Nonlocal transport models for capturing solute transport in one-dimensional sand columns: Model review, applicability, limitations and improvement
Corresponding Author
Yong Zhang
Department of Geological Sciences, University of Alabama, Tuscaloosa, Alabama, USA
Correspondence
Yong Zhang, Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487.
Email: [email protected]
Search for more papers by this authorDongbao Zhou
College of Mechanics and Materials, Hohai University, Nanjing, China
Search for more papers by this authorMaosheng Yin
School of Environmental Studies, China University of Geosciences, Wuhan, China
Search for more papers by this authorHongGuang Sun
College of Mechanics and Materials, Hohai University, Nanjing, China
Search for more papers by this authorWei Wei
School of Environment, Nanjing Normal University, Nanjing, China
Search for more papers by this authorShiyin Li
School of Environment, Nanjing Normal University, Nanjing, China
Search for more papers by this authorChunmiao Zheng
School of Environmental Science & Engineering, Southern University of Science and Technology, Shenzhen, China
Search for more papers by this authorCorresponding Author
Yong Zhang
Department of Geological Sciences, University of Alabama, Tuscaloosa, Alabama, USA
Correspondence
Yong Zhang, Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487.
Email: [email protected]
Search for more papers by this authorDongbao Zhou
College of Mechanics and Materials, Hohai University, Nanjing, China
Search for more papers by this authorMaosheng Yin
School of Environmental Studies, China University of Geosciences, Wuhan, China
Search for more papers by this authorHongGuang Sun
College of Mechanics and Materials, Hohai University, Nanjing, China
Search for more papers by this authorWei Wei
School of Environment, Nanjing Normal University, Nanjing, China
Search for more papers by this authorShiyin Li
School of Environment, Nanjing Normal University, Nanjing, China
Search for more papers by this authorChunmiao Zheng
School of Environmental Science & Engineering, Southern University of Science and Technology, Shenzhen, China
Search for more papers by this authorFunding information: National Natural Science Foundation of China, Grant/Award Number: 41931292; Guangdong Provincial Key Laboratory of Soil and Groundwater Pollution Control, Grant/Award Number: 2017B030301012; University of Alabama, Grant/Award Number: CARSCA program
Abstract
Modelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state-of-the-art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one-dimensional sand columns, which represents perhaps the simplest real-world application. Applications showed that, surprisingly, neither the popular time-nonlocal transport models (including the multi-rate mass transfer model, the continuous time random walk framework and the time fractional advection-dispersion equation), nor the spatiotemporally nonlocal transport model (ST-fADE) can accurately fit passive tracers moving through a 15-m-long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale-dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non-Fickian (where plume variance increases nonlinearly in time) and/or pre-asymptotic (with transition between non-Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale-dependency of the dispersion coefficient in the non-Euclidean space, and a two-region time fractional advection-dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non-ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.
Open Research
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Supporting Information
Filename | Description |
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hyp13930-sup-0001-supinfo.docxWord 2007 document , 501.6 KB | Figure S1 The measured BTCs (symbols) versus the best-fit results using the TF-SHF model (11) (blue line) and the TR-fADE model (16) (red line) at five positions (x = 400, 600, 800, 1,000, and 1,200 cm) in a 12.5-m-long heterogeneous sand column with constant parameters. Figure S2. The measured BTCs (symbols) versus the best-fit results using the TF-SHF model (11) (blue line) and the TR-fADE model (16) (red line) at five positions (x = 400, 600, 800, 1,000, and 1,200 cm) in a 12.5-m-long homogeneous sand column with constant parameters. Table S1. Parameters and RMSE for the TR-fADE model (16) after fitting the BTCs for the 12.5-m-long heterogeneous sand column. Table S2. Parameters and RMSE for the TR-fADE model (16) after fitting the BTCs for the 12.5-m-long homogeneous sand column. Table S3. Parameters and RMSE for the TF-SHF model (11) after fitting the BTCs for the12.5-m-long heterogeneous sand column. Table S4. Parameters and RMSE for the TF-SHF model (11) after fitting the BTCs for the 12.5-m-long homogeneous sand column. Table S5. Factor level using the CTRW framework (3) as an example Table S6. Calculation schemes and results of the orthogonal design for the CTRW framework (3) Table S7. Results of the variance analysis for the CTRW framework (3). In the legend, ‘Sum Sq.’ represents sum of square, ‘d.f.’ represents the degree of freedom, ‘MSE’ represents Mean square, ‘F’ denotes inspection level, and ‘P’ denotes significance level |
Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.
REFERENCES
- Abgaze, T. A., & Sharma, P. K. (2015). Solute transport through porous media with scale dependent dispersion and variable mass transfer coefficient. Journal of Hydraulic Engineering, 21(3), 298–311.
10.1080/09715010.2015.1021281 Google Scholar
- Adams, E. E., & Gelhar, L. W. (1992). Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis. Water Resources Research, 28(12), 3293–3307.
- Barry, D. A., & Sposito, G. (1989). Analytical solution of a convection dispersion model with time-dependent transport coefficients. Water Resources Research, 25(12), 2407–2416.
- Benson, D. A., & Meerschaert, M. M. (2009). A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Advances in Water Resources, 32, 532–539.
- Benson, D. A., Schumer, R., Meerschaert, M. M., & Wheatcraft, S. W. (2001). Fractional dispersion, Lévy motion, and the MADE tracer tests. Transport in Porous Media, 42(1/2), 211–240.
- Berkowitz, B., Cortis, A., Dentz, M., & Harvey, S. (2006). Modeling non-Fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics, 44(2), 1–49.
- Bianchi, M., & Zheng, M. C. (2016). A lithofacies approach for modeling non-Fickian solute transport in a heterogeneous alluvial aquifer. Water Resources Research, 52(1), 552–565.
- Boano, F., Harvey, J. W., Marion, A., Packman, A. I., Revelli, R., Ridolfi, L., & Wörman, A. (2014). Hyporheic flow and transport processes: Mechanisms, models, and biogeochemical implications. Reviews of Geophysics, 52, 603–679. https://doi.org/10.1002/2012RG000417
- Boggs, J. M., Young, S. C., & Beard, L. M. (1992). Field study of dispersion in a heterogeneous aquifer: 1. Overview and site description. Water Resources Research, 28(12), 3281–3291.
- Boggs, J. M., & Adams, E. E. (1992). Field study of dispersion in a heterogeneous aquifer: 4. Investigation of adsorption and sampling bias. Water Resources Research, 28(12), 3281–3291.
- Chen, W. (2006). Time-space fabric underlying anomalous diffusion. Chaos, Solutions and Fractals, 28(4), 923–929.
- Cirpka, O. A., & Valocchi, A. J. (2016). Debates-stochastic subsurface hydrology from theory to practice: Does stochastic subsurface hydrology help solving practical problems of contaminant hydrogeology? Water Resources Research, 52(12), 9218–9227.
- Coats, K. H., & Smith, B. D. (1964). Dead-end pore volume and dispersion in porous media. Society of Petroleum Engineers Journal, 3, 73–84.
- Cushman, J. H., Hu, B. X., & Ginn, T. R. (1994). Nonequilibrium statistical mechanics of preasymptotic dispersion. Journal of Statistical Physics, 75(5/6), 859–878.
- Cushman, J. H. (1997). The physics of fluids in hierarchical porous media: Angstroms to miles. New York: Springer.
10.1007/978-94-015-8849-2 Google Scholar
- Cushman, J. H., & Ginn, T. R. (2000). Fractional advection-dispersion equation: A classical mass balance with convolution-Fickian flux. Water Resources Research, 36(12), 3763–3766.
- Cvetkovic, V. (2011). The tempered one-sided stable density: A universal model for hydrological transport? Environmental Research Letters, 6, 034008.
- Dagan, G. (1982). Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, the solute transport. Water Resources Research, 18(4), 835–848.
- Dagan, G. (1988). Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resources Research, 24(9), 1491–1500.
- Dagan, G. (1989). Flow and transport in porous formations. Berlin: Springer-Verlag.
10.1007/978-3-642-75015-1 Google Scholar
- Dentz, M., & Berkowitz, B. (2003). Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resources Research, 39(5), 1111.
- Dentz, M., Cortis, A., Scher, H., & Berkowitz, B. (2004). Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport. Advances in Water Resources, 27, 155–173.
- Dentz, M., Icardi, M., & Hidalgo, J. J. (2018). Mechanisms of dispersion in a porous medium. Journal of Fluid Mechanics, 841, 851–882.
- Eab, C. H., & Lim, S. C. (2011). Fractional Langevin equations of distributed order. Physical Review E, 83(3), 031136.
- Engdahl, N. B., Ginn, T. R., & Fogg, G. E. (2013). Using groundwater age distributions to estimate the effective parameters of Fickian and non-Fickian models of solute transport. Advances in Water Resources, 54, 11–21.
- Fiori, A., Zarlenga, A., Gotovac, H., Jankovic, I., Volpi, E., Cvetkovic, V., & Dagan, G. (2015). Advective transport in heterogeneous aquifers: Are proxy models predictive? Water Resources Research, 51, 9577–9594. https://doi.org/10.1002/2015WR017118
- Fiori, A., Cvetkovic, V., Dagan, G., Attinger, S., Bellin, A., & Dietrich, P. (2016). Debates-stochastic subsurface hydrology from theory to practice: The relevance of stochastic subsurface hydrology to practical problems of contaminant transport and remediation. What is characterization and stochastic theory good for? Water Resources Research, 52(12), 9228–9234.
- Fiori, A., Zarlenga, A., Gotovac, H., Jankovic, I., Volpi, E., Cvetkovic, V., & Dagan, G. (2016). Reply to comment by S. P. Neuman on “Advective transport in heterogeneous aquifers: Are proxy models predictive?”. Water Resources Research, 52, 5703–5704. https://doi.org/10.1002/2016WR019209
- Fogg, G. E. (1986). Groundwater flow and sand-body interconnectedness in a thick, multiple-aquifer system. Water Resources Research, 22(5), 679–694.
- Fogg, G. E., & Zhang, Y. (2016). Debates-stochastic subsurface hydrology from theory to practice: A geologic perspective. Water Resources Research, 52(12), 9235–9245.
- Garrard, R. M., Zhang, Y., Wei, S., Sun, H. G., & Qian, J. Z. (2017). Can a time fractional-derivative model capture scale-dependent dispersion in saturated soils? Groundwater, 55(6), 857–870.
- Gelhar, L. W., & Axness, C. L. (1983). Three-dimensional stochastic analysis of macrodispersion in aquifers. Water Resources Research, 19(1), 161–180.
- Gelhar, L. W., Welty, C., & Rehfeldt, K. R. (1992). A critical review of data on field-scale dispersion in aquifers. Water Resources Research, 28(7), 1955–1974.
- Gelhar, L. W. (1993). Stochastic subsurface hydrology. New Jersey: Prentice Hall.
- Ginn, T. R., Schreyer, L. G., & Zamani, K. (2017). Phase exposure-dependent exchange. Water Resources Research, 53(1), 619–632.
- Ginn, T. R. (2018). Modeling bimolecular reactive transport with mixing-limitation: Theory and application to column experiments. Water Resources Research, 54(1), 256–270.
- Gorenflo, R., Luchko, Y., & Yamamoto, M. (2015). Time-fractional diffusion equation in the fractional Sobolev spaces. Fractional Calculus and Applied Analysis, 18(3), 799–820.
- Haggerty, R., & Gorelick, S. M. (1995). Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resources Research, 31(10), 2383–2400.
- Haggerty, R., McKenna, S. A., & Meigs, L. C. (2000). On the late-time behaviour of tracer breakthrough curves. Water Resources Research, 36, 3467–3479.
- Hansen, S. K., & Berkowitz, B. (2014). Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation. Advances in Water Resources, 74, 54–63.
- Hess, K. M., Wolf, S. H., & Celia, M. A. (1992). Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts 3. Hydraulic conductivity variability and calculate macrodispersivities. Water Resources Research, 28(8), 2011–2027.
- Huang, K., Toride, N., & van Genuchten, M. T. (1995). Experimental investigation of solute transport in large, homogeneous and heterogeneous, saturated soil columns. Transport in Porous Media, 18(3), 283–302.
- Istok, J. D., & Humphrey, M. D. (1995). Laboratory investigation of buoyancy-induced flow (plume sinking) during two-well tracer tests. Groundwater, 33(4), 597–604.
- Jiang, Z. R., & Wang, L. H. (2013). The affecting factors of slope stability based on orthogonal design. Advanced Materials Research, 690, 756–759.
- LaBolle, E. M., & Fogg, G. E. (2001). Role of molecular diffusion in contaminant migration and recovery in an alluvial aquifer system. Transport in Porous Media, 42, 155–179.
- Lee, Y., Filliben, J. J., Micheals, R. J., & Phillips, P. J. (2013). Sensitivity analysis for biometric systems: A methodology based on orthogonal experiment designs. Computer Vision and Image Understanding, 117(5), 532–550.
- Levy, M., & Berkowitz, B. (2003). Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. Journal of Hydrology, 64(3–4), 203–226.
- Liang, Y., Allen, Q. Y., Chen, W., Gatto, R. G., Luis, C., Mareci, T. H., & Magin, R. L. (2016). A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Communications in Nonlinear Science and Numerical Simulation, 39, 529–537.
- Liu, X. T., Sun, H. G., Zhang, Y., & Zheng, C. M. (2019). Simulating multi-dimensional anomalous diffusion in nonstationary media using variable-order vector fractional-derivative models with Kansa solver. Advances in Water Resources, 133, 103423.
- Llopis, A. C., & Capilla, J. E. (2009). Gradual conditioning of non-Gaussian transmissivity fields to flow and mass transport data: 3. Application to the macrodispersion experiment (MADE-2) site, on Columbus Air Force Base in Mississippi (USA). Journal of Hydrology, 371(1–4), 75–84.
- Lu, B. Q., Zhang, Y., Zheng, C. M., Green, C. T., Neill, C. O., Sun, H. G., & Qian, J. (2018). Comparison of time nonlocal transport models for characterizing non-Fickian transport: Form mathematical interpretation to laboratory application. Water, 10, 778. https://doi.org/10.3390/w10060778
- Lu, C. H., Wang, Z., Zhao, Y., Rathore, S. S., Huo, J., Tang, Y., … Luo, J. (2018). A mobile-mobile transport model for simulating reactive transport in connected heterogeneous fields. Journal of Hydrology, 560, 97–108.
- Mainardi, F., Mura, A., & Pagnini, G. (2008). Time-fractional diffusion of distributed order. Journal of Vibration and Control, 14(9–10), 1267–1290.
- Meerschaert, M. M., & Scheffler, H. P. (2001). Limit theorems for sums of independent random vectors. New York: John Wiley.
- Meerschaert, M. M., & Tadjeran, C. (2004). Finite difference approximations for fractional advection–dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1), 65–77.
- Meerschaert, M. M., Zhang, Y., & Baeumer, B. (2008). Tempered anomalous diffusion in heterogeneous systems. Geophysical Research Letters, 35(9), L17403. https://doi.org/10.1029/2008GL034889
- Metzler, R., & Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339, 1–77.
- Metzler, R., & Klafter, J. (2004). The restaurant at the end of the random walk: Recent development in fractional dynamics of anomalous transport processes. Journal of Physics A: Mathematical and General, 37, R161–R208.
- Neuman, S. P., Winter, C. L., & Newman, C. M. (1987). Stochastic theory of field-scale Fickian dispersion in anisotropic porous media. Water Resources Research, 23(3), 453–466.
- Neuman, S. P. (1990). Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resources Research, 26(8), 1749–1758.
- Neuman, S. P. (1993). Eulerian-Lagrangian theory of transport in space-time nonstationary velocity fields: Exact nonlocal formalism by conditional moments and weak approximation. Water Resources Research, 29(3), 633–645.
- Neuman, S. P., & Tartakovsky, D. M. (2009). Perspective on theories of non-Fickian transport in heterogeneous media. Advances in Water Resources, 32(5), 670–680.
- Neuman, S. P. (2016). Comment on “Advective transport in heterogeneous aquifers: Are proxy models predictive?” by A. Fiori, A. Zarlenga, H. Gotovac, I. Jankovic, E. Volpi, V. Cvetkovic, and G. Dagan. Water Resources Research, 52, 5701–5702. https://doi.org/10.1002/2016WR019093
- Peters, J. F., & Howington, S. E. (1997). Pre-asymptotic transport through porous media. In G. Delic & M. F. Wheeler (Eds.), Next generation environmental models and computational methods (p. 280). Philadelphia, PA: SIAM.
- Pickens, J. F., & Grisak, G. E. (1981). Scale-dependent dispersion in a stratified granular aquifer. Water Resources Research, 17(4), 1191–1211.
- Rehfeldt, K. R., Boggs, J. M., & Gelhar, L. W. (1992). Field study of dispersion in a heterogeneous aquifer: 3. Geostatistical analysis of hydraulic conductivity. Water Resources Research, 28(12), 3309–3324.
- Sanchez, V. X., & Fernandez, G. D. (2016). Debates-stochastic subsurface hydrology from theory to practice: Why stochastic modeling has not yet permeated into practitioners? Water Resources Research, 52(12), 9246–9258.
- Schumer, R., Benson, D. A., Meerschaert, M. M., & Baeumer, B. (2003). Fractal mobile/immobile solute transport. Water Resources Research, 39(10), 1296.
- Šimůnek, J., van Genuchten, M. T. H., Šejna, M., Toride, N., & Leij, F. J. (1999). The STANMOD computer software for evaluating solute transport in porous media using analytical solutions of convection-dispersion equation. Versions 1.0 and 2.0, IGWMC—TPS—71. International Ground Water Modeling Center, Colorado School of Mines, Golden, Colorado, p. 32.
- Sousa, R. G. U., de Oliveira, I. B., Machado, S. L., & Carvalho, M. F. (2020). Solute dispersion of organic compounds on undisturbed soil columns. Transport in Porous Media, 132, 267–282.
- Sun, H. G., Meerschaert, M. M., Zhang, Y., Zhu, J. T., & Chen, W. (2013). A fractal Richards' equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Advances in Water Resources, 52, 292–295.
- Sun, H. G., Zhang, Y., Chen, W., & Reeves, D. M. (2014). Capture the transient dispersion in heterogeneous media with a variable-index fractional-derivative model. Journal of Contaminant Hydrology, 157, 47–58.
- Sun, H. G., Li, Z. P., Zhang, Y., & Chen, W. (2017). Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos, Solitons & Fractals, 102, 346–353.
- Sun, H. G., Chang, A. L., Zhang, Y., & Chen, W. (2019). A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications. Fractional Calculus and Applied Analysis, 22(1), 27–59.
- van Genuchten, M. T., Šimůnek, J., Leij, F. L., Toride, N., & Šejna, M. (2012). STANMOD: Model use, calibration and validation, special issue standard/engineering procedures for model calibration and validation. Transactions of the ASABE, 55(4), 1353–1366.
- Vanderborght, J., & Vereecken, H. (2007). Review of dispersivities for transport modeling in soils. Vadose Zone Journal, 6(1), 29–52.
- Wang, M., Zhao, W. D., Garrard, R., Zhang, Y., Liu, Y., & Qian, J. Z. (2018). Revisit of advection-dispersion equation model with velocity-dependent dispersion in capturing tracer dynamics in single empty fractures. Journal of Hydrodynamics, 30(6), 1055–1063.
- Warrick, A. W., Kitchen, J. H., & Thames, J. L. (1972). Solutions for miscible displacement of soil water with time-dependent velocity and dispersion coefficient. Soil Science Society of America Journal, 36(6), 863–867.
- Yadav, R. R., & Kumar, L. K. (2019). Solute transport for pulse type input point source along temporally and spatially dependent flow. Pollution, 5(1), 53–70.
- Yin, M. S., Zhang, Y., Ma, R., Tick, G., Bianchi, M., Zheng, C. M., … Liu, X. T. (2020). Super-diffusion affected by hydrofacies mean length and source geometry in alluvial settings. Journal of Hydrology, 582, 124515.
- Zhang, Y., Benson, D. A., Meerschaert, M. M., & LaBolle, E. M. (2007). Space-fractional advection-dispersion equations with variable parameters: Diverse formulas, numerical solutions, and application to the macrodispersion experiment site data. Water Resources Research, 43, W05439. https://doi.org/10.1029/2006WR004912
- Zhang, Y., Benson, D. A., & Reeves, D. M. (2009). Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources, 32(4), 561–581.
- Zhang, Y., & Meerschaert, M. M. (2011). Gaussian setting time for solute transport in fluvial systems. Water Resources Research, 47, W08601.
- Zhang, Y., Meerschaert, M. M., & Packman, A. I. (2012). Linking fluvial bed sediment transport across scales. Geophysical Research Letters, 39, L20404.
- Zhang, Y., Meerschaert, M. M., Baeumer, B., & LaBolle, E. M. (2015). Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme. Water Resources Research, 51(8), 6311–6337.
- Zhang, Y., Sun, H. G., & Zheng, C. M. (2019). Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application. Fractional Calculus and Applied Analysis, 22(6), 1607–1640.
- Zheng, C., & Gorelick, S. M. (2003). Analysis of the effect of decimeter scale preferential flow paths on solute transport. Groundwater, 41(2), 142–155.