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Measuring and estimating coordinates of wetted bulb front ...

Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011 505 Journal of Food, Agriculture & Environment Vol.9 (1): 505-509. 2011 www.world-food.net

WFL Publisher Journal of Food, Agriculture & Environment Vol.9 (1): 505-509. 2011 www.world-food.net

Science and Technology

Meri-Rastilantie 3 B, FI-00980
Helsinki, Finland
e-mail: [email protected]

Measuring and estimating coordinates of wetted bulb front in soil under point surface
emission

Ahad Molavi *, Amir Hossein Nazemi, Aliashraf Sadraddini and Ahmad Fakheri Fard
Department of Water Engineering, Faculty of Agriculture, University of Tabriz, 5166616471, Tabriz, Iran.

*e-mail: [email protected]

Received 2 November 2010, accepted 4 January 2011.

Abstract
Trickle irrigation is one of the known techniques that can increase farm water use efficiency, provided that the emitter spacing, discharge and other
hydraulic characteristics are designed properly. Knowledge about development of wetted bulb through soil can be involved in designing optimum
arrangement of emitters and their flow rates. Correct usage of trickle irrigation system is a prerequisite for its progress and development, and perhaps
wrong design leads to this conclusion that trickle irrigation is not an efficient system. In this study, coordinates of the developing wetting fronts
through loam and sandy loam soils, resulting from surface drippers with inflow rates of 2 and 4 L h-1, were measured using trenching method and also
those were predicted with respect to the Green-Ampt potential parameter, the Green-Ampt water content parameter, saturated hydraulic conductivity,
radius of saturation part, and the reciprocal of the macroscopic capillary length scale, using Philip and Chu’s methods. Measured coordinates of the
developing wetting fronts were compared to those predicted by Philip and Chu’s methods. The results showed that in most cases the solutions of
Philip’s method were more accurate than those of Chu’s method, although solutions of the latter were also acceptable. The root mean square errors
between the observations and results of Philip and Chu’s methods were 15.88 and 32.06 mm in loam and also 12.755 and 26.722 mm in sandy loam
soils, respectively. Due to simplicity of the Chu’s method it is recommended as a suitable and applicable procedure. Final radii of the saturated zone
were measured for different discharges of emitters indidually and on the basis of these discharges equations were obtained for the both loam and sandy
loam using regression analysis.

Key words: Wetted bulb, wetting front coordinates, trenching, emitter.

Introduction of wetted bulb using Green-Ampt 13 analysis. In addition, in this
Proper design of trickle irrigation systems needs understanding of research the profile of wetted bulb’s front was measured and
the distribution of water in the soil that can be predicted by solving compared with that of Chu’s method.The purpose of this research
the related governing equations 1, 2. In comparison with other is to predict wetting front’s coordinates under surface trickle
irrigation systems, water-holding capacity of soil plays a less irrigation by Philip and Chu’s methods using soil physical
important role in drip irrigation. However, the hydraulic features specifications. Also the prediction abilities of these methods are
are critical because they control the infiltration phase 3, 4. The discussed on the basis of statistical comparison of the results with
advance of wetting front in a soil under an irrigation point source measured wetted bulb coordinates in two soil types.
is related to soil type, initial moisture content and water application
rate 5, 6. Ben-Asher 7 described an effective hemisphere model for Application of Green-Ampt analysis in estimation of wetted
determination of the infiltration rates under a surface emitter. Zur 8 bulb front: Chu14 derived Equation 1 to determine the coordinates
assumes that the volume and geometry of the wetted soil under a of a wedge-shaped wetted bulb front under a surface emitter with
point source can be represented by an incomplete ellipsoid. constant discharge using Green-Ampt 13, Darcy, and mass
Schwartzman and Zur 9 assume that the vertical and horizontal conservation equations (see Fig. 1).
advances of the wetted soil volume under an emitter are related to
hydraulic conductivity, emitter discharge and applied water volume. Emitter
Thus, they have introduced some equations for calculation of the
maximum width and depth of wetted bulb. Schmitz et al. 10 have R0 r0
pointed out that using numerical solution of the equations to E z
simulate the moisture distribution through the soil is an effective
technique in optimization of irrigation systems.Amin and Ekhmaj 11 R
presented some empirical relations to calculate wetting front’s depth
and radius and compared them with the measured ones in sandy, R1
loamy sand and silty clay soils. The results showed that the
differences between the measured and predicted values increased dD x
with time. Sepaskhah and Chitsaz 12 conducted a research on surface
trickle irrigation in six soil series and determined depth and radius Figure 1. General shape of a hypothetical wetted bulb profile for constant
discharge of surface emitter 14.

Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011 505

1 ¨¨§©ro H ¹·¸¸ R H ¨§¨©ro H ¹·¸¸ ln¨©§¨ RsinE H ¹¸¸· sinE Kro t(1) Emitter
sinE sinE sinE ro sinE H M
2
R2 ro2 ro

He also derived Equations 2 and 3 to estimate the radius (R0) and
depth (R1) of the wetted soil, respectively [L].

1 ro R1 Hro Hln©¨¨§ R1 H ¸¸·¹ kro t
ro H M
2
R12 ro2 H ro (2)
(3)
1 ro kHro t
2 M
3
Ro3 ro3 Ro2 ro2 Figure 2. Schematic representation of wetting front in soil under a surface
emitter.

where R is the radial distance of wetting pattern [L], r0 is the 5 the amounts of dimensionless depth and width of wetted bulb
saturated radius around the emitter [L], H is the Green-Ampt are calculated. Philip derived the following equation to estimate
wetting front coordinates:
potential parameter [L], M is the Green-Ampt water content
parameter [L3L-3] (difference between soil-saturated water content ^T R,I exp>R1 CosI @ R2 ¨§1 1 CosI ¸· R
and initial soil water content), β is the angle between ground ©2 ¹
surface and any radial distance R, K is saturated hydraulic
conductivity [L T-1] and t is the time of application [T]. ªR1 CosI ln1 CosI º

Philip’s method: An analytical approach to determine the wetting 1 ««¬«ln§¨©¨ 1 CosI exp>RCosI 1@ ·¸¹¸»¼»»
depth and radius was proposed by Philip 15: CosI
1 CosI (10)

T Z 2 Z ln1 Z (4) L1 CosI exp>RCosI 1@ L1 CosI `


2

> @T 2exp(R) 1 R R2 / 2 2 (5) 0I 1S
2
Z D z (6)
2 (7) where T(R,φ) is the dimensionless time parameter that it is a function
of the angle (φ) between direction of wetted bulb depth (z) and any
R Dr radial distance (r) and L(x) is the dilogarithm defined by:
2

where T is the dimensionless travel-time defined by Revol et al. 16: Lx ³x ln x dx (11)

1 x 1

T D 3qt (8) The amounts of R for different quantities of T(R,φ) and φ can be

16S'T

calculated using Equation 10. Then coordinates of wetting front

In Equations 7 and 8 t is the time of application [T], Z+ is the can be obtained using Equation 7 and relations of z = r.cos(φ),
dimensionless vertical distance to the wetting front below the s = r.sin(φ) (Fig. 2).

source, R is the dimensionless horizontal radius of the wetting

front, z+ is the vertical distance downward [L], r is the radial Materials and Methods

distance [L], á is the reciprocal of the macroscopic capillary length Field experiments were conducted in two different locations in
scale [L-1], q is the emitter discharge [L3T-1] and ∆θ is the average Tabriz suburb: 1) Khalatpooshan and 2) Arpadarasi. Values of

change in volumetric water content behind the wetting front [L3L-3] physical properties of soils are given in Table 1.

which is obtained by Ben-Asher 7.

'T Ts Experiments: In the experimental sites several emitters were
2 (9) installed on lateral pipes of 16 mm diameter. These lateral pipes

were connected to a 200-litre water reservoir through a main pipe

line. In order to decrease the turbulence in reservoir, first the

where ès is the saturated volumetric water content of soil. water was conveyed into a preliminary reservoir before entering

Fig. 2 shows a view of wetting front resulted from a surface into the main reservoir. The main reservoir was equipped with a

emitter. spillway to supply a constant head in emitters during the tests.

The depth of the wetting front and radius of the wetted soil The emitter discharges were set in two rates of 2 and 4 L h-1. The

can be estimated using Equations 4-8. For this aim, first the volume of the applied water during each test by an emitter was

dimensionless travel-time (T) is

determined for different application Table 1. Physical properties of soils used in the research.

times, using the known parameters such Experiment Sand Clay Silt Soil texture șs* ș0* ȡb* Ks
as emitter discharge and saturated and place (%) (%) (%) (%) (%) (g cm-3) (m day-1)
initial water content of the soil. Then
by substitution of T in Equations 4 and Khalatpooshan 70 12 18 Sandy loam 38 10 1.62 0.3878
14.2 1.53
Arpadarasi 44 24 32 Loam 43.5 0.1874

*ρb= bulk density [M L-3], θ and θs = initial and saturated soil water content [L3 L-3], respectively.
0

506 Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011

totally 4 or 8 litres. Coordinates of wetting front were measured Table 2. Values of physical parameters of loam and sandy loam
from the emission point by trenching in different times. soils obtained from Shani et al. 17 method.

Determining reciprocal of the macroscopic capillary length scale Soil Coefficient S C µ Į(m-1) H(mm)
and the Green-Ampt potential parameter: Two cocentric nearly texture b (mm s-1) (mm s-1/2)
circular saturated and unsaturated areas are formed around an Sandy loam 1.429 1.944 3.833 126.75
emission point on the ground surface. The inner area is saturated Loam 1.30 0.564 1.166 1.749 2.468 173.62
and surrounded by an unsaturated outer area. At the beginning of
water application by a surface dripper these wetted areas gradually 1.12 0.593
develop in time and finally the inner area reaches a constant position
at the saturation state. The final diameter of the saturated part Coordinates of wetting front were predicted with respect to
varies with soil texture and flow rate. The macroscopic capillary the Green-Ampt potential parameter, the Green-Ampt water
length scale α [L-1] can be calculated as 17: content parameter(M), saturated hydraulic conductivity (Ks),
radius of saturation part (r0), and the reciprocal of the macroscopic
D 4Ks/S.b (12) capillary length scale (α), using Philip and Chu’s methods. The
derived data from both methods were compared with the measured
ones and the results have been shown in Figs 3 and 4. The values
of RMSE and R2 between the measured radii of a wetted bulb
under emission flow rate of 2 L h-1 in sandy loam soil and those

where coefficient b is the slope of regression line of q (ratio of 00 HorHiozroiznontatall aaddvavnacen(cmem)(m(m1) ) (1) Horizontal advance (mm)
Horizontal advance (mm)
emitter discharge to area of saturated zone with radius of r0) and 0
1/r0. Also Shani et al. 17 derived the following equation to estimate 5500 110000 151500 202000 252500 330000 00 5500 101000 151500 202000 250250
the Green-Ampt potential parameter: -50-50 0

H b.ʌ>1 P @.> 4Ks . P @1 (13) Vertical advance (mm) -100100 Q=2lit/h (2(2)) Vertical advance (mm) -50-50 Q=2lit/h
t=4h (3) Vertical ad van ce (mm) -10-0100 t=2h
-150150
(3)
-200200
(4)
(4)

where: -250250 (5)(5) MeaMseuarseudred -15-0150 MeMaesausurreedd
-300300 CChhuu -20-0200 CChhuu
P 0.5 C 1.25 C 1.252 4C 1/2 (14) (6)(6) -25-0250 PPhhiliiplip
(7(7)) PPhhiiliipp
-350350 Q=2 lit/hr Q=2 lit/hr
t=2 hr
t=4 hr

C 1.25bTs T0 S2 (15) Vertical advance (mm) 00 Horizontal advance (mm) 00 Horizontal advance (mm) 225500
S Xr.Ts T0 / t0.5 (16) Horizontal advance (mm) 0 Horizontal advance (mm)
0 5500 101000 151500 202000
5500 110000 151500 202000 252500 330000 -50-50
-500 Vertical advance (mm)
In the above equations H is the Green-Ampt potential parameter Q=4lit/h Vertical ad van ce (mm)-100-100 Q=4lit/h
[L] and Xr is the horizontal distance from the ponded radius to -1000 t=2h -150-150 t=1h
the wetting front at different times [L].
-1500 Q=4 lit/hr MeMaesausrueredd -200-200 MeMaseuasrueredd
To evaluate the above-mentioned methods we used two t=2 hr CChhuu -250-250 CChhuu
statistical measures namely root mean square error (RMSE) and -2000 PPhhiliiplip -300-300 PPhhiililpip
the goodness of fit ratio (R2) described by:
-2500 Q=4lit/hr
t=1 hr
-3000

-3500

¦ RMSE§¨ n 2 / n ¸·1/ 2 (17) Figure 3. Comparison between measured and predicted coordinates of
©i1 Pi M i ¹ wetting front in sandy loam soil under emission rates of 2 and 4 L h-1.

Horizontal advance (mm) 00 Horizontal advance (mm)
5500 110000 151500 220000 252500
¨§ n¦ ¦ ¦ R22 n 2 ·¸¨§ n 2 ¸·1 (18) 00 0 5500 101000 151500 202000 225500 303000
©i1 Mi M Pi M i ¹© i 1 Mi M ¹ 0
i1 Vertical advance (mm) -50-50
() -50-50 Vertical advance (mm)
-10-0100 Q=4lit/h Vertical advan ce (m m)-100-100 Q=4lit/h
-15-0150 t=1h -150-150 t=2h
-20-0200
where P and M are the predicted and measured radial distances, -25-0250 MeMaesuasruerded -200-200 MeMaesausurreedd
ii -30-0300 CChhuu -250-250 CChhuu
PPhhiililpip -300-300 PPhhiliiplip
n is the number of data and M is the mean value of the measured -350-350

radial distances.

Q=4 lit/hr Q=4 lit/hr
t=1 hr t=2 hr

Results and Discussion 0 HHoorriizzonotanl taadvlaancdev(mamn) ce (mm) HoHroirzizoonntatlaadlvaandcev(amnm)ce (mm)
To estimate parameter of α, radii of the saturated zone were 5500 110000 11550 0 220000 225500 330000
measured for different discharges individually. Then the following Vertical advance (mm) 0 5050 100100 150 150200 220050 320500 00
equations were obtained for the both loam and sandy loam soils 0 Q=2lit/h
respectively, using regression analysis. -50 Vertical advance (mm) t=4h
Vertical advance (mm) -50-50
-100
r = 86.67 ln (Q) - 567.64 R2 = 0.96 (19) Q=2lit/h PPhhi l iiplip -100 PPhhilipilip
0 (20) -150 t=2h MMeaesausurreedd MMeaeassuureed d
-100
r = 45.95 ln (Q) - 285.97 R2 = 0.97 -200 Q=2 lit/hr CChhuu CC hhu u
0 t=2 hr -150
-250
-150 Q=2 lit/hr
t=4 hr
-200
where ro is the final radius of saturated area (mm) and Q is
discharge of emitter (cm3 h-1 ). Finally, the parameters H and α -200
were calculated, using Shani et al. method 17 (see Table 2). -250

-25-3000

-30-3050

Figure 4. Comparison between measured and predicted coordinates of
wetting front in loam soil under emission rates of 2 and 4 L h-1 .

Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011 507

Table 3. R2 and RMSE values in sandy loam soil under emission rate of 2 L h-1.

Radial distance of wetting front from emitter (mm)

Time of point 12 3 4 5 6 7 RMSE R2
application (mm)
(min) measured 245 269.25 259.42 250 291.54 308 320 0.846
240 Philip 242 245.4 254 264 283.12 301.16 306 12.62 0.7
Chu 202 216 230.35 239.2 253.8 260.53 261.51 42.68
120 195 216.5 210.8 220.4 214.65 226.16 240 0.601
measured 195 199.32 208.9 218.49 226.16 235.74 239 9.535 0.539
120, 240 Philip 168 180.16 193.57 201.24 205 205.07 208 24.26 0.923
Chu 11.195 0.862
Philip 34.717
Chu

Table 4. R2 and RMSE values in Philip and Chu’s methods. Figs 5 and 6 show the relationship between the measured and
predicted radial distance of wetting front by Chu and Philip’s
Sandy loam Loam Loam and methods in loam and sandy loam soils. In Fig. 5 the slope of
regression line is close to 1, indicating a good agreement between
Discharge Method sandy loam the estimated and measured values.
(L h-1)
R2 RMSE R2 RMSE R2 RMSE
(mm) (mm) (mm)
0.923 11.195 0.913 10.787
2 Philip 0.862 34.717 0.873 12.165 0.92 10.993
0.877 14.145 0.652 19.97
Chu 0.775 14.93 0.455 44.45 0.665 26.512 350 Pphilip
0.874 12.755 0.747 15.88 300 RMRM==00..99442222RRE +E +121.223.9239
4 Philip 0.62 26.722 0.406 32.06 0.781 17.199

Chu 0.378 32.66 Measured radial
distance, mm
2,4 Philip 0.837 14.378
Chu
0.372 29.465 250

200

predicted with Philip and Chu’s methods are represented in Table 150
3. Also corresponding values of RMSE and R2 for emission flow
rates of 2 and 4 L h-1 in both loam and sandy loam soils are 100 150 200 250 300 350
presented in Table 4. 100 Estimated radial distance, mm

Although the estimated values of the wetting front radial Figure 5. Relationship between measured (R ) and predicted (R ) radial
distances in emission flow rate of 2 L h-1 via Chu’s method are ME
less than the measured ones, still their accuracy is acceptable.
The maximum difference between these values and measured distance of wetting front by Philip’s method in loam and sandy loam soils.
ones was 5 cm and values of R2 and RMSE were calculated as
0.862 and 34.717 mm, respectively. 350 Chu

Under the above mentioned condition the predicted values of Chu
the wetting front radial distances via Philip’s method were more
accurate than those predicted using Chu’s method. Especially in Measured radial R = 0.6557R + 80.148
the case of 4-litre water application, the agreement between the distance, mm 300 RMM = 0.6557RE +E80.148
measured values and the results of Philip’s method was very
good, yielding R2 and RMSE equal 0.923 and 11.195 mm, 250
respectively.
200
The predicted wetting front coordinates via both methods of
Chu and Philip, showed good agreement with measured values, 150
when the emission flow rate of 4 L h-1 with total applied water
amounts of 4 or 8 litres, were provided. The RMSE values for Chu 100 150 200 250 300 350
and Philip’s methods in this case were 14.145 and 14.93 mm, 100 Estimated radial distance, mm
respectively.
Figure 6. Relationship between measured (R ) and predicted (R ) radial
The model results in loam soil were very accurate in the case of ME
q= 2 L h-1, and the values of RMSE calculated were 10.187 and
12.165 mm for Philip and Chu’s solutions, respectively. Also the distance of wetting front by Chu’s method in loam and sandy loam soils.
closeness between the two methods of Philip and Chu in the
cases of applying 8 litres water was higher than those with Conclusions
application of 4 litres water. The predicted wetting front According to the presented results in Figs 3-6 and also Tables 3
coordinates via Chu’s method were greater than measured ones and 4, Philip’s method for both emitter discharge of 2 and 4 L h-1 in
in loamy soil, when the emission rate of 4 L h-1 was used. The loam and sandy loam soils is confirmed as an efficient procedure
RMSE calculated value was 44.45 mm in this case while the R2 and is suggested to estimate coordinates of wetting front under
and RMSE values were 0.652 and 19.97 mm for Philip’s method in surface trickle irrigation.Chu’s method that uses Green-Ampt
this soil. analysis is a fairly suitable method to estimate coordinates of
wetting bulb front under surface emitter in both loam and sandy
loam soils. Of course the differences between predicted values
by this method and measured values are negligible if the temporal
and special varieties of soil physical properties in the field are
considered. Due to the simplicity of Chu’s method, we can use
this method to predict coordinates of wetting front of wetted
bulb under surface trickle irrigation. The ability of Chu’s method
to estimate the depth and radius of wetted bulb under surface

508 Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011

trickle irrigation also has been approved by Sepaskhah and Nomenclature
Chitsaz 12. Advantage of using Philip and Chu’s methods as β Angle between ground surface and any radial distance R
compared with other methods such as Schwartzman-Zur and α Reciprocal of the macroscopic capillary length scale [L-1]
Amin-Akhmaj is that Philip and Chu’s methods can anticipate ∆θ Average change in volumetric water content behind the wetting front
the profile of wetting front but the other mentioned methods just
can estimate width and depth of wetting front. [L3L-3]
Θs Saturated volumetric water content of soil
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Journal of Food, Agriculture & Environment, Vol.9 (1), January 2011 509


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