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.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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