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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

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.

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

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.

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

(a)

(c)

(b)

(d)

(e)

S. Xu et al. Pure Appl. Geophys.

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

Effect of Physical Processes and Sampling Resolution on Fault Displacement

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.

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

Effect of Physical Processes and Sampling Resolution on Fault Displacement

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.

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

Effect of Physical Processes and Sampling Resolution on Fault Displacement

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.

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

Effect of Physical Processes and Sampling Resolution on Fault Displacement

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.

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.

Effect of Physical Processes and Sampling Resolution on Fault Displacement

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(Received March 17, 2015, revised August 11, 2015, accepted September 2, 2015)

S. Xu et al. Pure Appl. Geophys.

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