Skip to main content
SpringerLink
Log in

Modeling, with a unified level-set representation, of the expansion of a hollow in the ground under different physical phenomena

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

This paper builds on the flexibility of the level-set representation to model in a unified manner the expansion of a hollow in the ground under different physical phenomena. In particular, the dissolving action of a flow of water in a saturated soil, and that of a jet of particles of water in a non-saturated one, are represented in a common framework. In that manner, the complex geometrical evolutions of the hollow can be followed without the need for remeshing and this approach allows for a smooth transition between saturated and non-saturated models of the soil. Implementation and numerical difficulties are discussed and two applications of industrial interest are considered. The first one describes the modeling of the piping phenomenon, and the second one the evolution of an excavation created by a leaking duct.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Crank J (1984) Free and moving boundary problems. Oxford University Press, Oxford

    MATH  Google Scholar 

  2. Unverdi SO, Tryggvason G (1992) A front-tracking method for viscous, incompressible, multi-fluid flows. J Comput Phys 100(1): 25–37. doi:10.1016/0021-9991(92)90307-K

    Article  MATH  Google Scholar 

  3. Gorczyk W, Gerya TV, Connolly JA, Yuen DA, Rudolph M (2006) Large-scale rigid-body rotation in the mantle wedge and its implications for seismic tomography. Geochem Geophys Geosyst 7(5):Q05,018. doi:10.1029/2005GC001075

  4. Jan YJ (2007) A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems. Comput Mech 40(2): 201–216. doi:10.1007/s00466-006-0098-x

    Article  MATH  Google Scholar 

  5. van Keken PE, King SD, Schmeling H, Christensen UR, Neumeister D, Doin MP (1997) A comparison of methods for the modeling of thermochemical convection. Phys Fluids A 102(B10):22, 477–22, 496. doi:10.1029/97JB01353

  6. Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2): 463–502. doi:10.1006/jcph.2000.6636

    Article  MATH  MathSciNet  Google Scholar 

  8. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1): 12–49. doi:10.1016/0021-9991(88)90002-2

    Article  MATH  MathSciNet  Google Scholar 

  9. Sussman M, Smereka P, Osher S (1994) A level set approach for computing solutions to incompressible two-phase flow. J Comput Phys 114(1): 146–159. doi:10.1006/jcph.1994.1155

    Article  MATH  Google Scholar 

  10. Mulder W, Osher S, Sethian JA (1992) Computing interface motion in compressible gas dynamics. J Comput Phys 100(2): 209–228. doi:10.1016/0021-9991(92)90229-R

    Article  MATH  MathSciNet  Google Scholar 

  11. Karlsen KH, Lie KA, Risebro NH (2000) A fast marching method for reservoir simulation. Comput Geosci 4(2): 185–206. doi:10.1023/A:1011564017218

    Article  MATH  MathSciNet  Google Scholar 

  12. Nielsen LK, Li H, Tai XC, Aanonsen SI, Espedal M (2008) Reservoir description using a binary level set model. Comput Vis Sci. doi:10.1007/s00791-008-0121-1

  13. Zlotnik S, Díez P, Fernández M, Vergés J (2007) Numerical modelling of tectonic plates subduction using X-FEM. Comput Methods Appl Mech Eng 196(41–44): 4283–4293. doi:10.1016/j.cma.2007.04.006

    Article  MATH  Google Scholar 

  14. Zlotnik S, Fernández M, Díez P, Vergés J (2008) Modelling gravitational instabilities: slab break-off and Rayleigh–Taylor diapirism. Pure Appl Geophys 165: 1–20. doi:10.1007/s00024-004-0386-9

    Article  Google Scholar 

  15. Sethian JA, Popovici AM (1999) 3-D traveltime computation using the fast marching method. Geophysics 64(2): 516–523. doi:10.1190/1.1444558

    Article  Google Scholar 

  16. Ito K, Kunisch K, Li Z (2001) Level-set function approach to an inverse interface problem. Inverse Probl 17(5): 1225–1242. doi:10.1088/0266-5611/17/5/301

    Article  MATH  MathSciNet  Google Scholar 

  17. Burger M (2001) A level set method for inverse problems. Inverse Probl 17(5): 1327–1355. doi:10.1088/0266-5611/17/5/307

    Article  MATH  MathSciNet  Google Scholar 

  18. Burger M, Osher SJ (2005) A survey of level set methods for inverse problems and optimal design. Eur J Appl Math 16(2): 263–301. doi:10.1017/S0956792505006182

    Article  MATH  MathSciNet  Google Scholar 

  19. Bonelli S, Brivois O, Borghi R, Benahmed N (2006) On the modelling of piping erosion. Comptes-Rendus Mécanique 334(8–9): 555–559. doi:10.1016/j.crme.2006.07.003

    Article  MATH  Google Scholar 

  20. Wan CF, Fell R (2004) Investigation of rate of erosion of soils in embankment dams. J Geotech Geoenviron Eng ASCE 130(4): 373–380. doi:10.1061/(ASCE)1090-0241(2004)130:4(373)

    Article  Google Scholar 

  21. Wan CF, Fell R (2004) Laboratory tests on the rate of piping erosion of soils in embankment dams. Geotech Testing J 27(3): 1–9. doi:10.1520/GTJ11903

    Google Scholar 

  22. Zlotnik S, Díez P (2009) Hierarchical X-FEM for n-phase flow (n > 2). Comput Methods Appl Mech Eng 198(30-32): 2329–2338. doi:10.1016/j.cma.2009.02.025

    Article  Google Scholar 

  23. Darcy H (1856) Les Fontaines Publiques de la Ville de Dijon. Dalmon, Paris

    Google Scholar 

  24. Bear J, Bachmat Y (1999) Introduction to mmodeling of transport phenomena in porous media. In: Bear J (eds) Theory and applications of transport in porous media, vol 4. Kluwer Academic Publishers, Dordrecht, p 136

    Google Scholar 

  25. Knapen A, Poesen J, Govers G, Gyssels G, Nachtergaele J (2007) Resistance of soils to concentrated flow erosion: a review. Earth-Science Rev 80(1–2): 75–109. doi:10.1016/j.earscirev.2006.08.001

    Article  Google Scholar 

  26. Vardoulakis I, Stavropoulou M, Papanastasiou P (1996) Hydro-mechanical aspects of the sand production problem. Transp Porous Media 22(2): 225–244. doi:10.1007/BF01143517

    Article  Google Scholar 

  27. Vardoulakis I, Papanastasiou P, Stavropoulou M (2001) Sand erosion in axial flow conditions. Transp Porous Media 45(2): 267–280. doi:10.1023/A:1012035031463

    Article  Google Scholar 

  28. Giménez R, Govers G (2002) Flow detachment by concentrated flow on smooth and irregular beds. Soil Sci Soc Am J 66: 1475–1483

    Article  Google Scholar 

  29. Woodward DE (1999) Method to predict cropland ephemeral gully erosion. CATENA 37(3-4): 393–399. doi:10.1016/S0341-8162(99)00028-4

    Article  Google Scholar 

  30. Sidorchuk A (2005) Stochastic components in the gully erosion modelling. CATENA 63(2–3): 299–317. doi:10.1016/j.catena.2005.06.007

    Article  Google Scholar 

  31. Donea J, Huerta A (2003) Finite element methods for flow problems. John Wiley & Sons Ltd, Chichester

    Book  Google Scholar 

  32. Adalsteinsson D, Sethian JA (1999) The fast construction of extension velocities in level set methods. J Comput Phys 148(1): 2–22. doi:10.1006/jcph.1998.6090

    Article  MATH  MathSciNet  Google Scholar 

  33. Russo G, Smereka P (2000) A remark on computing distance functions. J Comput Phys 163(1): 51–67. doi:10.1006/jcph.2000.6553

    Article  MATH  MathSciNet  Google Scholar 

  34. Rouy E, Touring A (1992) A viscosity solutions approach to shape-from-shading. SIAM J Numer Anal 29(3): 867–884. doi:10.1137/0729053

    Article  MATH  MathSciNet  Google Scholar 

  35. Budyn E, Hoc T, Jonvaux J (2008) Fracture strength assessment and aging signs detection in human cortical bone using an X-FEM multiple scale approach. Comput Mech 42(4): 579–591. doi:10.1007/s00466-008-0283-1

    Article  MATH  Google Scholar 

  36. Yvonnet J, Quang HL, He QC (2008) An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput Mech 42(1): 119–131. doi:10.1007/s00466-008-0241-y

    Article  MATH  MathSciNet  Google Scholar 

  37. Gross S, Reusken A (2007) Finite element discretization error analysis of a surface tension force in two-phase incompressible flows. SIAM J Numer Anal 45(4): 1679–1700. doi:10.1137/060667530

    Article  MATH  MathSciNet  Google Scholar 

  38. Legrain G, Moës N, Huerta A (2008) Stability of incompressible formulations enriched with X-FEM. Comput Methods Appl Mech Eng 197(21–24): 1835–1849. doi:10.1016/j.cma.2007.08.032

    Article  Google Scholar 

  39. Bligh WG (1910) Dams, barrages and weirs on porous foundations. Eng News 64: 708

    Google Scholar 

  40. Lane EW (1935) Security from under-seepage masonry dams on earth foundations. Trans ASCE 100: 1235–1351

    Google Scholar 

  41. Koenders MA, Sellmeijer JB (1992) A mathematical model for piping. J Geotechn Eng 118(6): 943–946. doi:10.1061/(ASCE)0733-9410

    Article  Google Scholar 

  42. Ojha CS, Singh VP, Adrian DD (2003) Determination of critical head in soil piping. J Hydraulic Eng ASCE 129(7): 511–518. doi:10.1061/(ASCE)0733-9429

    Article  Google Scholar 

  43. Bonelli S, Brivois O (2008) The scaling law in the hole erosion test with a constant pressure drop. Int J Numer Analytical Methods Geomech 32(13): 1573–1595. doi:10.1002/nag.683

    Article  Google Scholar 

  44. Lachouette D, Golay F, Bonelli S (2008) One-dimensional modeling of piping flow erosion. Comptes-Rendus Mécanique 336(9): 731–736. doi:10.1016/j.crme.2008.06.007

    Article  MATH  Google Scholar 

  45. Burns B, Barker R, Ghataora GS (2006) Investigating internal erosion using a miniature resistivity array. NDT & E Int 39(2): 169–174. doi:10.1016/j.ndteint.2004.12.009

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. International Center for Numerical Methods in Engineering (CIMNE), E.T.S. d’Enginyers de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya, Barcelona, Spain

    Régis Cottereau

  2. Laboratoire MSSMat, CNRS UMR 8579, École Centrale Paris, Châtenay-Malabry, France

    Régis Cottereau

  3. Laboratori de Càlcul Numèric, Escola Tècnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya, Barcelona, Spain

    Pedro Díez & Antonio Huerta

Authors
  1. Régis Cottereau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Díez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Huerta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Huerta.

Additional information

The support of Gas Natural Distribución SDG S.A. and Ministerio de Educación y Ciencia de España, through grants DPI2007-62395 and BIA2007-66965, is gratefully acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cottereau, R., Díez, P. & Huerta, A. Modeling, with a unified level-set representation, of the expansion of a hollow in the ground under different physical phenomena. Comput Mech 46, 315–327 (2010). https://doi.org/10.1007/s00466-009-0443-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-009-0443-y

Keywords

Search

Navigation