Abstract
Based on differential equations, integral equations can be formulated. A well-known example is the relation between the Laplace equation and Green’s identities. In this chapter, we will use the same procedure to develop integral equations to the quasistatic equations of poroelasticity. In this process, we also derive the adjoint equations to the quasistatic equations of poroelasticity. In order to obtain boundary integral formulations, it is necessary to calculate fundamental solutions to the quasistatic equations of poroelasticity. For this purpose, we use a solution approach by Biot. The derived fundamental solutions clearly reflect the relations of the quasistatic equations of poroelasticity to the heat equation, the Cauchy-Navier equation of linear elasticity and the system of Stokes’ equations. We show how these fundamental solutions are related to the fundamental solutions of the adjoint equations, which are needed to establish boundary integral formulations, from which the equivalents to single- and double-layer potentials of classical potential theory can be deduced formally.
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Augustin, M.: On the role of poroelasticity for modeling of stress fields in geothermal reservoirs. Int. J. Geomath. 3, 67–93 (2012)
Augustin, M., Bauer, M., Blick, C., Eberle, S., Freeden, W., Gerhards, C., Ilyasov, M., Kahnt, R., Klug, M., Michel, I., Möhringer, S., Neu, T., Nutz, H., I., Punzi, A.: Modeling deep geothermal reservoirs: recent advances and future perspectives. In: W. Freeden, Z. Nashed, T. Sonar (eds.) Handbook of Geomathematics, 2nd edn. Springer, New York (2015). Accepted for publication
Augustin, M., Freeden, W., Gerhards, C., Möhringer, S., Ostermann, I.: Mathematische Methoden in der Geothermie. Math. Semesterber. 59, 1–28 (2012)
Biot, M.A.: General solutions of the equations of elasticity and consolidation for a porous material. J. Appl. Mech. 78, 91–96 (1956)
Chen, J.: Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part I: two-dimensional solution. Int. J. Solid. Struct. 31, 1447–1490 (1994)
Chen, J.: Time domain fundamental solution to Biot’s complete equations of dynamic poroelasticity. Part II: three-dimensional solution. Int. J. Solid. Struct. 31, 169–202 (1994)
Cheng, A.H.D., Detournay, E.: On singular integral equations and fundamental solutions of poroelasticity. Int. J. Solid. Struct. 35, 4521–4555 (1998)
Costabel, M.: Boundary integral operators for the heat equation. Integral Equ. Operat. Theory 13, 498–552 (1990)
Costabel, M.: Time-dependent problems with the boundary integral equation method. In: E. Stein, R. de Borst, T.J.R. Hughes (eds.) Encyclopedia of Computational Mechanics, chap. 25. Wiley, Chichester (2004)
Detournay, E., Cheng, A.H.D.: Fundamentals of poroelasticity. In: C. Fairhurst (ed.) Comprehensive Rock Engineering: Principles, Practice and Projects. Analysis and Design Method, vol. II, chap. 5, pp. 113–171. Pergamon Press, Oxford (1993)
Ehrenpreis, L.: On the theory of kernels of Schwartz. Proc. Am. Math. Soc. 7, 713–718 (1955)
Freeden, W., Michel, V.: Multiscale Potential Theory with Applications to Geoscience. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2004)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Hazewinkel, M. (ed.): Encyclopedia of Mathematics. Kluwer Academic, Dordrecht (2002)
Ionescu-Casimir, V.: Problem of linear coupled thermoelasticity. I. Theorems on reciprocity for the dynamic problem of coupled thermoelasticity. Bull. Acad. Pol. Sci., Sér. Sci. Tech. 12, 473–488 (1964)
Kaynia, A.M., Banerjee, P.K.: Fundamental solutions of Biot’s equations of dynamic poroelasticity. Int. J. Eng. Sci. 31, 817–830 (1993)
Malgrange, B.: Éxistence et Approximation des Solutions des Équations aus Dérivées Partielles et des Équations de Convolution. Ann. Inst. Fourier 6, 271–355 (1956)
Manolis, G.D., Beskos, D.E.: Integral formulation and fundamental solutions of dynamic poroelasticity and thermoelasticity. Acta Mech. 76, 89–104 (1989)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Mineola (1994)
Mayer, C.: A wavelet approach to the Stokes problem. Habilitation Thesis, University of Kaiserslautern, Geomathematics Group (2007)
Mayer, C., Freeden, W.: Stokes problem, layer potentials and regularizations, multiscale applications. In: W. Freeden, Z. Nashed, T. Sonar (eds.) Handbook of Geomathematics, 2nd edn. Springer, New York (2015). Accepted for publication
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Nowacki, W.: Thermoelasticity. Aeronautics and Astronautics. Pergamon Press, Oxford (1986)
Pan, E.: Green’s functions in layered poro-elastic half-spaces. Int. J. Numer. Anal. Method. Geomech. 23, 1631–1653 (1999)
Predeleanu, M.: Reciprocal theorem in the consolidation theory of porous media. An. Univ. Bucur. Ser. Ştiinţ. Nat. Mat.-Mec. 17, 75–79 (1968)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Direct Laplace Transforms. Integrals and Series, vol. 4. Gordeon and Breach Science Publishers, Amsterdam (1992)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Inverse Laplace Transforms. Integrals and Series, vol. 5. Gordeon and Breach Science Publishers, Amsterdam (1992)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Schanz, M., Pryl, D.: Dynamic fundamental solutions for compressible and incompressible modeled poroelastic continua. Int. J. Solid. Struct. 41, 4047–4073 (2004)
Widder, D.: The Laplace Transform. Princeton University Press, Princeton (1941)
Wiebe, T., Antes, H.: A time domain integral formulation of dynamic poroelasticity. Acta Mech. 90, 125–137 (1991)
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Augustin, M.A. (2015). Boundary Layer Potentials in Poroelasticity. In: A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-17079-4_4
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DOI: https://doi.org/10.1007/978-3-319-17079-4_4
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