During a volcanic unrest period with dike injection, one of the main scientific tasks is to assess the geometry and the propagation path of the dike and, in particular, the likelihood of the dike reaching the surface to erupt. Currently, the dike path and geometry (including depth and opening/aperture) are both partly determined from geodetic surface data using mostly dislocation models that assume the volcanic zone/volcano to be an elastic half space of uniform mechanical properties. By contrast, field observations of volcanic zones/volcanoes (active and extinct) show that they are composed of numerous layers whose mechanical properties (primarily Young's modulus) vary widely and whose contacts commonly arrest dikes. Here we provide field observations and numerical models on the effects of a typical variation in Young's modulus in an active volcanic zone on the internal and surface stresses and displacements induced by a dike whose tip is arrested at 0.5 km depth below the surface of the volcanic zone. Above the layer or unit hosting the dike are four layers of equal thickness. We vary the Young's modulus or stiffness of the fourth layer (the one adjacent to the layer or unit hosting the dike) from 10 GPa to 0.01 GPa, while all the other layers/units maintain their Young's moduli in the model runs. The results show that as the fourth layer becomes more compliant or soft (0.1–0.01 GPa) dike-induced stresses and displacements (lateral and vertical) above the layer, including those at the surface, become suppressed but the stresses and displacements of the layer/unit hosting the dike increase and their peaks do not coincide in location or magnitude with those of the other layers. Thus, the dike-induced internal deformation of the volcanic zone increases as the fourth layer becomes softer. Also, the tensile-and-shear stress peaks at the surface occur at locations widely different from those of maximum surface uplift. More specifically, for a comparatively stiff fourth layer (1–10 GPa), the surface tensile and shear stresses peak at lateral distances of 0.5–0.7 km from the projection of the dike to the surface. (Essentially no tensile/shear stresses reach the surface when the fourth layer is as soft as 0.1–0.01 GPa, so that there are no stress peaks). By contrast the maximum surface displacements (uplift) peak at lateral distances of 2.8–3.3 km from the dike projection to the surface. If tension fractures or faults – in particular the boundary faults of a graben – are induced by the dike, they should form at the tensile/shear stress peaks and not, as is commonly suggested, at the location of the surface displacement peaks. Our results thus suggest that any dike-induced graben is likely to be of a width about twice the depth to the tip/top of the arrested dike. The results demonstrate that elastic half-space models overestimate the dike-induced surface stresses, and thus the depth to the tip/top of the associated dike. In particular, the models presented here indicate that, for typical dikes little or no dike-induced surface deformation would be expected until the dike tip propagates to depths below the surface of less than a kilometre.
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