A Three-Dimensional Analytical Solution for Reservoir Expansion, Surface Uplift and Caprock Stress Due to Pressurized Reservoirs

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 \(2\pi \) 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 \(2\pi \) 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 given where the analytical uplift is compared with the results of a numerical simulation and the agreement is excellent.