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IV. Modeling of Water and Salt Flows

IV. Modeling of Water and Salt Flows

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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.


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 frequency. As the frequency increases, the infiltration period becomes more important 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 2 Y , 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 independent unit in the sense that there is no flow from one element to another.

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



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.


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 perpendicular to the mean direction of flow. T h s indicates that for such media the

transport of the dispersed material can be defined by two characteristic dispersion 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 6t~(A, ~ - hT) vpy~vi t D p ( e )


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,



and D p ( e ) is the soil diffusion coefficient as defined by Bresler (1973) using Eq.

(57) of Olsen and Kemper (1968).


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

Nx,y,z,t) as



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

a e / a y = O a t y = 0 and y = Y f o r t > O

ae/& = 0 at z = z

for t > 0


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


e = e,

q e ) + E -K(s)&=




o < x 2 t y z G [p(t)]

x2 + y 2 > [p(t)l2




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


(1 1)




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


where 0, is the initial soil-water content.


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-dimensional 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 t o 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



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


ae 0;












r = O , r = R ; O G z < Z ; O










(1 7d)




[E -K(B,)$


rdr = Q(t)


where T is the end time of infiltration.

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

Finally, the initial conditions are

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


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



ae = a [~(e):]




aza [zqe)Y







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


at ax






where the subscripts x and z denoting the flow direction x l and x 3 , 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 precipitation, 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.


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 =


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.

(c G)l’’;

2. Linearized Water Flow Solutions


Solutions for transient and steady infiltration from point, line, strip, and disk

sources which can be applied to simulate trickle irrigation cases have recently


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





when po is a reference pressure defined by p o = 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 ,


K07) = K , exp (ap)


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. (15) and (19) resulted in differential

equations in the form


-ae= - +

- - a -a2s

ax2 a 2



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 solutions and the possibilities of adding single solutions together to represent

complex distribution of sources (Warrick and Lomen, 1974). The main limitaat



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 infiltration 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 considered 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(O)=J K ( p ) d p = S , = -Ks - K O




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








From Eq. (22) it follows that

and therefore


l K



$ Kdp=-J


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


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 approximately









If one also assumes that at large distances from the source, the low initial water

content remains constant, one has




It is generally convenient to take the radius pu as the length unit and t o 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


The problem was solved by Wooding (1968), who reduced the boundary

conditions t o 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 characterization, a,and on the distance between the laterals or trickle lines, X.


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 comparisons 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



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).

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IV. Modeling of Water and Salt Flows

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