- Email: [email protected]
Pesticide leaching in heterogeneous soils with oscillating flow: approximations
The Science of the Total Environment, 132 (1993) 167-179 Elsevier Science Publishers B.V., Amsterdam
167
Pesticide leaching in heterogeneous soils with oscillating flow: approximations S.EoA.T.M. van der Zee a and J.J.T.I. Boesten b
~Dept. Soil Science and Plant Nutrition, Agricultural University, P.O. Box 8005, 6700 EC Wageningen, Netherlands i'The Winand Staring Centre for Integrated Land, Soil and Water Research, P.O. Box 125, 6700 A C Wageningen, Netherlands ABSTRACT Pesticide screening models are used to assess which pesticides may leach in significant emounts into ground water. In this paper we consider the effects of transient flow and spatial variability of soil properties on the leached fraction of pesticides. Transient flow is approximated with sinusoidal variations of the flow velocity. We show that the main effect is due to the difference with respect to the residence time in soil, compared with the case of steady flow. This effect i~ most profound for pesticides with small retardation factors. Spatial variability of physical and (bio)chemical parameters is shown to lead to larger leached fractions than in the case of a homogeneous flow domain. Several effects are illustrated with analytical approximations.
Key words: pesticides; leaching; soil; heterogel city; oscillating flow
INTRODUCTION
To protect the quality of drinking water, standards have beea set for the maximum permissible pesticide concentration by the Council of the European Communities (1980). Concentr~Ltions may not exceed 0.1 mg m -3. This constraint implies that both old and, newly developed pesticides should be screened to assess whether their leaching potential may threaten the quality of water intended for human consumption. If that appears to be the case, such pesticides may have to be prohibited. Screening models as developed by Jury et al. (1983), Rao et al. (1985) and Boesten and Van der Linden (1991) characterize the leaching potential in terms of adsorption and transformation parameters. In view of spatial variability, Jury and Gruber (1989) took randomness of flow and pore scale dispersion into account in an analytical screening model. Because also adsorption and uansformation rates may be expected to vary spatially (Walker O348-9697/93/$06.00 © 1993 Elsevie: Science Publishers B.V. All fights reserved
168
S,E.A.T.M. VAN DER ZEE AND J.J.T.I. BOESTEN
and Brown, 1983; Elabd et al., 1986; Wood et al., 1987; Boekhold et al., 1991), random variation of the (bio)chemical parameters was considered in the screening model of Van der Zee and Boesten (1991). They illustrated that due to spatial variability, significantly more pesticide may leach than is expected when heterogeneity is ignored. In this paper we reconsider the model developed earlier (Van der Zee and Boesten, 1991) to evaluate the appropriateness of the assumed steady state flow. Because of daily and seasonal fluctuations of infiltration and evapotranspiration, the assumption is obviously an approximation. THEORY
Mathematical description Solute transport can be described for the one-dimensional case with the mass balance equation given by
OOc O[ ~c] Opq + ~ =~ OD O---t- Ot Oz -ff-Z-Z
OOvc Oz
S~
(1)
In Eqn. 1, p is the dry bulk density, q is the adsorbed amount, t is time, 0 is the volumetric water fraction, c is the concentration, D is the diffusion/dispersion coefficient, z is depth, v is velocity and Ss is the pesticide sink term. Usually, pesticide adsorption can be described with the Freundlich equation
q = KFc n
(2)
where the parameter n is usually smaller than 1 (Calvot et al., 1980) and Kr is the Freundlich coefficient. The diffusion/dispersion coefficient (D) accounts for molecular diffusior.~ corrected for the tortuosity and for convective dispersion. Hence, we obtain for this coefficJient (D)
D = M)o + Ldv
(3)
where h is the tortuosity factor, Do is the diffusion coefficient in water and Ld is the dispersivity. The loss rate of a pesticide (Ss) may be described with a first-order transformation rate (St) of dissolved as well as adsorbed pesticide in combination with a first order (passive) crop uptake rate (Su). This leads to Ss = S, + Su =
+ Oc) + gS c
(4)
PESTICIDE LEACHING IN HETEROGENEOUSSOILS WITH OSCILLATING FLOW
169
where the half life (~1/2) is equal to ln(2)/~0, and describes (bio)chemical transformations. In Eqn. 4,/~0 is the transformation rate parameter, g is the transpiration stream concentration factor and Sw is the water uptake rate of plants. Effect o f oscillating flow For a non-uniform porous medium, such as most soils, oscillation of flow as well as the presence of immobile water leads to additional dispersion. In that case flow does not occur of all water present (0) but only of a mobile fraction (On). The remainder (i.e., O~= 0 - Ore) does not move. To estimate the effect oa dispersion only, a non-reacting solute may be considered in a flow field subject to sinusoidal fluctuations of v. Denoting the mean velocity in the medium by v and the mobile zone velocity by u (= Ov/Om), we assume a sinusoidal fluctuation with amplitude vA and frequency ,o. This yields v(t) = V + VA sin(~0t)= V + P
(5)
where P is the time-dependent part of the velocity. For spherical aggregates a quasi-steady state approximation was obtained for the correction factor (De) of the dispersion coefficient by Farrell and Larson (1973). The correction factor due to immobile water and fluctuating flow effects equals 2 2
D~ = -OiO r4D-~ oVA g2(w) w 2 {[Oig2(w)] 2 + [Oigl(w) + 0m]2]
(6)
where we have defined w = ro(to/2Dm) 1~2. Although it may not be obvious from Eqn. 6, (De) is positive. The functions gl(w) and g2(w) are given in the Appendix and Din= M)0. The volumetric water fractions in the mobile and immobile zones are denoted by Om and 0~, respectively, and inside the spherical immobile zones with radius (r0) we assume only diffusion occurs (with diffusion coefficient Din). Setting for the mean squared velocity of the oscillations:
< p 2 > = (2,x/tz)-i
v 2 sin2 (~0t) dt = 0.5 v2a
(7)
we may rewrite Eqn. 6 as
D~,= Oi 1"20 ElOm,O,w] o
Dm
(8)
170
S.E.A.T.M. VAN DER ZEE A N D J.J.T.i. BOESTEN
The function E{. } in Eqn. 8 is given by E{Om,O,w} - 21 g2(w): w" 02 { [Oig2(w)]2 + [Oigl(w)
+
0m]2} -l
(9)
The total effective dispersion coefficient (DE) is given by 0_
(10)
D E = ~ - D + De
when we assume all dispersional effects are additive using the velocity (u) for D in Eqn. 3. Completely in agreement with analyses by Bolt (1982) and Valocchi (1985), immobile water effects lead to enhanced dispersion. The correction term to Bolt's (!982) immobile water dispersion term (i.e. E {. }) is shown for some interesting cases in Fig. 1. We observe that E = 1/15 for steady flow at arbitrary values of 0 and 0m, but decreases as fluctuations have a higher frequency (or as w increases). Immobile water effects on dispersion therefore are most profound when flow is steady instead of oscillating.
Et
O.Ol
.~
t
~
,.
,\ k
o.~.\.. , o,ool
-
~. I"
!
o.5~
q~--~,.~~_.=.' 0.9 ~
I,
,': t
"~
ix'
:
I= i !
I
lO
1 o0
w Fig. I. Correction term El 8m; e; w} as a function of w = r0(~/2Dm)1,2, for various values of 0i indicated at the curves. The thin horizontal line corresponds with E = 1/15.
171
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
This could be expected as the immobile water by-pass becomes smaller when flow is oscillating and water comes in contact with the same soil several times. The largest dispersion in terms of peclet numbers (P) for a ground water level, L, is given by
vL
P=~ De
(11)
is for steady flow and yields for aggregated media P = 10-20 (Beek, 1979). For cracked soils P-values may be significantly smaller (e.g., P = 1-10). But for those soils Eqn. 8 does not commonly hold because in The Netherlands the ground water level L is usually shallow (0 < L < 1 m). Therefore quasisteady state is not attained.
Leachedfractionfor a homogeneouscolumn In the sequel we assume that we are primarily interested in the leached fraction of the applied quantity of pesticides. Some pesticides constitute a problem with regard to screening because leached fractions are close to the permissible level. For those pesticides uptake may be disregarded (Boesten and Van de~ Linden, 1991). The leached fraction depends on the residence time in the soil layer under consideration and the transformation rate parameter (~) For a homogeneous column, the residence time of pesticide is distributed due to pore scale dispersion (DE). The leached fraction (mF) is given by Van der Zee and Boesten (1991) for a layer of thickness (L) and a retardation factor (R) by
mE = exp
I --~-I P [~ 1 + 4I~oRL vP
1] 1
(12)
In Eqr.. 12 we recognize the mean residence time (r) in a layer of thickness (L), i.e., LR/V. Due to adsc~tion non-linearity and first-order transformation, c decreases as contact time increases and the concentration-dependent averaged retardation factor/~(z) is not easily evaluat~ ~ analytically. The retardation factor (/~(z)) is an average value experienced by a solute slug that has reached depth z. This factor increases when z increases and may be evaluated numerically with the implicit expression given by Bosma and Van der Zee (1991): P [ -- (n-l)t~R(z)z]l R(L)=-~1 1 L o [ I+-~KFC~-Iexp r
dz
assuming D = 0. Evaluation of Eqn. 13 has to be done numerically.
(13)
172
S.E.A.T.M. VAN DER ZEE AND £J.T.i. BOESTEN
The leached fraction (mF) depends on the residence time and is theret0re a complicated function of the flow velocity, depth and retardation. Fluctuations in v may give rise to smaller or larger residence times. To assess when such fluctuations are (relatively) most important we make an approximation by linearizing sorption. We set for c = 1 and a linear adsorption coefficient (ki) that (14a)
q = KF c '~ = kic
Because usually 0 < n < 1, the values of KF and ki differ. The linear retardation factor is p RI = 1 + -~- ki
(14b)
and is independent of the concentration. We need to know when th~ difference between the residence times with and without oscillating flow is relatively large. The residence time (70) for oscillating flow, starting at t = 0 at an arbitrary phase (b) of the oscillation, is given by
V"+ v~ sin(tot + b)
(15)
0
The value of 70 does not need to be equal to 7, because ~- is the long term (t ~. 70) velocity average. Hence,
I
7/7o = (V'7o)-~ T,'7o-
VAcos (W7o+ b) + VAcos (b)1
(16a)
This yields: 7/7o = 1
vA [cos (O:7o+ b) - cos (b)] ¢o~'T0
'
(16b)
Due to oscillating flow, the actual travel time (70) may be larger or smaller than the estimated travel time (7) assuming steady flow. i~" 7o is larger than 7, the time period during which degradation occur~ is larger than estimated for a steady flow screening model. Hence, this ~:ase does not pose problems in screening. However, when 7o is smaller ~lan 7, degradation will be smaller than in the case of steady flow, which implies a larger leached fraction than expected. This means ~hat pesticidc~ that are classified as safe may in practise lead to an unacceptable ground water quality. Such errors should
! 73
PESTICIDE LEACHING IN HETEROGEI~EOUS SOILS WITH OSCILLATING FLOW
1; / 'i;o 2-
---b=O "
"
b= ~2 ....... b = ~
1 / il
!I ."
."
---
b=3~/2
•
O
0
I
I
~
I
1
2
3
4
Fig. 2. The ratio ~/~0 as a function of ¢o~/2~r for different values of b as calculated from Eqn (I 6b) with v,~/~" = 1.
ratio 6 25
20
,o
5
0
0
!
I
I
0.5
1
1.5
2
Fig. 3. The ratio (6) of leached fractions as a function of (vAIF), V = 2 m/year. The ratio, 8, is defined as the leached fraction in case of oscillating flow divided by this fraction for steady flow (and equal to m r = 0.~)1). Parameters were Ri = 1, L = I m, ~ = 23.4 yeal-~, ~0 = 2~ year -= and b = 0.
174
S.E.A.T.M. VAN DER ZEE AND J.J.T.I. BOESTEN
be avoided. The situations in which ~'0 is much smaller than • may be identified from Fig. 2 (which gives the quotient ~'/~'0as a function of ~07/2~r/br vA/v = 1). The general trend in Fig. 2 is that the effect of oscillating flow decreases with increasing ~or/2~r. The quotient 7"/~'0is madmal (which implies maximal increase of leaching due to oscillating flow) when b = 7r/2 and co approaches zero. However, this is the not relevant limiting case of no oscillations. So most severe relevant effects occur roughly in the range 0. ! < ~0~'/27r < 1. So the effects of oscillating flow are most profound, when the frequency of oscillations (i.e., c0/2~-)is ef the order of magnitude of the inverse residence time. When we are interested in seasonal flow fluctuations (effect of application time), the maximal effect may be expected for ~ < 1 year. For a ground water level at 1 m this implies maximal effect for values of R~ close to unity. In this context it is worthwhile to observe that an important group of pesticides are relatively mobile. It may be derived from ~ig. 2 that daily flow fluctuations due to rainfall have only a very small effect: assuming ~ to be in the order of I year or higher, to~/2~r is in the order of 100 or higher. The effects can be readily evaluated for the case of convective-dispersive flow where we assume P = 10, rewriting Eqn. 12 according to Van der Zee and Boesten (1991)
mF=exp
Il
--~-P
P
+l-
]}
1
(17)
using for ~-* either ~ (no oscillations) or ~0 (given by Eqn. 15). We take as an example the effect of seasonal flow fluctuations (~0 = 2~ year ~) for a mobile pesticide (Rt = 1) under Dutch conditions ~- = 2 m/year based on an excess of rainfall of 400 mm and T = 0.2); we assume b = 0 which corresponds with application in autumn. By choosing # 0 = 23.4 year -~ we obtain a leached fraction mr = 0.001 for the steady flow situation. For the case of oscillating flow we may calculate the leached fraction for different values of vA. Observe that a seasonal reversal of the flow direction, which is relevant for Dutch conditions, requires that VA/V" > 1. For those situations the relative error is large (Fig. 3). This error is calculated as the leached fraction that results for oscillating flow diwded by the leached fraction (0.001) found assuming steady flow. That for mobile pesticides the transientness of flow may be important was shown by Boesten (1991) and Boesten and Van der Linden (1991). Their results revealed the importance of the moment of application, which would be far less important if steady flow conditions had been assumed.
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
l '7~
Leached fiaction for a heterogeneous field Whereas in the previous sections the variations of flow in time were shown to affect the leached fraction, the spatial variations may also have a significant ,effect. Assuming steady flow, it is still likely that soil propez~ies vary spatially in a field. This is the case with regard to, e.g. hydraulic conductivity, adsoq)tion and transformation/uptake parameters (Van der Zee, 1990). Due to this variability, the residence time varies spatially and also the contact time for transformation varies in different parts of the field. Assuming that k~ and ~ are both normally distributed with variation coefficients of 0.25 and f~.10, respectively, we can calculate the expel:ted leached fraction, In Fig. 4 we show tha~ these fractions may be quite different from those calc~!ated in the absence of spatial variability. An analytical approximation may be given for the case that X = #RL/v is normally distributed, Using the same approximations (steady flow, uniform
200 KOm (L/kg)l I O.OOt
/ //
t 1 I I I I I I i I I I /
100 -
I/ I
°'2/ 0
/
,
,
100
200
half-life (d) Fig. 4. Fraction of the applied amount of pesticide leached beyond I m depth as a function of the organic m~tter/water-distribution coefficient (Kom,vertical axis in L/kg) and the halflife due to trar,sfi)rmation (horizontal axis in days). Solid lines give the contours (fraction as a percentage indi,:ated at the lines) for zero variability as calculated by Boesten and Van der Linden (1991). Dashed lines represent the fraction using the same parameter values but with normally distributed £o,,, and ~o (variation coefficients of 25% and It~'/o,respectivelyl.
176
S.E.A.T.M.VANDERZEEANDJ.J.T.I. BOESTEN
properties in the vertical direction, no transversal interaction between different steamtubes) as were made by Van dcr Zee and Boesten (1991) we obtain
m~ = exp {-0.SPi
[~(4m.vlPf)
+ 1 -1]}
(18)
which represents the aerial average of the leached fraction. In Eqn. 18, mx is the mean (or expected) value of X. The parameter Pf is an apparent 'field ~cale' peclet-number that represents variability of X. It is related to the variation coefficient of X (i.e., ~x) by 2
vt~x= -~f [Pf -
1 + exp (-Pf)]
(19)
To evaluate the accuracy of the approximation of Eqns. 18 and 19, we show Q" as a function of the exponential argument of Eqn. 18 in Fig. 5. The ratio Q* ix defined as the ratio of leached fractions with (Eqn. 18) and without (Eqr,. i2) spatial variability (but whh pore scale dispersion as given by Boest,:~ and Van der Linden (1991)). Due to heterogeneity, Q* exceeds unity. ~" Also shown (Fig° 5)• is Q* as calculated from the numerical data of Boe..,en
Qt 6r
sL
/
/
4-
/ i /
$ //"
I
|
0 ~--
I
L
'
t
0
2
4
6
8
.....
J
10
ARG Fig. 5. Multiplication ;~ctor (Q*) for atrazine {(K~,,) = 60 dm3/kg; half.life 60 days) according to the analy~ica! solution (Eqns (18,19)). The parameter, Q*, is defined by the leached fraction estimated with a variationcoefficientof ,g.,,~of 25%, divided by the leached fraction for
zero variabilityoPore scale dis~rsion was accounted for.
PESTICIDE LEACHING IN HETEROGENEOUSSOILS WITH OSCILLATING FLOW
177
and Van der Linden (1991). The agreement appears to be rather good in view of the complexity of their numerical analysis. CONCLUSION
In thi~ paper we considered to what extent spatial variability and t~,ansient flow affect the leached fraction of pesticides applied to the soil surface. The leached fraction depends on ,the residence time in the vadose zone and the transformation rate of the pesticide. Transient flow was approximated with regular oscillations of the flow velocity around a mean (~') value. Compared with the case of steady flow in an aggregated medium, a decrease of the effective dispersion coefficient may be expected. Hence, in view of Eqn. 12, the leached fraction is expected to be larger for steady flow conditions. However, oscillating flow may also affect the mean residence time. This effect is larger when the pesticide is less subject to sorption and the retardation factor is closer to one. Also, this effect increases when the frequency is closer to the inversed mean residence time. Depending on the variations in v as a function of time, the effects may be large, leading to large//-values (being the ratio of leached fraction for oscillating |low divided by the leached fraction for steady flow). When we consider not a single soil column but a spatially variable field (with respect to v, Rt and/or LL we may expect profound differences between the expected leached fraction and the estimated leached fraction that neglects such spatial variations. This assessment, based on numerical results by ~oesten and Van der Linden (199i) that account for sorPtion aon~.inearity, transient flow. dispersion and variations of properties wi~h depth, is adequately approximated with a relatively simple analytical solution. As we see from the results the effects of transient flow (i.e., oscillations) and of spatial variability are very similar (because ~ and Q* are similar), even though the transient flow character is profound only for small Rt. Although the effect of flow oscillations is minor on pore scale dispersion, they contribute to the apparent field scale dispersion (Py) that accounts for residence fitLle variability. In view of the calculated &values, the values of P / f o r mobile pesticides may be similar to the values for retarded pesticides due to variability of (bio)chemical parameters (R, #0). The results of this paper may b e o f practical use in as far as they reveal when various complications such as transientness of flow and heterogeheity should be taken into account.
This work has been partly funded by the Commission of the EuroI::4~.an
•
178
S.E.A.T.M. V A N DER Z E E A N D J.J.T.I. BOESTEN
Communities via the STEP-Research Program (S.v.d.Z.) and by the Netherlands Integrated Soil Research Program (J.B.). APPENDIX: FUNCTIONS OF EQN. 6
The functions gl(w) and g2(~9 are given by &(w) = 3[sinh(2w) - sin(2w)]/{ 2w[cosh(2w) - cos(2w)] ]
(I.1)
g2 (w) = 3/(2w 2) - 3[sinh(2w) + sin(2w)]/[ 2w[cosh(2w) - cos(2w)] ~
(I.2)
REFERENCES Beek, J., 1979. Phosphate retention by soil, in relation to waste disposal, PhD Thesis, Agric. Univ., Wageningen, 162 pp. Boekhold, A.E., S.E.A.T.M. van der Zee and F.A.M. de Haan, 1991. Spatial patterns of cadmium contents related to soil heterogeneity, Water, Air Soil Pollut., 479-488. Boesten, JJ.T.I., 1991. Sensitivity analysis of a ~nathematical model for pesticide leaching to ground water. Pestic. Sci., 31: 375-388. Boesten, J.J.T.I. and A.M.A. van der Linden, 1991. Modeling the influence of sorption and transformation on pesticide leaching and persistence. J. Environ. Qual., 20 (2): 425-435. Bolt, G.H., 1982. Movement of solutes in soil: Principles of adsorption/exchange chromatography. In: G.H. Bolt (Ed.), Soil Chemistry B, Physico-Chemical Models, Elsevier, New York, pp. 259-284. Bosma, W.J.P. and S.E.A.T.M. van der Zee, 1991. Analytical approximation for nonlinear adsorbing solute transport and first-order degradation. T=amp. Porous Media, submitted. Calvet, R., M. Terc6 and J.C. Arvieu, 1980. Adsorption des pesticides par les sols et leur con' stituants, III, Characteristiques g6n6rales de i'ad~Jrption des pesticides. Ann. Agron., 31: 239-257. Council of the European Communities, 1980. Directive of the Council of 15 July 1980 relating to the quality of water intended for human consumption, The Cgciai Journal of the European Communities, L 229, 30 August, pp. 11-29. Elabd, H., W.A. Jury and M.M. Cliath, 1986. Spatial variability of p~ticide adsorption parameters. Environ. Sci. Technoi., 20: 256-260. Farrell, D.A. and W.E. Larson, 1973. Effect of intra-aggregate diffusion on oscillatory flow dispersion in aggregated media. Water Resour. Res., 9 (1): 185-193. Jury, W.A. and J. Gruber, 1989. A stochastic analysis of the influence of soil and climatic var~:.~bil~, on the estimate of pesticide ground water pollution potential. Water Resour. Res., 25: 2465-2474. Jury, W.A, W.F. Spencer and W.J. Farmer, 1983. Behaviour assessment model for trace organics in soil, I, Model description. 3. Environ. Qual., 12: 558-~;64. P.ao, P.S.C., A.G. Hornsby and R.F. Jessup, 1985. Ind~ceb tbr ranking the potential for pesticide contamination of grom~dwater. Proc. Soil Crop Sci. Fla, 44: 1-8. Valocchi, A.J., 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res., 21: 808-820.
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
i 79
Van der Zee, S.E.A.T.M., 1990. Transport of reactive solutesin spatially variable unsaturated soils. In: K. Roth et al. (Eds.), Field Scale Water and Solute Flux in Soils, Birkhauser Verlag, Basel, pp. 269-279. Van der Zee, S.E.A.T.M. and J.J.T.I. Boesten, 1991. Effects of soil heterogeneity on pesticide leaching to ground water. Water Resour. Res., 27: 3051-3063. Walker, A. and P. Brown, 1983. Spatial variability in herbicide degradation rates and residues in soil. Crop Protect., 2 (l): 17-25. Wood, L.S., H.D. Scott, D.B. Marx and T.L. Lary, 1987. Variability in sorption coefficients of Metolachlor on a Captina silt loam. J. Environ. Qual., 16 (3): 251-256.
167
Pesticide leaching in heterogeneous soils with oscillating flow: approximations S.EoA.T.M. van der Zee a and J.J.T.I. Boesten b
~Dept. Soil Science and Plant Nutrition, Agricultural University, P.O. Box 8005, 6700 EC Wageningen, Netherlands i'The Winand Staring Centre for Integrated Land, Soil and Water Research, P.O. Box 125, 6700 A C Wageningen, Netherlands ABSTRACT Pesticide screening models are used to assess which pesticides may leach in significant emounts into ground water. In this paper we consider the effects of transient flow and spatial variability of soil properties on the leached fraction of pesticides. Transient flow is approximated with sinusoidal variations of the flow velocity. We show that the main effect is due to the difference with respect to the residence time in soil, compared with the case of steady flow. This effect i~ most profound for pesticides with small retardation factors. Spatial variability of physical and (bio)chemical parameters is shown to lead to larger leached fractions than in the case of a homogeneous flow domain. Several effects are illustrated with analytical approximations.
Key words: pesticides; leaching; soil; heterogel city; oscillating flow
INTRODUCTION
To protect the quality of drinking water, standards have beea set for the maximum permissible pesticide concentration by the Council of the European Communities (1980). Concentr~Ltions may not exceed 0.1 mg m -3. This constraint implies that both old and, newly developed pesticides should be screened to assess whether their leaching potential may threaten the quality of water intended for human consumption. If that appears to be the case, such pesticides may have to be prohibited. Screening models as developed by Jury et al. (1983), Rao et al. (1985) and Boesten and Van der Linden (1991) characterize the leaching potential in terms of adsorption and transformation parameters. In view of spatial variability, Jury and Gruber (1989) took randomness of flow and pore scale dispersion into account in an analytical screening model. Because also adsorption and uansformation rates may be expected to vary spatially (Walker O348-9697/93/$06.00 © 1993 Elsevie: Science Publishers B.V. All fights reserved
168
S,E.A.T.M. VAN DER ZEE AND J.J.T.I. BOESTEN
and Brown, 1983; Elabd et al., 1986; Wood et al., 1987; Boekhold et al., 1991), random variation of the (bio)chemical parameters was considered in the screening model of Van der Zee and Boesten (1991). They illustrated that due to spatial variability, significantly more pesticide may leach than is expected when heterogeneity is ignored. In this paper we reconsider the model developed earlier (Van der Zee and Boesten, 1991) to evaluate the appropriateness of the assumed steady state flow. Because of daily and seasonal fluctuations of infiltration and evapotranspiration, the assumption is obviously an approximation. THEORY
Mathematical description Solute transport can be described for the one-dimensional case with the mass balance equation given by
OOc O[ ~c] Opq + ~ =~ OD O---t- Ot Oz -ff-Z-Z
OOvc Oz
S~
(1)
In Eqn. 1, p is the dry bulk density, q is the adsorbed amount, t is time, 0 is the volumetric water fraction, c is the concentration, D is the diffusion/dispersion coefficient, z is depth, v is velocity and Ss is the pesticide sink term. Usually, pesticide adsorption can be described with the Freundlich equation
q = KFc n
(2)
where the parameter n is usually smaller than 1 (Calvot et al., 1980) and Kr is the Freundlich coefficient. The diffusion/dispersion coefficient (D) accounts for molecular diffusior.~ corrected for the tortuosity and for convective dispersion. Hence, we obtain for this coefficJient (D)
D = M)o + Ldv
(3)
where h is the tortuosity factor, Do is the diffusion coefficient in water and Ld is the dispersivity. The loss rate of a pesticide (Ss) may be described with a first-order transformation rate (St) of dissolved as well as adsorbed pesticide in combination with a first order (passive) crop uptake rate (Su). This leads to Ss = S, + Su =
+ Oc) + gS c
(4)
PESTICIDE LEACHING IN HETEROGENEOUSSOILS WITH OSCILLATING FLOW
169
where the half life (~1/2) is equal to ln(2)/~0, and describes (bio)chemical transformations. In Eqn. 4,/~0 is the transformation rate parameter, g is the transpiration stream concentration factor and Sw is the water uptake rate of plants. Effect o f oscillating flow For a non-uniform porous medium, such as most soils, oscillation of flow as well as the presence of immobile water leads to additional dispersion. In that case flow does not occur of all water present (0) but only of a mobile fraction (On). The remainder (i.e., O~= 0 - Ore) does not move. To estimate the effect oa dispersion only, a non-reacting solute may be considered in a flow field subject to sinusoidal fluctuations of v. Denoting the mean velocity in the medium by v and the mobile zone velocity by u (= Ov/Om), we assume a sinusoidal fluctuation with amplitude vA and frequency ,o. This yields v(t) = V + VA sin(~0t)= V + P
(5)
where P is the time-dependent part of the velocity. For spherical aggregates a quasi-steady state approximation was obtained for the correction factor (De) of the dispersion coefficient by Farrell and Larson (1973). The correction factor due to immobile water and fluctuating flow effects equals 2 2
D~ = -OiO r4D-~ oVA g2(w) w 2 {[Oig2(w)] 2 + [Oigl(w) + 0m]2]
(6)
where we have defined w = ro(to/2Dm) 1~2. Although it may not be obvious from Eqn. 6, (De) is positive. The functions gl(w) and g2(w) are given in the Appendix and Din= M)0. The volumetric water fractions in the mobile and immobile zones are denoted by Om and 0~, respectively, and inside the spherical immobile zones with radius (r0) we assume only diffusion occurs (with diffusion coefficient Din). Setting for the mean squared velocity of the oscillations:
< p 2 > = (2,x/tz)-i
v 2 sin2 (~0t) dt = 0.5 v2a
(7)
we may rewrite Eqn. 6 as
D~,= Oi 1"20
Dm
(8)
170
S.E.A.T.M. VAN DER ZEE A N D J.J.T.i. BOESTEN
The function E{. } in Eqn. 8 is given by E{Om,O,w} - 21 g2(w): w" 02 { [Oig2(w)]2 + [Oigl(w)
+
0m]2} -l
(9)
The total effective dispersion coefficient (DE) is given by 0_
(10)
D E = ~ - D + De
when we assume all dispersional effects are additive using the velocity (u) for D in Eqn. 3. Completely in agreement with analyses by Bolt (1982) and Valocchi (1985), immobile water effects lead to enhanced dispersion. The correction term to Bolt's (!982) immobile water dispersion term (i.e. E {. }) is shown for some interesting cases in Fig. 1. We observe that E = 1/15 for steady flow at arbitrary values of 0 and 0m, but decreases as fluctuations have a higher frequency (or as w increases). Immobile water effects on dispersion therefore are most profound when flow is steady instead of oscillating.
Et
O.Ol
.~
t
~
,.
,\ k
o.~.\.. , o,ool
-
~. I"
!
o.5~
q~--~,.~~_.=.' 0.9 ~
I,
,': t
"~
ix'
:
I= i !
I
lO
1 o0
w Fig. I. Correction term El 8m; e; w} as a function of w = r0(~/2Dm)1,2, for various values of 0i indicated at the curves. The thin horizontal line corresponds with E = 1/15.
171
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
This could be expected as the immobile water by-pass becomes smaller when flow is oscillating and water comes in contact with the same soil several times. The largest dispersion in terms of peclet numbers (P) for a ground water level, L, is given by
vL
P=~ De
(11)
is for steady flow and yields for aggregated media P = 10-20 (Beek, 1979). For cracked soils P-values may be significantly smaller (e.g., P = 1-10). But for those soils Eqn. 8 does not commonly hold because in The Netherlands the ground water level L is usually shallow (0 < L < 1 m). Therefore quasisteady state is not attained.
Leachedfractionfor a homogeneouscolumn In the sequel we assume that we are primarily interested in the leached fraction of the applied quantity of pesticides. Some pesticides constitute a problem with regard to screening because leached fractions are close to the permissible level. For those pesticides uptake may be disregarded (Boesten and Van de~ Linden, 1991). The leached fraction depends on the residence time in the soil layer under consideration and the transformation rate parameter (~) For a homogeneous column, the residence time of pesticide is distributed due to pore scale dispersion (DE). The leached fraction (mF) is given by Van der Zee and Boesten (1991) for a layer of thickness (L) and a retardation factor (R) by
mE = exp
I --~-I P [~ 1 + 4I~oRL vP
1] 1
(12)
In Eqr.. 12 we recognize the mean residence time (r) in a layer of thickness (L), i.e., LR/V. Due to adsc~tion non-linearity and first-order transformation, c decreases as contact time increases and the concentration-dependent averaged retardation factor/~(z) is not easily evaluat~ ~ analytically. The retardation factor (/~(z)) is an average value experienced by a solute slug that has reached depth z. This factor increases when z increases and may be evaluated numerically with the implicit expression given by Bosma and Van der Zee (1991): P [ -- (n-l)t~R(z)z]l R(L)=-~1 1 L o [ I+-~KFC~-Iexp r
dz
assuming D = 0. Evaluation of Eqn. 13 has to be done numerically.
(13)
172
S.E.A.T.M. VAN DER ZEE AND £J.T.i. BOESTEN
The leached fraction (mF) depends on the residence time and is theret0re a complicated function of the flow velocity, depth and retardation. Fluctuations in v may give rise to smaller or larger residence times. To assess when such fluctuations are (relatively) most important we make an approximation by linearizing sorption. We set for c = 1 and a linear adsorption coefficient (ki) that (14a)
q = KF c '~ = kic
Because usually 0 < n < 1, the values of KF and ki differ. The linear retardation factor is p RI = 1 + -~- ki
(14b)
and is independent of the concentration. We need to know when th~ difference between the residence times with and without oscillating flow is relatively large. The residence time (70) for oscillating flow, starting at t = 0 at an arbitrary phase (b) of the oscillation, is given by
V"+ v~ sin(tot + b)
(15)
0
The value of 70 does not need to be equal to 7, because ~- is the long term (t ~. 70) velocity average. Hence,
I
7/7o = (V'7o)-~ T,'7o-
VAcos (W7o+ b) + VAcos (b)1
(16a)
This yields: 7/7o = 1
vA [cos (O:7o+ b) - cos (b)] ¢o~'T0
'
(16b)
Due to oscillating flow, the actual travel time (70) may be larger or smaller than the estimated travel time (7) assuming steady flow. i~" 7o is larger than 7, the time period during which degradation occur~ is larger than estimated for a steady flow screening model. Hence, this ~:ase does not pose problems in screening. However, when 7o is smaller ~lan 7, degradation will be smaller than in the case of steady flow, which implies a larger leached fraction than expected. This means ~hat pesticidc~ that are classified as safe may in practise lead to an unacceptable ground water quality. Such errors should
! 73
PESTICIDE LEACHING IN HETEROGEI~EOUS SOILS WITH OSCILLATING FLOW
1; / 'i;o 2-
---b=O "
"
b= ~2 ....... b = ~
1 / il
!I ."
."
---
b=3~/2
•
O
0
I
I
~
I
1
2
3
4
Fig. 2. The ratio ~/~0 as a function of ¢o~/2~r for different values of b as calculated from Eqn (I 6b) with v,~/~" = 1.
ratio 6 25
20
,o
5
0
0
!
I
I
0.5
1
1.5
2
Fig. 3. The ratio (6) of leached fractions as a function of (vAIF), V = 2 m/year. The ratio, 8, is defined as the leached fraction in case of oscillating flow divided by this fraction for steady flow (and equal to m r = 0.~)1). Parameters were Ri = 1, L = I m, ~ = 23.4 yeal-~, ~0 = 2~ year -= and b = 0.
174
S.E.A.T.M. VAN DER ZEE AND J.J.T.I. BOESTEN
be avoided. The situations in which ~'0 is much smaller than • may be identified from Fig. 2 (which gives the quotient ~'/~'0as a function of ~07/2~r/br vA/v = 1). The general trend in Fig. 2 is that the effect of oscillating flow decreases with increasing ~or/2~r. The quotient 7"/~'0is madmal (which implies maximal increase of leaching due to oscillating flow) when b = 7r/2 and co approaches zero. However, this is the not relevant limiting case of no oscillations. So most severe relevant effects occur roughly in the range 0. ! < ~0~'/27r < 1. So the effects of oscillating flow are most profound, when the frequency of oscillations (i.e., c0/2~-)is ef the order of magnitude of the inverse residence time. When we are interested in seasonal flow fluctuations (effect of application time), the maximal effect may be expected for ~ < 1 year. For a ground water level at 1 m this implies maximal effect for values of R~ close to unity. In this context it is worthwhile to observe that an important group of pesticides are relatively mobile. It may be derived from ~ig. 2 that daily flow fluctuations due to rainfall have only a very small effect: assuming ~ to be in the order of I year or higher, to~/2~r is in the order of 100 or higher. The effects can be readily evaluated for the case of convective-dispersive flow where we assume P = 10, rewriting Eqn. 12 according to Van der Zee and Boesten (1991)
mF=exp
Il
--~-P
P
+l-
]}
1
(17)
using for ~-* either ~ (no oscillations) or ~0 (given by Eqn. 15). We take as an example the effect of seasonal flow fluctuations (~0 = 2~ year ~) for a mobile pesticide (Rt = 1) under Dutch conditions ~- = 2 m/year based on an excess of rainfall of 400 mm and T = 0.2); we assume b = 0 which corresponds with application in autumn. By choosing # 0 = 23.4 year -~ we obtain a leached fraction mr = 0.001 for the steady flow situation. For the case of oscillating flow we may calculate the leached fraction for different values of vA. Observe that a seasonal reversal of the flow direction, which is relevant for Dutch conditions, requires that VA/V" > 1. For those situations the relative error is large (Fig. 3). This error is calculated as the leached fraction that results for oscillating flow diwded by the leached fraction (0.001) found assuming steady flow. That for mobile pesticides the transientness of flow may be important was shown by Boesten (1991) and Boesten and Van der Linden (1991). Their results revealed the importance of the moment of application, which would be far less important if steady flow conditions had been assumed.
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
l '7~
Leached fiaction for a heterogeneous field Whereas in the previous sections the variations of flow in time were shown to affect the leached fraction, the spatial variations may also have a significant ,effect. Assuming steady flow, it is still likely that soil propez~ies vary spatially in a field. This is the case with regard to, e.g. hydraulic conductivity, adsoq)tion and transformation/uptake parameters (Van der Zee, 1990). Due to this variability, the residence time varies spatially and also the contact time for transformation varies in different parts of the field. Assuming that k~ and ~ are both normally distributed with variation coefficients of 0.25 and f~.10, respectively, we can calculate the expel:ted leached fraction, In Fig. 4 we show tha~ these fractions may be quite different from those calc~!ated in the absence of spatial variability. An analytical approximation may be given for the case that X = #RL/v is normally distributed, Using the same approximations (steady flow, uniform
200 KOm (L/kg)l I O.OOt
/ //
t 1 I I I I I I i I I I /
100 -
I/ I
°'2/ 0
/
,
,
100
200
half-life (d) Fig. 4. Fraction of the applied amount of pesticide leached beyond I m depth as a function of the organic m~tter/water-distribution coefficient (Kom,vertical axis in L/kg) and the halflife due to trar,sfi)rmation (horizontal axis in days). Solid lines give the contours (fraction as a percentage indi,:ated at the lines) for zero variability as calculated by Boesten and Van der Linden (1991). Dashed lines represent the fraction using the same parameter values but with normally distributed £o,,, and ~o (variation coefficients of 25% and It~'/o,respectivelyl.
176
S.E.A.T.M.VANDERZEEANDJ.J.T.I. BOESTEN
properties in the vertical direction, no transversal interaction between different steamtubes) as were made by Van dcr Zee and Boesten (1991) we obtain
m~ = exp {-0.SPi
[~(4m.vlPf)
+ 1 -1]}
(18)
which represents the aerial average of the leached fraction. In Eqn. 18, mx is the mean (or expected) value of X. The parameter Pf is an apparent 'field ~cale' peclet-number that represents variability of X. It is related to the variation coefficient of X (i.e., ~x) by 2
vt~x= -~f [Pf -
1 + exp (-Pf)]
(19)
To evaluate the accuracy of the approximation of Eqns. 18 and 19, we show Q" as a function of the exponential argument of Eqn. 18 in Fig. 5. The ratio Q* ix defined as the ratio of leached fractions with (Eqn. 18) and without (Eqr,. i2) spatial variability (but whh pore scale dispersion as given by Boest,:~ and Van der Linden (1991)). Due to heterogeneity, Q* exceeds unity. ~" Also shown (Fig° 5)• is Q* as calculated from the numerical data of Boe..,en
Qt 6r
sL
/
/
4-
/ i /
$ //"
I
|
0 ~--
I
L
'
t
0
2
4
6
8
.....
J
10
ARG Fig. 5. Multiplication ;~ctor (Q*) for atrazine {(K~,,) = 60 dm3/kg; half.life 60 days) according to the analy~ica! solution (Eqns (18,19)). The parameter, Q*, is defined by the leached fraction estimated with a variationcoefficientof ,g.,,~of 25%, divided by the leached fraction for
zero variabilityoPore scale dis~rsion was accounted for.
PESTICIDE LEACHING IN HETEROGENEOUSSOILS WITH OSCILLATING FLOW
177
and Van der Linden (1991). The agreement appears to be rather good in view of the complexity of their numerical analysis. CONCLUSION
In thi~ paper we considered to what extent spatial variability and t~,ansient flow affect the leached fraction of pesticides applied to the soil surface. The leached fraction depends on ,the residence time in the vadose zone and the transformation rate of the pesticide. Transient flow was approximated with regular oscillations of the flow velocity around a mean (~') value. Compared with the case of steady flow in an aggregated medium, a decrease of the effective dispersion coefficient may be expected. Hence, in view of Eqn. 12, the leached fraction is expected to be larger for steady flow conditions. However, oscillating flow may also affect the mean residence time. This effect is larger when the pesticide is less subject to sorption and the retardation factor is closer to one. Also, this effect increases when the frequency is closer to the inversed mean residence time. Depending on the variations in v as a function of time, the effects may be large, leading to large//-values (being the ratio of leached fraction for oscillating |low divided by the leached fraction for steady flow). When we consider not a single soil column but a spatially variable field (with respect to v, Rt and/or LL we may expect profound differences between the expected leached fraction and the estimated leached fraction that neglects such spatial variations. This assessment, based on numerical results by ~oesten and Van der Linden (199i) that account for sorPtion aon~.inearity, transient flow. dispersion and variations of properties wi~h depth, is adequately approximated with a relatively simple analytical solution. As we see from the results the effects of transient flow (i.e., oscillations) and of spatial variability are very similar (because ~ and Q* are similar), even though the transient flow character is profound only for small Rt. Although the effect of flow oscillations is minor on pore scale dispersion, they contribute to the apparent field scale dispersion (Py) that accounts for residence fitLle variability. In view of the calculated &values, the values of P / f o r mobile pesticides may be similar to the values for retarded pesticides due to variability of (bio)chemical parameters (R, #0). The results of this paper may b e o f practical use in as far as they reveal when various complications such as transientness of flow and heterogeheity should be taken into account.
This work has been partly funded by the Commission of the EuroI::4~.an
•
178
S.E.A.T.M. V A N DER Z E E A N D J.J.T.I. BOESTEN
Communities via the STEP-Research Program (S.v.d.Z.) and by the Netherlands Integrated Soil Research Program (J.B.). APPENDIX: FUNCTIONS OF EQN. 6
The functions gl(w) and g2(~9 are given by &(w) = 3[sinh(2w) - sin(2w)]/{ 2w[cosh(2w) - cos(2w)] ]
(I.1)
g2 (w) = 3/(2w 2) - 3[sinh(2w) + sin(2w)]/[ 2w[cosh(2w) - cos(2w)] ~
(I.2)
REFERENCES Beek, J., 1979. Phosphate retention by soil, in relation to waste disposal, PhD Thesis, Agric. Univ., Wageningen, 162 pp. Boekhold, A.E., S.E.A.T.M. van der Zee and F.A.M. de Haan, 1991. Spatial patterns of cadmium contents related to soil heterogeneity, Water, Air Soil Pollut., 479-488. Boesten, JJ.T.I., 1991. Sensitivity analysis of a ~nathematical model for pesticide leaching to ground water. Pestic. Sci., 31: 375-388. Boesten, J.J.T.I. and A.M.A. van der Linden, 1991. Modeling the influence of sorption and transformation on pesticide leaching and persistence. J. Environ. Qual., 20 (2): 425-435. Bolt, G.H., 1982. Movement of solutes in soil: Principles of adsorption/exchange chromatography. In: G.H. Bolt (Ed.), Soil Chemistry B, Physico-Chemical Models, Elsevier, New York, pp. 259-284. Bosma, W.J.P. and S.E.A.T.M. van der Zee, 1991. Analytical approximation for nonlinear adsorbing solute transport and first-order degradation. T=amp. Porous Media, submitted. Calvet, R., M. Terc6 and J.C. Arvieu, 1980. Adsorption des pesticides par les sols et leur con' stituants, III, Characteristiques g6n6rales de i'ad~Jrption des pesticides. Ann. Agron., 31: 239-257. Council of the European Communities, 1980. Directive of the Council of 15 July 1980 relating to the quality of water intended for human consumption, The Cgciai Journal of the European Communities, L 229, 30 August, pp. 11-29. Elabd, H., W.A. Jury and M.M. Cliath, 1986. Spatial variability of p~ticide adsorption parameters. Environ. Sci. Technoi., 20: 256-260. Farrell, D.A. and W.E. Larson, 1973. Effect of intra-aggregate diffusion on oscillatory flow dispersion in aggregated media. Water Resour. Res., 9 (1): 185-193. Jury, W.A. and J. Gruber, 1989. A stochastic analysis of the influence of soil and climatic var~:.~bil~, on the estimate of pesticide ground water pollution potential. Water Resour. Res., 25: 2465-2474. Jury, W.A, W.F. Spencer and W.J. Farmer, 1983. Behaviour assessment model for trace organics in soil, I, Model description. 3. Environ. Qual., 12: 558-~;64. P.ao, P.S.C., A.G. Hornsby and R.F. Jessup, 1985. Ind~ceb tbr ranking the potential for pesticide contamination of grom~dwater. Proc. Soil Crop Sci. Fla, 44: 1-8. Valocchi, A.J., 1985. Validity of the local equilibrium assumption for modeling sorbing solute transport through homogeneous soils. Water Resour. Res., 21: 808-820.
PESTICIDE LEACHING IN HETEROGENEOUS SOILS WITH OSCILLATING FLOW
i 79
Van der Zee, S.E.A.T.M., 1990. Transport of reactive solutesin spatially variable unsaturated soils. In: K. Roth et al. (Eds.), Field Scale Water and Solute Flux in Soils, Birkhauser Verlag, Basel, pp. 269-279. Van der Zee, S.E.A.T.M. and J.J.T.I. Boesten, 1991. Effects of soil heterogeneity on pesticide leaching to ground water. Water Resour. Res., 27: 3051-3063. Walker, A. and P. Brown, 1983. Spatial variability in herbicide degradation rates and residues in soil. Crop Protect., 2 (l): 17-25. Wood, L.S., H.D. Scott, D.B. Marx and T.L. Lary, 1987. Variability in sorption coefficients of Metolachlor on a Captina silt loam. J. Environ. Qual., 16 (3): 251-256.