II DeformaGon about a pressurized spherical cavity in an infinite body

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1 SUBSIDENCE IN THREE DIMENSIONS: CENTER OF DILATION (MOGI SOURCE) (4) I Main Topics A DefomaGon about a pessuized spheical cavity in an infinite body B Cente of dilagon (contacgon) in full- space C Refeences * PaUened ave Segall, 00 GG454 A displacement about a cavity of adius a The equilibium (foce balance) equagon in the adial diecgon is d dr + R d dr R Expess in the fom of a powe seies (any congnuous funcgon can be expessed that way). Also, we expect to decease with distance fom the cente, so! 0 as R!. The powes of R cannot be posigve, othewise! ± as R!. So and its fist and second deivagves can be expessed as + C 3 R 3 + C R + C R + C 0 R 0 d dr 3C 3R 4 C R 3 C R + 0C 0 d dr C 3 R 5 + 6C R 4 + C The tems C - i ae coefficients GG454

2 d dr + R d dr R InseGng the expessions fo and its deivagves into the equilibium equagon yields C 3 R 5 + 6C R 4 + C + ( R 3C 3 R 4 C R 3 C R + 0C 0 ) ( C R 3 R 3 + C R + C R + C 0 R 0 ) Now mulgply by the leading adial tems C 3 R 5 + 6C R 4 + C 6C 3 R 5 4C R 4 C C 3 R 5 C R 4 C R 3 C 0 Sum tems of like powes of R 4C 3 R 5 + 0C R 4 C R 3 C 0 GG C 3 R 5 + 0C R 4 C R 3 C 0 R 0 The equilibium equagon holds fo all values of R. Since all the tems in the powe seies wee linealy independent (one tem cannot be expessed as combinagons of the othes), the only way the equagon above can hold fo all values of R is if each tem equals zeo. That means all the coefficients except C - must equal zeo. Hence, u C R If the adial displacement at the wall of the hole, whee R a, is u 0, then ( R a) u 0 C a Solving fo C- yields C u 0 a Hence u 0 a R u 0 a GG454 4 R

B Pincipal stains and pincipal stesses in tems of u 0 u 0 a R ε RR R u 0 a ε φφ ε θθ R u 0 a Δ ε RR + ε φφ σ RR Gν ν Δ + Gε RR 4Gu 0a σ φφ σ θθ Gν ν Δ + Gε φφ σ RR Gu 0 a E G shea modulus +ν ( ) If p

3 B Pincipal stains and pincipal stesses in tems of u 0 u 0 a R ε RR R u 0 a ε φφ ε θθ R u 0 a Δ ε RR + ε φφ σ RR Gν ν Δ + Gε RR 4Gu 0a σ φφ σ θθ Gν ν Δ + Gε φφ σ RR Gu 0 a E G shea modulus +ν ( ) If p is the pessue in the cavity, then at the wall of the cavity σ RR ( R a) 4Gu 0 a 4Gu 0 p a 3 a u 0 p 4G a This allows the displacement, stains, and stesses to be expessed in tems of p. GG454 5 C Pincipal stains and pincipal stesses in tems of p Fom the pevious slide, u 0 pa/4g u 0 a R pa3 4GR ε RR pa3 R 4GR pa3 3 G ε φφ ε θθ R pa3 G Δ ε RR + ε φφ σ RR Gν ν Δ + Gε RR pa3 u 0 σ φφ σ θθ Gν ν Δ + Gε φφ σ RR pa3 p 4G a p is the pessue in the cavity at its walls GG

4 D Pincipal stains and pincipal stesses in tems of ΔV The change in volume of the sphee is ΔV 4πa u 0 πa 3 p/g u 0 a R ΔV 4π R ε RR R ΔV π ε φφ ε θθ R ΔV 4π Δ ε RR + ε φφ σ RR Gν ν Δ + Gε RR GΔV π u 0 σ φφ σ θθ Gν ν Δ + Gε φφ σ GΔV RR π These solugons with ΔV ae independent of a and apply as a! 0 p 4G a ΔV 4πa ΔV is the change in volume of the cavity esulgng fom the adial displacement of its walls GG454 7 u z u ε zz ε ( z d) z d ( z d) z d E Displacements and stains in tems of ΔV in cylindical coodinates along a plane above a pessuized cavity ( ) d R ΔV 4π R ( ) R ΔV ( z d) u z z ΔV 3z 4π + z ( z d) u ΔV 4π + z ε θθ ( z d) u ΔV 4π + d Δ z d ( ) ε zz + ε z R ΔV z 4π ( + z ) ΔV d 3 4π + d 4π R R ΔV 4π + z 4π + d ( ) + 5 ( + z ) 3 ( ) + 5 ( + z ) 3 ( ) 3 ( ) 3 ( ) ΔV 3 ( ) 3 ΔV 3d 4π + d ( ) + 5 ( + d ) 3 ΔV 4π + d ( ) + 5 ( + d ) 3 GG454 8 Cylindical coodinates These solugons with ΔV ae independent of a and apply as a! 0 4

5 ε zz ε E Stains and stesses in tems of ΔV in cylindical coodinates along a plane above a pessuized cavity (cont.) ( z d) u z z ΔV 3d 4π + d ( z d) u ΔV 4π + z ε θθ ( z d) u ΔV 4π + d Δ( z d) ε zz + ε σ zz σ ( ) + 5 ( + d ) 3 ( ) + 5 ( + d ) 3 ( ) 3 ( z d) Gν ν Δ + Gε zz GΔV 3d π + d ( z d) Gν ν Δ + Gε GΔV π + d ( ) + 5 ( + d ) 3 ( ) + 5 ( + d ) 3 σ θθ ( z d) Gν ν Δ + Gε θθ GΔV These solugons with ΔV ae π ( + d ) 3 independent of a and apply as a! 0 GG454 9 Cylindical coodinates F Key points Displacements ae adial Displacements decay as /R 3 Stains decay as / 4 Stesses decay as / 5 Displacements, stains, and stesses scale with ΔV 6 No volumetic dilagon pedicted anywhee in the linea elasgc full space outside the pessuized sphee R GG

III Cente of dilagon in full- space A Cente of dilagon A point fom which neaby displacements adiate outwad equally in all diecgons An infinitely small spheical hole (a!

6 III Cente of dilagon in full- space A Cente of dilagon A point fom which neaby displacements adiate outwad equally in all diecgons An infinitely small spheical hole (a!0) with a singula pessue fo which the solugons of the pevious slide apply, with ΔV > 0 3 A nucleus of stain obtained by diffeengagng and supeposing the effect of a foce at a point 4 Can epesent fluid accumulagon at geat depth B Cente of contacgon A point fom which neaby displacements convege equally in all diecgons An infinitely small spheical hole (a!0) containing a singula sucgon fo which the solugons of the pevious slide apply, with ΔV < 0 3 Can epesent fluid withdawal at geat depth Cente of dilagon Cente of contacgon GG454 C Nomal stains and displacements in a vegcal plane though a cente of dilagon Displacement field fo a cente of dilagon in full space VeGcal contacgon VeGcal VeGcal contacgon contacgon VeGcal contacgon Displacements not shown within dashed cicle fo diagammagc easons GG454 6

7 D Nomal stains and displacements in a vegcal plane though a cente of contacgon Displacement field fo a cente of dilagon in full space VeGcal contacgon VeGcal contacgon VeGcal contacgon VeGcal contacgon Displacements not shown within dashed cicle fo diagammagc easons GG454 3 Refeences Amelung, F., S. Jonsson, H. Zebke and P. Segall, 000, Widespead upliv and tap doo faulgng on Galápagos volcanoes obseved with ada intefeomety: Natue, v. 407, p Davies, J.H., 003, ElasGc field in a semi- infinite solid due to themal expansion o a coheently misfitng inclusion: Jounal of Applied Mechanics, v. 70, p Geetsma, J., 973, Land subsidence above compacgng oil and gas esevois: Jounal of Petoleum Technology, v. 5, p Mayuga, M.N., and Allen, D.R., date unknown, Subsidence in the Wilmington oil field, Long Beach, Califonia, U.S.A., p , hup:// McTigue, D.F., 987, ElasGc stess and defomagon nea a finite spheical magma body: esolugon of the point souce paadox: Jounal of Geophysical Reseach, v. 9, p.,93-,940. Mindlin, R.D., 936, Foce at a point in the inteio of a semi- infinite solid: Physics, v. 7, p Mindlin, R. D., and Cheng, D. H., 950, Nuclei of stain in the semi- infinite solid, Jounal of Applied Physics, v., p Mogi, K., 958, RelaGons between the eupgons of vaious volcanoes and the defomagons of the gound sufaces aound them, BulleGn of the Eathquake Reseach InsGtute, v. 36, p Segall, P., 989, Eathquakes Tiggeed by Fluid ExtacGon: Geology, v. 7, p Segall, P., 00, Eathquake and volcano defomagon: Pinceton Univesity Pess, Pinceton, New Jesey, 43 p. Wang, H., 000, Theoy of linea pooelasgcity: Pinceton Univesity Pess, Pinceton, New Jesey, 87 p. GG

II Center of dilaion in half- space (Mogi source)

II Center of dilaion in half- space (Mogi source) SUBSIDENCE IN THREE DIMENSIONS: CENTER OF DILATION (MOGI SOURCE) (43) I Main Topics A Center of ilaion (contracion) in a half- space B Case histories C References * PaNerne aoer Segall, 010 5/6/15 GG454