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Numerical modeling of free-drainage water samplers in the shallow vadose zone
Advances in Water Resources 15 (1992) 251-258
Numerical modeling of free-drainage water samplers in the shallow vadose zone R. Zhang*, A.W. Warrick & J.F. Artiola Department of Soil and Water Science, University of Arizona, Tucson, Arizona 85721, USA
A finite element model was used to simulate the performance of a free-drainage sampler. This device may be used to collect soil water by free drainage in the shallow vadose zone. A unique feature of this sampling device is that it serves as a barrier to flow and creates a locally saturated region in an otherwise unsaturated system. Steady and unsteady flows were modeled assuming the hydraulic functions of van Genuchten and Gardner. Results include flux into the sampler and distribution of hydraulic head as functions of soil hydraulic properties, background soil flux and sampler size. Key words. free-drainage water sampler, unsaturated flow, modeling.
INTRODUCTION
Recently, the authors have used a hollow glass brick sampler to collect water by free drainage. The sampler has dimensions of 0.3m x 0-3m x 0.1 m and is placed at about 1-5 m depth in an agricultural field. Water enters through perforations on the top (one of the 0.3 m x 0.3 m faces) by gravity and is removed from the hollow chamber when desired by applying a temporary suction. Preliminary data indi-cate that the sampler may indeed collect significant amounts of liquid in sand layers underlying heavier (loamy) agricultural soils. This may be due to the relatively fast water fluxes that can occur in sand under unsaturated conditions and low tensions. As the heavier textured upper layers become saturated, the underlying sand provides easy vertical flow to the samplers. When properly installed, the free-drainage samplers have several major advantages over porous cups. They provide continuous liquid collection, intercept preferential flow paths, simplify sample retrieval (it is less time-dependent), and give larger sample volumes (thereby allowing for the analysis of more parameters), and result in minimal chemical alteration of soil-pore liquid sample. The objective of this study is to model the flow performance of a free-drainage water sampler in the unsaturated zone. Both steady and unsteady flow regimes will be considered. The sampling device itself serves as an obstacle to flow which raises the soil water pressure and allows water entry into the sampling device by gravity. The analysis is limited to Darcian flow and preferential flow zone such as macropores are not considered in this study. Direct field validation is beyond the scope of this paper.
Monitoring devices are used to detect contaminants in the shallow vadose zone. Such instruments are of value for both shallow and deep water table conditions, Sampling in the unsaturated region above a deep water table can result in years of advance warning since the associated contaminant travel time can be substantial, The design, use and performance of various types of vadose monitoring equipment have been reviewed by Everett et al. 4 and Litaor. 8 Tension infiltrometers have been m o d e l e d b y W a r r i c k and Amoozegar-Fard, 18 Van der Ploeg and Beese 16 and Narasimhan and Dreiss. 12 Their results include travel time and radius of influence around the sampler. Experimental results by Morrison and Lowery II showed the radius of influence to be less than 50cm in medium grain sand. A free-drainage water sampler has features in c o m m o n with the 'seepage entry problem' of Babu I and Philip) 4 In this context, the primary consideration was water-tightness of subterranean cavities, such as for tunnels or waste burial. Generally, pressure will increase more for flow around bluntly shaped objects than for objects sharply pointed into an oncoming flow. Increases in pressure will result in a tendency for saturated regions to develop (such as at the top of a sampling pan) and the potential for free-drainage to occur, *Present address: US Salinity Laboratory, USDA, ARS,4500 Glenwood Drive, Riverside, California 92501, USA. Advances in Water Resources 0309-1708/92/$05.00
~"5 1992 Elsevier Science Publishers Ltd. 251
R. Zhang, A.W. Warrick & J.F. Artiola
252 THEORY
and the van G e n u c h t e n (VG) type is:
Consider Richards' equation for unsaturated water flow expressed as:
ot - r Or
-Or + Oz
Oz
Oz
(1)
where 0 is the volumetric water content, h is matric potential (as a length), r a radial coordinate, z depth, t time and K the unsaturated hydraulic conductivity. B o u n d a r y conditions are (see Fig. 1): h
h0 or q
q = 0
on AB
(2)
across BC, DE, EF, F G and A H
O(h + z) _ 1 Oz q
K(ho)
0orh =0
on G H on C D
K~ exp h*
@ = [1 + Ih*l ~/] ,,
(9)
where K s is the saturated hydraulic conductivity. The h* and O are defined by: h* = ~h
10)
0-Or (-) - 0s - 0~
11)
In eqns (10) and (11), (~ is a constant, 0s the saturated water content and Or the residual water content.
(4)
CALCULATIONS
(5)
A finite element formulation was chosen to approximate the Richards equation because of the ease and flexibility for assigning b o u n d a r y conditions and the refinement of the numerical approximations. The procedure utilized the Galerkin m e t h o d of weighted residuals and closely followed the p r o g r a m s developed by DaSilva 3 for infiltration from an irrigation furrow. Features of the c o m p u t e r p r o g r a m included a mixed f o r m a t i o n for water content and pressure head
(6)
O = [ e x p ( h * / 2 ) ( 1 - h * / 2 ) ] :/(m+2)
B
(8)
(3)
Hydraulic functions used in this p a p e r are given by G a r d n e r 5 and Russo 15 and by van Genuchten. 17 The G a r d n e r and Russo ( G R ) f o r m is: K
K = KsO° 5[1 - (1 - o1/m)m]:
(7)
A
q = const
B -f
*'7
AND RESULTS
q : const --I~I--TI ' , , "II I,'
, Zm~×
~
I ft f
•
Origi I"1
z
,
C
[
i
I
i
r
D
Zc
,/
F
/
r
G~
H
h = const f'rnax
(a)
~ Zmax
•
1
G
h const (b)
Fig. 1. The flow region without flow (a) and with flow into the brick (b).
2H
Numerical modeling of free-drainage samplers suggestedbyCeliaetal. 2 forwhichmassconservation was found to be generally easier to maintain than using
z, /
programa purely pressure-head C up a formulation. n tops boundary eThe algorithm t be run in either transient or steady-state conditions. The is for the condition of fixed head or fixed flux (along AB of Fig. 1). For the following simulations, only the fixed flux boundary is used because the two types of boundary conditions give nearly identical results. Most of the simulations are for a space domain 0 to 1 m (rmax) in the r direction and in the z direction from l m (Zmax) above the brick to l m below the brick (see Fig. 1). The sizes of rmax and Zma× are chosen large enough to satisfy the boundary conditions. For example, the setup of the domain leads to a flow rate at the right boundary which is at least 4 orders of magnitude smaller than the input flow rate. The brick is simulated by a cylinder with radius 0.15 m and height 0.10 m for the most part, although some calculations are with other sizes. The number of nodes on the top, at the bottom and on the side of the brick was specified as well as the total number of horizontal and vertical nodes, The vertical nodes were computed as an arithmetic progression by specifying the minimal spacing (taken just above the top and below the bottom of the brick) and Zma× with a specified number of nodes. The horizontal nodes to the right of brick varied by an arithmetic progression within rm~x. Triangular elements were formed in the flow region by drawing diagonals across the rectangles formed by the nodes (see Fig. 2). Typically 400 nodes and 700 elements were used. Various types of soil were chosen for this study. Soil parameters used are listed in Table 1. Four sets of hydraulic functions are of the van Genuchten form and three of the Gardner and Russo form according to the given references. Soils 3 and 4 are actually one soil fit with alternative hydraulic properties; likewise, soils 5, 6 and 7 are one soil described by three alternative hydraulic functions or parameters,
~ C~
253
/ /
/
/
~
6
c
/////// '
Hollow Cylinder
/ / / ( /"
f" //
F¢~~/ $
/
/
/
/
'///
/
~
/ /
•
h =O
o NO F l o w Fig. 2. The elements and nodes around the brick.
by using q = 0 across CD (see Fig. 1(a)). If the input flux is large enough at the top of the studied region, a positive matric head will develop at the top of the glass brick and a pressure deficit will form at the bottom of the brick. Figure 3 shows matric heads at top of the brick which change with time for a non-steady state simulation. Parameters of the hypothetical sandy loam soil (soil 3) are used with an input flux along AB equal to 7-8 × 1 0 - 6 m s I (90% of Ks). Within five hours, a positive matric head results along 16% of the top area. After 20h, a steady state is established and a positive matric head is developed in 81% of the area. The different streamline patterns are compared in Figs l(a) and (b) schematically without and with flow into the brick, respectively. Example 2. Steady state with flow into the brick To simulate the hollow glass brick (see Fig. l(b)), the authors initially set the boundary condition at the top of the brick as no flow entering (q = 0). During the iterations, if the matric head becomes positive at some of the nodes at the top of the brick, the boundary condition at these nodes was changed to fixed matric head of 0, while other nodes remained as a no flow
Example 1. Steady and non-steady state without flow into the brick If flow cannot enter the glass brick, it is set as a barrier
Table 1. Soil hydraulic functions used Soil
Functions
~
m
0r
0s
(m -1 )
Ks (ms -1 )
1. 2. 3.
Yolo light clay 1°19 Glendale clay loam 6 Hypothetical sandy loam 7j5
VG VG VG
1.50 1"04 1'00
0-500 0"283 0500
0.124 0106 0-170
0.495 0"469 0"470
1.23 x 10 7 1-52 x 10 6 8-69 x 10 6
4. 5. 6. 7.
Hypothetical sandy loam :15 Silt loam soil :uS Silt loam soil v15 Silt loam soil 715
GR VG GR GR
5"31 3"37 0-397 2'38
0 0"500 0 5"14
0'186 0'115 0.152 0"154
0"470 0"338 0-388 0-388
8-69 x 1"50 x 1.50 x 1-50 x
10-6 10 5 10 s 10 5
The Gardner 5 and R u s s o 15 relationships ofeqns (6) and (7) are denoted by GR; those of van Genuchten w by VG. Soils 3 and 4 are for a soil but with different parameters; likewise, soils 5, 6 and 7 represent different models for the same soil.
R. Zhang, A.W. Warrick & J.F. Artiola
254 0.1 0-
oo5 ~. ,~ ~_
~
~ ~
ooo ~
...... ~
\_ - ~ _
-0.05
~. 5 hours
Steady -0.10
0.00
overall soil flow rate for soil 2. Flow occurs into some of the six nodes at the top of the brick, depending upon the input flux. Higher input fluxes produce more saturated nodes. The lowest pressure appears in the left corner at the bottom of the brick (node F in Fig. 2). To examine the effects of truncation errors, the authors doubled the number of nodes across the top of the brick and refined other nodes. Several tests indicate that the differences of the rate into the brick are within 8% due to refinements of nodes. Table 3 shows the results with different sizes of bricks (radii of 0.1, 0.15 and 0.2m for soil 1). As the size of a brick increases, more flow goes into the brick and a larger pressure deficit develops below the brick for the
,
o.o4
State v
, 0.12
0.o8
\.
o. 6
r (m) Fig. 3. Matric heads at the top of the brick of non-steady state for soil 3 when the brick is impervious,
same input flux. When the input flux decreases, there is no flow into a brick with smaller radius.
boundary. If a node has a resulting flow back into the flow system, the node is switched back to that for a noflow boundary. Figure 2 gives an example showing nodes with h - 0 (nodes 1-4) and nodes without flow (nodes 5 and 6) at the brick top, also showing the elements around the brick, Table 2 gives flow rate into the brick as a function of
Example 3. Simulations with different hydraulic functions
Here the authors illustrate how the choice of hydraulic properties affects simulation results. Soils 3 and 4 are actually the same sandy loam soil but fit by different parameters (RussolS). Based on the parameters, simulation results for soil 3 and soil 4 are compared in Table 4.
Table 2. Rate into the brick as function of input flux for soil 2
Input flux
Input rate
Rate into brick
(xl0 -6)
(xl0 7)
3"818 3"343 2"865 2'388 1.910 0"955
0'792 0"599 0"432 0'215 0'048 0'000
(m3 s-l)
80% Ks 70% Ks 60% K~ 500/0 K~ 40'% K~ 20% K~
(m3s 1)
Nodes for h = 0
Lowest h (m)
6 6 5 5 3 0
-0"108 0" 114 0'127 -0" 146 -0" 177 0"331
Table 3. Rate into the bricks of different size as function of soil flux for soil 1 (brick radii are 0.1, 0"15 and 0"2m)
Input rate (m3s-1) (x10-7) 3-671 3"478 3"284 3"091 2"898 2'705
Rate into brick (m3s I)(xlO 8)
Lowest h (m)
0"1
0'15
0"2
0"203 0' 104 0"034 0"000 0'000 0"000
0' 529 0"366 0"221 0" 111 0-029 0-000
1"00 0"778 0"561 0' 367 0"216 0"090
0'1
0"15
0'2
0"091 --0" 105 --0" 121 --0"138 --0" 157 --0" 177
--0"129 0"143 0"157 --0"173 0"190 --0'209
0' 166 --0' 180 --0' 194 --0'209 --0'225 0'242
Table 4. Comparison of simulated results for different hydraulic models of the same sandy loam (soils 3 and 4)
Input rate (m3s 1)
Rate into brick (m3s -1) (xl0 6)
Lowest h (m)
(xl0 5)
Soil 3
Soil 4
Soil 4c
Soil 3
Soil 4
Soil 4c
4.241 3'770 3'299 2.827 2.356
0"158 0"000 0-000 0-000 0-000
0.334 0"201 0"080 0'003 0"000
0.199 0"015 0"000 0"000 0"000
-0.159 -0"206 -0"264 -0.327 -0"399
-0128 0"147 0"169 -0" 196 -0"230
0-151 -0-193 -0.248 0.311 0.385
Numerical modeling of free-drainage samplers
255
Table 5. Comparison of simulated results for different hydraulic models of the same silt loam (Soils 5, 6 and 7) Input flux
90% 80% 70% 60% 50% 40% 30% 20%
K~ Ks Ks Ks K~ K~ K~ K~
Rate into brick (m 3 s-1 ) (xl0 -6)
Lowest h (m)
Soil 5
Soil 6
Soil 7
Soil 5
Soil 6
Soil 7
0"937 0"842 0"707 0"620 0.434 0'350 0"236 0'094
0 0 0 0 0 0 0 0
0"331 0"020 0"000 0"000 0"000 0"000 0'000 0"000
-0"043 -0'061 -0"089 -0"094 -0"099 -0"106 -0" 117 -0"135
-0"390 -0"678 - 1"00 - 1'38 - 1"82 -2'41 -3"13 -4"16
- 0 ' 152 - 0 ' 196 -0.251 -0'316 -0"393 -0"486 -0"607 -0"778
It is clear that although the two hydraulic functions are used to characterize the same soil, the simulation results are very different, especially the matric head distribution. Significant differences of simulation results are presented in Table 5 for three hydraulic functions (given as soils 5, 6, and 7 in Table 1) for the same soil. Figures 4(a), (b) and (c) show matric heads for soils 5, 6 and 7, with soil flux 6 x 10 6 m s 1. The three hydraulic functions give noticeably different distribution patterns of matric head. One explanation for these differences follows, For arbitrary hydraulic functions, Philip 13 and White and Sully 2°'21 have suggested a macroscopic capillary length Ac defined by: hwetK(h) dh Ac = IK(hwet) - K(hdry)]-I Jhdr•
(12)
If Ac calculated for two hydraulic properties is similar, then qualitatively their behavior is expected to be more alike than if not. If K(h) is of van Genuchten's form (eqn (8)), then 1/A c is an integral equivalent to c~ of Gardner's hydraulic function. Putting the hydraulic conductivity function of soil 3 into eqn (12), we have 0.41 for Ac. N o w using l/Ac = 2.44 for c~ and the other parameters of soil 4 in Table 1 (we refer to the soil as soil 4c), we compared simulation results with those of soil 3 and soil 4. Results for soil 4c in Table 4 are much closer to those of soil 3 than those of soil 4. In the same way, putting parameters of soil 5 (~ and m of v a n Genuchten's form) into eqn (12), the authors obtained an equivalent Gardner's c~ as 29.3. Comparing the c~ with those of soils 6 and 7 (0"397 and 2"38) indicates a discrepancy and suggests the three hydraulic functions may be expected to give different results.
Example 4. Wetting and drying Using soil 3 with an input flux of 7-8 × 1 0 - 6 m s -1 and initial matric head of - 1 m, simulations of non-steady state flow (wetting) were performed up to 20h. After 10 h, the flow field approaches steady state flow. For the wetting process, the lowest matric head appears at the bottom of the brick (left node), which ranges from - 1 . 1 0 m (1 h simulation) to - 0 . 1 5 8 m (20h simulation),
Before 5 h, there was no flow into the brick. After about 10 h, the flux rate into the brick is 1.6 x 10 -v m 3 s -l, i.e. 13.8 liters day 1. Figure 5 is a time series plot of water content versus depth at r = 0-15 m (recall that the radius of the brick is 0-15 m). A sharp decrease of water content on the edge of the brick is observed until a steady state moisture content distribution is achieved. Also with soil 3, simulations of non-steady state drying process were performed for zero input flux and initially saturated condition in the studied field. After about 0.2 h, there was no flow into the brick. Figure 6 shows water content as a function of depth and time at r = 0.15m. Again water content decreases sharply on the edge of the brick. Example 5. Sensitivity analyses The simulated results depend on the hydraulic functions. The following results show how sensitive the results are with variations of parameters in the hydraulic functions. For the VG type function, we choose soil 3. For the input parameters given in Table 1 and an input flux of 0.9Ks, the flux into the brick (Q) is 1.58 × 10-Vm3s l and the smallest matric head (h) is - 0 . 1 5 9 m in steady state. If the input parameters are varied, the flux into the brick and the lowest matric head may change. In order to measure the relative changes in Q and h with respect to relative changes in m and c~, the relative sensitivities, SQ,m, Sh,m, SQ,~ and Sh,~ are defined: 9
AQm SQ'm- AmQ
(13)
Ah m
Sh,m - Am h
(14)
SQ,~ - AQ c~
(15)
Sh,~ - Ah c~ (16) Ac~ h The numerical experiments show that as m increases, Q and h decrease. When m changes from 0-4 to 0"6, h changes f r o m - 0 . 1 2 1 m t o - 0 - 2 4 0 m ; SQ,m is within the range of -7-5 to - 9 . 5 and Sh,m changes from - 1 . 2 to
R. Zhang, A.W. Warrick & J.F. Artiola
256
1.0
xe-~ D--a 0.5- ~ ~--~
~
~.
~-E
O.O-
N
-0.5-
1 hr 2 hrs 5 hrs 10 hrs
-1.0-
0.30
oX
o.b
0.45
WATER CONTENT 2
(a)
"
"~'"~
Fig.
"~.~'x.
~'~"
/.~.~
i~
5.
Water content as a function of depth with time at r=0.15m(wetting).
05
"
T i° N
--1
,,,<'.~
. . " ~ ' ~ " Z~-~ <
" • .- ~,-
(b)
CO
T
Fig. 6. Water content as a function of depth with time at r = 0"15 m (drying).
I
-1.0
-2.2-2.6-
~e °
-3.0
(c)
"" '~"
<
Fig. 4. Matric heads of soil 5 (a), soil 6 (b) and soil 7 (c) for steady conditions.
'
0.35
i
0.45
,
I
0.55
'
0.65
m Fig. 7. Relationship between Sh,,,, and m (VG type).
Numerical modeling of free-drainage samplers -2"5. Figure 7 presents the relationship between Sh,m and m. As c~ increases, Q and h increase, but SQ,~ and Sh,~ decrease. When c~ changes from 0-7 to 1'3, Q changes from 0.656 x 10 -7 m 3 s -1 to 0-223 x 1 0 - 6 m 3 s - l ; h changes from - 0 . 1 7 8 m to - 0 . 1 4 8 m ; SQ,e¢ from 1-9 to 1-4; and Sh,~ from 0"40 to 0"23. For G R type function and a given input flux in steady state, values of m do not affect Q and h. However, increasing m makes h increase for the non-steady state, As ~ increases, h increases; Q also increases if there is flux into the brick. Figure 8 gives the relationships between ~ and h with input flux rates of 0"95 K s and 0-4 K s based on parameters of soil 6. For a smaller input flux, the rate of increase of h versus ~ is larger, i.e. Sh,~ is larger,
S U M M A R Y AND C O N C L U S I O N S Numerical simulations were used to evaluate the performance of a free-drainage water sampler. This type of sampler can be a hollow glass brick or alternatively an open pan. Under unsaturated conditions, hydraulic functions of van Genuchten 17 as well as Gardner s and Russo 15 were considered for steady and unsteady flow. The flux into the brick sampler and the distribution of matric heads in the studied domain depends on the input flux at the soil surface, the hydraulic functions, and the size of the brick. Generally, as input flux increases, the flow into the brick increases. As the radius of the brick increases, the flux into the brick increases and the smallest matric head in the profile shows an algebraic decrease. Sensitivity analysis of the hydraulic functions was discussed. For instance, with the van Genuchten model, SQ,m is between - 7 . 5 and - 9 . 5 which indicates that as m increases the flow rate into the brick will decrease. In 0
- 1 ~. / / /
"~ z-2
--5
@
)
1
J
2
~-~ q = 0.95 Ks ~< )K~ q = 0.4 Ks I 5 4 5 O~
Fig. 8. Relationship between the smallest matric head and c~ with different input flux (GR type).
257
other words, a relative change in the input m gives an 8to 9-fold relative change in the output of the flux if other parameters are unchanged. The results show that the water collection is sensitive to the hydraulic properties of the soil. This is to be expected for any collection device. That different soil hydraulic functions for the same soil give different numerical results, points to the need for criteria to choose equivalent parameters of alternative forms in modeling. Here we have made an effort to quantify the effects. A unique feature of the modeled system is that the flow regime would be totally unsaturated if the sampler were not present. The sampler induces the saturated conditions necessary for collection. The device itself is effective for intercepting preferential flow although that effect is not included in the present simulations. A significant amount of soil water may be collected with the sampler under certain conditions, such as 14 liters per day shown in the simulation which supports anecdotical data. Although such devices are in use, the validation by carefully controlled field experiments was beyond the scope of this study.
ACKNOWLEDGMENTS Contribution from the Arizona Agricultural Experiment Station. Support in part by Western Regional Research Project W-128.
REFERENCES 1. Babu, D.K., Steady state moisture flows in unsaturated soil containing a plane, isolated, impermeable barrier. Adv. Water Resources, 2, (1979) 35- 46. 2. Celia, M.A., Bouloutas, E.T. & Zarba, R.L., A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res., 26, (1990) 1483 96. 3. DaSilva, E.M., Analysis of furrow irrigation uniformity as affected by furrow spacing. PhD dissertation, Tucson, University of Arizona, 1990, 449 pp. 4. Everett, L.G., Wilson, L.G. & Hoylman, E.W., Vadose zone monitoring for hazardous wastes sites. US Environmental Protection Agency. Office of Research and Development, Las Vegas, NV 89114, 1983, KT-82-018(R). 5. Gardner, W.R., Some steady state solutions of the unsaturated moisture flow equation with application to evaporation with a water table. Soil Sci., 85, (1958) 228-32. 6. Hills, R.G., Porro, I., Hudson, D.B. & Wierenga, P.J., Modeling one-dimensional infiltration into very dry soils. 1: Model development and evaluation, Water Resour. Res., 25, (1989) 1259-69. 7. Kool, J.B., Parker, J.C. & van Genuchten, M.Th., Determining soil hydraulic properties from one-step outflow experiments by parameter estimation. I. Theory and numerical studies. Soil Sci. Soc. Am. J., 49, (1985) 1348 54. 8. Litaor, M.I., Review of soil solution samplers. Water Resour. Res, 24, (1988) 727-33.
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R. Zhang, A . W . Warrick & J.F. Artiola
9. McCuen, R.H., A sensitivity and error analysis of procedures used for estimating evaporation and evapotranspiration models. Water Resources Bull., 12, (1974) 486-98. 10. Moore, R.E., Water conduction from shallow water tables, Hilgardia, 12, (1939) 383-425. 11. Morrison, R.D. & Lowery, B., Sampling radius of porous cup sampler: Experimental results. Ground Water, 28, (1990) 262-7. 12. Narasimhan, T.N. & Dreiss, S.J., A numerical technique for modeling transient flow of water to a soil water sampler. Soil Sci., 141, (1986) 230-6. 13. Philip, J.R., Reply to comments on 'Steady infiltration from spherical cavities'. Soil Sci. Soc. Am. J., 49, (1985) 788 9. 14. Philip, J.R., The scattering analog for infiltration in porous media. Rev. Geophy., 27, (1989)431 48. 15. Russo, D., Determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties. Water Resour. Res., 24, (1988) 453--9. 16. Van der Ploeg, R.R. & Beese, F., Model calculations for
17. 18. 19. 20. 21.
the extraction of soil water byceramic cups and plates. Soil Sci. Soc. Am. J., 41, (1977) 466-70. van Genuchten, M.Th., A close-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 41, (1980) 466 70. Warrick, A.W. & Amoozegar-Fard, A., Soil water regimes near porous cup water samplers. Water Resour. Res., 13, (1977) 203 7. Warrick, A.W., Lomen., D.O. & Yates, S.R., A generalized solution to infiltration. Soil Sci. Soc. Am. J., 49, (1985) 34 8. White, I. & Sully, M.J., Macroscopic and microscopic capillary length and time scales from infiltration. Water Resour. Res., 23, (1987) 1514-22. White, I. & Sully, M.J., Field characteristics of the macroscopic capillary length or alpha parameter. In Int. Conf. and Workshop on the Validation of Flow and Transport Models for the Unsaturated Zone, Las Cruces, New Mexico, ed. P.J. Wierenga & D. Bachelet. New Mexico State University, Las Cruces, New Mexico, 1988, pp. 517 24.
Numerical modeling of free-drainage water samplers in the shallow vadose zone R. Zhang*, A.W. Warrick & J.F. Artiola Department of Soil and Water Science, University of Arizona, Tucson, Arizona 85721, USA
A finite element model was used to simulate the performance of a free-drainage sampler. This device may be used to collect soil water by free drainage in the shallow vadose zone. A unique feature of this sampling device is that it serves as a barrier to flow and creates a locally saturated region in an otherwise unsaturated system. Steady and unsteady flows were modeled assuming the hydraulic functions of van Genuchten and Gardner. Results include flux into the sampler and distribution of hydraulic head as functions of soil hydraulic properties, background soil flux and sampler size. Key words. free-drainage water sampler, unsaturated flow, modeling.
INTRODUCTION
Recently, the authors have used a hollow glass brick sampler to collect water by free drainage. The sampler has dimensions of 0.3m x 0-3m x 0.1 m and is placed at about 1-5 m depth in an agricultural field. Water enters through perforations on the top (one of the 0.3 m x 0.3 m faces) by gravity and is removed from the hollow chamber when desired by applying a temporary suction. Preliminary data indi-cate that the sampler may indeed collect significant amounts of liquid in sand layers underlying heavier (loamy) agricultural soils. This may be due to the relatively fast water fluxes that can occur in sand under unsaturated conditions and low tensions. As the heavier textured upper layers become saturated, the underlying sand provides easy vertical flow to the samplers. When properly installed, the free-drainage samplers have several major advantages over porous cups. They provide continuous liquid collection, intercept preferential flow paths, simplify sample retrieval (it is less time-dependent), and give larger sample volumes (thereby allowing for the analysis of more parameters), and result in minimal chemical alteration of soil-pore liquid sample. The objective of this study is to model the flow performance of a free-drainage water sampler in the unsaturated zone. Both steady and unsteady flow regimes will be considered. The sampling device itself serves as an obstacle to flow which raises the soil water pressure and allows water entry into the sampling device by gravity. The analysis is limited to Darcian flow and preferential flow zone such as macropores are not considered in this study. Direct field validation is beyond the scope of this paper.
Monitoring devices are used to detect contaminants in the shallow vadose zone. Such instruments are of value for both shallow and deep water table conditions, Sampling in the unsaturated region above a deep water table can result in years of advance warning since the associated contaminant travel time can be substantial, The design, use and performance of various types of vadose monitoring equipment have been reviewed by Everett et al. 4 and Litaor. 8 Tension infiltrometers have been m o d e l e d b y W a r r i c k and Amoozegar-Fard, 18 Van der Ploeg and Beese 16 and Narasimhan and Dreiss. 12 Their results include travel time and radius of influence around the sampler. Experimental results by Morrison and Lowery II showed the radius of influence to be less than 50cm in medium grain sand. A free-drainage water sampler has features in c o m m o n with the 'seepage entry problem' of Babu I and Philip) 4 In this context, the primary consideration was water-tightness of subterranean cavities, such as for tunnels or waste burial. Generally, pressure will increase more for flow around bluntly shaped objects than for objects sharply pointed into an oncoming flow. Increases in pressure will result in a tendency for saturated regions to develop (such as at the top of a sampling pan) and the potential for free-drainage to occur, *Present address: US Salinity Laboratory, USDA, ARS,4500 Glenwood Drive, Riverside, California 92501, USA. Advances in Water Resources 0309-1708/92/$05.00
~"5 1992 Elsevier Science Publishers Ltd. 251
R. Zhang, A.W. Warrick & J.F. Artiola
252 THEORY
and the van G e n u c h t e n (VG) type is:
Consider Richards' equation for unsaturated water flow expressed as:
ot - r Or
-Or + Oz
Oz
Oz
(1)
where 0 is the volumetric water content, h is matric potential (as a length), r a radial coordinate, z depth, t time and K the unsaturated hydraulic conductivity. B o u n d a r y conditions are (see Fig. 1): h
h0 or q
q = 0
on AB
(2)
across BC, DE, EF, F G and A H
O(h + z) _ 1 Oz q
K(ho)
0orh =0
on G H on C D
K~ exp h*
@ = [1 + Ih*l ~/
(9)
where K s is the saturated hydraulic conductivity. The h* and O are defined by: h* = ~h
10)
0-Or (-) - 0s - 0~
11)
In eqns (10) and (11), (~ is a constant, 0s the saturated water content and Or the residual water content.
(4)
CALCULATIONS
(5)
A finite element formulation was chosen to approximate the Richards equation because of the ease and flexibility for assigning b o u n d a r y conditions and the refinement of the numerical approximations. The procedure utilized the Galerkin m e t h o d of weighted residuals and closely followed the p r o g r a m s developed by DaSilva 3 for infiltration from an irrigation furrow. Features of the c o m p u t e r p r o g r a m included a mixed f o r m a t i o n for water content and pressure head
(6)
O = [ e x p ( h * / 2 ) ( 1 - h * / 2 ) ] :/(m+2)
B
(8)
(3)
Hydraulic functions used in this p a p e r are given by G a r d n e r 5 and Russo 15 and by van Genuchten. 17 The G a r d n e r and Russo ( G R ) f o r m is: K
K = KsO° 5[1 - (1 - o1/m)m]:
(7)
A
q = const
B -f
*'7
AND RESULTS
q : const --I~I--TI ' , , "II I,'
, Zm~×
~
I ft f
•
Origi I"1
z
,
C
[
i
I
i
r
D
Zc
,/
F
/
r
G~
H
h = const f'rnax
(a)
~ Zmax
•
1
G
h const (b)
Fig. 1. The flow region without flow (a) and with flow into the brick (b).
2H
Numerical modeling of free-drainage samplers suggestedbyCeliaetal. 2 forwhichmassconservation was found to be generally easier to maintain than using
z, /
programa purely pressure-head C up a formulation. n tops boundary eThe algorithm t be run in either transient or steady-state conditions. The is for the condition of fixed head or fixed flux (along AB of Fig. 1). For the following simulations, only the fixed flux boundary is used because the two types of boundary conditions give nearly identical results. Most of the simulations are for a space domain 0 to 1 m (rmax) in the r direction and in the z direction from l m (Zmax) above the brick to l m below the brick (see Fig. 1). The sizes of rmax and Zma× are chosen large enough to satisfy the boundary conditions. For example, the setup of the domain leads to a flow rate at the right boundary which is at least 4 orders of magnitude smaller than the input flow rate. The brick is simulated by a cylinder with radius 0.15 m and height 0.10 m for the most part, although some calculations are with other sizes. The number of nodes on the top, at the bottom and on the side of the brick was specified as well as the total number of horizontal and vertical nodes, The vertical nodes were computed as an arithmetic progression by specifying the minimal spacing (taken just above the top and below the bottom of the brick) and Zma× with a specified number of nodes. The horizontal nodes to the right of brick varied by an arithmetic progression within rm~x. Triangular elements were formed in the flow region by drawing diagonals across the rectangles formed by the nodes (see Fig. 2). Typically 400 nodes and 700 elements were used. Various types of soil were chosen for this study. Soil parameters used are listed in Table 1. Four sets of hydraulic functions are of the van Genuchten form and three of the Gardner and Russo form according to the given references. Soils 3 and 4 are actually one soil fit with alternative hydraulic properties; likewise, soils 5, 6 and 7 are one soil described by three alternative hydraulic functions or parameters,
~ C~
253
/ /
/
/
~
6
c
/////// '
Hollow Cylinder
/ / / ( /"
f" //
F¢~~/ $
/
/
/
/
'///
/
~
/ /
•
h =O
o NO F l o w Fig. 2. The elements and nodes around the brick.
by using q = 0 across CD (see Fig. 1(a)). If the input flux is large enough at the top of the studied region, a positive matric head will develop at the top of the glass brick and a pressure deficit will form at the bottom of the brick. Figure 3 shows matric heads at top of the brick which change with time for a non-steady state simulation. Parameters of the hypothetical sandy loam soil (soil 3) are used with an input flux along AB equal to 7-8 × 1 0 - 6 m s I (90% of Ks). Within five hours, a positive matric head results along 16% of the top area. After 20h, a steady state is established and a positive matric head is developed in 81% of the area. The different streamline patterns are compared in Figs l(a) and (b) schematically without and with flow into the brick, respectively. Example 2. Steady state with flow into the brick To simulate the hollow glass brick (see Fig. l(b)), the authors initially set the boundary condition at the top of the brick as no flow entering (q = 0). During the iterations, if the matric head becomes positive at some of the nodes at the top of the brick, the boundary condition at these nodes was changed to fixed matric head of 0, while other nodes remained as a no flow
Example 1. Steady and non-steady state without flow into the brick If flow cannot enter the glass brick, it is set as a barrier
Table 1. Soil hydraulic functions used Soil
Functions
~
m
0r
0s
(m -1 )
Ks (ms -1 )
1. 2. 3.
Yolo light clay 1°19 Glendale clay loam 6 Hypothetical sandy loam 7j5
VG VG VG
1.50 1"04 1'00
0-500 0"283 0500
0.124 0106 0-170
0.495 0"469 0"470
1.23 x 10 7 1-52 x 10 6 8-69 x 10 6
4. 5. 6. 7.
Hypothetical sandy loam :15 Silt loam soil :uS Silt loam soil v15 Silt loam soil 715
GR VG GR GR
5"31 3"37 0-397 2'38
0 0"500 0 5"14
0'186 0'115 0.152 0"154
0"470 0"338 0-388 0-388
8-69 x 1"50 x 1.50 x 1-50 x
10-6 10 5 10 s 10 5
The Gardner 5 and R u s s o 15 relationships ofeqns (6) and (7) are denoted by GR; those of van Genuchten w by VG. Soils 3 and 4 are for a soil but with different parameters; likewise, soils 5, 6 and 7 represent different models for the same soil.
R. Zhang, A.W. Warrick & J.F. Artiola
254 0.1 0-
oo5 ~. ,~ ~_
~
~ ~
ooo ~
...... ~
\_ - ~ _
-0.05
~. 5 hours
Steady -0.10
0.00
overall soil flow rate for soil 2. Flow occurs into some of the six nodes at the top of the brick, depending upon the input flux. Higher input fluxes produce more saturated nodes. The lowest pressure appears in the left corner at the bottom of the brick (node F in Fig. 2). To examine the effects of truncation errors, the authors doubled the number of nodes across the top of the brick and refined other nodes. Several tests indicate that the differences of the rate into the brick are within 8% due to refinements of nodes. Table 3 shows the results with different sizes of bricks (radii of 0.1, 0.15 and 0.2m for soil 1). As the size of a brick increases, more flow goes into the brick and a larger pressure deficit develops below the brick for the
,
o.o4
State v
, 0.12
0.o8
\.
o. 6
r (m) Fig. 3. Matric heads at the top of the brick of non-steady state for soil 3 when the brick is impervious,
same input flux. When the input flux decreases, there is no flow into a brick with smaller radius.
boundary. If a node has a resulting flow back into the flow system, the node is switched back to that for a noflow boundary. Figure 2 gives an example showing nodes with h - 0 (nodes 1-4) and nodes without flow (nodes 5 and 6) at the brick top, also showing the elements around the brick, Table 2 gives flow rate into the brick as a function of
Example 3. Simulations with different hydraulic functions
Here the authors illustrate how the choice of hydraulic properties affects simulation results. Soils 3 and 4 are actually the same sandy loam soil but fit by different parameters (RussolS). Based on the parameters, simulation results for soil 3 and soil 4 are compared in Table 4.
Table 2. Rate into the brick as function of input flux for soil 2
Input flux
Input rate
Rate into brick
(xl0 -6)
(xl0 7)
3"818 3"343 2"865 2'388 1.910 0"955
0'792 0"599 0"432 0'215 0'048 0'000
(m3 s-l)
80% Ks 70% Ks 60% K~ 500/0 K~ 40'% K~ 20% K~
(m3s 1)
Nodes for h = 0
Lowest h (m)
6 6 5 5 3 0
-0"108 0" 114 0'127 -0" 146 -0" 177 0"331
Table 3. Rate into the bricks of different size as function of soil flux for soil 1 (brick radii are 0.1, 0"15 and 0"2m)
Input rate (m3s-1) (x10-7) 3-671 3"478 3"284 3"091 2"898 2'705
Rate into brick (m3s I)(xlO 8)
Lowest h (m)
0"1
0'15
0"2
0"203 0' 104 0"034 0"000 0'000 0"000
0' 529 0"366 0"221 0" 111 0-029 0-000
1"00 0"778 0"561 0' 367 0"216 0"090
0'1
0"15
0'2
0"091 --0" 105 --0" 121 --0"138 --0" 157 --0" 177
--0"129 0"143 0"157 --0"173 0"190 --0'209
0' 166 --0' 180 --0' 194 --0'209 --0'225 0'242
Table 4. Comparison of simulated results for different hydraulic models of the same sandy loam (soils 3 and 4)
Input rate (m3s 1)
Rate into brick (m3s -1) (xl0 6)
Lowest h (m)
(xl0 5)
Soil 3
Soil 4
Soil 4c
Soil 3
Soil 4
Soil 4c
4.241 3'770 3'299 2.827 2.356
0"158 0"000 0-000 0-000 0-000
0.334 0"201 0"080 0'003 0"000
0.199 0"015 0"000 0"000 0"000
-0.159 -0"206 -0"264 -0.327 -0"399
-0128 0"147 0"169 -0" 196 -0"230
0-151 -0-193 -0.248 0.311 0.385
Numerical modeling of free-drainage samplers
255
Table 5. Comparison of simulated results for different hydraulic models of the same silt loam (Soils 5, 6 and 7) Input flux
90% 80% 70% 60% 50% 40% 30% 20%
K~ Ks Ks Ks K~ K~ K~ K~
Rate into brick (m 3 s-1 ) (xl0 -6)
Lowest h (m)
Soil 5
Soil 6
Soil 7
Soil 5
Soil 6
Soil 7
0"937 0"842 0"707 0"620 0.434 0'350 0"236 0'094
0 0 0 0 0 0 0 0
0"331 0"020 0"000 0"000 0"000 0"000 0'000 0"000
-0"043 -0'061 -0"089 -0"094 -0"099 -0"106 -0" 117 -0"135
-0"390 -0"678 - 1"00 - 1'38 - 1"82 -2'41 -3"13 -4"16
- 0 ' 152 - 0 ' 196 -0.251 -0'316 -0"393 -0"486 -0"607 -0"778
It is clear that although the two hydraulic functions are used to characterize the same soil, the simulation results are very different, especially the matric head distribution. Significant differences of simulation results are presented in Table 5 for three hydraulic functions (given as soils 5, 6, and 7 in Table 1) for the same soil. Figures 4(a), (b) and (c) show matric heads for soils 5, 6 and 7, with soil flux 6 x 10 6 m s 1. The three hydraulic functions give noticeably different distribution patterns of matric head. One explanation for these differences follows, For arbitrary hydraulic functions, Philip 13 and White and Sully 2°'21 have suggested a macroscopic capillary length Ac defined by: hwetK(h) dh Ac = IK(hwet) - K(hdry)]-I Jhdr•
(12)
If Ac calculated for two hydraulic properties is similar, then qualitatively their behavior is expected to be more alike than if not. If K(h) is of van Genuchten's form (eqn (8)), then 1/A c is an integral equivalent to c~ of Gardner's hydraulic function. Putting the hydraulic conductivity function of soil 3 into eqn (12), we have 0.41 for Ac. N o w using l/Ac = 2.44 for c~ and the other parameters of soil 4 in Table 1 (we refer to the soil as soil 4c), we compared simulation results with those of soil 3 and soil 4. Results for soil 4c in Table 4 are much closer to those of soil 3 than those of soil 4. In the same way, putting parameters of soil 5 (~ and m of v a n Genuchten's form) into eqn (12), the authors obtained an equivalent Gardner's c~ as 29.3. Comparing the c~ with those of soils 6 and 7 (0"397 and 2"38) indicates a discrepancy and suggests the three hydraulic functions may be expected to give different results.
Example 4. Wetting and drying Using soil 3 with an input flux of 7-8 × 1 0 - 6 m s -1 and initial matric head of - 1 m, simulations of non-steady state flow (wetting) were performed up to 20h. After 10 h, the flow field approaches steady state flow. For the wetting process, the lowest matric head appears at the bottom of the brick (left node), which ranges from - 1 . 1 0 m (1 h simulation) to - 0 . 1 5 8 m (20h simulation),
Before 5 h, there was no flow into the brick. After about 10 h, the flux rate into the brick is 1.6 x 10 -v m 3 s -l, i.e. 13.8 liters day 1. Figure 5 is a time series plot of water content versus depth at r = 0-15 m (recall that the radius of the brick is 0-15 m). A sharp decrease of water content on the edge of the brick is observed until a steady state moisture content distribution is achieved. Also with soil 3, simulations of non-steady state drying process were performed for zero input flux and initially saturated condition in the studied field. After about 0.2 h, there was no flow into the brick. Figure 6 shows water content as a function of depth and time at r = 0.15m. Again water content decreases sharply on the edge of the brick. Example 5. Sensitivity analyses The simulated results depend on the hydraulic functions. The following results show how sensitive the results are with variations of parameters in the hydraulic functions. For the VG type function, we choose soil 3. For the input parameters given in Table 1 and an input flux of 0.9Ks, the flux into the brick (Q) is 1.58 × 10-Vm3s l and the smallest matric head (h) is - 0 . 1 5 9 m in steady state. If the input parameters are varied, the flux into the brick and the lowest matric head may change. In order to measure the relative changes in Q and h with respect to relative changes in m and c~, the relative sensitivities, SQ,m, Sh,m, SQ,~ and Sh,~ are defined: 9
AQm SQ'm- AmQ
(13)
Ah m
Sh,m - Am h
(14)
SQ,~ - AQ c~
(15)
Sh,~ - Ah c~ (16) Ac~ h The numerical experiments show that as m increases, Q and h decrease. When m changes from 0-4 to 0"6, h changes f r o m - 0 . 1 2 1 m t o - 0 - 2 4 0 m ; SQ,m is within the range of -7-5 to - 9 . 5 and Sh,m changes from - 1 . 2 to
R. Zhang, A.W. Warrick & J.F. Artiola
256
1.0
xe-~ D--a 0.5- ~ ~--~
~
~.
~-E
O.O-
N
-0.5-
1 hr 2 hrs 5 hrs 10 hrs
-1.0-
0.30
oX
o.b
0.45
WATER CONTENT 2
(a)
"
"~'"~
Fig.
"~.~'x.
~'~"
/.~.~
i~
5.
Water content as a function of depth with time at r=0.15m(wetting).
05
"
T i° N
--1
,,,<'.~
. . " ~ ' ~ " Z~-~ <
" • .- ~,-
(b)
CO
T
Fig. 6. Water content as a function of depth with time at r = 0"15 m (drying).
I
-1.0
-2.2-2.6-
~e °
-3.0
(c)
"" '~"
<
Fig. 4. Matric heads of soil 5 (a), soil 6 (b) and soil 7 (c) for steady conditions.
'
0.35
i
0.45
,
I
0.55
'
0.65
m Fig. 7. Relationship between Sh,,,, and m (VG type).
Numerical modeling of free-drainage samplers -2"5. Figure 7 presents the relationship between Sh,m and m. As c~ increases, Q and h increase, but SQ,~ and Sh,~ decrease. When c~ changes from 0-7 to 1'3, Q changes from 0.656 x 10 -7 m 3 s -1 to 0-223 x 1 0 - 6 m 3 s - l ; h changes from - 0 . 1 7 8 m to - 0 . 1 4 8 m ; SQ,e¢ from 1-9 to 1-4; and Sh,~ from 0"40 to 0"23. For G R type function and a given input flux in steady state, values of m do not affect Q and h. However, increasing m makes h increase for the non-steady state, As ~ increases, h increases; Q also increases if there is flux into the brick. Figure 8 gives the relationships between ~ and h with input flux rates of 0"95 K s and 0-4 K s based on parameters of soil 6. For a smaller input flux, the rate of increase of h versus ~ is larger, i.e. Sh,~ is larger,
S U M M A R Y AND C O N C L U S I O N S Numerical simulations were used to evaluate the performance of a free-drainage water sampler. This type of sampler can be a hollow glass brick or alternatively an open pan. Under unsaturated conditions, hydraulic functions of van Genuchten 17 as well as Gardner s and Russo 15 were considered for steady and unsteady flow. The flux into the brick sampler and the distribution of matric heads in the studied domain depends on the input flux at the soil surface, the hydraulic functions, and the size of the brick. Generally, as input flux increases, the flow into the brick increases. As the radius of the brick increases, the flux into the brick increases and the smallest matric head in the profile shows an algebraic decrease. Sensitivity analysis of the hydraulic functions was discussed. For instance, with the van Genuchten model, SQ,m is between - 7 . 5 and - 9 . 5 which indicates that as m increases the flow rate into the brick will decrease. In 0
- 1 ~. / / /
"~ z-2
--5
@
)
1
J
2
~-~ q = 0.95 Ks ~< )K~ q = 0.4 Ks I 5 4 5 O~
Fig. 8. Relationship between the smallest matric head and c~ with different input flux (GR type).
257
other words, a relative change in the input m gives an 8to 9-fold relative change in the output of the flux if other parameters are unchanged. The results show that the water collection is sensitive to the hydraulic properties of the soil. This is to be expected for any collection device. That different soil hydraulic functions for the same soil give different numerical results, points to the need for criteria to choose equivalent parameters of alternative forms in modeling. Here we have made an effort to quantify the effects. A unique feature of the modeled system is that the flow regime would be totally unsaturated if the sampler were not present. The sampler induces the saturated conditions necessary for collection. The device itself is effective for intercepting preferential flow although that effect is not included in the present simulations. A significant amount of soil water may be collected with the sampler under certain conditions, such as 14 liters per day shown in the simulation which supports anecdotical data. Although such devices are in use, the validation by carefully controlled field experiments was beyond the scope of this study.
ACKNOWLEDGMENTS Contribution from the Arizona Agricultural Experiment Station. Support in part by Western Regional Research Project W-128.
REFERENCES 1. Babu, D.K., Steady state moisture flows in unsaturated soil containing a plane, isolated, impermeable barrier. Adv. Water Resources, 2, (1979) 35- 46. 2. Celia, M.A., Bouloutas, E.T. & Zarba, R.L., A general mass-conservative numerical solution for the unsaturated flow equation. Water Resour. Res., 26, (1990) 1483 96. 3. DaSilva, E.M., Analysis of furrow irrigation uniformity as affected by furrow spacing. PhD dissertation, Tucson, University of Arizona, 1990, 449 pp. 4. Everett, L.G., Wilson, L.G. & Hoylman, E.W., Vadose zone monitoring for hazardous wastes sites. US Environmental Protection Agency. Office of Research and Development, Las Vegas, NV 89114, 1983, KT-82-018(R). 5. Gardner, W.R., Some steady state solutions of the unsaturated moisture flow equation with application to evaporation with a water table. Soil Sci., 85, (1958) 228-32. 6. Hills, R.G., Porro, I., Hudson, D.B. & Wierenga, P.J., Modeling one-dimensional infiltration into very dry soils. 1: Model development and evaluation, Water Resour. Res., 25, (1989) 1259-69. 7. Kool, J.B., Parker, J.C. & van Genuchten, M.Th., Determining soil hydraulic properties from one-step outflow experiments by parameter estimation. I. Theory and numerical studies. Soil Sci. Soc. Am. J., 49, (1985) 1348 54. 8. Litaor, M.I., Review of soil solution samplers. Water Resour. Res, 24, (1988) 727-33.
258
R. Zhang, A . W . Warrick & J.F. Artiola
9. McCuen, R.H., A sensitivity and error analysis of procedures used for estimating evaporation and evapotranspiration models. Water Resources Bull., 12, (1974) 486-98. 10. Moore, R.E., Water conduction from shallow water tables, Hilgardia, 12, (1939) 383-425. 11. Morrison, R.D. & Lowery, B., Sampling radius of porous cup sampler: Experimental results. Ground Water, 28, (1990) 262-7. 12. Narasimhan, T.N. & Dreiss, S.J., A numerical technique for modeling transient flow of water to a soil water sampler. Soil Sci., 141, (1986) 230-6. 13. Philip, J.R., Reply to comments on 'Steady infiltration from spherical cavities'. Soil Sci. Soc. Am. J., 49, (1985) 788 9. 14. Philip, J.R., The scattering analog for infiltration in porous media. Rev. Geophy., 27, (1989)431 48. 15. Russo, D., Determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties. Water Resour. Res., 24, (1988) 453--9. 16. Van der Ploeg, R.R. & Beese, F., Model calculations for
17. 18. 19. 20. 21.
the extraction of soil water byceramic cups and plates. Soil Sci. Soc. Am. J., 41, (1977) 466-70. van Genuchten, M.Th., A close-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J., 41, (1980) 466 70. Warrick, A.W. & Amoozegar-Fard, A., Soil water regimes near porous cup water samplers. Water Resour. Res., 13, (1977) 203 7. Warrick, A.W., Lomen., D.O. & Yates, S.R., A generalized solution to infiltration. Soil Sci. Soc. Am. J., 49, (1985) 34 8. White, I. & Sully, M.J., Macroscopic and microscopic capillary length and time scales from infiltration. Water Resour. Res., 23, (1987) 1514-22. White, I. & Sully, M.J., Field characteristics of the macroscopic capillary length or alpha parameter. In Int. Conf. and Workshop on the Validation of Flow and Transport Models for the Unsaturated Zone, Las Cruces, New Mexico, ed. P.J. Wierenga & D. Bachelet. New Mexico State University, Las Cruces, New Mexico, 1988, pp. 517 24.