(fs) > /t(0s),then^wA0/D(rs) > l.Thus,Eq. [11] different water application rate patterns,can be approximately written as: For the soils and rainfall patterns studied, it was found

that for rainfall patterns with equal depths of application,D(ts) • R(ts) = k(6s) • tyw • A0. [12] lower peak rate patterns result in less runoff than higher

peak rate patterns.Foragiven dt, the product in the righthand side of Eq. [12] The amount of runoff from a nearly symmetrical convex

is practically constant. Therefore, the following system variable rainfall pattern was very close to that which re-properties that might be of potential value are suggested: suited from its equivalent average constant application rate

pattern. This is true regardless of the significant differencesTI = k(6s) • Vw • A0 [12a] between the tinn:s at which runoff begins and the character-

istics of infiltration rates thereafter.T2 = D(ts) • R(ts) [12b] One undesirable feature of the proposed model is the im-

plicit nature of describing the infiltration after the pondingin which TJ = the predicted Surface Saturation Index, and time using Eq. [7] and [8]. Hachum and Alfaro (1976) haveT2 = the measured Surface Saturation Index. overcome this difficulty by presenting a graphical solution

Table 5 shows the measured Surface Saturation Index along with explicit approximate equations and compared thevalues, rz, for MSL soil. In Table 5, R(ts) is greater than performance of the model with that of Smith (1972).4 k(0s), which is common in practice. The predicted Sur- Research for extending the present study to considerface Saturation Index value, TI; as defined by Eq. f!2a] is more complex situations such as layered soil profiles is un-equal to 75.6 mm2/hour; for fc(0s) = 4.2 mm/hour, ^w = derway. Further study on considering the effect of the fall-75.0 mm, and A0 = 0.24. This value is almost equal to the ing raindrop impact on infiltration is of great value and sig-average value of T2 shown in Table 5. This supports the con- nificance.cept of the Surface Saturation Index.

For a known average 0j; one test involving the measure-ment of tt, D(ts) and R(ts) will be enough to estimate thevalue of T2 from which ts can be predicted under any otherrainfall pattern. Furthermore, Eq. [12] can be used to adjustthe Surface Saturation Index for any other initial water con-tent. Therefore, the model could predict the system behav-

966 SOIL sci. soc. AM. J., VOL. 41, 1977