The competition between folding and faulting in...

Proc. R. Soc. A (2012) 468, 1280–1303doi:10.1098/rspa.2011.0392

Published online 6 January 2012

The competition between folding and faultingin the upper crust based on the maximum

strength theoremBY G. KAMPFER AND Y. M. LEROY*

Laboratoire de Géologie, CNRS, École Normale Supérieure, Paris, France

It is proposed to complement the numerous geometrical constructions of fault-relatedfolds relevant to fold-and-thrust belts by the introduction of mechanical equilibriumand of the rock limited strength to discriminate between various deformation scenarios.The theory used to support this statement is the maximum strength theorem that isrelated to the kinematic approach of limit analysis known in soil mechanics. The classicalgeometrical construction of the fault-propagation fold (FPF) is proposed for illustrationof our claim. The FPF is composed of a kink fold with migrating axial surfaces ahead ofthe region where the ramp propagates. These surfaces are assigned frictional propertiesand their friction angle is found to be small compared with the usual bulk friction angleto ensure the full development of the FPF, a first scenario. For larger values of the axialsurface friction angle, this development during overall shortening is arrested by the onsetof fault breaking through the front limb, a second scenario. The amount of shortening atthe transition from folding to break-through faulting is established.

Keywords: fault-propagation fold; faulting; maximum strength; limit analysis;fold-and-thrust belts; optimization

1. Introduction

Some fault-propagation folds (FPFs) become detached, and their growth arrested,by the development of underlying break-through faults (BTFs), while othersdo not. The difference between these two scenarios observed in nature is usedto interpret the frictional properties at the time of deformation of sedimentaryrocks. The method considered to compare these scenarios relies on the maximumstrength theorem combined with the geometrical construction proposed bySuppe & Medwedeff (1984). Such coupling with mechanics could be applied tonumerous geometrical constructions of fault-related folds found in the literature.

The historical development of the subject of fault-related folds from thegeological point of view (Buil 2002) can be traced back to Willis (1893) and tothe idea that faulting occurs at the latest stage of deformation of folds, resulting

*Author for correspondence ([email protected]).

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2011.0392 or viahttp://rspa.royalsocietypublishing.org.

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Folding and faulting in the upper crust 1281

from bending rupture of the most stretched plane. The presence of ramps linkingweak horizons or décollements before any folding were discovered by Rich (1934)in a work in the Cumberland Mountain (USA), proving that the order of causeand effect, folding before faulting, could be reversed. Concomitant folding andfaulting were recognized by Dahlstrom (1970) in a study of the Canadian RockyMountains. Weak horizons and the concomitant developments of the ramp andthe fold formed the basis of geometrical constructions of fault-bend folds (Suppe1983) and FPFs (Suppe & Medwedeff 1984, 1990). The latter construction (FPF)is central to this contribution.

The initial work of Suppe & Medwedeff (1984) has attracted a lot of attentionfrom the structural geology community and the following review is certainly notcomplete. Jamison (1987) amended the geometrical rules to produce forelimbthickness changes motivated by a comparison of the final fold geometries andfield examples. Chester & Chester (1990) extended the FPF construction to startfrom a ramp of finite length. Mosar & Suppe (1992) proposed to include simpleshear within the internal layers of the FPF, resulting in a rotation of the kinkfold ahead of the propagating fault. The generalization to three dimensions of theFPF construction with lateral variations in the fault geometry and slip rates wasproposed by Wilkerson et al. (1991). Salvini & Storti (2001) studied the migrationof the axial surfaces in terms of local history of the activated deformationmechanisms, preparing the grounds for a comparison between field observationsand theoretical predictions.

Mechanical analyses should provide some insight on the governing propertiesand processes controlling faulting and folding. Hill & Hutchinson (1975) haveextended the seminal work of Biot (1965) and established the bifurcationconditions for strain-rate independent elasto-plastic materials, relevant to thefolding of brittle rocks in the upper crust. The local conditions for the onset ofshear banding or faulting are essentially due to Rudnicki & Rice (1975). Thebifurcation conditions applied to layered structures provide the stress conditionsfor the onset of folding, which can only prevail if the shear banding conditionsare not met (Triantafyllidis & Leroy 1997). Computational methods are requiredto follow the fold and fault developments, as carried out by Barnichon & Charlier(1996) to reproduce the thrusting in sand boxes and Cardozo et al. (2003) toinspect the velocity field ahead of a propagating fault. The difficulty of thenumerical approaches is that the solution search occurs close to or beyond thelocal conditions of shear banding for which the strain is localized on a lengthscale not resolved by the discretization. New algorithms are required to adaptthis discretization (Braun & Sambridge 1994) or a characteristic length shouldbe included, as with the discrete-element technique (Hardy & Finch 2007).

The approach followed here is very different from these numerical methodsand relies on the overall shape of the structures that is proposed by thestructural geologists based on simple geometrical rules. The structure shape isindeed not sought as the solution of boundary-valued problems. Faults areintroduced explicitly and most of the calculations remain analytical withoutthe need for numerical interpolation. The new idea of this contribution, comparedwith Maillot & Leroy (2006) and Kampfer & Leroy (2009), is to show that anygeometrical construction of folding could become mechanically balanced in thesense that mechanical equilibrium and material strength are accounted for. Thenow classical example of the FPF is considered for that purpose. Two sets of

Proc. R. Soc. A (2012)



1282 G. Kampfer and Y. M. Leroy

rules are proposed for the construction of this asymmetric fold (Jamison 1987;Suppe & Medwedeff 1990), and we choose here the set based on conservationof bed thickness and bed length. The deformation has to be accommodatedby bedding plane slip as material crosses axial surfaces of the fold. The foldis characterized by a kink fold, bounded by several propagating axial surfaces,ahead of the region where the ramp propagates.

Concepts of mechanics required to complement the geometrical constructionsare included by application of the maximum strength theorem. It corresponds tothe kinematic approach of the limit analysis (Salençon 1974, 2002) developedinitially for predicting bearing capacities of civil engineering structures. Thistheorem relies on a weak form of the equilibrium equations and on the existenceof the strength domain, convex in the appropriate stress space. This domain isthe set of admissible stresses that can be sustained by a given material. Thecase of frictional and cohesive materials is considered here, and the strengthdomain is bounded by the Coulomb criterion. These two concepts, mechanicalequilibrium and maximum material strength, provide an upper bound of theinternal dissipation and subsequently of the applied force.

The paper contents are as follows. Section 2 is devoted to the presentationof the geometrical construction of the FPF, which is further analysed in theelectronic supplementary material, §1 and §3. It is shown that the velocity fieldassociated with the FPF could be estimated in a rather systematic manner withthe help of Hadamard’s jump condition (electronic supplementary material, §2).This condition relates the differences in velocities and in transformation gradientsbetween the two sides of the discontinuity and its propagation (non-material)velocity. Section 2 also contains a description of the geometry of the BTF. Themaximum strength theorem, which is presented with the help of a simple examplein the electronic supplementary material, §4, is applied to the development of theFPF and its potential arrest by BTFs in §3. Analytical expressions are providedfor the upper bounds of the forces necessary for the FPF and the BTF and themaximum strength theorem ensures that the smaller of two loads determineswhich deformation mode dominates. Also, restrictions on the geometry of theFPF based on the frictional properties of the axial surfaces are established. Thecompetition between FPFs and BTFs is explored in §4 by comparing the leastupper bound to the force for BTFs and FPFs. An estimation of the amount ofshortening the FPFs can accumulate prior to the onset of the BTFs is proposed.In particular, it is shown that the friction angle of the fold axial surfaces is thekey parameter to determine this critical shortening.

2. The geometry

The objective of this section is to describe the geometry of the FPF during itsdevelopment and also the geometry of the BTF, which could occur at any stepof this development.

(a) The geometry of the fault-propagation fold

The two-dimensional structure includes an initially horizontal, competentlayer A′′OG ′B ′′ of length L, thickness Hs composed of a material of density rs





(figure 1a). The segment GG ′ acts as a décollement. Shortening is initiated bydisplacing the back wall towards the left. An inviscid fluid-like material of densityrf is on top of the competent layer to account for the effect of burial. This fluid-likematerial may represent poorly consolidated sediments such as shales: its weightis important for the underlying rock, but it cannot resist the lateral tectonicforces. The fluid-like layer has a constant thickness Hf with time because it isconnected to an external reservoir of infinite extent, illustrated on the left-handside of figure 1a.

The ramp dips at the pre-determined angle gB and emanates from point G;its length is zero at the onset. The material to the right of this point G, in theback-stop, is displaced towards the left by the quantity d increasing with timeand denoting the shortening at the back wall. The left boundary of the back-stopis the axial surface B. This surface is dipping at the angle qB , which is half thecomplementary angle of gB (2qB = p − gB). This assumption on the angle qB isclassical in the geometrical constructions of folds and implies the conservation ofthe thickness of the beds crossing the axial surface. Material points coming fromthe back-stop and crossing the axial surface B are entering the region GBB ′RS,which is the sub-domain HW1 of the hanging wall. Note that all the geometricalattributes are defined in table 1.

The development of the fold is marked by the propagation of the ramp withthe same constant dip gB . Point S in figure 1b marks the limit of the materialinitially on the décollement now found on the ramp after shortening by d. Thedotted line SS ′′ dipping at qB was initially on the axial surface B and is boundingto the left by all the material coming from the back-stop and now in the hangingwall. The selection of the angle qB to conserve bed thicknesses means also thatthe distance SG is equal to the shortening d. The ramp has propagated furtherthan this distance and its tip is now at point P. The bed passing through pointP is marked by the dashed line in figure 1b. Material points above and belowthis line have different kinematics. Region P ′PRS ′S ′′B ′AA′A′′ deforms in a kinkmode by the propagation of axial surfaces. The axial surface A′ is dipping at qA,an angle chosen to be half the dip of the layering within the kink band PRAA′,a choice made again to preserve bed thickness (2qA = p − gA). Axial surfaces Aand B′ are parallel to the surfaces A′ and B, respectively. The relation betweenthe angle gA and the propagation ratio k = PG/SG in terms of gB is provided inthe electronic supplementary material, §1.

The velocity field of the FPF is also presented in the electronic supplementarymaterial by applying in a systematic manner Hadamard’s jump condition fornon-material interfaces. This condition is found particularly useful to justify thatthe sub-domains HW1 and HW4 of the hanging wall have the same velocityand thus that the segment RS is not a dissipative interface. This information isimportant for the determination of the upper bound of the force necessary forthe FPF development.

(b) The geometry of the break-through fault

The BTF is defined by the instantaneous propagation of the ramp GP, whichis extended by the new fault, segment PP ′, upon further shortening of the wholestructure (figure 1d). The dip of this last segment could differ from the dip ofthe fold ramp gB and is denoted as gBTF. The faulted structure is composed





A B

e2

e1

HW1

HW3

HW4

(a)

(b)

(c)

(d)

propagation of the ramp

B

A

B

A

8

back

wal

l

= ' = '

A'' A=A' B=B' B''

G'G=S=P=Q=R

L

Hs

Hf

A''

P'

A B''

RS''

SP

A' B B''

Q'

G'G

Q

L–

h

Hs

Hf

A'

A B'

B B''

G'GO

G'GO

S

S

P

R

foot wall

foot wall

ramp

hanging wall

back-stop

back-stop

HW2

S'

D

O

O

8

B

B

HWlower

A'

A B'

B B''

P ramp

hanging wall

HWupperP' G''

P''

A''

A''

BTFl

BTFu

BTF

d d

back

wal

l

'

Figure 1. (a) The geometry of the structure at the onset and (b) during the development of theFPF. (c) Definition of the adopted terminology and of the four sub-domains composing the hangingwall of the FPF. (d) The geometry of the BTF, which could arrest the FPF development.





Table 1. Material properties and geometrical characteristics of the FPF and the BTF. The interfacesand axial surface cohesions are not indicated; they are used in the definitions of the upper boundsand denoted by the letter C , with the same subscript as that of the friction angle. The data thatare varied are indicated as ‘var.’.

symbol value unit

FPF

d shortening var. mh current height of ramp tip P var. mgB ramp dip var. ◦qB axial surfaces B and B′ dip var. ◦gA interface PR dip var. ◦qA axial surfaces A and A′ dip var. ◦f∗ friction angle of the axial surfaces var. ◦fB friction angle of the bedding 10 ◦

BTFgBTF break-through fault PP ′ dip var. ◦qBTFu upper axial surface dip var. ◦qBTFl lower axial surface dip var. ◦fs friction angle of competent layer 30 ◦

common propertiesL initial length of structure 1500 mHs thickness of competent layer 250 mHf thickness of fluid-like layer var. mg gravity acceleration 9.81 m s−2

rs material density for solid layer 2500 kg m−3

rf material density for fluid-like layer 2000 kg m−3

fi friction angle of a generic interfacefR friction angle of ramp GP 20 ◦fD friction angle of décollement GG ′ 10 ◦

of four regions, the back-stop (region GG ′B ′′G ′′ denoted BS), the lower andupper hanging walls (region PGG ′′P ′′ and region P ′PP ′′ denoted HWl and HWu,respectively) and the foot wall (region OGPP ′A′′ denoted FW). The boundarybetween the upper and the lower parts of the hanging wall is the axial surfacePP ′′ dipping at the unknown angle qBTFu. The transition from the back-stop tothe hanging wall is the axial surface GG ′′ dipping at the unknown angle qBTFl,which could differ from qB (the FPF axial surface) because the points G ′′ and Bdo not necessarily coincide.

3. Bounds of the applied forces

The objective of this section is to obtain the upper bounds of the force necessaryfor the development of the FPF and the BTF. These bounds are obtained byapplication of the maximum strength theorem that is presented in the electronicsupplementary material, §4, with the help of a simple example, the sliding ofa block up an inclined ramp. This example, corresponding to the geometry ofthe central region of our structure, should help the reader understand why the





sn

p >

n

U

U

U

nsn

(a)

(b)

(c) t

fi

h > p/2− fi

h = p/2− fi

h < p/2− fi

case 3

case 2

case 1

+

Ci

−

− +

T

t

t

t

h

Figure 2. (a) The stress vector acting on an interface i and (b) the construction of the velocitydiscontinuity oriented by the angle h from the normal. (c) The strength domain is bounded bythe Coulomb criterion. The internal power or dissipation on the interface is the scalar product ofthe velocity jump and the stress vector. Its maximum value is the function of the angle h andthree cases are considered. This maximum power or support function is infinite for case 3, and thevelocity jump should be oriented according to case 1 or 2 to be pertinent.

exact velocity field is not a pertinent velocity field for the application of themaximum strength theorem. This point is discussed in §3a. Section 3b,c presentsthe expressions for the bounds for the FPF and the BTF. Section 3d reveals thatFPFs may be impossible if the axial surfaces are assigned large friction angles.

(a) Strength domain and support function

Dissipation occurs over ramps, axial surfaces and décollement, which arevelocity discontinuities within the structures or on their boundaries. The set ofthese surfaces is denoted SU . Each member of this set labelled i is a Coulombinterface and has a limited strength defined by the domain

Gi = {T | |t| + sn tan fi ≤ Ci}, (3.1)

in which fi and Ci are the friction angle and the cohesion, respectively. The stressvector T in (3.1) is the sum tt + snn, in which t and sn are the shear and thenormal components in the orthonormal right-handed basis {n, t}, where n is thenormal vector (figure 2a). Note that the adopted sign convention is that stressesin tension are positive.

We shall work with kinematically admissible (KA) velocity fields that, bydefinition, satisfy the essential boundary conditions. These conditions in ourproblem stipulate that the foot wall remains at rest and that the materialpoints of the back-stop in contact with the back wall are displaced at the samevelocity. The set of KA velocity fields includes the exact field. A KA field isdenoted by a superposed hat: U . The velocity jump over any dissipative interface[[U ]] ≡ U

+ − U−

is defined as the difference between the velocity on the + and− side of the discontinuity (the normal points towards the + side), figure 2b.





The exact stress field required to define the internal power [[U ]] · T is unknown,and no attempt is made to determine it in this contribution. It is proposed to findan upper bound of the internal power and, for that purpose, we search for themaximum power that can be dissipated at every point along the surfaces in SU .It can be shown (Salençon 1974, 2002 and more recently Maillot & Leroy 2006)that there is indeed a maximum power because the strength domain Gi in (3.1) isconvex in the two-dimensional stress space. This maximum is called the supportfunction and is defined for the Coulomb strength domain by

case 1 : 0 ≤ |h| < p

2− fi , pi([[U ]]) = JCi cotan(fi) cos h,

case 2 : |h| = p

2− fi , pi([[U ]]) = JCi cos fi ,

and case 3 :p

2− fi < |h| ≤ p, pi([[U ]]) = +∞,

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭

(3.2)

in which J is the norm of the velocity jump on the interface of interest. Threecases are defined in (3.2) and illustrated in figure 2c depending on the angle h

between the velocity jump and the normal ([[U ]] = J (cos hn + sin ht)). Cases 1and 2 are of interest because the support function is finite. The velocity jumpvector for those two cases is oriented within (case 1) or at the boundary (case 2)of the cone of direction n and internal angle p/2 − fi . Choosing a velocityjump outside this cone leads to an upper bound that is infinite and thus of nointerest, case 3 in (3.2). The velocity jumps that are oriented within or at theboundary of the cone are the only ones of interest and are said to be pertinent.Note that the exact velocity field is not pertinent for our analysis since in thatinstance h = p/2.

The support function is integrated over all the interfaces composing SU toobtain the maximum, resisting power that is bounding the unknown, internalpower. The theorem of virtual power (Malvern 1969) stipulates that theinternal power and the external power are equal for any KA velocity field. Itis the combination of this latter theorem and the bounding proposed abovethat constitutes the maximum strength theorem and provides the boundsof the applied force. There is an optimum velocity field (uniqueness is notensured) that yields the least upper bound of the tectonic force. This optimumvelocity field defines the dominant collapse mechanism, a term often used instructural engineering. In this contribution, two collapse mechanisms are studied,the FPF and the BTF. The interested reader is referred to the electronicsupplementary material, §4) to find further information on this theorem andalso the expressions for the maximum resisting power and the external powerfor the two collapse mechanisms. Only the relevant KA velocity fields andthe expression for the upper bounds of the tectonic forces are provided inwhat follows.

(b) Faulting breaking through the fault-propagation fold

The boundaries between the different regions of the faulted structure definedin §2b together with the segmented ramp and the décollement are dissipativeinterfaces and they constitute the set of surfaces SU .





qBTFlgB + fR

qBTFu

UBS

UHWu

UHWl

U = 0FW

UHWu

UHWl

UBS JGG"

qBTFuJPP"

UHWl

qBTFl

g

gB

BTF

A B'

B

GO

P

(a) (b)

QBTF

G'

BTFg + fs

fDG

+

+

P+

+

fs

fs −

−

P' B"A'

− −

A"

P"

G" G" P"

Figure 3. (a) The KA velocity field for the BTF and (b) the hodograms of the velocity jumps acrossthe axial surfaces GG ′′ and PP ′′.

The KA velocity field of the faulted structure is constructed as follows, withrespect to the foot wall that is at rest (U FW = 0, figure 3a). The velocity UBSof the back-stop is oriented to be pertinent with respect to the décollement GG ′.Its angle h with respect to the normal to the décollement needs to be less thanor equal to p/2 − fD. It is proposed to set h = p/2 − fD, case 2 for the supportfunction in (3.2), to minimize the power against gravity, as shown in the electronicsupplementary material of Cubas et al. (2008). The same reasoning is applied tothe lower and the upper parts of the hanging wall. The velocity UHWl is orientedin the direction dipping at gB + fR, and the upper part is assigned a velocityUHWu dipping at gBTF + fs. The subscripts ‘D’, ‘s’ and ‘R’ to the friction anglesare referring to the décollement, the solid layer and the ramp, respectively. Theramp could have a lower friction than the bulk material of the solid layer becauseof damage accumulation upon shortening. These three velocities (UBS, UHWu andUHWl) define the KA field for the BTF and are presented in figure 3a, includingtheir orientations.

Attention is now turned to the intensity of these three velocity vectors. Theintensity of the back-stop velocity is arbitrary and will be used later on as anormalizing factor. The intensity of the velocity in the lower part of the hangingwall is determined by inspection of the axial surface GG ′′, which is assigned thefriction angle fs as for the new part of the ramp. This interface accommodatesthe jump in virtual velocity (UHWl − UBS) = JGG ′′ , a vector that is oriented tobe pertinent according to case 2 of the support function (3.2), as illustrated in thehodogram of figure 3b. The law of sines applied to this triangular constructionprovides

U HWl

sin(fD + qBTFl + fs)= U BS

sin(gB + fR + qBTFl + fs)

= J GG ′′

sin(fR + gB − fD). (3.3)

The same reasoning is considered for the upper region of the hanging wallseparated from the lower part by the axial surface PP ′′. The velocity of the upper





region UHWu makes the angle fs with the ramp and its norm U HWu is chosen suchthat the jump in velocity across the axial surface PP ′′ is also oriented accordingto case 2 of the support function for the friction angle fs. The correspondinghodogram is presented in figure 3b and reveals

U HWu

sin(gB + fR + qBTFu + fs)= U HWl

sin(gBTF + 2fs + qBTFu)

= J PP ′′

sin(gBTF + fs − gB − fR). (3.4)

We now apply the maximum strength theorem, and the expressions for themaximum resisting power and the external work are found in the electronicsupplementary material, §4. The upper bound of the force necessary for theBTF reads

Q ≤ QBTFu , (3.5)

with

QBTFu cos fD = (pA′LP ′A′ sin(fs + gBTF)

+ pA′ + pA

2LAA′ sin(fs + gBTF − gA) + pALAB sin(fs + gBTF)

+ pB′ + pP ′′

2LB′P ′′ sin(fs + gBTF − gB)

)U HWu

+(

pP ′′ + pB

2LP ′′B sin(fR) + pBLBG ′′ sin(fR + gB)

)U HWl

+ pBLG ′′B′′ sin fD + SHWuU HWursg sin(fs + gBTF)

+ SHWlU HWlrsg sin(fR + gB) + SBSU BSrsg sin(fD)

+ U HWuCs cos(fs)LPP ′ + J PP ′′Cs cos(fs)LPP ′′

+ U HWlCR cos(fR)LPG + J GG ′′Cs cos(fs)LGG ′′

+ U BSCD cos(fD)LGG ′ ,

in which the jumps and the velocities with a superposed tilde have been obtainedby the normalization with the arbitrary velocity of the back-stop U a = U a/U BS.The first four lines and the first term of the fifth line in the expression for the upperbound are due to the external work and correspond to the work of the velocityfield on the fluid pressure at the boundary between the competent and the fluid-like layer. The pressure at any given point M is denoted as pM , and the lengthbetween points M and M ′ as LMM ′ . The pressure at any point M of the top ofthe competent layer is rfgDM , where DM is the depth of this point, measuredfrom the stress-free top surface. The depth of each point is found directly fromthe geometrical description of the FPF given in §2. The second term in the fifthline and the next line in the expression for the upper bound in (3.5) correspondalso to the external work: the work of the velocity field on the gravity field.Surfaces are denoted by S , and the subscript identifies the sub-domain considered.





The last three lines correspond to the maximum resisting power over the fivediscontinuities composing the set SU . All these interfaces are in case 2 of thesupport function. Note that equation (3.5) was set up for the faulting eventdefined in figure 3a. The new fault extending the ramp reaches the competentlayer at point P ′ between A′′ and A′. Similarly, points P ′′ and G ′′ are presented tothe right of points B ′ and B, respectively. The results presented in the next sectioncould rely on different geometries, and proper modifications of this expression areaccounted for in a straight-forward manner, which is not discussed further here.Note also that the contribution of the fluid-like pressure on the back wall overthe depth Hf has been disregarded from the bound in (3.5) for the BTF. Thereason is that the same term contributes to the least upper bound determinednext for the FPF, and what is of interest is not the exact value of these bounds,but their difference.

Note finally that (3.5) is an upper bound that can be minimized by selectingthe values of the the dips qBTFu, qBTFl and gBTF. The minimum upper bound isreferred to as the least upper bound.

(c) The least upper bound for the fault-propagation fold

The frictional properties of the interfaces composing the FPF are firstdiscussed. The ramp has the friction angle fR assumed again to be lower thanthe friction angle of the pristine material because of damage accumulation.It is also assumed that the competent rock is layered. The rock composingeach layer has the friction angle fs considered in the previous section. Thelayers are separated by weak interfaces corresponding to the bedding planes,typically composed of shales and having the smaller friction angle fB ≤ fs.These weak planes facilitate the activation of the axial surfaces, and a detaileddiscussion of the axial surface mechanism is given by Maillot & Leroy (2003). Itis assumed here that the effective properties of the axial surface are cohesiveand frictional, the friction angle being denoted as f∗ and assumed smallerthan fs.

The velocities of the back-stop and of the region HW1 in the hanging wall areconstructed similar to the velocities of the back-stop and of the lower hangingwall of the BTF, respectively. The velocities UBS and UHW1 are oriented by theangles fD and fR from the directions of the décollement GG ′ and of the ramp,respectively. The norm of UHW1 is found from the application of the law of sinesto the hodogram for the velocity jump across the surface B, of norm JB, presentedin figure 4b,

U HW1

sin(fD + qB + f∗)= U BS

sin(gB + fR + f∗ + qB)

= JBsin(fR + gB − fD)

. (3.6)

The velocity of the region HW3 of the hanging wall is oriented by the angle f∗from the axial surface A′. Its norm is found by requiring that the velocity jumpover the axial surface PR dips at fB + gA. It is indeed the bedding friction angle





= 0

UHW3

UHW3

UHW2

UHW2

UHW3

UFW

UBS

UHW1

UHW1

UHW1

qA

qA

qB

+ f*

qA + f*

gA + fB

UBS

J

J

JPRqB + f*

qB + f*

gA

UHW2

gB + fR Q

gB

A

P

FPF

~

~

OG G'

(a)

(b)

fD

B'

B

+

+ −

− PR+

+ − −

−

A'' R

− −

UHW1 −

A'

J

'

'

''

'

'

'

Figure 4. (a) The virtual velocity field of the FPF and (b) the hodograms of the velocity jumpsover the axial surfaces B, PR A and B′.

that has to be used here because the surface PR is parallel to the bedding inHW3. The application of the law of sines to the hodogram corresponding to thejump across PR, of norm J PR, presented in figure 4b, yields

U HW3

sin(gB + fR + gA + fB)= U HW1

sin(qA + f∗ + fB + gA)

= J PR

sin(qA + f∗ − gB − fR). (3.7)

The velocity of the triangular region HW2 in the hanging wall is found bystudying the axial surfaces A and B′. The jump in velocity across the axial surfaceA is oriented with the dip qA + f∗. This vector is thus parallel to the velocityvector UHW3 and, consequently, the unknown velocity vector UHW2 has to havethe same direction, as illustrated in figure 4b. The norm of this last vector remainsunknown and is determined by inspection of the axial surface B′. The velocityjump over the axial surface B′ has the norm JB′ and is dipping at qB + f∗. The





corresponding hodogram is presented in figure 4b and the application of the lawof sines reveals

U HW2

sin(gB + fR + qB + f∗)= U HW1

sin(qB + 2f∗ + qA)

= JB′

sin(qA + f∗ − gB − fR), (3.8)

providing the unknown U HW2. The velocity field just constructed only existsunder certain constraints that are deduced from the four hodograms of figure 4b.These constraints will be presented in §3d.

The maximum strength theorem is now applied, and the expressions for themaximum resisting power and the external work are presented in the electronicsupplementary material, §4. The least upper bound is

Q ≤ QFPFlu , (3.9)

with

QFPFlu cos fD = pA′ + pA

2LAA′ sin(f∗ + gA + qA)U HW3

+ pALAB′ sin(f∗ + qA)U HW2 + pB′ + pB

2LB′B sin(f∗)U HW1

+ pBLBB′′ sin(fD)U BS + SHW3U HW3rsg sin(f∗ + qA)

+ SHW2U HW2rsg sin(f∗ + qA) + SHW1U HW1rsg sin(fR + gB)

+ SBSrsg sin(fD) + U HW3C∗ cos(fs)LA′P + J PRCB cos(fB)LPR

+ JAC∗ cos(f∗)LAR + JB′C∗ cos(f∗)LRB′ + U HW1CR cos(fR)LPG

+ JBC∗ cos(f∗)LGB + U BSCD cos(fD)LGG ′ ,

in which again all velocities have been normalized by the arbitrary velocity of theback-stop. The bound in (3.9) is the least because there is no degree of freedom,either in the geometry or in the velocity field that could be optimized.

(d) Locking of the fault-propagation fold

This section is devoted to the constraints due to the construction of the KAvelocity field of the FPF. The hodograms presented in figure 4b were plotted forrealistic orientations of the sense of shear on all the interfaces. The reversal of asense of shear would not be consistent with the development of the structure.It is said that in this instance, the FPF would lock. The condition for thislocking to occur is, typically, that two of the three vectors used for one of thehodograms become linearly dependent. These conditions are now presented andthe consequences for the development of the FPF discussed.





0 10 20 30 40 50 60

ramp dip, gB (º)

15

30

45

60

75

axia

l sur

face

fri

ctio

n an

gle,

f*

(º)

fB = 0∞fB = 0∞

fB = 10∞fB = 10∞

fB = 20∞fB = 20∞

for

PR2

PR1 admissible domainf*

'

Figure 5. The various constraints obtained from the hodograms of the FPF KA velocity jumps(figure 4b) defining an admissible domain for the selection of the ramp angle and of the axialsurface friction angle.

For the hodogram for the axial surface B in figure 4b, the three conditions are

fD − fR ≤ gB ,

f∗ + fR ≤ p + gB

2

and f∗ + fD ≤ p + gB

2.

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(3.10)

The décollement friction angle is less than the ramp friction angle, and the firstcondition is always met. Small values are expected for f∗ and fR so that thesecond condition is also not restrictive. Consequently, only the third conditionremains. This is presented in figure 5, where the axial surface friction angle isplotted as a function of the ramp dip, as a dotted line labelled with the letter B.The series of barbs mark the region that is restricted.

The second hodogram in figure 4b corresponds to the interface PR, and thethree conditions to ensure its validity are

fR − f∗ ≤ p − gA

2− gB ,

f∗ + fB ≤ p − gA

2and fR + fB ≤ p − gA − gB .

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

(3.11)





The first and second conditions correspond to the dotted curve PR1 and PR2 infigure 5, with the second being plotted for three values of the bedding frictionangle. The third condition is always respected.

The third hodogram in figure 4b for the axial surface A is based on threecollinear vectors and the appropriate sense of shear requires that U HW3 ≥ U HW2.Comparing the second and third hodograms in figure 4b, it is concluded that thiscondition is equivalent to

qB + f∗ ≥ gA + fB . (3.12)

This constraint is presented in figure 5 where it is labelled A and plotted for threevalues of fB .

The fourth hodogram is for axial surface B′ and the first of the three conditionsfor its validity is identical to the first in (3.11). The two others read

f∗ ≤ gA + gB

4= fc

∗

and fR + f∗ ≤ p − gB

2,

⎫⎪⎬⎪⎭ (3.13)

defining the critical friction angle fc∗. The first of these two conditions is the onlyrelevant one and is plotted in figure 5 with the label B′.

These various inequalities define the admissible domain for the couple (gB , f∗)in figure 5. It is found that for a ramp dip between 17 and 29◦, there is a minimumand a maximum value for f∗ to avoid the fold locking. The minimum value iszero for gB ≥ 29◦, and the maximum decreases significantly for larger gB .

The physical interpretation of these constraints is better explained in the spaceof forces than in the KA velocity space. The example of the first constraint in(3.13), which is characteristic of the region HW2, is chosen to illustrate thispoint. For f∗ > fc∗, the forces on the two axial surfaces A and B′ cannot balancethe vertical weight of the region HW2. There is thus a locking of this region ifthe friction angle is larger than the critical, maximum value fc∗. The proof of thisstatement in terms of forces is presented in the electronic supplementary material,§5, with a force balance analysis of the free body diagram of this region HW2.The fact that the restrictions based on forces and on KA velocities are the sameprovides a non-trivial illustration of the equivalence between the force approachand the maximum strength theorem.

4. Competition between folding and faulting

The main outcome of §3 is the estimation of upper bounds QBTFu and of the least

upper bound QFPFlu of the forces necessary for the BTF and for the development

of the FPF, respectively. Only bounds were obtained for the force required for theBTF because several angles necessary to describe this deformation mode and theassociated velocity field are not yet optimized. This section has two objectives.The first consists of conducting this optimization to obtain the least upper boundfor faulting QBTF

lu . The second objective is to compare QBTFlu and QFPF

lu to decideon the dominant mode, faulting or folding.





(a) Optimizing the break-through fault

The need to optimize the BTF is first assessed by considering the upperbound QBTF

u obtained for a fault PP ′ extending the initial ramp GP with thesame dip (gBTF = gB) and the upper and lower axial surfaces coinciding withthe segments PB ′ and GB, respectively (qBFTu = gA, qBFTl = gB). The optimizedBTF is obtained by numerical means searching among all possible orientationsgBTF, qBTFu and qBTFl the set of three angles leading to the smallest upperbound according to (3.5). Results, presented in Kampfer (2010) show that thereis a reduction by 45 per cent of the bound by optimizing the BTF at everystep of the development of the fold and in the absence of the fluid layer. Themagnitudes of these forces are rather independent of the amount of shortening.This reduction between the upper and the least upper bound for thrustingis of the order of 15 per cent only for a 500 m thick overburden. This issimply the consequence of the dominant role played by the fluid-like pressureon the top of the competent layer. These numbers lead to the conclusion thatoptimization of the BTF is necessary to warrant a precise comparison withthe FPF.

The geometry of the optimized BTF is presented in figure 6 with dotted, solidand dashed curves, for gBTF, qBTFu and qBTFl, respectively. The dimensionlessshortening is d = d/L. Results were obtained with gB , the fold ramp dip, beingset to 30◦. The three angles are found to have the same value, corresponding top/4 − fs/2, for zero shortening. This result is not influenced by the friction angleof the décollement nor by the length L − D of the back-stop. An analytical proof ofthis surprising result is presented in the electronic supplementary material, §6. Itis during the fold development that these various angles change drastically. Thecase with no overburden (Hf = 0) in figure 6a is first discussed. The evolutionof the angles is partitioned in two phases. In phase 1, the upper axial surfaceand the thrust ramp are conjugate faults with the same dip decreasing to reachthe critical value of 20◦, marking the beginning of phase 2. At the transitionand in phase 2, the upper hanging wall velocity UHWu and the velocity ofthe lower part of the hanging wall UHWl (see figure 3 for these definitions) areidentical. It implies that gBTF + fs = gB + fR and means that the hanging wallis acting as a single region. Thus, the upper axial surface does not exist anymore in this phase 2, as it can be seen by the two insets in figure 6a. In phase2, the thrust ramp keeps the same dip. The lower axial surface dip does notappear to be influenced by the transition between these two phases. Its dipdecreases steadily during the fold development until it intersects the back wall,resulting in the kink in the curve of figure 6a for the dimensionless shortening of0.15, approximately.

The evolution during the FPF development of the three dips of the BTFgeometry is more complex in the presence of an overburden, figure 6b, and ispartitioned into four phases. Phase 1 is identical to the one described above,except that the lower axial surface is now increasing its dip. In phase 2, theupper axial surface and the new segment of the ramp are not conjugate any morein terms of dip, the upper axial surface dip decreases more than the new ramp dip.It is found that the upper axial surface intersects the top of the competent layerexactly at the boundary between the back-stop to the hanging wall of the fold(point B in figure 3). In phase 3, the selection of this special point B ceases, and





0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

20

25

30

35

40

45

optim

ized

ang

les

for

the

BT

F (°

)

20

25

30

35

40

45(a)

(b)

optim

ized

ang

les

for

the

BT

F (º

)

1

4

2 3

1 2 3 4

o

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

normalized shortening, d ~


gBTF

gBTF

1 2qBTFu

qBTFu

qBTFu

qBTFl

qBTFl

qBTFl

oHf = 0 m

Hf = 500 m

phase 1 phase 2

gBTF

Figure 6. The optimized dip of the fault (gBTF) extending the ramp, the lower and the upper axialsurface dips (qBTFl and qBTFu) of the BTF during the shortening of the FPF. The fluid-like layerthickness in (a) 0 and (b) 500 m, respectively. The two insets in (a) and the four in (b) illustratethe geometry of the optimum BTF at different phases of the FPF development.

the upper axial surface remains within the hanging wall. Its dip increases sharply.During the same phase, the dip of the new segment of the ramp continues todecrease to reach the conditions gBTF + fs = gB + fR, for which the upper axialsurface does not exist. Phase 4 is thus identical to phase 2 in figure 6a where thehanging wall of the BTF behaves as a single region.





0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.050

0.075

0.100

0.125

0.150

0.175

0.200

0.225

norm

aliz

ed te

cton

ic f

orce

, Q ~

Hf = 500 m

Hf = 200 m

Hf = 100 m

Hf = 0 m

Hf = 300 m

Hf = 400 m

o

o

o

o

o

o


Figure 7. The least upper bounds for the FPF (solid curves) and for the BTF (dashed curves) fordifferent values of the fluid-like layer thickness. The circles indicate the intersection of the twocurves and signal the end of the fold development.

(b) The dominant mode of deformation

The objective is now to compare the two least upper bounds to determinethe lifespan of the FPF. This lifespan is defined by the amount of accumulatedshortening at the transition from folding to the BTF.

This comparison in terms of least upper bounds is presented in figure 7 wherethe forces for the FPF and the BTF are presented as solid curves and dashedcurves, respectively. Forces are normalized by rsgL2. Results are obtained forf∗ = 16◦. The bounds for the FPF are initially always below the bounds for theBTF and the circles mark the transition with increasing shortening from theformer to the latter. The FPF least upper bound without any overburden is anincreasing function of the shortening because of the increasing relief during folddevelopment. For Hf = 100 m, the same least upper bound first decreases withthe accumulated shortening, corresponding to a phase where the fluid pressureon the top of the competent layer decreases on average. The kink on that curvemarks the shortening at which the relief pierces the fluid layer. The FPF leastupper bounds for thicker overburdens are all similar: a slow decrease of the FPFbound with increasing shortening. The critical shortenings at the intersectionsof the bounds, marking the transition from FPFs to BTFs, are pinpointed bycircles. This critical shortening is a slightly decreasing function of Hf if largerthan 200 m. This dependence is more pronounced and of opposite sense for Hf lessthan 200 m.

These comparisons are now summarized in figure 8, which can be seen asa deformation mechanism map, in the space spanned by the dimensionlessshortening and thickness of the overburden. The map is partitioned into two





0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160

0.05

0.10

0.15

0.20

0.25

0.30

norm

aliz

ed b

uria

l dep

th, H

f

~

fault-propagationfolding

break-throughfaulting

f* = 16°

f* = 12°

f* = 8°

g B = 30°


Figure 8. The competition between the FPF and the BTF are presented in the map spanned bythe shortening and the initial burial depth defined by the thickness of the fluid-like layer. Theboundary between the domains of the fold and the thrust is drawn for three values of the fold axialsurfaces friction angle.

domains, one to the left (where the FPF develops) and one to the right (wherethe BTF has stopped this development). The boundary between the two regionsis the collection of the circles found in figure 7. The boundary between the BTFand the FPF domains is established for different values of the axial surface frictionangle f∗: the larger the f∗, the smaller the region of dominance of the fold. Notethe kink in the domain boundary for f∗ = 16◦ corresponding, for the smallervalues of Hf , to conditions for which the fold pierces the weak overburden. Theinfluence of Hf is weak if this piercing does not take place.

5. Conclusion

The first of the two objectives of this contribution was to show that geometricalconstructions of folds could be complemented by the account of mechanicalequilibrium and of the material limited strength. The second objective was toshow that the resulting mechanically balanced constructions could be comparedso as to explain, for example, the possible development of the FPF or its arrestby faulting through its fore limb (BTF).

The FPF is composed of a kink fold bounded by several axial surfaces that aremigrating during the fold development ahead of the region where the ramp haspropagated. It is shown how these migrations can be characterized by applicationof Hadamard’s jump condition, providing a rather systematic construction of theexact velocity field (electronic supplementary material). The application of themaximum strength theorem to the FPF shows that there are some restrictions





Q

Q

Q

Q d max = 0

(a)

(b)

(c)

(d)

BTF at

f* = 20°

˜

d max = 0.062

BTF at

f* = 16°

˜

d max = 0.120

BTF at

f* = 12°

˜

d max = 0.162

FPF complete

f* = 8°

˜

Figure 9. (a–d) The BTF interrupts the FPF development at different amounts of shorteningdepending on the axial surface friction angle, except for case (d) for which the fold development iscompleted before faulting could occur. The faults and the associated axial surfaces are the dashedand the dotted lines, respectively. Note in (c) that the fault dip is less than the fold ramp dipand that there is no upper axial surface. The dotted line is a marker and the top solid line is thestress-free surface of the fluid-like overburden. The dotted-dashed segment in (c) illustrates theposition of the faulting breaking through the FPF fore limb of the Puri anticline (Medd 1996).

on the range of admissible fault dips according to the possible values for the axialsurface friction angle. There are no ramp dips less than approximately 20◦. Forramp dips varying between 20◦ and 60◦, the maximum axial surface friction angledecreases from 30◦ to 20◦, approximately.

The comparison between the least upper bounds of the force to drive the BTFand to develop the FPF is directly interpreted in terms of the lifespan of thelatter defined by the accumulated shortening prior to the onset of the former.These verdicts on the lifespan are summarized in figure 9 for various values ofthe fold axial surfaces friction angle f∗. The top layer is composed of a fluid-likelayer (poorly consolidated sediments) and its top surface remains at a constantlevel during compression. The dotted line is a marker. For the largest value of theaxial surface friction angle, the BTF dominates as soon as shortening is initiatedand the dashed and the dotted segments in figure 9a mark the fault and the





associated axial surface, respectively, both dipping at 30◦. This particular anglecorresponds to the classical orientation for faulting p/4 − fs/2 in compression,where fs is the friction angle of the bulk material. This orientation of 30◦ isindependent of the friction on the décollement, as shown analytically in theelectronic supplementary material, §6. The interpretation of figure 9a is that afault-bent fold will take place instead of the FPF. Figure 9b,c shows that thefold develops until it is stopped by the onset of the BTF at a critical shortening,which increases with decreasing values of f∗. Note the difference between theBTF mechanisms in these two examples. In figure 9b, the top segment of theramp (dip gBTF) is sub-parallel to the lower ramp (dip gB) used by the fold andis associated with two axial surfaces, the first at the base of the fold ramp andthe second at its tip. In figure 9c, a single axial surface exists at the base of thefold ramp and the hanging whole is sliding as a single unit. In that instance,the dips are related by gBTF + fs = gB + fR, in which fs and fR are the frictionangles of the upper and lower segment of the ramps, respectively. In the lastexample (figure 9d), the fold develops completely without being interrupted bythe onset of faulting because of the small value of f∗. In the last three examples offigure 9, the fold pierces the fluid-like overburden. The thickness of thisoverburden has little influence on the shortening, marking the end of the FPFdevelopment once piercing of the overburden has occurred.

One should note that our predicted break-through ramp is dipping less thanthe ramp of the FPF because its friction angle is larger than the friction angle ofthe FPF ramp. Such a dip is certainly satisfactory in the early stage of the FPFdevelopment for predicting the ramp linking, for example, two weak décollementsas seen by Rich (1934), but it is not consistent with the later stage of the FPFlife. Consider for example the Puri anticline (Papua, New Guinea; Medd 1996),several faults are breaking through the front limb at a dip larger than the FPFramp dip (see the illustration with a dotted-dashed segment in figure 9c). Threereasons could justify this discrepancy. First, one could argue that the faults arenot emanating from the tip of the FPF ramp, as is done in this contribution, butrather initiate somewhere along the ramp, as is illustrated in figure 9c. A secondreason could be related to the overburden considered as a fluid-like material,whereas the Puri anticline is within a triangular zone and the overburden mustexert a certain amount of shear forces on the upper boundary of the FPF. Thirdly,and more theoretically, is this difference in dip an outcome of our choice to favourthe dissipation along the ramp instead of the work done by the fault propagation?A comparison with the work-budget approach proposed by Cooke & Murphy(2004) could clarify this issue.

One of the main outcomes of the comparisons between the FPF and the BTFis the importance of the friction angle of the axial surfaces that compose theformer. It is argued that this friction angle could be significantly smaller thanthe classical Coulomb friction angle of a pristine rock if slip parallel to thebedding is activated during the propagation of the axial surfaces. This mechanismhas been studied in detail by Kampfer & Leroy (2009) for the developmentof a kink fold as well as by Maillot & Leroy (2003) for the axial surface of afault-bend fold. The key to the weak axial surfaces of the kink fold in thesetwo references was the introduction of a compacting deformation mechanism inthe intrados of every bed crossed by the axial surface. Here, we assume simplythat the axial surface friction angle is smaller than the pristine rock friction





angle without explicitly introducing the details of the axial surface mechanisms.Extending the present analysis to include the detailed axial surface geometryand properties could certainly be carried out in the future, especially if fielddata were available to constrain the spacing of the bedding and laboratoryexperiments to quantify the bedding friction and the exact nature of thiscompaction mechanism.

In the meantime, it could be argued that the necessity to reduce our axialsurface friction angle is needed to palliate the fact that the kinematics of theFPF is not yet optimized. Such optimization is certainly important becausewe have seen here that an optimized BTF could lead to the reduction inthe applied force by up to 45 per cent. An optimized FPF could thus bemore competitive with respect to the BTF with a large axial surface frictionangle. Optimizing the FPF is beyond the scope of the present contribution,and it will require us to relax some of the assumptions adopted for thegeometrical constructions. Choosing which assumption to relax, or which setof geometrical rules to select, should be guided by comparisons with fieldexamples (Jamison 1987; Suppe & Medwedeff 1990). The first assumption thatis known not to be optimum is the choice of orienting some of the axial surfacesat half the complementary angle of the ramp. This point was discussed byMaillot & Leroy (2003) theoretically and by Maillot & Koyi (2006) and Koyi &Maillot (2007) from the results of analogue laboratory experiments and fieldobservations. The second assumption is the rate of propagation of the ramp.The rate in this contribution is either controlled by the FPF construction oris instantaneous if the BTF is dominant. A continuous transition from foldcontrol to fault control would be of interest to pursue this comparison: foldingor faulting?

This work is part of the doctoral thesis of G.K. at Ecole Normale Supérieure, Paris, France,which was supported by Shell IEP, Rijswijk, The Netherlands. The interest and the continuousencouragement of Dr N. Yassir and Dr L. Bazalgette were very much appreciated. Commentson several versions of this manuscript by Prof. D. Frizon de Lamotte, Dr B. Maillot, bothat the Université de Cergy-Pontoise, and by J. Raphanel, Ecole Polytechnique, are gratefullyacknowledged. The constructive comments of three anonymous reviewers were essential to preparethe final version of this manuscript. Authorization to publish was granted by Shell IEP.

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