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Effects of Physical Processes and Sampling Resolution on Fault Displacement Versus Length
Scaling: The Case of the Cantarell Complex Oilfield, Gulf of Mexico
SHUNSHAN XU,1 ANGEL F. NIETO-SAMANIEGO,1 GUSTAVO MURILLO-MUNETON,2 SUSANA A. ALANIZ-ALVAREZ,1
JOSE M. GRAJALES-NISHIMURA,3 and LUIS G. VELASQUILLO-MARTINEZ2
Abstract—In this paper, we first review some factors that may
alter the fault Dmax/L ratio and scaling relationship. The three main
physical processes are documented as follows: (1) The Dmax/L ratio
increases in an individual segmented fault, whereas it decreases in
a fault array consisting of two or more fault segments. This effect
occurs at any scale during fault growth and in any type of rock. (2)
Vertical restriction decreases the Dmax/L ratio along the fault strike
due to mechanical layers. (3) The Dmax/L ratio increases or
decreases due to fault reactivation depending on the type of reac-
tivation. Thus, using data from the normal faults of the Cantarell
oilfield in the southern Gulf of Mexico, we document that the
displacement (Dmax) and length (L) show a weak correlation of
linear or power-law scaling, with exponents that are much less than
1 (n & 0.5). This scaling relation is due to the combination of the
physical processes mentioned above, as well as sampling effects,
such as technique resolution. These results indicate that sublinear
scaling (n & 0.5) can occur as a result of more than one physical
process during faulting in a studied area. In addition to the physical
processes associated with brittle deformation in the studied area,
the sampling resolution dramatically affects the exponents of the
Dmax–L scaling.
Key words: Fault segmentation, mechanical layering, fault
reactivation, sublinear scaling, Gulf of Mexico.
1. Introduction
The relationship between the maximum displace-
ment (Dmax) and length (L) of faults is commonly
represented as a scaling power law in the form
Dmax ¼ cLn; ð1Þ
where c is a constant related to the material properties
of the rocks within which the faults develop. The
scatter and limited scale range of published fault
datasets, which come from different tectonic and
lithological settings, have impeded the ability to
discern whether a universal value of n exists (e.g.
HATTON et al. 1994; BONNET et al. 2001). Some works
have proposed that the Dmax relation displays super-
linear scaling, where n equals 2 (WATTERSON 1986;
NICOL et al. 1996) or 1.5 (MARRETT and ALLMENDINGER
1991; WILKINS and GROSS 2002). However, an
increasing amount of field and experiment-based data
shows that the relationship between the maximum
displacement and the fault trace length is approxi-
mately linear in a single tectonic environment with
uniform mechanical properties (e.g. GUDMUNDSSON
1987; GILLESPIE et al. 1992; DAWERS et al. 1993;
SCHLISCHE et al. 1996; BOHNENSTIEHL and KLEINROCK
2000; MANSFIELD and CARTWRIGHT 2001; GUD-
MUNDSSON 2004). The fracture mechanics theory
suggests that it should be linear (see Chapter 9 in
GUDMUNDSSON 2011), but that linear relation applies
to individual slip events—not necessarily to fault
growth in many slip events over long periods of time.
The linear scaling relation is also reported by
observing the planetary faults on Mars and Mercury
(WATTERS et al. 2002; SCHULTZ et al. 2006). Rupture
length and earthquake slip are known to obey linear
scaling (e.g. WELLS and COPPERSMITH 1994; SCHOLZ
1994; MANIGHETTI et al. 2001; DAVIS et al. 2005;
GUDMUNDSSON et al. 2013). The elastic and elastic–
plastic fracture models have been used to explain the
linear Dmax–L scaling relation (COWIE and SCHOLZ
1992; SCHULTZ et al. 2006, 2008; GUDMUNDSSON 2011;
GUDMUNDSSON et al. 2013). The results of the analysis
of global data also suggest that the relation between
1 Centro de Geociencias, Universidad Nacional Autonoma de
Mexico (UNAM), Blvd. Juriquilla No. 3001, 76230 Queretaro,
Mexico. E-mail: [email protected] Instituto Mexicano del Petroleo, Eje Central Lazaro
Cardenas No. 152, Col. San Bartolo Atepehuacan 07730 Mexico
D.F., Mexico.3 Instituto de Geologıa, Universidad Nacional Autonoma de
Mexico, C.P. 04510 Mexico D.F., Mexico.
Pure Appl. Geophys.
� 2015 Springer Basel
DOI 10.1007/s00024-015-1172-0 Pure and Applied Geophysics
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displacement and length is approximately linear (e.g.
CLARK and COX 1996; XU et al. 2006). Therefore,
while no consensus exists, many researchers accept
that n = 1.
Recently, a sublinear scaling of the Dmax–L rela-
tion with an exponent of n = 0.5 was proposed
(OLSON 2003; SCHULTZ et al. 2008), which implies
that maximum displacement depends on the square
root of length. This square root scaling relation is
obtained from data on opening-mode or closing-mode
discontinuities, including joints, veins, deformation
bands and compaction bands (e.g. FOSSEN et al. 2007;
SCHULTZ et al. 2008). However, linear relationships
have also been reported from studies on tension
fractures (e.g. JOHNSTON and MCCAFFREY 1996; GUD-
MUNDSSON et al. 2000). Conversely, the scaling
exponent of deformation bands can be equal to 1 or
0.5, depending on the degree of the closing dis-
placement (SCHULTZ et al. 2008). A sublinear scaling
relation with n\ 0.75 has been reported in shear
discontinuities (e.g. ACOCELLA and NERI 2005; BER-
GEN, and SHAW 2010). This scaling behaviour has not
been completely explained by the mechanism of fault
growth.
In nature, complicated physical processes modify
the ideal conditions of fault growth. To explain the
factors affecting the power-law exponent, we syn-
thesize the following special mechanism of fault
growth: (1) detailed explanations of the physical
processes of fault segmentation are provided in
Sect. 2.1. (2) Mechanical layering commonly causes
the vertical restriction of fault growth. A detailed
analysis of the effect of mechanical layering is pro-
vided in Sect. 2.2. (3) The effect of fault reactivation
on Dmax–L scaling is analysed in Sect. 2.3. In Sect. 3,
we provide a detailed example of the normal faults
from the Cantarell oilfield in the southern Gulf of
Mexico. All of the physical factors mentioned above
will be examined in this example. We document that
sublinear scaling (n & 0.5) in the studied area is due
to a combination of the physical processes mentioned
above. Specifically, we express that sampling effects,
such as technique resolution, significantly influence
the Dmax–L scaling. Therefore, sublinear scaling in
the studied area can occur in response to various
physical processes during faulting, and it could be
highly biased by sampling resolution as well.
In this paper, we emphasize factors such as fault
segmentation, mechanical layering and fault reacti-
vation in relation to the Dmax–L scaling. Other
factors, such as Young’s modulus, dilatant displace-
ment or the controlling dimension of a fault, would
also decrease or increase the power-law exponents,
but are beyond the analysis of this paper.
2. Physical processes affecting fault displacement
versus length scaling
2.1. Fault Segmentation
Fault interaction and linkage have a significant
impact on the dimension and displacement accumu-
lation of fault systems over a long period of time (e.g.
GUDMUNDSSON 1987; GUPTA and SCHOLZ 2000). The
segmented structure of faults is a basic characteristic
of most natural arrays showing fault interactions at
different scales (e.g. PEACOCK and SANDERSON 1994).
Two segment linkage models are proposed in the
literature—the isolated and the coherent models (e.g.
CHILDS et al. 1995; WALSH et al. 2003). The isolated
model shows how over time, initially isolated fault
segments grow by tip propagation and will experi-
ence eventual, incidental, lateral overlap and
interaction. In the coherent model, the kinematically
related segments, initially belonging to the same
structure, link into a single array in the final stage
(e.g. CHILDS et al. 1995). Generally, there are two
ways by which fractures link, through curved hook-
shaped fractures (mostly extension fractures), or
through connecting transfer fractures (e.g. FERRILL
et al. 1999; GUDMUNDSSON 2011).
For fault segments in the relay zone, the reorien-
tation, or tilt of bedding, transfers displacement
between the segments. This may produce higher
Dmax/L ratios when one segment enters the relay zone
(e.g. DAWERS and ANDERS 1995; MANIGHETTI et al.
2001; WALSH et al. 2003). However, closer to the
fault tips in the relay zone, the displacement gradient
decreases (WOJTAL 1994; WILLEMSE et al. 1996).
Generally, the Dmax/L ratio increases in the relay
segment for the isolated model (Fig. 1a). This
increase in Dmax/L ratio is due to a smaller segment
length and a greater maximum displacement than in
the case of isolated faults, and is based on the
S. Xu et al. Pure Appl. Geophys.
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displacement transfer by a local perturbation of the
stress field, which diminishes the growth of tip zones
near the relay zone (e.g. WILLEMSE et al. 1996;
Fig. 1a). In the displacement–length plot (Fig. 1b),
the position of each segment is located to the left
compared to isolated faults. In this sense, the
exponent deviates from that of an isolated fault with
n\ 1 or n[ 1, depending on the degree of fault
interaction—as relates to spacing and the amount of
fault overlap—for each segment. This case is differ-
ent from fault arrays associated by fault linkage, for
which the position of the total fault array moves to
the right. For isolated soft linkage, if the segments are
taken as isolated faults, the position of each segment
in the Dmax–L plot moves to the left, whereas a
physical linkage results in the position of the fault
array shifting to the right (Fig. 1b, c). For both cases,
the Dmax–L exponent can be less than 1 or greater
than 1.
2.2. Mechanical Layering
GROSS (1993) defines a mechanical layer as ‘‘…a
unit of rock that behaves homogenously in response
to an applied stress and whose boundaries are
located where changes in lithology mark contrasts in
mechanical properties’’. The mechanical layering of
host rocks has considerable effects on the develop-
ment of faults (e.g. BENEDICTO et al. 2003; SOLIVA
and BENEDICTO 2005). Whether a propagating frac-
ture becomes arrested by a layer interface or
penetrates a layer interface is determined by three
related parameters: the induced tensile stress ahead
of the propagating fracture tip; the rotation of the
principal stresses at the interface and the material
toughness or critical energy release rate of the
interface in relation to that of the adjacent rock
layers (GUDMUNDSSON et al. 2010). The mechanism
of a bedding-parallel slip may play an important
role in transferring and accommodating slips within
fault zones that cut across heterogeneous stratigra-
phy (e.g. GROSS et al. 1997; NEMSER and COWAN
2009). In large-scale extensional systems, fault
blocks rotate with progressive extension and bed-
ding rotates to steeper dips. The tendency for
slipping on bedding increases with increasing
extension and block rotation. The actual occurrence
of slips on a bedding surface depends on the
frictional resistance to sliding and cohesion on the
surface. Weak horizons may slip or shear at
relatively low slip tendencies (e.g. FERRILL et al.
1998; ALANIZ-ALVAREZ et al. 1998).
A four-stage conceptual growth model due to the
effect of lithological contacts is illustrated in Fig. 2.
(a)
AB
Dmx
Distance
Dis
plac
emen
tIsolated fault
Fault segment
(b)
Length
Dis
plac
emen
t
n = 1
n 1n 1
(c)
Length
Dis
plac
emen
t
n = 1
n 1
n 1
Figure 1aMaximum displacement increases and fault length decreases for a
fault segment. For isolated faults, linear Dmax–L scaling is assumed.
b Dmax–L data points move towards the left for soft-linked
segments. For isolated faults, linear Dmax–L scaling is assumed.
c Dmax–L data points move towards the right for hard-linked fault
arrays. In both cases, the power-law exponents are less than 1 or
greater than 1, depending on the distribution of the points (hollow
squares or hollow circles). Note that for isolated faults, linear
Dmax–L scaling is assumed
Effect of Physical Processes and Sampling Resolution on Fault Displacement
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At stage 1, the fault vertical dimension (H) is less
than the layer thickness, and there is no vertical
restriction. In this case, the fault plane has an ideal
elliptical shape, and the displacement profiles along
both the strike and dip show a triangular shape. This
allows the Dmax–L linear scaling to be ascertained
(e.g. GROSS et al. 1997; SOLIVA and BENEDICTO 2005).
At stage 2, further deformation causes the fault to
approach the interfaces between competent and
incompetent layers, and the fault plane becomes a
quasi-rectangle due to vertical constraints. The fault
growth along the strike is different from that along
the dip. Vertical (dip-parallel) fault growth is char-
acterized by increasing displacement and constant
length, whereas lateral (strike-parallel) fault growth is
characterized by a decreasing Dmax/L ratio. Although
the displacement profiles along the strike and dip are
similar to a mesa (plateau) shape, the mechanism of
displacement accumulation along each is distinct.
Along the fault strike, the displacement accumulation
in the restricted part (central part) is less than that of
areas far from the restricted part (Fig. 2a). Along the
fault dip, the parts near the fault tips are restricted and
displacement accumulation decreases (Fig. 2c). In the
Dmax–L plots, the dip-parallel growth line is vertical
and the strike-parallel growth line is below the linear
line (n\ 1) (Fig. 2b, d; SOLIVA and BENEDICTO, 2005).
At stage 3, a restoring stage occurs when the fault
breaches the mechanical layer. At this stage, fault
growth along the strike follows a constant length
model and the data points in the Dmax–L plot show a
vertical growth path. However, the vertical fault
growth is characterized by a decreasing Dmax/L ratio.
Stage 4 is called the restored stage. In this stage, the
fault plane once again demonstrates an ideal elliptical
shape and linear Dmax–L scaling is expected. The
cycle of this four-stage model may be repeated if the
fault propagates across the mechanical layer bound-
ary and begins to grow within the next larger
mechanical layer.
2.3. Fault Reactivation
Pre-existing faults affect sequent deformation in
two ways. First, the pre-existing faults serve as
nucleation sites for new faults. Second, the pre-
existing faults act as obstacles to the propagation of
the second-phase normal faults (HENZA et al. 2011).
According to the Mohr–Coulomb theory, for a pre-
existing plane, the critical condition for a slip is
(a)
Length
Dis
plac
emen
t
12
3
4
12
3
4
12
23
43
1
23
4(b)
(c)
(d)
Distance
Dis
plac
emen
t
Dis
plac
emen
tD
ispl
acem
ent
Distance
Length
3
4
2
Figure 2a Evolution stage of a fault plane before and after a vertical restriction within a brittle layer. The shaded area indicates the decrease of the
displacement increment at stage 2. b Model of along-strike displacement profiles at different stages. c Along-strike fault growth. The shaded
area indicates the decrease of the displacement increment at stage 2. d Model of vertical displacement profiles at different stages. e Vertical
fault growth
S. Xu et al. Pure Appl. Geophys.
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s ¼ C þ lsðrn�PÞ; ð2Þ
where s is the magnitude of shear stress, rn is the
magnitude of normal stress on the pre-existing plane,
C is the shear strength of the pre-existing plane when
rn is zero, ls is the coefficient of friction of the pre-
existing plane and P is fluid pressure (e.g. JAEGER
et al. 2007; GUDMUNDSSON 2011). The value of rn is
dependent on the orientation of the plane relative to
the principal stresses responsible for the reactivation
(e.g. Jaeger and Cook 1969). C and ls depend on
lithology. This means that reactivation processes are
selective and only occur on some portions of faults
(e.g. MORRIS et al. 1996; KELLY et al. 1999; BAUDON
and CARTWRIGHT 2008; LECLERE and FABBRI 2013). If
fault surfaces have a reduced or negligible cohesive
strength, their reactivation is controlled by the coef-
ficient of friction, the state of stress, the fault
orientation and the pore fluid pressure.
Based on the relative slip sense of a reactivation
event on a fault, three types of fault reactivation can
be distinguished, namely, normal reactivation,
reverse reactivation and oblique reactivation
(Fig. 3a). Normal reactivation occurs when the angle
between the new slip sense and previous slip sense
(h) is between 0� and 45�. Reverse reactivation refers
to reactivated faults with an opposite slip
(135�\ h\ 180�) response to changing stress con-
ditions or tectonic settings. When 45�\ h\ 135�,the fault is known as an oblique-reactivated fault
(Fig. 3a). Fault reactivation is an important factor for
modifying fault displacement geometries and for
controlling the pattern of deformation (e.g. CART-
WRIGHT et al. 1995; WALSH et al. 2002; VETEL et al.
2005). A reverse reactivated fault can exhibit a lower
displacement-to-length ratio compared to un-reacti-
vated faults (e.g. PEACOCK 2002; VETEL et al. 2005;
KIM and SANDERSON 2005). A possible result of a
change in the Dmax/L ratio is that the exponent
n becomes less than 1 (Fig. 3b). Normal-reactivated
faults can accumulate more displacement while
maintaining a near constant fault trace length (e.g.
BAUDON and CARTWRIGHT 2008). This disproportion-
ate increase of maximum displacement against length
shifts the growth path in a plot of displacement to
length to a path with a higher Dmax/L ratio (Fig. 3c).
3. Scaling of the Fault Displacement Versus
the Length of the Normal Faults
from the CANTARELL Oilfield in the Southern
Gulf of Mexico
3.1. The Geological Background of the Study Area
The Cantarell oilfield is located in the southern
part of the Gulf of Mexico, 85 km offshore from
Ciudad del Carmen, Yucatan Peninsula (Fig. 4a, b).
The Gulf of Mexico was formed as a result of Middle
Jurassic rifting, which produced passive margins
flanking a small area of oceanic crust in the central
part of the basin (e.g. SAWYER et al. 1991). The
counterclockwise rotation of the Yucatan Peninsula
block away from the North American plate took place
(b)
Length
Dis
plac
emen
t
n = 1
n 1
(c)
Length
Dis
plac
emen
t
n = 1n 1
45°45°
45°45°
Normal
reactivation
Reverse
reactivation
Obliquereactivation
Obliquereactivation
(a)
Figure 3a Classification of fault reactivation. The filled arrow indicates
previous slickenline senses on the fault. The hollow arrows indicate
the reactivated slickenline senses on the fault. b Dmax–L relationship
with a decreasing Dmax–L ratio due to fault reactivation.
c Dmax–L relationship with an increasing Dmax–L ratio due to fault
reactivation
Effect of Physical Processes and Sampling Resolution on Fault Displacement
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(a)
(c)
(b)
(d)
(e)
S. Xu et al. Pure Appl. Geophys.
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during the formation of the Gulf of Mexico (e.g. Bird
et al. 2005; PINDELL and KENNAN 2009). In Campeche
Bay, three primary superimposed tectonic regimes
are recorded (ANGELES-AQUINO et al. 1994): the
extensional regime initiated in the Middle Jurassic;
the compressional regime during the middle Mio-
cene; and the extensional regime extended
throughout the middle and late Miocene. During the
last two regimes, salt tectonics occurred in all of the
areas, overprinting structures and disturbing the
regional stress field.
Various studies have been published regarding the
structural features of the Cantarell oilfield (e.g.
SANTIAGO and BARO 1992; PEMEX-Exploracion Pro-
duccion 1999; MITRA et al. 2005). The Cantarell Field
has an overthrusted structure and shows an upright
cylindrical fold with gently plunging conical termi-
nations (MANDUJANO and KEPPIE 2006). The western
boundary is a normal fault with a minor strike-slip
component, whereas to the north and east, the field is
limited by reverse faults (Fig. 4d). The oilfield is
composed of a number of sub-fields or fault blocks.
These are the Akal, Chac, Kutz and Nohoch blocks.
The faults in the Akal block are normal faults, but the
observed slickensides in minor faults from the core
samples are generally oblique (XU et al. 2004), which
implies a strike component of displacement on the
faults. Recent interpretation of the geophysical data
suggested that Cantarell is a fold-thrust belt and a
duplex structure related to the Sihil thrust. These
faults in the Cantarell were not formed by simple
shear related to the movement of a larger fault (MITRA
et al. 2005; GARCIA-HERNANDEZ et al. 2005).
The stratigraphic records in this oil field are
shown in Fig. 4e (PEMEX-Exploracion Produccion
1999). The main units include Callovian salt, Oxfor-
dian siliciclastic strata and evaporites, Kimmeridgian
carbonates and terrigenous rocks, Tithonian silty and
bituminous limestone, Cretaceous dolomites, and
dolomitized breccias in the Cretaceous-Tertiary
boundary and Lower Paleocene. The Tertiary system
includes siltstone, sandstone and carbonate rocks.
The producing formation was created when the
Chicxulub meteor impacted the earth (GRAJALES-
NISHIMURA et al. 2000). The upper reservoir is a
brecciated dolomite of the uppermost Cretaceous age.
The breccia is from a shelf failure due to an
underwater landslide when the meteor hits. The
lower producing formation is a Lower Cretaceous
dolomitic limestone.
3.2. Relationship Between Fault Displacement
and Length
For analysis of the relationship between fault
displacement and length, structural contour maps are
used to measure the data of fault displacement and
the fault trace length. Four structural maps of a
1:50,000 scale were selected: the dolomitized brec-
cias located at the top of the Cretaceous/Tertiary
boundary (S1), the top of the Lower Cretaceous (S2),
the top of Tithonian (S3) and the top of Kimmerid-
gian (S4). We measured fault displacements from the
structural contour maps, applying the method pro-
posed by XU et al. (2004). According to this method,
fault vertical displacement (Dv) is related to the
dislocation of contour lines (Dc) across the fault trace
and dip of the corresponding layer (b):
Dv ¼ Dc tan b ð3Þ
To measure the value of Dc, two conditions are
required. First, the contour lines must be approxi-
mately perpendicular to the strike of the fault. If this
is not the case, the value of the bedding dip (b) needsto be corrected (XU et al. 2004). Second, the bedding
dip must not be larger than 35�. The tilts of
stratigraphic units in our study area are consistent
with this condition. For example, the average dip at
the top of S1 is 17.3� (XU et al. 2007).
3.2.1 Effect of Fault Interactions
To study the effect of fault interactions, we analysed
two types of datasets for all of the faults in the studied
area. One type of dataset is from the two-tip faults,
which either do not cut other faults or are not cut by
other faults. Another type of dataset is measured from
the one-tip or no-tip faults, which have branching
Figure 4a Location of the study area. b Sketch map of the Campeche Bay.
c Rose diagram of fault direction in the Campeche Bay. d Structural
contour map of the top of the Cretaceous-Tertiary carbonate
breccias (S1) in the Cantarell oilfield. e Integrated stratigraphic
column in the Cantarell oilfield
b
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faults or are branching faults themselves. The results
(Fig. 5a, b) indicate that the coefficients of determi-
nation (R2) for both linear and power-law
relationships are low, both are less than 0.6. For
two-tip faults, the R2 values for linear regression are
larger than those for power-law analysis (Fig. 5a),
suggesting, even weakly, a linear D–L relationship.
The power-law exponent is approximately equal to
0.5 (n = 0.61), but with a fairly low coefficient
(R2 = 0.42).
However, for one-tip or no-tip faults, the R2 = 0.3
for both linear and power-law relationships is much
lower than the R2 for two-tip faults (R2 & 0.5). The
scaling exponent is 0.49, which is far from the linear
scaling law. Although this low slope is poorly defined
and may not provide meaningful geological
y = 0.7403x 0.61
R² = 0.423y = 0.0307x + 21.139
R² = 0.5205
020406080
100120140160180200
(a)
y = 1.7622x 0.4958
R² = 0.3098
y = 0.0215x + 40.08R² = 0.329
0
50
100
150
200
250
300
350
00 500 1000 1500 2000 2500 3000 3500 4000 1000 2000 3000 4000 5000 6000 7000 8000
(b)
Bedding 1
2
Hard linkage
Soft linkage
up
1
2
3
4
5
6
6
3
3
4
56
Interlayer sliding
(c) (d)
Length (m)Length (m)
Dis
plac
emen
t (m
)
Dis
plac
emen
t (m
)
Figure 5a D–L relationship of two-tip faults combined from four horizons. b D–L relationship of one- and no-tip faults combined from four horizons.
c and d One core sample of well 3026D showing vertical linkage of minor normal faults and interlayer sliding due to normal faulting in the
Cantarell oilfield
S. Xu et al. Pure Appl. Geophys.
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information about the growth of the fault system, at
least it indicates that the fault interaction significantly
affects the displacement–length scaling relationship.
In the map view of the study area, terminating or
branching fault geometries are common (Fig. 4d).
Generally, terminating interactions are weaker than
crosscutting interactions (HENZA et al. 2011). For
parallel faults, a soft linkage may exist when spacing
to total system length ratios less than 12.5–15 % (e.g.
HUS et al. 2005; WILLEMSE 1997). Additionally,
beyond the observation dimension, an isolated fault
may connect with other faults (CHILDS et al. 1995;
MANSFIELD and CARTWRIGHT 2001; WILLEMSE 1997;
DAVIS et al. 2005). For example, both soft and hard
fault linkages for small faults are observed from
vertical sections in the core samples (Fig. 5c, d; XU
et al. 2011).
Fault intersections exist among most of the faults
in the studied area. The mechanical interaction occurs
when a fault intersects other faults. For this type of
linkage, the stress state around the intersection line is
perturbed (MAERTEN et al. 1999). Accordingly,
displacement profiles generally exhibit multiple slip
maxima near the line of intersection between two
faults (e.g. NICOL et al. 1996; MAERTEN et al. 1999).
For the restricted fault in the intersecting fault
system, the displacement maximum is located near
the intersection line with a restricting fault and
steeper slip gradient towards the line of intersection
(MAERTEN et al. 1999). In detail, the slip on the
hanging wall side of the restricted fault (fault B in
Fig. 6a) is always greater than or equal to that of an
isolated planar fault, whereas the slip on the footwall
side of fault B is always less (Fig. 6a). If the
intersection line is located near the centre of a fault,
the displacement/length ratio will increase, thus
altering the displacement–length scaling (Fig. 6b).
3.2.2 Effect of Mechanic Layering
To study the effect of mechanic layering, we analysed
the datasets of the one- or no-tip faults from four
reflection layers in the studied area. For reflection
layers S1 and S2, the coefficients for the power-law
distribution are larger than those for the linear
distribution (Fig. 7a, b). This seems to indicate that
the power law (n & 0.5) is more acceptable for these
two datasets. However, the coefficients (R2) for both
the linear and power-law relationships are not high
enough for reflection layers S1–S3, and it is not easy
to determine which scaling law fits for these datasets.
For layer S4, the coefficients (R2) for both the linear
and power-law relationships are quite low
(R2 & 0.1), indicating that neither of the scaling
laws is obeyed for this scatter dataset. These results
allow us to infer an exception for the fault interaction;
the mechanic layering between the lower layers may
play an important role on Dmax–L scaling because of
a highly scattered dataset for the lower reflection
layer. The more highly scattered dataset may be due
to a stronger mechanic layering between the lower
layers.
The lower strata in the Gulf of Mexico have more
salt and evaporites (e.g. BIRD et al. 2005). Salt in the
0 0.5-0.5-1 1
Dis
plac
emen
t on
faul
t A
Strike dimension
Intersction with fault FB
Plane of fault FA
Slip
(dis
plac
emen
t)
(a) (b)
Length
A A´
A A´
Figure 6a Displacement along the strike of fault A (dotted line) intersected by fault B. Displacement increases on the hanging wall of fault B and
decreases on the footwall of fault B. Black line is slip profile of an isolated fault (modified from Maerten et al. 1999). b Possible movement of
D–L points on D–L plot due to fault intersecting
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Gulf of Mexico was coeval with the rift sediments
during the Jurassic age (SALVADOR 1991). Campeche
salt was deposited in the Callovian and mobilized
during the Oligocene–Miocene (ANGELES-AQUINO
et al. 1994; MITRA et al. 2007). Salt in Campeche
Bay was squeezed from diapirs and extruded over the
eroded and uplifted fold belts (GOMEZ-CABRERA and
JACKSON 2009). Salt structures can be triggered by a
variety of mechanisms. In the Cantarell area, salt
might be triggered by compressive stress during the
Miocene. As it is a very weak layer, the salt acts as a
very efficient decollement between the salt contacts.
During deformation, the decollement layer undergoes
simultaneous layer-parallel shear and stretching (e.g.
FORT et al. 2004). The continuous propagation of the
Sihil thrust front in the Cantarell area resulted in the
forward and upward migration of the salt beds
(Fig. 4d).
At the core scale, there is clear evidence for layer-
parallel sliding. There are three types of relationships
between fractures and beddings: fracture crossing bed
contacts, fracture termination at contacts and fracture
incorporation into bedding interfaces (Fig. 8a, b).
Crosscutting fractures (or faults) do not cause contact
sliding. Fracture termination at a bedding contact is
due to slipping or opening along the contact (e.g.
COOKE and UNDERWOOD 2001). Incorporation of
fractures into layer interfaces formed further, larger
normal faults by using the interfaces as parts of their
paths (e.g. GRAHAM et al. 2003; AGOSTA and AYDIN
2006; LARSEN et al. 2010; GUDMUNDSSON 2011). The
linkage of fractures with layer contacts may cause
y = 1.4165x 0.5155
R² = 0.4634
y = 0.0191x + 34.31R² = 0.3758
0
50
100
150
200
250
0 2000 4000 6000 8000 0 2000 4000 6000 8000
0 2000 4000 6000 8000 0 2000 4000 6000 8000
(a)
y = 0.6128x0.5923
R² = 0.436
y = 0.0235x + 12.602
R² = 0.4833
0
50
100
150
200
250
300 (b)
y = 4.8127x 0.4251
R² = 0.3883
y = 0.0271x + 68.483R² = 0.4453
0
50
100
150
200
250
300
350y = 24.383x0.1796
R² = 0.0806
y = 0.0099x + 81.253R² = 0.1065
0
50
100
150
200
250
300(c) (d)
Length (m)
Length (m) Length (m)
Length (m)
Dis
plac
emen
t (m
)
Dis
plac
emen
t (m
)
Dis
plac
emen
t (m
)
Dis
plac
emen
t (m
)
Figure 7Plots of Displacements (D) versus Length (L) from data of one- and no-tip faults of different reflection beddings. a Data from the top of
Cretaceous/Tertiary boundary (S1); b the top of Lower Cretaceous (S2); c the top of Tithonian (S3); d the top of Kimmeridgian (S4)
S. Xu et al. Pure Appl. Geophys.
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contact sliding and/or opening (Figs. 5c, 8b). How-
ever, the slickensides on mechanical layers are
indicative of interlayer slip. One set of slickenlines
is visible on the bedding shown in Fig. 8c. Three sets
of slickenlines on a layer contact represent three
events of plane movement (Fig. 8d).
3.2.3 Evidence of Fault Reactivation
The more scattered datasets in the lower layers in
Cantarell shown in Fig. 7 may also be due to stronger
reactivation from more salt deposits in the lower
layers. Salt has a dramatically low yield strength, and
therefore, it is easy to deform under low strain rates
and low differential stresses (e.g. DAVISON et al.
1996). Beddings containing salt, therefore, evolve
and deform more complexly than those where salt is
absent. Nevertheless, salt structures in Campeche Bay
could have produced a withdraw extension during
minor later tectonic events (GOMEZ-CABRERA and
JACKSON 2009).
Three main episodes of deformation occurred in
the Cantarell oilfield (AQUINO-LOPEZ 1999; MITRA
et al. 2005). First, an extension during the Jurassic
to Early Cretaceous resulted in normal faults that
primarily affected Tithonian, Kimmeridgian and
Lower Cretaceous units. Second, in the Miocene,
the stress pattern turned from extensional to com-
pressional and the northwest trending Cantarell
thrust system was formed. The Sihil thrust fault
separates the allochthonous and autochthonous
blocks. Third, during the Pliocene to Holocene
extension, several of the pre-existing Jurassic nor-
mal faults were reactivated and new NS to NW
trending normal faults were formed. The normal
faults were grown under the last extensional tectonic
regime that followed the earlier compressional
regime.
Interlayer sliding
13
(b)
(c) (d)
(a)
Bedding A
Bedding B
2
Figure 8a Minor faults crosscut bedding A but terminate on bedding B. b Two minor normal faults incorporate and use the bedding plane. c The
slickenlines indicate a bedding-parallel sliding. d Three sets of slickenlines on a bedding plane, representing three movements. The order of
movement is 1, 2 and 3
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Evidence for reactivation is observed from the
core samples at the core scale (Fig. 9). In Fig. 9a, the
plane of sliding contains two sets of striations.
Additionally, this sliding plane is crosscut by another
vein-faulted plane. These observations also imply
that three tectonic events occurred. Two sets of
striations on a bedding plane, as shown in Fig. 9b,
indicate two episodes of bed-parallel movement. The
pressure solution seams also represent tectonic prin-
cipal directions (e.g. AGOSTA and AYDIN 2006;
LAVENU et al. 2014). Additionally, tectonic events
can be a result of the crosscutting relationship among
the fractures. For example, the fractures on the plane
in Fig. 9c imply three tectonic episodes. However,
three sets of stylolites are visible in a vertical section
of the core sample (Fig. 9d). The first is a bed-
parallel one, suggesting development under a per-
pendicular r1 to the bed plane. The second set is
perpendicular to and crosscuts the first one, formed in
response to the sequent compressive regime. The
third set is at angle of approximately 30� with the firstone, suggesting a post-compressional extension after
the tilting of the beddings.
3.2.4 Effect of Sampling Resolution
Measurement methods influence the values of the
power-law exponent of the Dmax–L data (e.g. MAN-
ZOCCHI et al. 2009). Many datasets of fault size are
from seismic reflection surveys, primarily from
hydrocarbon exploration (e.g. CHILDS et al. 2003;
PARRISH and ARSDALE 2004) or satellite imagery (e.g.
VETAL et al. 2005). Each sampling method will only
resolve faults above some limit. In the case of 3-D
seismic data, the resolution may be at displacements
of 10 m or more. Therefore, only parts of faults above
1
2
(a)
(b)
(c)
(d)
1
2
3
12
3
2
Figure 9a A fault plane has two sets of slickenlines. One faulted vein crosscuts the fault plane. The width of this figure is 10 cm. b Two sets of
slickenlines represent two events of movement of the fault plane. c Three generations of fractures are visible. d Three sets of stylolites imply
three tectonic regimes. The width of this figure is 8 cm
S. Xu et al. Pure Appl. Geophys.
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this limit can be observed and the true fault length
will be underestimated. PICKERING et al. (1997) argue
that the fault length may be underestimated by
250–1000 m, and thus, faults with observed lengths
of less than a few kilometres are significantly
underestimated.
To study the effect of the sampling resolution, we
analysed the datasets of the two-tip faults in Fig. 5a
by adding 300 and 500 m to the fault length, and the
analysis results are shown in Fig. 10a, b, respectively.
Compared with the results of Fig. 5a, the coefficients
for both linear and power-law relationships do not
evidently change, suggesting weak linear scaling.
Although the power-law coefficients are low and do
not change significantly, the power-law exponents
increase and are closer to 1 with an increase in the
added fault length (Fig. 10c). These results indicate
that the sampling resolution should be the one of the
factors decreasing the Dmax–L power-law exponent.
4. Discussion
4.1. Other factors relating Dmax–L scaling
In addition to those factors mentioned above, the
following can explain the changes of Dmax–L scaling.
(a) During the evolution of an active fault, the
effective Young’s modulus normally decreases
(e.g. GUDMUNDSSON et al. 2013). By contrast, for a
pre-existing fault, the effective Young’s modulus
may increase because of the healing and sealing
of the associated fault rocks and fractures (e.g.
GUDMUNDSSON et al. 2010, 2013). Young’s mod-
ulus of the rocks hosting the faults easily varies
by one or two orders of magnitude. However,
Young’s modulus of the damage zone and core of
the faults gradually changes with development.
(b) Some faults are primarily mode II cracks,
whereas others are mode III cracks. The dimen-
sion along the slip direction is shorter than that
perpendicular to the slip direction. For normal
and reverse faults, the dip dimension is longer
than the strike dimension. For strike-slip faults,
the strike dimension is longer than the dip
dimension. Generally, for the same fault system,
y = 0.0307x + 12.377R² = 0.5167
y = 0.1739x0.7873
R² = 0.4397
0
20
40
60
80
100
120
140
160
180
200 (a)
y = 0.0307x + 5.7692R² = 0.5205
y = 0.0708x0.8928
R² = 0.4447
0
20
40
60
80
100
120
140
160
180
200 (b)
0.50.55
0.60.65
0.70.75
0.80.85
0.90.95
1
0 1000 2000 3000 4000 5000
0 1000 2000 3000 4000 5000
0 100 200 300 400 500 600
(c)
Added length (m)
Length (m)
Length (m)
Dis
plac
emen
t (m
)D
ispl
acem
ent (
m)
Pow
er-la
w e
xpon
ent
0
0.2
0.4
0.6
0.8
1
Pow
er-law cooefficient
Figure 10a Power-law relationship between displacements (D) and length
(L) by adding 300 m to each original length of two-tip faults.
b Power-law relationship between displacements (D) and length
(L) by adding 500 m to each original length of two-tip faults.
c Relationship between power-law exponent (black line), coeffi-
cient (dotted line) and added length
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the fault trace length measured in map view is
different from that measured in the vertical
section. Thus, the Dmax/L ratio from a shorter
dimension is larger than that from a longer
dimension.
(c) The dimension that controls the displacement
is referred to as the controlling dimension
(GUDMUNDSSON et al. 2000). For some faults,
the controlling dimension is the strike dimen-
sion; for others, it is the dip dimension.
According to GUDMUNDSSON et al. (2013), the
fault displacement (D) and length (L) com-
monly obey linear scaling, as shown in the
following equation:
D ¼ 2sdð1þ mÞE
L: ð4Þ
If the controlling dimension is used, Eq. (4) is written
as
D ¼ 4sdð1þ mÞE
R; ð5Þ
where v is Poisson’s ratio, E is Young’s modulus
and sd is the driving shear stress (shear stress
drop). This equation is especially appropriate for
the through-crack mode III model of a seismo-
genic fault. The dip dimension R is the
controlling dimension of the fault displacement
for Eq. (5) (GUDMUNDSSON et al. 2013). The
controlling dimension may alternate between the
dip dimension and the strike dimension with
growth of a fracture (GUDMUNDSSON et al. 2000).
Therefore, the Dmax/L ratio also changes with
time according to Eq. (5).
(d) There are three basic geometric crack models:
through cracks; part-through cracks and interior
cracks (GUDMUNDSSON 2011). In the through
crack model, the cracks cut through the elastic
body hosting them. The part-through cracks
extend partly into the elastic body from its
surface. The interior cracks do not terminate at a
free surface. These three types of cracks result
in different displacements for the same fault
length and same Young’s modulus (GUD-
MUNDSSON 2011). If a Dmax–L dataset includes
three types of cracks, a scatter produces a
change of Dmax–L scaling.
4.2. Energy Considerations on Dmax–L Scaling
and Fracture Growth
4.2.1 Dmax–L Scaling Exponent and Elastic Energy
Fault zones are open thermodynamic systems. A fault
zone receives input energy, primarily elastic energy
from its surroundings, and partly transforms it into
surface energy by fracture propagation and heat due
to friction during fault slipping (GUDMUNDSSON 2014).
For plane strain conditions and model II fractures
(normal faults), the elastic energy is
Ue ¼ED2pA
16ð1� m2ÞLh
; ð6Þ
where Lh = L/2, which is half of the strike dimension
of the slip surface, and D is the maximum or average
displacement (GUDMUNDSSON 2014). If the maximum
displacement is considered, by combining Eqs. (1)
and (6), we obtain
Ue ¼22nc2EL2n�1
h pA
16ð1� m2Þ : ð7Þ
However, the relationship between the maximum
displacement (Dmax) and average displacement (Dav)
is in the form
Dmax ¼ qDav; ð8Þ
where q is a constant of magnitude less than 1,
commonly ranging from 0.6 to 0.7 (XU et al. 2014). If
the average displacement is considered in Eq. (6), by
combining Eqs. (1), (6) and (8), the elastic energy is
in the form
Ue ¼22nc2EL2n�1
h pA
16q2ð1� m2Þ : ð9Þ
Equations (7) and (9) indicate that there is a
positive correlation between the release elastic
energy (Ue) and the Dmax–L scaling exponent (n).
This relationship provides the possibility of analysing
different fault populations using the ‘‘n’’ value,
instead of a single fault using Dmax and L. As we
described above, the physical processes, such as fault
segmentation, mechanical layering and fault reacti-
vation, alter the fault Dmax/L ratio and the scaling
exponent; thus, it is expected that they also affect the
release elastic energy (Ue) during fracturing.
S. Xu et al. Pure Appl. Geophys.
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4.2.2 Population Scaling Exponent of Fracture
Length and Entropy
The power-law distribution of fault size (displace-
ment and/or length) has the form
NðxÞ / x�b; ð10Þ
where N is the number of fractures equal to or larger
than ‘‘x’’ size and b is the scaling exponent. We can
obtain the scaling exponents from maximum dis-
placements (b1) and lengths (b2). Then, the Dmax–L
scaling exponent (n) can be estimated by (MARRETT
and ALLMENDINGER 1991; XU et al. 2006)
n ¼ b2=b1: ð11Þ
From natural data, where the maximum displace-
ment and trace length populations are given, the
values of n can be either larger or less than b2/b1 dueto different deviations in the measured fault size (XU
et al. 2006).
GUDMUNDSSON and MOHAJERI (2013) showed a
positive linear correlation between the population
scaling exponents (b2) or length range (the difference
between the longest and the shortest fracture) and the
associated entropies. This correlation is explained
because the power-law size distributions of fractures
are a consequence of energy requirements during
fracture growth. As the fracture network grows, the
material damage increases and more fractures form
and link together and the scaling exponent of the
fracture population increases, as does the energy
release and the estimated entropy of the fracture
population (XIE 1993; LU et al. 2005; GUDMUNDSSON
and MOHAJERI 2013).
5. Conclusions
The published displacement–length datasets for
the various types of geologic faults in different
regimes indicate that, in some cases, fault displace-
ment and length obey a sublinear scaling relationship
with a power-law exponent between 0.75 and 0.35.
This sublinear scaling feature has been found in the
cases of opening-mode and closing-mode fractures.
In this paper, we first explained the sublinear scaling
relation where the value of n deviates from 1 by using
some physical processes of faulting. Multiple physi-
cal factors play a role in the alteration of the
D–L scaling relation. These factors include the fol-
lowing: (1) Fault relay structures—Fault relay
structures play an important role in fault growth. The
Dmax–L ratio may increase for segmented faults. As a
result, fault segmentation may decrease the Dmax–L
scaling exponent; (2) The restriction of fault propa-
gation by mechanic layers—Vertical restriction due
to mechanic layering decreases the displacement to a
long-strike length ratio and alters the Dmax–L scaling
relation and (3) Fault reactivation—Fault reactivation
can increase or decrease the displacement/length
ratio. It is one factor in the decrease of the Dmax–L
scaling exponent.
To study the effects of all factors mentioned
above, we analysed the data from the normal faults
of the Cantarell oilfield in the southern Gulf of
Mexico. The results of the analysis indicate that only
two-tip faults obey weak linear or weak power-law
sublinear scaling with n & 0.5. We further docu-
mented that sublinear scaling (n & 0.5) may be
derived from a combination of physical processes
such as fault segmentation, mechanical layering, and
fault reactivation. Nevertheless, sampling effects,
such as technique resolution, strongly affect Dmax–L
scaling. This studied example indicates that an
individual dataset may be subject to errors in mea-
surement and deviations to the ideal faulting
conditions due to special physical processes.
Accordingly, sublinear scaling may occur in shear
fractures, such as normal faults, with the exception
of opening-mode or closing-mode fractures such as
joints, veins and so on.
Acknowledgments
This work was supported by the PAPIIT Project
IN107610, the Conacyt projects 08967 and 80142,
and the Sener-Conacyt Project (No. 143935). The
helpful comments from an anonymous reviewer are
appreciated. Also, the authors thank A. GUDMUNDSSON
for his comments on an earlier version of the
manuscript.
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