Steady Infiltration from Single and Periodic Strip Sources1

VEDAT BATUZ

ABSTRACT and disc sources. Also Batu (1977) has presented bothThe matric flux potential and horizontal and vertical flux dis- theoretical and experimental results for steady infiltration

tributions were obtained using Fourier analysis techniques for single from a dltch uslni different boundary conditions at the soiland periodic strip sources located on the soil surface. The theory is surface.based on the assumption that the hydraulic conductivity is an This paper presents analytical solutions for Steady in-exponential function of the soil water potential. The matric flux filtration from single and periodic strip sources. Thepotential distributions were compared with the results obtained by analysis is based on a linearization of the flow equation inWarrick and Lomen (1976) who used different mathematical tech- terms of the matric flux potential, originally proposed bynia.ues- Gardner (1958). The solutions are of interest in connection

with sprinkler and furrow irrigation.Additional Index Words: matric flux potential, soil water, sprinkler

irrigation.Batu, Vedat. 1978. Steady infiltration from single and periodic strip r-WWWAT irr»TTATTr»wcsources. Soil Sci. Soc. Am. J. 42:544-549. Uli/rNliKAL fcy.UAllUIMS

———————————————— With the exponential relationship (Gardner, 1958)

RECENTLY, PHILIP (1968, 1969, 1971); Wooding (1968); . . _ , ,. r ,Raats (1970, 1971); and Zachmann and Thomas (1973) A w *" *wm> I1-!

have presented solutions for steady infiltration from pointline, and areal sources. Warnck (1974), Lomen and *Warrick (1974), and Warrick and Lomen (1976) have fh .presented, time-dependent solutions for point, line, strip, © = I K(h) dh = —— [2]

J— oo &

'Contribution from the Dept. of Civil Eng., Karadeniz (Blacksea) Tech.Univ., Trabzon, Turkey; and the Dept. of Soil Science, Univ. of Wis., where K(h) (units LIT) is the hydraulic conductivity of theMadison. Received 15 Sept. 1977. Approved 16 Feb. 1978. » , A - \ v t - i r AT\ .u u j i- j .- •."Assistant Professor of Civil Eng, Dept. of Civil Engineering, unsaturated soil; K, (units LIT) the hydraulic conductivityKaradeniz (Blacksea) Tech. Univ., Trabzon, Turkey. of the saturated soil; h(L) is the soil water potential; and a

BATU: STEADY INFILTRATION FROM A SINGLE AND PERIODIC STRIP SOURCE 545

(L-1) is an empirical constant. The units of @ areL2/T. Thelinearized form of the steady infiltration equation is

[3]

with the z axis positively downward. The horizontal andvertical components of the flux, as functions of matric fluxpotential, are:

u = — d®~dx

„ 30v = a® - ^— .

[4]

[5]

THE SOLUTION FOR A SINGLE STRIP SOURCELet us consider the solution for a single strip source of

width 1L, lying on the soil surface centered over they axisin the x-y plane (see Fig. 1). The boundary conditions areexactly those used by Warrick and Lomen (1976). Theboundary condition for the soil surface is that of no flowoutside the strip

a (ita&-~ = 0;z = 0, \x\>L.

The boundary condition over the source

[6]

v = v(jc,0), -L<x<L . [7]

Also it is assumed that @, d®/dx and d ®/dz vanish as x2 +

From the symmetry in Eq. [7] it follows that 9 is even inx which motivates one to take the Fourier cosine transformof Eq. [3] (Churchill, 1941).Let

TOO

= ®(*,z) cos Xxdx [8]Jo

[9]

and Eq. [3] imply

The general solution of Eq. [10]:

[10]

where C, and C2 are constants. Using the boundaryconditions at infinity, 9, and dQ/dz = 0, the constant C\must vanish. From Eq. [6] and [7], by letting /, theargument of x corresponding to z=0,

-®x= {L

Jo[12]

and substituting Eq. [11] in this equation

2L -HV W W v V

soil surface

Fig. 1 — Single strip source geometry.1/2-1-1 CL„ / n2\ 1/2-1

2" + \X2 + TJ J

By putting this in Eq. [11]

f r n / n2 \ 1/2 1 1 fL

expj |- U2 + -^H J z j J v( / ,0)cosW<«. [14]

Taking the inverse Fourier cosine transform we have2 f°° f a / a2\ i'2]"1

0 = — TT + X + -r-1TrJo [2 V 4 / J

v(/,0)cos X/rf / cos \xd\. [15]

This is the general solution for any form of v(/,0). Forthe special case of constant flux, by letting v(l,0)=v0 andXL=u and defining the dimensionless variables

7T0

Eq. [15] will have the following form:

f °°*= I /(«) cos(tu)duJo

[16]

[17]

where112

{X0 + (u2 + X/ [18]

and ; =

By using Filon's numerical integration method (Tranter,1951) Eq. [17] can be written as

sin

546

where

SOIL sci. soc. AM. ]., Vol. 42, 1978

- [2 T; sin

<">

From Eq. [4], [5], and [16] the dimensionless fluxesmay be expressed as

v0aL dX

-vnaL

[23]

[24]

By using Eq. [17] and again applying Filon's method, thefollowing relationships can be obtained:

? F»-1 sin [(2w ~ I}^T] sin[(2n ~2r, cos [25]

cos 2y

COS

sin )[2r, sin . [26]

In these equations 17 (|), j8(|) and y(^) are functions off andthe values are tabulated (Abramowitz and Stegun, 1964;Tranter, 1951).

The solution for a single line source, by letting X0 — > 0 inEq. [19] is

- cos ff +

F2ncos(2n|)+ F2f,_lCos[(2n - l)f]. [27]

THE SOLUTION FOR PERIODIC STRIP SOURCESIn this case it is assumed that there are an infinite number

of equally-spaced parallel strip sources at the soil surface.The spacing is 2D (see Fig. 2). It is assumed that the watertable is located at infinity.

From the geometry of the problem there are no flowperpendicular to the vertical planes having abscissae of x =0, ± (L + D), . . . The shaded portion of Fig. 2 is theregion to obtain a solution. The boundary conditions are:

v = v2(jc,0)

r'lf-l-^ = 0dx

[28]

[29]

[30]L

[31]- -dx

By applying the method of separation of variables to thetwo-dimensional form of Eq. [3], the general solution maybe obtained as

nz) + Bn exp (k2nz) sin knx

Dn exp (fcznz)] cos Knx} [33]

where

1/2

[34]

with Xn 's constants.According to Eq. [30] if dO/dx = 0, An and B,, must

vanish. From Eq. [31]

{An exp-

[Cn exp (fc lnZ

..= - X +

Fig. 2—Periodic strip source geometry.

sin A (L + D) = 0, \n =

BATU: STEADY INFILTRATION FROM A SINGLE AND PERIODIC STRIP SOURCE

nir , „ , „ . , \ rte-i where

547

(« = 0,1,2,3, . . . ) [35]

To satisfy Eq. [32], Cn must vanish and from Eq. [33],[34], and [35]

By using Eq. [5] and [36] the value of v when z = 0 is

V(JC,0)= ^CnCOS-3^ [37]n = 0

where

Therefore, consideration of the right hand side of Eq. [37]as a Fourier cosine series for the interval x = 0 to L + Dyields (Churchill, 1941)

'""IT"

That is,

= 2^-^sin L + D '

By substituting Eq. [40] and [41] into Eq. [36]

[40]

[41]

a(L TT

•2

1 .sin cos ,..„-,[42]

where

_ a _ I"/ mt y a* V*n ~ 2 [\L + D) 4 J '

For D»L and letting qreduces to

/SWv ,\ - 1 j. 4°

= 2v{L and v2 = 0 Eq. [42]

2a£> -n-2

[44]

Eq. [44] is, except for a change of variables, the sameequation obtained by Raats (1970, Eq. [42]) for an array ofequally-spaced line sources at the soil surface, q is thesource strength corresponding per unit length. For v2 = 0,Eq. [42] reduces to the solution of equally-spaced parallelstrip sources. Using the dimensionless variables denned byEq. [16] andX) = aD

2Eq. [42] takes the form

7T 2 Y ^•", n

sin [46]

where

[47]

The dimensionless horizontal and vertical flux distribu-tions defined by Eqs. [23] and [24] are

U = 2n r.0,[48]

By letting X0 -* 0 in Eq. [48] and [49] the flux distributionsfor equally-spaced parallel line sources, first derived byRaats (1970), can be obtained.

NUMERICAL RESULTS AND DISCUSSIONThe results for the matric flux potential distributions are

compared with the results obtained by Warrick and Lomen(1976). The authors, firstly obtained the time-dependentsolution for a single strip source based on Philip's theorem(1971) and expressed in an integral form including errorfunctions. The authors then obtained results for infinite setsof equally-spaced parallel strip sources by applying su-perposition to the solution for single strip source. Alsousing the same numerical parameters, the results forhorizontal and vertical flux distributions are presented.

Numerical analysis carried out on a computer for singlestrip source to determine the convenient values of N, g, -q,j8, and y for sufficient accuracy. The value of N was foundaround 100 and the coefficients were selected as£ = 0.05,

548 SOIL sci. soc. AM. J., Vol. 42, 1978

MODEL

WARRICK AND LOMEN'S MODEL

.3 .4

.1

.2

= .O(LINE SOURCE)

PRESENTED MODEL

WARRICK AND LOMEN'S MODEL

Fig. 3—Results for single strip sources for X0 =compared with the Warrick and Lomen model.

and 0.25 Fig. 4—Results for a single line source (X0 = 0.0) compared with theLomen and Warrick model.

ri = 0.00000555, /3 = 0.66699976, y = 1.33300003(Abramowitz and Stegun, 1964) for <J>, U, and V cal-culations. For the solutions of equally-spaced strip source,around 25 terms was found enough for sufficient ac-curacy.

In Fig. 3 the matric flux potential distribution for singlestrip sources (Eq. [19]) with X0 = 0.1 and 0.25 arecompared with the solution obtained by Warrick andLomen (1976, Eq. [14]). Dimensional values for strip

widths can be found by using the limiting values of a givenby Philip (1968). Fora = 0.002 cm"' the semi-widths ofX0 = 0.1 and 0.25 would be 100 and 250 cm, respectively,and for a = 0.05 cm"1 the semi-widths would be 4 and 10cm, respectively. As can be seen from Fig. 3, the resultsare approximately the same. In Fig. 4 the results for asingle line source computed from Eq. 27 are compared withLomen and Warrick's line source model (1974, Eq. [18]).Again, the agreement is reasonably good.

.2

.2

.3

.4

.3 .4

______PRESENTED MODEL

____ _ WARRICK AND LOMEN'S MODEL

I I I IFig. 5—Results for an infinite array of parallel strip sources for X0 =

0.1 and Xi = 0.4, and Xa = 0.25 and Xi = 0.25 compared with theWarrick and Lomen model.

Fig. 6—Distributions of 17 at X = 0.05 and 0.3, and of V at Zand 0.3 for a single strip source by taking X0 =0.1

:0.05

BATU: STEADY INFILTRATION FROM A SINGLE AND PERIODIC STRIP SOURCE 549

)( Z = 0.05. Also the horizontal component of the flux•——. yjj*.————!.?————; l~-———s4.—————- decreases with depth.

| /^U = 9.504 3^3.800 In Fig- 7 the flux distributions for equally-spaced strip• 1111 =B= '• i 1 1 1 1 1 1 1 1 1 1 1 1 1 iMrTT 'E 5J' i sources are presented. For the small values of X and Z the

EH' HJJJJJJ-Li-u-^ 3 / profiles are approximately the same as compared witht1 . F Tfl \y^ H L single strip source profiles. The other properties of these

if ' 3 / profiles are similar.

:fl/ °~'1 |j / ACKNOWLEDGMENT,2 \ • • 3 v H / — I would like to thank Dr. W. R. Gardner for helpful discussions of this

.̂ J X, = .4 J / paper. I also wish to thank Dr. P.A.C. Raats for his suggestions.

.4-g <D -3> ?>CO .»

" ^^__.307 j .637

Fig. 7—Distributions of [7 at AT = 0.05 and 0.3, and of V at Z = 0.05and 0.3 for an infinite array of parallel strip sources by taking X0 =0.1 and Xj = 0.4.

Figure 5 is for an infinite number of equally-spacedparallel strip sources. The semi-spacing is 21 a. If a =0.002 cm"1, the semi-spacing is 500 cm. Because of thegeometry,X = 0, ±0.5, ±1.5, ±2.5, . . . are symmetricalvertical planes. Again the results agree reasonably withthose based on the solution by Warrick and Lomen (1976).Since the differential equation and boundary conditions for9 are linear, Warrick and Lomen could superpose solutionsfor single strip sources.

The flux profiles are presented in Fig. 6 and 7. Fig. 6 isfor single strip source taking X0 = 0.1. Near the source thevertical component of the flux is large at small depths, butdecreases rapidly with depth. For example, the flux atX =0,Z = 0.05 is approximately 137 times the flux siX = 0.5,

ERRATASteady Infiltration from Single and Periodic StripSources

VEDAT BATUSOIL Sci. Soc. AM. J. 42(4):544-549. 1978 (July-Aug. issue).

An error occured in publication of the above article:

p. 545- Column 2, Eq. [19], F2n-\lX0 should be