This article was downloaded by: [MacEwan University Libraries] On: 16 December 2014, At: 14:23 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Environmental Science and Health, Part A: Toxic/Hazardous Substances and Environmental Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lesa20
Analytical and experimental analysis of solute transport in heterogeneous porous media a
a
a
Lei Wu , Bin Gao , Yuan Tian & Rafael Muñoz-Carpena
a
a
Department of Agricultural and Biological Engineering , University of Florida , Gainesville , Florida , USA Published online: 26 Nov 2013.
Click for updates To cite this article: Lei Wu , Bin Gao , Yuan Tian & Rafael Muñoz-Carpena (2014) Analytical and experimental analysis of solute transport in heterogeneous porous media, Journal of Environmental Science and Health, Part A: Toxic/Hazardous Substances and Environmental Engineering, 49:3, 338-343, DOI: 10.1080/10934529.2014.846686 To link to this article: http://dx.doi.org/10.1080/10934529.2014.846686
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Journal of Environmental Science and Health, Part A (2014) 49, 338–343 C Taylor & Francis Group, LLC Copyright ISSN: 1093-4529 (Print); 1532-4117 (Online) DOI: 10.1080/10934529.2014.846686
Analytical and experimental analysis of solute transport in heterogeneous porous media ˜ LEI WU, BIN GAO, YUAN TIAN and RAFAEL MUNOZ-CARPENA
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Department of Agricultural and Biological Engineering, University of Florida, Gainesville, Florida, USA
Knowledge of solute transport in heterogeneous porous media is crucial to monitor contaminant fate and transport in soil and groundwater systems. In this study, we present new findings from experimental and mathematical analysis to improve current understanding of solute transport in structured heterogeneous porous media. Three saturated columns packed with different sand combinations were used to examine the breakthrough behavior of bromide, a conservative tracer. Experimental results showed that bromide had different breakthrough responses in the three types of sand combinations, indicating that heterogeneity in hydraulic conductivity has a significant effect on the solute transport in structured heterogeneous porous media. Simulations from analytical solutions of a two-domain solute transport model matched experimental breakthrough data well for all the experimental conditions tested. Experimental and model results show that under saturated flow conditions, advection dominates solute transport in both fast-flow and slow-flow domains. The sand with larger hydraulic conductivity provided a preferential flow path for solute transport (fast-flow domain) that dominates the mass transfer in the heterogeneous porous media. Importantly, the transport in the slowflow domain and mass exchange between the domains also contribute to the flow and solute transport processes and thus must be considered when investigating contaminant transport in heterogeneous porous media. Keywords: Solute transport, heterogeneous porous media, preferential flow, two domain, analytical model.
Introduction Preferential flow in soils has received considerable attention in the past few decades because it may provide a fast path for many pollutants to reach the groundwater.[1-3] It has been noted that preferential flow phenomena can occur in different types of porous media under various conditions.[4,5] The most common preferential flow, however, often arises from the heterogeneities of porous media resulting from either biological activity (e.g., root channeling, worm burrowing), hydrogeologic processes (e.g., secondary dissolution, desiccation cracking, and stress-induced fracturing), or agrotechnical practices (e.g., plowing, boring, and digging).[2,6,7] Thus, it is very important to investigate the transport behavior of solute in heterogeneous porous media. Both laboratory and field experimental studies have been conducted to evaluate the effect of heterogeneities on flow and transport in porous media.[8–12] A number of field studies have confirmed the adverse role for preferential flow in accelerating contaminant transport in heterogeneous ˜ Address correspondence to Bin Gao or Rafael Munoz-Carpena, Agricultural & Biological Engineering, University of Florida, P.O. Box 110570, Gainesville, FL 32611-0570, USA; E-mail: [email protected] or [email protected] Received May 23, 2013.
soils.[13,14] Packed or undisturbed soils columns are often used in laboratory as mimics in the laboratory to help to better identify mechanisms stemming from physical heterogeneities affecting the transport of solutes.[15–17] It has been demonstrated that laboratory columns are useful tools that have provided insight to advance current understanding and prediction capacity of water and solute transport in heterogeneous media.[15] Findings from experimental investigations, particularly laboratory column observations, have guided the development of mathematical models simulating solute transport in heterogeneous soils.[18] Most of the current models of preferential flow are complicated and rely heavily on numerical computations.[19-21] Some analytical models of flow and transport in heterogenious porous media have been developed for special conditions or with bounded approximations. Those models are very convienent and have been suggested to be robust tools for analying solutiion transport in heterogeneous soils.[22,23] Nevertheless, relatively much less research attention has been paid to analytical models of flow and transport in heterogeneous porous media, particularly with respect to experimental evaluations of the models. Here we report new measurements from laboratory experiments and theoretical calculations on solute transport through heterogeneous porous media. The overall objective of this work is to evaluate an analytical model
339
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Solute transport in heterogeneous porous media
Fig. 1. Columns with continuous structured heterogeneity packed with different types of sand combinations: column 1 packed with 0.1–0.2 mm (right) and 1.4–1.6 mm (left) sand, column 2 packed with 0.3–0.4 mm (right) and 0.7–0.9 mm (left) sand, and column 3 packed with 0.3–0.4 mm (left) and 0.4–0.5 mm (right) sand (color figure available online).
of solute transport in heterogeneous porous media. Laboratory experiments were conducted to determine the transport behavior of a tracer (bromide) in sand columns pakced with two types of porous media of different grain sizes. A mathematical model based on the two-domain concept was developed analytically, and model simulations were tested against the experimental data.
Materials and methods Porous media Quartz sand (Standard Sand & Silica Co.) was used as the porous media in the column experiments. Stainless steel sieves were used to separate the sand into five size classes: 0.1–0.2 mm, 0.3–0.4 mm, 0.4–0.5 mm, 0.7–0.9 mm, and 1.4–1.6 mm. Impurities, such as iron hydroxide and organic coatings, and very fine particles were removed from the sand-grain surfaces according to the method reported in Tian et al. [24,25]
acrylic plate was then slowly withdrawn. Three types of saturated, heterogeneous columns packed with different types of sand combinations were used in the experiments (i.e., three cases): (1) KFFD /KSFD ≈ 10, (2) KFFD /KSFD ≈ 2, and (3) KFFD /KSFD ≈ 1 (Fig. 1). Table 1 lists the basic properties of the columns. A peristaltic pump (Masterflex L/S, Cole Parmer Instruments, Vernon Hills, IL) was connected to the inlet at the top of the column to regulate the downward flow at a constant specific discharge of 0.4 cm/min. The influent concentrations of bromide used in the experiments equaled 40 mg/L. DI water was first pumped through the saturated column for 1 h to flush and equilibrate the column. The breakthrough experiment was then initiated by switching from DI water to the bromide solution for 30 min; then the column was flushed with DI water again for 90 min. Effluent samples were collected from base of the column with a
Table 1. Basic properties of the structured heterogeneous columns used in the laboratory experiments.
Column experiments
Column #
dSFD (mm)
dFFD (mm)
θ SFD
θ FFD
KSFD (cm/s)
KFFD (cm/s)
The sand was packed into an acrylic column measuring 2.5 cm in diameter and 15 cm in height. Columns with a single structured, heterogeneity were constructed by placing a thin-walled acrylic plate (thickness < 1 mm) in the center of the column to separate it into two identical regions. To make a structured, heterogeneous porous medium column, sand of two different size classes was carefully wet-packed into the two regions of the column using a procedure similar to Tian et al.[26,27] After the column was packed, the
1 2 3
0.1–0.2 0.3–0.4 0.3–0.4
1.4–1.6 0.7–0.9 0.4–0.5
0.34 0.34 0.34
0.38 0.35 0.34
0.11 0.27 0.27
1.21 0.60 0.32
d, θ, and K represent diameter of sand, moisture content, and hydraulic conductivity, respectively; subscript SFD and FFD represent slow flow domain and fast flow domain, respectively. The moisture contents were determined volumetrically (volume of water/volume of column).The saturated hydraulic conductivities were determined by the constant head method based on Darcy’s Law.
340
Wu et al.
fraction collector. Bromide concentrations in the samples were measured with an Ion Chromatograph (Dionex ICS-90, Sunnyvale, CA, USA). Mathematical model
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Based on the two-domain (dual-permeability) concept,[28,29] we hypothesize that the transport of solute within the heterogeneous columns can be described with two advection-dispersion equations coupled by a term characterizing the exchange of solute between a fast and a slow flow domain: ∂ 2 CFFD ∂CFFD ∂CFFD = DFFD 2 − vFFD − α(CFFD − CSFD ) ∂t ∂ x ∂x (1) ∂ 2 CSFD ∂CSFD ∂CSFD = DSFD 2 − vSFD − α(CSFD − CFFD ) ∂t ∂ x ∂x (2) where the subscripts FFD and SFD refer to the fast flow domain and slow flow domain, respectively; C is the solute concentration in pore water (M L−3); D is the dispersion coefficient (L2 T−1); v is the velocity of pore water (L T−1); α is a first-order mass transfer coefficient for solute exchange between the fast-flow and slow-flow domains (L T−1). When α is small, Eqs. 1 and 2 can be solved using a regular perturbation technique for zero initial concentration, first-type inlet boundary, and semi-infinite outlet boundary conditions.[28] Because of the symmetry in the equations for CFFD and CSFD, the solution is presented here only for CFFD : CFFD =
l
α j CFFDj
(3)
j =0
For a pulse input (C0 ), the first term (j = 0) and the subsequent terms (j > 0) of the CFFD can be written as: x − vFFD t 1 erfc √ CFFD0 = C0 2 4D t FFD vFFD x x + vFFD t (4) + exp erfc √ DFFD 4DFFD t v2 t vFFD CFFDj = exp − FFD 2D 4DFFD t FFD ∞ × h j (y, τ ) G FFD (x, t; y, τ ) dydτ (5) 0
0
C −CFFDj−1 where h j (x, t) = SFDj−1 , θFFD (x−y)2 (x+y)2 exp[− 4D (t−τ ) ]−exp[− 4D (t−τ ) ] FFD FFD √ , and y 2 π DFFD (t−τ )
G FFD (x, t; y, τ ) = and τ are imaginary
time variables. Then the effluent (breakthrough) concentration then can be determined from the general solution of each domain by: C=
CFFD θFFD vFFD + CSFD θSFD vSFD θFFD vFFD + θSFD vSFD
(6)
where θ FFD and θ SFD are water content in the FFD and SFD, respectively. In this work, subsequent terms (j > 0) of the CFFD were obtained recursively and expressed as 2j multiple integral. Because the contributions from the higher terms (j > 1) were negligible (i.e., )only the first two terms (j = 0, 1) of the CFFDj were numerically evaluated. The Levenberg–Marquardt algorithm was used to estimate the value of the model parameters that minimized the sumof-the-squares differences between model-calculated and measured breakthrough concentrations.
Results and discussion Figure 2 shows the influence of physical heterogeneity factor (i.e., sand size and hydraulic conductivity (K)) on solute transport in structured heterogeneous columns. For the case (1), the transport of bromide in the column exhibited almost two separated breakthrough curves, corresponding to the quick breakthrough from the FFD and the slow breakthrough from the SFD (Fig. 2a). Bromide showed a different transport behavior in the case (2) and the breakthrough curve showed a period of rapidly increasing concentrations to reflect the delivery of solute through the FFD flow path followed by a brief period of slowly increasing concentrations, reflecting the transfer of solute from SFD to FFD (Fig. 2b). The breakthrough curve then climbed again for a second period of rapid concentration increasing, reflecting the delivery of solute through the SFD flow path. The descending limbs of the breakthrough curves also exhibited this stairstep pattern, presumably due to the flushing of the coarse-grained domain flow path with bromide-free water, followed by displacement of the bromide-containing water from the SFD. For the case (3), the bromide exhibited the typical breakthrough behaviors observed in homogenous porous media: a brief period of rapidly increasing concentrations followed by a plateau, and then rapidly reduced when the input of bromide was switched to electrolyte solution (Fig. 2c), indicating the porous medium heterogeneity was not sufficient enough to generate the preferential flow in the column. The column experimental results indicated that heterogeneity in hydraulic conductivity is one of the dominant physical factors governing the preferential flow and transport behavior in structured, heterogeneous porous media. The analytical model simulations reproduced these breakthrough characteristics closely (Fig. 2), with computation of coefficients of determination (R2) exceeding 0.96 for the all three cases tested. For the case (1), which was characterized by the greatest contrast in the hydraulic conductivity corresponding to the sand size in the two domains, the best-fit values of the pore-water velocity were near five times greater in the fast flow domain (vFFD ) than in the slow flow domain (vSFD ) (Table 2). Pore-water velocities were less than twice greater in the fast flow domain than
341
Solute transport in heterogeneous porous media 1
1
Model
Model Obs
0.8
0.8
0.6
C/C0
C/C0
0.6
0.4
0.4
0.2
0.2
(a)
(b)
0
0 0.0
2.0
4.0
6.0
8.0
0.0
10.0
2.0
4.0
6.0
8.0
10.0
PV
PV 1 Model Obs
0.8
0.6
C/C0
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Obs
0.4
0.2 (c) 0 0.0
2.0
4.0
6.0
8.0
10.0
PV
Fig. 2. Solute transport in saturated heterogeneous columns: (a) column 1, (b) column 2 and (c) column 3.
in the slow flow domain for the case (2), as the two pore water velocities were similar for the case (3), which was also in agreement with the trend of hydraulic conductivity comparisons. The model-estimated dispersion coefficients (D) varied by a factor of two ranging from 0.06 to 0.13 cm2 min−1 (Table 2).
For all three cases, D was greater for the fast flow domain than that for the slow flow domain, which is consistent with published observations that D varies proportionately with flow velocity.[30] Calculations of the Peclet number (Pe = vDH ), where H is the height of the column for three cases ranged from 92 to 225, 133 to 134 and 16 to 152 for the
Table 2. Best-fit model parameters and calculated mass recovery rates for solution transport in the heterogeneous columns. Column # vFFD (cm/min) vSFD (cm/min) DFFD (cm2/min) DSFD (cm2/min) α (cm/min) MRFFD (%) MRSFD (%) MRTotal (%) 1 2 3
1.65 1.21 0.99
0.37 0.46 0.79
0.11 0.13 0.10
0.06 0.08 0.09
0.0021 0.0053 0.0064
82.1 56.3 51.5
17.2 42.9 48.1
99.3 99.2 99.6
v, D, α, and MR represent pore velocity, dispersion coefficient, first-order mass transfer coefficient, and mass recovery rate, respectively; subscript SFD, FFD represent slow flow domain and fast flow domain, respectively.
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
342 three cases, respectively, which indicate that advection dominated solute transport in both domains for all conditions tested. The best-fit values of the mass-transfer coefficient (α) were sensitive to the experimental conditions tested and decreased with the ratio of KFFD /KSFD (Table 2), which indicated that the heterogeneity in the structured soils may decrease the efficiency of solute transport between two domains. Calculations of the Damkohler number for mass transfer (Da = αvH . ) for three cases ranged from 0.017 to 0.07, 0.06 to 0.11 and 0.12 to 0.21, respectively, which revealed that inter-domain exchange showed little impact on the final fate of bromide in the columns. Model simulations showed that although transport in the FFD dominated the solute mass transfer processes, solute transport within the SFD also contributed depending on the experimental conditions (Table 2). In the case (1), bromide recovered from the FFD (82.1%) account for most part of the total mass recovery (99.3%). In other two cases, however, less bromide was transported through the fast flow domains (Table 2). The success of the analytical model also provided an opportunity to examine the influences of inter-domain exchanges on solute transport in structured heterogeneous porous media. In the model, the mass exchange between the domains can be “turned off” by letting α be zero. The influences of mass exchange on solute transport then can be quantified by running the model with/without the exchange term. Simulations using experimental conditions of case (2) did not have much effect on the total bromide mass trans-
Wu et al. fer through the heterogeneous column, which is consistent with the analysis of the Damkohler number as discussed before. Notable differences, however, were observed when comparing the two breakthrough curves with/without the exchange term (Fig. 3), suggesting that mass exchange between the two domains should not be ignored when modeling solute transport in heterogeneous porous media.
Conclusions Results from this study indicated that the analytical solution of the two-domain model can be applied to describe solute transport in heterogeneous porous media. Experimental breakthrough data showed that heterogeneity in hydraulic conductivity (i.e., KFFD /KSFD ) is one of the dominant physical factors governing the preferential flow and solute transport in structured heterogeneous porous media. Model results suggested that advection dominated solute transport in both fast-flow and slow-flow domains for all the tested conditions. Findings from this study also indicated that inter-domain mass exchanges should be considered when investigating contaminant transport in heterogeneous porous media.
Acknowledgments This work was partially supported by the NSF through grant CHE-1213333.
References
Fig. 3. Model simulated breakthrough curves of solute transport through structured heterogeneous porous media (column 2) with/without solute exchange.
[1] Morales, V.L.; Parlange, J.Y.; Steenhuis, T.S. Are preferential flow paths perpetuated by microbial activity in the soil matrix? A review. J. Hydrol. 2010, 393, 29–36. [2] Baveye, P.; Boast, C.W.; Ogawa, S.; Parlange, J.Y.; Steenhuis, T. Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils. Water Resour. Res. 1998, 34, 2783–2796. [3] Selker, J.; Parlange, J.Y.; Steenhuis, T. Fingered flow in 2 dimensions. 2. Predicting finger moisture profile. Water Resour. Res. 1992, 28, 2523–2528. [4] Hill, D.E.; Parlange, J.Y. Wetting front instability in layered soils. Soil Sci. Soc. Am. Proc. 1972, 36, 697–699. [5] Parlange, J.Y.; Steenhuis, T.S.; Li, L.; Barry, D.A.; Stagnitti, F. Column flow in stratified soils and fingers in hele-shaw cells: A review. In Environmental Mechanics: Water, Mass and Energy Transfer in the Biosphere; Raats, P.a.C., Smiles, D., Warrick, A.W. Eds.; AGU Geophysical Monograph: Washington, DC, 2002; 79–85. [6] Germann, P.F.; DiPietro, L. When is porous-media flow preferential? A hydromechanical perspective. Geoderma 1996, 74, 1–21. [7] Beven, K.; Germann, P. Macropores and water-flow in soils. Water Resour. Res. 1982, 18, 1311–1325. [8] Darnault, C.J.G.; Steenhuis, T.S.; Garnier, P.; Kim, Y.J.; Jenkins, M.B.; Ghiorse, W.C.; Baveye, P.C.; Parlange, J.Y. Preferential flow and transport of cryptosporidium parvum oocysts through the vadose zone: Experiments and modeling. Vadose Zone J. 2004, 3, 262–270. [9] Liu, Y.P.; Bierck, B.R.; Selker, J.S.; Steenhuis, T.S.; Parlange, J.Y. High-intensity x-ray and tensiometer measurements in rapidly
Solute transport in heterogeneous porous media
[10]
[11]
[12]
[13]
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
[14]
[15]
[16] [17]
[18]
[19]
changing preferential flow-fields. Soil Sci. Soc. Am. J. 1993, 57, 1188–1192. Selker, J.S.; Parlange, J.Y.; Steenhuis, T.S. Comments on laboratory tests of a theory of fingering during infiltration into layered soils. Soil Sci. Soc. Am. J. 1991, 55, 896–896. Steenhuis, T.S.; Parlange, J.Y.; Andreini, M.S. A numerical-model for preferential solute movement in structured soils. Geoderma 1990, 46, 193–208. Wallach, R.; Parlange, J.Y. Applying the boundary layer concept to model transport of dissolved chemicals in preferential flow paths. Water Resour. Res. 2000, 36, 2845–2851. Zheng, C.; Bianchi, M.; Gorelick, S.M. Lessons learned from 25 years of research at the made site. Ground Water 2011, 49, 649–662. Ward, A.L.; Kachanoski, R.G.; Vonbertoldi, A.P.; Elrick, D.E. Field and undisturbed-column measurements for predicting transport in unsaturated layered soil. Soil Sci. Soc. Am. J. 1995, 59, 52–59. Dousset, S.; Thevenot, M.; Pot, V.; Simunek, J.; Andreux, F. Evaluating equilibrium and non-equilibrium transport of bromide and isoproturon in disturbed and undisturbed soil columns. J. Contam Hydrol. 2007, 94, 261–276. Cortis, A.; Berkowitz, B. Anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J. 2004, 68, 1539–1548. Bosma, W.J.P.; Vanderzee, S. Transport of reacting solute in a onedimensional, chemically heterogeneous porous-medium. Water Resour. Res. 1993, 29, 117–131. Simunek, J.; Jarvis, N.J.; van Genuchten, M.T.; Gardenas, A. Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J. Hydrol. 2003, 272, 14–35. Hu, B.X.; Huang, H.; Zhang, D.X. Stochastic analysis of solute transport in heterogeneous, dual-permeability media. Water Resour. Res. 2002, 38, 14-1–14-16.
343 [20] Hu, B.X.; Huang, H. Stochastic analysis of reactive solute transport in heterogeneous, fractured porous media: A dual-permeability approach. Transport Porous Med. 2002, 48, 1–39. [21] Zhang, D.X.; Sun, A.Y. Stochastic analysis of transient saturated flow through heterogeneous fractured porous media: A doublepermeability approach. Water Resour. Res. 2000, 36, 865–874. [22] Kim, Y.J.; Darnault, C.J.G.; Bailey, N.O.; Parlange, J.Y.; Steenhuis, T.S. Equation for describing solute transport in field soils with preferential flow paths. Soil Sci. Soc. Am. J. 2005, 69, 291–300. [23] Steenhuis, T.S.; Boll, J.; Shalit, G.; Selker, J.S.; Merwin, I.A. A simple equation for predicting preferential flow solute concentrations. J. Environ. Qual. 1994, 23, 1058–1064. [24] Tian, Y.; Gao, B.; Ziegler, K.J. High mobility of SDBS-dispersed single-walled carbon nanotubes in saturated and unsaturated porous media. J. Hazard. Mater. 2011, 186, 1766–1772. [25] Tian, Y.A.; Gao, B.; Silvera-Batista, C.; Ziegler, K.J. Transport of engineered nanoparticles in saturated porous media. J. Nanopart. Res. 2010, 12, 2371–2380. [26] Tian, Y.; Gao, B.; Morales, V.L.; Wu, L.; Wang, Y.; MunozCarpena, R.; Cao, C.; Huang, Q.; Yang, L. Methods of using carbon nanotubes as filter media to remove aqueous heavy metals. Chem. Eng. J. 2012, 210, 557–563. [27] Tian, Y.; Gao, B.; Wang, Y.; Morales, V.L.; Carpena, R.M.; Huang, Q.; Yang, L. Deposition and transport of functionalized carbon nanotubes in water-saturated sand columns. J. Hazard. Mater. 2012, 213, 265–272. [28] Skopp, J.; Gardner, W.R.; Tyler, E.J. Solute movement in structured soils – 2-region model with small interaction. Soil Sci. Soc. Am. J. 1981, 45, 837–842. [29] Gerke, H.H.; Vangenuchten, M.T. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous-media. Water Resour. Res. 1993, 29, 305–319. [30] Marino, M.A. Longitudinal dispersion in saturated porous-media. J. Hydraul. Div.-Asce. 1974, 100, 151–156.
Journal of Environmental Science and Health, Part A: Toxic/Hazardous Substances and Environmental Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lesa20
Analytical and experimental analysis of solute transport in heterogeneous porous media a
a
a
Lei Wu , Bin Gao , Yuan Tian & Rafael Muñoz-Carpena
a
a
Department of Agricultural and Biological Engineering , University of Florida , Gainesville , Florida , USA Published online: 26 Nov 2013.
Click for updates To cite this article: Lei Wu , Bin Gao , Yuan Tian & Rafael Muñoz-Carpena (2014) Analytical and experimental analysis of solute transport in heterogeneous porous media, Journal of Environmental Science and Health, Part A: Toxic/Hazardous Substances and Environmental Engineering, 49:3, 338-343, DOI: 10.1080/10934529.2014.846686 To link to this article: http://dx.doi.org/10.1080/10934529.2014.846686
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
Journal of Environmental Science and Health, Part A (2014) 49, 338–343 C Taylor & Francis Group, LLC Copyright ISSN: 1093-4529 (Print); 1532-4117 (Online) DOI: 10.1080/10934529.2014.846686
Analytical and experimental analysis of solute transport in heterogeneous porous media ˜ LEI WU, BIN GAO, YUAN TIAN and RAFAEL MUNOZ-CARPENA
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Department of Agricultural and Biological Engineering, University of Florida, Gainesville, Florida, USA
Knowledge of solute transport in heterogeneous porous media is crucial to monitor contaminant fate and transport in soil and groundwater systems. In this study, we present new findings from experimental and mathematical analysis to improve current understanding of solute transport in structured heterogeneous porous media. Three saturated columns packed with different sand combinations were used to examine the breakthrough behavior of bromide, a conservative tracer. Experimental results showed that bromide had different breakthrough responses in the three types of sand combinations, indicating that heterogeneity in hydraulic conductivity has a significant effect on the solute transport in structured heterogeneous porous media. Simulations from analytical solutions of a two-domain solute transport model matched experimental breakthrough data well for all the experimental conditions tested. Experimental and model results show that under saturated flow conditions, advection dominates solute transport in both fast-flow and slow-flow domains. The sand with larger hydraulic conductivity provided a preferential flow path for solute transport (fast-flow domain) that dominates the mass transfer in the heterogeneous porous media. Importantly, the transport in the slowflow domain and mass exchange between the domains also contribute to the flow and solute transport processes and thus must be considered when investigating contaminant transport in heterogeneous porous media. Keywords: Solute transport, heterogeneous porous media, preferential flow, two domain, analytical model.
Introduction Preferential flow in soils has received considerable attention in the past few decades because it may provide a fast path for many pollutants to reach the groundwater.[1-3] It has been noted that preferential flow phenomena can occur in different types of porous media under various conditions.[4,5] The most common preferential flow, however, often arises from the heterogeneities of porous media resulting from either biological activity (e.g., root channeling, worm burrowing), hydrogeologic processes (e.g., secondary dissolution, desiccation cracking, and stress-induced fracturing), or agrotechnical practices (e.g., plowing, boring, and digging).[2,6,7] Thus, it is very important to investigate the transport behavior of solute in heterogeneous porous media. Both laboratory and field experimental studies have been conducted to evaluate the effect of heterogeneities on flow and transport in porous media.[8–12] A number of field studies have confirmed the adverse role for preferential flow in accelerating contaminant transport in heterogeneous ˜ Address correspondence to Bin Gao or Rafael Munoz-Carpena, Agricultural & Biological Engineering, University of Florida, P.O. Box 110570, Gainesville, FL 32611-0570, USA; E-mail: [email protected] or [email protected] Received May 23, 2013.
soils.[13,14] Packed or undisturbed soils columns are often used in laboratory as mimics in the laboratory to help to better identify mechanisms stemming from physical heterogeneities affecting the transport of solutes.[15–17] It has been demonstrated that laboratory columns are useful tools that have provided insight to advance current understanding and prediction capacity of water and solute transport in heterogeneous media.[15] Findings from experimental investigations, particularly laboratory column observations, have guided the development of mathematical models simulating solute transport in heterogeneous soils.[18] Most of the current models of preferential flow are complicated and rely heavily on numerical computations.[19-21] Some analytical models of flow and transport in heterogenious porous media have been developed for special conditions or with bounded approximations. Those models are very convienent and have been suggested to be robust tools for analying solutiion transport in heterogeneous soils.[22,23] Nevertheless, relatively much less research attention has been paid to analytical models of flow and transport in heterogeneous porous media, particularly with respect to experimental evaluations of the models. Here we report new measurements from laboratory experiments and theoretical calculations on solute transport through heterogeneous porous media. The overall objective of this work is to evaluate an analytical model
339
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Solute transport in heterogeneous porous media
Fig. 1. Columns with continuous structured heterogeneity packed with different types of sand combinations: column 1 packed with 0.1–0.2 mm (right) and 1.4–1.6 mm (left) sand, column 2 packed with 0.3–0.4 mm (right) and 0.7–0.9 mm (left) sand, and column 3 packed with 0.3–0.4 mm (left) and 0.4–0.5 mm (right) sand (color figure available online).
of solute transport in heterogeneous porous media. Laboratory experiments were conducted to determine the transport behavior of a tracer (bromide) in sand columns pakced with two types of porous media of different grain sizes. A mathematical model based on the two-domain concept was developed analytically, and model simulations were tested against the experimental data.
Materials and methods Porous media Quartz sand (Standard Sand & Silica Co.) was used as the porous media in the column experiments. Stainless steel sieves were used to separate the sand into five size classes: 0.1–0.2 mm, 0.3–0.4 mm, 0.4–0.5 mm, 0.7–0.9 mm, and 1.4–1.6 mm. Impurities, such as iron hydroxide and organic coatings, and very fine particles were removed from the sand-grain surfaces according to the method reported in Tian et al. [24,25]
acrylic plate was then slowly withdrawn. Three types of saturated, heterogeneous columns packed with different types of sand combinations were used in the experiments (i.e., three cases): (1) KFFD /KSFD ≈ 10, (2) KFFD /KSFD ≈ 2, and (3) KFFD /KSFD ≈ 1 (Fig. 1). Table 1 lists the basic properties of the columns. A peristaltic pump (Masterflex L/S, Cole Parmer Instruments, Vernon Hills, IL) was connected to the inlet at the top of the column to regulate the downward flow at a constant specific discharge of 0.4 cm/min. The influent concentrations of bromide used in the experiments equaled 40 mg/L. DI water was first pumped through the saturated column for 1 h to flush and equilibrate the column. The breakthrough experiment was then initiated by switching from DI water to the bromide solution for 30 min; then the column was flushed with DI water again for 90 min. Effluent samples were collected from base of the column with a
Table 1. Basic properties of the structured heterogeneous columns used in the laboratory experiments.
Column experiments
Column #
dSFD (mm)
dFFD (mm)
θ SFD
θ FFD
KSFD (cm/s)
KFFD (cm/s)
The sand was packed into an acrylic column measuring 2.5 cm in diameter and 15 cm in height. Columns with a single structured, heterogeneity were constructed by placing a thin-walled acrylic plate (thickness < 1 mm) in the center of the column to separate it into two identical regions. To make a structured, heterogeneous porous medium column, sand of two different size classes was carefully wet-packed into the two regions of the column using a procedure similar to Tian et al.[26,27] After the column was packed, the
1 2 3
0.1–0.2 0.3–0.4 0.3–0.4
1.4–1.6 0.7–0.9 0.4–0.5
0.34 0.34 0.34
0.38 0.35 0.34
0.11 0.27 0.27
1.21 0.60 0.32
d, θ, and K represent diameter of sand, moisture content, and hydraulic conductivity, respectively; subscript SFD and FFD represent slow flow domain and fast flow domain, respectively. The moisture contents were determined volumetrically (volume of water/volume of column).The saturated hydraulic conductivities were determined by the constant head method based on Darcy’s Law.
340
Wu et al.
fraction collector. Bromide concentrations in the samples were measured with an Ion Chromatograph (Dionex ICS-90, Sunnyvale, CA, USA). Mathematical model
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Based on the two-domain (dual-permeability) concept,[28,29] we hypothesize that the transport of solute within the heterogeneous columns can be described with two advection-dispersion equations coupled by a term characterizing the exchange of solute between a fast and a slow flow domain: ∂ 2 CFFD ∂CFFD ∂CFFD = DFFD 2 − vFFD − α(CFFD − CSFD ) ∂t ∂ x ∂x (1) ∂ 2 CSFD ∂CSFD ∂CSFD = DSFD 2 − vSFD − α(CSFD − CFFD ) ∂t ∂ x ∂x (2) where the subscripts FFD and SFD refer to the fast flow domain and slow flow domain, respectively; C is the solute concentration in pore water (M L−3); D is the dispersion coefficient (L2 T−1); v is the velocity of pore water (L T−1); α is a first-order mass transfer coefficient for solute exchange between the fast-flow and slow-flow domains (L T−1). When α is small, Eqs. 1 and 2 can be solved using a regular perturbation technique for zero initial concentration, first-type inlet boundary, and semi-infinite outlet boundary conditions.[28] Because of the symmetry in the equations for CFFD and CSFD, the solution is presented here only for CFFD : CFFD =
l
α j CFFDj
(3)
j =0
For a pulse input (C0 ), the first term (j = 0) and the subsequent terms (j > 0) of the CFFD can be written as: x − vFFD t 1 erfc √ CFFD0 = C0 2 4D t FFD vFFD x x + vFFD t (4) + exp erfc √ DFFD 4DFFD t v2 t vFFD CFFDj = exp − FFD 2D 4DFFD t FFD ∞ × h j (y, τ ) G FFD (x, t; y, τ ) dydτ (5) 0
0
C −CFFDj−1 where h j (x, t) = SFDj−1 , θFFD (x−y)2 (x+y)2 exp[− 4D (t−τ ) ]−exp[− 4D (t−τ ) ] FFD FFD √ , and y 2 π DFFD (t−τ )
G FFD (x, t; y, τ ) = and τ are imaginary
time variables. Then the effluent (breakthrough) concentration then can be determined from the general solution of each domain by: C=
CFFD θFFD vFFD + CSFD θSFD vSFD θFFD vFFD + θSFD vSFD
(6)
where θ FFD and θ SFD are water content in the FFD and SFD, respectively. In this work, subsequent terms (j > 0) of the CFFD were obtained recursively and expressed as 2j multiple integral. Because the contributions from the higher terms (j > 1) were negligible (i.e., )only the first two terms (j = 0, 1) of the CFFDj were numerically evaluated. The Levenberg–Marquardt algorithm was used to estimate the value of the model parameters that minimized the sumof-the-squares differences between model-calculated and measured breakthrough concentrations.
Results and discussion Figure 2 shows the influence of physical heterogeneity factor (i.e., sand size and hydraulic conductivity (K)) on solute transport in structured heterogeneous columns. For the case (1), the transport of bromide in the column exhibited almost two separated breakthrough curves, corresponding to the quick breakthrough from the FFD and the slow breakthrough from the SFD (Fig. 2a). Bromide showed a different transport behavior in the case (2) and the breakthrough curve showed a period of rapidly increasing concentrations to reflect the delivery of solute through the FFD flow path followed by a brief period of slowly increasing concentrations, reflecting the transfer of solute from SFD to FFD (Fig. 2b). The breakthrough curve then climbed again for a second period of rapid concentration increasing, reflecting the delivery of solute through the SFD flow path. The descending limbs of the breakthrough curves also exhibited this stairstep pattern, presumably due to the flushing of the coarse-grained domain flow path with bromide-free water, followed by displacement of the bromide-containing water from the SFD. For the case (3), the bromide exhibited the typical breakthrough behaviors observed in homogenous porous media: a brief period of rapidly increasing concentrations followed by a plateau, and then rapidly reduced when the input of bromide was switched to electrolyte solution (Fig. 2c), indicating the porous medium heterogeneity was not sufficient enough to generate the preferential flow in the column. The column experimental results indicated that heterogeneity in hydraulic conductivity is one of the dominant physical factors governing the preferential flow and transport behavior in structured, heterogeneous porous media. The analytical model simulations reproduced these breakthrough characteristics closely (Fig. 2), with computation of coefficients of determination (R2) exceeding 0.96 for the all three cases tested. For the case (1), which was characterized by the greatest contrast in the hydraulic conductivity corresponding to the sand size in the two domains, the best-fit values of the pore-water velocity were near five times greater in the fast flow domain (vFFD ) than in the slow flow domain (vSFD ) (Table 2). Pore-water velocities were less than twice greater in the fast flow domain than
341
Solute transport in heterogeneous porous media 1
1
Model
Model Obs
0.8
0.8
0.6
C/C0
C/C0
0.6
0.4
0.4
0.2
0.2
(a)
(b)
0
0 0.0
2.0
4.0
6.0
8.0
0.0
10.0
2.0
4.0
6.0
8.0
10.0
PV
PV 1 Model Obs
0.8
0.6
C/C0
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
Obs
0.4
0.2 (c) 0 0.0
2.0
4.0
6.0
8.0
10.0
PV
Fig. 2. Solute transport in saturated heterogeneous columns: (a) column 1, (b) column 2 and (c) column 3.
in the slow flow domain for the case (2), as the two pore water velocities were similar for the case (3), which was also in agreement with the trend of hydraulic conductivity comparisons. The model-estimated dispersion coefficients (D) varied by a factor of two ranging from 0.06 to 0.13 cm2 min−1 (Table 2).
For all three cases, D was greater for the fast flow domain than that for the slow flow domain, which is consistent with published observations that D varies proportionately with flow velocity.[30] Calculations of the Peclet number (Pe = vDH ), where H is the height of the column for three cases ranged from 92 to 225, 133 to 134 and 16 to 152 for the
Table 2. Best-fit model parameters and calculated mass recovery rates for solution transport in the heterogeneous columns. Column # vFFD (cm/min) vSFD (cm/min) DFFD (cm2/min) DSFD (cm2/min) α (cm/min) MRFFD (%) MRSFD (%) MRTotal (%) 1 2 3
1.65 1.21 0.99
0.37 0.46 0.79
0.11 0.13 0.10
0.06 0.08 0.09
0.0021 0.0053 0.0064
82.1 56.3 51.5
17.2 42.9 48.1
99.3 99.2 99.6
v, D, α, and MR represent pore velocity, dispersion coefficient, first-order mass transfer coefficient, and mass recovery rate, respectively; subscript SFD, FFD represent slow flow domain and fast flow domain, respectively.
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
342 three cases, respectively, which indicate that advection dominated solute transport in both domains for all conditions tested. The best-fit values of the mass-transfer coefficient (α) were sensitive to the experimental conditions tested and decreased with the ratio of KFFD /KSFD (Table 2), which indicated that the heterogeneity in the structured soils may decrease the efficiency of solute transport between two domains. Calculations of the Damkohler number for mass transfer (Da = αvH . ) for three cases ranged from 0.017 to 0.07, 0.06 to 0.11 and 0.12 to 0.21, respectively, which revealed that inter-domain exchange showed little impact on the final fate of bromide in the columns. Model simulations showed that although transport in the FFD dominated the solute mass transfer processes, solute transport within the SFD also contributed depending on the experimental conditions (Table 2). In the case (1), bromide recovered from the FFD (82.1%) account for most part of the total mass recovery (99.3%). In other two cases, however, less bromide was transported through the fast flow domains (Table 2). The success of the analytical model also provided an opportunity to examine the influences of inter-domain exchanges on solute transport in structured heterogeneous porous media. In the model, the mass exchange between the domains can be “turned off” by letting α be zero. The influences of mass exchange on solute transport then can be quantified by running the model with/without the exchange term. Simulations using experimental conditions of case (2) did not have much effect on the total bromide mass trans-
Wu et al. fer through the heterogeneous column, which is consistent with the analysis of the Damkohler number as discussed before. Notable differences, however, were observed when comparing the two breakthrough curves with/without the exchange term (Fig. 3), suggesting that mass exchange between the two domains should not be ignored when modeling solute transport in heterogeneous porous media.
Conclusions Results from this study indicated that the analytical solution of the two-domain model can be applied to describe solute transport in heterogeneous porous media. Experimental breakthrough data showed that heterogeneity in hydraulic conductivity (i.e., KFFD /KSFD ) is one of the dominant physical factors governing the preferential flow and solute transport in structured heterogeneous porous media. Model results suggested that advection dominated solute transport in both fast-flow and slow-flow domains for all the tested conditions. Findings from this study also indicated that inter-domain mass exchanges should be considered when investigating contaminant transport in heterogeneous porous media.
Acknowledgments This work was partially supported by the NSF through grant CHE-1213333.
References
Fig. 3. Model simulated breakthrough curves of solute transport through structured heterogeneous porous media (column 2) with/without solute exchange.
[1] Morales, V.L.; Parlange, J.Y.; Steenhuis, T.S. Are preferential flow paths perpetuated by microbial activity in the soil matrix? A review. J. Hydrol. 2010, 393, 29–36. [2] Baveye, P.; Boast, C.W.; Ogawa, S.; Parlange, J.Y.; Steenhuis, T. Influence of image resolution and thresholding on the apparent mass fractal characteristics of preferential flow patterns in field soils. Water Resour. Res. 1998, 34, 2783–2796. [3] Selker, J.; Parlange, J.Y.; Steenhuis, T. Fingered flow in 2 dimensions. 2. Predicting finger moisture profile. Water Resour. Res. 1992, 28, 2523–2528. [4] Hill, D.E.; Parlange, J.Y. Wetting front instability in layered soils. Soil Sci. Soc. Am. Proc. 1972, 36, 697–699. [5] Parlange, J.Y.; Steenhuis, T.S.; Li, L.; Barry, D.A.; Stagnitti, F. Column flow in stratified soils and fingers in hele-shaw cells: A review. In Environmental Mechanics: Water, Mass and Energy Transfer in the Biosphere; Raats, P.a.C., Smiles, D., Warrick, A.W. Eds.; AGU Geophysical Monograph: Washington, DC, 2002; 79–85. [6] Germann, P.F.; DiPietro, L. When is porous-media flow preferential? A hydromechanical perspective. Geoderma 1996, 74, 1–21. [7] Beven, K.; Germann, P. Macropores and water-flow in soils. Water Resour. Res. 1982, 18, 1311–1325. [8] Darnault, C.J.G.; Steenhuis, T.S.; Garnier, P.; Kim, Y.J.; Jenkins, M.B.; Ghiorse, W.C.; Baveye, P.C.; Parlange, J.Y. Preferential flow and transport of cryptosporidium parvum oocysts through the vadose zone: Experiments and modeling. Vadose Zone J. 2004, 3, 262–270. [9] Liu, Y.P.; Bierck, B.R.; Selker, J.S.; Steenhuis, T.S.; Parlange, J.Y. High-intensity x-ray and tensiometer measurements in rapidly
Solute transport in heterogeneous porous media
[10]
[11]
[12]
[13]
Downloaded by [MacEwan University Libraries] at 14:23 16 December 2014
[14]
[15]
[16] [17]
[18]
[19]
changing preferential flow-fields. Soil Sci. Soc. Am. J. 1993, 57, 1188–1192. Selker, J.S.; Parlange, J.Y.; Steenhuis, T.S. Comments on laboratory tests of a theory of fingering during infiltration into layered soils. Soil Sci. Soc. Am. J. 1991, 55, 896–896. Steenhuis, T.S.; Parlange, J.Y.; Andreini, M.S. A numerical-model for preferential solute movement in structured soils. Geoderma 1990, 46, 193–208. Wallach, R.; Parlange, J.Y. Applying the boundary layer concept to model transport of dissolved chemicals in preferential flow paths. Water Resour. Res. 2000, 36, 2845–2851. Zheng, C.; Bianchi, M.; Gorelick, S.M. Lessons learned from 25 years of research at the made site. Ground Water 2011, 49, 649–662. Ward, A.L.; Kachanoski, R.G.; Vonbertoldi, A.P.; Elrick, D.E. Field and undisturbed-column measurements for predicting transport in unsaturated layered soil. Soil Sci. Soc. Am. J. 1995, 59, 52–59. Dousset, S.; Thevenot, M.; Pot, V.; Simunek, J.; Andreux, F. Evaluating equilibrium and non-equilibrium transport of bromide and isoproturon in disturbed and undisturbed soil columns. J. Contam Hydrol. 2007, 94, 261–276. Cortis, A.; Berkowitz, B. Anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J. 2004, 68, 1539–1548. Bosma, W.J.P.; Vanderzee, S. Transport of reacting solute in a onedimensional, chemically heterogeneous porous-medium. Water Resour. Res. 1993, 29, 117–131. Simunek, J.; Jarvis, N.J.; van Genuchten, M.T.; Gardenas, A. Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J. Hydrol. 2003, 272, 14–35. Hu, B.X.; Huang, H.; Zhang, D.X. Stochastic analysis of solute transport in heterogeneous, dual-permeability media. Water Resour. Res. 2002, 38, 14-1–14-16.
343 [20] Hu, B.X.; Huang, H. Stochastic analysis of reactive solute transport in heterogeneous, fractured porous media: A dual-permeability approach. Transport Porous Med. 2002, 48, 1–39. [21] Zhang, D.X.; Sun, A.Y. Stochastic analysis of transient saturated flow through heterogeneous fractured porous media: A doublepermeability approach. Water Resour. Res. 2000, 36, 865–874. [22] Kim, Y.J.; Darnault, C.J.G.; Bailey, N.O.; Parlange, J.Y.; Steenhuis, T.S. Equation for describing solute transport in field soils with preferential flow paths. Soil Sci. Soc. Am. J. 2005, 69, 291–300. [23] Steenhuis, T.S.; Boll, J.; Shalit, G.; Selker, J.S.; Merwin, I.A. A simple equation for predicting preferential flow solute concentrations. J. Environ. Qual. 1994, 23, 1058–1064. [24] Tian, Y.; Gao, B.; Ziegler, K.J. High mobility of SDBS-dispersed single-walled carbon nanotubes in saturated and unsaturated porous media. J. Hazard. Mater. 2011, 186, 1766–1772. [25] Tian, Y.A.; Gao, B.; Silvera-Batista, C.; Ziegler, K.J. Transport of engineered nanoparticles in saturated porous media. J. Nanopart. Res. 2010, 12, 2371–2380. [26] Tian, Y.; Gao, B.; Morales, V.L.; Wu, L.; Wang, Y.; MunozCarpena, R.; Cao, C.; Huang, Q.; Yang, L. Methods of using carbon nanotubes as filter media to remove aqueous heavy metals. Chem. Eng. J. 2012, 210, 557–563. [27] Tian, Y.; Gao, B.; Wang, Y.; Morales, V.L.; Carpena, R.M.; Huang, Q.; Yang, L. Deposition and transport of functionalized carbon nanotubes in water-saturated sand columns. J. Hazard. Mater. 2012, 213, 265–272. [28] Skopp, J.; Gardner, W.R.; Tyler, E.J. Solute movement in structured soils – 2-region model with small interaction. Soil Sci. Soc. Am. J. 1981, 45, 837–842. [29] Gerke, H.H.; Vangenuchten, M.T. A dual-porosity model for simulating the preferential movement of water and solutes in structured porous-media. Water Resour. Res. 1993, 29, 305–319. [30] Marino, M.A. Longitudinal dispersion in saturated porous-media. J. Hydraul. Div.-Asce. 1974, 100, 151–156.
