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An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil
Journal of Hydrology (2006) 328, 614– 619
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An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil Xiaoxian Zhang
a,b,*
, Xuebin Qi
a,c
, Xinguo Zhou a, Hongbin Pang
a
a
Farmland Irrigation Research Institute, The Chinese Academy of Agricultural Sciences, Xinxiang 453003, Henan Province, China b SIMBIOS Centre, University of Abertay Dundee, Bell Street – Kydd Bldg, Dundee DD1 1HG, United Kingdom c Northwest A&F University, Yangling 712100, Shanxi Province, China Received 2 February 2005; received in revised form 8 September 2005; accepted 7 January 2006
KEYWORDS Longitudinal dispersion; Transverse dispersion; In situ method; Solute transport; Analytical solution; Triple-ring infiltrometer
Summary The knowledge of hydraulic conductivity and solute transport parameters of topsoil is important in a variety of fields and their measurement has been an interest in both theory and practice. In this paper we present an in situ method to measure the longitudinal and transverse dispersion coefficients of solute movement by modifying the double-ring infiltrometer into a triple-ring infiltrometer. Water flow in the apparatus is controlled in one dimension and solute movement in three dimensions. The solute transport parameters can be measured simultaneously with the hydraulic conductivity. Analytical solutions are derived to describe the solute movement, and field experiment was carried out to calculate the solute parameters in homogeneous soil using a simple method developed based on the analytical solutions. Simulating results using these estimated parameters predict the observed breakthrough curves reasonably well. ª 2006 Elsevier B.V. All rights reserved.
Introduction Solute transport in top-soil plays an important role in a variety of fields including leaching of agrochemicals to groundwater, nutrient uptake by plants and remediation of * Corresponding author. Tel.: +44 1382 308611; fax: +44 1382 308117/308537. E-mail address: [email protected] (X. Zhang).
contaminated soils. Its mathematical modelling is usually based on the convection–dispersion equation (CDE). In CDE, solute movement is assumed to comprise a convective flux and a dispersive flux (Bear, 1979). The convective flux describes the average movement of solute with soil water and the dispersive flux describes the impact of inherent spatial variation of water velocity induced by soil heterogeneity at scales that are not resolved in CDE. CDE has been questioned to describe chemical transport in natural soils
0022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2006.01.004
An in situ method to measure the longitudinal and transverse dispersion coefficients characterised by heterogeneity that exists across various scales. As a result, alternative approaches, ranging from stochastic model (Gelhar, 1992; Zhang, 2002), continuous time random walks (Berkowitz et al., 2000; Cortis and Berkowitz, 2004; Dentz et al., 2004), lattice gas-based model (Zhang and Ren, 2003), to fractional convection–dispersion equation (Benson et al., 2000; Zhang et al., 2005) have been proposed and compared (Bromly and Hinz, 2004; Ptak et al., 2004) for modelling solute transport in soils at different scenarios. In practice, however, CDE with constant or scale-dependent dispersion coefficients still remains, and is likely to continue to remain, the main approach for modelling chemical transport in soils and aquifers because of its simplicity and that, in most cases, it describes the chemical transport process satisfactorily well. Solving CDE either numerically (Zhang et al., 2002) or analytically (Leij et al., 1991) needs to know the values of solute transport parameters, including the longitudinal dispersion coefficient and the transverse dispersion coefficient. The longitudinal dispersion coefficient measures the spreading of solute along the water flow direction, and the transverse dispersion coefficient measures the spreading of solute in direction perpendicular to the water flow direction. Their values for a specific soil are generally obtained through displacement experiment. There has been extensive research on the dispersion coefficients, especially the longitudinal dispersion and its dependence on water velocity and soil heterogeneity, over the past few decades in both repacked and undisturbed soil columns (Nielsen and Biggar, 1961; Jardine et al., 1993) and in field soils (Russo, 1989a,b; Butters and Jury, 1989; Butters et al., 1989; Ellsworth and Jury, 1991; Ellsworth et al., 1991). Compared to the longitudinal dispersion, the transverse dispersion is difficult to measure as calculating it needs to know the solute concentration in the direction perpendicular to water flow direction. Like the longitudinal dispersion, the mechanism of the transverse dispersion is also dominated by pore structure, grain size and unresolved heterogeneity by CDE (Grane and Gardner, 1961). A number of indoor studies had been carried out to measure the dependence of the transverse dispersion coefficient on flow rate (Harleman and Rumer, 1963; Yule and Gardner, 1978). Practical applications, however, often need the parameter to be measured in situ because the unresolved heterogeneity dominating the dispersion in field soil differs from the heterogeneity that dominates solute movement in soil columns. The purpose of this paper is to present an in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in top-soil. The method is to modify the double-ring infiltrometer into a triple-ring infiltrometer. Water flow is controlled in one dimension and solute movement in three dimensions. The solute transport parameters can be measured simultaneously with the hydraulic conductivity. Analytical solutions for the solute movement in homogenous soil are derived and a simple analytical method is presented to calculate the longitudinal and transverse dispersion parameters. In situ experiments were carried out to measure the two dispersion coefficients.
615
Materials and methods The apparatus The apparatus designed to measure the longitudinal and transverse dispersion coefficients is a triple-ring infiltrometer as shown in Fig. 1. It comprises an internal ring, a middle ring and an outer ring. The diameter of the outer ring is 120 cm, and of the middle and internal rings is 100 and 10 cm, respectively. Water flow is in saturated condition and the flow rate is kept a constant during the experiment by controlling the water depth over the soil surface in all the three rings unchanged using the Marriott apparatus shown in Fig. 1. The tracer is applied to the internal ring, and the lateral movement of the tracer in soil outer the region controlled by the internal ring reveals the significance of the transverse dispersion. The tracer can be applied either as an instant pulse or as a continuous source. For ease of control, we applied the tracer in an instant pulse in this paper. Continuously supplying tracer may be needed when studying the movement of reactive chemicals in soils.
Experiment Field experiment was carried out in Shangqiu, eastern Henan Province of China. The soil is loamy sand and the soil profile is relatively uniform until depth of 150 cm, revealed by a soil profile in a drainage ditch few meters away from the experimental site; the soil taken from the augers when installing suction cups in the site also confirmed this. Prior to the installation of the apparatus on the selected site, the top 15 cm soil was trimmed off with a spade to make the soil surface level. Suction cup samplers were then placed into soil at different positions to measure the breakthrough curves. The samplers were assembled by sealing porous ceramic suction cup samplers (diameter 1.5 cm, length 4 cm) to the end of plastic pipes of the same diameter. Following carefully digging holes with augers, the suction cup samplers were
Plane view
4
4
1
2
3
3
2
1
Cross section 5
u
1. Outer ring, 2. Middle ring, 3. Internal ring, 4. Marriott apparatus, 5 Suction cup
Figure 1 Schematic view of the in situ apparatus for measuring the longitudinal and transverse dispersion coefficients.
616 inserted vertically into the soil. Slurry of sieved soil was used to ensure good contact between the cups and soil. Once the cup samplers were installed, the triple-ring infiltrometer was pushed into the soil on the site. The site was first irrigated with water pumped from groundwater from a borehole 20 m away from the site in attempts to bring the soil saturated and make the initial solute concentration profile uniform. The water depth over soil surface in all the three rings was kept unchanged during the experiment using two Marriott apparatuses, one controls the internal ring and the other controls the water in the outer and middle rings as shown in Fig. 1. The outer and the middle rings were made hydraulically connected through a hole drilled in the middle ring. Measuring the amount of water flowing out the Marriott apparatus that controls the water supply to the internal ring allows the average infiltration rate to be calculated. Once the flow rate reached steady state, we measured the water depth over the soil surface inside the internal ring and used this measurement to calculate the volume of water in the internal ring (denoted by V) for reason to be seen below. We then switched off the water supply to the internal ring and quickly dried the internal ring using pumps and bowls. After that, 300 g CaCl2 dissolved in water of volume V was immediately poured into the internal ring, and the water supply to the internal ring was then switched on. We applied tracer in this way in an attempt to minimize the disturbance to the water velocity field. Soil water samples were taken from the cup samplers in a time interval of approximately 2 h. The concentration of Cl1 in the soil water samples were taken to laboratory for analysis. During the experiment, care was made to ensure that water depth over soil surface in the internal, outer and the middle rings was the same. In the experiment, five suction cups were installed, two were at depth of 30 and 10 cm from the origin of the internal ring, two were at depth of 50 and 20 cm from the origin of the internal ring, and one was right through the centre of the internal ring at depth of 24 cm.
Calculating the transport parameters In the absences of non-equilibrium physical and chemical reactions and that the soil profile is uniform where porewater velocity and solute dispersion coefficients can be approximated as constants, the solute movement in the soil beneath the apparatus shown in Fig. 1 can be described by the following equation: ! oc o2 c o2 c 1 oc oc þ ¼ DL 2 þ DT u ; ð1Þ ot oz or2 r or oz where t is time, c is concentration, z and r are coordinates with z pointing downward and r originating from the centre of the internal ring, DL and DT are the longitudinal and transverse dispersion coefficients, respectively, u is pore-water velocity. Eq. (1) can account equilibrium physical and chemical reactions, and in this case the two dispersion coefficients and the average pore water velocity need to be scaled by a retardation factor. The diameter of the middle ring was made much larger than the diameter of the internal ring in order to safely assume that during the experiment, the tracer in the soil did not move out the region controlled by the middle ring.
X. Zhang et al. Therefore, the associated initial and boundary conditions with Eq. (1) for the solute movement in the apparatus shown in Fig. 1 are cðz; r; 0Þ ¼ 0; cðz; 1; tÞ ¼ 0; cð1; r; tÞ ¼ 0; oc m uc DL Hðr R0 ÞdðtÞ; ¼ oz z¼0 R20 pn
ð2Þ
where n is soil porosity, m is the mass of tracer being applied into the internal ring, R0 is the radius of the internal ring, H(r R0) is the Heaviside function in that H(r R0) = 1 if r < R0 and H(r R0) = 1 otherwise, and d(t) is the Dirac delta function. As shown in Appendix, the analytical solution of Eq. (1) associated with the initial and boundary conditions given by Eq. (2) is Z 1 cðr; z; tÞ ¼ C0 Fðz; tÞ J1 ðpR0 ÞJ0 ðprÞ expðDT p2 tÞ dp; ð3Þ 0
where Jm(x) is the Bessel function of the first kind, m is the order of the Bessel function, C0 = m/R0pn, and ! 1 ðz utÞ2 u Fðz; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2D 4D t DL pt L L uz z þ ut erfc pffiffiffiffiffiffiffi . exp ð4Þ DL 2 DL t Solution given by Eq. (3) can be further simplified if the radius of the internal ring is much smaller than the diameter of middle ring in that the cross-section of the internal ring can be approximated as a point. In this limitation, when x is far away from the soil surface. Eq. (3) reduces to ! m ðz utÞ2 r2 . ð5Þ cðr; z; tÞ ¼ exp pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2 There is a very simple way to calculate the transport parameters using Eq. (5). We denote the peaking concentration in a measured breakthrough curve at location (r, z) by cmax and the time that the concentration peaks by tmax. Dividing the both sides of Eq. (5) by cmax yields c cmax ! 1:5 tmax ðz utmax Þ2 r2 ðz utÞ2 r2 . ¼ exp þ t 4DL tmax 4DT tmax 4DL t 4DT t
c0 ¼
ð6Þ Taking logarithm to both sides of Eq. (6) gives " # 1 ðz utmax t0 Þ2 2 0 0 ln c ¼ 1:5 ln t þ ðz utmax Þ 4DL tmax t0 1 r2 ð7Þ r2 0 ; þ 4DT tmax t where t 0 = t/tmax is a dimensionless time. Eq. (7) can be rewritten into the following form: y¼
x 1 þ ; 4DL 4DT
ð8Þ
An in situ method to measure the longitudinal and transverse dispersion coefficients where
(a)
1
2
R =0.98
0.8
ð9Þ
c'
1 r2 0 0 2 ; y ¼ ½lnðc Þ þ 1:5 lnðt Þtmax r 0 t " # 1 ðz utmax t0 Þ2 r2 2 2 . r 0 x ¼ ðz utmax Þ t0 t
617
0.6 0.4 0.2
DL ¼
N P 2
ai
b2i
N P
ai bi
N P
i¼1
N P
N P 2
i¼1
N P
i¼1
i¼1
i¼1
N P
1
80
2
R =0.95
0.8 0.6 0.4 0.2 0 0
20
40
60
80
Time (hours) (c)
1 R =0.91
0.6 0.4 0.2 0 0
20
40
60
80
Time (hours)
i¼1
where N is total number of the measurements taken from one suction cup and d i ¼ 1:5 ln t0i ; ai ¼ ðz utmax Þ2 t0 Þ 2
60
2
ð10Þ
ðzut
40
0.8
ai b2i ai bi ai bi 1 i¼1 i¼1 i¼1 i¼1 DT ¼ ; N N N N P P P 4tmax P 0 2 ðbi d i þ bi ln ci Þ ai ai bi ðai d i þ ai ln c0i Þ i¼1
(b)
ai bi
1 i¼1 i¼1 i¼1 i¼1 ; N N N N P P P 4tmax P 2 ðai d i þ ai ln c0i Þ bi ai bi ðbi d i þ bi ln c0i Þ i¼1
20
Time (hours)
c'
N P
0 0
c'
Eq. (8) reveals that y, which can be calculated using measured data, is a linear function of x that can also be calculated using measured data. The reciprocals of the longitudinal and transverse dispersion coefficients DL and DT determine the slope and intercept of the line, respectively. We fitted the experimental data to Eqs. (8) and (9) to estimate the values of DL and DT, but the results did not prove very successful. The reason is that when t 0 approaches 1, both x and y defined in Eq. (9) diverge. Therefore, instead of using Eqs. (8) and (9), we directly applied Eq. (7) to calculate the two parameters. We fitted the experimental data to Eq. (7) using the least square method. Since ln(c 0 ) is a linear function of the reciprocals of the DL and DT, the values of DL and DT can be calculated straightforwardly as follows provided that the pore water velocity is known
Figure 2 Comparison of the observed (symbols) and simulated (solid lines) breakthrough curves for positions: z = 30 cm and r = 10 cm (a, b), and z = 50 cm and r = 20 cm (c).
2
i tmax ; bi ¼ r2 rt0 , in which t0i represents the dimen0 i i sionless time where the ith measurement c0i is taken. The use of Eq. (10) needs to know the pore-water velocity. Theoretically, the pore-water velocity can be estimated using the measured porosity (0.44) and flow rate (0.51 cm/h); this gives u = 1.16 cm/h. However, the velocity estimated in this way often failed to predict the observed breakthrough curves in most of our experiments. We therefore used a method developed by Zhang (1989) to estimate the pore water velocity from the breakthrough curve obtained from a purposely installed suction cup in soil right through the centre of the internal ring at r = 0. From Eq. (5) we know that when r = 0, the change of concentration at depth of z with time is given by ! m ðz utÞ2 cðr; z; tÞ ¼ . ð11Þ exp pffiffiffiffiffi 4DL t 4nDT DL ðptÞ3=2
From Eq. (11) we know that a new variable Y defined as follows ! m ðz utÞ2 3=2 ð12Þ exp Yðx; tÞ ¼ cðz; tÞt ¼ pffiffiffiffiffi 4DL t 4nDT DL ðpÞ3=2 peaks at z = ut, that is, when t* = z/u. The pore water velocity u can therefore be estimated from the peaking time t* of
Y, which can be calculated using the measured data, and the depth z where the suction cup is located. From the breakthrough curve measured from the cup located at z = 24 cm and r = 0, the estimated pore-water velocity is approximately u = 1.4 cm/h. The estimated pore-water velocity was used to calculate the two dispersion coefficients using the breakthrough curves obtained from other suction cups. One cup located at depth of 50 cm was fault, we therefore analysed the data obtained from other three cups, two at position z = 30 cm and r = 10 cm, and one at position of z = 50 cm and r = 20 cm. The estimated values of the DL and DT using Eq. (10) are DL = 4.5 cm2/h DT = 0.87 cm2/h. To evaluate the accuracy of these estimated parameters, we used them to predict the breakthrough curves obtained from the three suction cups by solving Eq. (1) with the associated initial and boundary conditions given in Eq. (2) and R0 = 5 cm. Fig. 2 compares the simulations and the measurements. Overall they match well.
Summary The work reported in this paper presents an in situ method to measure the longitudinal and transverse dispersion
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X. Zhang et al.
coefficients of solute transport in soil by modifying the double-ring infiltrometer into a triple-ring infiltrometer. Water flow is controlled in one dimension and solute transport in three dimensions. An analytical solution is derived to describe solute transport, and a simple method based on the analytical solution is proposed to estimate the two parameters. Field experiment was carried out to measure the parameters. The proposed analytical method for calculating the two parameters assumes that soil is homogenous where the pore-water velocity and the dispersion coefficients can be approximated as constants. The method is invalid for heterogeneous soil where the pore-water velocity and the solute dispersion coefficients may vary spatially. In this situation, it needs numerical solution to solve the transport equation. Also, the apparatus is designed to measure the longitudinal and transverse dispersion coefficients in saturated conditions. It might be possible to improve and modify the apparatus to measure the two parameters in unsaturated condition in a combination with the disc and ring infiltrometers (Angulo-Jaramillo et al., 2000). This, however, needs further work.
Acknowledgements We like to thank the two anonymous reviewers for their constructive comments to improve the manuscript. We also thank the Chinese Agro-Ecological Experimental Station in Shangqiu, Henan, for their financial support to this work.
Applying the inverse Laplace transform to Eq. (A3) and using the following formula: 2 pffiffi 1 1 x pffiffi h expðhx þ th2 Þ expðx sÞ ! pffiffiffiffiffi exp 4t sþh pt pffiffi x erfc pffiffi þ h t ; ðA4Þ 2 t we have ~c ¼
C0 J1 ðpR0 Þ expðDT p2 tÞ Fðz; tÞ; p
ðA5Þ
where ! 1 ðz utÞ2 u Fðz; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2DL 4DL t DL pt uz z þ ut ffiffiffiffiffiffiffi p . erfc exp DL 2 DL t
ðA6Þ
Applying the inverse Hankel transform to Eq. (A6) gives Z 1 cðr; z; tÞ ¼ C0 Fðz; tÞ J1 ðpR0 ÞJ0 ðprÞ expðDT p2 tÞ dp. ðA7Þ 0
A special case of Eq. (A7) is when the radius of the internal ring is small and its cross-section can be seen as a point. Under this condition, we have Z 1 C0 JðpR0 Þ expðDT p2 tÞ dp lim R0 !0 0 m r2 exp ; ðA8Þ ¼ 4DT t 4pnDT t Eq. (A7) is therefore simplified to
Appendix Applying the Laplace transform to Eqs. (1) and (2) with respect to t gives ! o2 ~c o2 ~c 1 o~c o~c u ; þ s~c ¼ DL 2 þ DT 2 oz or r or oz ~cðx; 1; tÞ ¼ 0; ðA1Þ ~cð1; r; tÞ ¼ 0; o~c C0 ¼ Hðr R0 Þ; u~c DL oz z¼0 R0 R1 where ~c ¼ 0 cest dt and C0 ¼ R0mpn. Applying the Hankel transform to Eq. (A1) with respect to r yields o2 ~c o~c ~c ¼ DL 2 DT p2 ~c u ; s oz oz ~cð1; r; tÞ ¼ 0; ðA2Þ c ~ o C ¼ 0 J1 ðpR0 Þ; u~c DL p oz z¼0 R 1 where ~c ¼ 0 r~cJ0 ðrpÞ dr. Solving Eq. (A2) for ~c yields ~c ¼
2C0 J1 ðpR0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p u2 þ 4DL ðDT p2 þ sÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ux u2 þ 4DL ðDT p2 þ sÞ exp . 2DL ðu þ
ðA3Þ
! m ðz utÞ2 r2 cðr; z; tÞ ¼ exp pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2 2 um uz r z þ ut exp erfc pffiffiffiffiffiffiffi . 8pnDT DL t DL 4DT t 2 DL t ðA9Þ Like the analytical solution for solute transport in an infinite domain under a prescribed concentration, the second term on the right-hand side of Eq. (A9) is smaller than the first term when z is far away from soil surface. Under this condition, Eq. (A9) can be further simplified to ! m ðz utÞ2 r2 . ðA10Þ exp cðr; z; tÞ ¼ pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2
References Angulo-Jaramillo, R., Vandervaere, J.P., Roulier, S., Thony, J.L., Gaudet, J.P., Vauclin, M., 2000. Field measurement of soil surface hydraulic properties by disc and ring infiltrometers – a review and recent developments. Soil Till. Res. 55 (1–2), 1–29. Bear, J., 1979. Hydraulics of Groundwater. McGraw-Hill. Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M., 2000. The fractional-order governing equation of Levy motion. Water Resour. Res. 36 (6), 1413–1423. Berkowitz, B., Scher, H., Silliman, S.E., 2000. Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36 (1), 149–158.
An in situ method to measure the longitudinal and transverse dispersion coefficients Bromly, M., Hinz, C., 2004. Non-Fickian transport in homogeneous unsaturated repacked sand. Water Resour. Res. 40 (7), art. no. W07402. Butters, G.L., Jury, W.A., 1989. Field scale transport of bromide in an unsaturated soil. 2. Dispersion modelling. Water Resour. Res. 25 (7), 1583–1589. Butters, G.L., Jury, W.A., Ernst, F.F., 1989. Field scale transport of bromide in an unsaturated soil. 1. Experimental methodology and results. Water Resour. Res. 25 (7), 1583–1589. Cortis, A., Berkowitz, B., 2004. Anomalous transport in ‘‘classical’’ soil and sand columns. Soil Sci. Soc. Am. J. 68 (5), 1539–1548. Dentz, M., Cortis, A., Scher, H., Berkowitz, B., 2004. Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27 (2), 155–173. Ellsworth, T.R., Jury, W.A., 1991. A three-dimensional field study of solute transport through unsaturated, layered, porous media. 2. Characterization of vertical dispersion. Water Resour. Res. 27 (5), 967–981. Ellsworth, T.R., Jury, W.A., Ernst, F.F., Shouse, P.J., 1991. A threedimensional field study of solute transport through unsaturated, layered, porous media. 1. Methodology, mass recovery and mean transport. Water Resour. Res. 27 (5), 951–965. Gelhar, L.W., 1992. Stochastic Subsurface Hydrology. Prentice Hall College. Grane, F.E., Gardner, G.H.F., 1961. Measurements of transverse dispersion in granular media. J. Chem. Eng. 6, 283–287. Harleman, D.R.F., Rumer, R.R., 1963. Longitudinal and lateral dispersion in an isotropic porous medium. J. Fluid. Mech. 16, 385–394. Jardine, P.M., Jacobs, G.K., Wilson, G.V., 1993. Unsaturated transport processes in undisturbed heterogeneous porous media. I. Inorganic contaminants. Soil Sci. Soc. Am. J. 57, 945–953. Leij, F.J., Skaggs, T.H., van Genuchten, M.T., 1991. Analytical solutions for solute transport in three-dimensional semi-
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infinite porous media. Water Resour. Res. 27 (10), 2719– 2733. Nielsen, D.R., Biggar, J.W., 1961. Miscible displacement in soils. I. Experimental information. Soil Sci. Soc. Am. Proc. 25, 1–5. Ptak, T., Piepenbrink, M., Martac, E., 2004. Tracer tests for the investigation of heterogeneous porous media and stochastic modelling of flow and transport – a review of some recent developments. J. Hydrol. 294 (1–3), 122–163. Russo, D., 1989a. Field-scale transport of interacting solutes through the unsaturated zone. 1. Analysis of the spatial variability of the transport properties. Water Resour. Res. 25 (12), 2475–2485. Russo, D., 1989b. Field-scale transport of interacting solutes through the unsaturated zone. 2. Analysis of the spatial variability of the field repose. Water Resour. Res. 25 (12), 2487–2495. Yule, D.F., Gardner, W.R., 1978. Longitudinal and transverse dispersion coefficients in unsaturated plainfield sand. Water Resour. Res. 14, 582–588. Zhang, X.X., 1989. Measuring the dispersion coefficients of solute transport in soil. J. Hydraulic Eng. 20 (1), 1–6 (in Chinese with English abstract). Zhang, D.X., 2002. Stochastic Methods for Flow in Porous Media: Coping with Uncertainties. Academic Press, San Diego. Zhang, X.X., Bengough, A.G., Deeks, L.K., Crawford, J.W., Young, I.M., 2002. A novel three-dimensional lattice Boltzmann model for solute transport in variably saturated porous media. Water Resour. Res. 38 (9), art. no. 1167. Zhang, X.X., Ren, L., 2003. Lattice Boltzmann model for agrochemical transport in soils. J. Contam. Hydrol. 67, 27–42. Zhang, X.X., Crawford, J.W., Deeks, L.K., Stutter, M.I., Bengough, A.G., Young, I.M., 2005. A mass balance based numerical method for the fractional advection-dispersion equation: theory and application. Water Resour. Res. 41, art. no. W07029.
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/jhydrol
An in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in soil Xiaoxian Zhang
a,b,*
, Xuebin Qi
a,c
, Xinguo Zhou a, Hongbin Pang
a
a
Farmland Irrigation Research Institute, The Chinese Academy of Agricultural Sciences, Xinxiang 453003, Henan Province, China b SIMBIOS Centre, University of Abertay Dundee, Bell Street – Kydd Bldg, Dundee DD1 1HG, United Kingdom c Northwest A&F University, Yangling 712100, Shanxi Province, China Received 2 February 2005; received in revised form 8 September 2005; accepted 7 January 2006
KEYWORDS Longitudinal dispersion; Transverse dispersion; In situ method; Solute transport; Analytical solution; Triple-ring infiltrometer
Summary The knowledge of hydraulic conductivity and solute transport parameters of topsoil is important in a variety of fields and their measurement has been an interest in both theory and practice. In this paper we present an in situ method to measure the longitudinal and transverse dispersion coefficients of solute movement by modifying the double-ring infiltrometer into a triple-ring infiltrometer. Water flow in the apparatus is controlled in one dimension and solute movement in three dimensions. The solute transport parameters can be measured simultaneously with the hydraulic conductivity. Analytical solutions are derived to describe the solute movement, and field experiment was carried out to calculate the solute parameters in homogeneous soil using a simple method developed based on the analytical solutions. Simulating results using these estimated parameters predict the observed breakthrough curves reasonably well. ª 2006 Elsevier B.V. All rights reserved.
Introduction Solute transport in top-soil plays an important role in a variety of fields including leaching of agrochemicals to groundwater, nutrient uptake by plants and remediation of * Corresponding author. Tel.: +44 1382 308611; fax: +44 1382 308117/308537. E-mail address: [email protected] (X. Zhang).
contaminated soils. Its mathematical modelling is usually based on the convection–dispersion equation (CDE). In CDE, solute movement is assumed to comprise a convective flux and a dispersive flux (Bear, 1979). The convective flux describes the average movement of solute with soil water and the dispersive flux describes the impact of inherent spatial variation of water velocity induced by soil heterogeneity at scales that are not resolved in CDE. CDE has been questioned to describe chemical transport in natural soils
0022-1694/$ - see front matter ª 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2006.01.004
An in situ method to measure the longitudinal and transverse dispersion coefficients characterised by heterogeneity that exists across various scales. As a result, alternative approaches, ranging from stochastic model (Gelhar, 1992; Zhang, 2002), continuous time random walks (Berkowitz et al., 2000; Cortis and Berkowitz, 2004; Dentz et al., 2004), lattice gas-based model (Zhang and Ren, 2003), to fractional convection–dispersion equation (Benson et al., 2000; Zhang et al., 2005) have been proposed and compared (Bromly and Hinz, 2004; Ptak et al., 2004) for modelling solute transport in soils at different scenarios. In practice, however, CDE with constant or scale-dependent dispersion coefficients still remains, and is likely to continue to remain, the main approach for modelling chemical transport in soils and aquifers because of its simplicity and that, in most cases, it describes the chemical transport process satisfactorily well. Solving CDE either numerically (Zhang et al., 2002) or analytically (Leij et al., 1991) needs to know the values of solute transport parameters, including the longitudinal dispersion coefficient and the transverse dispersion coefficient. The longitudinal dispersion coefficient measures the spreading of solute along the water flow direction, and the transverse dispersion coefficient measures the spreading of solute in direction perpendicular to the water flow direction. Their values for a specific soil are generally obtained through displacement experiment. There has been extensive research on the dispersion coefficients, especially the longitudinal dispersion and its dependence on water velocity and soil heterogeneity, over the past few decades in both repacked and undisturbed soil columns (Nielsen and Biggar, 1961; Jardine et al., 1993) and in field soils (Russo, 1989a,b; Butters and Jury, 1989; Butters et al., 1989; Ellsworth and Jury, 1991; Ellsworth et al., 1991). Compared to the longitudinal dispersion, the transverse dispersion is difficult to measure as calculating it needs to know the solute concentration in the direction perpendicular to water flow direction. Like the longitudinal dispersion, the mechanism of the transverse dispersion is also dominated by pore structure, grain size and unresolved heterogeneity by CDE (Grane and Gardner, 1961). A number of indoor studies had been carried out to measure the dependence of the transverse dispersion coefficient on flow rate (Harleman and Rumer, 1963; Yule and Gardner, 1978). Practical applications, however, often need the parameter to be measured in situ because the unresolved heterogeneity dominating the dispersion in field soil differs from the heterogeneity that dominates solute movement in soil columns. The purpose of this paper is to present an in situ method to measure the longitudinal and transverse dispersion coefficients of solute transport in top-soil. The method is to modify the double-ring infiltrometer into a triple-ring infiltrometer. Water flow is controlled in one dimension and solute movement in three dimensions. The solute transport parameters can be measured simultaneously with the hydraulic conductivity. Analytical solutions for the solute movement in homogenous soil are derived and a simple analytical method is presented to calculate the longitudinal and transverse dispersion parameters. In situ experiments were carried out to measure the two dispersion coefficients.
615
Materials and methods The apparatus The apparatus designed to measure the longitudinal and transverse dispersion coefficients is a triple-ring infiltrometer as shown in Fig. 1. It comprises an internal ring, a middle ring and an outer ring. The diameter of the outer ring is 120 cm, and of the middle and internal rings is 100 and 10 cm, respectively. Water flow is in saturated condition and the flow rate is kept a constant during the experiment by controlling the water depth over the soil surface in all the three rings unchanged using the Marriott apparatus shown in Fig. 1. The tracer is applied to the internal ring, and the lateral movement of the tracer in soil outer the region controlled by the internal ring reveals the significance of the transverse dispersion. The tracer can be applied either as an instant pulse or as a continuous source. For ease of control, we applied the tracer in an instant pulse in this paper. Continuously supplying tracer may be needed when studying the movement of reactive chemicals in soils.
Experiment Field experiment was carried out in Shangqiu, eastern Henan Province of China. The soil is loamy sand and the soil profile is relatively uniform until depth of 150 cm, revealed by a soil profile in a drainage ditch few meters away from the experimental site; the soil taken from the augers when installing suction cups in the site also confirmed this. Prior to the installation of the apparatus on the selected site, the top 15 cm soil was trimmed off with a spade to make the soil surface level. Suction cup samplers were then placed into soil at different positions to measure the breakthrough curves. The samplers were assembled by sealing porous ceramic suction cup samplers (diameter 1.5 cm, length 4 cm) to the end of plastic pipes of the same diameter. Following carefully digging holes with augers, the suction cup samplers were
Plane view
4
4
1
2
3
3
2
1
Cross section 5
u
1. Outer ring, 2. Middle ring, 3. Internal ring, 4. Marriott apparatus, 5 Suction cup
Figure 1 Schematic view of the in situ apparatus for measuring the longitudinal and transverse dispersion coefficients.
616 inserted vertically into the soil. Slurry of sieved soil was used to ensure good contact between the cups and soil. Once the cup samplers were installed, the triple-ring infiltrometer was pushed into the soil on the site. The site was first irrigated with water pumped from groundwater from a borehole 20 m away from the site in attempts to bring the soil saturated and make the initial solute concentration profile uniform. The water depth over soil surface in all the three rings was kept unchanged during the experiment using two Marriott apparatuses, one controls the internal ring and the other controls the water in the outer and middle rings as shown in Fig. 1. The outer and the middle rings were made hydraulically connected through a hole drilled in the middle ring. Measuring the amount of water flowing out the Marriott apparatus that controls the water supply to the internal ring allows the average infiltration rate to be calculated. Once the flow rate reached steady state, we measured the water depth over the soil surface inside the internal ring and used this measurement to calculate the volume of water in the internal ring (denoted by V) for reason to be seen below. We then switched off the water supply to the internal ring and quickly dried the internal ring using pumps and bowls. After that, 300 g CaCl2 dissolved in water of volume V was immediately poured into the internal ring, and the water supply to the internal ring was then switched on. We applied tracer in this way in an attempt to minimize the disturbance to the water velocity field. Soil water samples were taken from the cup samplers in a time interval of approximately 2 h. The concentration of Cl1 in the soil water samples were taken to laboratory for analysis. During the experiment, care was made to ensure that water depth over soil surface in the internal, outer and the middle rings was the same. In the experiment, five suction cups were installed, two were at depth of 30 and 10 cm from the origin of the internal ring, two were at depth of 50 and 20 cm from the origin of the internal ring, and one was right through the centre of the internal ring at depth of 24 cm.
Calculating the transport parameters In the absences of non-equilibrium physical and chemical reactions and that the soil profile is uniform where porewater velocity and solute dispersion coefficients can be approximated as constants, the solute movement in the soil beneath the apparatus shown in Fig. 1 can be described by the following equation: ! oc o2 c o2 c 1 oc oc þ ¼ DL 2 þ DT u ; ð1Þ ot oz or2 r or oz where t is time, c is concentration, z and r are coordinates with z pointing downward and r originating from the centre of the internal ring, DL and DT are the longitudinal and transverse dispersion coefficients, respectively, u is pore-water velocity. Eq. (1) can account equilibrium physical and chemical reactions, and in this case the two dispersion coefficients and the average pore water velocity need to be scaled by a retardation factor. The diameter of the middle ring was made much larger than the diameter of the internal ring in order to safely assume that during the experiment, the tracer in the soil did not move out the region controlled by the middle ring.
X. Zhang et al. Therefore, the associated initial and boundary conditions with Eq. (1) for the solute movement in the apparatus shown in Fig. 1 are cðz; r; 0Þ ¼ 0; cðz; 1; tÞ ¼ 0; cð1; r; tÞ ¼ 0; oc m uc DL Hðr R0 ÞdðtÞ; ¼ oz z¼0 R20 pn
ð2Þ
where n is soil porosity, m is the mass of tracer being applied into the internal ring, R0 is the radius of the internal ring, H(r R0) is the Heaviside function in that H(r R0) = 1 if r < R0 and H(r R0) = 1 otherwise, and d(t) is the Dirac delta function. As shown in Appendix, the analytical solution of Eq. (1) associated with the initial and boundary conditions given by Eq. (2) is Z 1 cðr; z; tÞ ¼ C0 Fðz; tÞ J1 ðpR0 ÞJ0 ðprÞ expðDT p2 tÞ dp; ð3Þ 0
where Jm(x) is the Bessel function of the first kind, m is the order of the Bessel function, C0 = m/R0pn, and ! 1 ðz utÞ2 u Fðz; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2D 4D t DL pt L L uz z þ ut erfc pffiffiffiffiffiffiffi . exp ð4Þ DL 2 DL t Solution given by Eq. (3) can be further simplified if the radius of the internal ring is much smaller than the diameter of middle ring in that the cross-section of the internal ring can be approximated as a point. In this limitation, when x is far away from the soil surface. Eq. (3) reduces to ! m ðz utÞ2 r2 . ð5Þ cðr; z; tÞ ¼ exp pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2 There is a very simple way to calculate the transport parameters using Eq. (5). We denote the peaking concentration in a measured breakthrough curve at location (r, z) by cmax and the time that the concentration peaks by tmax. Dividing the both sides of Eq. (5) by cmax yields c cmax ! 1:5 tmax ðz utmax Þ2 r2 ðz utÞ2 r2 . ¼ exp þ t 4DL tmax 4DT tmax 4DL t 4DT t
c0 ¼
ð6Þ Taking logarithm to both sides of Eq. (6) gives " # 1 ðz utmax t0 Þ2 2 0 0 ln c ¼ 1:5 ln t þ ðz utmax Þ 4DL tmax t0 1 r2 ð7Þ r2 0 ; þ 4DT tmax t where t 0 = t/tmax is a dimensionless time. Eq. (7) can be rewritten into the following form: y¼
x 1 þ ; 4DL 4DT
ð8Þ
An in situ method to measure the longitudinal and transverse dispersion coefficients where
(a)
1
2
R =0.98
0.8
ð9Þ
c'
1 r2 0 0 2 ; y ¼ ½lnðc Þ þ 1:5 lnðt Þtmax r 0 t " # 1 ðz utmax t0 Þ2 r2 2 2 . r 0 x ¼ ðz utmax Þ t0 t
617
0.6 0.4 0.2
DL ¼
N P 2
ai
b2i
N P
ai bi
N P
i¼1
N P
N P 2
i¼1
N P
i¼1
i¼1
i¼1
N P
1
80
2
R =0.95
0.8 0.6 0.4 0.2 0 0
20
40
60
80
Time (hours) (c)
1 R =0.91
0.6 0.4 0.2 0 0
20
40
60
80
Time (hours)
i¼1
where N is total number of the measurements taken from one suction cup and d i ¼ 1:5 ln t0i ; ai ¼ ðz utmax Þ2 t0 Þ 2
60
2
ð10Þ
ðzut
40
0.8
ai b2i ai bi ai bi 1 i¼1 i¼1 i¼1 i¼1 DT ¼ ; N N N N P P P 4tmax P 0 2 ðbi d i þ bi ln ci Þ ai ai bi ðai d i þ ai ln c0i Þ i¼1
(b)
ai bi
1 i¼1 i¼1 i¼1 i¼1 ; N N N N P P P 4tmax P 2 ðai d i þ ai ln c0i Þ bi ai bi ðbi d i þ bi ln c0i Þ i¼1
20
Time (hours)
c'
N P
0 0
c'
Eq. (8) reveals that y, which can be calculated using measured data, is a linear function of x that can also be calculated using measured data. The reciprocals of the longitudinal and transverse dispersion coefficients DL and DT determine the slope and intercept of the line, respectively. We fitted the experimental data to Eqs. (8) and (9) to estimate the values of DL and DT, but the results did not prove very successful. The reason is that when t 0 approaches 1, both x and y defined in Eq. (9) diverge. Therefore, instead of using Eqs. (8) and (9), we directly applied Eq. (7) to calculate the two parameters. We fitted the experimental data to Eq. (7) using the least square method. Since ln(c 0 ) is a linear function of the reciprocals of the DL and DT, the values of DL and DT can be calculated straightforwardly as follows provided that the pore water velocity is known
Figure 2 Comparison of the observed (symbols) and simulated (solid lines) breakthrough curves for positions: z = 30 cm and r = 10 cm (a, b), and z = 50 cm and r = 20 cm (c).
2
i tmax ; bi ¼ r2 rt0 , in which t0i represents the dimen0 i i sionless time where the ith measurement c0i is taken. The use of Eq. (10) needs to know the pore-water velocity. Theoretically, the pore-water velocity can be estimated using the measured porosity (0.44) and flow rate (0.51 cm/h); this gives u = 1.16 cm/h. However, the velocity estimated in this way often failed to predict the observed breakthrough curves in most of our experiments. We therefore used a method developed by Zhang (1989) to estimate the pore water velocity from the breakthrough curve obtained from a purposely installed suction cup in soil right through the centre of the internal ring at r = 0. From Eq. (5) we know that when r = 0, the change of concentration at depth of z with time is given by ! m ðz utÞ2 cðr; z; tÞ ¼ . ð11Þ exp pffiffiffiffiffi 4DL t 4nDT DL ðptÞ3=2
From Eq. (11) we know that a new variable Y defined as follows ! m ðz utÞ2 3=2 ð12Þ exp Yðx; tÞ ¼ cðz; tÞt ¼ pffiffiffiffiffi 4DL t 4nDT DL ðpÞ3=2 peaks at z = ut, that is, when t* = z/u. The pore water velocity u can therefore be estimated from the peaking time t* of
Y, which can be calculated using the measured data, and the depth z where the suction cup is located. From the breakthrough curve measured from the cup located at z = 24 cm and r = 0, the estimated pore-water velocity is approximately u = 1.4 cm/h. The estimated pore-water velocity was used to calculate the two dispersion coefficients using the breakthrough curves obtained from other suction cups. One cup located at depth of 50 cm was fault, we therefore analysed the data obtained from other three cups, two at position z = 30 cm and r = 10 cm, and one at position of z = 50 cm and r = 20 cm. The estimated values of the DL and DT using Eq. (10) are DL = 4.5 cm2/h DT = 0.87 cm2/h. To evaluate the accuracy of these estimated parameters, we used them to predict the breakthrough curves obtained from the three suction cups by solving Eq. (1) with the associated initial and boundary conditions given in Eq. (2) and R0 = 5 cm. Fig. 2 compares the simulations and the measurements. Overall they match well.
Summary The work reported in this paper presents an in situ method to measure the longitudinal and transverse dispersion
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X. Zhang et al.
coefficients of solute transport in soil by modifying the double-ring infiltrometer into a triple-ring infiltrometer. Water flow is controlled in one dimension and solute transport in three dimensions. An analytical solution is derived to describe solute transport, and a simple method based on the analytical solution is proposed to estimate the two parameters. Field experiment was carried out to measure the parameters. The proposed analytical method for calculating the two parameters assumes that soil is homogenous where the pore-water velocity and the dispersion coefficients can be approximated as constants. The method is invalid for heterogeneous soil where the pore-water velocity and the solute dispersion coefficients may vary spatially. In this situation, it needs numerical solution to solve the transport equation. Also, the apparatus is designed to measure the longitudinal and transverse dispersion coefficients in saturated conditions. It might be possible to improve and modify the apparatus to measure the two parameters in unsaturated condition in a combination with the disc and ring infiltrometers (Angulo-Jaramillo et al., 2000). This, however, needs further work.
Acknowledgements We like to thank the two anonymous reviewers for their constructive comments to improve the manuscript. We also thank the Chinese Agro-Ecological Experimental Station in Shangqiu, Henan, for their financial support to this work.
Applying the inverse Laplace transform to Eq. (A3) and using the following formula: 2 pffiffi 1 1 x pffiffi h expðhx þ th2 Þ expðx sÞ ! pffiffiffiffiffi exp 4t sþh pt pffiffi x erfc pffiffi þ h t ; ðA4Þ 2 t we have ~c ¼
C0 J1 ðpR0 Þ expðDT p2 tÞ Fðz; tÞ; p
ðA5Þ
where ! 1 ðz utÞ2 u Fðz; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2DL 4DL t DL pt uz z þ ut ffiffiffiffiffiffiffi p . erfc exp DL 2 DL t
ðA6Þ
Applying the inverse Hankel transform to Eq. (A6) gives Z 1 cðr; z; tÞ ¼ C0 Fðz; tÞ J1 ðpR0 ÞJ0 ðprÞ expðDT p2 tÞ dp. ðA7Þ 0
A special case of Eq. (A7) is when the radius of the internal ring is small and its cross-section can be seen as a point. Under this condition, we have Z 1 C0 JðpR0 Þ expðDT p2 tÞ dp lim R0 !0 0 m r2 exp ; ðA8Þ ¼ 4DT t 4pnDT t Eq. (A7) is therefore simplified to
Appendix Applying the Laplace transform to Eqs. (1) and (2) with respect to t gives ! o2 ~c o2 ~c 1 o~c o~c u ; þ s~c ¼ DL 2 þ DT 2 oz or r or oz ~cðx; 1; tÞ ¼ 0; ðA1Þ ~cð1; r; tÞ ¼ 0; o~c C0 ¼ Hðr R0 Þ; u~c DL oz z¼0 R0 R1 where ~c ¼ 0 cest dt and C0 ¼ R0mpn. Applying the Hankel transform to Eq. (A1) with respect to r yields o2 ~c o~c ~c ¼ DL 2 DT p2 ~c u ; s oz oz ~cð1; r; tÞ ¼ 0; ðA2Þ c ~ o C ¼ 0 J1 ðpR0 Þ; u~c DL p oz z¼0 R 1 where ~c ¼ 0 r~cJ0 ðrpÞ dr. Solving Eq. (A2) for ~c yields ~c ¼
2C0 J1 ðpR0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p u2 þ 4DL ðDT p2 þ sÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! ux u2 þ 4DL ðDT p2 þ sÞ exp . 2DL ðu þ
ðA3Þ
! m ðz utÞ2 r2 cðr; z; tÞ ¼ exp pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2 2 um uz r z þ ut exp erfc pffiffiffiffiffiffiffi . 8pnDT DL t DL 4DT t 2 DL t ðA9Þ Like the analytical solution for solute transport in an infinite domain under a prescribed concentration, the second term on the right-hand side of Eq. (A9) is smaller than the first term when z is far away from soil surface. Under this condition, Eq. (A9) can be further simplified to ! m ðz utÞ2 r2 . ðA10Þ exp cðr; z; tÞ ¼ pffiffiffiffiffi 4DL t 4DT t 4nDT DL ðptÞ3=2
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