Skip to main content

The Numerical Solution of the Biot Equations

  • Chapter
  • First Online:
Land Subsidence Analysis in Urban Areas

Part of the book series: Springer Environmental Science and Engineering ((SPRINGERENVIRON))

  • 1661 Accesses

Abstract

In this chapter, first a review of the iterative methods proposed for the numerical solution of the coupled Biot equations is presented. Then, the numerical scheme based on the decoupling method of the solution of the Biot model is presented. The numerical method, called the compartmental model is similar to the finite-volume method. We describe the treatment of the different types of boundary conditions.

Eliyahu Wakshal (Deceased)

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Aguilar G, Gaspar F, Lisbona F, Rodrigo C (2008) Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int J Numer Method Eng 75:1282–1300

    Article  Google Scholar 

  • Babuska I, Oden JT, Lee JL (1977) Mixed-hybrid finite element method approximations of second-order elliptic boundary value problems. Comput Method Appl M 1:175–206

    Article  Google Scholar 

  • Bachmat Y (1995) A compartmental model for the Yarkon Taninim aquifer. Report of the Hydrological Service of Israel - Israel

    Google Scholar 

  • Barbeiro S, Wheeler MF (2008) A priori error estimates for the numerical solution of a coupled geomechanics and reservoir flow model with stress-dependent permeability. Comput Geosci 14:755–766

    Article  Google Scholar 

  • Barth T, Ohlberger M (2004) Finite volume methods: foundation and analysis. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics. Wiley, New York

    Google Scholar 

  • Bear J (1979) Hydraulics of groundwater. McGraw-Hill, New York

    Google Scholar 

  • Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods, vol 15, Springer series in computational mathematics. Springer, Berlin

    Book  Google Scholar 

  • Campana ME, Harrington GA, Tezcan L (2001) Compartmental model approaches to groundwater flow simulation. In: Mook W (ed) Environmental isotopes in the hydrological cycle: principles and applications, vol vi. UNESCO, Paris, pp 37–73

    Google Scholar 

  • Carcione JM, Morency C, Santos JE (2010) Computational poroelasticity- a review. Geophysics 75:229–243

    Article  Google Scholar 

  • Carroll RWH, Pohll G, Earman S, Hershey RL (2008) A comparison of groundwater fluxes computed with MODFLOW and a mixing model using deuterium: application to the eastern Nevada test site and vicinity. J Hydrol 361:371–385

    Article  Google Scholar 

  • Chavent G, Roberts JE (1991) A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv Water Resour 14:329–348

    Article  Google Scholar 

  • Cordes C, Kinzelbach W (1996) Comment on: Mose et al. Application of the mixed hybrid finite approximation in a groundwater flow model: luxury or necessity? Water Resour Res 32:1905–1909

    Article  Google Scholar 

  • Darlow BL, Ewing RE, Wheeler MF (1984) Mixed finite elements methods for miscible displacement problems in porous media. Soc Pet Eng J 24:391–398

    Google Scholar 

  • Dassargues A, Li XL (1991) Computing the land subsidence of Shanghai by a finite element method. In: Proceedings of the fourth international symposium on land subsidence, May 1991. IAHS Publication – 200

    Google Scholar 

  • Durlofsky LJ (1994) Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resour Res 30:965–973

    Article  Google Scholar 

  • Ern A, Guermond JL (2004) Theory and practice of finite elements. Springer, New York

    Google Scholar 

  • Eymard R, Gallouet T, Herbin R (2006) Finite volume methods, update of the preprint no 97-19. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol 7, pp 713–1020

    Google Scholar 

  • Ferronato M, Castelletto N, Gambolati G (2010) A fully coupled 3-D mixed finite element model of Biot consolidation. J Comput Phys 229:4813–4830

    Article  Google Scholar 

  • Gambolati G, Teatini P, Bau D, Ferronato M (2000) The importance of poro-elastic coupling in dynamically active aquifers of the Po River basin, Italy. Water Resour Res 36:2443–2459

    Article  Google Scholar 

  • Geertsma J (1973) Land subsidence above compacting oil and gas reservoirs. J Pet Technol 25:734–744

    Google Scholar 

  • Girault V, Raviart PA (1986) Finite element methods for Navier–Stokes equations: theory and algorithms, vol 5, Springer series in computational mathematics. Springer, Berlin

    Book  Google Scholar 

  • Guvanasen V, Chan T (1995) A new three-dimensional finite-element analysis of hysteresis thermo-hydromechanical deformation of fractured rock mass with dilatance in fractures. The second conference on mechanics of jointed and faulted rocks, Vienna, Austria

    Google Scholar 

  • Hicks TW, Pine RJ et al (1996) A hydro-thermo-mechanical numerical model for HDR geothermal reservoir evaluation. Int J Rock Mech Min 33:499–511

    Article  Google Scholar 

  • Hörlin NE (2010) A symmetric weak form of Biot’s equations based on redundant variables representing the fluid, using a Helmholtz decomposition of the fluid displacement vector field. Int J Numer Method Eng 84:1613–1637

    Article  Google Scholar 

  • Hoffmann J, Leake SA, Galloway DL, Wilson AM (2003) MODFLOW-2000 ground-water model – user guide to the subsidence and aquifer-system compaction (SUB) package. US Geological Survey Open-File Report 03–233

    Google Scholar 

  • Hsieh PA (1996) Deformation-induced changes in hydraulic head during ground-water withdrawal. Ground Water 34:1082–1089

    Article  Google Scholar 

  • Kaasschieter EF, Huijben AJM (1992) Mixed-hybrid finite elements and streamline computation for the potential flow problem. Numer Method Partial D E 8:221–266

    Article  Google Scholar 

  • Lippmann MJ, Narasimhan TN, Witherspoon PA (1976) Numerical simulation of reservoir compaction in liquid dominated geothermal systems. In: Proceedings of the second international symposium on land subsidence, Anaheim, CA, December 1976

    Google Scholar 

  • Mose R, Siegel P, Ackerer P, Chavent G (1994) Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity? Water Resour Res 30:3001–3012

    Article  Google Scholar 

  • Narasimhan TN, Witherspoon PA (1976) Numerical model for land subsidence in shallow groundwater system. In: Proceedings of the second international symposium on land subsidence, Anaheim, CA, December 1976

    Google Scholar 

  • Ohnishi Y, Shibata H et al (1987) Development of finite element code for the analysis of coupled thermo-hydro-mechanical behavior of a saturated-unsaturated medium. Coupled processes associated with nuclear waste repositories. Academic, Orlando, pp 679–696

    Google Scholar 

  • Raviart PA, Thomas JM (1977) A mixed finite element method for second order elliptic problems. In: Mathematical aspects of finite element method. Lecture notes in mathematics, vol 606, pp 292–315. Springe, New York

    Google Scholar 

  • Shimizu M (1991) Finite element analysis of land subsidence due to the variation of groundwater level. In: Proceedings of the fourth international symposium on land subsidence, IAHS Publication – 200, May 1991

    Google Scholar 

  • Szostak-Chrzanowski A, Chrzanowski A (1991) Modeling and prediction of ground subsidence using an iterative finite element method. In: Proceedings of the fourth international symposium on land subsidence, May 1991. IAHS Publication – 200

    Google Scholar 

  • Toh KC, Phoon KK (2007) Comparison between iterative solution of symmetric and non-symmetric forms of Biot’s FEM equations using the generalized Jacobi preconditioner. Int J Numer Anal Method Geomech 32:1131–1146

    Article  Google Scholar 

  • Younes A, Ackerer PH, Chavent G (2004) Int J Numer Method Eng 59:365–388. doi: 10.1002/nme.874

    Article  Google Scholar 

  • Zeitoun DG (2012) A compartmental model for stable time- dependent modeling of surface water and groundwater. Environ Model Assess 17:673–697

    Article  Google Scholar 

  • Zeitoun DG, Laible JP, Pinder GF (1995) A weighted least squares method for first-order hyperbolic systems. Int J Numer Method Fl 20:191–212

    Article  Google Scholar 

  • Zeitoun DG, Laible JP, Pinder GF (1997) An iterative penalty method for the least squares solution of boundary value problems. Numer Method Partial D E l3:257–281

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Zeitoun, D.G., Wakshal, E. (2013). The Numerical Solution of the Biot Equations. In: Land Subsidence Analysis in Urban Areas. Springer Environmental Science and Engineering. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5506-2_6

Download citation

Publish with us

Policies and ethics