مدل سازی عددی انتقال آلاینده در آبراهه‌های با پهنه ماندابی و جریان غیریکنواخت با استفاده از معادله جابه‌جایی-پراکندگی کسری

نوع مقاله : مقالات پژوهشی

نویسندگان

دانشگاه تربیت مدرس

چکیده

پیش‌بینی انتقال آلاینده‌ها در پهنه‌های آبی در مدیریت و جلوگیری از آلودگی آبهای سطحی از اهمیت ویژه‌ای برخوردار است. ناهمگونی و عدم یکنواختی در مورفولوژی و خصوصیات فیزیکی در تمام طول رودخانه‌، که به پهنه‌های ماندابی شناخته می‌شوند، انتقال یکنواخت آلاینده‌ها به پائین‌دست را با مشکل مواجه می‌کند. مدل‌های یک‌بعدی که بر پایه معادله جابه‌جایی-پراکندگی کلاسیک قرار دارند، قادر به شبیه‌سازی دقیق پدیده انتقال در این‌گونه رودخانه‌ها نیستند. یکی از روش‌های جایگزین که در صورت به‌کارگیری صحیح می‌تواند منجر به پیش‌بینی مناسب از پدیده انتقال در حالت مذکور گردد، معادله جابه‌جایی-پراکندگی کسری است. در این تحقیق با بهره‌گیری از روش‌های حل عددی مشتقات جزئی‌کسری، یک روش حل برای معادله جابه‌جایی‌-پراکندگی کسری که حالت عمومی معادله جابه‌جایی‌-پراکندگی کلاسیک است، در حالت جریان غیریکنواخت ارائه شده‌است. برای برآورد پارامترهای تاثیرگذار در پیش‌بینی انتقال ماده‌ آلاینده حل‌شده، روش بهینه‌سازی ریاضی مورد استفاده‌ قرار گرفت. جهت صحت‌سنجی، مدل ارائه شده با داده‌های واقعی برداشت شده از نهر یواس‌کریک در کالیفرنیا مقایسه شده است. بر اساس آزمایش مذکور ماده ردیاب کلراید بصورت پیوسته در این نهر تزریق شده و در ایستگاه‌های پایین‌دست غلظت آن اندازه‌گیری شده است. خروجی مدل انطباق قابل‌قبولی با داده‌های مشاهداتی داشته و نشان می‌دهد که روش حل ارائه‌شده، روشی دقیق و قابل قبول در شبیه‌سازی انتقال ماده حل‌شده با استفاده از معادله جابه‌جایی-پراکندگی کسری در آبراهه‌های دارای پهنه ماندابی است.

کلیدواژه‌ها


عنوان مقاله [English]

Numerical Solution for Non-Uniform Fractional Advection Dispersion Equation

نویسندگان [English]

  • Mohammad Hatamjafari
  • Mehdi Mazaheri
  • Jamal Mohammad Vali Samani
Tarbiat Modares University
چکیده [English]

Introduction: Solute transport modeling in water bodies and especially rives, provides a useful tools for decision-makers and consumers, to take a timely decisions in order to prevent or avoid any catastrophic consequences. Many factors such as velocity, cross-section area, bedform and etc. affect the transport of any solute particle in the media. The aforementioned factors vary both in width and depth, which cause some of the particles to deviate from the main channel to areas in which the particle stays for a longer time. Those particles are released to the main channel gradually causing a heavy-tailed breakthrough curve in downstream of the injection point. The areas where solute spends more time than the main channel is called deadzones. Any area in the channel that reduce the velocity field to relatively zero could be categorized as deadzones, such as areas behind a boulder, in pocket beside the river, behind or inside a vegetation cluster. In streams containing deadzones the classical adevection-dispersion equation (ADE) cannot meet the requirements. Scholars proposed many different approaches to capture the heavy-tailed behavior. One of the renowned and novel approaches to quantify the role of deadzones in natural streams is using Fractional Calculus. This field of mathematics helps to change the fundamental assumption of ADE, which is laid upon “central limit theorem” and subsequently generate a new equation based on fractional partial derivatives, The Fractional Advection –Dispersion Equation (FADE). The FADE provides a better description of solute movement in natural streams, because of its general form and non-locality.
Materials and Methods: In this research, FADE for a non-uniform but steady condition in natural streams has been investigated. FADE is a product of central limit theorem generalization and hence can capture the anomalous behavior of solute transport in streams containing deadzones. One of the important points that can be inferred from tracer study in such streams is that the concentration of a point is influenced by the point in far upstream; that is either the solute particle comes faster than the main cloud or slower and defy the basic assumption. Since classic ADE just considers the points in the closest proximity of the study point to calculate the bearkthrough curve (BTC), it cannot be used in cases that the deadzones are involved; which calls for a more comprehensive consideration of the stream. Interestingly, factional derivatives are non-local; fractional derivatives consider the state of the function in its whole domain. Hence, FADE can accurately describe and calculate the BTC in streams affected by deadzones. The big difference between FADE and classic ADE is that, the former is based on Levy motion while the later developed and shows Brownian motion. Also, the order of spatial derivative in FADE is a value , unlike the classic ADE in which the order is fixed to . To investigate the FADE in depth standard Grunwald definition of fractional derivatives were employed for discretization, and an algorithm is proposed. The more the order of the fractional partial derivative decreases, the more would be the effect of the deadzones, this means that while the order is equal to , the streamflow do not experience any deadzones in its way. In order to fit the data, two variables in the FADE needs to be calibrated; the unknown parameters were estimated using Particle Swarm Optimization (PSO) technique.
Conclusion: The model was validated using a set of data from a small stream. The comparison shows that the model can accurately recreate the BTC and perform dramatically better than classic ADE which even in some stations is incapable of predicting the peak concentration and the arrival time of the particle to the station. Quantitatively, the difference between the FADE and observation represented by RMSE in average is 0.236 while for classic ADE is around 4 times higher. The difference in the first two stations is negligible for the stream does not contain any deadzones. This shows that not only the FADE is capable of considering deadzones, but also if needed it can easily reduce to classic ADE. This flexibility offers vast opportunity for application. In addition, the sensitivity of the model to the calibration parameter was analyzed, the analysis shows that both variables (order of the derivative and skewness parameter) should be treated with great care, since a little variation in their value drastically alter the shape of the BTC, and decrease the accuracy of the prediction. In this research, the capability of FADE were investigated and proved that this model performing far better in predicting solute transport in natural streams in which the variability of the morphology may hinder the solutes movement.

کلیدواژه‌ها [English]

  • Deadzones
  • Fractional advection dispersion equation
  • Numerical Solution
  • pollutant transport
  • Stream
1- Buffington J.M., and Tonina D. 2009. Hyporheic exchange in mountain rivers II: effects of channel morphology on mechanics, scales, and rates of exchange. Geography Compass, 3(3):1038-1062.
2- Jones J.B., and Mulholland P.J. 1999. Streams and ground waters. Academic Press.
3- Fleckenstein, J.H., Krause S., Hannah D.M., and Boano, F. 2010. Groundwater-surface water interactions: New methods and models to improve understanding of processes and dynamics. Advances in Water Resources, 33(11):1291-1295.
4- Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J. and Brooks, N.H. 1979. Mixing in inland and coastal waters.
5- Bencala, K.E. 1983. Simulation of solute transport in a mountain pool‐and‐riffle stream with a kinetic mass transfer model for sorption. Water Resources Research, 19(3):732-738.
6- Wörman, A., Packman, A.I., Johansson, H. and Jonsson, K. 2002. Effect of flow‐induced exchange in hyporheic zones on longitudinal transport of solutes in streams and rivers. Water Resources Research, 38(2):1-15.
7- Gooseff, M.N., Bencala, K.E., Scott, D.T., Runkel, R.L. and McKnight, D.M. 2005. Sensitivity analysis of conservative and reactive stream transient storage models applied to field data from multiple-reach experiments. Advances in Water Resources, 28(2):479-492.
8- Haggerty, R., McKenna, S.A. and Meigs, L.C. 2000. On the late‐time behavior of tracer test breakthrough curves. Water Resources Research, 36(5):3467-3479.
9- Haggerty, R., Wondzell, S.M. and Johnson, M.A. 2002. Power‐law residence time distribution in the hyporheic zone of a 2nd‐order mountain stream.Geophysical Research Letters, 29(18):1-4.
10- Metzler, R. and Klafter, J. 2000. Subdiffusive transport close to thermal equilibrium: From the Langevin equation to fractional diffusion. Physical Review, 61(1):6308- 6311.
11- Berkowitz, B., Cortis, A., Dentz, M. and Scher, H. 2006. Modeling non‐Fickian transport in geological formations as a continuous time random walk. Reviews of Geophysics, 44.
12- Boano, F., Packman, A., Cortis, A., Revelli, R. and Ridolfi, L. 2007. A continuous time random walk approach to the stream transport of solutes. Water resources research, 43(5):12-21.
13- Marion, A. and Zaramella, M. 2005. A residence time model for stream-subsurface exchange of contaminants. Acta Geophysica Polonica, 53(1):527-534.
14- Marion, A., Zaramella, M. and Bottacin‐Busolin, A. 2008. Solute transport in rivers with multiple storage zones: The STIR model. Water resources research, 44(4):38-50.
15- Meerschaert, M.M. and Sikorskii, A., 2012. Stochastic models for fractional calculus (Vol. 43). Walter de Gruyter.
16- Blank, L., 1996. Numerical treatment of differential equations of fractional order. University of Manchester, Department of Mathematics.
17- Deng, Z.Q., Singh, V.P. and Bengtsson, L. 2004. Numerical solution of fractional advection-dispersion equation. Journal of Hydraulic Engineering, 130(3):422-431.
18- Shen, C. and Phanikumar, M.S. 2009. An efficient space-fractional dispersion approximation for stream solute transport modeling. Advances in Water Resources, 32(10):1482-1494.
19- Schumer, R., Benson, D.A., Meerchaert, M.M. and Wheatcraft, S.W. 2001. Eulerian derivation of the fractional advection-dispersion equation. Journal of Contaminant Hydrology, 48(5):69-88.
20- Oldham, K.B., Spanier, J. and Ross, B. 1974. Fractional calculus.
21- Meerschaert, M.M. and Tadjeran, C. 2004. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics, 172(1):65-77.
22- Eberhart, R.C. and Kennedy, J. 1995, October. A new optimizer using particle swarm theory. In Proceedings of the sixth international symposium on micro machine and human science, 1(1):39-43.
23- Zand, S.M., Kennedy, V.C., Zellweger, G.W. and Avanzino, R.J. 1976. Solute transport and modeling of water quality in a small stream. United States Geological Survey.