An Unsteady Subsurface Drainage Equation IncorporatingVariability of Soil Drainage Properties

Ashok Kumar Pali & Praful Katre & Dhiraj Khalkho

Received: 2 December 2013 /Accepted: 15 April 2014 /Published online: 12 May 2014# Springer Science+Business Media Dordrecht 2014

Abstract Almost all unsteady subsurface drainage equations developed so far use constantvalue of drainable porosity and hydraulic conductivity which may not be representative ofentire drainage flow region. A drainage equation was, thus, developed incorporating depth-wise variability of drainable porosity (f) and hydraulic conductivity (K) of saline soils ofHaryana state in India. The drain spacing with measured hydraulic heads at different periods ofdrainage were estimated by the developed equation and compared with the correspondingdrain spacing estimated by commonly used unsteady drainage equations. The study revealedthat the developed equation estimated the drain spacing that was nearest to the actual drainspacing of the existing subsurface drainage system, when a generally used design criterion of30 cmwater table drop in 2 days is considered. For a criterion of desiredwater table drop in 3 daysand beyond, Glover equation was found to be the most superior. Hence, both the developedequation and Glover equation can be readily used with the associated design criteria fordesigning unsteady subsurface drainage systems in saline soils of the state of Haryana, India.

Keywords Bouwer and Schilfgaarde . Boundary conditions . Heterogeneity .ModifiedintegratedDagan equation . Variability

1 Introduction

In arid and semi-arid regions, low rainfall with uncertainly of its occurrence results in moisturedeficiency during most part of the crop season. Introduction of irrigation in such areas has beenconsidered as the most effective way for improving crop production. However, failures ofirrigation projects in arid and semi-arid regions due to development of water-logging and soilsalinity by rising water tables have also been noticed world wide. Such repeated failures ofagriculture in several irrigated areas have led to a frequently propounded theory that civiliza-tion can no longer endow when dependent upon irrigated agriculture only.

The agricultural drainage thus, takes a place of prominence among the inputs that promisequick avoidance of water-logging and salinity condition. Drainage has been identified as theforgotten factor in sustaining irrigated agriculture (Scheumann and Freisem 2002). The

Water Resour Manage (2014) 28:2639–2653DOI 10.1007/s11269-014-0631-1

A. K. Pali (*) : P. Katre : D. KhalkhoFaculty of Agricultural Engineering, Indira Gandhi Agricultural University, Raipur, Chhattisgarh, Indiae-mail: akpali@hotmail.com

drainage of excess water caused by usual practice of over irrigation to croplands, particularly inarid and semi-arid regions helps in establishment and maintenance of proper salt balance inthe soil. Since the 1950s, land drainage has been a technique well-established andelaborated in engineering science; it has attracted increasing attention in connection withthe development of water resources for agriculture. On the political international scene,however, many events that took place during the 1990s suggest that land drainage shouldhave become an essential issue in the global dialogue on food security (Amer 1996). Theestimates of Food and Agriculture Organization (FAO 1994) show that out of 250 millionhectares, about 30 million hectares are severely affected by salinity and an additional 60to 80 million hectares are affected to some extent. Soil salinity has adversely affectedabout 10 % of the total area irrigated in Mexico, 11 % in India, 21 % in Pakistan, 23 %in China, 28 % in the United States, and in some Central Asian republics, it is over 50 %.Amer (1996) reported that in India, 7 million hectares of land have been abandonedbecause of salinity.

As is established, the provision of drainage especially subsurface drainage is an importanttechnique for reclaiming water logged and saline soils. The mechanism, by which thisobjective is achieved by subsurface drainage, involves the physics of flow of fluids throughthe soil profile and consequently depends heavily on soil drainage properties. Thus, theeffective design of subsurface drainage system is characterized in terms of soil drainageproperties, drainage system parameters and boundary conditions. Hooghoudt (1940) devel-oped a steady state approach in the Netherlands that has been adopted for humid areasthroughout the world. The drainage design in humid areas generally is based on the idea ofa steady state system and the design criteria require the removable of a specified depth of waterin a given period of time to ensure adequate aeration of the soil. In arid irrigated areas,however, rainfall is a minimal consideration in drainage design, since the major source ofexcess water is a result of irrigation inefficiency that causes rise of groundwater table andconsequently water logging and soil salinity. (Datta et al. 1997). Therefore in irrigated arid andsemi arid lands, the prime objective is the control of rising water table, which is normally underunsteady state condition. Steady drainage theory therefore does not apply. Many drainageworkers have developed unsteady state drainage theories for irrigated lands. In developingthese theories, mostly Boussinesq equation has been used to understand unsteady water tablefluctuations in subsurface drainage problems (Rai and Singh 1987). Most of unsteady statedrainage equations require drainable porosity (f) and saturated hydraulic conductivity (K) asthe two soil properties that directly influence the drainage process. These drainage equationsuse constant values of these properties and may be useful for drainage design in homogeneousand isotropic soils but these soils are rarely encountered in natural systems. Heterogeneity andanisotropy are common soil features, and hence, the representation of such features in drainagedesign models is of considerable importance (Parissopoulos and Wheater 1990). Occurrence ofheterogeneity and anisotropy is particularly existent in saline alluvial soils of the state ofHaryan in India (Pali 1986). Field investigations have indicted a strong dependence of f and Kon water table depth i.e. the soil depth. Several workers have developed drainage designequations considering different values of f and K for different soil layers. But in the field suchclear stratification is seldom found to exist because the soil properties generally vary in agradual way. Use of constant values of f and K in subsurface drainage design often leads tovariation in theoretically estimated results and the experimental results. The soils in Haryanastate of India are the result of deposition of fine soil sediments of various natures brought fromShivalic foothills by floodwaters of Yamuna river. Due to ever changing sediment deposits, thesaline soils of the region exhibit inherent heterogeneity in their drainage properties. It is,therefore, necessary to incorporate this heterogeneity in drainage design equations. Keeping

2640 A.K. Pali et al.

this objective in mind, the present study was undertaken for developing an unsteady subsurfacedrainage equation incorporating variability of drainable porosity and hydraulic conductivity ofthe soil. The performance results of the developed equation were also validated with the fieldresults.

2 Materials and Methods

2.1 Location of the Study Area

The study area is situated in the village Mudlana which is about 29 km away from Sonepatdistrict of Haryana state in India. A location map along with drainage basins covering the stateof Haryana is shown in Fig. 1. Geographically, the study area is located at 290 10″ N latitudeand 760 43″ E longitudes with an altitude of 226 m above mean sea level. The area is nearlyflat with a gentle slope below 1 % and is a part of vast alluvial plain in the catchment of riverYamuna. The peripheral higher elevations form a closed basin which restricts the surface andsubsurface natural drainage resulting in inundation of the area for a few weeks by Yamunafloodwaters during heavy rains.

Fig. 1 Location map and drainage basins of Haryana state in India

Unsteady Subsurface Drainage Equation Incorporating Variability 2641

2.2 Climate

The climate is subtropical semiarid with high evaporation rates. The area receives an averageannual rainfall of 550 mm, 83 % of which pours down from July to September. From Octoberto June end, weather remains dry excepting a few showers received during winter fromwestern depressions. Pan evaporation generally exceeds precipitation throughout the yearexcept in rainy season. High evaporation rates (114–170 mm per month) just after the rainyseason cause accumulation of salts in root zone and on soil surface due to rise of salinegroundwater table. The summer season is very hot when atmospheric temperature shoots up to40 °C in May and in January temperature drops down to 4–6 °C.

2.3 Soils

The study area represents inline saline and water logged soils of Indo-Gangetic plains that areslightly heavier in nature. Top 25 cm soils are compact with bulk density of 1.75–1.78 g cm−3.The soils chiefly contain chlorides and sulphate salts of Na, Ca and Mg. Due to predominanceof sodium salts, SAR of the soil solution is invariably high (69.9 %) in top 15 cm depth. Thetop 15 cm soil has a pH as 9, ECe as 13.1 mmhoscm−1 and ESP as 69.2 %. The soils containabout 21 % clay, 17 % silt and 61 % sand, falling under textural class S1.

2.4 Groundwater

The groundwater is highly saline, with ECe up to 21.5 mmhoscm−1, rendering it not fully fit forirrigation and drinking. The water table almost reaches ground surface during monsoon seasonand begins receding from October declining to 1.5 to 2.0 m below ground surface in the monthof June. The rate of water table decline has a trend almost similar to the average evaporationrate from January to May.

2.5 Experimental Drainage System

The existing subsurface drainage system is installed in sandy loam soils with drain spacings as50, 67 and 84 m, each at a depth of 1.6 m below ground surface. The impermeable layer issituated at 3.4 m below drain centre. The internal diameter of lateral drains is 75 mm and thatof collector drain is 140 mm having a total length of 310 m. The present study was conductedon 50 m drain spacing only. Piezometer tubes are installed between lateral drains at an averagedepth of 2 m below ground surface to monitor water table fluctuation between drains.

2.6 The Field Data

The measured data of drain outflows and groundwater table fluctuations between 50 m drainspacing area were obtained fromHaryana LandDevelopment Corporation. The measured watertable depths were converted into hydraulic heads by subtracting them from drain depth of 1.6 m.

2.7 Theoretical Considerations

2.7.1 Drainable Porosity ( f )

Soil drainable porosity (f ) is defined as the volume of water that is drained (or taken up) by aunit volume of soil when the water table drops (or rises) over a unit distance. The value of f is

2642 A.K. Pali et al.

not generally constant but besides other things, it is a function of water table depth i.e. soildepth, Z (Taylor 1960). The time of drawdown and shape of the water table depend on theparticular way in which drainable porosity is related to water table depth. Thus, it is convenientand often necessary in drawdown studies to express f as a function of Z. The f valuescorresponding to different Z were determined from water table drawdown or hydraulic head(h) and drain discharge (q) measurements. A functional relationship between f and Z was thendeveloped by regression which is shown in Fig. 2 and is described by the following equation:

f ¼ aZb ð1ÞWhere ‘a’ and ‘b’ are the regression coefficients (a=0.138; b=0.550).An average value of f representing the entire drainage flow region can be obtained by

integrating Eq. (1) within the boundary conditions Z=0 to Y and then dividing by drain depthY. Thus,

f ave ¼

Z0

Y

aZbdz

Y

¼ aYbþ1

Y

¼ aYb

bþ 1ð Þ ð2Þ

2.7.2 Hydraulic Conductivity (K)

Soil hydraulic conductivity (K) is the drainage water flow per unit area of the soil mass per unithydraulic gradient. Like f, K also exhibits spatial variation. Therefore, it is equally important toincorporate the variability of K in drainage design equations. For determining K correspondingto different Z values, K/f ratios were first determined using the procedure given by Skaggs(1976) and then these were multiplied by f values obtained from Eq. (1). K values so obtainedwere plotted against Z values (Fig. 3). The following relation was found:

K ¼ αZβ r2 ¼ 0:99� � ð3Þ

Where ‘α’ and ‘β’ are regression constants (α=0.323; β=−0.51).

Fig. 2 Variation of drainable porosity with water table depth

Unsteady Subsurface Drainage Equation Incorporating Variability 2643

An average value of K can also be determined in a similar manner as used forfave. Therefore,

Kave ¼ αZβ

β þ 1ð Þ ð4Þ

2.8 Derivation of Unsteady Subsurface Drainage Equation

The water table recession during drainage of a soil profile is the slowest midwaybetween drains, hence the most critical. If the water table is assumed to be horizontal,the rate of fall of water table is uniform between drains. Therefore, a steady statedrainage relationship, which also assumes a uniform flux, can be used to describe therate of fall ( dhdt ) of the water table between drains (Bouwer and Van Schilfgaarde1963). However with a falling water table, the flux in general varies with distancefrom the drains. Hence in order to use steady state solutions, a correction factor Cneeds to be introduced. The expression for drain discharge or drainage flux can beexpressed as:

q ¼ −fCdh

dtð5Þ

In above expression, q = drain discharge or drainage flux, mday−1; f = drainableporosity of the soil, in fraction. Negative sign indicates that h decreases with increasein time, t. For rising water table, negative sign disappears. The correction factor (C)varies between 0.8 and 1.0 for 0.02<h0/S <0.08, except for the first stages ofrecession following a ponded case (when h0/S>0.08). Here, h0 is initial value ofhydraulic head and S is the drain spacing.

In Eq. (5), Bouwer and Shilfgaarde considered the drainable porosity, f as constant. But ithas been established that f varies with Z as described by Eq. (1).

Fig. 3 Variation of hydraulic conductivity with water table depth

2644 A.K. Pali et al.

Since Z=(Y−h), where Y is the depth of drain from ground surface (Fig. 4), Eq. (1) can beexpressed in terms of hydraulic head, h as,

f ¼ a Y−hð Þb ð6Þ

Substitution of Eq. (6) into Eq. (5), yields,

q ¼ −aC Y−hð Þb dhdt

ð7Þ

Dagan (1964) equation for steady state drainage is expressed as,

h ¼ qS

KFD ð8Þ

Fig. 4 Geometry and symbols used in sub surface drainage equations

Unsteady Subsurface Drainage Equation Incorporating Variability 2645

Where, FD is a function defined by Dagan as,

FD ¼ 1�4

S

2D−2

πln 2cosh

πrD

−2�� �

ð9Þ

FD can also be determined from the following relation:

FD ¼ 1

4

S

2D−γ

� ð10Þ

Where, γ ¼ 2π ln 2cosh πr

D −2� �

is to be determined from the nomograph developed by VanBeer (Dagan 1964); D is the depth of impermeable layer below drains; r is the radius of thedrain, and S is the drain spacing.

Dagan (1964) developed the steady state drainage equation (Eq. 8) by dividing the flowregion into horizontal and radial flow regions. Radial flow was assumed in the area between

the drains at a distance 12D

ffiffiffi2

paway from the drain and an intermediate, though mainly

horizontal, flow in the area between the 12D

ffiffiffi2

pplane and the mid plane between the drains.

Substituting q from Eq. (7) into Eq. (8), we get,

hk

SFD¼ −aC Y−hð Þb dh

dtð11Þ

Rearranging, we get,

Kdt

aCSFD¼ −

1

hY−hð Þbdh

Kdt

aCSFD¼ −

Yb

h1−

h

Y

� b" #

dh

The expression 1− hY

� �bfor h

Y ≤ 1 can be expanded using Binomial theorem as,

Kdt

aCSFDYb ¼ −1

h1−

bh

Yþ b b−1ð Þh2

2!Y 2 þ……::

� �dh

Kdt

aCSFDYb ¼ −1

h−b

Yþ b b−1ð Þh

2!Y 2

2

þ……:

" #dh

ð12Þ

Considering only first three terms of the series and integrating for the boundary conditions,h = h0 at t=0 and h = h at t = t, we get,

K

aCSFDYb ∫0

t

dt ¼ −∫h0

h 1

h−b

Yþ b b−1ð Þh

2!Y 2

� �dh ð13Þ

Minus sign before the integral on right hand side disappears, if limits are interchanged.Therefore,

K

aCSFDYb ∫0

t

dt ¼ ∫h

h0 1

h−b

Yþ b b−1ð Þh

2!Y 2

� �dh ð14Þ

2646 A.K. Pali et al.

K

aCSFDYb t½ �t0 ¼ lnh−bh

Yþ b b−1ð Þh2

4Y 2

� �h0h

Kt

aCSFDYb ¼ lnh0−lnhð Þ− b

Yh0−

b

hh

� þ b b−1ð Þ

4Y 2 h02−

b b−1ð Þ4Y 2 h2

�

Kt

aCSFDYb ¼ lnh0h−

b

Yh0−hð Þ þ b b−1ð Þ

4Y 2 h02−h2

� �ð15Þ

In Eq. (15), FD remains the same as in original Dagan equation (Eq. 8).

∴S ¼ Kt

aCFDYb lnh0h−b

Yh0−hð Þ þ b b−1ð Þ

4Y 2 h0−h2� �� ð16Þ

Substituting the expression of K from Eq. (3) in Eq. (16), we get,

S ¼ αZβ:t

aCFDYbþ1 lnh0h−b

Yh0−hð Þ þ b bþ 1ð Þ

4Y 2 h02−h2

� �� � ð17Þ

The above equation may be named as Modified Integrated Dagan Equation.

2.9 Comparison with Other Drainage Equations

Following five unsteady state subsurface drainage equations were chosen for comparison ofthe developed equation (Eq. 17):

(a) Luthin and Worstell EquationLuthin and Worstell (1959) suggested an unsteady subsurface drainage equation for

elliptical shaped water table between drains as:

S ¼ 4cKt

πf lnh0h

ð18Þ

In Eq. (18), c is a constant representing the slope of line obtained by plotting the rateof flow (q) into drains per unit length as a function of Kh (hydraulic conductivity timesthe vertical distance from the drain to the water table midway between drains).

(b) Glover EquationGlover as reported by Dumm (1954) proposed the following drain spacing equation for

homogeneous soils. The equation was developed assuming an initially flat water table:

S ¼ πKDa

f ln 4h0πh

� �" #1=2

for > D > h0 ð19Þ

According to Dumm, although Glover equation was developed under the assumptionthat h0 is small as compared to D but if one uses Da = D + h0/2 or Da = de + h0/2 as the

Unsteady Subsurface Drainage Equation Incorporating Variability 2647

effective thickness of flow region, the equation will give a fairly satisfactory approxima-tion even though the drains are placed at the bottom of the flow region.

(c) Van Schilfgaarde EquationVan Schilfgaarde (1963) proposed an unsteady subsurface drainage equation, which

corrected for Dupuit-Forchheimer assumptions and avoided the assumption of a constantthickness of the flow region. The following equation was proposed:

S ¼ 3AK de þ hð Þ de þ h0ð Þt

2 f h0−hð Þ� �1=2

ð20Þ

Where, A ¼ 1− dedeþh0

� �2� �1=2

(d) Modified Glover EquationVan Schilfgaarde (1965) extended the analysis of unsteady state drainage problem and

modified the Glover equation to correct for convergence near drains. He assumedinitially parabolic water table and depth of impermeable layer was not small. Theseassumptions were in contrast with those assumed by Glover. The following drainageequation was proposed:

S ¼ 3AKtde

f lnh0 2deþhð Þh 2deþhð Þ

24

351=2

ð21Þ

‘A’ is the same as in Eq. (20).(e) Integrated Hooghoudt Equation

Bouwer and Van Schilfgaarde (1963) presented a simplified procedure for predictingthe rate of fall of water table in tile drained or ditch drained land, based on steady statetheory and abrupt drainage of pore space. They developed the following unsteady statedrainage equation based on Hooghoudt steady state drainage equation:

Kt

f¼ CS2

8deln

h0 hþ 2deð Þh h0 þ 2deð Þ ð22Þ

Where, C is a correction factor for using steady state solution of Hooghoudt whichgenerally varies between 0.8 and 1.0 for 0.02<h0/S<0.08, except for the first stagesof water table recession following a ponded case (h0/S>0.08). In that case, C ishigher (C=1.1). The notations and symbols used in all drainage equations havebeen defined in Fig. 4.

3 Results and Discussion

The soil drainage properties viz. drainable porosity (f) and hydraulic conductivity (K) weredetermined from drain discharge (q) and water table drawdown (Z) measurements. Themethod of estimating drainage properties from observed drain discharge and water table

2648 A.K. Pali et al.

drawdown offers the advantage of integrating the effects of soil heterogeneity and anisotropyoccurring in the drainage flow region.

3.1 Drainable Porosity (f)

As said earlier, the value of f is not usually a constant but besides other things, it is a functionof water table depth as the time of water table decline and its shape depend on the relation of fwith Z. In this study also, a power relation was found to exist between f and Z as described byEq. (1) (Fig. 2). The relation shows that near soil surface, f appeared to be low and graduallyincreased with increasing Z. The value of f ranged between 0.06 near soil surface to 0.098 at50 cm below ground surface. The texture of the soil being sandy loam, the values of fcorresponded well to the values in the range of 6 to 9 % for medium textured soils asobtained by Dieleman and Trafford (1976). Lower f values near the soil surface wereresponsible for severe drainage problem leading to water logging condition in the study areabecause with lower f, addition of even small amount of rainfall or irrigation led to large rise ofwater table because of slow infiltration and percolation that restricted the natural drainage ofsoil profile.

3.2 Hydraulic Conductivity (K)

Similar to drainable porosity, the hydraulic conductivity of drainage flow region is not usuallyconstant. Because of heterogeneous nature of saline alluvial soils of the study area, formed as aresult of sediment deposits of various grades and nature brought by flood waters of Yamunariver, the vertical hydraulic conductivity (K) also showed a large variation as described byEq. (3) (Fig. 3). Contrasting to f, K was found higher near soil surface and gradually decreasedwith depth. Near soil surface, K was found to be about 0.70 mday−1 and decreased to0.40 mday−1 at a depth of 51 cm. Apparently, it looks that with increasing f, K must alsoincrease but this is not so here for saline soils, because K is controlled by the relativedistribution of different pore sizes in the soil mass not by the porosity alone. Thus, even withincreasing f, K for the lower depths was found to be low due to changing inherent nature ofsaline soils. The low values of K at deeper soil layers caused the prolonged discharge flowfrom the drainage system as also observed during the operation of the existing drainagesystem.

3.3 Performance of the Developed Drainage Equation

The measured field data on drain discharges (q) and hydraulic heads (h) corresponding todifferent drainage periods (t) was input to the drainage equations along with other necessaryparameters. The measured depth of impermeable layer (D) below drain level was 3.4 m thatwas transformed to its equivalent depth (de) as 2.5 m by using Van Beer’s nomograph. Theaverage value of f as 0.11 and of K as 0.61 mday−1 was estimated from q/h versus hrelationship. It was, however, observed that the average values of f and K obtained from q/hversus h relationship were quite different from those determined from the developed relationsdescribed by Eqs. (1) and (3). Moreover, the q/h versus h relation did not represent the wholedrainage flow region up to the drain depth (Y). So, the variable values of both f and K asdescribed by Eqs. (1) and (3), representing entire flow region was used in all the drainageequations to estimate the drain spacings and estimated drain spacings are given in Table 1. Thedeviations of the estimated drain spacings with respect to the actual spacing of 50 m are givenin Table 2.

Unsteady Subsurface Drainage Equation Incorporating Variability 2649

Tab

le1

Com

parisonof

drainspacingcomputedby

thedevelopedequationwith

theselected

equations

Drainage

period

(t)

indays

Drain

discharge

(q)in

mm

day−

1

Observed

hydraulic

head

(h)in

metres

h 0−h

inmetres

Drainable

porosity

(f)

infractio

n

Hydraulic

conductivity

(K)in

mday−

1

Average

rate

ofwater

table

drop

[(h 0−h

)/t]

inmetres

Drain

spacingestim

ated

bydrainage

equations

(m)

Luthin-

Worstell

equatio

n

Glover

equatio

nVan

Schilfgaarde

equatio

n

Modified

Glover

equatio

n

Integrated

Hooghoudt

equatio

n

Developed

equatio

n

010.50

1.57

(h0)

––

––

––

––

––

18.75

1.38

0.19

0.060008

0.699145

0.19

137.98

36.05

52.08

51.29

63.01

45.48

28.25

1.32

0.25

0.06852

0.618233

0.125

158.94

42.39

56.07

54.91

78.08

51.88*

37.95

1.28

0.29

0.073741

0.577532

0.10

175.76

46.67

59.09

57.65

88.84

56.34

47.50

1.23

0.34

0.079871

0.536315

0.085

168.12

47.81

57.96

56.26

94.64

56.5

57.40

1.20

0.37

0.08337

0.515409

0.074

175.71

50.03

59.37

57.44

101.31

58.68

67.20

1.17

0.4

0.086753

0.496745

0.066

178.48

51.48

59.95

57.82

106.57

59.91

77.00

1.13

0.44

0.091103

0.474715

0.063

169.44

51.40

58.57

56.23

109.48

59.05

86.50

1.06

0.51

0.098332

0.442264

0.064

139.94

48.41

53.51

50.94

108.17

54.22

Actualdrainspacingin

thefieldwas

50m

originally

designed

t=2days

andwater

tabledrop

(h0-h)=30

cm

*Closestdrainspacingw.r.t.actualdesigned

spacingof

50m

with

observed

drainage

period

=2days

andwater

tabledrop

[(h 0-h

=1.57−1

.32=0.25

mi.e.25

cm)

h 0istheinitialhydraulic

head

atmidpointbetweentwodrains

2650 A.K. Pali et al.

Combing Tables 1 and 2, we find that after 1st day of drainage, the developed equationestimated drain spacing of 51.88 m that was 9.04 % lower than the actual spacing of 50 m,whereas Modified Glover equation gave only 2.58 % higher drain spacing than the actual,followed by Van Schilfgaarde which estimated 4.17 % higher value of drain spacing. Thismeans, the field performance of the developed equation was little inferior to ModifiedGlover and Van Schilfgaared equations when a drainage period (t) of 1 day is consideredfor drainage design. But the criterion of t=1 day is generally not used in subsurfacedrainage design because this leads to closer value of drain spacing, that means highercapital investments. The criterion of lowering of water table to a prescribed depth in1 day of drainage is not practical too. Most field crops can withstand water logging for24 to 48 h (1 to 2 days). Therefore, a safe excess water removal period is generallyconsidered as 2 days (Bhattacharya and Michael 2003). If the excess water is removedwithin 2 days, there may not be any irreversible adverse impact on the crops and henceon the yields. With this criterion only, the existing drainage system with 50 m drainspacing was originally designed for a 30 cm water table drop in 2 days of drainage. If thesame criterion is strictly followed, it is clear that the developed equation estimated thedrain spacing which was closest to the actual designed spacing of 50 m (51.88 m for t=2 days and h0−h=0.25 m or 25 cm). But from t=3 days to t=8 days, the Glover equationwas found to be estimating the drain spacings in the closest agreement to the actualspacing of 50 m. From t=3 days and onwards, the developed equation estimated higherdrain spacings than the actual spacing and the deviation of the estimated spacings variedfrom +8.44 to +19.82 %. The reason for this over-estimation may be attributed to that att=3 days and onwards i.e. at soil depth Z=Y- h=0.32 m, the values of hydraulicconductivity (K) estimated by theoretically developed regression relation between Kand Z might be somewhat over-estimated K values as compared to the actual K due toprobable presence of plant roots, cracks and holes below 32 cm depth. Consequently, thedeveloped equation was perhaps predicting higher drain spacings. In a nutshell however,it can be well established that for a commonly used design criterion of 25 cm water tabledrop in 2 days of drainage, the developed equation was found to be the superior amongall the drainage equations.

Table 2 Deviation of estimated drain spacing by drainage equations w.r.t. the actual spacing of 50 m

Drainageperiod (t)in days

Hydraulichead (h)

Deviation from actual drain spacing of 50 m

Luthin-Worstellequation

Gloverequation

VanSchilfgaardeequation

ModifiedGloverequation

IntegratedHooghoudtequation

Developedequation

0 1.57 (h0) – – – – – –

1 1.38 175.97 −27.90 4.17 2.58 26.01 −9.042 1.32 217.89 −15.22 12.14 9.83 56.16 3.76a

3 1.28 251.53 −6.66 18.18 15.30 77.68 12.68

4 1.23 236.24 −4.38 15.92 12.52 89.28 13.00

5 1.2 251.41 0.06 18.73 14.88 102.61 17.36

6 1.17 256.95 2.97 19.90 15.63 113.13 19.82

7 1.13 238.89 2.81 17.15 12.47 118.96 18.10

8 1.06 179.87 −3.17 7.03 1.88 116.34 8.44

a Lowest deviation from actual designed spacing of 50 m for originally designed for t=2 days

Unsteady Subsurface Drainage Equation Incorporating Variability 2651

4 Conclusions

The correct estimation of subsurface drain spacing under unsteady water table regime plays akey role in drainage design and is necessary for the impact evaluation of the existing subsurfacedrainage system in terms of lowering of water table or for the calculation of parameters of newdrainage systems. The international drainage literature foresees many equations for unsteadystate regime with various degrees of complexity. Some of these equations become consecratedamong researchers and designers involved in drainage issues. The variability of the soil physicalproperties especially hydraulic conductivity and drainable porosity is the reason that many ofthese drainage design equations do not perform in conformity to the actual field performanceresults. This is an established fact that both soil hydraulic conductivity and drainable porosityshow lot of spatial variability that makes the selection of the most appropriate values of the twosoil properties very difficult. Most unsteady state drainage equations available assume constantvalues of drainable porosity and hydraulic conductivity. Since the point measurement of thesesoil properties involve a small amount of soil mass, the average soil drainage propertiesdetermined by point measurements usually do not represent the entire thickness of drainageflow region which contributes to drain outflow. In this study also, the verification of approx-imation of drain spacing estimated by well known unsteady subsurface drainage equations ofLuthin and Worstell, Glover, Van Schilfgaarde, Modified Glover, and Integrated Hooghoudtequations did not conform to the actual designed drain spacing of 50 m for a commonly usedcriterion of the desired water table drop in 2 days of drainage. The study revealed that thedeveloped equation was the most superior to other drainage equations from the view point ofthe above criterion. But for a criterion of drainage period more than 2 days, Gover equation wasfound to be the most superior drainage design equation. It can be thus concluded that, both theequations can be used safely according to the design criteria associated with them (as statedabove) for designing unsteady subsurface drainage in the areas resembling to the study area.

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