Steady Infiltration from a Ditch: Theory and Experiment1

VEDAT BATU2

ABSTRACT

The two-dimensional, steady-state water flow equation, which isbased on the assumption that the hydraulic conductivity is an exponen-tial function of the soil water pressure, is solved for a ditch on the soilsurface. The results include expressions describing the distributions ofmatric flux potential and water content. Measured water contentswere compared with this mathematical model and also with the modelsgiven by Lonien and Warrick (1974) for a line source, and Warrickand Lomen (1976) for a strip source.

Additional Index Words: ditches, hydraulic conductivity, plane flow,steady two-dimensional flow, soil water potential, unsaturated soils,water content distribution.

THE FIRST DETAILED example of .an analytical solution ofthe steady infiltration equation for two dimensions was

given by Philip (5) for buried point sources and sphericalcavities. Also, the solution for a single, horizontal linesource in an infinite region was briefly discussed by thesame author (6) using a continuous source and assumingthat the matric flux potential approaches zero at infinity.Wooding (12) used the same form as Philip and obtained anapproximate result for steady infiltration from a circularpond. Raats (9) presented an analytical result for steady in-filtration from an array of equally-spaced line sources orfurrows at the surface of a semiinfinite soil profile. Waterflow from line, strip, and disc sources were analyzed byWarrick and Lomen (4,11) using a linearized, time-depen-dence model of the water flow equation.

THEORY AND PROCEDUREThe equation for steady flow in homogeneous, unsaturated

porous media is:

where the matric flux potential 0 (units L2T~') is defined by thefollowing expression (6, 7):

6=re /-*

D(6)dO =J Sl, J *!)

[2]

where 6 is the volumetric water content, D (units L'ZT~1) is the dif-fusivity, K (units LT~l) is the hydraulic conductivity, and $ (unitsL) is the soil water potential; i//0 and 0,, are reference values. Theparameter a (units L~') in Eq. [1] is defined by the equation (5,6)

dK(0)K(6)

= a, i .e . , K(<]J) x exp (ai/ [3]

Let x-z be a coordinate system whose origin is at the center ofthe ditch with the z axis pointing downward in the direction of

'Adapted from the Ph.D. dissertation presented to the Technical Univer-sity of Istanbul (Turkey) in March 1974 and supported as research projectNo. MAG-299 by the Scientific and Technical Council of Turkey. Re-ceived 16 Aug. 1977. Approved 9 March 1977.

2Assistant Professor, Dep. of Civil Engineering, Karadeniz (Blacksea)Technical Un iv . , Trabzon, Turkey. Currently on leave at the Dep. of SoilScience, Univ. of Wisconsin, Madison, WI 53706.

gravity. The width of the ditch is 2L. The coordinates of the physi-cal problem are shown in Fig. 1.

It is assumed that there is a constant water content over a width2L at the soil surface. Equations [2] and [3] imply that the matricflux potential and the hydraulic conductivity are related by:

= K(0)/a [4]

If the matric flux potentials 0ri and 0S correspond to the water con-tents at the soil surface over a width 2L and at saturation, respec-tively, then

[5]

where K,, and Ks are the corresponding hydraulic conductivities,respectively.

The boundary condition at the ground surface, with the excep-tion 2L, is assumed to take the following form:

Q(x, Q) = 0;x>L, x< -L [6]

According to Eq. [4] this means that the hydraulic conductivity iszero in these domains. On the other hand, mathematically, this hasthe meaning that the soil water potential goes to minus infinity.

Because of the effects of capillarity and gravity, the water con-tents will decrease downward from the ditch. This situation is trueuntil the flow domain reaches zero water content which exists atinfinity. So the boundary conditions at infinity will have the fol-lowing form:

lim

lim

= 0

Also

Mm B(x,z) =

[7]

[8]

[9]

because the ditch is the only source of water.As the flow is considered to be two dimensional, it is sufficient

to investigate the problem in the x-z coordinate system. Taking theFourier transform of the two-dimensional version of Eq. [1]

Fig. 1—Geometry of the problem.

677

678 SOIL SCI. SOC. AM. J . , VOL. 41, 1977

Let the Fourier transform of the solution Q(x,z) with respect tox be (2,3)

,z)] = J B(x,z)e\p(-i\x)dx [11]

, z ) ] = - - t/(X,z)exp(-iAjt)<fc. [12]

then

The Fourier transform of the first term on the left of Eq. [10], afterpartial integration and using Eq. [7], becomes:

[13]

The transform of the second term on the left of Eq. [10] is:

d2U(x,z) [14]

And, the transform of the last term is:

&\a =a- [15]

Now by substituting Eq. [13], [14], and [15] into Eq. [10], the fol-lowing ordinary differential equation can be obtained:

'•"M-af^-X'tfM-O. [16]dz2 dz

This equation is of the form of Eq . [31] given in reference 10. Thegeneral solution of Eq. [16] is

[17]

where A(\) and B(X) are constants. When A(\) = 0, Eq. [17] willsatisfy both Eq. [8] and Eq. [9]. By letting z = 0 in Eq. [17], andt/(X,0) = 5(X) then

a2\1/2l ]T) JZJ

[18]

The Fourier transform of 9 (x, 0) is

Z7 (X , 0) = 3? [6 (*,())=[ &(x,0)e-'**dx [19]

by letting, u, the argument of A:, corresponding to z = 0 and put-ting Eq. [11] into Eq. [18], and by taking the inverse Fourier trans-form of Eq. [18]; that is

,z)=^-l f e(tt,2.TT J —oo J —oo

[20]

Using Euler's formula and Eq. [5] and [6], Eq. [20] can be writtenin the following form:

2 sin XL cos

2 s i n X L s i n X x

[21]

Since the function in the second integral is an odd function with re-spect to \, the imaginary part of Eq. [21] is zero (7). Furthermore,since the integrand of the first term is an even function, and by in-troducing the dimensionless variables

/~\ tr (ff\

- - [22]

Eq. [21] can be rearranged in the form:

where X/aL = t.To evalute this integral, Filon's numerical method is used (10).

The Filon's integral formula for the cosine term is

f /(«) cos tu du = h [TJ (ft/(b) sinJ «

2W_ ,][24]

In our case a = 0 and i> = 2/V/i. These show that the range dividedinto 2N equal parts with an interval h. Using this method:

0 =/2JV - [27, sin(2tfft - 0 cos7T

where

[27]

[28]

The dimensionless vertical flux component is

[29]

V =/2JV - [27, sin - J8 cosn 1-J

ll1 '2]

D0]

As can be seen from Eq. [25], the matric flux potential is aunique function of aL. The parameter of L is a characteristicdimension of the ditches . In Eq . [25] through [28] f, = ht and 17 , B ,

BATU: STEADY INFILTRATION FROM A DITCH: THEORY AND EXPERIMENT 679

.25 .50 .75 Xf- .——n- .....1 a ^^^^ i -••

.25

.50

.75

Z

-dL = 0.5- =1

3 X

\i\\\iimlCMI

—— dL=0.5—— =1

Fig. 2—The distribution of <£ foraL = 0.5 and 1.

and y are the functions of f. These values are tabulated (11 , p. 71)and the numerical analysis showed that by taking £ = 0.05, TJ =0.00001, /3 = 0.66700, and y = 1.33300, the results are suf-ficiently accurate.

Calculations for aL = 0.5, 1, 2.5, and 5 are presented in Fig.2A, 2B, and 3. Figure 2A gives the values near the ditches. Ac-cording to the boundary condition, given by Eq. [6], </> has zerovalues at the ground surface, except if -aL < X < aL. The valueof (j> equals 1 over a width 2L at the ground surface which is con-venient to the boundary condition. Therefore the equal $ lines arediscontinuous near the ground surface. The values of </> in the dis-continuous areas are smaller than shown on the equal </> lines if|X| > aL and larger if |X| < aL.

The vertical flux distribution at the ground surface is presentedin Fig. 4. As can be seen from this figure the theory gives anupward flux at the ground surface because of the boundary condi-tion. The numerical analysis showed that the total upward flux isapproximately 10% or less of the discharge from the ditch. AlsoFig. 4 shows that the infiltration rate over a width 2L is approxi-mately equal to Ks.

The solution for an inf ini ty array of ditches spaced D units apartis (4):

© array Q(X~aD,Z}. [31]

LABORATORY MODELWater contents were obtained using the method based on the

dependence of electrical resistivity in porous media upon the watercontent (1). The hydraulic conductivities and soil water potentialswere obtained from vertical cylindrical samples by measuring theflow rate and potential gradients. In the experiments, two kinds of

10

10 15ir OL=2.5= 5

&\X\

Fig. 3—The distribution of <f> foraL = 2.5 and 5.

-1.0r\

aL=1

.5

1.0

.z=o

-1.0

aL=5

0 2.5

1.0

.2=0

Fig. 4—The vertical flux distribution at the ground surface for aL = 1and 5.

soils were used. One of these was coarse sand, Turkish standardsand (TS.33), with grain sizes between 0.5 and 1.8 mm. The sec-ond material clayey sand, has grain sizes between 0.1 and 0.5 mmand also contains a mixture of clay and silt. The K(6) and i|/(0)relationships were obtained by using the method mentioned in (1).the conductivity-capillary potential relationships are shown in Fig.5. The conductivity-water content relationships of coarse sand andclayey sand are, respectively,

K(0) = 3.5057 02-912" cm/sec, 0.15 =s 0 =s 0.30 [32]

K(0) = (188.0530 - 39.271) X 10~3 cm/sec, 0.22 =S 6 0.38.[33]

The experimental model (see Fig. 6) was 230-cm long, 120-cmhigh, and 14.60-cm wide, made of plexiglass with 45 pairs of in-ternal probes. The bottom plate of the system was made of 5-mmthick plexiglass and included as many holes as possible to reduceits resistivity against the flow. The distances between the probes inthe vertical and horizontal directions were 15 cm and 20 cm, re-spectively. The brass probes were attached to the model with auniform spacing of 10 mm between adjacent probes. To increasethe accuracy of the measuring values, the distance between theends of the probes should be as small as possible. It must bepointed out that the minimum value of this distance is limited bythe structure and grain sizes of the soils. The 5-mm-diam probeswere insulated except for the ends. The experiments were carried

680 SOIL SCI. SOC. AM. J . , VOL. 41, 1977

I^COARSE SAND_<

•K = 0.30cm/secs _.

'a =0.8698 cm»

CLAYEYSAND_O^' K = 0.0331cm/sec

S

, a =3.9577cm~1

310

oHI2 <n

10 £o

O3ClZOo

DCo

10 >

out with alternating current having 50 Hz of frequency and usingthe calibration curve that corresponds to 10 V alternating current(1). The calibration curves were obtained on the same porousmedia. The system was filled with the dry soil up to x axis (seeFig. 5). Then, the system was saturated very slowly by supplyingwater at the bottom end. After the saturation was completed thesoil was drained very slowly to field capacity. After achieving ap-proximately uniformly wetted and settled soil, the experimentswere carried out. During all experiments a constant small depth ofwater was provided in the ditches to produce saturated conditionsover a width 2L.

COMPARISON OF EXPERIMENTAL DATA WITHCOMPUTED RESULTS AND DISCUSSION

The experimental data are compared with calculationsbased on the presented model and also with the line sourcemodel of Lomen and Warrick (1974, Eq. [20]) and the stripsource model of Warrick and Lomen (1976, Eq. [14]). Thewater contents were calculated from Eq. [32] and [33]. Thevalues of a were obtained from Fig. 5 and the values of Kswere measured using saturated soil columns.

When the water level in a ditch approaches zero, the flowrate q from the ditch is close to

q = 2LKS [34]

-15 -10 -5

CAPILLARY POTENTIALCcm H O )

Fig. 5—Hydraulic conductivity and soil water potential relationshipsfor soils.

which is given by Polubarinova-Kochina (9, p. 162). Thesource strengths computed from Eq. [34] are 3.000 cm3

cm"1 sec"1 and 0.331 cm3 cm"1 sec""1 for the coarse sandand the clayey sand, respectively.

The calculated and experimental water contents are givenin Fig. 7 and 8. Each data point is the mean between 15 and60 measured values. The standard deviations, which arealso shown in the figures, have values between 0.016 and

inA

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in4

in

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i

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Q-> =In • k 2° * 2o * 3s >M 1 ^ T* 1* ^1

. *£l VTSX:gpJ^^SKV^vWS^•>'.::;:'•' :•' |"•"' • • • i

«'.• • * J

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230 j cm \

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r-i^cz4= c=

*'';•*:•"'*•''&£<*£

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Fig. 6—The scheme of the laboratory model.

BATU: STEADY INFILTRATION FROM A DITCH: THEORY AND EXPERIMENT 681

•50rn————\————|————|————|————I .50|—|——r—T————|————|————J————I\ \* 2L = 10 cm \* COARSE SAND \

V ^ * q - 3.0 cm'/sec \^

V^^ \ •—xa.0cm ) -~ V_ • ^V"* I EXPERIMENTAL _ " ~^^ _

1 *>> °- *20>"Cm' ~ 4= ————---ll^L^,*l.°cm^

S r***8«8£«.-, _ . S I I T T T T • — ~*i 1 T5*?** -. T ¥ °____Jb Ol ATx=5'0<:m_5rS •" ^^^s^sss, u ———^----fc*—it"-—gf—4-1 4-> <,-''" -1-

C 25 - - C25~ '' 2L=10cmQ T O f CLAYEY SAND( \ ~ T O ( J s ' q - 0.331 cm!/sec

T" O I 7Jh T O 1 ..,_, —-•?* flN • -x-.Ocm 14i O •*• »-I°j£sS*»1*if"^-^~ S ^EXPERIMENTALCO T J. ~z£>^i^~''""" CO O— -5.0cm)

* "2 ^ - * ', X "——— PRESENTED MODEL ——— PRESENTED MODEL

*-y* ___LOMEN AND WARRICK'S LINE SOURCE MODELj-^^ --- LOMEN AND WARRICK'S LINE SOURCE MODEL

S'/f' -.-.- WARRICK AND LOMEN'S STRIP SOURCE MODEL

JQ____I____i____i____i____ _j____I i____i i_____5 20 35 50 65 80 5 20 35 50 65 80z(cm) zccm)

Fig. 7—Calculated and measured water contents (or coarse sand. Fig- 8—Calculated and measured water contents for clayey sand.

0.032 for the coarse sand; and 0.010 and 0.039 for the tance of these boundaries to the flow. The results show thatclayey sand. the flow is not affected by the presence of the side bounda-

Figure 7 shows that the water contents predicted by Eq. ries. The minimum distance of the probes from the bottom[25] and [32] for the coarse sand are 100 to 109% of the plate was 45 cm. The results showed that the effect of themeasured values at x = 0.0 cm and 23 to 85% of the bottom plate was negligible in the upper 65 cm. This ismeasured values at x = 20.0 cm. Figure 7 also shows that quite good agreement inasmuch as the lower boundary con-Warrick and Lomen's strip source model gives approxi- dition in the experiment is not identical with that in the theo-mately the same results as compared with the presented retical analysis.model. This model gives no results when z = Q and —L < x The other reasons for the quantitative discrepancies may< L, The water contents predicted by the authors' line be the result of one or several of the following: (i) poroussource model are 100 to 120% of the measured values at x media are not perfectly homogeneous and isotropic as as-= 0.0 cm and 18 to 80% of the measured values at x = 20.0 sumed in the theory; (ii) unsaturated flows have a nonlinearcm. character in reality, whereas the theoretical results were ob-

Figure 8 shows that the water contents predicted by Eq. tained from the linearized steady infiltration equation; (iii)[25] and [33] for the clayey sand are 100 to 120% of the inaccurate definition of the soil surface boundary condi-measured values at x = 0.0 cm and 100 to 144% of the tions; (iv) hysteresis during the infiltration process; andmeasured values at x = 5.0 cm. The water contents pre- (v) the effect of possible measurement errors cannot bedieted by Lomen and Warrick's line source model are 115 excluded,to 170% of the measured values at x = 0.0 cm and 100 to \r^v*jf\\\7i irrkr'iv/rc'ivrr, ,-„, ,. , . . _ „ .„. n, Av*lv[NUWl-<liiUvjlVlll,i>l 1145% of the measured values at x = 5.0 cm (Fig. 8).

There are several reasonable explanations for the dif- I am grateful to Dr K Cecen for suggesting the problem and for helpful,. , j j i i j • discussions. I also wish to thank Dr. W. R. Gardner for helpful discussionsterences between measured and calculated water contents in on tne manuscript and Dr. A. W. Warrick who provided the calculationsthe flow field. The theory assumes that the ditch is in a me- for 'he line and strip source models.dium which extends to infinity in all directions. However,in the laboratory model, the ditch is in a finite mediumbounded on both sides and at the bottom. It may be arguedthat flow at a point in a finite medium will be different fromthe flow at similar points in an infinite medium even whenthe chosen point is away from the boundaries. Here, the ef-fect of the presence of the side boundaries is investigated.For this purpose, the position of the ditch is changed (seeFig. 6) and the measuring values were collected by the pairsof probes buried in the medium. The ditch was never closerto the side boundaries than 55 cm (distance between thelongitudinal axis of the ditch and the nearest side boun-dary) . The probes are placed well away from the boundariesto eliminate the possibility of readings affected by the resis-

682 SOIL sci. soc. AM. J., VOL. 41, 1977