Models of thermoelasticity for porous materials with fractures taken into account

  • Alekseev Valentin N., alekseev.valen@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
  • Vasilyeva Maria V., vasilyevadotmdotv@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
  • Prokopiev Grigorii A., reilroot@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
  • Tyrylgin Aleksey A., kocglk@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 42 Kulakovsky Street, Yakutsk 677000, Russia
Keywords: thermoelasticity, fracture, porous medium, fuel element, double diffusion model, interface condition, discontinuous Galerkin method, numerical simulation

Abstract

We propose a mathematical model and a computational algorithm for solving thermoelasticity problems in fractured porous media. To simulate heat transfer in porous media, a mathematical model is constructed using double diffusion models. The heat transfer in the cracks is taken into account by setting the interface condition which allows modeling the temperature jump at the crack boundary. To calculate the stress-strain state, a linear elasticity model is used with an additional condition on the crack. For the numerical solution of the problem an approximation is constructed using the Galerkin discontinuous method which allows taking into account the interface condition in the variational formulation. We present the results of the numerical realization of the model problem using the proposed model of thermoelasticity.

References

[1]Vasil'eva M. V. and Stal'nov D. A., “Mathematical modelling of the thermodynamic state of the heat-inducing element”, Vestn. SVFU, 2016, no. 1, 45–59
[2]Hales J. D., Tonks M. R., Chockalingam K., Perez D. M., Novascone S. R., Spencer B. W., Williamson R. L., “Asymptotic expansion homogenization for multiscale nuclear fuel analysis”, Comput. Materials Sci., 99 (2015), 290–297  crossref  scopus
[3]Antic A., Hill J. M., “The double-diffusivity heat transfer model for grain stores incorporating microwave heating”, Appl. Math. Model., 27:8 (2003), 629–647  crossref  zmath  scopus
[4]Bai M., Elsworth D., Roegiers J.-C., “Multiporosity/multipermeability approach to the simulation of naturally fractured reservoirs”, Water Resources Research, 29:6 (1993), 1621–1634  crossref  scopus
[5]Lee A. I., Hill J. M., “On the general linear coupled system for diffusion in media with two diffusivities”, J. Math. Anal. Appl., 89:2 (1982), 530–557  crossref  mathscinet  zmath  scopus
[6]Showalter R. E., Visarraga D. B., “Double-diffusion models from a highly-heterogeneous medium”, J. Math. Anal. Appl., 295:1 (2004), 191–210  crossref  mathscinet  zmath  scopus
[7]Arbogast T., Douglas Jr J., Hornung U., “Derivation of the double porosity model of single phase flow via homogenization theory”, SIAM J. Math. Anal., 21:4 (1990), 823–836  crossref  mathscinet  zmath
[8]Barenblatt G. I., Zheltov Iu P., Kochina I. N., “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks (strata)”, J. Appl. Math. Mech., 24:5 (1960), 1286–1303  crossref  zmath  scopus
[9]Kazemi H., Merrill Jr L. S., Porterfield K. L., Zeman P. R., “Numerical simulation of water-oil flow in naturally fractured reservoirs”, Soc. Petroleum Engin. J., 16:6 (1976), 317–326  crossref
[10]Warren J. E., Root P. J., “The behavior of naturally fractured reservoirs”, Soc. Petroleum Engin. J., 3:3 (1963), 245–255  crossref
[11]Vabishhevich P. N. and Grigor'ev A. V, “Numerical modeling of fluid flow in anisotropic fractured porous media”, Numer. Anal. Appl., 9:1 (2016), 45–56  mathnet  crossref  mathscinet  mathscinet  elib  scopus
[12]Boal N., Gaspar F. J., Lisbona F., Vabishchevich P., “Finite-difference analysis for the linear thermoporoelasticity problem and its numerical resolution by multigrid methods”, Math. Modell. Anal., 17:2 (2012), 227–244  crossref  mathscinet  zmath  elib  scopus
[13]Brown D. L., Vasilyeva M., “A generalized multiscale finite element method for poroelasticity problems I: linear problems”, J. Comput. Appl. Math., 294 (2016), 372–388  crossref  mathscinet  zmath  elib  scopus
[14]Brown D. L., Vasilyeva M., “A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling”, J. Comput. Appl. Math., 297 (2016), 132–146  crossref  mathscinet  zmath  elib  scopus
[15]Kolesov A. E., Vabishchevich P. N., Vasilyeva M. V., “Splitting schemes for poroelasticity and thermoelasticity problems”, Comput & Math. Appl., 67:12 (2014), 2185–2198  crossref  mathscinet  zmath  scopus
[16]Vabishhevich P. N., Vasil'eva M. V., and Kolesov A. E., “Splitting scheme for poroelasticity and thermoelasticity problems”, Comput. Math. Math. Phys., 54:8 (2014), 1345–1355  mathnet  crossref  elib
[17]Akkutlu I. Y., Efendiev Y., Vasilyeva M., “Multiscale model reduction for shale gas transport in fractured media”, Comput. Geosci., 20:5 (2016), 953–973  crossref  mathscinet  zmath  elib  scopus
[18]Chen H.-Y., Teufel L. W., “Coupling fluid-flow and geomechanics in dual-porosity modeling of naturally fractured reservoirs-model description and comparison”, SPE International Petroleum Conference and Exhibition in Mexico (Villahermosa, Mexico, 1-3 February), Society of Petroleum Engineers, 2000
[19]Garipov T. T., Karimi-Fard M., Tchelepi H. A., “Discrete fracture model for coupled flow and geomechanics”, Comput. Geosci., 20:1 (2016), 149–160  crossref  mathscinet  scopus
[20]Goltsev A.S., “Using discontinuity method in plane thermoelastic problems of fracture mechanics”, J. Thermal Stresses, 35:12 (2012), 1108–1118  crossref  elib  scopus
[21]Hanowski K. K., Sander O., Simulation of deformation and flow in fractured, poroelastic materials, 2016, arXiv: 1606.05765
[22]Martin V., Jaffré J., Roberts J. E., “Modeling fractures and barriers as interfaces for flow in porous media”, SIAM J. Sci. Comput., 26:5 (2005), 1667–1691  crossref  mathscinet  zmath  scopus
[23]Duflot M., “The extended finite element method in thermoelastic fracture mechanics”, Intern. J. Numer. Meth. Engineering, 74:5 (2008), 827–847  crossref  mathscinet  zmath  scopus
[24]Sivtsev P. V., Vabishchevich P. N., Vasilyeva M. V., “Numerical simulation of thermoelasticity problems on high performance computing systems”, Intern. Conf. Finite Differ. Methods., Springer-Verl., Berlin, 2014, 364–370  mathscinet
[25]Vabishhevich P. N. and Vasil'eva M. V., “Numerical modelling of thermoelasticity problems”, Vestn. SVFU, 10:3 (2013), 5–10
[26]Chung E. T., Efendiev Y., Gibson Jr R. L., Vasilyeva M., “A generalized multiscale finite element method for elastic wave propagation in fractured media”, GEM-Intern. J. Geomath., 7:2 (2016), 163–182  crossref  mathscinet  zmath  elib  scopus
[27]Arnold D. N., Brezzi F., Cockburn B., Marini L. D., “Unified analysis of discontinuous Galerkin methods for elliptic problems”, SIAM J. Numer. Anal., 39:5 (2002), 1749–1779  crossref  mathscinet  zmath  scopus
[28]De Basabe J. D., Sen M. K., Wheeler M. F., “The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion”, Geophys. J. Intern., 175:1 (2008), 83–93  crossref  scopus
[29]Riviere B., Wheeler M. F., Girault V., “Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I”, Comput. Geosci., 3:3-4 (1999), 337–360  crossref  mathscinet  zmath
[30]Brenner S., Scott R., The mathematical theory of finite element methods, Springer Science & Business Media, New York, 2007  mathscinet
[31]Riviere B., Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation, Soc. Industr. Appl. Math., 2008, 201 pp.  mathscinet  zmath
[32]Samarskii A. A. and Vabishhevich P. N., Vychislitel'naya Teploperedacha, LIBROKOM, Moscow, 2009
How to Cite
Alekseev, V., Vasilyeva, M., Prokopiev, G. and Tyrylgin, A. (2017) “Models of thermoelasticity for porous materials with fractures taken into account”, Mathematical notes of NEFU, 24(3), pp. 19-37. doi: https://doi.org/10.25587/SVFU.2018.3.10887.
Section
Mathematical Modeling