Deformation due to a pressurized horizontal circular crack in an elastic half-space, with applications to volcano geodesy

Abstract

SUMMARY We consider deformation due to sill‐like magma intrusions using a model of a horizontal circular crack in a semi‐infinite elastic solid. We present exact expressions for vertical and horizontal displacements of the free surface of a half‐space, and calculate surface displacements for a special case of a uniformly pressurized crack. We derive expressions for other observable geophysical parameters, such as the volume of a surface uplift/subsidence, and the corresponding volume change due to fluid injection/withdrawal at depth. We demonstrate that for essentially oblate (i.e. sill‐like) source geometries the volume change at the source always equals the volume of the displaced material at the surface of a half‐space. Our solutions compare favourably to a number of previously published approximate models. Surface deformation due to a ‘point’ crack (that is, a crack with a large depth‐to‐radius ratio) differs appreciably from that due to an isotropic point source (‘Mogi model’). Geodetic inversions that employ only one component of deformation (either vertical or horizontal) are unlikely to resolve the overall geometry of subsurface deformation sources even in a simplest case of axisymmetric deformation. Measurements of a complete vector displacement field at the Earth's surface may help to constrain the depth and morphology of active magma reservoirs. However, our results indicate that differences in surface displacements due to various axisymmetric sources may be subtle. In particular, the sill‐like and pluton‐like magma chambers may give rise to differences in the ratio of maximum horizontal displacements to maximum vertical displacements (a parameter that is most indicative of the source geometry) that are less than 30 per cent. Given measurement errors in geodetic data, such differences may be hard to distinguish.

Segmented lateral dyke growth in a rifting event at Bárðarbunga volcanic system, Iceland

“…The covariance of the error for each pair of InSAR measurements is calculated assuming a one dimensional exponential covariance function: Cov = 0.0016exp(-h/5) m 2 , where h is the distance between the measurement points in km. The model parameters are opening and strike-slip for the dyke patches 37 , position and pressure decrease of a penny-shaped crack 38 or point pressure source 39 , a bilinear orbital error ramp for each interferogram, and the hyperparameter σ 2 . We allow for slip as well as opening, as dykes that are not perpendicular to the minimum compressive stress direction will be subject to shearing across the dyke walls 40 .…”

“…Median values (a) and standard deviations (b) of the posterior probability distribution, for a model with a deflating penny-shaped crack 38 (used to represent the top of a flattopped chamber 48 ), two dip-slip faults beneath the caldera and a dyke modelled as four segments divided into multiple rectangular patches 37 . The data used are detailed in Extended Data Fig.…”

SAR Data Analysis in Solid Earth Geophysics: From Science to Risk Management

“…The Okada model, however, can be also used to describe magma intrusion like sills or dykes [2], [3], [4], interseismic and post-seismic deformations (see Section 4), landslides [5] and ground subsidence induced by fluid extraction [6]. Source parameters are: East and North position, depth, length, width, strike angle, dip angle, dislocation (or slip), dislocation angle (rake), opening ( Figure 1); point pressure source [7]: it is one of the simplest and effective source used in volcanology, as its description requires only 4 parameters: depth, east and north position, volume variation or pressure variation 1 ( Figure 2) Several other sources have been proposed in the literature, with the aim of providing more realistic solutions to describe geophysical phenomena: dislocation over a finite triangular source [8]; volume variation of a dipping finite prolate spheroid [9]; inflation of an arbitrarily oriented triaxial ellipsoidal cavity [10]; pressure change in a penny-crack source [11]; closed vertical pipe [12]; stress induced by a finite spherical source [13]. A description of the differences among all these sources is beyond the scope of this article, and we refer the reader to the cited literature.…”

A Sea Floor Gravity Survey of the Sleipner Field to Monitor Co2 Migration

“…The first method approximates the CO 2 within the reservoir as a penny shaped pressurized crack. Our calculations were based on a semi-analytical solution from Fialko et al (2001). The pressure of the CO 2 was taken as the buoyancy pressure.…”