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Constructal design of permeable reactive barriers: groundwater-hydraulics criteria

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Abstract

Unidirectional, steady-state, Darcian flow in a confined homogeneous aquifer is partially intercepted by a permeable reactive barrier (PRB), the shape of which is optimized with the following hydraulic criteria: seepage flow rate through a PRB (equivalent to the width and frontal area of the intercepted part of the plume in 2-D and 3-D cases, correspondingly) and travel time of a marked particle through the PRB interior along streamlines. The wetted perimeter, cross-sectional area and volume of the reactive material are selected as isoperimetric constraints. The PRB contour is modeled as either a constant head line (if the reactive material is much more permeable than the aquifer) or as a refraction boundary (if the reactive material has an arbitrary permeability), on which the hydraulic head and normal flux components in the barrier and aquifer are continuous. In the former case, the complex potential domain of the flow is a tetragon and a broad class of PRBs can be studied. In the latter case, analytical solutions are available for ellipses and ellipsoids (only these classes of shapes are considered in optimization). In the 2-D case and constant head PRB, a novel shape-control technique through the kernels of singular integrals is implemented: the Zhukovskii function is introduced; a Dirichlet boundary-value problem is solved for this function by setting the orientation (with respect to the incident flow direction) of the Darcian velocity vector on the PRB contour as a control function. Unlike similar controls for impermeable airfoils in aerodynamic design, the kernel has two discontinuities, which reflect the flow topology near a hinge (stagnation) point and the PRB tip. The integral is evaluated for V-shaped and curve-shaped PRBs and parametric expressions for the contours are obtained resulting (for the latter case) in a “pointy banana” shape. In the class of a V-shaped PRB, it is proved that a straight-line barrier minimizes the perimeter if the plume width is fixed. In 2- and 3-D refracting PRBs, the Pilatovskii (ellipse) and Poisson (ellipsoid) solutions for the flow field inside and outside the PRB are used for obtaining explicit formulae for the magnitude of the velocity, which is uniform inside the PRB. Simple expressions for the longest travel time within the PRB and the discharge intercepted by it are obtained. The ellipse/ellipsoid axes ratio/ratios are used as control variables in optimization. Extrema are obtained and analyzed for different PRB-aquifer conductivity ratios and for varying angles between the incident velocity vector and the ellipse/ellipsoid axes.

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References

  1. Day SR, O’Hannesin SF, Marsden L (1999) Geotechnical techniques for the construction of reactive barriers. J Hazard Mater B67: 285–297

    Article  Google Scholar 

  2. Gavaskar AR, Gupta N, Sass BM, Janosy RJ, O’Sullivan D (1998) Permeable barriers for groundwater remediation design, construction, and monitoring. Battelle Press, Columbus

    Google Scholar 

  3. De Zwart BR (2007) Investigation of clogging processes in unconsolidated aquifers near water supply wells. PhD thesis, Delft University of Technology

  4. Barkle GF, Schipper LA, Burgess CP, Painter BDM (2008) In situ mixing of organic matter decreases hydraulic conductivity of denitrification walls in sand aquifers. Ground Water Monit Remediat 28: 57–64

    Article  Google Scholar 

  5. Bartlett TR, Morrison SJ (2009) Tracer method to determine residence time in a permeable reactive barrier. Ground Water 47(4): 598–604

    Article  Google Scholar 

  6. Li L, Benson CH, Lawson EM (2005) Impact of mineral fouling on hydraulic behavior of permeable reactive barriers. Ground Water 43(4): 582–596

    Article  Google Scholar 

  7. Anderson EI, Mesa E (2006) The effects of vertical barrier walls on the hydraulic control of contaminated groundwater. Adv Water Resour 29(1): 89–98

    Article  ADS  Google Scholar 

  8. Hudak PF (2007) Mass transport in groundwater near hanging-wall interceptors. J Environ Sci Health A 42: 317–321

    Article  Google Scholar 

  9. Klammler H, Hatfield K (2008a) Analytical solutions for flow fields near continuous wall reactive barriers. J Contam Hydrol 98(1-2): 1–14. doi:10.1016/j.jconhyd.2008.01.005

    Article  Google Scholar 

  10. Klammler H, Hatfield K (2008b) The problem of flow-by-pass at permeable reactive barriers. In: Mander U, Brebbia CA, Martin-Duque JF (eds) Geo-environment and landscape evolution III, WIT Transactions on the Built Environment, vol 100. Wessex Institute of Technology, Southampton, UK, pp 15–24. doi:10.2495/GEO080021

    Chapter  Google Scholar 

  11. Klammler H, Hatfield K (2009) Analytical solutions for the flow fields near funnel-and-gate reactive barriers with hydraulic losses. Water Resour Res 45: W02423. doi:10.1029/2008WR007452

    Article  Google Scholar 

  12. Klammler H, Hatfield K, Kacimov AR (2010) Analytical solutions for flow fields near drain-and-gate reactive barriers. Ground Water 48(3): 427–437. doi:10.1111/j.1745-6584.2009.00661.x

    Article  Google Scholar 

  13. McMahon PB, Dennehy KF, Sandstrom MW (1999) Hydraulic and geochemical performance of a permeable reactive barrier containing zero-valent iron, Denver Federal Center. Ground Water 37(3): 396–404

    Article  Google Scholar 

  14. Painter BDM (2004) Reactive barriers: hydraulic performance and design enhancements. Ground Water 42(4): 609–617

    Article  Google Scholar 

  15. Robertson WD, Yeung N, van Driel PW, Lombardo PS (2005) High-permeability layers for remediation of ground water; go wide, not deep. Ground Water 43(4): 574–581

    Article  Google Scholar 

  16. Strack ODL (1989) Groundwater mechanics. Prentice Hall, Englewood Cliffs

    Google Scholar 

  17. Polubarinova-Kochina PY (1977) Theory of ground-water movement. Nauka, Moscow (in Russian)

    Google Scholar 

  18. Ilyinsky NB, Kacimov AR (1992) Analytical estimation of ground-water flow around cutoff walls and into interceptor trenches. Ground Water 30(6): 901–907

    Article  Google Scholar 

  19. Henrici P (1974) Applied and computational complex analysis, vols 1, 3. Wiley, New York

    Google Scholar 

  20. Bejan A (2000) Shape and structure, from engineering to nature. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  21. Pironneau O (1984) Optimal shape design for elliptic systems. Springer, New York

    MATH  Google Scholar 

  22. Elizarov AM, Kacimov AR, Maklakov DV (2008) Optimal shape design problems in aerohydrodynamics. Fizmatlit, Moscow (in Russian)

    Google Scholar 

  23. Craig JR, Rabideau AJ, Suribhatlab R (2006) Analytical expressions for the hydraulic design of continuous permeable reactive barriers. Adv Water Resour 29(1): 99–111. doi:10.1016/j.advwatres.2005.05.006

    Article  ADS  Google Scholar 

  24. Kacimov AR (2006) Analytical solution and shape optimisation for groundwater flow through a leaky porous trough subjacent to an aquifer. Proc R Soc Lond A 462: 1409–1423

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Kacimov AR (2008) Optimization and analysis of advective travel times beneath hydraulic structures. J Hydraul Eng ASCE 134(9): 1311–1317

    Article  Google Scholar 

  26. Kacimov AR, Marketz F, Pervez T (2009) Analytical solutions for one-phase seepage flows impeded by wellbore seals. J Petrol Sci Eng 64(1–4): 67–76. doi:10.1016/j.petrol.2008.12.005

    Article  Google Scholar 

  27. Nughin MT, Ilyinsky NB (1963) Methods of constructing subsurface contours of hydraulic structures. Inverse boundary value problems in seepage theory. Kazan University Press, Kazan (in Russian)

    Google Scholar 

  28. Tumashev GG, Nughin MT (1965) Inverse boundary-value problems and their applications. Kazan University Press, Kazan (in Russian)

    Google Scholar 

  29. Driscoll TA, Trefethen LN (2002) Schwarz–Christoffel mapping. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  30. Gakhov FD (1963) Boundary value problems. GIFML, Moscow (in Russian)

    Google Scholar 

  31. Kacimov AR (1993) Inclusion shaping and extremal property of the Taylor–Saffman bubble. Fluid Dyn 28(5): 741–743

    Article  MathSciNet  ADS  Google Scholar 

  32. Polya G, Szego G (1951) Isoperimetric inequalities in mathematical physics. Princeton University Press, Princeton

    MATH  Google Scholar 

  33. Prudnikov AP, Brychkov YuV, Marichev OI (1986) Integrals and series, vol 1. Gordon and Breach, New York

    MATH  Google Scholar 

  34. Wolfram S (1991) Mathematica. A system for doing mathematics by computer. Addison-Wesley, Redwood City

    Google Scholar 

  35. Goldstejn RV, Entov VM (1989) Qualitative methods in continuum mechanics. Nauka, Moscow (in Russian)

    Google Scholar 

  36. Il’inskii NB, Kacimov AR, Yakimov ND (1998) Analytical solutions of seepage theory problems. Inverse methods, variational theorems, optimization and estimates (A review). Fluid Dyn 33(2): 157–168

    Article  MathSciNet  ADS  Google Scholar 

  37. Pilatovskii VP (1966) Basic hydromechanics of a thin formation. Nedra, Moscow (in Russian)

    Google Scholar 

  38. Kacimov AR, Obnosov YuV, Yakimov ND (1999) Groundwater flow in a medium with a parquet-type conductivity distribution. J Hydrol 226: 242–249

    Article  Google Scholar 

  39. Salle C, Debyster J (1976) Formation des gisements de petrole: etude des phenomenes geologiques fondamentaux. Edition Techip (in French)

  40. Kacimov AR, Obnosov YuV (1994) Minimization of ground water contamination by lining of a porous waste repository. Proc Indian Natl Sci Acad A 60: 783–792

    Google Scholar 

  41. Antontsev SN, Domanskii AV, Penkosvkii VI (1989) Seepage in the near-wellbore zone and problems of intensifying the inflow. Lavrentyev Institute of Hydrodynamics, Novosibirsk (in Russian)

    Google Scholar 

  42. Mamilov VA, Petrov RP, Shushaniya GR (1980) Uranium recovery by in-situ lixiviation. Atomizdat, Moscow (in Russian)

    Google Scholar 

  43. Gheorgita SI (1966) Metode matematice in hidrogazodinamica subterana. Academia Republica Socialista Romania, Bucuresti (in Romanian)

    Google Scholar 

  44. Kacimov AR, Obnosov YuV (2000) Steady water flow around parabolic cavities and parabolic inclusions in unsaturated and saturated soils. J Hydrol 238: 65–77

    Article  Google Scholar 

  45. Maxwell JC (1954) A treatise on electricity and magnetism, vol 2. Dover, New York

    Google Scholar 

  46. Zhao C, Hobbs B, Ord A, Peng S, Liu L (2008) Inversely-mapped analytical solutions for flow patterns around and within inclined elliptic inclusions in fluid-saturated rocks. Math Geosci 40: 179–197. doi:10.1007/s11004-007-9138-0

    Article  MATH  Google Scholar 

  47. Bakker M (2004) Modeling groundwater flow to elliptical lakes and through multi-aquifer elliptical inhomogeneities. Adv Water Resour 27: 497–506

    Article  ADS  Google Scholar 

  48. Dagan G, Fiori A, Jankovic I (2003) Flow and transport in highly heterogeneous formations, part 1. Conceptual framework and validity of first-order approximations. Water Resour Res 18: 1268. doi:10.1029/2002WR001717

    Article  Google Scholar 

  49. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids, 2nd edn. Clarendon Press, Oxford

    Google Scholar 

  50. Park E, Hongbin Z (2009) One-dimensional solute transport in a permeable reactive barrier–aquifer system. Water Resour Res 45: W07502. doi:10.1029/2008WR007155

    Article  Google Scholar 

  51. Poisson SD (1824) Second memoire sur la theorie du magnetisme. Lu l’Academie royale des Sciences le 27 decembre I824. Memoires de l’Academie des Sciences, annees I982 et I822, V:488–553 (in French)

  52. Scaife BKP (1989) Principles of dielectrics. Clarendon Press, Oxford

    Google Scholar 

  53. Houben G, Treskatis C (2007) Water well rehabilitation and reconstruction. McGraw-Hill Professional, New York

    Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Soils and Water and Agricultural Engineering, Sultan Qaboos University, PO Box 34, Al-Khod, 123, Sultanate of Oman

    Anvar R. Kacimov

  2. Department of Civil and Coastal Engineering, University of Florida, Gainesville, FL, USA

    Harald Klammler & Kirk Hatfield

  3. Inter-Disciplinary Program in Hydrologic Sciences, University of Florida, Gainesville, FL, USA

    Harald Klammler & Kirk Hatfield

  4. Department of Hydraulic Engineering and Water Resources Management, Graz University of Technology, Graz, Austria

    Harald Klammler

  5. Institute of Mathematics and Mechanics, Kazan University, Kazan, Russia

    Nikolay Il’yinskii

Authors
  1. Anvar R. Kacimov
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  2. Harald Klammler
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  3. Nikolay Il’yinskii
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  4. Kirk Hatfield
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Correspondence to Anvar R. Kacimov.

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Kacimov, A.R., Klammler, H., Il’yinskii, N. et al. Constructal design of permeable reactive barriers: groundwater-hydraulics criteria. J Eng Math 71, 319–338 (2011). https://doi.org/10.1007/s10665-011-9457-5

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  • DOI: https://doi.org/10.1007/s10665-011-9457-5

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