IV. Modeling of Water and Salt Flows
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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 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.
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 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 )
(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 D p ( 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
Nx,y,z,t) as
I.
{
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
(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
e = e,
q e ) + E -K(s)&=
aZ
o
for
o < x 2 t y z G [p(t)]
x2 + y 2 > [p(t)l2
for
aP
1
/[K(B,) + E - K(0,) az 1 dx dy = z Q ( t >
357
(1 1)
(12)
(13)
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-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
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
e=O,;
-=
ae 0;
ar
ae
-=O;
az
8=8,;
E-K(O,)--O;
aHaz
27r
O
O
t=O
r = O , r = R ; O G z < Z ; O
(174
(17b)
O
z=Z;
0
(17c)
O
z=O
0
(1 7d)
p(t)
0
(17e)
[E -K(B,)$
aH
rdr = Q(t)
0
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)
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):]
at
ax
t
aza [zqe)Y
az
ac
ac
a
ac
ac
a(ce)- a (Dxx- + D,, -- qxc) t - (LIZ,- t Dzx- - q,c)
(20)
at ax
ax
az
az
aZ
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.
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 =
+
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
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@)=?
Po
KO?)dP
(21)
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 ,
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. (15) and (19) resulted in differential
equations in the form
a2s
-ae= - +
- - a -a2s
ax2 a 2
as
ax
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
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 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(P),
0
z=O;
OGrGp,;
S(O)=J K ( p ) d p = S , = -Ks - K O
Po
a
(254
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
aP
K--K=O
as
--K=0.
or
a2
aZ
From Eq. (22) it follows that
and therefore
P
l K
Po
KO
$ Kdp=-J
dK
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 approximately
TRICKLE-DRIP IRRIGATION
atz=O;
pu
as
---orS=O
az
363
(25b)
If one also assumes that at large distances from the source, the low initial water
content remains constant, one has
zz
trz
+~,s=o
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
O0
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.
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 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
364
ESHEL BRESLER
a=0.983 crn’
cm-’ mi6’
L
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).