ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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3F - Corinth (Faults, Fractures & Fluids)
ENK6-2000-0056
WP5: GEOMECHANICAL MODELING
George Exadaktylos, Department of Mineral Resources, Technical University of Crete
January 2002
A. ACCOMPLISHED WORK
Task 5.2: Fracture mechanics analysis of faulting in the Gulf
The aim of this task is to develop a “Complex Variable Element” Model – that is essentially an
elegant type of a Boundary Element Method - for holes and fractures in thermo-porous elastic
media (double porosity media). The use of the complex variable formulation in computational
geomechanics has recently attracted the interest of researchers due to its effectiveness (Hromadka
and Lai, 1987; Strack and Verruijt, 2000; Strack, 2000; Schanz, 2001). The benefits of this complex
variable formulation is first that stress equilibrium and compatibility relations are automatically
satisfied, secondly we may use the results already developed in classical elasticity theory1, thirdly
we may effectively take into account singularities such as fault or notch tips, and finally we reduce
the considerably the computing time and memory. Complex variable representations for plane strain
and anti-plane strain boundary value problems of faulting in thermoporoelastic geologic media that
may be analytically or semi-analytically solved - in conjunction with finite elements - in space and
in time domains, have been developed. This model will be used as a tool for back-analysis and
understanding of the underlying basic mechanisms in (i) downhole hydraulic fracturing stress
measurements under Task 1.2 (back-analysis of hydrofracturing results in order to evaluate the in
situ stresses and stress variation with depth in regions exhibiting normal faulting), and (ii) for the
study of interaction and activation of normal faults and pore pressure changes in regions such as the
Gulf of Corinth.
Within this Report the derivation of integral formulae coupling fracture modes I, II and III with the
fluid pressure and temperature changes in plane strain and anti-plane strain conditions, as well as
the solution of the coupled nonlinear plane strain poroelastic fault problem have been accomplished
(Deliverable 5.2). The general solution of the complex potential functions - that are necessary for
the computation of stresses and displacements in the faulted region - of the mixed boundary value
problem with prescribed stress drop along a straight fault has been found to be
 (z, t )
1


(z, t )  2i z 2   2



2   2 p(, t )d
1

z
2i


q(, t )d
z
(1’)

where p’ and q’ are appropriate expressions for the tractions and pore pressure along the crack.
Also, the dynamic nonlinear pore pressure diffusion equation for the rock with spatially variable
and anisotropic intrinsic permeability takes the form

21  2   p
  k ij p 
41   cB  Re (z)
 c 
B  



31   
3
t

 t x i   x j 
1
Muskhelishvili N.I., Singular Integral Equations, P. Noordhoff N.V., 1946.
(2’)
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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where p is the pore pressure deviation from a reference pressure. Both equations (1’) and (2’)
evolve simultaneously in time. Using the complex variable method only the nonlinear diffusion
equation (2’) needs to be solved in space and time. This is performed by the finite element method.
In Fig. 1’ the Adaptive Finite Element scheme for the solution of (2’) in a domain with a straight
crack is shown.
Fig. 1’
Further the problem of notched borehole in elastic rock (Fig. 2’) has been formulated within the
frame of complex variables and conformal mapping methods and the closed-form solution is given
in terms of complex potential functions. For this problem a code in Matlab v.6.0 has been
developed for the presentation of results referring to stresses and displacements (Fig. 3’) (part of
Deliverable 5.2). The solution may be easily applied to any notch geometry by properly selecting
the respective conformal mapping function via a methodology that has been developed here. This
solution may be easily extended to attack the coupled poroelastic problem. Borehole breakouts are
valuable indicators of the direction of action of the minimum compressive stress, while their size
and shape, recorded via dipmeters and more precisely now by televiewers, may provide information
about the magnitudes of the maximum and minimum stresses relative to the strength of the rock. As
the in situ strength of rock and its state of stress are difficult to determine at great depth,
observations of the size and shape of the breakouts and conditions under which they form could
lead to estimation of these parameters via the proposed model.
Fig. 1’. Borehole breakout configuration.
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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It remains the extension of the solutions to three-dimensional half-space configurations by the
generalized plane strain technique and the comparison of hydraulic fracturing test results and of
possible post-seismic free surface displacements and pore pressure changes with the coupled
diffusion - complex variable model. This work will be accomplished in the 2nd year of the Project.
Fig. 2’. Plot of tangential stresses around a notched borehole subjected to internal pressure.
1. G.E. Exadaktylos, P.A. Liolios, M.C. Stavropoulou, A closed-form elastic solution for notched circular
openings, to be submitted to Int. J. Solids & Structures, 2002.
2. MATLAB, Version 4 User’s Guide, The Math Works Inc., 1997.
3. Strack, O.E. and Verruijt A. "A complex variable solution for the ovalization of a circular tunnel in an
elastic half-plane". GeoEng 2000: An International Conference on Geotechnical & Geological
Engineering, November 19-24, 2000, Melbourne Australia.
4. Strack, O.E. "Key design elements in an object-oriented approach to programming analytic elements:
suggestions based on programming experience with MLAEM." Proceedings of the 3rd International
Conference on the Analytic Element Method in Modeling Groundwater Flow, April 16-19, 2000,
Brainerd Minnesota.
5. Schanz, M.: Dynamic Poroelasticity Treated by a Time Domain Boundary Element Method. In
UTAM/IACM/IABEM Symposium on Advanced Mathematical and Computational Mechanic Aspects of the
Boundary Element Method . (Burczynski, T., Ed.), Kluwer Academic Publishers, Dodrecht, 303-314, 2001.
6. Hromadka II, T.V. and Lai, Chintu, 1987, The Complex Variable Boundary Element Method in
Engineering Analysis, Springer-Verlag Publishers.
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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Task 5.3: Thermo-poro-mechanical coupled model for faults: The role of clays
The aim of this task (Deliverable D5-3) is to provide a theoretical model for fault activation due to
thermo-poro-mechanical coupling and the importance of various physicochemical factors that affect
the performance of the formation fluids.
The observations from the active fault drilling operation of the Geological Survey of Japan at
Nojima Hirabayashi after the 1995 Hyogoken-nanbu (Kobe) Earthquake and the following analysis
by Otsuki et al. (2000) renewed the interest of the geophysics community on the role of temperature
and fluids in active faulting (Lachenbruch, 1980; Mase and Smith,1 985). Fault zones are often
characterized by large amounts of clay minerals, which form well-defined banded structures within
the fault zone. These are widely believed to affect significantly the mechanical behavior of faults.
In series of recent publications related to the project, Vardoulakis (2000, 2001, 2002a), reformulated the set of equations that govern the motion of a rapidly deforming shear-band, starting
from first principles. The fault gouge was considered as a two-phase mixture of solids and fluid, and
the governing equations were derived from the corresponding conservation laws of mass,
momentum and energy. The resulting governing equations are a set of coupled diffusion-generation
partial differential equations that contain three unknown functions, the pore-pressure, the
temperature and the velocity field inside the shear-band. This set of equations turns out to be in
general mathematically ill-posed unless some additional assumptions are met concerning the
dependency of the friction coefficient of the gouge on the strain-rate (Dieterich, 1992). Indeed
Vardoulakis (2001) could show that the original, mathematically ill-posed problem is regularized by
using a viscous-type together with a 2nd gradient regularization of the momentum balance equation
(Vardoulakis and Sulem, 1995).
Temperature and pore-pressure isochrons across an 1.4 mm clayey, shear-band under rapid shearing
(Vardoulakis, 2002a)
In a sequel paper Vardoulakis (2002b) discussed recently the possibility of thermal run-away
(Gruntfest, 1963) in clayey shear-bands. The analysis by Otsuki et al. (2000), in relation to the
Nojima fault, describes the slip velocity and shear stress mechanism as a spatio-temporal instability
of nucleation - thermal pressurization - fluidization and eventual subsequent melting of the gouge
material. These observations mean that one should expand the existing thermo-poro-mechanical
model in order to capture fluidization and melting. In both cases thermal runaway is also a
possibility.
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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1. Dieterich, J. H. (1992). Earthquake nucleation on faults with rate- and state-dependent strength.
Tectonophysics, 211, 115-134.
2. Gruntfest, I.J. (1963). Thermal feedback in liquid flow: plane shear aat constant stress. Trans. Soc. Rhel.,
7, 195-208.
3. Lachenbruch, A.H. (1980). Frictional heating, fluid pressure and the resistance tofault motion. J. of
Geophys. Res., 85, 6097-6112.
4. Mase, C.W. and Smith, L. (1985). Pore-fluid pressures and frictional heating on a fault surface.
Pageoph., 122, 583-607.
5. Otsuki,K., Monzawa,N. and Nagase,T., 2000, Thermal pressurization, Fluidization and Melting of Fault
Gouge During Seismic Slip Recorded In the Rock from Nojima Fault. Proceedings of The International
Workshop On The Nojima Fault Core And Borehole Data Analysis,43-49; JGR (submitted).
6. Vardoulakis, I. (2000). Catastrophic landslides due to frictional heating of the failure plane. Mech. Coh.
Frict. Mat., 5, 443-467.
7. Vardoulakis, I. (2001). Thermo-poro-mechanical analysis of rapid fault deformation, Powders & Grains
2001, Proceedings of the 4th International Conference on Micromechanics of Granular Media, May 2125, 2001, Sendai Japan, (Kishino ed.) 273-280, Balkema.
8. Vardoulakis, I. (2002a). Dynamic thermo-poro-mechanical Analysis of Catastrophic Landslides.
Gèotechnique, in print.
9. Vardoulakis, I. (2002b). Steady shear and thermal run-away in clayey gouges. Int. J. Solids and
Structures, in print.
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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Task 5.4: Near-hole stress field analysis: FE 2nd Gradient Elastoplasticity
The regional stresses in central and eastern part of the Gulf of Corinth suggest a N-S extension (see
also Task 5.5). The regional and global tectonic processes that control the considered extensional
environment are most probably related to an asymmetric detachment with exhumation of
metamorphic core complexes and subsequent rifting (see also Task 5.1).
Regional stress field as determined from focal mechanisms ("The World Stress Map Project" 2).
Possible tectonic model of N-S extensional mode with asymmetric detachment3
Borehole breakouts induced by unequal horizontal far-field in situ stresses are usually aligned
parallel to the minimum horizontal compressive stress and are formed by spalling of fragments of
the wellbore. Stress orientations, inferred from breakout azimuths, are consistent with data obtained
by other, independent methods of in-situ stress measurement (Plumb and Cox, 1987). Using data
from in-situ stress measurements among others, Bell and Gough (1982) demonstrated that breakout
azimuth is related to the compressive forces of unequal horizontal principal stresses near the
borehole. These findings have been studied in detail experimentally (Haimson and Herrick, 1985,
Ewy et al., 1990) analytically (Zoback and Healy, 1984; Vardoulakis I. et. al., 1988) and numerically
(Papanastasiou and Vardoulakis, 1991; Zervos et al., 2001 b).
The Deliverable D5-4 aimed at providing a robust Finite Element code for borehole breakouts
tracing in anisotropic far-field of stresses. Although elasto-plastic analysis can yield a fair estimate
for the stress level at which localization may initiate, progressive localization cannot be modeled
without incorporating strain-softening behavior in the constitutive relations. Strain softening,
however, renders the boundary value problem of elasto-plasticity mathematically ill-posed, as the
governing equations in the softening regime are no longer elliptic. Consequently, numerical
solutions either diverge in the softening regime or become pathologically mesh-dependent. Recently
Zervos et al. (2001a) developed a 2nd gradient elastoplasticity F.E. code capable of modeling postfailure rock behavior under different loading conditions. In particular, the formation of shear-bands
in biaxial loading conditions (Zervos et al., 2001a), failure of a borehole due to the formation of
breakouts (Zervos et al., 2001b) and spontaneous localization in an internally pressurized cavity
(Zervos et al., 2001c) were examined. In all cases the final localized failure mechanism was
obtained in a robust way. This F.E. code is based on a Mohr-Coulomb cohesion hardening-softening
yield criterion, and is currently being extended to incorporate different yield surfaces and flow rules.
2
http://www-wsm.physik.uni-karlsruhe.de/pub/stress_data/stress_data_frame.html.
Zagorchev, I.(2000). Neogene low-angle normal faults in the Sandanski graben. Rhodope Geodynamic Hazards, Late
Alpine tectonics and Neotectonics, International Conference - May 2001. http://www.geology.bas.bg/rgh/lowangle.html
3
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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Robust post-failure shear-banding computatons using Zervos' (2001) F.E. code
1. Bell, J.S., and Gough, D.I. (1982). The use of borehole breakouts in the study of crustal stress, in M.D.
Zoback and B.C. Haimson, eds., Workshop on hydraulic fracturing stress measurements [December 2-5],
proceedings: U.S. Geological Survey Open-file Report 82-1075, p. 539-557. Also published in 1983, as
Hydraulic fracturing and stress measurements: National Academy Press, p. 201-209.
2. Ewy, R.T., and Cook, N.G.W. (1990). Deformation and fracture around cylindrical openings in rock--II;
initiation, growth, and interaction of fractures: International Journal of Rock Mechanics and Mining
Sciences & Geomechanics Abstracts, 27, 409-427.
3. Haimson, B.C., and Herrick, C.G. (1985). In situ stress evaluation from borehole breakouts,
experimental studies, in E. Ashworth, ed., Research and engineering applications in rock masses [26th
U.S. symposium on rock mechanics, South Dakota School of Mines and Technology, Rapid City, 26-28
June, proceedings], volume 2: A.A. Balkema, Boston, 1207-1218.
4. Papanastasiou P. and Vardoulakis I. (1991). Numerical Analysis of progressive localization with
application to borehole stability. Int. J. Num. Anal. Meth. Geomechanics, 16, 389-424.
5. Plumb, R.A., and Cox, J.W. (1987). Stress directions in eastern North America determined to 4.5 km
from borehole elongation measurements: Journal of Geophysical Research, 92, no. B6, May 10, 48054816.
6. Vardoulakis I., Sulem J. and Guenot A. (1988). Stability analysis of deep boreholes. Int. J. Rock Mech.
Min. Sci. Geomech. Abstr., 25, 159-170.
7. Zervos, A., Papanastasiou, P., and Vardoulakis, I. (2001a). A finite element displacement formulation for
gradient elastoplasticity: Int. J. Numer. Meth. Engng., 50, 1369-1388
8. Zervos, A., Papanastasiou, P., and Vardoulakis, I. (2001b). Modelling of localisation and scale effect in
thick-walled cylinders with gradient elastoplasticity: Int. J. Solids Structures, 38, 5081-5095
9. Zervos, A., Papanastasiou, P., and Vardoulakis, I. (2001c). Shear localisation in thick-walled cylinders
under internal pressure based on gradient elastoplasticity: IUTAM Symposium on Analytical and
Computational Fracture Mechanics of Non-homogeneous Materials, University of Cardiff, 18-22 June
2001.
10. Zoback, M.D., and Healy, J.H. (1984). Friction, faulting and in situ stress: Annales Geophysicae, 2, no.
6, 689-698.
ENK6-2000-0056, 3F-Corinth, WP5, Task 5.2, Partner 5 NTUA, Report Jan. 2002
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B. DELIVERABLES
No
D5.2
Due Date
18
Deliverable
2D Code for fault activation and
propagation problem
D5.3
12
Model for fault activation due to thermoporo-mechanical coupling and the
importance of various physicochemical
factors that affect the performance of the
formation fluids
Robust FE code for borehole breakouts
tracing in anisotropoic far-field stress
field
D5.4
24
Plan
3D Code for fault
activation and
propagation problem
-
Under further
development
C. REFERRED TO THE PROJECT PUBLICATIONS
1. Vardoulakis, I. (2001). Thermo-poro-mechanical analysis of rapid fault deformation, Powders &
Grains 2001, Proceedings of the 4th International Conference on Micromechanics of Granular
Media, May 21-25, 2001, Sendai Japan, (Kishino ed.) 273-280, Balkema.
2. Vardoulakis, I. (2002a). Dynamic thermo-poro-mechanical Analysis of Catastrophic Landslides.
Gèotechnique, in print.
3. Vardoulakis, I. (2002b). Steady shear and thermal run-away in clayey gouges. Int. J. Solids
and Structures, in print.
PLANNING OF FUTURE PUBLICATIONS
1. G.E. Exadaktylos, P.A. Liolios, M.C. Stavropoulou, A closed-form elastic solution for notched
circular openings, to be submitted to Int. J. Solids & Structures, 2002.
2. A. Zervos, I. Vardoulakis, P. Papanastasiou, The influence of non-associativity on localization
and failure in geomechanics based on gradient elastoplasticity, 2002.
3. A. Zervos, I. Vardoulakis, P. Papanastasiou, Shear localization of deformation in hollow
cylinders under internal pressure, 2002.