Non-linear absorption of soil water from a disc source

Adm~c~

in W a t ~ Resources

16 (1993) 285-292

Non-finear absorption of soil water from a disc source A.W. Warriek & W.

Ojeda

Department of Soil and Water Science. Uni)~rsity of Arizona, Tucson, Arizona 85721. USA

IN'on-linear absorption of water from a surface-disc source is examined. The solution can be evaluated only by numerical techniques. However, results can be generalized in a dimensionless form invariant to disc radius, saturated hydraulic conductivity, and capillary length. Gravity effects are negligible for small time and for small disc radii. For the limiting effect of gravity for large times, Wooding's equation gives the ratio of excess flow due to gravity as ~r0/(4~) compared to 1, where ro is the disc radius and ~¢ the capillary length. Examples are presented by using two sets of empirical hydraulic functions as well as an estimate of the product of saturated hydraulic conductivity and capillary length from field data. Key word~: disc sources, soil water, inflltmdon.

~TRODUCTION

For a plot of I against t°s for small times, S is simply a slope. The steady-state flow rate QD for a disc source without gravity and valid for all hydraulic conductivities is: 16

Disc infiltrometers are useful for determining /n sire hydraulic properties of soil as discussed recently by White and Sully, Is Ankeny et al., l Smettem and Clothier, 14 Reynolds and Eh'ick, 12 Warrick, 1~ and Felton. s These are devices designed to measure intake for a carefully controlled water pressure within a circular interface at the soil surface. The water pressure can be slightly positive but more often is at a small tension equivalent to 0-0.2 m of water. By maintaining the entry head, flow into the larger macropores can be avoided. The analysis of data from disc inflltrometers for short times has often assumed equivalence to a one-dimensional absorption: ~s19 Q

~ 0-5St -°s

~D =

(3)

The relationship of eqn (3) is for flow from one side of the disc, such as when the disc is on the surface. Since gravity is ignored, a 'buried' source would have a discharge of 2~D. In eqn (3), K , ~ ----K(/~=) is the hydraufic conductivity corresponding to the source pressure head h~=, r0 is the radius, and ,~ is the capillary length:l° = .rK ( t ~ = j , - X ( h ~ ) : - I

(I)

'[i' J/~.,

X ( h ) dh

(4)

The effect of gravity may be examined by using Wooding's approximation 2° based on the hydraulicconductivity function of Gardner s as defined in Table 1. The Wooding flow rate Qw is

where Q is the flog' from the disc infiltrometer (e.g. m s s-l), r0 is the disc radius (e.g. in), S the sorptivity (e.g. m s-°s): and t time (e.g. s). The sorptivity is dependent upon the initial water content, supply-water content, and diffusivity function and may be calculated by using the quasi-analytical method of Philip. 9 Integration of each side of eqn (1) with respect to t results in the cumulative infiltration I: I = St °'s

4K,,=~r0

Qw = r.ro2K~=( 1 - 4 - ~ o )

(5)

Thus the ratio of Qw/QD can be expressed as: ~ w = 1 -4-~ "~¥0 QD 4~

(2)

(6)

The effects of gravity diminish as to~ ~e becomes smaller. Calculations by Warrick, ~6 using the hydraulic functions of van Genuehten Is (as defined in Table 1), resulted in steady-state Q values from 0-8 to 1-4 of Qw with the

Advances in Water Resources 0309-170894S07.00

1994 Elsevier Science Limited 285

286

A.W. Warrick. W. Ojeda Table 1. H.vdrlfie feaefiom eomldered in the examples

1. Gardner e and Russo 13 (GR)

exp h"

.'exp (h"/2~ (1 - h"/2)~2->- 2.-

2. van Genuchten Is (VG)

O°S'l - (I - O:!=)m] 2

(1-

3. Brooks and Corey2 (BC)

O 2-2~

'h" - : - m

h" :__--:1-m;-?-"

1

0.188, for m = 0"3" 0"405, for m = 0.5 0.637, for m = 0-7 0.875, for m -- 0-9 I - 0.5ma

°For the VG and BC functions, the values of ~ assume O¢~y= 0 and 0== = 1. same capillary length. Thus, eqn (6) can be used as an indication of the effects of gravity for at least two hydraulic-conductivity functions, those of Gardner 6 and of van Genuchten.I 5 Hussen and Warrick ~ compared field data against alternative algebraic models for the disc-tension infiltrometer. Sorptivity values based on linear diffusion without gravity were found to be generally reliable for discs of radii 5.2 and l l.8cm. Results were also compared to the liuearized model of Pullan: n /'T,ro~ f D t ' ~

e = e-- iN)giN

)

(7)

THEORY

Richard's Equation for unsaturated water flow- without gravity is: O0 l O frKOh~ O K-~ O"~=r~ ~r/+Oz

with g(u) = (,ru) -°

exp ( - u ) - erfc (u °5)

(8)

The Qss is equivalent to Qw above, D is a constant diffusivity value, and erfc is the complementary error function. By making comparisons of a numerical solution for small times to a linearized approximation, Warrick 16 concluded that the geometrical effects dominated over gravity for up to three hours for two specific examples. This would indicate that solutions developed for nonfinear sorption from a surface-disc source (without gravity) could be useful for an extension of the shorttime results for disc infiltrometers. Moreover, the characteristic times9 t__

approach will be similar to that of Warrick et al. r for one-dimensional infiltration in that a solution will be presented in terms of reduced variables. Results will also be compared with those with gravity in order to investigate the appropriateness of eqns (6) and (10) to determine when gravity effects should be included.

=

{ r 0 ( 0 ~ - 0dry)}2

(9)

where 0 is the volumetric water content, h the matric potential (as a length), r a radial co-ordinate, z the depth, t the time and K the unsaturated hydraulic conductivity. Dimensionless forms O, h', R, Z: T, and K" are defined by: O = (0 - Or) (0s -- Or) T=

R

=

Z =

tgav =

{ s }2 Kwet-- Kdry'

(io)

were found useful for estimating times for which gravity became important compared with geometrical effects. In eqns (9) and (10), r0 is the supply radius, and/(,,~ and are the hydraulic conductivifies corresponding to the supply water content 01,= and the initial water content 0dr>. Motivated by the above observations, the objective of this study is to develop generalized solutions for absorption from a disc source. That is, Richard's Equation will be solved by neglecting gravity. The

(12)

K~t ar~(Os - Or)

(13)

r/ro

(14)

zlro

(15)

Substitution of the dimensionless variables into eqn (11) results in: 00

and

(11)

1 a

( .oh') o

(16)

with K" = K/K.

(17)

h" = ah

(is)

We consider the large class of functions for which: K/Ks = K'(h=. m)

(19)

and

0-0r -0s- - - ~ r = O(h': m)

(20)

Three examples of such functional relationships are

Non-linear absorption of soil water from a disc source

given in Table I. These are by Brooks and Corey,2 Gardner, 6 and van Genuehten. Is Gardner defined a conductivity function in terms of exp (h'), and recently Russo 13 derived a corresponding soil-characteristic function. Van Genuehten's soil-water characteristic curve was formulated to give the correct behavior for h" either close to zero or of large magnitude and the form of his K is also consistent with the general hydraulic conductivity of Mualem. s Appropriate initial and boundary conditions for the tension infiltrometer are:

(21)

h'(R,Z.O) = h~. h'(R,O,T)=h~ Oh" =0. OZ Oh" =o. OR Oh" = o. OR Oh" OZ = 0,

0
R> 1.Z=0

(22) (23) (24)

R-.

(25)

Z ---, ~c

(26)

Q = - 2 ~ / ~ . t J~3 [~-~hz],= 0r dr

(27)

or

(2s) where (Oh'/OZ) is the average dimensionless pressureheed gradient, given by:

(~--~) =210

I

Oh" R[~-~]z.odR

(29,

The large-time flow rate Qn for all hydraulicconductivity functions is eqn (3). A dimensionless form of (2 may be expressed as (2":

Q'= Q QD

(30)

or

,,/oh'\ =

!F

Q ' ( T ' ) dT'

(32)

~J0

The dimensional I is related by

The solution ofeqn (16) subject to eqns (21)-(26) will be a function of R, Z, and T as well as m, h~., and h ~ . For conversion to dimensioned values, additional requirements are values of ch r0, K~t, 0,, and Or. In the case of the steady-state solution, the initial condition is not required and neither are 0s and Or. (For the Gardner function, m is not required for the steady-state relationship, either.) The flow rate Q (m3 s-1) from the disc is from

Q-

and m for the three hydraulic functions of Table 1. Moreover, a~¢ can be found directly if m is known. For the Gardner function, a ~ is trivially 1 by eqn (4). For van Genuchten's relationship, a~¢ is by numerical integration with sample results in Table 1. For the Brooks and Corey2 relationship, the integral from eqn (4) is analytical and recorded in Table ! for Odry = 0 and O,== 1.Thus, Q" can be found for any of the three hydraulic functions, given a value for T, h~y, h~=, and m. To recover the value of Q, additional geometrical and hydraulic properties are needed, namely, 0,, Or, Ks, c~, and r0 and with which hdry,/t~=, 0wet, 0dry, and K~,= are redundant. A corresponding dimensionless cumulative infiltration I" can be defined by: I" = -

R=o

287

(31)

The value (Oh'/OZ) is a function of only T, h~., h ~ ,

(33) where I is the cumulative flow divided by the area of the disc surface ~r~.

I~-t,'MERICAL CALCULATIONS In order to carry out the necessary numerical calculations, a finite-element program was used. The procedure utilized the Galerkin method of weighted residuals and closely followed the programs developed by DaSilva4 for infiltration from an irrigation furrow. A mixed formulation for water content and pressure head was used. This follows Celia et al., 3 who found mass conservation to be generally easier to maintain with the mixed formulation as compared with a purely pressure-head formulation The generation of nodes was chosen on a variably spaced grid. The horizontal spacing was fixed for 0 < r < r0 (beneath the disc) and then Ar varied by an arithmetic progression between r0 and rm~. The number of nodes on the disc was specified as well as the total number of horizontal nodes. The vertical nodes were chosen as an arithmethic progression by specifying the minimal spacing (taken at the surface), the number of vertical nodes, and zma. This defined all the nodes for the flow" region, after which triangular elements were formed by drawing diagonals across the rectangles formed by the nodes. The boundary conditions of eqns (25) and (26) cannot be met within a finite-flow domain. Thes" were approximated by choosing a 'no-flow' condition at R = Rmax and Z = Znm. The simulation was stopped when h at the boundaries increased. The program was coded in FORTRAN 77. Soil hydraulic properties are in Table 2. The loam and Yolo light clay are described with the van Genuchten is

288

A . W . Warrick. W. Ojeda Table 2. Sell ~

Functions

a (m-n)

m

O,

VG

1.0

0.5

0.45

0-1

6(10)-6

GR

2.38

5-14

0-388

0.154

1-5(10)-s

VG

1.5

0-5

0-495

0-124

1.23(10)--

1. Loam

Warrick et al. n" 2. Silt loam Russon3 3. Yolo fight cla~/ Warrick et al. 1--

function, and the silt loam is by Gardner 6 and Russo 13 functions.

Example 2: Co~lmrisen with and wiilmut gravity for ~lt loam sou

As a first comparison, we consider a linearized approximation to the silt loam (Soil 2) of Table 2. The comparison allows a check for consistency for the finiteelement algorithm and the analytical approximation of Warrick: 16 = 1 + 0.25r. ° S ( T ) - ° s

Results are compared for the non-linear case with and without gravity for the silt-loam soil (Soil 2 of Table 2). The results for r0 = 0.1 are given in Fig. 2(a). The effect of gravity is somewhat small, with an increase of 8% after 350s increasing to 13% after 3500s. In addition, for a reference, the analytical result for linear diffusion (eqn (34)) is repeated. The limiting effect of gravity is by eqn (6), which results in an estimated increase of 19%. The effect of gravity is more pronounced for r0 = 0-Sm. This is illustrated in Fig. 2(b). There is an increase of 15% after 350s and of 35% after 3500 s. The limiting effect of gravity given by the ratio QW/QD is estimated as 1.93 or an increase of 93%.

- 0-216(exp ( - 4 . 0 1 T ) }

(34) 2

(35)

K, a(0s - 0-)

(36)

T = z)t/

D=

K, (m---s)

differences occurring only at very small times (note that T is on a logarithmic scale, which tends to exaggerate discrepancies).

E x m p l e 1: l.inearized eomparkon for the i t loam sog

Qo

Or

Results are plotted in Fig. 1 for r0 = 0-1 m. Generally, the two results are in close agreement, with

g

?

6

8

.........

0 I

0.001

i

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l

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i

i i i

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0,010

i

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0.100

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|

i i i

i

1JDO0

i

i

|

i

I

i

i

i I

1~000

T Fig. 1. Plots of the linearized approximation for time-dependent diffusion (eqn (34)) and the finite-element solution as a function of dimensionless time 7".

Non-linear absorpzion of soil water from a disc source

289

"-. (a) 9

O/QD It

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1000 t

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(m:)

Fill. 2. Plots of Q/QD for Soil 2 of Table 2. The disc radius is r0 = 0-1 m for (a) and re = 0-5m for (b). Example 3: Ctleeladon for loam sad Yolo light clay ( w i e m t ~ravity) As a final comparison and an illustration o f the advantage o f scaling, Q/QD and the dimensionless cumulative intake for two contrasting soils are corn-

pared in Fig. 3. Since m = 0.5 for both the loam (Soil 1 o f Table 2) and Yolo light clay (Soil 3 o f Table 2), the dimensionless values Q/QD and dimensionless cumulative infiltration I " can be plotted as c o m m o n curves. Calculations for the two soils were performed separately. The small differences of Q/QD are due to

A.W. Warrick, W. Ojeda

290

9-~ "

o

o

o YOLO U O H T CLAY

•

•

•

•

80

LO/~ •

7

•

Q/Qo

" °o

p

Q/Qo

0o •

°°o ••

• ••

.............................................................. •cpo

0,01

'

'

0

' ' ' ' i n

0.10

0

o00 '

•

~JO

•

O°°~oe

'

40 •

o

8

•

I*

• •

0

,00

•

5

1

70

•

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•

w ........... t m u l

............

10

ooooal '

' ' ' ' ' i

'

'

"

' ' ' ' ' i

1.00

'

'

'

' ' ' ' ' i

10,00

•

100,00

•

| , | . =

0

1000,00

T Fig. 3. Plots of Q/QD and I" for two contrasting soils, a loam and Yolo fight clay, as a function of dimensionless time T. sensitivity in finding inflow rates for small times in the case o f the loam. The range o f times for the loam is from 6s for T = 0-01 to 16h for T = 100. The times are much longer for the Yolo light clay, corresponding to 452 s and 1256h for T = 0-01 and T = 100, respectively. E x m p l e 4: Generalized results Results for Q / Q D are ~iv~[l for van Genuchten ]5 functions as Table 3. Included are dimensionless T for 0-01-500 and m values o f 0"3, 0"5, and 0-7. The initial condition is assumed to be 0 = Or and the inlet boundary assumed to be maintained at h,,= = 0. As an example o f how Table 3 may be applied, suppose we wish to evaluate the cumulative intake at 3 h for the Yolo o f Table 2. By using Table 3 for T = 2.60 (3h), the corresponding values are Q/QD = 1-64 and I " = 1-60. We can also make a quick estimate o f the limiting (large-time) effects of gravity by eqn (5). The value of)~c for m = 0-5 from Table 1 is ~ = 0-405/1.5 = 0-270. By eqn (6), the large-time discrepancy is obtained from: Qw QD

-

1

,'rro [~4)~0-27j t

i t37,

For r0 = 0.1, the rate is Qw/QD = 1-29, or about 30% of the steady flow is due to the effects o f gravity. Example S: F_~Amafion of I . / Q and a~c from field data Hussen and Warrick" present field data as I against t for

a field tension inflltrometer with r0 = 0.052m. The supply water content was h ~ . = 0, 05 was 0-411, and Or was taken as 0.09. The scaled and dimensional times and infiltration depths may be written as: T = Bt

(38)

I" = CI

(39)

where, by eqns (13) and (33), with K , ~ = Ks: B=

C =

K, OW2(O, - - Or)

(40)

1

(41)

4a~cro(0s - 0:)

The field data are presented in Fig. 4. Also in Fig. 4 is a regression curve on the values from Table 3 for m = 0"5 with: [- = AlTO 5 _ A2 T - - A 3 ~] -

exp (-A4 T)~

(42)

with Al =0"7189, A2 =0"3249, A3 = - 0 " 4 9 5 3 , and A4 = 0.8639. Values o f B and C were determined to be (9.16)(10)-4s - ' and 59-51m - l , respectively. By eqns (40) and (41), this leads to estimates of Ks~o, = (7-95)(10) -7 m 2 s-" and c~¢ = 0-252. Laboratory core measurements 13 were reported as Ks/c~ = (6-02)(10)-6m 2 s -] and m = 0.428, which would lead to c ~ = 0-33.

Non-linear absorption of soil water from a disc source

291

Table 3. Valem of Q/Qn and !" for ~iffereat v a l m of m for the ~ u Geme.btea feaeflon." T

m = 0.3

0-0001 0-0005 0-001 0.005 0-01 0.05 0-1 0-5 1-0 2.0 4-0 6-0 8.0 10-0 15-0 20.0 30-0 40-0 50-0 60-0 80-0 lO0.O 200-0 500.0 ~c

m = 0-5

m = 0-7

Q/QI

l"

Q/QD

I"

Q/QD

328-0 ! 08.0 71-1 25-3 i 6.5 7.16 5-25 2-90 2.36 1-99 1.73 1-61 1-54 i-49 1.42 1.37 1.32 i-28 1-25 1.23 1-21 i.19 1-15 l-lO I'00

0.0308 0.0527 0-0662 0.115 0-147 0-273 0-369 0.830 1-24 1.92 3.09 4.15 5.16 6-12 8-43 10-7 14-9 19-1 23.1 27-1 34.1 42-6 79-8 187-0 ~c

98-1 44-8 30.9 13-7 9-87 4.86 3-74 2-26 1-91 1.67 1-56 1.42 1-37 1-34 1.29 1.25 1.22 1.19 1-17 1-16 1.14 1-13 1"09 1-06 1-00

0.00472 0.0126 0-0185 0.0422 0.0606 0-141 0.208 0-554 0-884 1.15 i.96 3-37 4.27 5-13 7-22 9.23 i 3.2 17-0 20.8 24-5 31-8 39.1 74-9 178.0 oc

26.9 19-0 15-6 9.85 7-89 4.03 3-14 1.99 1-72 1.53 1-40 1.34 1-30 i-27 1-23 1.20 1-17 i.15 1-14 i-13 1.12 1-11 1"08 1"04 1"00

I"

0-00096 0.00378 0.00651 0-0213 0.0354 0.101 0.157 0.458 0-749

1-26

2.19 3-06 3-89 4-71 6-70 8-64 12.4 16-1 19.8 23-4 30-6 37-6 72.4 174-0

~See paper by van Genuchten. I-~Values are invariant with respect to a, K,, 0,, 0-, and rc.

:.=

."

. i

I (am) 1.2 -

0.8"

I

i o.o--t

• I

0

'

I

,

1000

i

I

U

'

i

I

I

SO00

t (too) Fig. 4. Plot of I against t for field data of Hussen and Warrick'-- along with the theoretical curve for m = 0-5 of Table 3.

292

A . W . Warrick, W. Ojeda

DISCUSSION Non-finear absorption of soil water form a surface-disc source has been examined. The convenience of neglecting gravity is that results can be expressed in terms of dimensionless time to give solutions that are invariant with respect to the disc radius, saturated hydraulic conductivity, and capillary length ~c (or a) (cf. Table 3). The use of the solutions without gravity remains valid for (a) small times or (b) small-diameter discs. Precise estimates of the errors caused by neglecting ~ a v i t y are difficult, but the characteristic time of Phifip given b~ eqn (10) and the steady-state intake rate of Wooding ~ are related. The characteristic time for gravity, tsrav, is based on one-dimensional flow but nonetheless gives an indication of when gravity is expected to become important. The Wooding approximation, expressed as a ratio to the one-dimensional, steady-state flow rate, indicates the limiting effect of gravity occurring for large times by eqn (6). For small r0 the effect is less; for large r0, the effect is more pronounced: as is also indicated by Example 2. Although Wooding's approximation assumed K of the form K = K s - e x p ( h / ~ ) , other hydraulic functions are approximated by use of eqn (4). For tinter-textured soils, 3~ tends to increase, and the effect of gravity WIU tend to be less for equal disc radii r0. The results of Table 3 are invariant with respect to Ks, ,~, r0, 0s, and 0.. Furthermore, the range of m = 0.3-0"7 for the van Genuchten ~5parameter includes many, if not most, soils. For other saturated conditions at the supply surface or wetter initial conditions, other similar tables could be developed. Additional tables could be prepared for other forms of hydraulic function. An example was presented to demonstrate the estimation of/(.s/Ok and a'~c from field infiltration from tension-infihrometer data. The field data obviously are affected by gravity, but, for the small disc radius (0.05m), gravity is a secondary factor. The fact that Ks/,~ can be determined is analogous to results for onedimensional infiltration with gravity, which also can provide an estimate of the ratio rather than Ks and a independently, without further information. The estimate of ,~,~ is reasonable.

REFERENCES 1. Ankeny, M.D., Kaspar: T.C. & Horton, R., Design for an automated tension infiltrometer. Soil Sci. Am. J., 52 (1988) 893-6.

2. Brooks, R.H. and Corey, A.T., Hydraulic properties of porous media. Hydrology Paper 3, Colorado State University, Fort Collins, CO, USA, 1964. 3. Celia, M.A., Bouloutas, E.T. & Zarba, R.L., A general mass-conservative numerical solution for the unsaturated flow equation, Water Resour. Res., 7,6 (1990) 1483-96. 4. DaSilva, E.M., Analysis of furrow irrigation uniformity as affected by furrow spacing. Ph.D. dissertation, University of Arizona, USA, 1990. 5. Fehon, G.K., Soil water response beneath a tension infiltrometer: Computer simulation, Soil Sci., 154 (1992) 14-24. 6. Gardner, W.R., Some steady-state solutions of the unsaturated moisture flow equation with appfication to evaporation from a water table, Soil Sci., a$ (1958) 22832. 7. Hussen, A.A. and Warrick, A.W., Algebraic models for disc tension infiltrometer, Water Resour. Res., 29 (1993) 2779-86. 8. Mualem, Y.T., A new model for predicting the hydraulic conductivity of unsaturated porous media, Water Resour. Res.: 12 (1976) 513-22. 9. Philip, J.R., Theor;" of infiltration, Adv. Hydrosci.. $ (1969) 215-305. 10. Philip, J.R., Reply" to comments on "Steady infiltration from spherical cavities, Soil Sci. Soc. Amer. J., 49 (1985) 788-9. l 1. Pnilan, A.J., Linearized time-dependent infiltration from a shallow pond, Water Resour. Res., 7,8 (1992) 1041-6. 12. Re,molds, W.D. & Elrick, D.E., Determination of hydraulic conductivity" using a tension infihrometer. Soil Sci. Soc. Am. J., 55 (1991) 633-9. 13. Russo, D., Detcrminiug soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties, Water Resour. ICes., 24 (1988) 45369. 14. Smetten, K.R.J. & Clothier, B.E., Measuring unsaturated sorptivity and hydraulic conductivity using multiple disk permeameters, J. Soil Sci., 40 (1989) 563-8. 15. van Genuchten, M. Th., A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soils Sci. Soc. Am. J., 44 (1980) 892-8. 16. Warrick, A.W., Models for disc infiltrometers, Water Resour. Res., 28 (1992) 1319-27. 17. Warrick, A.W., Lomen, D.O. & Yates, S.R., A generalized solution to infiltration, Soil Sci. Soc. Am. J., 49 (1985) 348. 18. White, I. & Sully, M.J., Macroscopic and microscopic capillary length and time scales from field infiltration. Water Resour. Res., 23 (1987) 1514-22. 19. White, I. & Sully, M.J., Field characteristics of the macroscopic capillary" lengths or alpha parameter, IN. Wierenga, PJ. and D. Bachelet, Intern. Conf. and Workshop on the Validation of flow and Transport Models for the Unsaturated Zone, Las Cruces, New Mexico, pp. 517-524, 1988. 20. Wooding, R.A., Steady infiltration from a shallow circular pond, Water Resour. Res., 4 (1968) 1259-73.