Skip to main content
SpringerLink
Log in

Fixed Domain Methods for Free and Moving Boundary Flows in Porous Media

  • Chapter
Mathematical Modeling for Flow and Transport Through Porous Media

  • 236 Accesses

Abstract

A survey is presented concerning fixed domain methods used to solve mathematical models of free and moving boundary flow problems in porous media. These include the following: variational inequality or quasi-variational inequality formulations; general inequality formulations which have been set and solved in fixed domains; and the residual flow procedure. Finally, some parallel computing methods and mesh adaptation methods are discussed to demonstrate how these fixed domain formulations can be solved with current technology.

The fixed domain methods that are referenced herein can be classified into two groups: the variational inequality method and the extended pressure head method. Baiocchi was the first to apply the variational inequality method to free boundary problems of flows through porous media. This method in general also uses an extension of the pressure head but adds an application of an integral transformation (a Baiocchi transformation) to the problem. The method possesses a beautiful mathematical structure for its theory and yields simple numerical solution algorithms. However, application of the method is difficult if not impossible in some cases depending upon the regularity of the seepage domain.

The extended pressure head method is based on the concept that the pressure is extended “smoothly” across the free or moving boundary into the unsaturated region from the flow domain. The extension of the pressure head to the entire porous medium yields an extended coefficient of permeability of the medium which is equal to the saturated coefficient in the seepage region and is equal to zero or some small value (for computational purposes) in the unsaturated region.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Alt, H.W., 1977a, The fluid flow through porous media. Regularity of the free surface, Manuscripta Math. 21, 255–272.

    Article  Google Scholar 

  • Alt, H.W., 1977b, A free boundary problem associated with the flow of ground water, Arch. Rat. Mech. Anal. 64, 111–126.

    Article  Google Scholar 

  • Alt, H.W., 1979, Stromungen durch inhomogene porose Medien mit freiem Rand, J. Reine Angew. Math 305, 89–115.

    Google Scholar 

  • Alt, H.W., 1980a, Numerical solution of steady-state porous flow free boundary problems, Numer. Math. 36, 73–98.

    Google Scholar 

  • Alt, H.W., 1980b, A new numerical method for solving the dam problem, Proc. Seminar on Free Boundary Problems Pavia 1979, Ist. Naz. Alta Mat., Roma, 1, 89–108.

    Google Scholar 

  • Baiocchi, C., 1971, Sur un probleme a frontiere libre traduisant le filtrage de liquides a travers des milieux poreux, C.R. Acad. Sci. Paris 273, 1215–1217.

    Google Scholar 

  • Baiocchi, C., 1972, Su un problema di frontiera libera connesso a questioni di idraulica, Ann. Math. Pura Appl 92(4), 107–127.

    Google Scholar 

  • Baiocchi, C., 1975a, Free boundary problems in the theory of fluid flow through porous media, Proc. Int. Con gr. Math. Vancouver Vancouver, 1974, II, 237–243.

    Google Scholar 

  • Baiocchi, C., 1975b, Studio di un problema quasi-variazionale connesso a problemi di frontiera libera, Boll. UMI. 11, 589–613.

    Google Scholar 

  • Baiocchi, C., Brezzi, F. and Comincioli, V., 1976a, Free boundary problems in fluid flow through porous media, Proc. 2nd Int. Symp. Finite Element Methods in Flow Problems ICCAD, Santa Margherita Ligure, 409–420.

    Google Scholar 

  • Baiocchi, C., Comincioli, V. and Maione, U., 1975, Unconfined flow through porous media, Meccanica-J. Ital. Assoc. Theoret. Appl. Mech., 10, 151–155.

    Google Scholar 

  • Baiocchi, C., Comincioli, V., Guerri, L. and Volpi, G., 1973a, Free boundary problems in the theory of fluid flow through porous media: a numerical approach, Calcolo 10, 186.

    Google Scholar 

  • Baiocchi, C., Comincioli, V., Magenes, E. and Pozzi, G.A., 1973b, Free boundary problems in the theory of fluid flow through porous media: existence and uniqueness theorems, Ann. Mat. Pura Appl. 93(4), 1–82.

    Google Scholar 

  • Baiocchi, C., Comincioli, V., Magenes, E. and Pozzi, G.A., 19766, Fluid flow through porous media: a new theoretical and numerical approach, Proc. Int. Autumn Course, Math. Numerical Methods in Fluid Dynamics Int. Centre for Theoretical Physics, Trieste, 395–447.

    Google Scholar 

  • Baiocchi, C. and Friedman, A., 1977, A filtration problem in a porous medium with variable permeability, Ann. Mat. Pura Appl. 114(4), 377–393.

    Google Scholar 

  • Bathe, K.J. and Khoshgoftaar, M.R., 1979, Finite element free surface seepage analysis without mesh iteration, Int. J. for Num. and Anal. Meth. in Geomech. 3, 13–22.

    Google Scholar 

  • Bear, J., 1972, Dynamics of Fluids in Porous Media American Elsevier, New York.

    Google Scholar 

  • Bensoussan, A. and Lions, J.L., 1973, Nouvelle formulation de problemes de controle impulsionnel et applications, C.R. Acad. Sci. Paris, (A-B), 276 1189–1192.

    Google Scholar 

  • Bouwer, H., 1964, Unsaturated flow in ground-water hydraulics, J. of Hydraulics Div. ASCE, 90, 121–144.

    Google Scholar 

  • Brezis, H., Kinderlehrer, D., and Stampacchia, G., 1978, Sur une nouvelle formulation du probleme de l’ecoulement a travers une digue, C.R. Acad. Sci. Paris 287, 711–714.

    Google Scholar 

  • Bruch, J.C., Jr., 1980, A survey of free boundary value problems in the theory of fluid flow through porous media: variational inequality approach, Adv. in Water Resources Part I 3, 65–80; Part II 3, 116–124.

    Google Scholar 

  • Bruch, Jr., J.C., 1989, Bibliography of fixed domain methods in flow through porous media, Tech. Rept. UCSB-ME-89–5 Dept. of Mech. and Env. Engr., Univ. of Calif., Santa Barbara, Calif., 21 pp.

    Google Scholar 

  • Burkley, V.J., 1989, Engine optimization and adaptive error analysis, M.S. Thesis, Dept. of Mech. and Envr. Engr., Univ. of Calif., Santa Barbara, Calif., 139 pp..

    Google Scholar 

  • Caffarelli, L.A., 1976, The smoothness of the free surface in a filtration problem, Arch. Rat. Mech. Anal. 63, 77–86.

    Google Scholar 

  • Caffrey, J. and Bruch, J.C., Jr., 1979, Three-dimensional seepage through a homogeneous dam, Advances in Water Resources 2 (4), 167–176.

    Article  Google Scholar 

  • Chung, K.Y. and Kikuchi, N., 1987, Adaptive methods to solve free boundary problems of flow through porous media, Int. J. for Num. and Anal. Meth. in Geomech. 11, 1731.

    Google Scholar 

  • Comincioli, V., 1974, A comparison of algorithms for some free boundary problems, Publ. Lab. Analisi Numerica CNR Pavia 79, 38 pp.

    Google Scholar 

  • Comincioli, V. and Guerri, L., 1976, Numerical solution of free boundary problems in seepage flow with capillary fringe, Comput. Meth. Appl. Mech. Eng. 7, 153–178.

    Google Scholar 

  • Cryer, C.W., 1976a, A survey of steady state porous flow free boundary problems, MRC Techn. Summary Rep. 1657, Univ. Wisconsin, Madison, 135 pp.

    Google Scholar 

  • Cryer, C.W., 1976b, A survey of trial free-boundary methods for the numerical solution of free boundary problems, MRC Techn. Summary Rep. 1693, Univ. Wisconsin, Madison, 156 pp.

    Google Scholar 

  • Cryer, C.W., 1977, A bibliography of free boundary problems, MRC Techn. Summary Rep. 1793, Univ. Wisconsin, Madison.

    Google Scholar 

  • Desai, C.S., 1976, Finite element residual schemes for unconfined flow, Int. J. for Num. Meth. in, Short Communication, 10, 1415–1418.

    Google Scholar 

  • Desai, C.S. and Baseghi, B., 1989, Theory and verification of residual flow procedure for 3-D free surface seepage, Advances in Water Resources (in press).

    Google Scholar 

  • Desai, C.S. and Li, G.C., 1983, A residual flow procedure and application for free surface flow in porous media, Advances in Water Resources 6, 27–35.

    Google Scholar 

  • Duvaut, G. and Lions, J.L., 1976, Inequalities in mechanics and physics, Grund. der Math. Wiss. 219, Springer, Berlin.

    Book  Google Scholar 

  • Friedman, A., 1976a, The shape and smoothness of the free boundary for some elliptic variational inequalities, Ind. Univ. Math. J. 25, 103–118.

    Google Scholar 

  • Friedman, A., 1976b, A problem in hydraulics with non-monotone free boundary, Ind. Univ. Math. J. 25, 577–592.

    Google Scholar 

  • Friedman, A. and Jensen, R., 1977, Convexity of the free boundary in the Stefan problem and in the dam problem, Arch. Rat. Mech. Anal. 67, 1–24.

    Google Scholar 

  • Gilardi, G., 1976, Studio di una famiglia di disequazioni quasivariazionali connessa con un problema di frontiera libera, Boll. U.M.I. 13 — B 138–159.

    Google Scholar 

  • Glowinski, R., 1976, Introduction to the approximation of elliptic variational inequalities, No. 76006, Universite Paris VI, Laboratoire Analyse Numerique L.A., 189.

    Google Scholar 

  • Glowinski, R., Lions, J.L., and Tremolieres, R., 1976, Analyse Numerique des Inequations Variationnelles 2 vols., Dunod.

    Google Scholar 

  • Jensen, R., 1977, Structure of the non-monotone free boundaries in a filtration problem, Ind. Univ. Math. J. 26, 1121–1135.

    Google Scholar 

  • Kinderlehrer, D., 1976, The free boundary determined by the solution to a differential equation, Ind. Univ. Math. J. 25, 195–208.

    Google Scholar 

  • Lewy, H. and Stampacchia, G., 1969, On the regularity of the solution of a variational inequality, Comm Pure Appl. Math. 22, 153–188.

    Google Scholar 

  • Lewy, H. and Stampacchia, G., 1971, On the existence and smoothness of solutions of some non-coercive variational inequalities, Arch. Rat. Mech. Anal. 41, 241–253.

    Google Scholar 

  • Lin, C.S., Bruch, J.C., Jr., Sloss, J.M., and Comincioli, V., 1984, On the solution of transient free surface flow problems in porous media by a fixed domain method, J. of Hydrology 71 (3/4), 353–376.

    Google Scholar 

  • Lions, J.L. and Stampacchia, G., 1967, Variational inequalities, Comm. Pure Appt. Math. 20, 493–519.

    Google Scholar 

  • Marini, L.D. and Pietra, P., 1986, Fixed-point algorithms for stationary flow in porous media, Comp. Meth. in Appl. Mech. and Engr. 56, 17–45.

    Google Scholar 

  • Mosco, U., 1973, An introduction to the approximate solution of variational inequalities, Constructive Aspects of Functional Analysis Edizioni Cremonese, Roma, 499–685.

    Google Scholar 

  • Oden, J.T. and Kikuchi, N., 1980, Theory of variational inequalities with applications to problems of flow through porous media, Int. J. Engr. Sci. 18(10), 1173–1284.

    Google Scholar 

  • Pietra, P., 1982, An up-wind finite element method for a filtration problem, R.A.I.R.O. Anal. Num. 16(4), 463–481.

    Google Scholar 

  • Rank, E. and Werner, H., 1986, An adaptive finite element approach for the free surface seepage problem, Int. J. for Num. Meth. in Engr. 23, 1217–1228.

    Google Scholar 

  • Remar, J., Bruch, J.C., Jr., and Sloss, J.M., 1984, Numerical solutions to some free surface flows through nonhomogeneous media, Int. J. for Num. Meth. in Engr. 20, 143–167.

    Google Scholar 

  • Wang, K.-P. and Bruch, J.C., Jr., 1989, Solutions of a steady state free surface seepage problem on a hypercube concurrent computer, Engr. Comp. 6 (3), 225–236.

    Google Scholar 

  • Westbrook, D.R., 1985, Analysis of inequality and residual flow procedures and an iterative scheme for free surface seepage, Int. J. for Num. Meth. in Engr. 21, 1791–1802.

    Google Scholar 

  • White, R.E., 1987, Multisplittings and parallel iterative methods, Comp. Meth. in Appl. Mech. and Engr. 64, 567–577.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, Santa Barbara, CA, 93106, USA

    J. C. Bruch Jr.

Authors
  1. J. C. Bruch Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Tel Aviv, Israel

    Gedeon Dagan

  2. Neubiberg, Germany

    Ulrich Hornung

  3. Augsburg, Germany

    Peter Knabner

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Bruch, J.C. (1991). Fixed Domain Methods for Free and Moving Boundary Flows in Porous Media. In: Dagan, G., Hornung, U., Knabner, P. (eds) Mathematical Modeling for Flow and Transport Through Porous Media. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2199-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-2199-8_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4127-2

  • Online ISBN: 978-94-017-2199-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation