# Multiscale Modelling in Poroelasticity

## Multiscale Modelling in Poroelasticity

Poroelasticity is part of the material research discipline and describes the interaction between a solid material and a fluid. This physical process can be described by a set of partial differential equations, which were first established by Biot in the 1930s.

There are many applications one can think of for poroelasticity – our focus is geothermal research. Here the aim is to find an aquifer to use the hot water in it for electricity and heat generation. On the one hand mining this hot water and returning the cooled off water has an effect on the surrounding material and on the other hand displacement has an influence on the pore pressure. These effects can be modelled by poroelasticity.

Multiscale modelling was used for other problems and equations before, namely the Laplace (with the aspect of decorrelation of potential and density data), the Helmholtz and the d’Alembert equation. Furthermore it was also used for the Cauchy-Navier equation as a tensor-valued ansatz.

Our aim is to apply this multiscale modelling in poroelasticity for the two components displacement and pressure. For this purpose, we need the fundamental solution tensor of the partial differential equations and regularize their singularity (like it is done in the Laplace and the other cases mentioned above) to obtain scaling functions and corresponding wavelets. These scaling functions are convolved with the given displacement and pore pressure data to decompose these data. This gives us the opportunity to see underlying structures for different scales that cannot be seen in the whole data. We get more detailed structures and information of our data. We can show that the scaling functions fulfill the property of an approximate identity. Furthermore, numerical results will show the decomposition.

References to some Previous Research by Other Groups:

- M. A. Augustin:
*A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs.*Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015, Birkhäuser, New York, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015, Birkhäuser, New York, 2015 ( Link to the book) - C. Blick:
*Multiscale Potential Methods in Geothermal Research: Decorrelation Reflected Post-Processing and Locally Based Inversion.*Geomathematics Group, Department of Mathematics, University of Kaiserslautern, Geomathematics Group, Department of Mathematics, University of Kaiserslautern, 2015, ( Link to the book) - C. Blick, W. Freeden, H. Nutz:
*Feature extraction of geological signatures by multiscale gravimetry,*Int. J. Geomath., 2017, 8, 57-83. - C. Blick, S. Eberle:
*Multiscale density decorrelation by Cauchy--Navier wavelets,*GEM Int. J. Geomath., 2019, 10, 24. - W. Freeden, C. Blick:
*Signal Decorrelation by Means of Multiscale Methods.*World of Mining, 2013, 304-317.