Analytical solution and numerical validation of long-term evolution of pore pressure in anisotropic porous media with special emphasis on boundary effects

Accurate prediction of pore pressure evolution during fluid injection is essential for assessing reservoir performance and associated effective stress changes that may promote induced seismicity. Analytical solutions for pressure diffusion commonly assume isotropic permeability, whereas long-term numerical simulations often require large computational domains to reduce boundary effects. We investigate injection-induced pressure evolution over timescales comparable to geothermal project lifespans under hydraulic anisotropy, using analytical and numerical methods with finite and infinite elements. We derive a closed-form three-dimensional analytical solution for continuous point-source injection in a medium with permeability anisotropy. Our closed-form analytical solution shows that the permeability component along a given axis governs the temporal pressure evolution, while the perpendicular components control the pressure amplitude. To identify limitations of numerical modelling for long-term scenarios, we compare analytical and numerical results for isotropic permeability. Boundary effects in conventional finite element models manifest earlier in materials with higher permeability. For an 8 km cubic Berea sandstone model, deviations increase from below 1.4% after 0.1 years to more than 40% after 10 years. Infinite elements reduce the maximum deviation below 0.56% over 10 years while also lowering computational runtime. Finally, we modelled a geothermal doublet to demonstrate how doublet orientation and permeability anisotropy jointly control pore pressure evolution. The combined analytical and numerical framework provides rapid pressure estimates and practical guidance for long-term modelling of geothermal and other subsurface fluid-injection systems.