Water-table heights in drained anisotropic homogeneous soils

Agricultural Water Management, 11 (1986) 1--11 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

WATER-TABLE HOMOGENEOUS

HEIGHTS SOILS

IN DRAINED

1

ANISOTROPIC

E.G. Y O U N G S

Soils Division,Rothamsted Experimental Station, Harpenden, Herts. A L 5 2JQ (Great Britain) (Accepted 30 September 1985)

ABSTRACT

Youngs, E.G., 1986. Water-table heights in drained anisotropic homogeneous soils. Agric. Water Manage., 11 : 1--11. Water-table heights in drained homogeneous anlsotropic soils in which the vertical hydraulic conductivity K z differs from the horizontal hydraulic conductivity Kx, are presented in terms of the parameters Hm/N/'AD, q/AK x and d/N/'AD, where H m is the m a x i m u m water-table height midway between parallel drain channels that are spaced 2 D apart at a height d above an impermeable barrier, and A is the anlsotropy factor equal to Kz/K x. These parameters replace Hm/D, q/K and d/D, respectively, in drainage equations derived for isotropic soils.Analysis of the drainage problem in the two extreme situations with drains in infinitely deep soil and with drains laid on an impermeable barrier gives the range of the effect that anisotropy can have on water-table heights. In an infinitely deep soil the hodograph analysis shows that H m / D increases as A decreases for a given value of q/Kx, and therefore drains need to be installed closer together when A ~ 1 in order to obtain the same water-table control as in an isotropic soil with hydraulic conductivity equal to Kx, but further apart when A ~ 1. For drains laid on an impermeable barrier, seepage analysis gives bounds between which H m / D lies for a given value of q/K x. The lower bound is independent of A ; however, the upper bound, although little different from the lower bound when A ~ 1, increases as A decreases. Thus, for a given water-table control the spacing of drains laid on an impermeable barrier is found to be little affected by anisotropy when A ~ 1. W h e n A <~ 1, the increase in the upper bound for the water-table height as A decreases, whilst giving a lack of precision in determining the drain spacing, indicates some probable decrease in spacing.

INTRODUCTION The hydraulic conduct i vi t y of m a n y soils is different in t he vertical and horizontal directions. This a n i s o t r o p y influences the equipotential and streamline patterns in the gr oundw at er zone and is a fact or in determining the position o f the water table f o r given b o u n d a r y conditions. It thus affects the relationship be t w een water-table height and drain discharge in drained lands and should t her e f or e be taken into consideration in t he design o f drainage installations.

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The analysis of groundwater flows in anisotropic soils is given in textbooks (see, for example, Muskat, 1937, pp. 225--227) where it is shown that the use of transformed coordinates allows the flow to be considered in an equivalent isotropic region. Maasland (1957)discusses the theory of groundwater movement in anisotropic softs as it relates to land drainage and gives examples of the influence of anisotropy on the drain spacing required to maintain a given water-table height for parallel drain-pipe installations. Although water-table heights and drain flows in drained anisotropic soils can be obtained by means of transformations from drainage equations derived for isotropic soils, little attention is paid generally to anisotropy in discussions of land drainage. In this paper, after giving in parametric form, involving the degree of anisotropy, a comprehensive graphical presentation of the relationships between the water-table height and steady uniform rainfall rate in anisotropic soils drained by installations of parallel drain channels at different heights above an impermeable barrier, we discuss in detail the influence that anisotropy has on land drainage in the extreme situations when the drains are laid in infinitely deep soil and when they axe laid on an impermeable barrier. SOLUTIONS OF GROUNDWATER

F L O W P R O B L E M S IN ANISOTROPIC SOILS

The flow of water in homogeneous anisotropic soils is described by Darcy's law in tensor form: v = --K. grad qb

(1)

where v is the flow velocity, K the hydraulic conductivity tensor and cp the hydraulic potential. If, as generally is assumed to be the case, the principal axes are in the vertical and horizontal directions, then the potential distribution in the flow region under consideration is described by: a2cp 32~ ~2cp Kx 3x 2

+Ky

~

3y 2

+Kz

3z 2

-

0

(2)

subject to the imposed boundary conditions. In equation (2), Kx, Ky and K, are the components of hydraulic conductivity in the horizontal x- and y-directions, and in the vertical z-direction, respectively. Because of anisotropy, the streamlines in the flow region are not in general orthogonal to the equipotentials. With: x ' = v/-K-o/Kxx,

y' = x / g o / g ~ y ,

z' = x/-K-o/Kzz

(3)

where K0 is an arbitrary constant, (2) is transformed into Laplace's equation: ~ + ~ + 3x'2 3y'2

.... 0 3z '2

(4)

which describes the potential distribution in the transformed space, in which the equivalent isotropic hydraulic conductivity is x/Kx Ky K.. It is noted that the arbitrary constant K0 that appears in the transformed coordinates does not occur in the equivalent hydraulic conductivity (see Maasland, 1957, pp. 225--226, for a discussion on this point). Flow problems in anisotropic soils can thus be solved as problems in isotropic softs by using the transformations

of (3). LAND DRAINAGE IN ANISOTROPIC SOILS

In his account of anisotropy and land drainage, Maasland (1957) discusses the influence that anisotropy has on drain spacing, showing in his examples that for the same value of horizontal hydraulic conductivity (Kx = Ky ) the spacing needed to maintain the water table at a given height with a given rainfall rate is less when Kx > Kz, the case he considers most usual in the field, than when Kx = Kz. When Kx < K~, a situation that can be expected when there is vertical structuring in the soil caused by roots for example, the spacing is greater. Maasland's example is for an infinite depth of soil below the drains. Since water-table heights in drained lands depend on the depth of an impermeable barrier below the drains, a discussion on anisotropy and land drainage is incomplete without the consideration of the influence of anisotropy on the dependence of water-table heights on the depth of impermeable barrier. This influence was demonstrated by Edwards (1956) in some electric analogue experiments which also illustrated the nonorthogonality of the streamlines and equipotentials. The two
q/~

111111111/1111/11

Kz = 5Kx

(a)

(b)

~q

I I111

Kz =Kx/5

(c)

Fig. 1. (a) Groundwater region in anisotropic soils drained by parallel cylindrical drains. (b) Transformed flow region for drains in equivalent isotropic soil for A ~> 1 (A = 5). (c) Transformed flow region for drains in equivalent isotropic soil for A ~ 1 (A = 0.2).

such as that shown in Fig. l b when Kx > Kz and in Fig. lc when Kx < Kz through the use of (3). With the arbitrary constant K 0 made equal to Kz : x'

z' = z

(5)

where A = K,/Kx, the anisotropy factor, and the potential ¢ is given by the two
a2¢ ~x'2

a:¢ + -

~z '2

= 0

(6)

The hydraulic conductivity in the equivalent isotropic flow region is

%/KxKz = v ~ K x with a steady state rainfall rate q/.v~ . In Figs. l b and lc the vertical dimensions remain the same as in Fig. l a through the transformations of (5), whilst the horizontal dimensions are changed by a factor Vr)[. Thus a drain of circular cross section in the anisotropic flow region becomes elliptical in the transformed isotropic space. So long as the drain channel in the transformed isotropic space is greater than the o p t i m u m size found in the hodograph analysis of water-table heights in drained lands it may be assumed that this deviation from a circular cross-section has insignificant effect on the water-table height. Thus drainage equations derived for isotropic soils may be used to obtain water-table heights in drained anisotropic soils by applying these equations in the transformed space of the equivalent isotropic soils. The transformed dimensions, denoted by primed symbols, in the equivalent isotropic region are related to those in the anisotropic space by: H~ = Hm d' = d

(7)

D' --- x / ~ D where Hm is the m a x i m u m water-table height midway between drains spaced 2D apart above an impermeable barrier at depth d below, with the hydraulic conductivity K' =%/K~,K, and the steady rainfall rate q'= q/x/A in the equivalent isotropic space. Thus, in the transformed isotropic space, the dimensionless parameters used in the drainage equations are:

H~ /19' = H m ~/AD d'/D' = d~rAD q'/K' = q/K, = q/AKx

(8)

Thus the drainage equations discussed for land drainage installations in isotropic soils (see, for example, Lovell and Youngs, 1984) give for anisotropic soils relationships between the parameters Hm/.v/"AD and q/AK,, for given values of dA/AD. In Fig. 2 are shown the relationships calculated using Hooghoudt's (1940) equivalent depth drainage equation, an equation that Lovell and Youngs (1984) considered could be used with reasonable confidence for did < ~ 0.3, and the hodograph analysis for d/D ~ oo, both for

10 0 '

'

'

'

'

'''1

Hm

.~D 1 0-'

.Oo:

oO,

jT0

,?

-

1 0 "~'

1 0 "3

-

I 0 "4

-

I

I 0 "3

1 0 .2

I

I 0-1

I 0°

Cl AK,

Fig. 2. Water-table heights maintained by a steady rainfall rate in drained anisotropic soils with an impermeable barrier below in terms of the parameters Hm/N/~D, q/AK x and d/N/~D, given b~_(16) for d/,v/-AD - 0 (the range of bounds is shown by the shaded area, by (11) for d/3/AD----oo, and Hooghoudt's equivalent depth drainage equation for other values of dIN~tAD.

optimum drain size. For d / x / A D = 0 the bounds for the relationship obtained from seepage analysis (Youngs, 1965) are given. The graphs in Fig. 2 are plotted using logarithmic scales in order to accommodate the extended range of the parameters on account of the anisotropy factor. I N F L U E N C E O F A N I S O T R O P Y O N T H E DESIGN O F D R A I N A G E

INSTALLATIONS

The graphs in Fig. 2 may be used to determine the water-table height in lands drained by parallel drain lines for a given steady state rainfall rate, given the anisotropy factor A. More simply we can use Youngs' (1985) empirical drainage equation which becomes for anisotropic soils: (I/aa) (l/an) = (q/AKx) (9) H m/%I~D = (q/K=) where Ola =

2(dAc/AD)

= 1.36,

(dld'A-b)

dL/'~r~ -. oo

0 < d l- D < 0.35

(lO)

For values o f d/~/AD > ~ 0.3 the effect o f the depth o f soil below the drain on the water-table height is the same as for infinite soil depth and hence ~ , = 1.36 is to be used in (9).

In using equation (9) it must be stressed that it resulted from a fit with Hooghoudt's equation in the range 0.01 ( q/K ~ 0.1 for isotropic soils and outside this range the differences between the two becomes large. For anisotropic soils, the anisotropic factor A in (9) puts values of the parameters outside the fitted range, so that the errors in using (9) become unacceptable. We may, however, analyse the situations when there is infinite depth of soil below the drain and when the drain is laid directly on the floor, using physically based drainage equations. These two extreme situations show the range of the effect o f soil anisotropy on the performance of land drains.

Infinite soil depth (d/D ~ oo) The hodograph analysis o f the drainage problem for infinitely deep soils may be applied to the equivalent isotropic flow region with the dimensionless parameters of (8) provided it can be assumed that the elliptical drain in the equivalent isotropic transformed space makes negligible differences to the results of the analysis. Thus, for o p t i m u m conditions which apply for large enough drain channels when the size of the drains makes negligible difference to the water-table height, the analysis gives for anisotropic soils:

Hm/D = (x/A/Tr)[ln (1 + 2/7') + (2/3")1n(1 + 3"/2)]

(11)

where 3"= A K x / q - 1. The water-table height maintained with a given steady rainfall rate thus varies with the degree of anisotropy. Figure 3 shows the relationships between Hm/D and q/Kx for a range of values of A. Equation (11) can be used to provide information concerning the influence that soil anisotropy has on drain design specifications. Thus, if Di is the spacing of parallel drains required to maintain the water-table height at Hm with a steady rainfall rate q in an isotropic soil (A = 1) of hydraulic conductivity K = K~, and Da is that for an anisotropic soil with an anisotropy factor A, the ratio Da/Di m a y be easily calculated for a given q/K~ value. The variation o f Da/Di with A for various values of q/Kx is shown in Fig. 4. Also shown is the relationship obtained using Youngs' (1985) empirical relationship with cz = 1.36:

Da/D i = A °'236

(12)

which is independent of the value of q/K. Equation (11) can be used also to give the ratio of the water-table height in an anisotropic soil to that in an isotropic soil for the same drain spacing and steady rainfall rate. This ratio is the reciprical of that obtained for the drain spacing ratio shown in Fig. 4 required to maintain the water-table heights at the same level in the two situations. Another assessment of the differences in drainage performances in anisotropic and isotropic soils can be made by calculating the ratio of steady rainfall rates q~/qi required to maintain the same water-table height Hm with the same drain spacing 2D in the two situations, q~/qi is shown as a

0.

Hm D

O.

0.

0.

0

0.05 q K,

o. 10

Fig. 3. Relationship between water-table height, expressed as a fraction o f the half-drain spacing, and steady rainfall rate, expressed as a fraction o f the horizontal hydraulic cond u c t i v i t y , in drained anisotropic soils of infinite depth for a range of values of the anisotropy factor A.

101, 0-0 l x . ~

Q Da

o.0o1~

~

D~

~ 0.1 . 0.1

10 c

0.001 O,0001

10 "1 0-2

i

,

,

,

,,,,I

i

10-'

i

i

i

iiiii

i

i

10 °

,

,

i i,,I

101

i

,

,

,

,ii

02

A

Fig. 4. Influence of soil anisotropy on drain spacing expressed as the ratio Da/Di of drain spacings in anisotropic and isotropic soils required to maintain a given water-table height with a given rainfall rate in infinitely deep soils. The dashed line is (12) derived from Youngs' empirical drainage equation (9).

10' :

2_'q,

0.0s

1o°

~

.

~

O.Ol :

os

/ /"/0.5

10 .2 ........ 10 -2

1

I 10-'

, .... ,,I 10 °

,

.......

I 10'

.......

10 2

A Fig. 5. Influence of soil anisotropy on the steady rainfall rate expressed as the ratio

qa/qi of the rates in anisotropic and isotropic soils required to maintain the water-table height with a given drain spacing in infinitely deep soils. The dashed line is (13) derived from Youngs' empirical drainage equation (9).

function of A for various values of Hm/D in Fig. 5. Youngs' (1985)empirical drainage equation gives: qa/qi = A°'32

(13)

which is independent of H=/D. Drains on the floor ( d / D = O)

Youngs (1965) calculated bounds to the water-table height midway between ditches that extended in depth to a horizontal impermeable barrier, for a steady rainfall rate q. He noted that the vertical hydraulic head gradient at the mid-drain position lay between two values such that: 0 < ahm < q ~z K

(14)

where hm is the hydraulic head at a height z above the impermeable barrier. For anisotropic soils K refers to the vertical hydraulic conductivity K,. The horizontal seepage towards the drain is given in terms of the horizontal hydraulic conductivity K= as: ; q d x = ; K= -aax -h d z 0

0

(15)

"°'°' I

0.4

I

i o'

Hm

5--

/ o.,/o.2

0.5

"

0.3

0.2

0.1

0

I

I

I

0

I

q Kz

I

0.05

I

I

I

I

0.10

Fig. 6. Lower bound (heavy line) and upper bounds given by (16) for water table heights in drained anisot~opic soils with drains laid on an impermeable barrier.

at any position x, where h is the hydraulic head at (x, z). Thus Youngs' (1965) seepage analysis leads to the inequality:

~x ~ Hm ~ f D

~]1

---'-~-K:

-- ~1 qlKx -- q l A K x

(16)

to describe water-table heights in drained anisotropic soils. The relationship between HmID and q/Kx given by (16) is shown in Fig. 6; the lower bound is independent of the anisotropy factor A, but the value of the upper bound is larger the smaller the value ofA. ForA = 1 and for qfKx = 0.I0, the difference between the upper and lower bounds is 5.41% and for qfKx = 0.05 it is 2.60%; as A increases, these differences decrease, being respectively 2.60% and 1.27% for A = 2, 1.01% and 0.50% for A = 5, and 0.50% and 0.25% for A = 10. As A decreases below A--1.0, the difference becomes increasingly larger, being 11.8% for qfKx = 0.i0 and 5.4% for qfKx = 0.05 when A = 0.5, and 41.4% for q]Kx = 0.10 and 15.5% for q]Kx = 0.05 when A=0.2. DISCUSSION

The water-table height in anisotropic soils, drained by parallel drains of larger than o p t i m u m size, maintained by steady rainfall on the soil surface can be expressed generally in the dimensionless form:

10

Hm/19 = f ( q / K , ` , d/D, A )

(17)

and more simply in parametric terms: Hm/x/AD

= g ( q / A g , ` , d/'~/AD)

(18)

The analyses of the drainage situations for the two extreme cases of infinite soil depth below the drain (d/D-+ ¢,o) and of the drain lying on an impermeable bed (d/D = 0) show the range of the effect that anisotropy has on drain performance and the extent that it must be considered when designing land-drainage schemes. Water-table heights are lower the larger the anisotropy factor A when there is infinite depth o f soil below the drain, as seen in Fig. 3. In contrast, water-table heights are little affected when the drains lie on an impermeable barrier when A ~ 1, as seen b y the b o u n d s calculated using Youngs' (1965) seepage analysis and shown in Fig. 6, although for A ~ 1 the bounds become increasingly farther apart as A decreases so that the precision of predicted water-table heights becomes less. For intermediate situations with a finite depth of soil below the drain to an impermeable barrier, the effect of anisot r o p y lies between these two extreme situations and can be obtained from the graphs given in Fig. 2. The effect that soil anisotropy can have on the drain spacing needed to effect the same drainage as in an anisotropic soil is shown in Fig. 4 for the situation of infinite soil depth. F o r A ~ 1 the drain spacing is increased whilst for A < 1 it is decreased for the same water-table height and rainfall rate. The hodograph analysis used for this situation shows that for a given anisotropy factor A ~ 1.0 the spacing is less the larger the value o f q/K,,. Thus for A = 10, Da/Di, the ratio o f the drain spacing for drains in the anisotropic soil to that in an isotropic soil with the hydraulic conductivity equal to K,`, is 1.85 for q/K,, = 0.1, 2.17 for q/Kx = 0.01, 2.40 for q/K,, = 0.001 and 2.55 for q/Kx = 0.0001. W h e n A < 1.O, Da/Di is greater (but < 1.0) the larger the value of q/K~,. Thus f o r A = 0.10, D , / D i = 0.54 for q/K,` = 0.01, 0.46 for q/Kx = 0.001 and 0.42 for 0.0001; no value can be given for q/K,, = 0.1 since then q/Kz = 1 and hence water ponds on the soil surface. For the other extreme case with the drain on an impermeable barrier, Youngs' (1965).seepage analysis gives a lower b o u n d for the water-table height independent of the anisotropy factor A while the upper b o u n d is close for A > 1.0 b u t becomes m u c h farther apart as A decreases below 1.0. This indicates that with Kz > Kx (A > 1.0), drainage performance and the drain spacing needed to a c c o m m o d a t e given specifications are practically independent o f A . However, for A < 1.0 with the precision of the water-table height given by the bounds becoming worse as A becomes less, it would appear that anisotropy becomes a factor for consideration. For intermediate depths o f the impermeable floor, the effect on drain spacing lies between the extreme values calculated for diD = ~ and d i d = 0, and again m a y be f o u n d from the graphs of Fig. 2.

11

Drain performances may also be compared by obtaining values of the ratio

qa/qi, where qa is the steady rainfall required to maintain the water-table height at a given level in the anisotropic soil and qi that for an isotropic soil with hydraulic conductivity Kx. This is shown in Fig. 5 for infinite soil depth. The ratio approaches 1.0 as the depth of soil below the drain is reduced to zero. The results of the analyses, presented here in terms of the horizontal hydraulic conductivity component Kx, show that performances of drains laid on an impermeable barrier are influenced mainly by the horizontal conductivity, with the vertical hydraulic conductivity becoming more important as the soils become deeper. Thus, while it is sufficient just to measure the horizontal component of hydraulic conductivity for the design of a drainage installation when drains are close to an impermeable barrier, the vertical component needs also to be measured in deep soils.

REFERENCES Edwards, D.H., 1956. Water tables, equipotentials and streamlines in drained soils with anisotropic permeability. Soil Sci., 81: 3--18. Hooghoudt, S.B., 1940. Bijdragen tot de kennis van eenige natuurkundige grootheden van den grond. 7, Algemeene beschouwing van her probleem van de detail ontwatering en de infiltratie door middel van parallel loopende drains, greppels, slooten en kanalen (Contributions to the knowledge of some physical problems of the soil. 7, General discussion of the problem of drainage and infiltration by means of parallel drains, trenches, ditches and canals). Versl. Landbouwkd. Onderz., 46: 515--707. Lovell, C.J. and Youngs, E.G., 1984. A comparison of steady-state land-drainage equations. Agric. Water Manage., 9: 1--21. Maasland, M., 1957. Soil anisotropy and land drainage. In: J.N. Luthin (Editor), Drainage of Agricultural Lands. Monogr. 7, American Society of Agronomy, Madison, WI, pp. 216--285. Muskat, M., 1937. The Flow of Homogeneous Fluids through Porous Media. McGraw-Hill, New York, NY, 763 pp. Youngs, E.G., 1965. Horizontal seepage through unconfined aquifers with hydraulic conductivity varying with depth. J. Hydrol., 3: 283--296. Youngs, E.G., 1985. A simple drainage equation for predicting water-table drawdowns. J. Agric. Eng. Res., 31: 321--328.