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Methods of Fundamental Solutions in Poroelasticity

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A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

Part of the book series: Lecture Notes in Geosystems Mathematics and Computing ((LNGMC))

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Abstract

Although single- and double-layer potentials can be used to reduce the dimension of a problem, this comes at the prize of having to deal with integrals over the boundary of a domain. Moreover, these boundary integrals are singular. In order to remove the singularity, we use an approach by Runge which shifts integration towards the boundary of a larger domain. Based on this approach, we introduce the method of fundamental solutions in which a solution to a given partial differential equation is approximated as a linear combination of fundamental solutions to said equations with singularities at suitable points. Subsequent to a short historical overview on the method of fundamental solutions, we show how such a method can be formulated in the context of quasistatic poroelasticity. The main goal of this chapter is to prove density of suitable sets of fundamental solutions in a certain space of solutions to the quasistatic equations of poroelasticity. In a first step, this is done under the assumption of vanishing initial conditions. In order to allow for non-vanishing initial conditions, density results for the method of fundamental solutions in the context of the heat equation are used. This leads to an alternative solution scheme which is also based on the method of fundamental solutions.

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Notes

  1. 1.

    The diffusion equation is basically the same as the heat equation. Both can be converted into a Helmholtz equation by applying Fourier transformation with respect to time.

  2. 2.

    Burger’s equation is non-linear when using spatial coordinates, but can be transferred into an uncoupled system of scalar diffusion equations by using material coordinates (cf. Definition 2.50).

  3. 3.

    There are two theorems numbered as 3 in [144]. We refer here to the second one.

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    Matthias Albert Augustin

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Augustin, M.A. (2015). Methods of Fundamental Solutions 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_5

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