Representing subsurface drains in a finitedifference model

J. GALLICHAND

Departement de genie rural, Universite Laval, Quebec, QC, Canada, GIK 7P4. Received 3 November 1992; accepted 14 May1993.

Gallichand, J. 1993. Representing subsurface drains in a finitedifference model. Can. Agric. Eng. 35:105-112. Because of thesimplifyingassumptionsrequired to obtain analytical solutions to thesubsurface drainage problem, numerical methods are increasinglyused to simulate complex soil hydraulic conditions. MODFLOW isa widelyusedfinite differencemodel that can simulategroundwaterflow between subsurface lateral drains. In this study, subsurfacedrains were represented, in a MODFLOW grid cell network, bycorrecting thehydraulic conductivity of the draincell basedon gridsize, effective drain radius, and surrounding soil hydraulic conductivity. Results from the numerical solution of MODFLOW werecompared with Kirkham's analytical solution and Hooghoudt'sequation, fora variety of drainage configurations with grid sizes of0.1 and 0.2 m. Outflow rates predicted by MODFLOWwere alwayswithin 1.23% of the Kirkham's solution, whereas hydraulic headsdidnotdiffer bymore than 11 mmfor the0.1 m grid,and36mmforthe 0.2 m grid. On the average, watertable depths predicted byMODFLOW were within 21 mm of those calculated with Hooghoudt's equation. A calibration curve was developed to determinethe corrected hydraulic conductivity of the drain cell to simulatedrains of various radii in a square MODFLOW grid cell network.

La solution analytique aux problemes de drainage souterrainnecessite souvent d'avoir recours a des hypotheses simplificatrices.Les solutions numeriques sont done utilisees de plus en plus poursimuler des conditions complexes d'ecoulement souterrain. MOD-FLOW estun modeletres repandu de differences finiesqui peutetreutilise pour simuler l'ecoulement entre deux lateraux de drainagesouterrain. Danscetteetude, lesdrains souterrains onteterepresenteden corrigeant laconductivite hydraulique de lacellule contenant ledrain selon l'espacement de la grille de discretization, le rayoneffectif du drain, et la conductivite hydraulique du sol entourant ledrain. Les resultats numeriques obtenus avec MODFLOW ont etecompares a la solution analytique de Kirkham et a 1'equation deHooghoudt pour plusieurs configurations de drainage avec desespacements de grille de 0.1 et0.2 m. Les debits predits par MOD-FLOW n'ont jamais differe de plus de 1.23% de la solutionanalytique, alors que la charge hydraulique totale n'a pas varie deplus de 11 mm pour un espacement de 0.1m, et de 36 mm pour unespacement de 0.2 m. En moyenne, les profondeurs de nappe ob-tenues avec MODFLOW correspondaient a 21 mm pres a cellescalculees avec 1'equation deHooghoudt. Pour lecas oudes cellulescarrees sont utilisees, une courbe de calibration a ete developpeepour predire la conductivite hydraulique de la cellule contenant ledrain, ce qui permet de simuler des drainsde differents rayons.

INTRODUCTION

Most analytical solutions to theproblem of subsurface drainage are based on simplifying assumptions about the natureand variation of soil hydraulic conductivity. Therefore, numerical methods are increasingly used to investigate the

effect of soil conditions for which analytical solutions arenearly impossible. Finite element and finite difference methods have been used to simulate steady and transient-statetwo-dimensional flow between lateral drains under eithersaturated (Hwang et al. 1974; Mervaet al. 1983; Rogers et al.1988) or saturated-unsaturated conditions (Tang and Skaggs1980; Fipps and Skaggs 1986).

The accuracy of numerical solutions to drainage problemsdepends heavily on the precision with which the subsurfacedrain boundary is implemented. Several researchers havemodeled the subsurface drain by fixing the hydraulic head toa single node of the finite difference or finite element grid(Khanjani and Bloomsburg 1978; Tang and Skaggs 1978).Garaaty-Sani andKing (1978) found that large errors on thedrainage outflow rate may result when using this method.Theaccuracy of finite element solutions canbe improved bymodeling the complete drain using many elements. Thismethod has been used successfully by Zaradny and Feddes(1979) and Zissis and Stuyt (1991), butmodeling the drain isa tedious and time consuming task. Consequently, it is notconceivable that this method be used to obtain solutions on aroutine basis.

The single node representation ofa drain can beimprovedby assuming radial flow near the drain. In that case, the radialflow equation is used to compute the hydraulic head to befixed at some distance from the drain (Gureghian and Youngs1975). Alternatively, the flow rate is calculated with theradial flow equation and thepredicted hydraulic head atsomedistance from the drain (Fipps et al. 1986; Martinez et al.1989).

To simulate subsurface drains of various radii with a finitedifference model, Skaggs andTang(1979) useda correctionfactor (Cd) derived from anelectrical analogue network study(Vimoke etal. 1962; Vimoke and Taylor 1962). This methodis easy to implement and consists inadjusting the hydraulicconductivity between the drain grid node and adjacent nodesbased on the size of the drain in relation to the grid size. Fippset al. (1986) compared several methods of implementing thedrain boundary condition. They found that drain outflowdetermined usingthe Cd correction factor was in closeagreement with the analytical solution of Kirkham (1949) for theponded water case. However, large differences inthe hydraulic head were observed within 0.2 m of the drain.

Rogers and Fouss (1989) found that drain outflows predicted when using theCd correction factors gaveinconsistentresults forgridsystems andmeshsizes different from theone

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 2, April/May/June 1993 105

used by Fipps et al. (1986). Their investigation showed thataccurate drain outflow and hydraulic head could be obtainedby using one-half of the correction factors of Vimoke andTaylor (1962).

The finite difference model of McDonald and Harbaugh(1988), known as MODFLOW, is a widely used groundwater flow model that can solve a variety of subsurfacehydrology problems. MODFLOW allows simulation of one,two, or three-dimensional problems in either steady or transient-state, and isotropic or anisotropic conditions.Fixed-head or constant-flux boundary conditions are particularly easy to implement, and a range of preprocessors isavailable to prepare the input files required by MODFLOW.It has been used by Millette et al. (1992) to study seepagefrom an irrigation canal to interceptor drains. Recent modifications to this model (McDonald et al. 1991) make possibleits application to the study of water flow between lateraldrains.

The objective of this paper is to investigate the possibilityof using MODFLOW for simulating steady-state two-dimensional ground water flow between subsurface drains, with amethod similar to that of Vimoke and Taylor (1962) to represent the drain boundary condition.

PROCEDURE

Numerical method and drain representation

The partial differential equation solved by MODFLOW(McDonald and Harbaugh 1988) is:

o( dh"XX -.

OX

d (.. oh\

Kyydy fcrfVwdx ay

dh

dt

az

(1)

where:

Kxx, Kyy, Kzz = hydraulic conductivityalong x, y and zaxes (LT'1),

h = hydraulic head (L),W= volumetric flux per unit volume (T1),Ss =specific storage of the porous medium (L1),t = time (T),

x, y = horizontal coordinates (L), andz = vertical coordinate (L).

For vertical two-dimensional steady-state flow with isotropic conditions, Eq. 1 reduces to:

_3_dx

where:

K= hydraulic conductivity (LT1).

The discretization scheme used by Vimoke and Taylor(1962) is point-centered, which means that the flow resistance is assignedbetweentwo nodes of the discretization grid(Fig. la). However, MODFLOW uses a block-centered discretization, in which the flow domain is divided into gridcells to which the desired hydraulic properties are assigned

106

f, 3/0 a ( dh\K-- + ^T~ K—

ox dz axV J \ J

-w=o (2)

o Grid node

- - Boundary of grid cells

Area representing the drain

Ks

(a) Point-centered (b) Block-centered

Fig. 1. Comparison between point and block-centereddiscretization grid.

(Fig. lb). For point centered discretization, the drain can berepresented by correcting the hydraulic conductivity of thearea representing the drain (Rogers and Fouss 1989):

Kd = 2Cd

(3)

where:

Kd = hydraulic conductivity of the area representing thedrain,

Ks = hydraulic conductivity of the surrounding soil, andCd = correction factor of Vimoke and Taylor (1962).

The same principle can be applied for block-centered discretization:

Kdc =Cdc (4)

where:

Kdc = hydraulic conductivity of the drain cell, andCdc = correction factor for the drain cell.

Because of the difference in the geometry of the arearepresenting the drain (Fig. 1), there is no directequivalencebetween Cdc and Cd. Therefore, Cdcfactors need to be determined for the block-centered discretization scheme used inMODFLOW.

Figure 2 is a schematic representation of the flow domainbetween two lateral drains showing the symbols used in thisstudy. This situation was modeled with MODFLOW, asshown in Fig. 3, usinga uniform gridcell size throughout theflow domain. To conserve symmetry, the block-centereddiscretization imposes that the entire spacing be modeled withhalf the spacing on either side of the drain. Two grid cell sizeswere used: 0.1 and 0.2 m. Rogers and Fouss (1989) found thata 0.1 m grid spacing yielded accurate outflow rates andhydraulic heads. Large spacings and deep permeable soilsmay, however, require the use of 0.2 m cells to reduce computation time.

Calibration procedure

Values of Cdc were calibrated with the unit flow rate predicted by the solution of Kirkham (1957) for the ponded

GALLICHAND

Soil surface

^"T""

!WTD

1

t

DD

1

i7

^ Watertable ^

r

////////////// Impermeable layer //x/ ///y y/// "^i «L "

Fig. 2. Schematic diagram of geometrical parameters insubsurface drainage.

Soil surface

— s—-

• • e • 11 s •

\• • • •

//// //// //// Impermeable layer

fH Drain cell (K^) [•] Soil cell (Ks)

Fig. 3. Schematic diagram of the subsurface drainageflow domain discretization used in this study.

water case using reference drainage configurations of 25 mdrain spacing, 1 m depth to drain center, 1 m/d hydraulicconductivity, and depths to the impermeable layer of 4 and12mforthe0.1 and0.2 m gridspacing, respectively. Forthereference drainage configuration with a depth to the impermeable layer of12 m, the 0.2 mcells were used toreduce thesize ofthe problem. For each grid size, a Cdc value resultingin the outflow rate predicted by the analytical solution wasdetermined by trials and adjustments for effective drain radiiof50, 10, and 1mm. This range covers most of the effectivedrain radii that may beobserved ina field situation (Mohammad and Skaggs 1983). Additional Cdc values were obtainedwith the 0.1 mgrid size drainage configuration for ratios ofthe grid size to the drain radius (s/r) ranging from 2 to 200.These values were used to define a polynomial regressionequation relating the s/r ratio to Cdc

Outflow rates and watertable depths

Hydraulic heads predicted by the analytical solution ofKirkham (1957) were compared to those obtained for thereference drainage configuration with grid sizes of 0.1 and0.2 m and drain radii of 50,10, and 1 mm. Outflow rates werealso compared using the same grid sizes and drain radii forother drainage configurations. Mid-spacing watertabledepths obtained with MODFLOW for two recharge rates (1and 10 mm/d) were compared to those predicted by thesolution of Hooghoudt (Luthin 1978). Thedrainage configu

rations tested were for depths to the impermeable layer of 3,7, and 12 m, drain radii of 50, 10, and 1 mm, and drain depthsof 1.4 m. The drain spacing used for each comparison wasdetermined by computing, with Hooghoudt's equation, thespacing required to maintain a watertable depth of 1 m. Theresulting calculated spacing was rounded to the nearest integer and watertable depth was recomputed with this roundedspacing.

RESULTS AND DISCUSSION

Determination of Cdc values

The values of Cdc calibrated with the flow rates predicted bythe analytical solution of Kirkham (1957) are presented inTable I for grid sizes of 0.1 and 0.2 m, and effective drainradii of 50, 10, and 1 mm. These values are smaller than theCd/2 values suggested by Rogers and Fouss (1989). Forexample, with a ratio of the grid size to the drain radius (s/r)of 4.0, the Cd value presented by Vimoke and Taylor (1962)is 1.8395 and that of Cd/2 is 0.9198, compared to 0.5842 forCdc- For the network of square cells used in this study,changing the hydraulic conductivity of the drain cell inMODFLOW is equivalent to modifying the hydraulic resistance of half the distance between the grid node representingthe drain and an adjacent grid node. Therefore, correctionfactors for the drain cell need to be smaller than the Cd/2values to achieve the same internodal hydraulic resistance.

Outflow rates and hydraulic heads

Simulations were run to compare the hydraulic heads computed with MODFLOW with those predicted with Kirkham'sanalytical solution, using the same drainage configurationthat was used for calibrating Cdc values with the 0.1 m gridsize (L=25 m, Z=4 m, DD=1 m, and K=l m/d). Results forthe 0.1 m grid size are presented in Table II for variouselevations above and below the drain. Differences in hydraulic heads between the numerical and analytical solutions aresummarized in Table III for both grid sizes. This table showsthat, for the0.1 m gridsize, the absolute difference betweenthetwosolutions was always lessthanorequal to 11 mm. Forthe 0.2 mgrid size, differences were larger with a maximumabsolute difference of 36 mm. These differences are in thesame order of magnitude as those obtained by Rogers andFouss (1989) with finite elements. It is also consistent withtheir results showing that accuracy decreases for a grid sizelarger than0.1 m near the drain.

Additional simulations were run to determine if the Cdcvalues obtained with the reference drainage configurations(Table I) would yield accurate drain outflows when appliedtoother configurations. Preliminary analyses showed that thedifference between outflow rates from the analytical andnumerical solutions was not affected by either drain spacingorhydraulic conductivity. Therefore, drain spacing was fixedat 25 m and hydraulic conductivity at 1 m/d. Results arepresented in Table IV for drain radii of 50, 10, and 1 mm,depths to the impermeable layer of2, 4, 7, 10, and 15 m, anddrain depths of 1.0 and 1.6 m. This table shows that theabsolute error on the outflow rates was small ( <1.23%) forthe various drainage configurations tested. Absolute errorson the outflow rates, resulting from using the Cdc values of

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 2, April/May/June 1993 107

Table I: Results of calibration for two cell sizes (s) and three effective drain radii (r)

s = 0.1mz s = 0.2 my

r

(mm)

s/r Qx(nrW1.^1)

Cdc s/r Qx(mW^d'1)

Cdc

50

10

1

2

10

100

1.6071

1.1635

0.8201

0.0049

1.3225

3.4120

4

20

200

1.6246

1.1727

0.8247

0.5842

1.9135

3.9757

(z) with L = 25 m, Z = 4 m, DD = 1 m, and K = 1 m/d(y) with L = 25 m, Z = 12 m, DD = 1 m, and K = 1 m/d(x) calculated withanalytical solution of Kirkham (1957)

Table II: Comparison between hydraulic head at the drain obtained from analyticaland finite difference solutions fora grid cell size of 0.1 m and three effective drain radii

r (mm)

50 10 ]I

Elevation Hany Hfdx Hany Hfdx Hany Hfdx(m)z (m) (m) (m) (m) (m) (m)

4.0 4.0W 4.0W 4.0W 4.0W 4.0W 4.0W3.8 3.894 3.905 3.923 3.931 3.946 3.9513.6 3.778 3.788 3.839 3.846 3.887 3.8913.4 3.637 3.646 3.737 3.743 3.815 3.8183.2 3.427 3.434 3.585 3.584 3.708 3.7053.0 3.r 3.0W 3.1v 3.0W 3.1v 3.0W2.8 3.370 3.366 3.544 3.533 3.679 3.6692.4 3.602 3.606 3.712 3.714 3.797 3.7981.6 3.737 3.741 3.810 3.812 3.866 3.8670.9 3.782 3.785 3.842 3.844 3.889 3.8900.0 3.794 3.797 3.851 3.853 3.895 3.896

Note: withL = 25 m, Z = 4 m, DD = 1 m, andK = 1m/d(z) Elevation above impermeable layer(y) Han: analytical solution (Kirkham 1957)(x) Hfd: finite difference solution (McDonald and Harbaugh 1988)(w) boundary condition(v) assumed boundary condition

Table I, were always less than0.78% except with the50 mmdrain radius for which the maximum error was -1.23% withthe 0.1 m grid size. These values compare well with the0.66% error on the outflow rate reported by Martinez et al.(1989) with finite elements and using a correction based onthe radial flow assumption.

Drain depth seems to affect the error. In cases for whichthe drain depth was the same asthat used for determining theCdc values (DD=1.0 m), the error on the outflow rate wasnegligible (maximum of 0.02%), whereas for the 1.6 m draindepth, the absoluteerror percentage rangedbetween 0.02and1.23 of the analytical solution. This suggests that the draindepth used for calibration of the Cdc values may affect theprecision of the drain outflow obtained with the numerical

108

solution. The drain radius also seems to affect the outflowrate error. With the 0.1 m grid spacing, maximum absoluteerrors were 1.23, 0.16, and 0.36% for the 50, 10, and 1 mmdrain radius, respectively, whereas for the 0.2 mgrid spacing,maximum absolute errors of 0.79, 0.35, and 0.78% wereobtained for the same radii. Larger errors with the 50 mmdrain radius were, most likely, caused by the small s/r ratio,which resulted in small Cdc values (e.g.,0.0049 for the 0.1 mgrid) and high hydraulic conductivities of the drain cells. Forhigh hydraulic conductivities, small errors on the hydraulicheads can cause large errors of the calculated outflow rate.Rogers and Fouss (1989) found that the error on outflow ratesincreased for s/r ratios going from 3 to 1. Therefore, theoptimal grid spacing is dictated by two extremes: 1) the s/r

GALLICHAND

Table III: Summary of hydraulic head differences at the drain between analytical and finite difference solutions forgrid sizes of 0.1 and 0.2 m

Head differencez(mm)

S := 0.1my s ==0.2 mx

r (mm) r (mm)

Elw 50 10 1 Elw 50 10 1

(m) (m)

4.0 0 0 0 12.0 0 0 0

3.8 11 8 5 11.8 22 16 12

3.6 10 7 4 11.6 19 13 9

3.4 9 6 3 11.4 14 8 5

3.2 7 -1 -3 11.2 10 -1 -3

3.0 NAV NA NA 11.0 NA NA NA

2.8 -4 -11 -10 10.8 -30 -36 -32

2.4 4 2 1 10.4 -1 -4 -4

1.6 4 2 1 10.0 4 1 0

0.8 3 2 1 8.0 3 3 2

0.0 3 2 1 4.0 3 2 2

0.0 -8 2 1

(z) finite difference solution - analytical solution(y) with L= 25m,Z = 4 m,DD = 1 m, and K= 1m/d(x) with L= 25 m,Z= 12 m,DD = 1m, and K= 1m/d(w) elevation above impermeable layer(v) not applicable

ratio must not be less than two to three, and 2) the gridspacing near the drain must be small enough to yield accuratehydraulic heads. Skaggs and Tang (1979) used agrid spacingof 0.2 and0.25 m, whereas Rogers and Fouss (1989) foundthat0.1 m grid spacing wouldbe adequate.

Watertable depths

Watertable depths obtained with MODFLOW were compared against values predicted by Hooghoudt's equation.Hooghoudt's equation was selected because itis incorporatedin widely used drainage models (Skaggs 1978; Chung et al.1992). Table Vpresents, for the various configurations listed,the grid size used (s), drainage rates (q), watertable depthspredicted by Hooghoudt's equation (WTD-HE) and byMODFLOW (WTD-FD), and the difference between thetwo. From Table V it can be seen that, on the average, theabsolute difference on the watertable depth predicted by thetwo methods increased with increasing drainage rate, depthto impermeable layer, and grid size, but decreased with increasing drain radius. The absolute difference varied between1and 54mm, which represents, respectively, 0.25 and 13.5%ofthe height ofthe watertable above the drain center. On theaverage, an absolute difference of 21 mm was observed,which represents 5.3% of the height of the watertable abovedrain centerline. Watertable depths computed from the finitedifference method implemented in MODFLOW were, therefore, close to those predicted from Hooghoudt's equation.The difference between the two solutions cannot be posi

tively attributed to an error due to the finite differencemethod since Hooghoudt's solution is notanexact analyticalsolution to Eq. 2, but usessimplifying assumptions.

Because results of Table IV showed that the drainageconfiguration used for calibration may affect the Cdc values,other calibrations were performed with Kirkham's equationtodetermine theCdc* for theexactdrainage configuration ofselected casesof TableV.The results,presented in TableVI,showed thatwatertable depths predicted by MODFLOW didnot vary by more than 1mm from those presented inTable V.From the comparison of Cdc values ofTable VI and Table I,it canbe concluded that, for the casesstudied, differences inCdc as large as 0.045 have negligible effect on the watertabledepth determined by MODFLOW.

Calibration curve

A calibration curve to predict thedrain cell correction factorofEq. 4was developed for s/r ratios between 2and 200, usingthereference drainage configuration ofTable I fora depth tothe impermeable layer of 4 m, and a grid size of 0.1 m.Results are shown in Fig. 4. This figure shows the s/r ratiosused (square symbols), and the regression line correspondingtothe equation presented inthe figure. This equation yields amaximum error of 0.004 unit of the Cdc value. The Cdc valuesofFig. 4 can be used to simulate drains with MODFLOW ifsquare cells are used for the drain and the eight surroundingcells. Rectangular cells, other than square, will change theinternodal distance for which the hydraulic resistance is cor-

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 2, April/May/June 1993 109

Table IV: Error on drain outflow when using Cdc valuesof Table I with different drainageconfigurations

Grid

size r Z DD Qz Error3'

(m) (mm) (m) (m) (mV-d"1) (%)

0.1 50

10

0.2 50

10

2.0

4.0

7.0

2.0

4.0

7.0

2.0

4.0

7.0

10.0

15.0

10.0

15.0

10.0

15.0

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.0

1.6

1.5340

1.9519

1.6071

2.2736

1.6216

2.3246

1.1244

1.5021

1.1635

1.6905

1.1711

1.7187

0.8005

1.1206

0.8201

1.2229

0.8239

1.2375

1.6242

2.3335

1.6248

2.3358

1.1725

1.7236

1.1728

1.7248

0.8245

1.2401

0.8247

1.2407

Note: with L = 25 m and K = 1 m/d

C)analytical solution( ) [ (Qnumerical - Qanalytical) / Qanalytical ] X100

0.00

-1.23

0.00

-0.98

-0.02

-1.01

0.00

-0.16

0.00

0.02

-0.02

-0.03

0.00

0.20

0.00

0.35

-0.01

0.36

0.00

-0.78

0.00

-0.79

0.00

0.35

0.00

0.35

0.01

0.78

0.00

0.78

rected, and erroneous outflow rates and watertable depthswill result.

SUMMARY AND CONCLUSIONS

Numerical experiments were performed with the finite difference ground water flow model MODFLOW to simulatesteady-state two-dimensional flow between subsurface lateral drains. Subsurface drains were represented by a singlenode for which the hydraulic conductivity was correctedbased on the grid size, drain radius, and hydraulic conductivity of the surrounding soil. Solutions were obtained for twogrid sizes (0.1 and 0.2 m), three effective drain radii (50, 10,and 1 mm), and a range of drain depths, drain spacings, anddepths to the impermeable layer.

no

0)13

o

Fig. 4.

Cdc = -0.50054 + 0.68652 £n(s/r)

+ 0.05505 [Jen(s/r)]2- 0.00424 [jen(s/r)]°

80 120

s/r ratio

Calibration curve for determining drain cellhydraulic conductivity from effective drainradius, grid size, and soil hydraulicconductivity.

200

Numerical results of drain outflow were compared withKirkham's analytical solution for the ponded water case. Forthe range of drainage configurations tested, the numericalsolution for drain outflow was always within 1.23% of theanalytical solution, whereas hydraulic heads did not vary bymore than 11 mm for the 0.1 m grid size, and 36 mm for the0.2 m grid. The errors on the drain outflow rate were thelargest for small ratios of the grid size to the drain radius (s/r).

Mid-spacing watertable depths obtained by MODFLOWwere compared with those predicted by Hooghoudt's equation. Differences in watertable depth ranged from 1 to 54 mmwith an average of 21 mm, which represents only 5.3% of theheight of the watertable above drain centerline.

To represent drains of various radii in square grid MOD-FLOW models, a calibration curve was developed to predictthe correction to be applied to the hydraulic conductivity ofthe drain cell, for s/r ratios from 2 to 200.

GALLICHAND

Table V: Comparison between mid-spacing watertable depth obtained by Hooghoudt's equation and finite differencewith Cdc values of Table I

q z L s

WTDZ (m)

r HEy FDX DifT

(mm) (mm/d) (m) (m) (m) (m)

50 1 3 72 0.1 1.002 0.992 -0.010

7 115 0.2 1.002 1.001 -0.001

12 141 0.2 0.999 1.003 0.004

10 3 21 0.1 0.988 0.971 -0.017

7 26 0.1 1.003 1.004 0.001

12 27 0.2 0.997 1.014 0.017

10 1 3 70 0.1 0.996 0.995 -0.001

7 107 0.2 0.999 1.016 0.017

12 127 0.2 0.999 1.026 0.027

10 3 19 0.1 0.985 0.992 0.007

7 22 0.1 0.997 1.032 0.035

12 22 0.2 0.997 1.036 0.039

1 1 3 66 0.1 1.001 1.008 0.007

7 96 0.2 1.000 1.031 0.031

12 110 0.2 1.000 1.046 0.046

10 3 16 0.1 1.009 1.034 0.025

7 18 0.1 1.000 1.054 0.054

12 18 0.2 0.997 1.043 0.046

Note: with DD = 1.4 m and K = 1 m/d

(z)WTD : watertable depth(y)HE : from Hooghoudt's equation(x)FD : with finite difference(w) Diff = WTD(FD)- WTD(HE)

Table VI: Comparison between mid-spacing watertable depth obtained by Hooghoudt's equation and finite differencewith Cdc values calibrated for the specificdrainage configurations

q Z L s Cdc

WTDZ (m)

rHEy FDX Diffw

(mm) (mm/d) (m) (m) (m) (m)

50 1 3 72 0.1 0.0015 1.002 0.992 -0.010

12 141 0.2 0.5610 0.999 1.004 0.005

10 3 21 0.1 0.0015 0.988 0.971 -0.017

12 27 0.2 0.5615 0.997 1.015 0.018

10 1 3 70 0.1 1.3235 0.996 0.995 -0.001

12 127 0.2 1.9268 0.999 1.026 0.027

10 3 19 0.1 1.3235 0.985 0.992 0.007

12 22 0.2 1.9280 0.997 1.036 0.039

1 1 3 66 0.1 3.4330 1.001 1.007 0.006

12 110 0.2 4.0200 1.000 1.045 0.045

10 3 16 0.1 3.4330 1.009 1.033 0.024

12 18 0.2 4.0210 0.997 1.042 0.045

Note: with DD = 1.4 m and K = 1 m/d

(z) WTD : watertable depth(y)HE : from Hooghoudt's equation(x)FD : with finite difference(w)Diff = WTD(FD) - WTD(HE)

CANADIAN AGRICULTURAL ENGINEERING Vol. 35, No. 2, April/May/June 1993 111

REFERENCES

Chung, S.O., A.D. Wardand C.W. Schalk. 1992. Evaluationof the hydrologic component of the ADAPT water tablemanagement model. Transactions of the ASAE35:571-579.

Fipps, G. and R.W. Skaggs. 1986. Effect of canal seepage ondrainage to parallel drains. Transactions of the ASAE29:1278-1283.

Fipps, G., R.W. Skaggs and J.L. Nieber. 1986. Drain asboundary condition in finite elements. Water ResourcesResearch 22:1613-1621.

Garaathy-Sani, R. and L. King. 1978. Interceptor drains onsloping land. ASAE Paper No. 78-2028. St. Joseph, MI:ASAE.

Gureghian, A.B. and E.G. Youngs. 1975. The calculation ofsteady-state water-table heights in drained soils by meansof the finite-element method. Journal of Hydrology27:15-32.

Hwang, R.B., J.N. Luthin and G.S. Taylor. 1974. Effect ofbackfill on drain flow in layered soils. Journal of theIrrigation and Drainage Division, ASCE 100:267-276.

Khanjani, M.J. and G.L. Bloomsburg. 1978. Drainage in anatural layered profile by finite element analysis. ASAEPaper No. 78-2036. St. Joseph, MI: ASAE.

Kirkham, D. 1949. Flow of ponded water into drain tubes insoil overlying an impervious layer. Transactions of theAmerican Geophysical Union 30:369-385.

Kirkham, D. 1957. The ponded water case. In Drainage ofAgricultural Lands, ed. J.N. Luthin, Monograph 7,Chapter 11-111:139-181. American Society of Agronomy,Madison, WI.

Luthin, J.N. 1978. Drainage Engineering. Huntington, NY:Krieger Publishing Company.

Martinez, M.A., E.J. Monke and E.J. Kladivko. 1989. Drainflow simulation using finite element technique. ASAEPaper No. 89-2099. St. Joseph, MI: ASAE.

McDonald, M.G. and A.W. Harbaugh. 1988. A modularthree-dimensional finite-difference ground-water flowmodel. In U.S. Geological Survey Techniques of WaterResources Investigations, Book 6, Chapter Al. U.S.Geological Survey, Denver, CO.

McDonald, M.G., A.W. Harbaugh, B.R. Orr and D.J.Ackerman. 1991. A method of converting no-flow cells tovariable-head cells for the U.S. Geological Surveymodular finite-difference ground-water flow model.Open-File report 91-536. U.S. Geological Survey,Denver, CO.

112

Merva,G.E.,L. Segerlind,H. Muraseand N.R.Fausey. 1983.Finite element modeling for depth and spacing of drainsin layeredsoils. Transactions of the ASAE 26:452-456.

Millette, D., G.D. Buckland, W.R. Galatiuk and C.A.Madramootoo. 1992. Groundwater-flow modeling ofdeep interceptor and grid drainage for interception ofcanal seepage and groundwater. Canadian AgriculturalEngineering 34:7-15.

Mohammad, F.S. and R.W. Skaggs. 1983. Effect of draintube openings on transient drainage. In Proceedings 2ndInternational Drainage Workshop, 185-197. CorrugatedPlastic Tubing Association, Carmel, IN.

Rogers, J.S. and J.L. Fouss. 1989. Hydraulic conductivitydetermination from vertical and horizontal drains in

layered soil profiles. Transactions of the ASAE32:589-595.

Rogers, J.S., H.M. Selim and J.L. Fouss. 1988. Steady stateflow to drains in multilayered anisotropic soils. ASAEPaper No. 88-2616. St. Joseph, MI: ASAE.

Skaggs, R.W. 1978. A water management model for shallowwater table soils. Technical Report No. 134. WaterResources Research Institute of the University of NorthCarolina. North Carolina State University, Raleigh, NC.

Skaggs, R.W. and Y.K. Tang. 1979. Effect of drain diameter,openings and envelope on water table drawdown.Transactions of the ASAE 22:326-333.

Tang, Y.K. and R.W. Skaggs. 1978. Subsurface drainage insoils with high hydraulic conductivity layers.Transactions of the ASAE 21:515-521.

Tang, Y.K. and R.W. Skaggs. 1980. Drain depth andsubirrigation in layered soils. Journal of the Irrigationand Drainage Division, ASCE 106:113-122.

Vimoke, B.S. and G.S. Taylor. 1962. Simulating water flowin soil with an electrical resistance network. ARS 41-65.Agricultural Research Service, USDA.

Vimoke, B.S., T.D. Tyra, T.J. Thiel and G.S. Taylor. 1962.Improvements in construction and use of resistancenetworks for studying drainage problems. Soil ScienceSociety ofAmerica Proceedings 26:203-207.

Zaradny, H. and R.A. Feddes. 1979. Calculation ofnon-steady flow towards a drain in saturated-unsaturatedsoil by finite elements. Agricultural Water Management2:37-53.

Zissis, T. and L.C.P.M. Stuyt. 1991. Effect of radial soilheterogeneity around a subsurface drain on the watertable height computed using a finite element model.Agricultural Water Management 20:47-62.

GALLICHAND