SEEPAGE FROM SHALLOW CHANNELS AND IRRIGATION FURROWS

E. N. Bereslavskii and V. V. Matveev UDC 532.546

Two cases of two-dimensional steady seepage from broad channels and irrigation furrows through a layer of homogeneous isotropic soil into a highly permeable pressurized aquifer in the presence of soil capil- larity are examined in the hydrodynamic formulation~ A uniform method of solution in which the solution of the problem of seepage from an irrigation furrow can be obtained from the solution for a broad channel is given. The special and limiting cases studied in [1--3] and elsewhere are noted. Computer calculations are used to analyze the dependence of the rate of seepage from channels and irrigation furrows and the capillary spread on the profile and width of the channel or irrigation furrow, the thickness of the layer, the head and the capillarity of the soil.

The effect of capillary rise on seepage from channels of horizontal cross section with a low water level (broad channels) was studied in [i, 2]. However, the method proposed is suitable only for problems in which the free surface is associated with impermeable boundaries~

In [3, 4] it was shown to be necessary to introduce zones of escape of Capillary water to the earth's surface, intervals of escape with evaporation and infiltration and intervals of capillary flow were distinguished, and a new analytic function was proposed for the efficient solution of the problem [4]. A theory of flow through capil- lary soils for any type of external boundary was thereby created.

In [3] the effect of soil capillarity on seepage from channels with a small near- semicircular cross section (irrigation furrows) was investigated. It should be noted that in [1--3] and then in [5] free seepage, when a water-permeable layer of soil is underlain at a large depth by a layer of soil of much greater permeability not contain- ing ground water under pressure, was considered. As noted in [3], it is difficult to apply the results obtained for broad channels when calculating irrigators. The case of seepage from a broad channel with a head, when at a certain finite depth below the channel bottom a highly permeable pressurized horizon is located, was examined in [6] and also in [7], where this horizon was transferred to infinity. However, in [6, 7] the capillary rise effect was neglected.

i. Broad Channel

Figure i is a schematic representation of the right half of the region of flow from a channel through a layer of soil of thickness T into an underlying, highly water- permeable horizon with pressure head H, reckoning from the interface with the layer in which the seepage is investigated. The channel is simulated by a horizontal segment of length s the depth of the water in the channel is assumed to be infinitely small.

We take ~=0 along the bottom of the channel AB and ~ = 0 along the line of symmetry AF, where ~=~+h~ is the complex potential divided by the permeability of the soilo On the interval BC we set ~ = Q/2, where Q is the unknown rate of seepage from the chan- nel (per unit length), also divided by the permeability. Thus, the interval BC is assumed to be an impermeable line of escape of capillary water to the earth's surface [3, 4]. Then at the free surface CE the conditions ~=--y+hk, ~=Q/2 should be satisfied, where z = x + iy is the complex coordinate of points in the seepage zone, and h k is the static height of capillary rise of the ground water. At the base of the formation FE the con- dition ~=T-H is satisfied. In addition to the flow rate Q, we are interested in the capillary spread L.

Leningrad, Kiev. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 96-102, January-February, 1989. Original article submitted December 10, 1987.

80 0015-4628/89/2401-0080512.50 �9 1989 Plenum Publishing Corporation

, L /2 L

0,5 m

. . . . .

8 C E

:F

T-H

a

8

A

F

b

- , x= , 0 a b 1 d . ' ~ '

c

B " ~ C E

o ~

e

Fig. i Fig. 2

d

We carry out the conformal mapping of the domain of the complex potential m (Fig. 2a) and the domain of dz/dm -- the inversion of the velocity hodograph (Fig. 2b) -- onto the upper half-plane of the variable ~ (Fig. 2c). From the Schwarz--Christoffel transformation we find

(1.i)

~ - ~ ~ " ' ~1--~ - - 1--~ - = ' r ( l - - a ) b ' ~ = a r c m n v ( l - - S ) a ' (1

H e r e , F ( X , k ) i s a n e l l i p t i c i n t e g r a l o f t h e f i r s t k i n d w i t h m o d u l u s k ; , \ , a , b (0 < a < b < 1 ) , d (1 <_ d < ~ ) , a n d A = d -- 1 a r e c o n f o r m a l m a p p i n g p a r a m e t e r s ( F i g . 2 c ) .

F rom t h e c o n d i t i o n o f c o r r e s p o n d e n c e o f t h e p o i n t s F a n d C i n t h e p r o c e s s o f c o n - f o r m a l m a p p i n g ( F i g s . 2a a n d 2 c ) a n d r e l a t i o n ( 1 . 1 ) we o b t a i n t h e f o l l o w i n g e x p r e s s i o n s f o r t h e f l o w r a t e Q a n d t h e m o d u l u s k i n t e r m s o f t h e p a r a m e t e r s a a n d b :

2 h ~ Q= (1.3) F ( a , k ' )

2(r-H) K' @ =~ (1.4)

The direct elimination of the parameter ~ from relations (I.I) and (1.2) does not enable us to solve the problem completely.*

We introduce the new variable ~, setting

*The same applies to the irrigator problem in Sec. 2. We note that in [8] the direct elimina- tion of the parameter ~ led to expressions (30)--(32), (35)--(37), which contain unre- movable singularities.

81

~ = s i n ~ (1 .5)

2d), The substitution (1.5) transforms the half-plane ~ into the half-strip ~ (Fig. and the differentiation of (i.I) with respect to ~ using (1.5) gives

do) iq? ( i-a) b d~ 2KY (a--sin' ~) (b--sin' ~)

(1.6)

dz Q]/ (i--aib('c+A tg ~) - - = (1.7) d~ ~KY (a-s in ' ~) (b-s in ' ~)

Express ions (1 .5) and (1 .7 ) c o n t a i n t he t h r e e unknown c o n s t a n t s a, b, and A, which de te rmine the t h i c k n e s s of t he l a y e r , the wid th of the channel s and the p r e s s u r e head H. The w r i t i n g of r e p r e s e n t a t i o n (1 .7 ) f o r v a r i o u s i n t e r v a l s of t h e boundary of t he

domain wi th subsequent i n t e g r a t i o n leads to t he pa r ame t r i c equa t i ons of the co r r e - sponding i n t e r v a l s of the model. As a r e s u l t , a f t e r t he removal of t h e s i n g u l a r i t i e s a t t he ends of c e r t a i n i n t e g r a l s , as in [9] , we o b t a i n

~ j 2

l q?(i-a)b ~ arosin(~/asint)+A]/asint(i-asin't) -u' -'2 = nK o ] / ( i - a sin ~ t)' (bia sin z t) dt ( 1 .9 )

The monotonicity of the functions On the right of Eqs. (1.8) and (1.9) and, moreover, in the equation hk/(T -- H) = F(I, k)/K', which follows from (1.3) and (1.4), was verified numerically; in this way it is possible to establish the unique solvability of the problem for the unknown constants.

After the parameters a, b, and A have been found, together with the flow rate Q from Eq~ (1.3)~ it remains to determine the capillary spread:

4q~ (i-a)b ~ t[-m t+A (i-t') (~+~)-']dt L = ~ r . Y('i~-(4a-2)~+P] [ t + ( 4 b - 2 ) t % ~ ] (1 .10)

and the coordinates of the points on the depression curve t

l 4Q]/(i-a)b [ t[-lnt+A(i+t ~) ( i - tz ) - l ]d t x = - - + L + t (1 .11)

2 ~K Jo 1/[ i + (2-4a) t%t ' ] [ i + (2-4bi t%t ' ]

= ~__~v [p(p, k')-F(~, ~') ]

~=arctg Yb/(i-b), ~=arctg [ ( t - t 2) ( t+ t ~)-Wb/(1-b) ], 0 ~ t ~ i

From the exp re s s ion f o r y (1 .12) when t = 1, which cor responds to the p o i n t E, it follows that

~-~rv.. F (~, k') T-H-h~

(1.12)

(1.13)

Then from (1.13) and (1.3) it follows that

2(T-H) F(a,k')WF(~,k') q QK

Since co t a co t ~ ffi k, u s ing the known r e s u l t F (a , k ' ) + F(8, k ' ) = K' [10] , we aga in a r r i v e a t r e l a t i o n ( 1 . 4 ) .

In F ig . 1 the con t inuous curve r e p r e s e n t s t he dep re s s ion curve c a l c u l a t e d fo r T = i, s = 0.4, hk = 0.i, H ffi 0.i.

We note certain cases associated with the limiting values of the mapping parameters. When d = i we have A ffi 0; this case corresponds to seepage without a pressure head,

82

when H = 0 [Ii].

If b = i, then the modulus k = 0 and, consequently, the lower, highly permeable layer is located at infinite depth (T = =). Then from (1.3) we arrive at the expres- sion Q = ~hk/arsh[#(l -- a)/a], i.e., a = i/cosh=(~hk/Q), which coincides with Eq. (4.3) of [12]. We can also distinguish the following two cases: free seepage and seepage with a pressure head. The first case corresponds to the value A = 0 [2, 12]; in the second case the solution contains one unknown constant A ~ 0, which can be determined if the coordinates of a point on the depression curve are given [12, 13].

2. Irrigation Furrow

We first replace the irrigator by a point source located at the point A (Fig. I). We then take one of the lines of equal head, say BG, represented by the broken line in Fig. I, as the semicircular profile of an irrigation furrow of diameter Z and on this line set ~=0. We also assume that ~IAF=0, ~I,z=T--H, and $Ic=Q/2.

The domain of the complex potential is represented in Fig. 2e; the dz/dm domain has the same form as in Sec. i. Carrying out the conformal mapping of the ~ domain (Fig. 2e) onto the upper half-plane ~ (Fig. 2c), we obtain

r_H_Q_arsh~ b-~ # (i--b)~

In this case too A = d -- 1 and

b-- th ~ Q

Transformation (1.5) leads to the solution of the problem in parametric form:

d_t~ = Q~ --=dz 2iQ~ (~+A tg ~) ( 2.1 )

dr ~ sin ~Y b--sin z �9 d~ ~z sin ~? b- -s i~

Equations (2.1) contain thetwo unknown constants b and A. Comparing the solution for a source (2.1) with the solution (1.6), (1.7) for a broad channel, we see that it is obtained from the latter when a = 0, i.e., when the point B in Fig. 1 merges with the point A. Expressions for the constant A, the capillary spread L, and the abscissa of points on the depression curve x are obtained from relations (1.8), (i.i0), and (i.ii), respectively, by setting a = 0. Since in this case k = 0 and k' = i, for the ordinates of points on the depression curve we obtain

O Y b (l-g) y = - ( T - H - h ~ ) + - : - a r s h ___- - , O ~ t ~ l

Y l - - b ( t + ~ )

The expressions for the radius of the irrigation furrow and the flow rate take the form:

- S 2 #~ o sh t~b+sh2 t ~ - b c h ~

where U is the ordinate of the point B in the �9 plane (Fig. 2d), which is also subject to determination.

In Fig. I the broken curve represents the depression curve calculated for the same values of the parameters as in Sec. i. We note certain particular and limiting cases. As in Sec. i, when d = i we have A = 0; this case corresponds to seepage without a pressure head [14]. If b = i, then T = ~ and after the evaluation of the correspond- ing indeterminate form in (2.2) we arrive at the expression Q = ~hk/in cot ~, which co- incides with expression (15) of [3].

3. Analysis of the Numerical Results

Table 1 gives the results of calculating the seepage characteristics L and Q for broad channels (upper row) and irrigation furrows (lower row) for various values of s h k, and H. The table consists of two series corresponding to the cases h k > H (top

83

TABLE 1

l L Q H L Q h k r. Q

0,3

0,5

0,7

0,9

t,0

0,3

0.5

0,7

0,9

t,0

0,t404 0,2364 0,t504 0,2716 0,155t 0,2933 0,1575 0.3077 0,1582 0,3i31

0,0456 0,0996 0,0478 0,ti6i 0,0489 0.t267 010495 0.1338 0,0496 0,t365

0,8487 1,t707 t,0659 t,5692 t,2631 1,9834 t,45t6 2.4368 t,5442 2,6832

0,5508 0,8067 0,7254 t,tt50 0,88t4 1,4306 t,0293 t,773t 1.i0t6 t,9585

0,1

0,3

0,6

0,65

0,69

0,t

0,5

0,8

0,85

0,89

0,1464 0.2563 0.t907 0,3t75 0,5857 0.7589 0.9168 1.0986 t,8494 2,0348

0.0440 0,0999 0,0566 0.i308 0.2099 0,3472 0,41t3 0.5748 tA919 1,3765

0,9612 i,3701 0.8079 1,t275 0,4910 0,6745 0.4306 0,5912 0.3816 0,5238

0,7006 t,1042 0.5279 0,757t 0.2433 0.3350 0.t842 0.2530 0,t354 0,t859

0,1

0,5

0,8

0,85

0,89

0,t

0,3

0,6

0,65

0,69

0,0440 0,0999 0,3t49 0,4640 t,0833 t,2669 t,5112 1,6969 1,9423 2,713t

0,0469 0A088 0A907 0,3i75 0,9264 t,t308 1.3520 t,5372 2,359O 2,5532

0,7006 t,t042 1,0680 IA795 t,1072 t,5202 t,1077 t,5207 t,1078 1,5208

0,64t7 0,96t8 0,8079 1,t275 0,6809 t,1821 0,86t5 t,t927 0,86t9 1,1828

TABLE 2

r ~-I 1,5 2 2.5 3 3,5 4 4,5 5 L 0,0433 0.0445 0,0455 0,0464 0.0472 0.0480 0,0487 0,0495 0,0499

0,0923 0,0980 0,1025 0,t06t 0.t092 0.i119 0,it43 0,1165 0.1t85 0,5881 0.5675 0.5520 0,5397 0.5295 0.5208 0,5t33 0.5043 0.5006

Q 0,9079 0,824~ 0,7735 0,7378 0:7t08 0,6894 0,6714 0,6565 0,6438

o f t a b l e ) and h k < H, and e a c h s e r i e s c o n s i s t s o f t h r e e s e c t i o n s , i n w h i c h one o f t h e p a r a m e t e r s s H o r h k i s v a r i e d ( t h e l a s t two i n s u c h a way t h a t H + h k < T ) , w h i l e t h e others take fixed values: T = I, s = 0.4; for the first series H = 0.I, h k = 0.3, and for the second H = 0.3~ h k ffi 0.1.

We will analyze the effect of the shape of the source profile. In the case of a furrow-type irrigator the capillary spread and the flow rate are greater than in the case of a broad channel, the difference reaching a factor of 1.5--2 for relatively small (<0.01) and large (>0.9) values of s The greatest difference (by a factor of more than 2) is observed for structured soils possessing weak capillarity (h k < 0.i). As for the differences hL and AQ between the corresponding values for a broad channel and an irrigation furrow, they increase with increase in s Thus, as s increases from 0.3 to i, the quantity AL/s is more than halved.

An increase in the channel width s by a factor of 3.3 leads to only a slight in- crease in the capillary spread L (by a factor of 1.1--1.3), the flow rate Q increasing by approximately 100%.

As H varies from 0.1 to T -- h k, the quantities L and Q vary by factors of 12.6 and 2.5, respectively, in the case of a broad channel and by factors of 8 and 2.6, respec- tively, in the case of an irrigation furrow. For the second series the changes are more considerable - 27 and 5.2 for a broad channel and 13.8 and 6 for furrow-type ir- rigators; the greatest increases in L are observed at values of H close to T -- h k. Thus, a variation in H from 0.85 to 0.89, i.e., by 4.7%, leads to a 189% increase in L. The closer the ground water level to the surface, i.e., the greater the value of H, the less the rate of seepage [I]. When H = 0 for the first series L = 0.1336 and Q = 1.0188 in the case of a broad channel and L = 0.2400 and Q = 1.4713 for an irriga- tion furrow.

When h k > H as h k varies from 0.i to 0.89 the quantity L increases by a factor of 44 for broad channels and 27.2 for irrigation furrows, which indicates that the capil- larity of the soil has an important influence on the seepage.

When h k = 0 and h k ~ T -- H the capillary spread exceeds the capillary rise hk, the difference being greatest for values of h k close to T -- H. Thus, for a broad channel when h k = 0.89 we have L = 1.942 and hence s k = 2.2. This confirms the importance of horizontal absorption noted in [3] (including for soils with weak capillarity); for irrigation furrows the effect is even more considerable (see Table I).

84

The results of calculating L and Q for a variable layer thickness T satisfying the condition T -- H -- h k = 0.8 when ~ = 0~3 and h k = 0.i are presented in Table 2.

Clearly, the presence of an underlying formation has an important influence on L and Q only when T < 3.0. At large T the differences between the corresponding values of L and Q does not exceed 1.7 and 1.9% for a broad channel and 2.9 and 3.6% for an irrigation furrow.

The authors are grateful to N. N. Verigin for valuable comments.

LITERATURE CITED

i. V. V. Vedernikov, Theory of FLow Through Porous Media and its Application to Irriga- tion and Drainage Problems [in Russian]; Gosstroiizdat, Moscow (1939).

2. B. K. Rizenkampf, "Ground water hydraulics," Uch. Zap. Sarat. Univ., 15, 3 (1940)~ 3. N. N. Verigin, "Seepage of water from an irrigation channel," Dok!. Akad. Nauk SSSR,

66, 589 (1949). 4. N. N. Verigin, "Some cases of ground water rise in the presence of general and local

intensified infiltration," Inzh. Sb., ~, 21 (1950). 5. S. N. Numerov, "A method of solving problems of flow through porous media," Izv.

Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 4, 133 (1954). 6. V. I. Aravin and S. N. Numerov, Theory of Motion of Liquids and Gases in a Nonde ~-

formable Porous Medium [in Russian], Gostekhizdat, Moscow (1953). 7. A. R. Tsitskishvili, "Seepage from a channel of trapezoidal cross section," Izv.

Akad. Nauk SSSR, Otd. Tekh. Nauk, No. 3, 125 (1957). 8. V. A. Vasil'ev, "Seepage from a shallow water channel with allowance for capillarity,"

Tr. Sredneaziat. Univ., 83, 43 (1958). 9. P. Ya. Polubarinova-Kochina, V. G. Pryazhinskaya, and V. N. Emikh, Mathematical

Methods in Irrigation Problems [in Russian], Nauka, Moscow (1969). i0. A. M. Zhuravskii, Handbook of Elliptical Functions [in Russian], Izd. Akad. Nauk

SSSR, Moscow (1941). ii. E. N. Bereslavskii and V. V. Matveev, "Ground water regime in a layer of soil in

the presence of seepage from a channel with allowance for the capillarity of the soil," in: Computational and Applied Mathematics, No. 61 [in Russian], Kiev (1987), p. 43.

12. P. Ya. Polubarinova-Kochina, Theory of Ground Water Motion [in Russian], Nauka, Moscow (1977).

13. N. B. Ii'inskii and A. Ro Kasimov, "Inverse problem of seepage from a channel in the presence of a pressure head," in: Proceedings of Seminar on Boundary-~alue Problems, No. 20 [in Russian], Izd. Kazan. Univ., Kazan' (1983), p. 104.

14. E. N. Bereslavskii, "Note on the problem of seepage from an irrigation channel," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 105 (1987).

85