A computational framework for time dependent deformation in viscoelastic magmatic systems

Time-dependent ground deformation is a key observable in active magmatic systems, but is challenging to characterize. Here we present a numerical framework for modeling transient deformation and stress around a subsurface, spheroidal pressurized magma reservoir within a viscoelastic half-space with variable material coefficients, utilizing a high-order finite-element method and explicit time-stepping. We derive numerically stable time steps and verify convergence, then explore the frequency dependence of surface displacement associated with cyclic pressure applied to a spherical reservoir beneath a stress-free surface. We consider a Maxwell rheology and a steady geothermal gradient, which gives rise to spatially variable viscoelastic material properties. The temporal response of the system is quantified with a transfer function that connects peak surface deformation to reservoir pressurization in the frequency domain. The amplitude and phase of this transfer function characterize the viscoelastic response of the system, and imply a framework for characterizing general deformation time series through superposition. Transfer function components vary with the frequency of pressure forcing and are modulated strongly by the background temperature field. The dominantly viscous region around the reservoir is also frequency dependent, through a local Deborah number that measures pressurization period against a spatially varying Maxwell relaxation time. This near-reservoir region defines a spatially complex viscous/elastic transition whose volume depends on the frequency of forcing. Our computational and transfer function analysis framework represents a general approach for studying transient viscoelastic crustal responses to magmatic forcing through spectral decomposition of deformation time series, such as long-duration geodetic observations. Plain Language Summary Ground motions associated with subsurface magma reservoirs are the result both of magma movement and time-dependent deformation of crustal rocks. We have developed a new computational framework to help interpret surface deformations associated with magmatic systems embedded within viscoelastic rocks as expected in volcanic regions. This framework is general in the sense that a broad range of scientific studies can be explored by specifying particular conditions at domain boundaries or magma reservoir geometries, and we perform rigorous numerical tests to ensure credible solutions. We then apply the model to study a simple but highly generalizable type of transient behavior-the cyclic pressurization and depressurization of a spherical reservoir. We develop a theoretical approach to simply analyze the time-dependent output, and find that temporal lag and amplification of surface deformation with respect to the reservoir pressure is explained by an aureole of material surrounding the chamber with a dominantly viscous response, whose size is frequency-dependent. Our results can be extended to many transient deformation scenarios because a sinusoidal response forms the basic element of general pressure time-series.