Inscrição na biblioteca: Guest
Portal Digital Begell Biblioteca digital da Begell eBooks Diários Referências e Anais Coleções de pesquisa
International Journal for Multiscale Computational Engineering
Fator do impacto: 1.016 FI de cinco anos: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Imprimir: 1543-1649
ISSN On-line: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2019027066
pages 201-221

ELASTO-PLASTIC PHASE-FIELD MODEL OF HYDRAULIC FRACTURE IN SATURATED BINARY POROUS MEDIA

Mangesh Pise
Institut für Mechanik, Fachbereich für Ingenieurwissenschaften/Abtl. Bauwissenschaften
Joachim Bluhm
Institut für Mechanik, Fachbereich für Ingenieurwissenschaften/Abtl. Bauwissenschaften
Jörg Schröder
Institut für Mechanik, Fachbereich für Ingenieurwissenschaften/Abtl. Bauwissenschaften; Universitat Duisburg-Essen, Germany

RESUMO

In many fields of engineering, especially in geo sciences and rock mechanics, the theoretical and numerical modeling of hydraulic fracturing of porous materials plays an important role. Hydraulic fracturing is a well-known technology in which porous materials are fractured by a pressurized liquid. The process involves the pressure injection of a fracking fluid (primarily water, often enriched with filling materials and thickening agents) and accompanied by crack nucleation and propagation, as well as mass transport. This article presents a macroscopic model based on the Theory of Porous Media (TPM). For simplification, an incompressible binary model consisting of the solid and liquid phases is used. The development of the damage of the elastic-plastic solid phase is controlled by an evolution equation, which corresponds to known diffusive phase-field models within a continuum mechanical framework. A numerical example shows that the simplified model is indeed capable of simulating hydraulic fracturing of porous media.

Referências

  1. Alessi, R., Ambati, M., Gerasimov, T., Vidoli, S., and De Lorenzis, L., Comparison of Phase-Field Models of Fracture Coupled with Plasticity, Comput. Methods Appl. Sci., vol. 46, pp. 1-21, 2018.

  2. Ambati, M., Gerasimov, T., and De Lorenzis, L., Phase-Field Modeling of Ductile Fracture, Comput. Mech, vol. 55, no. 5, pp. 1017-1040, 2015a.

  3. Ambati, M., Gerasimov, T., and De Lorenzis, L., A Review on Phase-Field Models of Brittle Fracture and a New Fast Hybrid Formulation, Comput. Mech, vol. 55, pp. 383-405, 2015b.

  4. Ambrosio, L. and Tortorelli, V., Approximation of Functionals Depending on Jumps by Elliptic Functionals via y-Convergence, Commun. Pure Appl. Math, vol. 43, pp. 999-1036, 1990.

  5. Atkin, R. and Craine, R., Continuum Theories of Mixtures: Basic Theory and Historical Development, Quart. J. Mech. Appl. Math., vol. 29, pp. 209-244,1975.

  6. Barenblatt, G., The Mathematical Theory of Equilibrium Cracks in Brittle Fracture, Adv. Appl. Mech., vol. 7, pp. 55-129,1962.

  7. Biot, M., General Theory of Three-Dimensional Consolidatio, J. Appl. Phys., vol. 12, pp. 155-164,1941.

  8. Biot, M., Generalized Theory of Acoustic Propagation in Porous Dissipative Media, J. Acoust. Soc. Am., vol. 35, no. 5, Part 1, pp. 1254-1264, 1962.

  9. Bluhm, J., Ein Modell zur Berechnung GeometrischNichtlinearer Probleme der Elastoplastizitat mit Anwendung auf Ebene Stab- tragwerke, PhD, Universitat-GH Essen, Forschritt-Berichte VDI, Reihe 18, No. 94, VDI-Verlag, Diisseldorf, 1991.

  10. Bluhm, J., Modelling of Saturated Thermo-Elastic Porous Solids with Different Phase Temperatures, in Porous Media: Theory, Experiments and Numerical Applications, W. Ehlers and J. Bluhm, Eds., Springer, pp. 87-118, 2002.

  11. Borden, M., Hughes, T., Landis, C., Anvari, A., and Lee, I., A Phase-Field Formulation for Fracture in Ductile Materials: Finite Deformation Balance Law Derivation, Plastic Degradation, and Stress Triaxiality Effects, Int. J. Numer. Methods Eng., vol. 312, pp. 130-166,2016.

  12. Bourdin, B., Francfort, G., and Marigo, J., The Variational Approach to Fracture, J. Elasticity, vol. 91, pp. 5-148, 2008.

  13. Bowen, R., Theory of Mixtures, in Continuum Physics, A.C. Eringen, Ed., vol. III, New York: Academic Press, pp. 1-127, 1976.

  14. Bowen, R., Incompressible Porous Media Models by Use of the Theory of Mixtures, Int. J. Eng. Sci., vol. 18, pp. 1129-1148, 1980.

  15. Bowen, R., Compressible Porous Media Models by Use of the Theory of Mixtures, Int. J. Eng. Sci, vol. 20, pp. 697-735,1982. Braides, D., Approximation of Free Discontinuity Problems, Springer, 1998. Braides, D., r-Convergence for Beginners, Oxford: Oxford University Press, 2002.

  16. Cheng, A.D., Poroelasticity, vol. 27 of Theory and Applications of Transport in Porous Media, Springer, 2016. Dal Maso, G., An Introduction to F-Convergence, Basel, Switzerland: Birkhauser, 1993.

  17. Dastjerdy, F., Barani, O., and Kalantary, F., Modeling of Hydraulic Fracture Problem in Partially Saturated Porous Media Using Cohesive Zone Model, Int. J. Civil Eng., vol. 13, nos. 3-4B, pp. 185-194,2015.

  18. de Boer, R., Constitutive Equations for Granular and Brittle Materials in Plastic Range - A Kinematic Hardening Model, Report MECH 88/6, FB 10/Mechanik, Universitat-GH-Essen, 1988a.

  19. de Boer, R., On Plastic Deformation of Soils, Int. J. Plasticity, vol. 4, no. 4, pp. 371-391, 1988b.

  20. de Boer, R., Theory of Porous Media-Highlights in the Historical Development and Current State, Springer, 2000.

  21. de Boer, R. and Ehlers, W., Theorie der Mehrkomponentenkontinua mit Anwendung auf Bodenmechanische Probleme, Forschung-berichte aus dem Fachbereich Bauwesen, Heft 40, Universitat-GH-Essen, 1986.

  22. de Borst, R., Rethore, J., and Abelian, M.A., A Two-Scale Approach for Fluid Flow in Fracturing Porous Media, in Comput. Modell. Concrete Struct., N. Bicanic, R. de Borst, H. Mang, and G. Meschke, Eds., London: Taylor and Francis, pp. 451-459, 2010.

  23. de Laguna, W., Disposal of Radioactive Wastes by Hydraulic Fracturing: Part I. General Concept and First Field Experiments, Nuclear Eng. Design, vol. 3, no. 2, pp. 338-352, 1966a.

  24. de Laguna, W., Disposal of Radioactive Wastes by Hydraulic Fracturing: Part II. Mechanics of Fracture Formation and Design of Observation and Mentoring Wells, Nuclear Eng. Design, vol. 3, no. 3, pp. 431-438, 1966b.

  25. Detournay, E. and Cheng, A.D., Plane Strain Analysis of a Stationary Hydraulic Fracture in a Poroelastic Medium, Int. J. Solids Struct., vol. 27, no. 13, pp. 1645-1662, 1991.

  26. Drucker, D. andPrager, W., Soil Mechanics and Plastic Analysis or Limit Design, Quart. Appl. Math., vol. 10, pp. 157-165,1952.

  27. Ehlers, W., Porose Medienein Kontinuummechanisches Modell auf der Basis der Mischungstheorie, Habilitationsschrift, Universitat - GH Essen, 1989.

  28. Ehlers, W., Torward Finite Theories of Liquid Saturated Elasto-Plastic Porous Media, Int. J. Plasticity, vol. 7, pp. 433-475,1991.

  29. Ehlers, W., Constitutive Equations for Granular Materials in Geomechanical Context, in Continuum Mechanics in Environmental Sciences and Geophysics, K. Hutter, Ed., CISM Courses and Lectures No. 339, Springer, 1993.

  30. Ehlers, W., Grundlegende Konzepte in der Theorie Poroser Medien, Tech. Mechanik, vol. 16, pp. 63-76, 1996.

  31. Ehlers, W., Porose Medienein Kontinuummechanisches Modell auf der Basis der Mischungstheorie, Neuaulage der Orginalar-beit: Habilitationsschrift, Forschungsberichte aus dem Fachbereich Bauwesen, Universitat-Gesamthochschule-Essen, Heft 47, Essen, 1989, Report No.: II-22, Institut fur Mechanik (Bauwesen), Lehrstuhl fur Kontinuumsmechanik, Prof. Dr.-Ing. W. Ehlers, Universitat Stuttgart, 2012.

  32. Ehlers, W. and Luo, C., A Phase-Field Approach Embedded in the Theory of Porous Media for the Description of Dynamic Hydraulic Fracturing, Comput. Methods Appl. Mech. Eng., vol. 315, pp. 348-368,2017.

  33. Francfort, G. and Marigo, J., Revisiting Brittle Fracture as an Energy Minimization Problem, J. Mech. Phys. Solids, vol. 46, pp. 1319-1342, 1998.

  34. Griffith, A., The Phenomenon of Rupture and Flow in Solids, Philosoph. Trans. Royal Soc. London, Series A, vol. 221, pp. 163-198, 1921.

  35. Hakim, V. and Karma, A., Laws of Crack Motion and Phase-Field Models of Fracture, J. Mech. Phys. Solids, vol. 57, pp. 342-368, 2009.

  36. Haupt, P., Continuum Mechanics and Theory ofMaterials, Springer, 2002.

  37. Heider, Y. and Markert, B., A Phase-Field Modeling Approach of Hydraulic Fracture in Saturated Porous Media, Mech. Res. Commun., vol. 80, pp. 38-46, 2017.

  38. Heinrich, G. and Desoyer, K., Hydromechanische Grundlagen fur die Behandlung von Stationaren und Instationaren Grundwasser-stromungen, Ingenieur-Archiv, vol. 23, pp. 182-185, 1955.

  39. Heinrich, G. and Desoyer, K., Hydromechanische Grundlagen fur die Behandlung von Stationaren und Instationaren Grundwasser-stromungen, II. Mitteilung, Ingenieur-Archiv, vol. 24, pp. 81-84,1956.

  40. Heinrich, G. and Desoyer, K., Theorie Dreidimensionaler Setzungsvorgange in Tonschichten, Ingenieur-Archiv, vol. 30, pp. 225-253, 1961.

  41. Hesch, C. and Weinberg, K., Thermodynamically Consistent Algorithms for a Finite-Deformation Phase-Field Approach to Fracture, Int. J. Numer. Methods Eng., vol. 99, pp. 906-924,2014.

  42. Hofacker, M. and Miehe, C., A Phase Field Model of Dynamic Fracture: Robust Field Updates for the Analysis of Complex Crack.

  43. Patterns, Int. J. Numer. Methods Eng., vol. 93, pp. 276-301, 2013.

  44. Irwin, G., Elasticity and Plasticity: Fracture, Encyclopedoa ofPhys., S. Flugge, Ed., Springer, vol. 6, pp. 551-590, 1958.

  45. Karma, A., David, A., and Levine, H., Phase-Field Model of Mode III Dynamic Fracture, Phys. Rev. Lett., vol. 87, p. 045502, 2001.

  46. Kuhn, C., Noll, T., and Muller, R., On Phase Field Modeling of Ductile Fracture, GAMM-Mitteilungen, vol. 39, no. 1, pp. 35-54, 2016.

  47. Kuhn, K. and Muller, R., A Continuum Phase Field Model for Fracture, Eng. Fracture Mech., vol. 77, pp. 3625-3634, 2010.

  48. Lee, E., Elasto-Plastic Deformation at Finite Strains, J. Appl. Mech, vol. 36, pp. 1-6, 1969.

  49. Lee, E. and Liu, D., Finite-Steain Elasto-Plastic Theory with Application to Plane-Wave Analysis, J. Appl. Phys, vol. 38, no. 1, pp. 19-27,1967.

  50. Li, L., Tang, C., Li, G., Wang, S., Liang, Z., and Zhang, Y., Numerical Simulation of 3D Hydraulic Fracturing based on an Improved Flow-Stress-Damage Model and a Parallel FEM Technique, Rock Mech. Rock Eng., vol. 45, pp. 801-818, 2012.

  51. Luo, C. and Ehlers, W., Hydraulic Fracturing based on the Theory of Porous Media, Proc. Appl. Math. Mech. (PAMM), vol. 15, pp. 401-402,2015.

  52. Markert, B. and Heider, Y., Coupled Multi-Field Continuum Methods for Porous Media Fracture, Recent Trends in Computational Engineering-CE2014, M. Mehl, M. Bischoff, and M. Schafer, Eds., vol. 105 of Lecture Notes Comput. Sci. Eng., Springer, 2015.

  53. Mendelsohn, D., A Review of Hydraulic Fracture Modeling-I: General Concept, 2D Models, Motivation for 3D Modeling, J. Energy Resour. Technol., vol. 106, pp. 369-376, 1984a.

  54. Mendelsohn, D., A Review of Hydraulic Fracture Modeling-II: 3D Modeling and Vertical Growth in Layered Rock, J. Energy Resour. Technol, vol. 106, pp. 543-553, 1984b.

  55. Miehe, C., Aldakheel, F., and Raina, A., Phase Field Modeling of Ductile Fracture at Finite Strains: A Variational Gradient-Extended Plasticity-Damage Theory, Int. J. Plasticity, vol. 84, pp. 1-32, 2016.

  56. Miehe, C., Hofacker, M., and Welschinger, F., A Phase Field Model for Rate-Independent Crack Propagation: Robust Algorithmic Implementation based on Operator Splits, Comput. Methods Appl. Mech. Eng., vol. 199, pp. 2765-2778, 2010a.

  57. Miehe, C., Welschinger, F., and Hofacker, M., Thermodynamically Consistent Phase-Field Models of Fracture: Variational Principles and Multi-Field FE Implementations, Int. J. Numer. Methods Eng., vol. 83, pp. 1273-1311,2010b.

  58. Mikelic, A., Wheeler, M.F., and Wick, T., A Phase-Field Method for Propagating Fluid-Filled Fractures Coupled to Surrounding Porous Medium, Multiscale Model. Simul., vol. 13, no. 1, pp. 367-398, 2015.

  59. Mumford, D. and Shah, J., Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems, Commun. Pure Appl. Math, vol. 42, pp. 577-685, 1989.

  60. Ricken, T., Schwarz, A., and Bluhm, J., A Triphasic Model of Transversely Isotropic Biological Tissue with Application to Stress and Biological Induced Growth, Comput. Mater. Sci., vol. 39, pp. 124-136, 2007.

  61. Truesdell, C., Thermodynamics of Diffusion, Rational Thermodynamics, C. Truesdell, Ed., Springer, pp. 219-236,1984.

  62. Truesdell, C. and Toupin, R.A., The Classical Field Theories, Handbuch der Physik, S. Flugge, Ed., vol. III/1, Springer, pp. 226-902, 1960.

  63. Volk, W., Untersuchung Des Lokalisierungsverhaltens Mikropolarer Poroser Medien Mit Hilfe Der Cosserat-Theorie, PhD, Universitat Stuttgart, Institut furMechanik (Bauwesen), Lehrstuhl II, Prof. Dr.-Ing. W. Ehlers, BerichtNr. II-2, 1999.

  64. von Mises, R., Mechanik Der Festen Korper Im Plastisch-Deformablen Zustand, Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen, Mathematisch-Physikalische Klasse, pp. 582-592, 1913.

  65. Zienkiewicz, O. and Taylor, R., The Finite Element Method, 6th Ed., vols. 1-3, Elsevier, 2005.


Articles with similar content:

MULTISCALE MODEL FOR DAMAGE-FLUID FLOW IN FRACTURED POROUS MEDIA
International Journal for Multiscale Computational Engineering, Vol.14, 2016, issue 4
Mahdad Eghbalian, Richard Wan
EVAPORATION FROM THIN POROUS MEDIA WITH MIXED WETTABILITY
International Heat Transfer Conference 16, Vol.21, 2018, issue
Rui Wu, Changying Zhao
CALCULATION OF THE INTERGRANULAR ENERGY IN TWO-LEVEL PHYSICAL MODELS FOR DESCRIBING THERMOMECHANICAL PROCESSING OF POLYCRYSTALS WITH ACCOUNT FOR DISCONTINUOUS DYNAMIC RECRYSTALLIZATION
Nanoscience and Technology: An International Journal, Vol.7, 2016, issue 2
Peter V. Trusov, Nikita S. Kondratev
COUPLED COHESIVE ZONE REPRESENTATIONS FROM 3D QUASICONTINUUM SIMULATION ON BRITTLE GRAIN BOUNDARIES
International Journal for Multiscale Computational Engineering, Vol.9, 2011, issue 4
Carsten Konke, Torsten Luther
OPEN-CELL METAL FOAM MESH GENERATION FOR LATTICE BOLTZMANN SIMULATIONS
Journal of Porous Media, Vol.21, 2018, issue 5
A. Festuccia, Gino Bella, D. Chiappini