### Sammendrag

An analytical solution is presented for the displacement, strain and stress
of a three-dimensional poro-elastic model with three layers, where the three layers are
an underburden, a reservoir with a given fluid pressure, and an overburden. The fluid
pressure in the reservoir is assumed symmetrical around the z-axis and represented
by a Fourier cosine series. The poro-elastic solution is expressed as a superposition
of the solutions for each term in the Fourier series. It is shown that the bulk strain
in the reservoir layer is proportional to the fluid pressure and that the bulk strain in
the underburden and overburden is zero. Using these properties of the bulk strain,
a solution is derived for the three-layer model where the fluid flow and mechanics
are fully coupled. A particular aim of the model is to study the surface uplift from a
given reservoir pressure. The expansion of the reservoir and the uplift of the surface
are studied in terms of the wavelengths in the Fourier representation of the pressure.
It is shown that the surface uplift can be written in a similar form to the 1D vertical
expansion of the reservoir layer, but where the fluid pressure is based on the Fourier
series. It is shown that the amplitudes with average wavelengths longer than 2π times
the thickness of the reservoir give expansion of the reservoir, but average wavelengths
much shorter than this limit do not. Similarly, amplitudes with average wavelengths
much longer than 2π times the thickness of the overburden produce surface uplift, but
wavelengths much shorter do not. The stress in the overburden, which is generated
by the reservoir fluid pressure, is also analysed in terms of the wavelengths. A case is
givenwhere the analytical uplift is compared with the results of a numerical simulation
and the agreement is excellent.

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