THEORY OF PIPE DRAINAGE ASSISTED BY MOLE DRAINAGE

By Helmi M. Hathoot,l Member, ASCE

ABSTRACT: This paper addresses the problem of draining a heavy soil of low hydraulic conductivity overlyingan impermeable layer by means of a combined system of pipe and mole drains. The analysis uses the complexfunctions and the theory of images to develop formulas for pipe and mole drain discharges and for pipe drainspacing design. A flow pattern predicted on the basis of these formulas shows that boundary conditions aresatisfied and the stream line distribution is logical. An equation for the unsteady vertical movement of the watertable is presented, from which an approximate but satisfactory spacing equation is established. A numericalexample is provided and it is shown that although the spacing formula is implicit, only a small number of trialsare needed. It is also shown that introducing mole drains may allow the spacing of pipe drains to be increasedseveral times. A technique is proposed for choosing the number of mole drains between two pipe drains suchthat the optimal spacing design is achieved.

in which C3 = a real constant. The complex potential of thefictitious sinks of strength ml is given by

in which ml = sink strength; 2c = spacing between sinks; D+ Dl = soil thickness below mole drains; and C2 = a realconstant.

The impermeable layer can be replaced by the two rows ofimaginary sinks shown in Fig. 2. The complex potential of thefictitious sinks of strength m is given by

(3)

(5)

. lI'(z + iD)W2 =m In Sill L + C3

The complex potential of the system is obtained by simplyadding complex potentials (I) through (4).

[. lI'(z - iD) I . 1I'(z + iD)]

W =m In Sill + n Sill ~-_...::.L L

{. 1I'[z - C - i(D + D I)]

+ m. In Sill2c

I. lI'[z - C + i(D + DI)]} C+ nsm + s

2c

INTRODUCTION

• The soil is homogenous and isotropic.• The effect of the capillary fringe is neglected.• On the phreatic surface the pressure is atmospheric.• The flow is radial near the drain boundary.

MATHEMATICAL MODEL

The ideal system of drainage, which would secure the max­imum benefit with the minimum overall cost, is a system ofsubsurface drainage pipes because it saves a considerable areaof land that may be consumed in the traditional system of opendrains (Schilfgaarde 1974). The problem of the design of sub­surface pipe drainage has been investigated by researchers us­ing different approaches (Youngs 1985). Design formulas fordischarge, spacing, and depth of tiles have been successfullyused in providing rational drainage system designs (Kirkham1966; Youngs 1985; Hathoot et al. 1993). However, in casesof heavy soils with low hydraulic conductivity these designformulas provide narrow spacings between drains, which isuneconomical and sometimes impractical. In practice, one ofthe proposed solutions to such a problem is to use mole drains,which are cheap and shallow, together with relatively deeppipe drains to increase the spacing between pipe drains andget more economical designs. This paper presents design for­mulas for a combined system of pipe and mole drains. Theproblem dealt with here is that of draining a heavy clay layeroverlying an impermeable barrier (Fig. 1). The following as­sumptions apply throughout the paper:

The complex potential of an infinite number of equallyspaced sinks (Liggett 1994) that represent pipe drains (Fig. 2)is

in which m = sink strength; Z = complex coordinate = x +iy; D = soil thickness below drains; L = spacing betweendrains; i = V-I; and C1 = a real constant. The complex po­tential of an infinite number of equally spaced sinks that rep­resent mole drains (Fig. 2) is given by

'Prof.• Dept. of Math .• Coli. of Sci., King Saud Univ., P.O. Box 2455.Riyadh 11451. Saudi Arabia.

Note. Discussion open until September 1, 1998. To extend the closingdate one month. a written request must be filed with the ASCE Managerof Journals. The manuscript for this paper was submitted for review andpossible publication on December 5. 1996. This paper is part of the Jour­nal of Irrigation and Drainage Engineering. Vol. 124, No.2. MarchiApril. 1998. ©ASCE. ISSN 0733-9437/98/0002-0102-0107/$4.00 +$.50 per page. Paper No. 14342.

lI'(Z - iD)WI =m In sin L + CI (1)

in which Cs = a real constant. Substituting z = x + iy, thensimplifying and rearranging, produces

{ [lI'X 11' 1I'X. 11' ]w=m In sin-cosh-(y - D) + icos-smh-(y - D)L L L L

[. 1I'X 11' . lI'X. 11' ] }+ In sm - cosh - (y + D) + I cos - smh - (y + D)

L L L L

{ [lI'(x - c) 11' .+ ml In sin cosh - (y - D - D,) + I cos

2c 2c

11'~-~ 11' ] [11'~-~. sinh-(y - D - D1) + In sin cosh2c 2c 2c

11' lI'(x-c) 11' ]}'-(y+D+Dt)+icos sinh-(y+D+D\) +Cs2c 2c 2c

(6)

Setting w = ep + ilil. in which ep = velocity potential and iii =stream function, then rearranging, produces

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(10)

(11)

(12)

(13)

L =2e(n)

FLOW PATTERN

IjI =m {tan- 1 [cot '7 tanh Z(y - D)]

+ tan-I [cot '7 tanh Z(y + D)]}+ ml {tan-I [cot 7 (x - ~) tanh 7 (y - D- D I)]

+ tan-I [cot: (x - ~) tanh: (y + D + D I )]}

in which n = number of mole drains installed between twopipe drains. Substitution from (10) into (8) and (9) yields

<I> =~ In {[sin2 '7 + sinh

2~ (y - D)]

. [sin2 '7 + sinh2Z(y + D)]}mt {[. 2 1Tn ( L ) . 2 1Tn ]+ "2 In sm L x - 2n + smh L (y - D - D I )

[1Tn ( L ) . 1Tn ] }. sin2 L x - 2n + Slnh2 L (y + D + D t ) + Cs

IjI = m {tan-' [cot '7 tanh ~ (y - D)]+ tan- t [cot '7 tanh ~ (y + D)]}

+ ml {tan-1 [cot ~ (x - e)tanh ~ (y - D- D,)]

+ tan-I [cot ~ (x - c)tanh;: (y + D+ DI)]} (9)

As it is made clear by Figs. 1 and 2, pipe drain spacing, L,and mole drain spacing, 2e, are interrelated by

It is evident that in Fig. 1 the lines, SIS:, S2S~, S3Si, ...are vertical lines of symmetry. Accordingly, a block such asS3S4S~Si completely represents the flow system. To demon­strate the general features of the flow pattern, the followingcase is considered: L = 16.0 m; 2e = 4.0 m; n = 4; D = 2.0m; D 1 = 1.0 m; and ml/m = 0.1. Flow is considered to besteady and the soil is assumed to be completely waterlogged.

To facilitate computation of the flow pattern, (12) is put inthe following form:

IjI I[ ~ 1T ]- =tan- cot - tanh - (y - 2)m 16 16

[1TX 1T ]+ tan-I cot 16 tanh 16 (y + 2)

+ 0.1 {tan-' [cot (:x -~) tanh~(Y - 3»)

+ tan-I [cot (~x - ~) tanh ~ (y + 3)J}According to (13), ljI/m values, in degrees, are evaluated andthe streamlines of Fig. 3 are plotted at intervals of 15°. Asshown in Fig. 3, mole drains and pipe drains share drainage

s;

(8)

S'2FIG. 1. Geometry of Problem

o Sink of J' 0 0 T'strength t---2c -I· 2c ---I 0,m,

"'. (mole dr.;ns) /0 t 0

Sinks o.f .tr.n~th m ~(pipe drains) Y 0

.. L -l----L P--...------_.o X

Im.ginary sinkS 0,/ of str.nth m ""'" 1

(I (Imagin.ry pipe dr.insl"w:, t c

/Im.ginary. 1---2c - ..-i......-2c --I 0,

• sinks of\: ,) 1" strength m, (\

(imaginary pipe dr.ins)

FIG. 2. Mathematical Model

S'1

I I Ground ~5.\ Sal 5,} 54 Surfan

-.If MJle '1:i'W.i"~.~ ;AWW; I'i 1 / drains....... . 0 BI h .I d, :0 . 1:"0 .~ 0- I'1 1 I I I-~ c -.. 2i:--I '\

'0,· .

I Pipe U-±t--!i IA I Icfdr.in I d '.Q . c;>~ L' T ~~~~. 2 • 2 .

I · 0 i I II y . .I' er~e.ble La e,!

<I> + iljl = ~ In [sin21TX + sinh2~ (y - D)] + im tan-I2 L L

· [cot~tanh~(Y - D)] + ~In [sin2~ + sinh2~(y + D)]L L 2 L L

+ im tan- t [cot '7 tanh ~ (y + D)] + ~t In [sin2~ (x - e)

+ sinh2~ (y - D- DI)] + im t tan-'

· [cot~ (x - c)tanh ;: (y - D - D I ) ]

+ mt In [sin2..!. (x - e) + sinh2..!. (y + D+ Dl)]2 2c 2c

+ imt tan-I [cot~ (x - e)tanh~ (y + D + Dt)] + Cs (7)

Equating real to real and imaginary to imaginary on both sidesof (7) and rearranging, produces

<I> = ~ In {[sin2 '7 + sinh2~ (y - D)]

· [Sin2 :x + sinh2 ~ (y + D)]}

+ ~l In {[sin2;: (x - c) + sinh2~ (y - D - D 1) ]

· [sin2~ (x - e) + sinh2~ (y + D+ DI)]} + Cs

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o·

(17)

K(D + D , + h) =mIn [COShZ(D , + h)' coshZ(2D + D , + h)]

[1Tnh 1Tn ]+ m,ln coshT' coshT(2D + 2D, + h) + Cs

Eliminating Cs between (16) and (17) produces

K (D' + h - ~) =mF + m,G (19)

Finally, at point E in Fig. 1, at the bottom of a mole drain,the pressure is practically atmospheric since the drain is run­ning nearly empty. Applying (15) to point E (L/2 - c, D +D, - d,I2), and simplifying produces

( d') m {[ 2 1T 21T ( d.)]K D + D, - 2" = 2" In cos 2n + sinh L D, - 2"

-[cos2;: + sinh

2 Z(2D + D , - i) ]}[

1Tndl 1Tn ( d')]+ ml In sinh 2L . sinh -L 2D + 2D, - -2 + Cs (18)o· o· S'

42 3 4 5 6 7 8 x

8·00m --IFIG. 3. Example of Flow Pattern

2~~+--+----:::l~~-+--+--~-+---;~o

~--+--I----+---4--+---+--+---1~o·

S'O~"--"""'~;;...I",,_--l._+-""_~_~_~_~~""

o1-

y4 FilI....-...!40U:I~...!.lC'£.-W'!~I='M~:.r-,---'~.;:;&~~'-if-~·.,..

flow on the basis of the flow pattern. It is also evident that onlines S3S~ and S4S~ the streamlines are vertical, whereas theline S~S; the streamline is horizontal. This satisfies the pre­scribed boundary conditions.

DRAIN DISCHARGE

in which

[

COSh z: (D, + h)-cosh z: (2D + D, + h)]F = In . 1Td . 1T ( d)

smh --smh - 2D + ­2L L 2

(20)

In flow through porous media (Harr 1962; Polubarinova­Kochina 1952) the velocity potential, <1>, is given by

(14)

in which K = hydraulic conductivity of soil; p = gauge pres­sure; p =density of water; g =acceleration due to gravity; andy = vertical coordinate of a point. Combination of (11) and(14) yields

K(;g + Y) =~ In {[sin2 7+ sinh2 Z(y - D)]

. [ sin2 :x + sinh2 Z(y + D)]}m, {[. 2 1Tn ( L) 2 1Tn ]+ - In sm - x - - + sinh - (y - D - D )2 L ~ L '

-[sin2 ~ (x - ~) + sinh2 ~ (y + D+ D,)]} + Cs

(15)

On the top of a pipe drain at point A in Fig. I, the pressureis atmospheric (zero) (Hathoot 1987). Applying (15) to pointA (0, D + d/2) and simplifying yields

K (D + ~) = m In [sinh ;:. sinh Z(2D + ~)] + m,

·In [COSh 7(g -D I ) • cosh :n (2D + D, + g)] + Cs

(16)

[

1T~ ~ ]cosh -·cosh - (2D + 2D, + h)G = In L L

cosh :n (~- DI) .cosh 7 (2D +D, +~)

Eliminating Cs between (17) and (18) produces

K ( h + i) =mH + mIl

in which

H = In {COSh Z(D. + h)' cosh Z(2D + D, + h)/

{[cos2;: + sinh

2 Z(D' - ~)]

[ ( )]}"2}2 1T • 2 1T d,- cos 2n + smh L 2D + D, - "2

[

1Tnh 1Tn ]cosh ['cosh L (2D + 2D, + h)

I =In . 1Tnd, . 1Tn ( d.)smh ---smh - 2D + 2D -­

2L L I 2

Solving (19) and (22) for m and ml yields

K [ h(H - F) + H (D. - ~) - F ~]

ml = GH - FI

(21)

(22)

(23)

(24)

(25)

On the phreatic surface at point B in Fig. 1, (U2, D + D, +h), the pressure is atmospheric. Applying (15) to point Bandsimplifying yields

(26)

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in which

(33)

(34)

(35)

(36)

(37)

B =GH - Fl

E =2(G - I) + n(H - F)

d, (d)J ="2 (2G - nF) - V, - 2 (21 - nH)

in which 1, and 12 = initial and final times, respectively. It isworthy to note that F, G, H, and I are functions of the waterhead, h. However, h is contained in natural logarithm terms inthe expressions for the aforesaid quantities and, therefore, aphysical variation of h is expected to yield relatively smallvariations in F, G, H, and I. Better results can be obtained ifh is replaced by the arithmetic average of hi and h2• Eq. (32)can be put in the form

t2 - t, =A [' [Eh B+ J] dh

7rK [ h(H - F) + H (VI - g) - F ~]

Q, = GH - FI (27)

The pipe drain strength m is multiplied by 27r since thesedrains are sufficiently below the water table that water is re­ceived from the whole circumference of a pipe drain.

If the water table exists in the upper soil layer above the moledrains, the discharge reaching each unit length of a mole drainis given by

The strength m, is multiplied by 7r because a mole drain isconsidered to receive water from one-half of its perimeter(Hathoot and Rezk 1989). This assumption is based on theobservation that the water table generally intersects the moledrains in its upper stage. On the other hand, the dischargereaching each unit length of a pipe drain is given by

27rK [h(G - I) + ~ G - 1 (D' - g)]Q = GH - FI (28)

The unsteady downward movement of the water table is char­acterized by (38).

As the practical dimensions and soil constants of subsurfacedrainage are considered, changing h from 0.6 to 0.2 m (66.7%)corresponds to changes on the order of 4% in B, E, or J. It isworthy to note that a chosen practical pipe drain spacing andthe spacing obtained from direct calculation may differ bymore than 4%. Accordingly B, E, and J can be practicallyconsidered to be constants and (33) can be integrated and re­duced to

UNSTEADY MOVEMENT OF WATER TABLE

The unsteady movement of the water table has been con­ventionally assumed to be the same as a continuous successionof steady states, with the flux through the water table assumedto be uniform and given by the drain discharge rate dividedby the surface area (Schilfgaarde 1974; Childs 1969). The wa­ter table can be assumed to be a horizontal straight line withmore or less local depressions above the drains (Hathoot1979). The differential equation for the downward movementof the water table can be written as

(29)

in which IJo =drainable porosity of soil; and 1 =time. Substi­tuting from (27) and (28) into (29) and rearranging produces

t2 _ t, =(A:)'ln rh

• + ~]h +­

2 E

(38)

DESIGN OF PIPE DRAIN SPACING

The drawdown drainage requirement is the deciding crite­rion in many cases of pipe drainage design. The root zone ofplant should be cleared from water in a specified period oftime, T, which depends upon the kind of plant; otherwise, theplant will die. In the worst case the soil is assumed to becompletely waterlogged after excessive irrigation or afterheavy rainfall. The drain spacing L is to be chosen such thatthe root zone, ho - h (see Fig. 1), should be drained withintime period T. For convenience (38) is written in the form

17K { h[2(G - I) + n(H - F)) +~ (2G - nF) - (D' - ~) (21- nH)}

GH-Fl(30)

Separation of variables yields

dl= _vL17K

{

GH- F/ }. ~ d ~

h[2(G -I) + n(H - F)] + "2 (2G - nF) - (D' -2) (2/- nH)

(31)

An estimate of the time necessary for lowering the water tablefrom hI to h2 can be made by integrating (31).

7rKETL =------..;;,;.:,.--

[J]h +-

1JoB' In ~h+­

E

(39)

1',

t, - I, = vL17K "

{

GH- Fl }. d, d dh

h[2(G - I) +n(H - F)] +"2 (2G - nF) - (DI - 2) (2/- nH)

(32)

Although the spacing, c, does not appear in (39), it is indirectlyincluded through (lO).

For a rationally chosen mole drain spacing, 2c, the corre­sponding pipe drain spacing, L, can be evaluated from (39). Itis worthwhile to note that L is included in the parameters B,E, and J, appearing in the right-hand side of (39). Accordingly,a suitable initial value of L is assumed and (39) is to be solvedthrough a trial-and-error procedure.

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TABLE 1. Trial Cycles of Spacing L

L L(assumed) (calculated)

(m) F G H I E J B (m)(1 ) (2) (3) (4) (5) (6) (7) (8) (9)

60.00 6.250 0.711 0.0315 4.041 -81.28 -10.11 -25.23 50.3850.38 6.011 0.820 0.0364 3.93 -77.92 -9.73 -23.60 51.6951.69 6.044 0.804 0.0356 3.95 -78.40 -9.79 -23.83 51.51

CONCLUSIONS

A flow pattern plotted on the basis of the derived formulasshows that boundary conditions are satisfied. The solution ofa numerical example shows that although the pipe drain spac­ing formula is implicit, only a few trial cycles are needed toattain the final design spacing. It is also shown that introducingmole drains increases pipe drain spacing, which is more eco­nomical. A technique for choosing the number of mole drainsbetween two pipe drains is proposed so that an economicaldesign that satisfies drainage requirements may be achieved.

nFIG. 4. Levellzed Net Annual Cost per Square Meter ofDrained Land (LNACM) versus Number of Mole Drains

NUMERICAL EXAMPLE

A heavy agricultural soil (K = 0.1 m1day, j.L = 0.05) liesover an impermeable layer 21.6 m from the ground surface. Adesign is needed for a combined system of pipe and moledrains to clear the root zone (0.4 m) within 48 hours. Pipedrains are 0.1 m diameter tubes and 1.6 m deep, whereas moledrains are 0.076 m diameter channels 0.6 m below groundlevel.

Solution

An initial spacing L =60 is assumed, with n = 12 (2c =5.0m). Eq. (39) is applied, in which D =20.0 m, D 1 =1.0 m, ho

= 0.6 m, h =0.2 m, and hay = 0.4 m. Successive trial resultsof applying (39) are listed in Table 1.

For convenience the tile drain spacing is taken as L = 48 mand, therefore, the spacing between mole drains is

482c =-= 4.0 m

12

To show the advantage of using mole drains together with pipedrains, it is interesting to estimate the spacing between pipesin the absence of mole drains. For the heavy soil used in thisexample, the pipe drain spacing (Hathoot 1979, 1985) is L =3.2 m. It is obvious that using mole drains increases tile drainspacing 15 times.

OPTIMAL DESIGN OF COMBINED SYSTEM

In example 1, the number of mole drains installed betweentwo tile drains was arbitrarily chosen (n = 12). Changing thenumber of mole drains n has the effect of changing the designspacing Land 2c. In fact, there are an infinite number of nvalues, each of which satisfies the drawdown requirements ofa drainage problem. The selection of n should be based oneconomic considerations. As n increases, L increases andtherefore the levelized net annual cost of pipe drainage, persquare meter, (LNACM) decreases. On the other hand, increas­ing n decreases the spacing between moles, 2c, and thus in­creases the levelized net annual cost of mole drainage persquare meter. The optimal value of n is that in which the totalcost of pipe and mole drainage is at a minimum (Fig. 4).

APPENDIX I. REFERENCES

Childs, E. C. (1969). Introduction to the physical basis of soil waterphenomena. John Wiley & Sons, London, U.K.

Harr, M. (1962). Groundwater and seepage. McGraw-Hili Inc., NewYork, N.Y.

Hathoot, H. M. (1979). "New formulas for determining discharge andspacing of subsurface drains." Int. Commission on Irrigation andDrainage (ICID) Bull., 28(2), 82-86.

Hathoot, H. M. (1985). "Graphical design of tile and mole drainage sys­terns." ICID Bull., 34(1), 43-48.

Hathoot, H. M. (1987). "Drainage of agricultural soils by subsurfacedrains." Bull. Fac. Engrg., Alexandria Univ., Alexandria, Egypt,XXVI, 1-18.

Hathoot, H. M., and Rezk, M. A. (1989). "Modification of drain dischargeand spacing formulas." Alexandria Engrg. J., Alexandria, Egypt, 28(4),231-245.

Hathoot, H. M., AI-Amoud, A. I., Mohammad, F. S., and Abo Ghobar,H. M. (1993). "Design criteria of drain tube systems in the centralregion of Saudi Arabia." J. King Saud Univ., Engrg. Sc., 5(2), 191­212.

Kirkham, D. (1966). "Steady state theories for land drainage." J. Irrig.and Drain. Div., ASCE, 92, 19-39.

Liggett, J. A. (1994). Fluid mechanics. McGraw-Hili Inc., New York,N.Y.

Polubarinova-Kochina, P. Y. (1952). Theory ofthe motion ofgroundwater.Gostekhizdat, Moscow, U.S.S.R.

Schilfgaarde, J. V., ed. (1974). "Drainage for agriculture." No. 17, Am.Soc. of Agronomy, Madison, Wis., 115-143.

Youngs, E. G. (1985). "A simple drainage equation for predicting water­table drawdowns." J. Agric. Engrg. Res., 31, 321-328.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A = quantity defined by (34);B = quantity defined by (35);c = half the spacing between mole drains;

CI , C2, ••• = real constants;D = thickness of soil layer between pipe drains and im­

permeable layer;d = pipe drain diameter;

D I = thickness of soil layer between pipe and moledrains;

d l = mole drain diameter;E = quantity defined by (36);F = quantity defined by (20);G = quantity defined by (21);

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g = acceleration due to gravity;H = quantity defined by (23);h water table height above mole drains midway be­

tween two mole drains;ho = depth of mole drains;hi = initial height of water table above mole drains mid­

way between two mole drains;h2 final height of water table above mole drains mid-

way between two mole drains;I = quantity defined by (24);i = V-I;J = quantity defined by (37);K = hydraulic conductivity of soil;L = spacing between pipe drains;

LNACM = levelized net annual cost per square meter;m strength of a sink representing pipe drain;

m. = strength of a sink representing mole drain;

n = number of mole drains installed between two pipedrains;

p = pressure;Q = discharge reaching each unit length of pipe drain;

Q, = discharge reaching each unit length of mole drain;T = specified time period required for lowering water

table from ho to h above mole drains;t = time;

I, = initial time;12 = final time;w = complex potential;x horizontal coordinate of point;y = vertical coordinate of point;z = complex coordinate (=x + iy);

I.t = drainable porosity of soil;p = density of water;

<I> velocity potential; and'" = stream function.

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nal u

se o

nly;

all

righ

ts r

eser

ved.