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

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.

[1]  John J. Dvorak,et al.  Volcano geodesy: The search for magma reservoirs and the formation of eruptive vents , 1997 .

[2]  K. N. Srivastava,et al.  The effect of penny-shaped crack on the distribution of stress in a semi-infinite solid , 1969 .

[3]  L. Delves,et al.  Computational methods for integral equations: Frontmatter , 1985 .

[4]  D. Dzurisin,et al.  Vertical surface displacements at Yellowstone Caldera, Wyoming, 1976–1986 , 1987 .

[5]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[6]  R. Voo.  Comments on ‘Palaeomagnetism of the Lisbon Volcanics’ by N. D. Watkins and A. Richardson , 1968 .

[7]  Y. Okada.  Surface deformation due to shear and tensile faults in a half-space , 1985 .

[8]  D. Pollard,et al.  On the mechanical interaction between a fluid-filled fracture and the earth's surface , 1979 .

[9]  D. Pollard.  Derivation and evaluation of a mechanical model for sheet intrusions , 1973 .

[10]  D. Pollard,et al.  8 – THEORETICAL DISPLACEMENTS AND STRESSES NEAR FRACTURES IN ROCK: WITH APPLICATIONS TO FAULTS, JOINTS, VEINS, DIKES, AND SOLUTION SURFACES , 1987 .

[11]  Mark Simons,et al.  Deformation and seismicity in the Coso geothermal area, Inyo County, California: Observations and modeling using satellite radar interferometry , 2000 .

[12]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions , 1920, Nature.

[13]  M. Wu,et al.  Deformations and Inferred Stress Field for Ellipsoidal Sources at Long Valley, California, 1975–1982 , 1988 .

[14]  R. Sun,et al.  Theoretical size of hydraulically induced horizontal fractures and corresponding surface uplift in an idealized medium , 1969 .

[15]  K. Mogi.  Relations between the Eruptions of Various Volcanoes and the Deformations of the Ground Surfaces around them , 1958 .

[16]  James H. Dieterich,et al.  Deformation from Inflation of a Dipping Finite Prolate Spheroid in an Elastic Half‐Space as a Model for Volcanic Stressing , 1988 .

[17]  Paul M. Davis,et al.  Surface deformation due to inflation of an arbitrarily oriented triaxial ellipsoidal cavity in an elastic half-space, with reference to Kilauea volcano, Hawaii , 1986 .

[18]  D. Mctigue,et al.  Volume of magma accumulation or withdrawal estimated from surface uplift or subsidence, with application to the 1960 collapse of Kilauea volcano , 1994 .

[19]  Angioletta Coradini,et al.  Modeling of surface deformation in volcanic areas: The 1970–1972 and 1982–1984 crises of Campi Flegrei, Italy , 1987 .

[20]  Mark Simons,et al.  Finite source modelling of magmatic unrest in Socorro, New Mexico, and Long Valley, California , 2001 .

[21]  Yuri Fialko,et al.  Fracture criteria at the tip of fluid‐driven cracks in the Earth , 1995 .

[22]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[23]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[24]  D. Mctigue.  Elastic stress and deformation near a finite spherical magma body: Resolution of the point source paradox , 1987 .

[25]  K. Ágústsson,et al.  Mechanism of the 1991 eruption of Hekla from continuous borehole strain monitoring , 1993, Nature.

[26]  J. C. Savage,et al.  Deformation near the Long Valley Caldera, eastern California, 1982–1986 , 1987 .

[27]  A. Rubin.  Dike‐induced faulting and graben subsidence in volcanic rift zones , 1992 .

[28]  Agust Gudmundsson.  Possible effect of aspect ratios of magma chambers on eruption frequency , 1986 .

[29]  James H. Dieterich,et al.  Finite element modeling of surface deformation associated with volcanism , 1975 .

[30]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[31]  F. Sigmundsson,et al.  Comment on " Volume of magma accumulation or withdrawal estimated from surface uplift or subsidence, with application to the 1960 collapse of Kīlauea volcano" by P. T. Delaney and D. F. McTigue , 2000 .

[32]  Paul M. Davis,et al.  Surface deformation associated with a dipping hydrofracture , 1983 .

[33]  Y. Fialko,et al.  What controls the along-strike slopes of volcanic rift zones? , 1999 .

[34]  Ian Parsons,et al.  Surface deformation due to shear and tensile faults in a half-space , 1986 .

[35]  Riccardo Lanari,et al.  Dynamic deformation of Etna Volcano observed by satellite radar interferometry , 1998 .