~N6~N~ER~NG
ELSEVIER
Engineering Geology 49 (1998) 53-60
A graphical technique to predict slip along a pre-existing
plane of weakness
Susana A. Alaniz-Alvarez a,,, fi~ngel Francisco Nieto-Samaniego a, Gustavo Tolson b
a Estaci6n Regional del Centro, Instituto de Geologia, Universidad Nacional Aut6noma de Mkxico, Apdo. Postal 376,
Guanajuato, Gto., 36 000, Mexico
b Instituto de Geologia, Universidad Nacional Aut6noma de Mkxieo, Ciudad Universitaria, Delegaci6n Coyoac6n 04510,
Mkxico, D.F., Mexico
Received 1 November 1996; accepted 8 August 1997
Abstract
A graphical technique is proposed to determine whether a pre-existing plane of weakness will be reactivated by slip
under a stress field. This technique is based on Coulomb-Navier criteria and the method of Yin and Ranalli (Yin,
Z., Ranalli, G,. 1992. Critical stress difference, fault orientation and slip direction in anisotropic rocks under nonAndersonian stress systems. J. Struct, Geol. 14, 237-244). It consists of calculating which mechanism, rupture or
sliding, needs the smaller stress difference to liberate the deformation. Using the results of calculations over a wide
range of plane orientations, we plotted, in an equal-area net, the line which separates the orientation fields where
rupture needs less stress difference from the fields where slip on pre-existing planes is favored. We named these plots
slip-rupture graphs. For the three Andersonian fault regimes, the graphs are presented as dendrograms. These
dendrograms show the variation of the range of orientations favorable for reactivation as a function of cohesion and
friction of the plane of weakness, depth, pore fluid pressure and the stress ratio. The slip-rupture graphs are compared
with the Mohr diagram and slip-tendency graphs (Morris et al., 1996). Relative to Mohr diagrams, our graphs have
the advantage that it is possible to work with geographic orientations of planes and principal stresses, and it is not
necessary to transform the field data to a stress space. The slip-rupture graphs are similar to slip-tendency graphs;
however, the former can lead to estimate physical parameters that make reactivation possible along planes with
unfavorable orientations. © 1998 Elsevier Science B.V.
Keywords: Brittle deformation; Faults; Pre-existing structures; Stress; Structural analysis
1. Introduction
The reactivation o f pre-existing planes o f weakness is a c o m m o n p h e n o m e n o n in nature and it
has m a n y implications in structural and engineering geology at all scales. In order to avoid the
* Corresponding author. E-mail: [email protected]
oo 13-7952/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved.
PH S0013-7952(97)00071-9
collapse o f constructions, engineering geologists
must consider the reactivation o f pre-existing
planes o f weakness because it is necessary to
maintain the rock bodies below the critical stress
needed to initiate sliding. Reactivation is implicit
in some new geological concepts such as inversion
tectonics (e.g. Gillcrist et al., 1987), collapse o f
orogens (e.g. Dewey, 1988), weakness o f m a j o r
54
S.A. Ahmiz-Alvare: el al.
Engi, eering Geoh~gy 49 (1998) 53 60
faults (e.g. Lachenbruch and Sass, 1992; Bird,
1995), or low-angle normal
faults (e.g.
Allmendinger et al., 1987). Furthermore, it is
important to mention that the total displacement
of a fault consists of many episodes of slip after
the formation of the fault plane. Thus, in a strict
sense, these are episodes of reactivation.
The M o h r - C o u l o m b circle has been used to
relate the stress field (stress tensor) and the orientation of the plane of rupture, or if it is the case, the
plane of sliding. We have developed a graphical
technique to calculate the effect of varying physical
parameters on reactivation, which also permits
determination of the orientations of pre-existing
planes prone to liberate the stress state by slip, We
follow the methods proposed by Wallace (1951)
and Jaeger and Cook (1969) using an equal-area
projection to represent the principal stresses. The
graphs provide an alternative tool to analyze the
relationships between stress field and faulting in
brittle deformation. In addition, they could be
very helpful in structural analysis to back-calculate
the mechanical properties of faults and in seismic
risk assessment.
Yin and Ranalli (1992) reformulated the
Coulomb-Navier slip and failure criteria in order
to apply them to general crustal conditions. They
considered the stress difference o-l-cr3 as the
dependent variable instead of the shear stress (r)
and, they determine the effective stress field,
introducing the overburden pressure (pgz), modilying it by a pore fluid factor )L (pore fluid
pressure/overburden pressure), where p is the mean
density of rocks, g is acceleration due to gravity,
and z is the depth. Yin and Ranalli (1992) also
introduced the stress ratio R = (c~2 - 0 3 ) / ( 0 1 -0"3).
the unit vector normal to the fault Ni with components Ni = cos 7i, where ~,,~is the angle between N~
and the X~-axis; and the unit vector normal to a
horizontal plane M~ (Fig. 1 ). These equations are:
2Ftpg:( 1 - 2) + 2C
0-t -0"3 -= (¢t2 + 1 )1/2 _ i t + 2 F l ( M 2 + R M 2 ) "
for an isotropic medium, and
61 --,9-3 =
ttopgz( 1 -- ).o ) + Co
2 +RN2)211:2'
[(N 2 + R Z N Z ) - ( N
+ ~,o[(m 2 + R M 2 ) - ( N
2. Theoretical considerations about reactivation
The following theoretical analysis is based on
four assumptions: ( l ) a crustal block is deformed
in the brittle regime and without volume change;
(2) the planes of weakness cut the block completely; (3) constant displacement along each
fault plane; and (4) the maximum principal
stress
(al) -> intermediate
principal
stress
(az) >_minimum principal s t r e s s ( 0 " 3 ) > 0 .
When a rock body which contains pre-existing
planes of weakness is subjected to a stress field,
strain by fault slip can be accomplished either by
sliding along the pre-existing planes or by development of new fractures and slip along them.
The faulting process can be described by the
Coulomb-Navier slip criterion for the former case
and by the Coulomb-Navier failure criterion for
the development of new fractures. The potential
of a pre-existing plane of weakness to be reactivated largely depends on its orientation relative to
the stress field.
( 1)
2 + RN~)]
(2)
to initiate slip along a pre-existing plane of
weakness.
3. Capability of planes of weakness for reactivation
A simple test to determine whether or not a
pre-existing plane of weakness will be reactivated
consists of calculating the stress difference
(o-1- a3) necessary to initiate the slip and comparing it with the stress difference needed to form a
new fracture. The mechanism that needs less stress
difference, sliding along the pre-existing plane or
the rupture, will be the one that accommodates
the deformation.
Defining the parameter F as the difference
between Eqs. (1) and (2), we get:
F = E q . ( 1 ) - E q . (2).
(3)
Positive values of F correspond to planes of
weakness that will be reactivated before failure (if
S.A. Alaniz-Alvarez et al. / Engineering Geology 49 (1998) 53-60
section, we discuss certain aspects o f our technique
to determine the slip-rupture b o u n d a r y and also
its advantages over the M o h r - C o u l o m b diagram
and the slip tendency analysis (Morris et al., 1996).
Xl; (~1
/'
a
55
/
MI /' B
4. The slip-rupture graph technique
X2, o2
--:::\ C
~
X~, o~
Xl, O"1
b
IM,
.......
/ / .......IN
. jj,
:::~i::i~i::;::i::ii!
!~::
X2,o" ~
~2
C
//~/~.
"
'"
Fig. 1. Geometric relationships used in the equations of the text.
(a) The reference axes and principal stresses coincide. Mi is the
vertical unit vector; Ni is the unit vector perpendicular to the
plane ABC; ~< is the angle between a 1 and Ni. (b) Block
diagram representing a plane oriented N10 ° W-35 ° SW.
Since al is considered vertical M1 = 1, N1 =cos 35°, N 2 =
sin 35° sin 10° and N 3 =sin 35° cos 10°. From the diagram, the
dip and dip direction format is obtained directly: dip = 35° and
dip direction = 260°. (c) Equal-area, lower hemisphere plot. The
star is the pole to the plane shown in (b). a~, a2 and a a represent
the orientation of the principal stresses regardless of their
magnitudes.
the stress difference is sufficient for slip to occur),
while negative values correspond to planes that
will not slide because failure is favored. In the next
The orientation o f any plane is uniquely defined
by the direction cosines o f the unit vector perpendicular to the plane (Fig. 1). In stereographic and
equal-area projections, the line perpendicular to a
plane is plotted as a point referred to the reference
sphere. In order to use commercial software for
handling orientation data, we transformed direction cosines o f the plane normals to a dip and dip
direction format. The cases when one o f the principal stresses is vertical are shown in Fig. 2, and the
orientation of a plane with respect to the principal
stress directions is plotted as a pole on the net.
We developed a c o m p u t e r p r o g r a m to obtain
values o f F for more than 2000 in the positive
q u a d r a n t o f the net. We plotted the orientations
with positive values o f F. Due to the symmetry o f
the stress tensor, we can directly obtain values in
the other three quadrants. For a given stress field
and fixed physical conditions, we can separate
orientations with positive values o f F, where slip
occurs, i.e. in the slip domain, and the negative
values, where fracture occurs obeying Eq. (1), i.e.
in the rupture domain.
Graphically, Fig. 2 shows the capability o f
planes o f weakness to have displacement in the
three Andersonian stress states (i.e. where one o f
the principal stress axes is vertical). Such a condition is indicated by nonfractional values o f M~.
The crustal conditions o f the reference graphs
assume the values: z = 10 k m in order to place our
model near the base o f the seismogenic crustal
zone (Sibson, 1986); 2 = ) ~ o = 0 . 4 to consider a
saturated rock b o d y with interconnected pore
spaces from the surface to depth; and we used
/~o=0.75 for the coefficient o f friction, which is
considered close to the "typical value" obtained
by Byerlee (1978), and the same value for the
coefficient o f internal friction, p = p o = 0 . 7 5 . We
used experimental values o f cohesion ( C =
4 0 M P a ) , reported by A f r o u z (1992) for intact
56
S.A. Aluniz-Ah'arez et al.
Engineering Geology 49 (1998) 53 -60
NORMAL FAULT REGIME
,~IKE-SLIP
FAULT REGIME
. . . . . . . . .
c.=o
r~mL~
~,=o.4
F-] Rupturedomain
\\
\ ~
\
\
~=o.I
~0~'5
!¢
REVERSE FAULT REGIME
~o.9
_
~--0.9
R=0
I~mw~
z~'lOkm
c,=o
~pO.4
Cn~
\
~.75
R=0.5
M,,,,1
Fig. 2. Slip-rupture graphs for the three Andersonian fault regimes. Each graph is an equal-area, lower hemisphere plot. The
parameters specified in reference graphs are the same in the three dendrograms. Each graph shows the changed parameter, the other
parameters remain unchanged with respect to the values from the graph to which it is joined by a line. C is the cohesion, /~ is the
coefficient of internal friction, 2 is the pore fluid factor (pore fluid pressure/overburden pressure), all of intact rock; C0 is the cohesion,
iio the coefficient of friction and ),o the pore fluid factor, all in the plane of weakness; z is the depth, R = a z %,,'a~ - a ~ , and Mi is
the vertical unit vector. The slip domain (stippled areas) contains the poles of planes on which slip will occur before the rupture of
the material (F>0). The rupture domain (blank areas) contains poles of the planes on which slip will not occur before rupture of
the material (F<0).
S.A. Alaniz-AIvarez et al. / Engineering Geology 49 (1998) 53-60
igneous rocks and suppose that faults are cohesionless ( C o = 0 M P a ) . R = 0 . 5 assuming o-2 is the
average value of ~1 and ~3. For instance, to
illustrate the use of the slip-rupture graphs, we
highlight the following features of the reference
graphs (Fig. 2):
( 1) In the normal fault regime, where crI is vertical,
it is clear that planes with a dip of less than
30 D will not be reactivated regardless of the
dip direction.
(2) In the reverse fault regime, where a3 is vertical,
the slip domain is restricted to planes oriented
nearly parallel to ~2 dipping between 10 and
40". According to these graphs, it is possible
to reactivate an ancient thrust (dipping
25-40 ° ) as a normal fault, but it is more
difficult to reactivate a normal fault dipping
50-75 ° in a reverse mode.
(3) In the strike-slip fault regime, where cr2 is
vertical, planes oriented nearly parallel to ~
with dips of > 50 ° are prone to reactivation.
(4) All planes oriented nearly parallel to o3 are in
the rupture domain for the three faulting
regimes (Fig. 2), meaning that the planes with
orientations unsuitable for reactivation are
determined by the orientation of ~r3, rather
than that of cr1 and Oz. This is explained by
the transfer of matter towards the direction of
lowest pressure.
4.1. Slip-rupture graphs varying crustal conditions
Using this graphical technique, we are able to
determine the changes in the boundary between
the two domains produced by variations in crustal
conditions. For example, we can simulate shallow
levels of depth by decreasing z, anomalous pore
fluid pressure on pre-existing planes making
20 > 2, or planes filled with montmorillonite gouge
by diminishing/~o. Based on Fig. 2, we emphasize
the following points.
4.1.1. Depth
At 1 km depth, all the planes dipping > l0 ° can
be reactivated under a normal fault regime. In the
reverse fault regime, the planes with dips <60 °
belong to the slip domain. In the strike-slip fault
regime, the planes subparallel to ~3 are not prone
57
to reactivation. It is observed that the normal fault
regime offers a larger slip domain, followed by the
strike-slip fault regime and for the reverse fault
regime.
4.1.2. Friction
When the coefficient of internal friction for the
intact rock and the coefficient of friction for preexisting plane are equal, there is little variation in
the domains in response to the change in the
coefficients of friction. For a variation from
p = p o = 0 . 5 to p = p 0 = 0 . 9 , the slip domain
increases moderately (Fig. 2). On the other hand,
the slip domain is reduced significantly when
/~o< #. For instance, simulating low-friction
hydrated clays in a fault gouge, we used # = 0 . 7 5
and po=0.3 and observed a dramatic increase of
the slip domain in the three regimes.
4.1.3. Cohesion
We considered distinct cohesion for intact rock
and pre-existing planes in order to simulate
different kinds of planes. A cohesionless plane
(Co=0) represents an " o p e n " fault or fracture;
higher values of cohesion could simulate a closed
fault, unconformity or bedding, and yet higher
cohesion could simulate a "welded" contact. By
increasing the cohesion of the plane of weakness,
the slip domains approximates to the ideal fault
orientation. For instance, in the three fault regimes,
the graphs with Co=39 MPa show slip domains
with the same orientations that the ideal fault
orientation 7=26.5 ° for ~=0.75 (Fig. 2).
4.1.4. Pore fluid pressure
The presence of pore fluid pressure causes a
rock to behave as though the depth was lower by
an amount equal to the pore fluid pressure. This
is because it produces a decrease in the effective
normal stress acting on a surface. For an
Andersonian fault regime, the lithostatic pressure
(Pl) corresponds to the vertical principal stress.
Thus, for a reverse fault regime, tr3=P~; in a
strike-slip fault regime, o"2 = P l; and in a normal
fault regime t~l =Pl. The pore fluid pressure in the
fault must be less than ~3 to avoid tension fractures. Sibson (1990) has suggested that the pore
fluid pressure notably increases the weakness of a
58
S.A. Alaniz-Ah'are: et al. / Engineerine Geology 49 (1998) 53 60
fault in the reverse fault regime because fluid
pressure can reach the value of the tithostatic
pressure. In the other two regimes, where a3 is
horizontal, the maximum value of the pore pressure is less than the lithostatic pressure. Our
slip-rupture graphs show that the pore pressure
has the biggest influence in the slip domain under
the normal fault regime and the smallest in the
reverse fault regime. We explained this behavior
by noting that in normal fault regime al = P~, and
recalling that 0-1 >62>0"3. Thus, normal fault
regimes are characterized by smaller absolute stress
magnitudes and, subsequently, smaller critical
stress difference. Consequently, the magnitude of
the pore fluid pressure, which is independent of
the stress field, represents a major portion of the
normal stress in this regime.
Fig. 2 shows that in a reverse fault regime at
: = 3 . 5 km, planes with a dip of > 5 0 cannot be
reactivated even if pore fluid pressure equals lithostatic pressure ()~0=1), but many planes with
such dips can be reactivated considering a high
pore fluid pressure and a low coefficient of friction
together. Similar behavior is observed under the
other regimes. We conclude that in order to reactivate planes with highly unfavorable orientation,
both tow normal stresses and low coefficients of
friction are necessary.
5. The Mohr diagram and the rupture-slip graphs
The Mohr diagram has many uses in the handling of stresses in brittle faulting because it shows
the relationship between the orientation of a plane
with the values of normal stress and shear stress
on that plane. Using Mohr diagrams, it is possible
to know the orientations of planes of weakness
susceptible to reactivation. Below the shear fracture envelope and above the slip envelope, there is
not enough stress difference to break the rock but
enough to induce slip for a range of plane orientations. In this region the states of stress are stable
for fracture and unstable for slip (Fig. 3).
In the two-dimensional Mohr diagram, planes
parallel to 0-2 are plotted on the circumference,
which contains a~ and a3. The range of orientations
of planes with other directions prone to slip cannot
T
"c = C + l.ta .
if3
if2 ff~
ffl ~n
ff~
ff 2
fit
T.
Fig. 3. (a) In the two-dimensional Mohr circle, the plane orientations which are parallel to a 2 are represented by the circumference with diameter cr~- a 3. The variation in the magnitude of
c~, does not produce a change in the range of orientation which
are over the envelope for sliding (gray arc). (b) Three-dimensional Mohr circle for failure in isotropic rock and failure along
a plane of weakness. Circles concentric with the circle with
diameter a t - a 3 represent stress states on planes with fixed
N2: circles concentric with the circle with diameter a2-a 3 represent stress states on planes with fixed N 1. The full range of
orientations is represented by the area between the three circles.
The upper envelope displays the condition for failure in isotropic rock, and the lower envelope the condition for sliding
along preexisting plane of weakness. The planes over the envelope for sliding will have reactivation.
be visualized. In the slip rupture graphs (Fig. 2),
the poles of planes parallel to a 2 are plotted on
the line which contains 0-1 and 0-3. For example, if
we modify the relative magnitude of 0-2 by varying
the R value, we can see the dramatic change in the
boundary between the two domains in three dimensions, but the planes parallel to 0-2 do not change
domain [Figs. 2 and 3(a)]. This example clearly
S.A. Alaniz-Alvarez et al. / Engineering Geology 49 (1998) 53-60
illustrates an advantage of the slip-rupture graphs
over the two-dimensional Mohr diagram.
In the three-dimensional Mohr diagram, it is
possible to visualize the full range of plane orientations [Fig. 3(b)]. However, it is necessary to transform the data from the stress space ( ~ , - z space)
to the physical space. Furthermore, the threedimensional Molar diagram does not consider geographic orientation, whereas using the slip-rupture
graphs, we can plot the geographic North to know
the geographic orientation of the data. That is
another advantage of slip-rupture graphs; they
permit a direct visualization of lines and planes in
a geographic frame, which is how they are seen in
the field.
6. Slip-tendency analysis and the slip-rupture
graphs
Morris et al. (1996) proposed another technique
to determine the likelihood that slip occurs along
a surface. Their technique is based on the principle
of variation in shear stress within the stress tensor.
The slip tendency, Ts = r/(a,), is determined by the
relative magnitudes of the shear and normal
stresses. The distribution of slip tendency with
fault orientation depends upon the relative values
of the principal stresses. Morris et al. (1996)
developed a computer program in which it is
necessary to specify the stress tensor in terms of
the principal stresses. The program calculates and
displays the slip tendency for planes of all orientations. The obtained slip-tendency graphs were
made using the ratio Ts/TstaA x for each of all
possible orientations. T~MAXis the maximum limiting value of slip tendency and it is equivalent to
/~0 in our notation. The value for each orientation
is plotted in an equal-angle net. The graphs show
the dependence of the reactivation from the plane
orientation relative to the principal stresses. The
slip tendency varies from 100% near the ideal fault
orientation toward 0% in the principal stress direction where the shear stress is zero. The slip tendency graphs are quite similar to those obtained
by us and the change in the graphs due to variations in stress ratio are equivalent. The main
difference between their technique and the
59
slip-rupture graphs is the reference used to obtain
the potential of reactivation. We consider as reference the rupture strength of the intact rock,
whereas Morris et al. (1996) consider the frictional
resistance to sliding on the fault. The slip-rupture
graph technique allows the physical conditions to
be varied until a graph which permits reactivation
is obtained. Following this, it is possible to predict
the parameters which permit activity of the considered fault. For instance, a pre-existing fault oriented N60 ° W-50 ° SW under a normal stress
regime with a2 trending North has a low slip
tendency (40%). It is possible to reactivate the
plane with either high pore fluid pressure in the
fault plane or low coefficient of friction (Fig. 2).
Furthermore, selecting parameters which favor
reactivation (shallow depth, high pore fluid factor
and low coefficient of friction), we observed that
it is possible to reactivate planes with a slip tendency lower than 20% (Fig. 2).
7. Conclusions
The slip-rupture graphs are an alternative tool
for the analysis of stress-plane orientation relationships. These graphs allow one to determine
whether an existing plane of weakness will be
reactivated by slip or whether new fractures will
form under different crustal conditions. It is shown
that by increasing pore fluid pressure and decreasing depth, cohesion and coefficient of friction in
the plane of weakness, the range of orientations
in which reactivation could occur increases toward
the planes parallel to t~3. Relative to Mohr diagrams, the graphs presented here have the advantage of working with geographic orientations of
planes and principal stresses and it is not necessary
to transform the data to a stress space. The
slip-rupture graphs are very similar to the sliptendency graphs; however, the former allow predictions to be made regarding the possible reactivation of slip along planes with unfavorable
orientations by varying different physical parameters. Furthermore, slip-rupture graphs address not
only the possible reactivation of pre-existing planes
of weakness as a function of the various parameters
discussed above, but also the possibility of fracture.
60
S.A. Alaniz-Alvarez et el/, / Engineering Geology 49 (1998) 53 60
Acknowledgment
This research was financed by CONACYT projects 3155T and 3159PT. We thank Randall
M a r r e t t f o r c o n s t r u c t i v e c o m m e n t s in a n e a r l i e r
version of the manuscript. The comments of David
A. Ferrill, G o o d l u c k O f o e g b u , A l a n M o r r i s a n d
a n a n o n y m o u s r e f e r e e h e l p e d us t o i m p r o v e t h e
manuscript.
References
Afrouz, A.A., 1992. Practical Handbook of Rock Mass Classification Systems and Modes of Ground Failure. CRC Press.
Boca Raton, FL, USA, p. 195,
Allmendinger, R.W., Hauge, T.A., Hauser, E,C., Potter, C.J.,
Klemperer, S.L., Nelson, K.D.. Knuepfer, P., Oliver, J.,
1987. Overvie,~, of the COCORP 40'N transect, western
United States: the fabric of an orogenic belt. Geol. Soc, Am.
Bull. 98. 308 319.
Bird, P., 1995. Lithosphere dynamics and continental detbrmation. Rev. Geophys. 33, 379 383.
Byerlee, J,, 1978. Friction of rocks. Pure Appl. Geophys. 116,
615 --626.
Dewey, J.F., 1988. Extensional collapse of orogens. Tectonics
7. 1123 1139,
Gillcrist, R., Coward, M., Mugnier, J., 1987. Structural inversion and its controls: examples from the Alpine foreland and
the French Alps, Geodinamica Acta 1, 5 34.
Jaeger, J.C., Cook, N.G.W., 1969. Fundamentals of Rock
Mechanics. Methuen, London, p. 513.
kachenbruch, A.H., Sass, J.H., 1992. Heat flow from Cajon
Pass, fault strenght, and tectonic implications.. J. Geophys.
Res. 97, 4995 5015.
Morris, A., Ferrill, D.A., Henderson, D.B.. 1996. Slip-tendency
analysis and fault reactivation. Geology 24, 275-278.
Sibson, R.H., 1986. Earthquakes and rock deformation m
crustal fault zones. Ann. Rev. Earth Planet. Sci. 14, 149-175.
Sibson, R.H., 1990. Conditions for fault valve behavior. In:
Knipe, R.J., Rutter, E.H. (Eds.), Deformation Mechanisms,
Rheology and Tectonics, Geological Society of London,
Special Publication No. 54, pp. 15--28.
Wallace, R.E., 1951. Geometry of shearing stress and relation
to lhulting. J. Geol. 59, 118-130.
Ym. Z.M.. Ranalli, G., 1992. Critical stress difference, fault
orientation and slip direction in anisotropic rocks under nonAndersonian stress systems. J. Struct. Geol. 14, 237- 244.