TRICKLE-DRIP IRRIGATION: PRINCIPLES AND APPLICATION TO SO1 L-WATER MANAGEMENT

Eshel Bresler

Division of Soil Physics. Institute of Soils and Water. Agricultural Research Organization. Volcani Center. Bet Dagan. Israel'

I . Introduction .................................................. I1 . Potential Advantages of Trickle Irrigation ............................

B . Minimizing the Salinity Hazard to Plants . . . . . . . . . . . . . . . . . . . . . . . . . . C . Partial Wetting of the Soil Volume ............................... D . Maintaining Dry Foliage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . Flexibility in Fertilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F . Possible Water Saving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G . Technical-Economical Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 . Problems in Practical Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Clogging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Salinity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . System Design Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Technical Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV . Modeling of Water and Salt Flows .................................. A . The Physical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B . Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Water Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Two-Dimensional Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F . Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soil-Water Regime during Trickle Infiltration .......................... A . Size of the Saturated Water Entry Zone B . Water-Content Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Wetting Fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D . Effect of Surface Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII . Application of Infiltration Models to the Design of Trickle Irrigation Systems A . Transient State Infiltration B . Steady Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . Estimation of Spacing between Emitters . . . . . . . . . . . . . . . . D . Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A . Improving Soil-Water Regime for Greater Crop Yield . . . . . . . . . . . . . . . . .

V . . . . . . . . . . . . . . . . . . . . . .

VI . Solute Distribution during Infiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.....................................

. . . E . Effect of Soil Hydraulic Properties on Spacing-Discharge Relationships F . Designing the Lateral System

VIII . Water Management in Marginal Soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

344 345 346 348 348 349 350 350 350 351 351 352 352 353 353 354 355 356 357 360 363 367 367 371 371 372 372 376 377 377 378 382 382 385 387 389

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

' This review was in part prepared during sabbatical leave at the Department of Agronomy. Cornell University. Ithaca. New York .

34 3

344 ESHEL BRESLER

I. Introduction

The use of trickling or dripping as a method of irrigating large fields has become quite common practice in agricultural production all over the world. The method is now one of the fastest growing new technologies in agriculture (Gustafson et al., 1974; Halevy et al., 1973). The acreage under trickle irrigation in the United States and throughout the whole world is steadily increasing. The area under drip irrigation in the United States was 100 acres in 1970 and over 70,000 acres in 1974. The anticipated 5-year projection is for more than 217,000 acres in the United States and for at least 100,000 acres in other countries (see Table I). Trickle irrigation is being adapted to almost all types of crop production (Boaz, 1973; Dan, 1974; Goldberg and Shmueli, 1970; Gornat e t aZ., 1973; Goldberg er ab, 1971; Halevy etal . , 1973; Hiler and Howell, 1973; Yagev and Choresh, 1974; Waterfield, I973), to all types of land (sand dunes to clay soils, level land to steep hilside), and to relatively saline water. New land which previously could not be used successfully for agriculture has been made available through trickle irrigation (Boaz, 1973; Goldberg and Shmueli, 1970; Gornat et aZ., 1973; Halevy et aL, 1973; Willens and Willens, 1974).

A trickle irrigation system consists of emitters which distribute the water for irrigation. These are usually attached to laterals to which water is delivered through the submains or main lines (Heller and Bresler, 1973; Rolland, 1973). The trickle emitter is essentially a water energy reducer. It acts generally as a resistor which dissipates the energy of water flowing through it and thus reduces the flow rate to a given discharge. The energy loss is controlled by means of an orifice, micro-holes, “button” nozzles, slits, or pores, and by the Iength of the

TABLE I Drip Irrigation Survey‘

Country

Australia British Columbia, Canada Central America Cyprus Israel Mexico New Zealand South Africa Other countries United States

Total

Acreage in 1974

25,000 500 700 400

15,000 16,000

2,000 8,600 3,000

72,000

143,000

5-Year estimate

60,000

-

3,000 -

-

30,000

>217,000

>3 10,000

-

‘After Gustafson er al. (1974).

TRICKLE-DRIP IRRIGATION 34 5

water-flow path. The length of the water-flow path is adjusted with a spiral or screw thread plastic tube, by “spaghetti” or maze emitters, or by means of a porous material. Vortex drippers in which water is forced tangentially into a circular chamber have been fairly recently designed (Halevy et al., 1973). The flow rate is controlled by the pressure distribution which in turn is determined by size and type of dripper and the water carrier (Karmeli, 1971; Halevy et al., 1973).

The surface approach, as opposed to underground application of trickling (subsurface irrigation), was initiated in Israel. It has been successfully and widely used for the past few years (Boaz, 1973; Heller and Bresler, 1973; Gornat etal., 1973). In the surface approach the irrigation emitter is placed directly on the soil surface so that the water infiltration takes place in an area which is small compared with the total soil surface of the irrigated field. As a result one has a case of three-dimensional irrigation. This differs from the usual one-dimensional flooded or sprinkler irrigation where the area across which irrigation water enters the soil is considered to be identical to the total area of the irrigated field. Optimal irrigation criteria are achieved by adjusting the trickling system to the soil hydraulic properties and to the water and nutritional requirements of the specific crop grown. This is done by selecting the best known combination of all constituents which comprise the trickle irrigation system.

Recent work on the mechanisms underlying the general principles of a trickle irrigation system is presented here. The basic concepts of transport phenomena in soils are related to a quantitative evaluation of the water and salt regime in trickle-irrigated soils. Emphasis is placed on applying these principles by agron- omists, soil specialists, and engineers to the design of trickle-drip irrigation systems for soils with different hydraulic characteristics. A general summary of the present knowledge of the subject is provided rather than a detailed review, interpretation, and evaulation of each individual publication that has appeared in the scientific literature.

I I. Potential Advantages of Trickle Irrigation

The traditional methods of water application to the soil have advanced from simple flood irrigation, through furrow irrigation, to a scientifically controlled sprinkler irrigation system. Each of the conventional irrigation methods has certain advantages and limitations when one considers their technical, economic, and crop production values. The trickle irrigation method has been developed for specific conditions of an intensive irrigated agriculture. Some of the technical and agronomic advantages that may be achieved by selecting the trickling method are described in the following sections.

346 ESHEL BRESLER

A. IMPROVING SOIL-WATER REGIME FOR GREATER CROP YIELD

The traditional irrigation cycle under furrow, flood, or portable sprinkler system consists of a relatively short period of infiltration followed by a long period of simultaneous redistribution, evaporation, and extraction of water by the growing plants. In t h s mode of irrigation there is a fixed cost associated with each water application and therefore one’s goal is to minimize the number of irrigations by increasing the time interval between two successive irrigations without causing an economic yield reduction. To minimize irrigation frequency the usual goal was .to maximize the quantity of available water stored in the soil for subsequent water use by the crop before the next irrigation. The economic constraint of the traditional irrigation methods, which cause extremely large time fluctuation in the soil-water potential (Bresler and Yaron, 1972), has been partly removed by the solid-set and central pivot irrigation systems and by the development of trickle irrigation systems capable of delivering water to the soil in small quantities as often as desired with no additional cost (Rawlins, 1973).

As the frequency of irrigation increases, the time-average soil-water potential increases and is restricted to a narrow range, hence eliminating low average soil-water content and high water fluctuations as factors affecting plant growth and crop yields. This increase in soil-water potential (or decrease in the total soil-water suction) at very frequent or continuous water application is a con- sequence of both high average matric potential and low soil solution concentra- tion resulting from the fact that the salinity of the soil solution approaches that of the irrigation water. Ayers el al. (1943), Wadleigh and Ayers (1945), and Wadleigh et al. (1951) suggested that matric and osmotic potentials are additive in their effect on plant growth. There is some evidence to support the view that crop yield of many crops is increased by maintaining the soil-water regime at high time-average values of soil-water potential in the effective root zone (Rawitz, 1970; Hillel, 1972; Childs and Hanks, 1975). The general increase in the total soil-water potential with irrigation frequency suggests that crop yield may be highly increased by very high irrigation frequency.

The maintenance of continuously high water potential, thus minimizing fluc- tuations in the soil-water contents during the irrigation cycle, may be an important and advantageous feature of trickle irrigation if the yield response curve to water is convex and the effect of fluctuation in the water status on the crop yield behaves similarly to the theory of Zaslavsky and Mokady (1966) and Zaslavsky (1972), as follows.

It is a well-established assumption (e.g., Taylor, 1952) that crop yield-irriga- tion relationships depend on both the average of some index of the soil-water regime 5 and the time deviations from 6. Generally, this can be expressed as

Y(@) = ml (1)

TRICKLE-DRIP IRRIGATION 34 7

where Y osmotic space), t

is crop yield, Qi is the soil-water regime index (e.g., sum of matric and potentials), & is its time average (assuming uniform 4, with respect to is time and @ is the time (daily) deviation of @(t) from &, i.e.,

@(t) = @(t) - 6 (2)

Note that when there are no fluctuations in Q, (i.e., q5 = 0), then @(t) = 6 and the yield response function to water becomes

Y(@) = Y(@ = 6,O) = Yo(&) (3)

where Q, is perfectly uniform at the 6 level. Expanding Y(@) function about Qi = & by Taylor’s series and neglecting

third-order and higher terms, we obtain

(4) ay [@Wl az Y ( 6 ) Y(6 + #) = Yo($) + #(t)- (&) t - - a@ 2 aQZ

Averaging the yield over time

- i @(t)dt + ,=l [ Ydt=Yo(@)+- 0

1 [@(t)] dt (5) 0

The first term on the right-hand side (RHS) of Eq. (5) is identical to Eq. (3), i.e., yield response function without fluctuation in @; the second term on the RHS of Eq. (5) vanishes upon averaging because average deviations are zero. Since the integral

2T T t = 0

T

T

0 T

is the mathematical definition of the variance u2 therefore

u2 a2y F = YO(*) t- - 2 aQ2

Here again Y is the yield, r is the average yield, @ is the water regime index (e.g., sum of matric and osmotic potentials), & is its average over time, and uz is the variance of @. Note that the first term on the RHS of Eq. (6) is the yield that would have been obtained with perfectly uniform levels of @ all at the average 6 (yield without fluctuations), and the second term is the first correc- tion to the yield due to fluctuations (fluctuation contributions). Note also that if the Y(@) function is convex, then the second derivative in Eq. (6 ) is negative and time fluctuations in the soil-water potential will cause a crop yield reduc- tion. Generally, a’ Y / M 2 < 0, and thus fluctuations of soil-water content or of total water potential with time have a negative effect on the crop yield. If this is indeed the general case, then the best irrigation policy is to apply the water as frequently as possible (Rawlins and Raats, 1975) as long as there are no aeration

348 ESHEL BRESLER

problems (Dasberg and Steinhardt, 1974). Referring to Eq. (6 ) it is clear that the crop yield increases with increasing irrigation frequency if the yield response to soil-water potential is a convex monotonic increasing function. With the trickle irrigation method, in addition to solid-set and central pivot irrigation systems capable of delivering water t o the soil as often as desirable with no additional costs (Rawlins and Raats, 1975; Heller and Bresler, 1973), this potentiality in increasing crop yield can be achieved. It is possible, therefore, to draw a conclusion that if the views expressed by Zaslavshy (1972) and given in Eq. ( 6 ) will consistently be supported by experimental evidence (such as that of Patter- son and Wierenga, 1974, in the experiments performed in 1972 but not in 1973), then optimizing the soil-water regime for greater crop yield may be an important advantage of the trickle-drip irrigation method.

B. MINIMIZING THE SALINITY HAZARD TO PLANTS

Recent work by Bernstein and Francois (1973) showed that brackish irrigation water (2450 mg/liter total salts) can be used successfully in drip irrigation to obtain almost the same yield as nonsaline good quality water. Using the same water for furrow and sprinkler irrigations caused yield reductions of 54 and 94%, respectively. Increasing the irrigation frequency caused only an 18 and 59% reduction in yield for furrow and sprinkler irrigation, respectively. Minimizing the salinity hazard to plants irrigated by trickling can be related to: (a) the displacement of salts beyond the main efficient root zone (Patterson and Wierenga, 1974, Fig. 10; Tscheschke et al., 1974; Yaron et al., 1973); (b) lower- ing the salt concentration by maintaining a relatively high soil water content due to the high frequency irrigation; and (c) avoiding leaf “burning” and damage due to salt accumulation on the surface of leaves which are in contact with irrigation water. Bernstein and Francois (1975) attributed higher yield loss and injury of bell peppers (Capsicum fmtescens) irrigated by sprinkling as compared with trickling primarily to foliar salt adsorption rather than to osmotic shock caused by flushing the salt-which had accumulated at the soil surface between two successive irrigations-into the root zone.

C. PARTIAL WETTING OF THE SOIL VOLUME

An additional feature of trickle irrigation is the possibility of restricting water supply to those parts of the soil where the activity of the root system, with respect to water and nutrients, is the most efficient (Dasberg and Steinhardt, 1974). Bernstein and Francois (1973) found that many of the roots under trickle irrigation occur in the surface 2.5 cm of soil, except when salt accumula-

TRICKLE-DRIP IRRIGATION 349

tion inhibits root development. Black and West (1974) tested the effect on water use of young apple trees when varying proportions of the root systems were supplied with an “optimum” water regime. They found that when the fraction of the wetted root system was decreased from 1.0 to 0.25, the relative transpira- tion reduced from 1.0 to only 0.75. Their results also suggested that wetting substantially less than the total root system daily would produce at least as good a regime for plant water supply as would wetting the entire root system with a 14-day interval between irrigations. Glasshouse and growth-cabinet experiments (Frith and Nichols, 1974) showed that local application of water to less than the total root volume did not affect the ability of the tree to take up sufficient water, and that roots in the wetted root zone increased their ability to take up water. These results are similar to those of Lunin and Gallatin (1965).

It was also found by Frith and Nichols (1974) that a portion of the root system could, if required, assimilate as much nitrate nitrogen as the whole root system. This experimental finding is of practical importance since satisfactory nutrition can be obtained by dissolving fertilizers in the trickle irrigation water and the roots in the wetted root zone should increase their efficiency of nutrient uptake in a manner similar t o their increasing efficiency in the water uptake. Thus, under trickle-drip irrigation it is possible for trees, under certain climatic conditions, to have considerably less than the total root system wetted. The ability of a peach tree quickly and profoundly to adapt its root system to the partial wetting pattern of trickle irrigation is also of interest. A whole new root system for large trees (5.8 meters wide and 5 meters high) was developed in a few months and the tree continued to produce heavy yields (WiUoughby and Cockroft, 1974).

Selective wetting of the soil surface has additional benefits, such as reducing water evaporation by preventing evaporation of water from outside of the wetted surface zone. The partial wetting also restricts the growth of weeds to the wetted region and thus reduces the cost of weed control by decreasing the need for weed control beyond the wetted region. Weeds at the wet spots may be controlled very efficiently by applying herbicides through the drip system. More convenient pest control is achieved by leaving dry strips on which the pest control machinery can be moved.

D. MAINTAINING DRY FOLIAGE

Dry foliage retards the development of leaf diseases that require humidity. It obviates the necessity for removing plant-protecting chemicals from the leaves by washing. This is, of course, in addition to the prevention of leaf burns due to the lack of direct contact of the leaves with saline irrigation water (Bernstein and Francois, 1975).

350 ESHEL BRESLER

E. FLEXIBILITY IN FERTILIZATION

Trickle irrigation offers flexibility in fertilization, a benefit unique to this system (Lindsey and New, 1974; Isob, 1974). Since fertilizers can easily be applied along with irrigation water, frequent or continuous application of nutrients at low concentrations is feasible and seems to be very good practice (Safran and Parnas, 1975). Optimizing the nutritional balance of the root zone is possible by supplying the nutrients directly to the most efficient part of the root zone. Other features include good fertilizer distribution with minimum leaching beyond the root zone and more options in the timing of fertilizer application than with any other distribution system. However, the fertilizer mixtures must be completely soluble in water, not leave any residues in the dispenser, and must not cause clogging of the emitters (Grobbelaar and Lourens, 1974).

F. POSSIBLE WATER SAVING

There are several ways by which water may be saved in using trickle irrigation as compared with other traditional irrigation methods (Patterson and Wierenga, 1974). Under trickle irrigation loss of water due to runoff in low permeable or crusted soils (Kemper and Noonan, 1970) is reduced. In addition, destruction of the surface-soil structure and the development of surface crust (Lemos and Lutz, 1957) is avoided and water infiltration into the soil is largely improved (Rose, 1961).

Much water saving may be achieved by restricting the water supply to the extent of the most efficient root zone (Dasberg and Steinhardt, 1974). By not wetting the entire inter-row or inter-tree space, especially in young crops or trees (Dan, 1974), direct evaporation from the soil surface and water uptake by weeds are drastically reduced (Lemon, 1956). On steep hills and/or under strong wind conditions, furrow and sprinkle irrigation methods are very inefficient with respect to water saving (Seginer, 1967). Under these conditions, the use of trickle irrigation prevents water loss beyond the border of the irrigated field by wind convection (Seginer, 1969) or runoff in contour cultivation on steep hills.

G. TECHNICAL-ECONOMICAL FEATURES

In many countries and instances where labor and water have become limiting and too expensive, the method of trickle irrigation has been developed. Automa- tion is one of the very reliable tools which can be easily used in trickle irrigation for accurate soil-water control, the supply of water as needed, and a large reduction in manpower. A simple automation system includes an automatic

TRICKLE-DRIP IRRIGATION 35 1

metering valve or an automatic timer that can be set for daily operation at certain hours of the day. More sophisticated and complicated units can be controlled by electrified tensiometers in the root zone, by having an electrical impulse trigger the system when the matric potential is reduced to a certain dryness.

These days, with the energy supply becoming more limited and expensive, an optimal irrigation method should also rely on a relatively low operational pressure. This may be an important technical feature of the trickling method as long as the energy losses in the “Control System Unit” (see Heller and Bresler, 1973, Fig. 5) are not too large.

An additional important technical-economical feature of trickle irrigation is the use of a small pipe diameter and the possibility of operating the system 24 hours a day, including during windy hours (Heller and Bresler, 1973). Frequent irrigation during the windy hours protects sensitive and high investment crops from cfesiccation without wasting water by wind convection (Seginer, 1969).

Another economical-technical feature of the trickle-irrigation system is the reduced cost of weed control. This is so because weeds grow only in the wetted spots and weed control may be achieved through the trickling system. The use of herbicides through the irrigation system offers an answer to weed problems under trickle irrigation, especially if one uses herbicides capable of killing weeds as they germinate (Lange et al., 1974). Another aspect of trickle irrigation in relation to the economy of pest control is the possibility of using the soil fungicides in the irrigation system to control root rot fungus (Zentmyer et al., 1974).

Ill. Problems in Practical Use

The technical and agronomic advantages utilized in selecting the trickle-drip method have been discussed. However, some problems in the practical imple- mentation of the method do still exist: some of the problems will be discussed here.

Operational difficu

A. CLOGGING

ies with the trickling metho- sometimes arise from clog- ging of the drippers, which is the most severe maintenance problem. Clogging affects nonuniform water distribution and requires frequent replacement of emitters, which is quite an expensive procedure. Clogging is caused by several factors (Peleg et al., 1974), such as: (a) root penetration; (b) blockage of orifice by sand, rust, leaves, small soil animals, microorganisms, etc.; and (c) precipita-

352 ESHEL BRESLER

tion of soluble salts such as carbonates, iron, aluminum, and phosphate com- pounds. Smaller particles can pass through the filtering system and act as crystallization nuclei in the trickling system. Overcoming the clogging-caused difficulties involves the use of a very efficient filtering system and periodic cleaning or replacement of the drippers (Boaz, 1973), reverse flushing (Rawlins, 1974), and chemical treatments (McElhoe and Hilton, 1974). All these, of course, involve extra expense and manpower, especially when a large number of emitters and laterals per unit area are needed.

B. SALINITY PROBLEMS

In arid regions where saline water must be used, there is a tendency for salt to accumulate close to the margins of the wetted zone (Tscheschke et al., 1974; Patterson and Wierenga, 1974; Yaron et al., 1973). The salt which tends to accumulate at the periphery of the wetted soil volume, midway between emit- ters (Gerard, 1974), may be washed by rain into the main effective root zone and may cause osmotic shock to plants (Bernstein and Francois, 1973). The accumulation of salt at the periphery between emitters could be a serious problem also for seasonal crops, because some of the newly sown plants may be found in regions of high salt concentration from the previous crop. This salt must therefore be leached before the next crop is planted (Patterson and Wierenga, 1974). In areas where there is not sufficient rainfall for the leaching process, then leaching must be accomplished by sprinkler or flood irrigation, which increases the cost considerably. Another expensive solution is to place the emitters at a much closer spacing in order to approach one-dimensional vertical flow. With high frequency irrigation it is also possible to apply water in excess of evapotranspiration in order to leach the accumulated salts out of the root zone and thus control soil salinity. Controlling the quantity of water passing through the root zone to avoid salinity buildup is one of the problems of trickle irrigation.

C. SYSTEM DESIGN PilOBLEMS

In the sprinkle irrigation method, when one-dimensional water flow takes place, there are no severe design problems. The spacing between sprinklers and the operational pressure are designed so as to meet certain uniformity criteria (Christiansen, 1942; Hart, 1961). Irrigation scheduling is designed by taking into account soil-water or plant-water criteria, one-dimensional evapotranspiration, and effective rooting depth (Hake and Hagen, 1967). In trickle irrigation, when

TRICKLE-DRIP IRRIGATION 353

three-dimensional flow occurs, the main design problem is to select the proper combination of emitter spacing and discharge for a given soil and crop (see Section VTI).

In cases where there are no salinity problems, the wetted volume of soil should be kept within the borders of the efficient root zone. In order to prevent losses of water beyond the efficient root zone, one must determine the proper combination of emitter spacing and discharge, irrigation frequency, and amount of water applied for a given set of soil-water characteristics, crop root distribu- tion, water distribution prior to each irrigation, as well as the pattern of water uptake by roots. Determination of the proper emitter spacing and discharge under given external conditions requires knowledge of soil hydraulic properties together with crop response to size and form of the wetted soil volume and to the water distribution and fluctuation within this volume.

D. TECHNICAL IMPROVEMENTS

Technical improvements include the following aspects: (1) mechanical im- provements of emitters, various fittings, and filters; (2) finding chemical and other treatments to eliminate mechanical and chemical clogging of emitters and failure of the operation system; and (3) development of a good fertilization system. In addition, development of a portable trickle irrigation system and of cheaper components (Rawlins et al., 1974; Wilke, 1974) would make the method more suitable for irrigating close-planted crops, which are sensitive to the water distribution in the soil.

IV. Modeling of Water and Salt Flows

Mathematical models are used to approximate physical conditions through mathematical equations. When the equations are too complicated to be solved analytically, their solutions may be obtained with the aid of a computer. Several phases are involved in the development of the model. These are: (a) identifica- tion of the relevant physical problems and the mathematical model that can simulate the physical problem; the physical parameters that are included in the model are determined separately and independently; (b) mathematical formula- tion of the governing equations and the boundary conditions of the problem; (c) development of a method of solution in order t o solve the mathematical problem; (d)verification of the model, first by a comparison between exact analytical solutions and the approximate model solution, and second by per- forming physical experiments and comparing the theoretical results with field or

354 ESHEL BRESLER

laboratory observations; (e) efficient implementation of the procedure on the computer if computer methods are used; (f) compatibility of the final product with existing models, if there are such models.

A. THE PHYSICAL SYSTEM

The irrigation system to be considered first consists of an emitter which distributes the water for irrigation. Water enters the emitter-which reduces its pressure-and discharges out as a trickle at a predetermined rate. The irrigation trickle emitter is placed directly on the soil surface, so that the area across which infiltration takes place is very small compared with the total soil surface. As a result, one has a case of three-dimensional infiltration of water into the soil. This differs from the usual one-dimensional case of flood or sprinkler infiltration, where the area across which water enters the soil is assumed to be identical to the entire soil surface.

As mentioned earlier, one of the potential advantages of trickle irrigation is to maximize the time-average soil water potential by increasing irrigation fre- quency. As the frequency increases, the infiltration period becomes more impor- tant and the irrigation cycle is changed from an extraction-dominated process to an infiltration-dominated process (Rawlins, 1973). Since, in trickle irrigation, when irrigation is sufficiently frequent, the irrigation cycle is dominated by the infiltration stage, the discussion here is limited to modeling of salt and water flows during infiltration only.

Consider a field (Brandt et aZ., 1971) that is irrigated by a set of emitters or trickle sources, spaced at regular intervals, 2X and 2Y, as shown in Fig. 1. Due to the symmetry of the pattern, one can subdivide the entire field into identical volume elements, W, of length X , width Y, and depth 2, where the latter always remains below the wetting front. Here, each volume element acts as an indepen- dent unit in the sense that there is no flow from one element to another.

FIG. 1. Schematic representation of a trickle-irrigated field.

TRICKLE-DRIP IRRIGATION 355

Therefore, in order to describe the salt and water flows in the entire field, it is sufficient to analyze their status in a single element, W. This, of course, is true only for the interior part of the field that is not too close to the margins.

B. GOVERNING EQUATIONS

The differential equations governing the flow of water and noninteracting solutes in an unsaturated soil system can be written in indicial notation as (cf. Neuman, 1973; Bear, 1972):

Here xi (i = 1,2,3) are spatial coordinates (xj the vertical considered to be positive downward), 0 < Kr < 1 is the relative hydraulic conductivity (Neuman, 1973), K:j is the hydraulic conductivity tensor at saturation, 0 is volumetric soil-water content, t is time, p is pore water pressure head, Dij is coefficient of hydrodynamic dispersion (combining the effects of diffusion and mechanical dispersion), q is specific flux of the soil solution, and c is solute concentration in the solution. Equations (7) and (8) are written in an indicial notation such that quantities with a single subscript, or index, represent components of vectors; quantities with two subscripts are components of tensors; and when an index appears twice in any given term this term must be summed over all admissible values of that particular index [such as i and j in Eqs. (7) and (8)] .

The form of the dispersion term Dij in Eq. (8) has been the subject of intense discussion. Recent experimental and theoretical studies (Ogata, 1970; Perkins and Johnston, 1963; Bear, 1972) suggest that in isotropic and homogeneous porous media the principal axes of dispersion are oriented parallel and perpen- dicular to the mean direction of flow. Ths indicates that for such media the transport of the dispersed material can be defined by two characteristic disper- sion components that are specified when the mean direction of flow is known. Thus the hydrodynamic dispersion coefficient Djj for isotropic media can be defined similarly to the definition by Bear (1972, p. 612) as

oij = hT1 v 1 6 ~ ~ t (A, - hT) vpy~ vi t Dp(e ) (9)

where X, is the longitudinal dispersivity of the medium, AT is the transversal dispersivity of the medium, 6 i j is Kronecker delta (i.e., 6ii = 1 if i = j and 6ij = 0 if i # j ) , & is the d-th component of the average interstitial solution velocity Y,

356 ESHEL BRESLER

and Dp(e ) is the soil diffusion coefficient as defined by Bresler (1973) using Eq. (57) of Olsen and Kemper (1968).

C. WATER FLOW BOUNDARY CONDITIONS

Referring to Fig. 1, we shall place the origin (O,O,O) of the coordinates at the center of a particular emitter (trickle source) and define W as the domain { W = 0 < x = x l < X , 0 < y = x2 < Y , 0 < z = x g 1 . I t is clear t h a t x = O , x = X , y = 0, and y = Y are planes of symmetry, for which the normal derivative of 0 must vanish, and where no flow exists across these boundaries. If one also assumes that below the wetting front (at the depth z = Z), M / a z = 0 is a good approximation for the period of the infiltration, or at least that imposing this condition would have a negligible effect on the region of interest (Brandt et al., 1971), one has the following no-flow boundary conditions formulated for

aO/ax= 0 a t x = 0 and x = X for t > O

ae/& = 0 at z = z for t > 0

< .I

Nx,y,z,t) as

ae/ay=O a t y = 0 and y = Y f o r t > O (10)

In order to define the boundary conditions at the soil surface (z = 0), the discharge from the trickle source must be known as a function of time. This rate of discharge is denoted by Q(t) and will be referred to as trickle discharge. In addition, an assumption must be made concerning the horizontal area across which water infdtration takes place. It has been observed (Bresler et al., 1971) that in general a radial area of ponded water develops in the vicinity of the trickle source. This area is initially very small, but its radius ( p ) becomes larger as time increases. Since the ponded body of water is usually very thin, one can safely neglect the effect of storage of water at the soil surface. This means that the water from the trickle source is able to infiltrate into the soil, or evaporate into the air, instantaneously. Obviously, the soil-water content immediately beneath the ponded area is always equal to the water content at saturation, €Is. This saturated area is the only place where water can.infiltrate into the soil element, W. Thus it will be referred to as the saturated area, or the zone of water entry. It: is assumed that the center of this disklike zone is at (O,O,O) (Fig. 1) and that its radius p(t) is a function of time. The only additional boundary condition that must be satisfied at the soil surface outside the saturated area of water entry is that the water flux be equal to a given rate of evaporation, E.

Therefore, the boundary conditions that must be satisfied at the soil surface are, “moving boundary conditions,” and they can be mathematically formulated for all t > 0 and at z = 0 as

TRICKLE-DRIP IRRIGATION 357

e = e, for o < x 2 t y z G [p( t ) ] (1 1)

(12)

(13)

q e ) + E -K(s)&= a Z o for x2 +y2 > [p(t)l2

/[K(B,) + E - K(0,) az 1 dx dy = z Q ( t > aP 1

G

Here, G is the quarter disk defined by x z + y 2 S [p(t)] 2 , x,y > 0 ; 0, is the water content of saturated soil; p(t) is the radius of the ponded or water saturated area as a function of time; E = E(t), is a given evaporation function; and Q(t) is the time-changing discharge of the trickle source.

The large-scale field flow problem is now confined to the cube

w= ( o < x s x , o ~ y < Y , o < z < z ) ,

on the sides of which it is possible to formulate suitable boundary conditions. To these we have to add the initial conditions

8 = 8, in W, for t = O (14)

where 0, is the initial soil-water content.

D. TWO-DIMENSIONAL APPROXIMATIONS

At present, the problem defined in Eqs. (7) and (10) through (14) can be solved only by numerical methods with the aid of a high-speed digital computer. Excessive amounts of computer time are needed to solve the four-dimensional grid (x,y,z,t) in this kind of problem. The expense of long computer runs can be greatly mitigated by considering special cases of the three-space dimensional problem which are amenable to two-space dimensional modeling. Also two-di- mensional experimental data for comparison with simulated results are more easily obtained. Such a two-dimensional flow is the cylindrical flow which takes place in the field as long as the wetting front has not reached the outer boundaries of the volume element W. Other cases of a two-dimensional problem are: (a) a line of trickle sources very close to one another, and (b) a line of perforated or porous tube. Here, the problem can be viewed essentially as one of a plane flow. Thus, two mathematical models will be considered in this review: (1) a “plane flow” model involving the Cartesian coordinates x and z, and (2) a “cylinder flow” model described by the cylindrical coordinates I and z.

1. Axisymmetric Cylindrical Flow Models

In the axisymmetric cylindrical flow model the case of a single trickle emitter (or a number of emitters spaced at sufficient distance apart) is considered. The

358 ESHEL BRESLER

origin is at the center of the epitter and c and 6’ are functions of the coordinates x = x3, t, and r = (x2 t y2)”’ = (x: t x$)”~ . The axial symmetry conditions and assuming the soil to be a stable, isotropic, and homogeneous porous medium, with Darcy’s law applied in both saturated and unsaturated zones, cause Eqs. (7) and (8) to take the form:

Here, Y is the radial coordinate, z is the vertical coordinate (considered to be positive downward), K(0) = Kr(B)KS is the hydraulic conductivity of the soil (depending on 0 alone), and H is the hydraulic head (sum of pressure head, p , and gravitation head, z ) . Note that Eq. (15) considers the water pressure head p and the hydraulic conductivity K to be single-valued, continuous functions of 0 and the hydraulic gradient to be the only force causing water to flow. Also note that in the development of Eq. (16) one usually assumes that solute transport is governed by convection and diffusion, and that DIP, D,, = D,, and D,, are given by Eq. (9) with r and z substituted for i and j , respectively.

The initial and boundary conditions for e(x ,r , f ) [Eq. (15)] in the cylindrical element W = (0 < r < R , 0 < z < 2) are as follows:

e = O , ; O < r < R ; O < z < Z ; t = O (174

r = O , r = R ; O G z < Z ; O<t<T (17b)

(17c)

8=8,; O < r < p ( t ) ; z = O 0 < t < T (1 7d)

(17e)

ae -= 0; ar

ae - = O ; O < r < R ; z = Z ; az 0 < t < T

0 < t < T E-K(O,)--O; aH- p ( t ) < r < R ; z = O az

aH [E -K(B,)$ rdr = Q(t) 27r

0

where T is the end time of infiltration.

metric flow conditions of infiltration from trickle sources (Bresler, 1975), are The boundary and initial conditions for c(r,z,t), [Eq. (16)] under the axisym-

Finally, the initial conditions are

c(r,z,O) = c,(r,z) in

Here, q,(r,O,t) is the specific downward flux of water at the soil surface as given by Darcy’s law, C, is the solute concentration at the inlet of the trickling water, cn(r,z) is the predetermined initial soil solution concentration, and Drz(r,O,t), D,, (r,O,t) are the hydrodynamic dispersion coefficients at the soil surface, the sum of diffusion and mechanical dispersion coefficients as given by Eq. (9).

2. Plane Flow Model

Consider: (a) An infinite straight line of perforated or porous tube, or (b) a set of drippers that are spaced very close to each other at equal intervals along an infinite straight line, so that their ponding areas overlap after only a short period of time. One can assume that the total ponding area has the form of an infinite strip of width p(t). Let this strip be oriented along they = x2 axis, and let x = x1 be the horizontal coordinate normal t o y then the flow becomes independent of the y coordinate. Considering the plane flow model. and the aforementioned assumptions the governing Eqs. (7) and (8) now become

360 ESHEL BRESLER

- ae = - a [ ~ ( e ) : ] t az a [zqe)Y at ax az

ac ac a ac ac at ax ax az az aZ ax

a(ce)- a (Dxx- + D,, - - qxc) t - (LIZ,- t Dzx- - q,c) (20)

where the subscripts x and z denoting the flow direction x l and x3 , respectively and D,,, Dxz = D,,, and D,, are given in (9). Note that Eqs. (15), (16), (19), and (20) do not include sinks or sources due to uptake by plants or precipita- tion, dissolution, and adsorption of solute. Note also that the variable y must now be eliminated from the boundary conditions (10) through (14), and that x is substituted for r, and X for R, in (18). Of course, 2n and r in the last term on the left-hand side (LHS) of (180 must also be omitted. Notice also that Q(t) is now the trickle discharge per unit length of the strip (cm3 cm-’ rnin-’ ).

This plane flow model is relatively easy to reproduce in the laboratory and can therefore serve as a convenient tool for comparison purposes. Such a comparison between laboratory and theoretical results will be described in Section IV, F of this chapter.

E. SOLUTIONS

1. Numerical Approaches

The problems defined by the nonlinear partial differential Eqs. (15) and (16) or (19) and (20) together with the associated boundary conditions given by Eqs. (17) and (18) can, at present, be solved only by numerical methods with the aid of a computer. Brandt et al. (1971) used an approach that combined the noniterative alternating-direction implicit (ADI) difference method with the iterative Newton method to solve numerically for values of @(y,z,t), q,(y,z,t), and 4,(y,z,t), where y is introduced for x when dealing with plane flow and for r when the flow is cylindrical. The calculated values can then be used to obtain c(y,z,t) from Eq. (16). To solve for c(y,z,t) by the second-order finite difference approximation to Eq. (16) or (20) and Eq. (18), the values of D,,(y,z,t), DYz(y,z,t), and D,(y,z,t) must be known (Bresler, 1975). After B(y,z,t) and q(y,z,t) are known, one may calculate V7(y,z,t) = q7(y,Z,t)/e(y,z,t) as well as V,(y,z,t) = q,(y,z,t)/O(y,z,t) and V = (c + G)l’’; and so estimatesD,,, D,,, and D,, from Eq. (9). The methods used to solve for O(y,z,t) (Brandt et al., 1971) and c(y;z,t) (Bresler, 1975) resulted in a numerical technique that is simple, efficient, unconditioned stable, and second-order accurate.

2. Linearized Water Flow Solutions L

Solutions for transient and steady infiltration from point, line, strip, and disk sources which can be applied to simulate trickle irrigation cases have recently

TRICKLE-DRIP IRRIGATION 36 1

been published (Wooding, 1968; Philip, 1971; Raats, 1971, 1972; Warrick, 1974; Warrick and Lomen, 1974, 1976; Lomen and Warrick, 1974). The linearization of the nonlinear differential Eq. (15) or (19) is attained by applying the transformed water content S(0) function similar to the matric flux potential SO?) (Gardner, 1958) as

S @ ) = ? KO?)dP (21) Po

when po is a reference pressure defined by po = p(0 , ) . Usually the reference value is chosen such that po + - w.

In practice when water content varies over a limited range it is sometimes possible to approximate the nonhysteritic hydraulic conductivity function , KO?), as

K07) = K , exp (ap) (22)

where K, is usually the saturated hydraulic conductivity and a is a constant characteristic of the soil. Talsma (1963) showed that hydraulic conductivity of some field soils can be represented by Eq. (22) with values of a within the range 0.002 to 0.2 cm-’ (Philip, 1968; Braester, 1973; Bresler, 1977). Substituting Eqs. (2 1) and (22) into the nonlinear Eqs. (1 5) and (19) resulted in differential equations in the form

ae a2s a2s as a t ax2 a 2 ax -=-+--a-

For the transient case, Eqs. (23) and (24) can be linearized by assuming that de/dS is a constant (Warrick, 1974; Lomen and Warrick, 1974). For the steady state flows the left-hand side is zero and thus Eqs. (23) and (24) reduce to linear differential equations as used by Wooding (1968) and Raats (1972) for circular and line sources, respectively.

Solutions of (23) and (24) for strip, disk, point, and line sources have been obtained by Warrick and Lomen (1974, 1976), Warrick (1974), and Lomen and Warrick (1974). They assumed zero initial conditions (S = 0 at t = 0), and S was required to vanish at large distances from the sources. The boundary conditions at the soil surface (z = 0) were that of no vertical flow (except for the source) at r # 0 or x # 0, and that at r = 0 or x = 0 the surface source is allowed to change discretely with time. The latter enables one to simulate “on”-“off’ pulses in the irrigation cycle (Warrick and Lomen, 1974).

The advantages of these linear solutions are the existence of analytical solu- tions and the possibilities of adding single solutions together to represent complex distribution of sources (Warrick and Lomen, 1974). The main limita-

362 ESHEL BRESLER

tions are the error involved in the linearization procedure and the assumptions concerning the soil surface boundary conditions which are inadequate for the trickle-drip irrigation problem. As mentioned before (Section IV,C), the actual trickle source does not behave as an idealized point source since the water discharging from the emitter spread over a finite saturated area of the soil surface. It is more appropriate then to simulate trickle irrigation by an infiltra- tion from a shallow circular pond. Solution to steady state infiltration from such a circular pond has been presented by Wooding (1968). He used the steady state form of Eq. (23), i.e., ae/ar = 0 with boundary conditions appropriate to steady infdtration (the time variable [t] is excluded) similar to Eqs. (17a-f), but con- sidered a semi-infinite region with fixed saturation water entry zone (instead of prescribed discharge). Thus at the soil surface (z = 0) over a constant ponded water entry area, the soil is saturated, or in terms of the transformed variable S(P),

(254 Ks - K O 0

z = O ; O G r G p , ; S(O)=J K ( p ) d p = S , = -

where S, = S(p = 0) is the value of S in saturated soil, and pu is the ultimate radius of the ponded area.

Beyond the water entry zone over the nonwetted area of the soil surface, the vertical water flux is zero if evaporation may be neglected. This means that in this region

Po a

aP as a2 aZ

K - - K = O or - - K = 0 .

From Eq. (22) it follows that

and therefore

P l K $ K d p = - J dK Po KO

Integrating the last equation using Eq. (2 I), yields

a

During steady infiltration from a trickle source most of the soil surface in the region r 2 pu is air dry, or at least at a water content sufficiently low so that KO = K(Oo) is negligibly small, yielding K = ols. It follows, therefore, that approxi- mately

TRICKLE-DRIP IRRIGATION

as az

a t z = O ; p u < r < m ; - - -o rS=O

If one also assumes that at large distances from the source, content remains constant, one has

zz t r z + ~ , s = o

363

(25b)

the low initial water

It is generally convenient to take the radius pu as the length unit and to define dimensionless cylindrical coordinates .$ = r / p , and 5' = z/p,. The boundary conditions (25a-c) can then be written in terms of [ and 5' as

S(t,O = s(e,>, t2 + t2 -b O0

The problem was solved by Wooding (1968), who reduced the boundary conditions to a system of dual integral equations and used a modification of Tranter's (1951) method. His solution is restricted to values of crp, < 10, which are sufficiently good for all purposes of practical application to the theory. For example, (Y = 0.1 the value of pu is less than 100, which allows at least 2 meters spacing between emitters. For any smaller values of cr the maximum possible spacing is obviously larger.

The problem of steady infiltration from a field of equally spaced line sources was analyzed by Raats (1971). He used Eqs. (21) and (22) to derive hisgoverning equations similar to Eq. (24) with &Ofat = 0. His analytical solution showed that the flow pattern is a unique function of aix. Thus, when steady state plane flow trickle irrigation is considered, the main flow features depend both on soil char- acterization, a, and on the distance between the laterals or trickle lines, X.

F. COMPARISON WITH EXPERIMENTAL DATA

The solutions of the aforementioned mathematical models must be compared with experimental results in order to establish the reliability of the theoretical models, to evaluate the physical assumptions involved in them, and to ascertain the validity of the views expressed in the theories. There have been no compari- sons made between analytical solution and laboratory or field results reported in the literature. Very little information has been obtained regarding a comparison between numerical solution results and experimental data (Bresler el al., 1971; Bresler and Russo, 1975).

Laboratory experiments conducted under conditions similar to those assumed in the two-dimensional plane flow models (i.e., with Y < X so that Y + 0 was practically correct) have been described by Bresler ef al. (1971) and by Bresler

364 ESHEL BRESLER

L a= 0.983 crn’ cm-’ mi6’

FIG. 2. Vertical water-content distribution, e(0,z) in the plane of symmetry (x = 0). Computed results (solid lines) are compared with experimental data of Gilat loam soil (scattered points). The numbers labeling the lines indicate infiltration time (0; Q indicates the discharge per unit length. Note that the zero vertical plane (z = 0) is shifted and the data are accordingly translated along the z axis. After Bresler et ul. (1971).

TRICKLE-DRIP IRRIGATION 365

and Russo (1975). Quantitative comparisons between the theoretical and experi- mental water-content results are given in Fig. 2 , where the water-content profiles in the z dimensions are shown. This figure shows several sets of experimental data for each constant value of discharge rate (Q) together with the correspond- ing theoretical continuous curves. The theoretical results are calculated by the numerical method of Brandt et at. (1971) with the aid of soil-water parameters K ( 0 ) and p ( 0 ) of Gilat loam soil (Bresler et al., 1971).

The comparison between the calculated and experimental data was made for all times and points for which there were water-content determinations, and at all the tested trickling rates. In Fig. 2 the water-content distribution along the center vertical plane [O(O,z)] is presented for five different measurement times (t) and at four discharge rates (Q) given as cm3 cm-' min-' due to the plane geometry. Note that the zero point ( z = 0) was shifted and the data were accordingly translated along the z axis, in order to distinguish between the different sets. The shifting of the data provides easier identification of the experimental points which corresponded to the calculated lines. The quality of the agreement between calculated and measured water-content distribution can be observed from Fig. 2 .

In the case of infiltration into an air-dry soil, a clear boundary exists between the wetted zone and the dry zone in the wetting front. The wetting front is easy to determine experimentally with a minimum error. Furthermore, the wetting front is important from a practical point of view since it shows the boundaries of the irrigated soil volume. In the laboratory experiment with the initially air-dry

HORIZONTAL DISTANCE X (cm)

5 10 15

L

20 1

20 1 FIG. 3. Computed wetting front position (solid lines) as compared with the observed

one (dashed lines) for four infiltration times (the numbers labeling the lines). After Bresler and Russo ( 1 975).

366 ESHEL BRESLER

Gilat soil (Bresler et aL, 1971; Bresler and Russo, 1975) it was easy to identify the wetting front visually through the Perspex plates. For comparison, the location of the wetting front in the corresponding numerical solution was defined as a ridge (or line of maxima) of lgrad 0 I.

Figure 3 shows a comparison, with a relatively good agreement, between the location of the calculated wetting front and the observed one as a function of the space coordinates (x,z) and the total infdtration time. The total infiltration time is indicated on each wetting front line that appears in the figure.

The fact that the agreement between calculated and observed water-content distribution (Fig. 2) and wetting front (Fig. 3) is generally good suggests that model computations of water flow have little effect on discrepancies or agree-

1 1 1 1 I I -0.2 0. 0.2 0.4 0.6 0.8

RELATIVE CONCENTRATION (C-Col /Cn

FIG. 4. Vertical salt concentration distribution in terms of (c - C,)/c, in horizontal planes. Computed “plane flow” results (solid lines) are compared with measured chloride data (dashed lines). Note that the zero vertical plane (z = 0) is shifted as in Fig. 2. After Bresler and Russo (1975).

TRICKLE-DRIP IRRIGATION 367

ments between measured and computed salt flow data (Fig. 4). The compari- son between measured chloride concentration and computed solute distribu- tion data after T = 160 minutes of infiltration is given in Fig. 4 (Bresler and Russo, 1975). To be consistent with Fig. 9 the salt distribution data are expressed in a dimensionless relative form, (c - Co)/cn, where c and c, are the concentration in the soil solution at t = T and t = 0, respectively, and C, is inlet concentration. In Fig. 4 the data are presented along the vertical axis ( z ) at five different horizontal distances from the source: x = 0, 3, 7, 11, and 15 cm. Note that here also the zero point (z = 0) was shifted and the data were accordingly translated along the z axis in order to facilitate the comparisons and to distin- guish between the different sets. The agreement between theoretical and measured salt distribution results is shown in Fig. 4.

The agreement between laboratory and numerical water and salt flow results, which is as good as one could expect for laboratory techniques (Bresler et al., 1971 ; Bresler and Russo, 1975), shows that two-dimensional transport of solute and water during nonsteady trickle infiltration may be quantitatively predicted by the theoretical models. However, the prediction capability of the model and its applicability are restricted to the conditions which prevailed in the experi- ment and should be tested against further observations. In addition, the experi- mental data from a limited number of laboratory studies might be insufficient for general detailed model verification.

V. Soil-Water Regime during Trickle Infiltration

The effect of any irrigation method on the soil-water regime of a given soil depends primarily on the conditions prevailing at the soil-surface boundary. In the case of trickle irrigation, these conditions may be defined by the trickle discharge (Q) measured as the amount of water per unit time (or amount per unit time per unit length of laterals), and by the hydraulic properties and rate of evaporation at the soil surface which determine the horizontal area across which infiltration takes place. The rate of evaporation becomes an important factor only when the potential evaporation is extremely high and the saturated hydrau- lic conductivity of the soil is very low.

A. SIZE OF THE SATURATED WATER ENTRY ZONE

As already mentioned, it was observed that during trickle irrigation in general, a radial area of ponded water develops in the vicinity of the trickle source. This area is initially very small, but its radius becomes larger with time. Water from the emitter is able to infiltrate through this area into the soil, or evaporate from

368 ESHEL BRESLER

.01

it into the air, instantaneously. The soil-water content immediately beneath the ponded area, A = n p 2 , is equal to the water content at saturation, OS. This saturated area, the size of which is-a function of time, is the only place where water can infdtrate into the soil from the surface.

Figure 5 shows the calculated vertical water flux at the surface across the saturated water entry zone, as a function of infiltration time ( t ) and distance from the center of source. Since 0 = 0, in the interval 0 < x < p(t) , the hydraulic conductivity of the saturated zone is constant and equal to the saturated conductivity. Thus, the decrease in vertical water flux with time (Fig. 5) is due only to the decrease in negative value of the vertical hydraulic gradient at the

--K@s)

I I I I I I I

.Of

.O i

.o 6

.o : - C

'i . .O.c -

x 3 2.03

.O i

.o 1

0

-

2 cm o=o.ge3 crn' cm-' min-'

I I I I I 1 1 2 0 0 400 600 eoo 1000 1200 1x10

TRICKLE-DRIP IRRIGATION 369

saturated zone. Within the time considered, the hydraulic gradient has always been less than unity (the vertical flux of Fig. 5 is always greater than K,) but tends to approach a limit. The limiting value of the hydraulic gradient and the length of time involved in obtaining it depends on both the distance from the trickle source and its rate of discharge (Fig. 5). The larger the trickle discharge and the smaller the distance from the source, the faster the hydraulic gradient as a function of time approaches its limiting value. If a value of 1 to this limit were approached throughout the whole ponded area, then we would have from Eq. (170 p(t -+ m) + [Q/K(0,)n]'/2. However, since at finite value of time, the limiting hydraulic gradient is less than 1, then p ( f ) < (Q/nK,)'12 in cylindrical flow [p(t ) < Q/(2K,) in plane flow], if evaporation is not too significant. (If evaporation is important, then p(t)< {Q/[K,n + E] "2.),Neglectingevaporation the difference between p( t ) and [Q/(K,n)]'/2 [or Q/(2K,)] depends on the average vertical water pressure gradient at z = 0 over the interval 0 G r (or x) < p ( t ) [Eq. (29)]. Since the rate of change of this average value decreases with time (Fig. S) , the rate of growth of the saturated water entry zone also decreases with time, as shown in Fig. 6. From this figure one can see that at a relatively large infiltration time ( t ) , the radius of the saturated water entry zone ( p ) is much less than [Q/K,n)] ' I 2 (see Table I1 for values of K,). Figure 6 further indicates that the radius of the saturated water entry zone will always be larger as K , decreases and Q increases. This fact has a very important practical implication (see Section VII).

I

I E ""- 0.4 0=20 l i ter /hour l i ter/hour -

____-----

0- 0 50 100 150 200

INFILTRATION TIME, t (minutes) i0

FIG. 6. Radius of the saturated water entry zone p ( t ) as a function of infiltration time ( t ) for two soils and two discharge rates (Q). After Bresler (1977).

3 70 ESHEL BRESLER

Q.0.495 cm3 cm-' min-' Q.0.983 cm3 cm-l min" HORIZONTAL DISTANCE X, cm

0 8 16 24 32 LO 0 8 16 24 32 40 48 I

8

16

-

- -

FIG, 7. Water content filed as a function of cumulative infiltration ( V = 4') for two cases of discharge (Q). The numbers labeling the curves indicate water content (0). After Brandt et al. (1971).

TRICKLE-DRIP 1RRIGATlON 371

B. WATER-CONTENT DISTRIBUTION

To illustrate the water-content distribution, the plane flow model is consid- ered. Examples of water-content distribution for two cases of trickle discharge (Q), expressed in terms of discharge per unit length, are given in Fig. 7, which shows how the soil-water content changes with time and position during trickle infiltration. The illustration also shows the effect of trickle discharge on the water-content field. The saturated water-entry zone is shown by the particular line of saturated water content at the surface, where B s = 0.44. This zone is larger as the rate of trickle discharge increases. In the vicinity of the source (x =

0, z = 0 in Fig. 7), the moisture gradients increase when the rate of discharge decreases. These gradients can be calculated from the distances between lines of equal water content. This condition is reversed as the wetting front is ap- proached. The overall shape of the wetted zone also depends on the trickle discharge. The vertical component of the wetted zone becomes larger and the horizontal component narrower as the rate of discharge decreases.

C. WETTING FRONTS

The wetting front is an important factor in trickle infiltration because it indicates the boundaries of the irrigated soil volume. Figure 8 shows the location of the wetting front, as a function of the space coordinates (Y,z) and the total amount of infiltrated water, for two different trickle discharges and two soils differing in their hydraulic characteristics (Table I1 and Fig. 1-3 of Bresler et al,

RADIAL DISTANCE, r ( c r n ) 40

Q=4

(2.20 I i ter/hour

I

N 30

FIG. 8. Wetting front position as a function of infiltration time and cumulative infiltra- tion water (in liters) indicated by the numbers labeling the lines, for two soils and two d i s charges. After Bresler (1977).

372 ESHEL BRESLER

1971). The total amount of infiltrated water (in liters) is indicated on the wetting front lines appearing in the figure. The trickle discharge, Q, in terms of volume per unit time, and the soil textures, are also indicated in the figure.

The data presented in Fig. 8 demonstrate clearly that the rate of trickle discharge and the hydraulic properties of the soil have a remarkable effect on the shape of the wetted soil zone. Increasing the rate of discharge and decreasing the saturated conductivity result in an increase in the horizontal component of the wetted area and a decrease in the vertical component of the wetted soil depth. This is probably affected by the changes in the size of the water-entry saturated zone for each soil type and rate of discharge (Fig. 6). This ponded zone becomes larger as the soil becomes less permeable and as the trickle rate increases.

The possibility of controlling the wetted volume of any particular soil, and the water content distribution within its boundaries, by regulating the trickle dis- charge according to the hydraulic properties of the soil (Figs. 7 and 8), is of practical interest in the design of field irrigation systems (see Section VII).

D. EFFECT OF SURFACE EVAPORATION

A possible effect of soil-water evaporation, which occurs simultaneously with infiltration, on the water distribution pattern can also be tested by numerical models (Brandt ef al., 1971; Bresler, 1975). The results of water distribution data, using the numerical procedure of Hanks er a/. (1969) to evaluate evapora- tion as a function of soil-water content E(B), as well as a high potential evaporation value of E, = 10 mm/day, remain essentially the same as if evaporation were completely neglected. It appears, therefore, that generally water evaporation which takes place during infiltration is not an important factor in infiltration from a trickle source. This is so because usually free water evaporation rate (E,) is on the order of 1 cm day-’ or less and K , of many “normal” soils is on the order of 1 cm hour-’, so that E,/K, is generally less than 0.04. The net water flux into the soil anywhere in the saturated water entry zone is given by qnet = K,(dH/dz) -E,. Since during trickle infiltration the value of dH/dz is generally less than -1, the ratio between outward flow due to evaporation and gross inward infiltration flow is, in most practical cases, less than 4%. However, under very dry conditions where E, is extremely high, and in a soil of very low permeability where K , is extremely low, the effect of evaporation during infiltration from a trickle source may be important.

VI. Solute Distribution during Infiltration

As mentioned before (Section III,B), the accumulation of salt between emit- ters may be a serious problem for crops irrigated by trickling. Control of soil

TRICKLE-DRIP IRRIGATION 373

salinity regime is therefore essential to the permanent operation of a trickle irrigation system. In addition, since fertilization simultaneously with irrigation seems to be a good practice, optimizing the nutritional regime in the root zone depends on the possibility of controlling the fertilizer salts in the wetted soil volume. Salt distribution in the soil under trickle irrigation will be demonstrated here by an example involving the case of low concentrated inflow solution in the irrigation water that miscibly displaces a highly concentrated solution originally present in the soil. Interactions between the solute and the soil matrix are ignored. Two soil media with widely different hydraulic properties and salt dispersion characteristics are discussed.

The results presented in Figs. 9 and 10 were obtained from the cylindrical flow model when the hydraulic soil functions K(e) and p(8) were those of Gilat (loam) and Nahal Sinai (sand) soils (Bresler et ul., 1971, Figs. 1, 2, and 3). The soil diffusion coefficient was taken as Dp(0) = 0.004 exp (loo), and the dispersivities were chosen to be AL = 0.2 cm, and AT = 0.01 cm [Eq. (9)] . The soils are assumed to have initially uniform water contents and solution concen- tration [c,(r,z) = 52.35 meqbiter and 0,(r,z) = 0.213 for Gilat; c,(r,z) = 300 meq/liter and 0,(r,z) = 0.0375 for Nahal Sinai] with 0, = 0.265 and 0, = 0.44 for Nahal Sinai and Gilat, respectively. The “water capacity” (0, - O n = 0.227), the initial volumetric salt content (cnO, = 11.25 meq/liter soil), and COO,, which is the minimum possible volumetric salt content at the soil inlet, are taken to be the same for the two soils. [Inlet salt concentration was C,(t) = 5 meq/liter for Nahal Sinai and C,(t) = 3.0 meq/liter for Gilat soil.] Two cases of constant trickle discharge, Ql = 4 liters/hour and Q, = 20 liters/hour, are considered.

The general pattern of salt distribution in the two soils and for the two trickle discharges is demonstrated in Figs. 9 and 10. In Fig. 9 the salt concentration field is expressed in terms of salt concentration in the soil solution (as meq/liter bulk soil). Both figures show how the position of the dissolved salt field in the cylindrical flow pattern is influenced by the trickle discharge and the hydraulic properties of the two soils (Bresler et ul., 1971).

The fact that the water capacity (0, - On), the initial volumetric salt content (c,0,), and the minimum inlet salt content (COO,) are identical in the two soils, makes it possible to compare the salt distribution data (Figs. 9 and 10) of these two soils. To be able to compare the effect of trickle discharges and to place the two different rates on an equal basis, the results are compared for an identical amount of cumulative infiltration (12 liters) and not for an equal time allowed for the irrigation. It should be noted that the values of OS, O n , c,, C,, Q and total infiltration used in the examples, are in the range of practical and actual field conditions.

Figure 9 demonstrates the shape of the actual salt-concentration distribution. In both soils and with both trickle discharges, the solution concentration of the saturated water entry zone is identical to that of the infiltrated water. The concentration rises as the wetting front is approached and reaches its initial value

374 ESHEL BRESLER

RADIAL DISTANCE r (crn )

t c NAHAL SINAI (SAND)

FIG. 9. Computed salt-concentration field for two cases of trickle discharges (Q, and Q , ) and two soils. The numbers labeling the curves indicate relative concentration in terms of (C - Co)/cn. The numbers in parentheses are salt concentrations (c), in meq/liter soil solution. The heavy solid lines indicate the saturated water entry radius, ( p ) and the peripheral heavy lines are the wetting fronts. After Bresler (1975).

TRICKLE-DRIP IRRIGATION

RADIAL DISTANCE r (cm)

375

0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 40 45 50 55 1

5

10

15

20

NAHAL SINAI (SAND1

FIG. 10. Computed volumetric salt-content field (ce) for two cases of trickle discharges and two soils. The numbers labeling the curves indicate volumetric salt content (ce) in meq/liter bulk soil. Heavy solid lines indicate the same as in Fig. 9. After Bresler (1975).

near the wetting front boundary. It is clear that the shape of the overall wetted zone, which lies between the heavy lines (Figs. 9 and lo), is soil- and discharge- dependent. Therefore, the salt distribution pattern should also be affected by these properties. Thus, the completely leached zone (c - C,)/c, = 0, remains at the water-saturated zone in the sandy soil but penetrates to a deeper depth in the loam soil. In addition, the leached part of the soil.in the vertical component of the wetted zone is deeper, and in the radial component it is narrower, as the soil becomes coarser (having higher hydraulic conductivities) and as the trickle discharge becomes slower.

376 ESHEL BRESLER

The picture for the volumetric salt-content field (Fig. 10) differs from that of Fig. 9 owing to the fact that the salt-content pattern is affected by the distribution of both water content (0) and salt concentration (c). Thus, although the pattern of the leached part, close to the saturated zone, is similar in Fig. 10 to that of Fig. 9, the accumulation part-close to the wetting front-is com- pletely different. Here (Fig. 10) the salt quantities from the leached part of the soil are accumulated and reach a maximum at a certain distance from the source. The location of this maximum salt accumulation zone is largely dependent on both soil hydraulic properties and the discharge rate of the trickler. The size of this maximum salt content, however, is affected mainly by the soil properties and not by the discharge rate.

The general pattern of salt distribution and its dependence on initial salinity, salinity of the irrigation inlet water, rate of trickler discharge, and the hydraulic characteristics of the soil (Figs. 9 and lo), are of practical interest for problems connected with the design of a field irrigation system, in order to control the soil salinity or fertility in the wetted root zone. Consider, for instance, an initially saline field being irrigated by a set of symmetrical emitters sufficiently far apart. Suppose that it is important to evaluate the leaching effectiveness of removing salts from a given root volume in which the main plant roots function. With an infiltration duration large enough, depending on soil and discharge rate, this leached zone may be sufficient for most of the roots to concentrate and function without disturbance (Tscheschke et al., 1974; Shdhevet and Bernstein, 1968). On the other hand, a large quantity of salt may accumulate to an appreciable level at a certain distance from the source close to the wetting fronts (Fig. 10). It is apparent (Fig. 10) that one can overcome this limitation to plant growth by changing the discharge of the trickler, its position, or both.

Applications of the data of Fig. 9 to fertilization problems are possible. In this case the complete leached zone (c - C,)/C, = 0 may be interpreted as the part of the root zone in which the concentration of the nutrient in the soil solution is identical to its concentration in the irrigation water. Knowing t h s region is of practical importance from the point of view of efficient use of fertilizers under trickle irrigation.

VII. Application of Infiltration Models to

the Design of Trickle Irrigation Systems

The problem of designing trickle irrigation systems involves the determination of trickler discharge rate, spacing between emitters, as well as diameter and length of the lateral system for any given set of soil, crop, and other field conditions.

TRICKLE-DRIP IRRIGATION 377

A. TRANSIENT STATE INFILTRATION

In the previous sections, analytical and numerical models of the more natural transient infiltration from trickle sources were discussed (Warrick and Lomen, 1974; Brandt ef ul., 1971). From these models wetting fronts and the pattern of two-dimensional distribution of a specific soil-water variable [S(O), p(B) , or B ] can be obtained for soils with various hydraulic properties and for different trickle discharges. The results, such as those presented in Figs. 7 and 8, can be applied in order to estimate the actual spacing between emitters for any given set of crop, soil, irrigation water quantity, and trickle discharge. Thus, for a particu- lar crop and given field conditions, the optimal combination of discharge rate, amount of irrigation water, and emitter spacing can be obtained. For these purposes comprehensive numerical models (Brandt et ul., 1971; Bresler et al., 1971; Bresler, 1975) are preferred. However, in some specific problems when analytical solutions exist, they may also be applied to a complex field problem. An additional useful application of linearized analytical solution to transient flow problems is that they can serve as a check for more comprehensive numerical models (Warrick and Lomen, 1974).

B. STEADY INFILTRATION

The conditions of steady infiltration are not generally met in the field but may be approximated when irrigation is continuous for a relatively long time, or in intermittent irrigation when irrigation is very frequent. Water content pulses resulting from intermittent infdtration may be damped out a few centimeters from the source for many soils (Rawlins, 1973). This makes it possible to consider flow beyond this region to be essentially steady for continuous irriga- tion when infiltration takes place during a relatively long time, and for inter- mittent irrigation when irrigation frequency is of the order of a day or less. Of course, a truly steady flow can never be achieved during trickle infiltration but it represents an asymptotic case which can be used for practical purposes of designing trickle irrigation.

The case of steady infiltration from a shallow circular ponded area on the horizontal surface of a semi-infinite soil, as treated by Wooding (1968), is considered to be a useful designing tool appropriate for application purposes. T h s is so because the actual trickle source does not behave as an idealized point source as assumed by other steady models, but the water is spaced over a finite circular water-saturated area on the soil surface. This radial saturated water entry zone is initially vely small but its radius p(t) becomes larger as time increases and approaches a constant value p u at some finite infiltration time (Fig. 6). The

378 ESHEL BRESLER

ultimate size of this zone depends primarily on the saturated hydraulic conduc- tivity of the soil and the discharge rate of the dripper. An estimate of the relationships between the ultimate radius (p , ) of the ponded-saturated water entry zone, the soil hydraulic properties K , and a, and trickle discharge Q, may be obtained from Eq. (64) of Wooding (1968) as

Q = T P ~ K , + 4K,pU/a (27)

Equation (27) is equivalent to the boundary conditions in Eq. (17f) for the time in which the ultimate values of vertical water fluxes and hydraulic gradients are approached throughout the ponded area (Fig. 5). For this ultimate time we have from Eq. (170 (neglecting E),

Q = npiK , -+ nK,pi Icl (28)

where Ic 1 is the absolute value of the average vertical component of the pressure head gradient in the ponded zone defined by

In Eqs. (27) and (28) the first term on the right-hand side (RHS) of both equations represents the flow as a result of gravitation only, while the second term on the RHS represents the flow as a result of the pressure head gradients in the saturated water entry zone. Solving the quadratic Eqs. (27) and (28) for the positive root of p u , the ultimate size of the saturated zone is obtained as

It is clear from Eq. (30) that p u becomes larger as Q increases and K , decreases and that: (a) when Q + 0 then p u + 0; (b) when Q is very large, p u is large and the effects of a are negligibly small; and (c) when vertical pressure gradients are negligibly small, pu reaches its largest value of (Q/zKs)"'. Also p u decreases as a increases.

C. ESTIMATION OF SPACING BETWEEN EMITTERS

To estimate the actual spacing between emitters for a given set of field conditions, the hydraulic properties of the soil [K(8), p (8 ) or a, and K,] , and the desired value of critical water pressure p = p c midway between emitters must be known or estimated for any particular case.

Using this information one can calculate wetted soil volume (Fig. 8) or the distribution of p , 0 , or S, i.e., p(r ,z) , 0(r,z) (Fig. 7), or S(g,a), from which the desired spacing can then be determined. For convenience and general designing

TRICKLE-DRIP IRRIGATION 379

purposes, it is sometimes sufficient to have the distribution of p , 8, or S at the soil surface, as given by 8(x,O) in Fig. 7 or by S(l,O) in Fig. 11. Figure 11 was reconstructed from Wooding (1968) and gives the calculated distribution of S(t,O)/S, for seven values of a = apu/2. Because a unique relationship between S and 8 and between 0 and p (nonhysteritic case) exists, the data in Fig. 11 represent the soil surface distribution of water content as well as the water

DIMENSIONLESS RADIUS E

FIG. 11. Soil surface values of S/S, as a function of 5 = r / p , for seven different values of a = aU/2 (the numbers labeling the lines).

380 ESHEL BRESLER

pressure head. To obtain p(r,O) and e(r,O) values from Fig. 11, when a K,, and pu are known, one has to use Fig. 11, Eq. ( 2 2 ) and the relationships

The values of 0 can then be taken from the O ( p ) relationship. It should be noted that due to the linear steady state form of Eq. (23),

solutions from a single source as given in Fig. 11 can be added together to solve for a more realistic situation of multiple sources field. Furthermore, the general solution to the linearized steady state form of Eq. (23) or the flow equation (1 5) can be used for many particular and practical field problems, although it includes some errors owing to the linearization procedure and to the steady flow assumption. The assumptions of no surface water flux away from the water entry zone and of a constant (Y value may also be inadequate for some field problems (Bresler, 1977). In addition, it cannot be claimed that Eq. (22) is universally exact, but it does model the observed rapid nonlinear decrease of K with p in unsaturated soil (Philip, 1968). Values of a are generally smaller in fine-textured material and larger in coarse-textured material (Bresler, 1977), where gravity may be relatively more important than capillarity for water movement. However, in spite of these limitations the linearized steady state solution may be a useful alternative to the nonsteady numerical solution in determining the spacing between emitters in the irrigated field according to the rate of discharge, the hydraulic properties of the soil, and the desired soil water pressure between emitters.

The general procedure of calculating the spacing between emitters as a func- tion of discharge and midway critical pressuree pc involves the following steps (Bresler, 1977):

(1) Estimate K(0) and p ( 0 ) or K , and a for any given soil (e.g., Figs. 2 and 3 of Bresler e l al., 1971 or Table I of Bresler, 1977).

(2 ) Estimate p u from Eq. (30) as a function of Q for any soil using the pre- determined value of K , and a, or K(0) and p(0) , and calculate a = p u a / 2 .

(3) Select a critical value of 0, to be used and estimate pc from the p(0 , ) characteristic curve (e.g., Bresler er aZ., 1971, Fig. 2) or select directly a value of pc to be used.

(4) Calculate K(p,) from Eq. (22), S, from Eq. (32), and S(p,)/S, from Eq. (3 1) for any soil.

(5) Obtain ,$(pc) from Fig. 11 using S(P,)/S, and a = apu/2 or from a figure similar to Fig. 7 or 8. Note that S@,)/S, = [K(p,) - K O ] / (K , - K O ) X K(pc) /K,

(6) Calculate the distance (half spacing) between emitters as a function of the = exp (olpc).

TRICKLE-DRIP IRRIGATION 38 1

I0

." DISCHARGE 0 f l i t G / h o i r r )

FIG. 12. Estimated spacing between emitters r = rc(Pc,Q) = d / 2 for sandy soil (K, = 0.13 cm minute-', cx = 0.09), as a function of trickle discharge, Q, and selected values of pore-water pressure head (p = p,) midway between emitters (the numbers labeling the lines).

selected pressure head and trickle discharge [d = 2r,(p,,Q)] from rc = t C p u , or obtain it directly from a figure similar to Fig. 7 or 8.

(7) Graph the results (e.g., Fig. 12). A figure such as Fig. 12 may be used as a practical nomogram to calculate the

distance between emitters if their rate of discharge is known, to select the desired discharge when spacing is given, or to find the best spacing-discharge combination when a certain economic criterion has to be achieved. In applying the results as given in Fig. 12 only one iso-p, line must be used, since any d-Q combination depends on the chosen value of p c . This value must therefore be determined before any d-Q selection is made. The selection of p c is somewhat arbitrary and depends on the safety factor that one would like to choose. This is so because of the uncertainty involved in the response of plants to p, and to the degree of partial wetting of their root zone. It should also be remembered that in calculating the data of Fig. 12 from Wooding's method a single emitter (no interaction) was assumed. With this assumption a safety factor has already been

382 ESHEL BRESLER

taken into account regardless of the value of p c chosen. As a general guideline it is recommended that the value of p c to be selected, be the one corresponding to the transition between the lowest and highest values of a p / & (i.e., when lgrad 0 I starts to increase steeply with r). This is, of course, an approximate average p c value which may be lower (more negative) for less sensitive crops and must be higher when a specific crop sensitive to soil-water stress is grown.

D. NUMERICAL EXAMPLE

Evaluation of the applicability of any estimation method must be obtained by comparing its results with actual trickle infiltration field data. Since a good agreement exists between the numerical method and experimental field and laboratory results (Bresler et al., 1971), the steady state results of Wooding (1968) were compared with simulated transient flow data obtained by the numerical method proposed by Brandt et al.. (1971). This is, of course, a first approach which should be fully examined in the field.

The parametric soil data given in Table I1 were taken from Bresler et al. (1971) for the two soils studied. In calculating K(p,) it was assumed that the relation- ship (22) holds for values of p which are smaller than the air entry value of pa of each soil. Values of S were calculated from Eq. (31), with K values substituted from Eq. (22) and S, values from Eq. (32), or simply by S/S, = exp(.p,).

The soil parameters given in Table I1 and Figs. 1 to 3 in Bresler el al. (1971) were used to calculate the midway distance r , corresponding to p c for the two soils and two different trickle discharges using the data of Fig. 11 and the results of the numerical solution (Num. Soln.) of Brandt et al. (1971).

The data presented in Table I11 show that with the described estimation methods one is able to calculate radial distances at the soil surface which correspond to a given soil-water pressure (or water content). It should be emphasized again that the proposed methods are good approximations to actual field conditions as long as the solutions simulate the actual field situations. This was previously shown in some limited, specific cases (Bresler et al., 1971).

E. EFFECT OF SOIL HYDRAULIC PROPERTIES O N SPACING-DISCHARGE RELATIONSHIPS

The effects of soil hydraulic properties on the spacing-discharge relationships are demonstrated in Figs. 6 and 13. Figure 6 , which represents the radius of the saturated-ponded water entry zone as a function of time for two discharge rates and the two soils, demonstrates mainly the effect of the saturated K , on the spacing-discharge relationships. The effect of (Y (or K(p) and p(O)] on the

TABLE I1 Parametric Soil Data

Gilat (loam) -40 0.014 0.56 0.025 0.28 -80 -40 5.15 X 0.367 Mahal-Sinai (sand) -15 0.142 3.15 0.062 0.11 -5 5 4 0 2.35 x lo-' 0.166

TABLE 111 Midway Distances (rc) between Trickle Emitters Calculated by Linearized Steady State Solution (Lin. Soln.) and Numerical Transient State

Solution (Num. Soh.) for Two Soils and Two Rates of Discharge

Q Pu Pu rcVc) 'c(Pc)

Soil hour) (cm) (cm) (a pu/2)' (Fig. 11)' (cm) (cm) a (Lin. Soln.) (Num. S o h ) (Uter/ (Lin. Soh.) (Num. Soln.) W C )

Cilat (loam) 4 21.1 21.0 0.32 1.49 31.4 33.1 20 65.2 61.1 0.98 1.22 79.5 66.1

Nahal-Sinai (sand) 4 5.7 5.9 0.18 3.55 20.2 20.7 20 18.9 17.9 0.59 2.10 39.7 30.3

aDimensionless.

TRICKLE-DRIP IRRIGATION 385

250 I I I I p c = -70 cm -

0 4 0 12 16 20

DISCHARGE Q (-) l i t e r

hour

FIG. 13. Distance between emitters (d) as a function of trickle discharge Q) for two soils and two values of pC After Bresler (1977).

spacing-distance (d-Q) relationships for a given p c is illustrated in Fig. 13. According to Philip (1968), a is a measure of the relative importance of gravity and capillary for water movement in a particular soil. In soil with a high a value gravity tends to dominate and in soil with low (Y values capillary tends to dominate. This tendency is more pronounced at smaller rates of discharge (Q). Figure 13 also illustrates the effects of soil-water properties and the rate of discharge on the selection of spacing between emitters. Larger spacing is per- mitted in soils with lower values of K , and a (see Table I1 and Fig. 13) and also when the crop grown is not sensitive to water stresses and/or to partial soil wetting (higher p c values are permitted). Closer spacing is required for soils having higher K , and 01 values when a sensitive crop is being grown. For any given soil the emitter spacing can be increased as the discharge rate becomes higher. However, since the rate of growth of rc or d with Q decreases as Q in- creases (Figs. 12 and 13), the proper choice of Q chiefly depends on some optimization criteria in the engineering design of the field irrigation system.

F. DESIGNING THE LATERAL SYSTEM

The data as given in Fig. 12 or 13 can be combined with principles of hydraulics in order to obtain diameter (0) and length ( L ) of the lateral system for design purposes (Bresler, 1977). For a given d-Q combination and uniform- ity criteria, the L-D design relationship and the necessary pressure head at the lateral inlet depend upon emitter discharge function, elevation changes, reduc-

386 ESHEL BRESLER

tion coefficient for dividing flow, and pipe roughness coefficient. The Hazen- Williams equation accounting for dividing flow between emitters is

HL = 2.78. F*L*D-4*8' (6N/C)1.85 (33)

where HL = friction head loss in laterals (meters)

N = number of emitters per lateral F = reduction coefficient for dividing flow between emitters along the

L = the lateral length (meters) D = inside diameter of the lateral pipe (meters) C = the Hazen-Williams roughness coefficient

= average rate of emitter discharge (meters3 hours-')

lateral

Equation (33) is an empirical equation. Care should be taken in using the units specified to each of the above-dimensioned variables. The empirical values of F and C have been tabulated (compare, e.g., Howell and Hiler, 1974). As the spacing between emitters along the lateral is given by d = 2r, (Figs. 12 and 13), it is possible to express N as

(34) N = - L =4

2r, d where d is the distance between emitters along the lateral. Substituting L / d for N in Eq. (33) and rearranging

L = 88.88 D'.708 (HL/F)0.351 (Cd/a)"649 (35)

Knowing the Q-d relationship (Fig. 12 or 13) and assuming F to be constant over a given range of L and d, Eq. ( 3 5 ) gives the L-D relationships for any preselected value of head loss HL .

To select an appropriate value of HL for Eq. (39, a proper criterion may be based on the differences between the emitter discharge at the lateral inlet and the downstream discharge, relative to the average discharge,

Q i - Q d G E Q

Here Qi is the inlet discharge, Qd is the downstream discharge, and E is a preselected error fraction, say 0.05 or similar.

The relationship between emitter discharge rate and the hydraulic head at the emitter may be given by the empirical expression

Q = b @ (37)

where b and p are constant characteristic of the flow regime in the emitter and H is the hydraulic head at the emitter. Data of b and are available from the

TRICKLE-DRIP IRRIGATION 387

manufacturer but also can easily be determined experimentally in the labora- tory.

Using the maximum value of e it follows from Eq. (36), using Eq. (37), that

(38) H f = - €0 +Pd

b

Since HL = Hi - H d , then

Knowing both the pressure head at the lateral inlet and the emitter constants b and 0, permits the calculation of HL from Eq. (39) for a given and Q. This value of HL is then substituted into Eq. (35) to obtain the D-L relationship needed for the lateral design. When the lateral length is given by the size of the plot and Hi is known, the diameter D is calculated from Eq. (35). Otherwise, the optimum economic 0-L-Hi combination has to be calculated for each field and soil condition.

In summary, it is suggested (Bresler, 1977) that steady and nonsteady infdtra- tion models, which are well suited for the analysis of unsaturated flow through porous media, can be applied to design a trickle irrigation system. The two modeling approaches make it possible to calculate the spacing between emitters as a function of their rate of discharge, soil hydraulic properties, and crop sensitivity to water stress. However, it is important to remember that each of the proposed methods has certain limitations. For example, some of them involve errors that arise from the linearization procedure and from the estimation of K , and a. The assumption concerning the steady state flow is a very restricting one. It should also be emphasized that problems are involved in selecting the correct hydraulic parameters of the soil: K(O) and p(O). In addition, seepage of unused water below the rooting zone is not considered when S/S, (Fig. 1 I), O(r, o), or p(r,o) is taken at the soil surface. It is also emphasized that problems are involved in seiecting the correct p c value. Additional research is needed to ascer- tain the validity of the views expressed in this chapter, to develop field methods for determining the necessary soil-water parameters, and to select the best mid- space pressure head for a given set of soil, climate, and crop growing conditions (Bresler, 1977).

VIII. Water Management in Marginal Soils

Hardpans of various types, different sands and sand dunes, desert pavement of many kinds, and saline and alkali soils are very common marginal soils in arid

388 ESHEL BRESLER

zones of the world (see, e.g., Gile, 1961; Litchfield and Mabbutt, 1962; Ives, 1959; Marbut, 1935; Burvill, 1956; Pennefather, 1951). In addition, in many areas of the humid tropics, soils are leached and become very acidic. Soil acidity is generally associated with aluminum toxicity, which limits the rooting depth, especially of those crops which are sensitive to aluminum (Charreau, 1974; Wolf and Drosdoff, 1974; Wolf, 1975).

Most of the above-mentioned marginal soils are characterized by low values of “available water” and/or “water-holding capacity,” properties often associated with limited rooting systems. Moreover, in many parts of the humid tropics and of semiarid zones, rains are either inadequate in total amount or irregular in annual distribution. This unfavorable climate-soil combination tends to produce soil-water deficits which in turn cause water management to be a critical factor for successful agriculture. This is so because of the unfavorable soil-water properties of these marginal soils in combination with the occurrence of long dry periods and the limited root growth owing to hardpans, desert pavement, salinity, aLkalinity or acidity, and aluminum toxicity. Obviously, a crop yield can not be obtained without irrigation during the dry seasion, in any kind of soil.

It appears that, since irrigation must be applied in these areas, a trickle irrigation system may be the preferred one in these marginal soils, for the following reasons. The method is capable of delivering water into the soil in small quantities as often as desired, so as to maximize irrigation frequency without any additional costs. As irrigation frequency increases, the infiltration period becomes the most important part of the irrigation cycle. When irrigation frequency is sufficiently high, so that the irrigation cycle is dominated by infiltration rather than by the extraction stage, the water-holding capacity or water-availability properties of any marginal soil become relatively unimportant. This is so because the soil-water regime is continuously maintained at a relatively high water-content level so that water is supplied to the crop as it is needed and there is no need to store water within the limited soil-root zone (Rawlins, 1973). Not having to bring water from storage to the limited root zone also eliminates the possible negative effect of fluctuations in soil-water content. The effect is probably more severe as the soil becomes more marginal with respect to its water-holding properties. Water management under high-frequency trickle irriga- tion therefore renders the unfavorable water-storage properties of many marginal soils essentially unimportant. Irrigation management therefore involves optimiza- tion in the design of spacing between emitters and the lateral system (Section VII), as well as control of the quantities of water to be applied in order to meet the crop requirements and to supply the amount of water needed to pass through the effective root zone to avoid salinity buildup (see Section VI).

The quantity of water to be applied by an emitter in each single irrigation for any day after planting may be calculated from

TRICKLE-DRIP IRRIGATION 389

4 '(t) = R '(r)&(t)I(t)m$ (40)

where 4' is the quantity of water to be applied by an emitter at the day t after irrigation, E, is the class A pan evaporation between the preceding and the present irrigation, Z is the irrigation interval, r , is the radius of area wetted by an emitter, and R'(t) = ET(t)/E,(t) is the ratio between evapotranspiration (ET) and E, for the period t. This ratio as a function of number of days after irrigation, must be obtained experimentally. Note that this calculation does not take into account the quantity that must pass the effective root zone to avoid the hazard of salinity, which was discussed in Section VI.

An additional management problem common to many marginal soils is that they may have a low fertility status-as in sands or sand dunes, or a high capacity to fuc phosphorus so as to make it unavailable to crops [e.g., marginal soils in the humid tropics (Wolf, 1975)l. The combination of low fertility status, high aluminum concentration, and phosphorus fixation may create an additional serious soil limitation to agricultural development. These soil fertility problems may be controlled by applying fertilizers simultaneously with irrigation through the trickling system (Shani, 1973). By using the proper management practice, one is able to optimize this system with respect to the nutritional balance and water status.

LIST OF SYMBOLS

The following symbols are used in this chapter:

a = ap,/2 = parameter proportional to the length scale, dimensionless b = constant, Lz T-' c = solute concentration in the soil solution, C = Hazen-Williams roughness coefficient, dimensionless C, = solute concentration of the irrigation water, I W L - ~ D = inside diameter of the lateral pipe, L d = distance between emitters, L D, = soil diffusion coefficient, L 2 T-I Dq = hydrodynamic dispersion tensor, L 2 T-' E = evaporation flux, LT-' E, = class A pan evaporation rate, LT-' ET = evapotranspiration rate, LT-' F = reduction coefficient for dividing flow, dimensionless (? = average vertical pressure head gradient over the ponded area at the soil surface,

dim ensionless H = hydraulic head, L Hi = inlet head, L Hd = downstream head, L HL = head loss in lateral, L I = irrigation interval, T

390 ESHEL BRESLER

K$ = saturated hydraulic conductivity tensor, LT-' K,.(e) = relative hydraulic conductivity, dimensionless K = K ( 0 ) = K @ ) = Kr(e)KS = capillary conductivity in isotropic media, LT-' L = lateral length, L N = number of emitters per lateral, dimensionless p = pore-water pressure head, L pa = air entry value of p , L Q = rate of discharge from emitter, L 3 T-' Qi = inlet discharge, L 3 T-' Qd = downstream discharge, L 3 T ' q = specific solution flux (Darcy's velocity), LT-' q' = quantity of water to be applied by an emitter, L 3 r = radial coordinate, L R = radial boundary of flow region, L R' = ratio between evapotransipration and class A pan evaporation, dimensionless rw = radius of the wetted area, L S = S(0) = S@) = transform water content, transform pore-water pressure head, L z T-' r = time, T T = end time of infiltration, T x i = Cartesian coordinate, L x, = vertical coordinate, L x,y = horizontal coordinates, L Y = crop yield per unit land area, V = average solution flow velocity, LT-' z = vertical coordinate, L Z = vertical boundary of flow region, L 01 = constant, L-' p = constant, dimensionless y = horizontal or radial coordinate, L f = relative vertical coordinate, dimensionless 0 = volumetric water content, dimensionless @ = soil-water regime index, dimensionless 5 = average value of 0, dimensionless @ = deviation of 0 from &, dimensionless E = discharge difference fraction, dimensionless uz = variance of @(t), dimensionless h~ = longitudinal dispersivity, L AT = transversal dispersivity, L 5 = relative radial coordinate, dimensionless p = radius of the ponded-water entry zone, L

Su bscrzp rs

o = reference value usually air-dry water content N = initial value s = value at saturation u = ultimate value, limiting value c = selected critical midway value

TRICKLE-DRIP IRRIGATION 39 1

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