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Self-similar fault slip in response to fluid injection

Published online by Cambridge University Press:  12 October 2021

Robert C. Viesca Open the ORCID record for Robert C. Viesca [Opens in a new window]
Robert C. Viesca*
Affiliation:
Department of Civil and Environmental Engineering, Tufts University, Medford, MA 02155, USA
*
†Email address for correspondence: robert.viesca@tufts.edu
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Abstract

There is scientific and industrial interest in understanding how geologic faults respond to transient sources of fluid. Natural and artificial sources can elevate pore fluid pressure on the fault frictional interface, which may induce slip. We consider a simple boundary value problem to provide an elementary model of the physical process and to provide a benchmark for numerical solution procedures. We examine the slip of a fault that is an interface of two elastic half-spaces. Injection is modelled as a line source at constant pressure and fluid pressure is assumed to diffuse along the interface. The resulting problem is an integro-differential equation governing fault slip, which has a single dimensionless parameter. The expansion of slip is self-similar and the rupture front propagates at a factor $\lambda$ of the diffusive length scale $\sqrt {\alpha t}$. We identify two asymptotic regimes corresponding to $\lambda$ being small or large and perform a perturbation expansion in each limit. For large $\lambda$, in the regime of a so-called critically stressed fault, a boundary layer emerges on the diffusive length scale, which lags far behind the rupture front. We demonstrate higher-order matched asymptotics for the integro-differential equation, and in doing so, we derive a multipole expansion to capture successive orders of influence on the outer problem for fault slip for a driving force that is small relative to the crack dimensions. Asymptotic expansions are compared with accurate numerical solutions to the full problem, which are tabulated to high precision.

JFM classification

Boundary Layers: Boundary layer structure Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A29
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Adachi, J.I. & Detournay, E. 2008 Plane strain propagation of a hydraulic fracture in a permeable rock. Engng Frac. Mech. 75, 4666–4694.CrossRefGoogle Scholar
Ball, T.V. & Neufeld, J.A. 2018 Static and dynamic fluid-driven fracturing of adhered elastica. Phys. Rev. Fluids 3, 074101.10.1103/PhysRevFluids.3.074101CrossRefGoogle Scholar
Barenblatt, G.I. 1956 On the formation of horizontal cracks in hydraulic fracture of an oil-bearing stratum. Prikl. Mat. Mekh. 20, 475–486.Google Scholar
Bhattacharya, P. & Viesca, R.C. 2019 Fluid-induced aseismic fault slip outpaces pore-fluid migration. Science 364, 464–468.CrossRefGoogle ScholarPubMed
Brantut, N. 2021 Dilatancy toughening of shear cracks and implications for slow rupture propagation. arXiv:2104.06475.Google Scholar
Bunger, A.P. & Cruden, A.R. 2011 Modeling the growth of laccoliths and large mafic sills: role of magma body forces. J. Geophys. Res. 116, B02203.Google Scholar
Ciardo, F. & Lecampion, B. 2019 Effect of dilatancy on the transition from aseismic to seismic slip due to fluid injection in a fault. J. Geophys. Res. 124, 3724–3743.CrossRefGoogle Scholar
Desroches, J., Detournay, E., Lenoach, B., Papanastasiou, P., Pearson, J.R.A., Thiercelin, M. & Cheng, A. 1994 The crack tip region in hydraulic fracturing. Proc. R. Soc. Lond. A 447, 39–48.Google Scholar
Detournay, E. 2016 Mechanics of hydraulic fractures. Annu. Rev. Fluid Mech. 48, 311–339.CrossRefGoogle Scholar
Dublanchet, P. 2019 Fluid driven shear cracks on a strengthening rate-and-state frictional fault. J. Mech. Phys. Solids 132, 103672.Google Scholar
Erdogan, F., Gupta, G.D. & Cook, T.S. 1973 Numerical solution of singular integral equations. In Methods of Analysis and Solutions of Crack Problems (ed. G.C. Sih), Mechanics of Fracture, vol. 1, chap. 7, pp. 368–425. Springer.CrossRefGoogle Scholar
Flitton, J.C. & King, J.R. 2004 Moving boundary and fixed-domain problems for a sixth-order thin-film equation. Eur. J. Appl. Maths 15, 713–754.10.1017/S0956792504005753Google Scholar
Garagash, D.I. 2012 Seismic and aseismic slip pulses driven by thermal pressurization of pore fluid. J. Geophys. Res. 117, B04314.10.1029/2011JB008889CrossRefGoogle Scholar
Garagash, D.I. 2021 Fracture mechanics of rate-and-state faults and fluid injection induced slip. Phil. Trans. R. Soc. Lond. A 379, 20200129.Google ScholarPubMed
Garagash, D.I. & Detournay, E. 2000 The tip region of a fluid-driven fracture in an elastic medium. Trans. ASME J. Appl. Mech. 67, 183–192.CrossRefGoogle Scholar
Garagash, D.I. & Germanovich, L.N. 2012 Nucleation and arrest of dynamic slip on a pressurized fault. J. Geophys. Res. 117, B10310.CrossRefGoogle Scholar
Garipov, T.T., Karimi-Fard, M. & Tchelepi, H.A. 2016 Discrete fracture model for coupled flow and geomechanics. Comput. Geosci. 20, 149–160.Google Scholar
Healy, J.H., Rubey, W.W., Griggs, D.R. & Raleigh, C.B. 1968 The Denver earthquakes. Science 161, 1301–1310.CrossRefGoogle ScholarPubMed
Hewitt, I.J., Balmforth, N.J. & de Bruyn, J.R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26, 1–31.CrossRefGoogle Scholar
Hinch, E.J. 1991 Perturbation Methods. Cambridge University Press.CrossRefGoogle Scholar
Hosoi, A.E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93, 137802.CrossRefGoogle Scholar
Jha, B. & Juanes, R. 2014 Coupled multiphase flow and poromechanics: a computational model of pore pressure effects on fault slip and earthquake triggering. Water Resour. Res. 50, 3776–3808.CrossRefGoogle Scholar
Khristianovic, S.A. & Zheltov, Y.P. 1955 Formation of vertical fractures by means of highly viscous fluid. In Proceedings of the 4th World Petroleum Congress (ed. A. Barbier), vol. 1, pp. 579–586. World Petroleum Council.Google Scholar
Kennedy, B.M., Kharaka, Y.K., Evans, W.C., Ellwood, A., DePaolo, D.J., Thordsen, J., Ambats, G. & Mariner, R.H. 1997 Mantle fluids in the San Andreas fault system, California. Science 278, 1278–1281.CrossRefGoogle Scholar
King, F.W. 2009 Hilbert Transforms, vols. 1 & 2. Cambridge University Press.Google Scholar
Lenoach, B. 1995 The crack tip solution for hydraulic fracturing in a permeable solid. J. Mech. Phys. Solids 43, 1025–1043.CrossRefGoogle Scholar
Lipovsky, B.P. & Dunham, E.R. 2017 Slow-slip events on the Whillans Ice PLain, Antarctica, described using rate-and-state friction as an ice stream sliding law. J. Geophys. Res. 122, 973–1003.CrossRefGoogle Scholar
Lister, J.R. 1990 a Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors. J. Fluid Mech. 210, 263–280.CrossRefGoogle Scholar
Lister, J.R. 1990 b Buoyancy-driven fluid fracture: similarity solutions for the horizontal and vertical propagation of fluid-filled cracks. J. Fluid Mech. 217, 213–239.CrossRefGoogle Scholar
Lister, J.R., Peng, G.G. & Neufeld, J.A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111, 154501.CrossRefGoogle Scholar
Lister, J.R., Skinner, D.J. & Large, T.M.J. 2019 Viscous control of shallow elastic fracture: peeling without precursors. J. Fluid. Mech. 868, 119–140.CrossRefGoogle Scholar
Marck, J., Savitski, A.A. & Detournay, E. 2015 Line source in a poroelasredtic layer bounded by an elastic space. Intl J. Numer. Anal. Meth. Geomech. 39, 1484–1505.CrossRefGoogle Scholar
Mason, J.C. & Handscomb, D.C. 2002 Chebyshev Polynomials. Chapman and Hall.CrossRefGoogle Scholar
McClung, D.M. 1979 Shear fracture precipitated by strain softening as a mechanism. J. Geophys. Res. 84, 3519–3526.CrossRefGoogle Scholar
Michaut, C.M. 2011 Dynamics of magmatic intrusions in the upper crust: theory and applications to laccoliths on Earth and the Moon. J. Geophys. Res. 116, B05205.CrossRefGoogle Scholar
Mushkhelishvili, N.I. 1958 Singular Integral Equations. Springer.Google Scholar
Noda, H., Dunham, E.M. & Rice, J.R. 2009 Earthquake ruptures with thermal weakening and the operation of major faults at low overall stress levels. J. Geophys. Res. 114, B07302.CrossRefGoogle Scholar
Palmer, A.C. & Rice, J.R. 1973 The growth of slip surfaces in the progressive failure of over-consolidated clay. Proc. R. Soc. Lond. A 332, 527–548.Google Scholar
Peacock, S.M. 2001 Are the lower planes of double seismic zones caused by serpentine dehydration in subducting oceanic mantle? Geology 29, 299–302.Google Scholar
Peng, G.G. & Lister, J.R. 2020 Viscous flow under an elastic sheet. J. Fluid Mech. 905, A30.CrossRefGoogle Scholar
Platt, J.D., Viesca, R.C. & Garagash, D.I. 2015 Steadily propagating slip pulses driven by thermal decomposition. J. Geophys. Res. 120, 6558–6591.CrossRefGoogle Scholar
Puzrin, A.M. & Germanovich, L.N. 2005 The growth of shear bands in the catastrophic failure of soils. Proc. R. Soc. Lond. A 461, 1199–1228.Google Scholar
Rice, J.R. 1968 Mathematical analysis in the mechanics of fracture. In Fracture (ed. H. Liebowitz), vol. 2, chap. 3, pp. 191–311, Academic.Google Scholar
Rice, J.R. 1973 The initiation and growth of shear bands. In Proceedings of the Symposium on the Role of Plasticity in Soil Mechanics (ed. A.C. Palmer), vol. 2, pp. 263–274. Cambridge University Department of Engineering.Google Scholar
Rice, J.R. 2006 Heating and weakening of faults during earthquake slip. J. Geophys. Res. 111, B05311.CrossRefGoogle Scholar
Rubin, A.M. 1995 Propagation of magma-filled cracks. Annu. Rev. Earth Planet. Sci. 23, 287–336.CrossRefGoogle Scholar
Rutqvist, J., Birkholzer, J., Cappa, F. & Tsang, C.-F. 2007 Estimating maximum sustainable injection pressure during geological sequestration of $\textrm {CO}_2$ using coupled fluid flow and geomechanical fault-slip analysis. Energy Convers. Manage. 48, 1798–1807.CrossRefGoogle Scholar
Schmitt, S.V., Segall, P. & Matsuzawa, T. 2011 Shear heating-induced thermal pressurization during earthquake nucleation. J. Geophys. Res. 116, B06308.CrossRefGoogle Scholar
Spence, D.A. & Sharp, P. 1985 Self-similar solutions for elastohydrodynamic cavity flow. Proc. R. Soc. Lond. A 400, 289–313.Google Scholar
Tsai, V.C. & Rice, J.R. 2010 A model for turbulent hydraulic fracture and application to crack propagation at glacier beds. J. Geophys. Res. 114, F03007.Google Scholar
Viesca, R.C. & Garagash, D.I. 2015 Ubiquitous weakening of faults due to thermal pressurization. Nat. Geosci. 8, 875–879.CrossRefGoogle Scholar
Viesca, R.C. & Garagash, D.I. 2018 Numerical methods for coupled fracture problems. J. Mech. Phys. Solids 113, 13–34.CrossRefGoogle Scholar
Viesca, R.C. & Rice, J.R. 2012 Nucleation of slip-weakeing rupture instability in landslides by localized increase of pore pressure. J. Geophys. Res. 117, B03104.CrossRefGoogle Scholar
Wang, Z.-Q. & Detournay, E. 2018 The tip region of a near-surface hydraulic fracture. Trans. ASME J. Appl. Mech. 85, 041010.CrossRefGoogle Scholar
Williams, M.L. 1957 On the stress distribution at the base of a stationary crack. Trans. ASME J. Appl. Mech. 79, 109–114.CrossRefGoogle Scholar
Yang, Y. & Dunham, E.M. 2021 Effect of porosity and permeability evolution on injection-induced aseismic slip. J. Geophys. Res. 126, e2020JB021258.Google Scholar
Zhu, W., Allison, K.L., Dunham, E.M. & Yang, Y. 2020 Fault valving and pore pressure evolution in simulations of earthquake sequences and aseismic slip. Nat. Commun. 11, 4833.CrossRefGoogle ScholarPubMed
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