Deformation of shallow thermo-poro-elastic spherical sources and the 2021 Vulcano Island (IT) unrest

Fluids play a fundamental role in controlling deformation, stress redistribution, and seismicity in volcanic and geothermal systems. Variations in pore pressure and temperature associated with hydrothermal circulation can significantly alter the mechanical state of the crust, particularly during unrest episodes in volcanic scenario. Classical analytical models, such as the Mogi point source, have been widely used to interpret surface deformation induced by magmatic intrusions. However, these formulations neglect thermo-poro-elastic coupling and predict an isotropic stress state within the source, thus failing to account for seismicity occurring inside the deformation source.

Thermo-poro-elastic (TPE) theory provides a physically consistent framework to describe the coupled effects of fluid pressurization and heating in porous media. Analytical thermo-poro-elastic inclusion models have recently demonstrated their effectiveness in reproducing stress heterogeneities and associated focal mechanisms both internal and external to the source.The inclusion represents a finite, permeable region affected by temperature and pore-pressure variations, while the surrounding medium is assumed to be in isothermal and drained conditions. Nonetheless, at present time, the available solutions for spherical inclusions are derived for an infinite medium, limiting their applicability when surface observations are considered, especially for shallow sources.

In this study, we develop new fully analytical solutions for spherical and spherical shell TPE inclusions embedded in a half-space, explicitly accounting for the presence of a free surface. Closed-form expressions are obtained for displacement, strain, and stress fields throughout the domain, including within the source.

The problem is formulated under an axisymmetric hypothesis using cylindrical coordinates. Free-surface boundary conditions are enforced through a combination of the image source method and the Galerkin approach. The methodology is first applied to a spherical TPE inclusion representing a pressurized and heated reservoir, and subsequently extended to a spherical magmatic source surrounded by a spherical TPE shell, modeling a mechanically distinct fractured zone surrounding a magma chamber.

The results show that the free surface strongly modifies deformation and stress fields compared to full-space solutions. For shallow sources significant differences arise in all mechanical fields. In the spherical shell configuration, thinner shells exhibit enhanced internal shear stress and reduced external deformation, suggesting a higher susceptibility to internal failure.

The model is applied to the 2021 unrest episode at Vulcano Island. Using source parameters constrained by previously published we found that significant shear stress concentrations are predicted within and around the source, providing a physically consistent explanation for the clustered shallow seismicity observed near the crater. These results highlight the importance of TPE coupling and free-surface effects in the interpretation of volcanic unrest processes and fluid-driven seismicity.