discrete element modelling of the influence of cover strength on basement-involved fault-propagation...

2006) 225–238www.elsevier.com/locate/tecto

Tectonophysics 415 (

Discrete element modelling of the influence of cover strength onbasement-involved fault-propagation folding

Stuart Hardy a,b,⁎, Emma Finch c,1

a ICREA (Institució Catalana de Recerca i Estudis Avançats), Spainb GGAC, Facultat de Geologia, Universitat de Barcelona, C/ Martí i Franquès s/n, 08028 Barcelona, Catalonia, Spain

c Basin and Stratigraphic Studies Group, Department of Earth Sciences, University of Manchester, Manchester M13 9PL, U.K.

Received 26 August 2004; received in revised form 12 January 2006; accepted 18 January 2006

Abstract

A discrete element model is used to investigate the influence of sedimentary cover strength on the development of basement-involved fault-propagation folds. We find that uniformly weak cover best promotes the development of classical, trishear-like fault-related folds showing marked anticlinal thinning and synclinal thickening, with cover dips increasing downwards towards the faulttip. Uniformly strong cover results in more rounded fold forms with only minor hinge thickening/thinning and significant basementfault-propagation into the sedimentary cover. Heterogeneous, layered, cover sequences with marked differences in strengthpromote the development of more complex and variable fold forms, with a close juxtaposition of brittle and macroscopicallyductile features, which diverge from the predictions of simple kinematic models. In these structures the upper layers are often poorindicators of deeper structure. In addition, we find that in layered cover sequences fault-propagation into the cover is a complexprocess and is strongly buffered by the weaker cover units.© 2006 Elsevier B.V. All rights reserved.

Keywords: Folding; Faulting; Trishear; Modelling; Basement

1. Introduction

Fault-propagation folds, and their associated blindfaults, have recently been recognized as extremelyimportant for their seismic hazard potential (e.g. Shawand Shearer, 1999; Allmendinger and Shaw, 2000) andfor their importance in controlling stratigraphic archi-

⁎ Corresponding author. ICREA (Institució Catalana de Recerca iEstudis Avançats), Universitat de Barcelona, C/ Martí i Franquès s/n,08028 Barcelona, Catalonia, Spain. Fax: +34 934021340.

E-mail addresses: [email protected] (S. Hardy),[email protected] (E. Finch).1 Fax: +44 1612753947.

0040-1951/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.tecto.2006.01.002

tectures in sedimentary basins (e.g. Ford et al., 1997;Gawthorpe et al., 1997). They are also the location ofmany oil and gas traps (e.g. Mitra and Mount, 1998).Where a faulted, rigid basement is involved in thedeformation, the folds are sometimes called “drape” or“forced” folds (Stearns, 1978; Fig. 1a,b) where theoverall shape and trend are dominated by the forcing(basement) member below, in contrast to other fault-related folds which are the result of fault movementwithin the cover rocks. Here we refer to such structuressimply as basement-involved fault-propagation folds.Evidence from well-exposed folds in the Laramideorogen (e.g. Erslev and Mayborn, 1997), the Bighorn

Fig. 1. Natural examples of basement-involved structures: (a) the Willow creek and (b) Rangely anticlines (after Mitra and Mount, 1998), in bothcases a fault rooted in deeper basement has propagated into the sedimentary cover causing folding. (c) Schematic diagram illustrating the trishearkinematic model. In the trishear model a triangular zone of distributed shear opens outwards and upwards from the fault tip. The hangingwall movesrigidly parallel to the fault at the slip rate (S) while the footwall is static, with the shear zone acting as a transition between them. The arrows indicatethe velocity vectors within the trishear zone. (d) The geometry of a trishear fault-propagation fold, redrawn from Zehnder and Allmendinger (2000),thin lines are bedding tops and thicker lines the fault and trishear zone boundaries. (e) The geometry of a basement-involved trishear fault-propagationfold with growth strata, redrawn from Hardy and Ford (1997).

226 S. Hardy, E. Finch / Tectonophysics 415 (2006) 225–238

and Uinta basins (Mitra and Mount, 1998), theCalifornian peninsular ranges (e.g. Allmendinger andShaw, 2000), and from analogue (e.g. Withjack et al.,1990; Mitra and Islam, 1994) and numerical modelling(Patton and Fletcher, 1995; Johnson and Johnson,2002a,b; Cardozo et al., 2003) has indicated that suchbasement-involved folds form as upward widening

zones of distributed deformation (monoclines) abovediscrete faults at depth. Field studies indicate that avariety of mechanisms are responsible for the distributeddeformation including “ductile” flow, bedding slip, rigidrotation and small extensional and thrust faults (e.g.Gawthorpe et al., 1997; Sharp et al., 2000). Analoguemodelling studies have shown that with increasing

227S. Hardy, E. Finch / Tectonophysics 415 (2006) 225–238

displacement (strain) the overlying fold may be cut bythe basement fault as it propagates upwards into thecover (Withjack et al., 1990).

Basement-involved fault-related folds have been thesubject of research for many years (e.g. Stearns, 1978;Narr and Suppe, 1993; Erslev and Mayborn, 1997;Bump, 2003) and a series of kinematic models havebeen proposed to explain their development (Erslev,1991; Narr and Suppe, 1993; McConnell, 1994; Mitraand Mount, 1998), of which the trishear model is nowthe most popular and well-accepted (Erslev, 1991;Hardy and Ford, 1997; Allmendinger, 1998; Fig. 1c,d).In the trishear model a triangular zone of distributedshear, with non-parallel shear planes oblique to bedding,opens outwards and upwards from the fault tip, whichmay propagate at a rate independent of the slip rate onthe basement fault (Fig. 1c). The hangingwall movesrigidly parallel to the fault at the slip rate while thefootwall is static, with the shear zone acting as atransition between them. The trishear velocity fieldrepresenting the shear zone must maintain continuityand satisfy boundary conditions, and a simple lineardecrease in magnitude and orientation of velocityvectors towards the static footwall is often assumed(e.g. Zehnder and Allmendinger, 2000). The geometriesof two typical trishear folds are shown in Fig. 1d and e,they illustrate several key characteristics of such folds:1) an asymmetrical fold pair verging in the direction ofthrusting comprising a footwall syncline and hanging-wall anticline, 2) fold limbs in which beds are notuniform in dip and heterogeneous deformation, partic-ularly in the steep forelimbs where local tectonicthickening and thinning is observed. Such geometriesare difficult to explain, or model, using kink bandkinematics based on flexural slip (see Hardy and Ford,1997; Allmendinger, 1998).

While the trishear kinematic model appears toexplain well the geometric development and finite strainof basement-involved fault-related folds (e.g. Erslev,1991; Hardy and Ford, 1997; Allmendinger, 1998;Hardy and McClay, 1999; Zehnder and Allmendinger,2000; Allmendinger and Shaw, 2000; Allmendinger etal., 2004), many aspects of their mechanics are only nowbeing investigated. Johnson and Johnson (2002a,b),using viscous folding theory, found that, despite itsempirical simplicity, the trishear kinematic modeladequately described forced folding in isotropic sedi-mentary cover. However, they found that anisotropicsedimentary cover produced more parallel, kink-likefolds and that the trishear model was inappropriate forfault-arrest folds (modelled as a stress-free crack in anelastic medium near a free surface). Cardozo et al.

(2003), using finite-element modelling, found that foldgeometries, finite strain and velocity fields in modelswith incompressible (either frictional or frictionless)sedimentary cover materials most closely resembled thetrishear kinematic model. Finch et al. (2003) used adiscrete element technique to investigate the mechanicalcontrol of sedimentary cover strength upon fault-propagation and fold development in forced folds.They found that a key parameter of the trishear model,the fault-propagation to slip (p / s) ratio, was a grossreflection of cover strength with high and low p / s ratioscorresponding to strong and weak sedimentary covers,respectively.

However, while confirming the applicability of thetrishear kinematic model to fault-propagation folding,previous studies have only considered the developmentof basement-involved folds in homogenous coversequences or within cover sequences which have aparameterized (uniform) viscous anisotropy. However,cover stratigraphy in nature is not uniform, oftenpresenting marked lateral and vertical changes inmechanical response. Indeed, in the brittle upper crustone would expect that differences in rock strength/competency between cover units (e.g. between lime-stone, shale and sandstone), and thus rock behaviour,would be a controlling factor in the development ofdifferent deformation styles (cf. Mitra and Mount, 1998;Johnson and Johnson, 2002b). Both experimental andobservational evidence suggests that this is the case, andthat the precise arrangement of different mechanicalunits within the cover is an important control on thedevelopment of basement-involved fault-propagationfolds (see Couples et al., 1994; Mitra and Mount, 1998).In particular, Mitra and Mount (1998), as a result of keyfield data and subsurface constraints, observed that if thecover contains both incompetent and competent units,the former are characterized by more penetrativedeformation (exhibiting marked thickness changesacross folds) whereas the latter are more simply faultedafter a small amount of penetrative deformation (andmaintain their thickness during folding). Such observa-tions are the motivation for this paper, whose objectiveis to better understand how basement-involved fault-related folds develop when the sedimentary covercontains marked strength contrasts and under whatconditions the trishear model is an appropriate kinemat-ic representation of the development of such fault-related folds. We do not address temperature-controlledchanges in deformation mode (brittle–ductile) that onewould expect in the deeper crust (cf. Rutter, 1986). Wedo this using a 2D discrete element modelling techniquethat we have previously used to investigate contractional


fault-propagation folding in homogeneous covers(Finch et al., 2003). We have previously demonstratedthat this technique has the advantage of allowing the‘unforced’ propagation of the basement fault into thecover and that it reproduces well many of the featuresobserved in natural structures (Finch et al., 2003).

2. Discrete element modelling

Discrete element models are becoming commonlyused in the description of the non-linear interaction of alarge number of particles (e.g. Donzé et al., 1996; Kuhn,1999; Camborde et al., 2000; Finch et al., 2003, 2004).Unlike continuum techniques, these discontinuummethods use simple particle interactions and thereforepermit the dynamic evolution of a system to be modelledand observed. In addition, it is a technique well-suited tostudying problems in which discontinuities (faults,joints, or fractures) are important as it allows deforma-tions involving large relative motion of individualelements, and does not require the complex re-meshingat moderate to high strains that, e.g., finite-elementtechniques typically require. The discrete elementtechnique was first applied to simulate the failure of asystem of semi-brittle jointed rock blocks by Cundall(1971) and is derived from the Particle DynamicsMethod, widely applied in chemistry and physics toimitate liquid and gas behaviours (Allen and Tildesley,1987; Gould and Tobochnik, 1988). Further develop-ments of the discrete element method have addressedproblems in soil mechanics and granular media (e.g.Cundall and Strack, 1979; Bardet and Proubet, 1992;Kuhn, 1999), and a number of geological andgeophysical problems in 2- and 3-dimensions (Saltzerand Pollard, 1992; Antonellini and Pollard, 1995; Scott,1996; Donzé et al., 1996; Strayer and Huddleston, 1997;Camborde et al., 2000; Iwashita and Oda, 2000; Toomeyand Bean, 2000; Strayer and Suppe, 2002; Place et al.,2002; Finch et al., 2003, 2004; Strayer et al., 2004).

2.1. Discrete element model

We have developed a 2D discrete element model ofsedimentary cover deformation in response to contrac-tional basement faulting in an effort to better understandthe development of basement-involved fault-propaga-tion folds. The discrete element model we describe hereis a variant of the Lattice Solid Model (LSM) (Mora andPlace, 1993, 1994; Place et al., 2002). A full, detaileddescription of the theory behind this modelling approachand its application to geological problems is given inMora and Place (1993) and Place et al. (2002). An

outline of the modelling approach, and our implemen-tation of it, is given here.

The discrete element model used here treats a rockmass as an assemblage (or lattice) of circular elementsthat interact in pairs as if connected by breakable elasticsprings and that undergo motion relative to one another.The behaviour of the elements assumes that the particlesinteract through a ‘repulsive–attractive’ force (Mora andPlace, 1993) in which the resultant force, Fs, is given by:

FS ¼Kðr−RÞ; r < r0; intact bondKðr−RÞ; r < R; broken bond0; rzR; broken bond

8<: ð1Þ

Here, K is the elastic constant (spring stiffness) of thebond, R is the equilibrium separation between theparticles, r0 is a breaking separation, and r is the currentseparation between the particle pair.

Particles within the model assemblage are bondeduntil the separation (r) between them exceeds a definedbreaking separation or threshold, r0, at which time thebond breaks. The force acting on a bond at this thresholdrepresents the force necessary for a bond to fail or yield,or, alternatively, can be cast as the stress acting on aparticle at failure. Mora and Place (1994) have shownthat at the macroscopic scale the breaking separation formost materials is typically much less than c. 0.11R.Hence, this value is considered an upper limit on thebreaking separation of our modelled materials. Ourprevious work has discussed in detail the effect ofsmaller and larger values of the breaking separation (seeFinch et al., 2003, 2004). Large values of the threshold(e.g. 0.1R) produce “strong” materials which fail bylocalised faulting, whereas low values (e.g. 0.01R)produce “weak” materials which deform in a macro-scopically ductile manner as a result of non-localiseddeformation caused by the relative motion of manyhundreds of elements. After this breaking threshold, theparticle pair experiences no further attractive force andthe bond is irreversibly broken. However, if the twoparticles return to a compressive contact (i.e. r<R), arepulsive force acts between them. Thus, healing ofbonds is not permissible in the present model.

The total elastic force, Fi,α , exerted on a particle isobtained by summing the forces on each bond that linksthe particle to its α neighbours (Fig. 2a), calculated by:

Fi;a ¼Xj¼1;a

fi;j ð2Þ

in which fi,j is the elastic force experienced by particle ifrom its neighbouring particle j. Additionally we includea viscous damping term (proportional to particle

Fig. 2. Illustration of the discrete element modelling technique used discussed in this paper: (a) relationship between a given particle i and its αneighbours, (b) packing of particles of four different radii in lattice, and (c) initial model set-up for the forced folding experiments. The model contains8906 elements, with radii of 0.5, 0.4, 0.3 and 0.2units, in a box that is 150×22 units. Displacement is incremented on the basement fault block at0.000005units per time step.


velocity) that acts to dampen reflected waves from therigid edges/boundaries of our model, preventing a build-up of kinetic energy within the closed system (cf. Moraand Place, 1994; Place et al., 2002). This artificialviscosity is frequency independent and does notfundamentally alter the dynamics if carefully chosen(Mora and Place, 1994). The viscous term has the effectof attenuating dynamic phenomena such as wavepropagation, thus making our model less dynamic andmore quasi-static, as is appropriate for modelling thedevelopment of tectonic structures over long time scales(cf. Donzé et al., 1994). This global damping approachis widely used in discrete element studies (Donzé et al.,1994; Wang et al., 2000).

In addition, gravitational forces, Fg, acting on eachelement are calculated in the y direction. Therefore, thetotal force on any particle is given by:

F ¼ Fi;a−m �xþ Fg ð3Þ

where ν represents the dynamic viscosity and x˙ is thevelocity of the particle.

At each discrete time step, the particles are advancedto their new positions within the model by integratingtheir equations of motion using Newtonian physics anda velocity–Verlat based scheme (Allen and Tildesley,1987). The positions (x(t)) and velocities (x˙(t)) of theparticles at the next discrete time step (t+Δ t), arecalculated from:

xðt þ DtÞ ¼ xðtÞ þ Dt �xðtÞ þ Dt2

2!x��ðtÞ

�xðt þ DtÞ ¼ �xðtÞ þ Dt2ðx��ðtÞ þ x��ðt þ DtÞ�

x�� ¼ FðtÞ=M

ð4Þ

where x is the acceleration and M is the mass of aparticle, and where Δt is calculated, to ensure numericalprecision and stability, as (Mora and Place, 1994; Placeet al., 2002):

Dt ¼ 0:2Dmin

Vp

� �ð5Þ

where Dmin is the minimum particle diameter in thelattice, and Vp is the compressional wave speed,


corresponding to the maximum speed of informationpropagation in the numerical experiment.

Previous discrete element simulations utilized aregular hexagonal lattice (where α=6) (Donzé et al.,1994). However, this imposed an unphysical, first-ordergeometric control in which the well-defined 60° planesof weakness in the lattice dominated the fault geometryof the resulting structure (e.g. Donzé et al., 1994). Tocounteract the influence of such isotropy, we generate amedium in which different particle sizes are distributedrandomly and which consequently reduces the likeli-hood for preferred planes of weakness (cf. Antonelliniand Pollard, 1995; Scott, 1996). The equilibriumseparation of a particle pair (i,j), Rij, in this randomassembly of particles is then defined as the initialdistance between a particle and its neighbour (Fig. 3b).Such random lattice solid models have been shown tohave the same elastic, and wave propagation, character-istics as regular lattice models (e.g. Place and Mora,2001; Place et al., 2002).

2.2. Model units and real world scaling

In discussing discrete element models such as ours,the mass (M), diameter (D) (or equilibrium separation,R0), P-wave velocity (Vp) and the spring stiffness (K) ofparticles are typically given in non-dimensionalized

Fig. 3. Results of the homogeneous sedimentary cover experiments after 15breaking strain of 0.1. See text for details.

model units that are often (but not necessarily) set tounity (see e.g. Donzé et al., 1994; Mora and Place,1994). However, in order to compare model results withreal geological or geophysical data in MKS units, themodels must be scaled to, or run using, real worldparameters (cf. Toomey and Bean, 2000). In this sectionwe detail the manner in which model results can bescaled to real world units and dimensions or vice versa.

Essentially scaling of the model involves choosingvalues of particle diameter (D), density (p) and P-wavevelocity (Vp) with which to convert the spring constantand time step increment. For a closely -packed lattice, Kin 2D is given by, or can be found from, (Place andMora, 2001):

K ¼ 8M9

� �Vp

D

� �2

N dm−1 ð6Þ

where M is the mass of a particle, D is the particlediameter, and Vp is the speed of P-wave propagationthrough the media. The mass (M) of a particle is givenby:

M ¼ qffiffiffi3

p=2

� �D2 kg ð7Þ

Using MKS units, and choosing values for D, ρ and Vp

we can derive appropriate scaled values for the elasticconstant (K) and thus interparticle forces/bond strengths

units of fault-parallel displacement. (a) breaking strain of 0.01 and (b)


within our model assemblage (for a full discussion onmodel scaling see Place et al. (2002)).

The viscosity damping used to damp kinetic energyout of the closed system is converted from model toMKS units using:

v ¼ v VMVp

D

� N s m−1 ð8Þ

where v′ is the viscosity coefficient in model units(Place et al., 2002).

2.3. Boundary and initial conditions

In this paper we apply the discrete element modeldescribed above to the problem of fault-propagationfolding in a contractional setting, in which a weakersedimentary cover is deformed in response to themovement of a strong, rigid basement block (‘forcedfolding’).

A particle assembly containing 8906 elements, withfour different radii of 0.5, 0.4, 0.3 and 0.2 lattice units, iscreated with elements positioned at random in anenclosed rectangular box, and then permitted to relaxto a stable equilibrium. Particles are then left to settleunder gravity for 100,000 time steps to further minimisevoid space.

In the models presented here, we assume that onelattice unit corresponds to 250m, and rock density isdefined as 2500kg m−2. The elastic spring constant (K)is set to be 2.7×109N m−1 (equivalent to 20 in modelunits) and the dynamic viscosity (v) is 4.0×107Nsm−1

(equivalent to 3 in model units). For this scaling of themodel, the equivalent value of Young's modulus (E) isapproximately 3GPa, about an order of magnitudesmaller than that observed in laboratory experiments onintact rocks specimens. These values are chosen as aresult of exhaustive model tests to determine valuesthat are appropriate for the media under consideration(i.e. no gaps, crevasses or cliffs form as a result ofimproper scaling of cover strength under the influenceof gravity). Our range of breaking thresholds of 0.01Rand 0.1R correspond (scale) to bond strengths of c.27and 270MPa, respectively. Like other discrete elementstudies of fault-related folding (e.g. Saltzer and Pollard,1992; Strayer et al., 2004) we have found thatappropriate/calibrated values of Young's modulus areat least an order of magnitude less than those seen inlaboratory experiments. This is hardly surprising in thatnatural systems involving larger volumes of rockcontain more joints and small faults, and thus Young'smodulus decreases with increasing sample size (Bien-

iawski, 1984). The behaviour of the modelled rockmass is broadly elasto-plastic and frictionless (seeDonzé et al., 1994; Place and Mora, 2001; Finch et al.,2004), which appears to be a good first approximationof brittle deformation in sedimentary rocks in the uppercrust (Braun and Sambridge, 1994; Cardozo et al.,2003).

The resulting particle assembly (cover stratigraphy)is 22 model units thick (5.5km) and 150 model unitswide (37.5km) (Fig. 2c). A pre-defined fault that dips at45°is created within the basement, the basement beingconsidered to be rigid and undeformable (Fig. 2c) withparticles effectively welded to the base of the box andfree to move at the side boundaries. Experiments are runfor 3,000,000 time steps with displacement on the faultincremented at 0.000005 lattice units per time stepresulting in a total fault-parallel displacement in allexperiments of 15units (3.75km) (Fig. 2c). Displace-ment parallel to the fault is applied to all hangingwallparts of the bounding box. Deformation is illustrated inrelation to 12 initially flat-lying layers which can havedifferent strengths as discussed in the next section. Wedefine faulting in the particle assembly as localised slipgreater than an element diameter along a planar surface,whereas fracturing is defined as bond breakage and slipless than an element diameter. Flow is defined asdistributed deformation resulting from the relativemotion of many hundreds of elements.

3. Experimental results

Under the experimental and boundary conditionsdescribed above we now investigate the influence ofcover strength on basement-involved fault-propagationfolding. We describe firstly the deformation of homog-enous cover sequences above a blind basement faultbefore investigating the effect of heterogeneous (lay-ered) cover sequences on fault-propagation and folding.

3.1. Homogeneous cover sequences

We examine firstly two models with homogeneouscover strengths (breaking separations between elements)of 0.01R and 0.1R, which scale to bond strengths of c.27and 270MPa, respectively. Apart from the variation incover strength all other model parameters are identical.These breaking separations cover the range of brittle andmacroscopically “ductile” behaviours expected in geo-logical materials in the upper crust (Mora and Place,1993; Finch et al., 2003). The effect of different(homogeneous) cover strengths is discussed in moredetail in Finch et al. (2003). The final configuration of


the two models after 15units of displacement is shownin Fig. 3.

The final configuration of the model with a weakcover (breaking separation of 0.01R, Fig. 3a) exhibits anupward-widening monocline above the blind thrust faultat depth, closely resembling a classical trishear fold (cf.Erslev, 1991; Hardy and Ford, 1997; Fig. 1d). At thesurface the monocline is broad and shallow, howeverbed dips in the cover increase progressively downwardsand towards the basement fault tip, becoming sub-vertical. Across the monocline individual beds exhibitsignificant synclinal thickening and anticlinal thinning(achieved by small-scale extensional faulting). The faultin the rigid basement does not propagate markedlyupwards into the cover from its initial position at thebasement-cover contact (the p / s is approximately 1.2)but cuts through the lower four units of the coverresulting in a marked footwall syncline (cf. Finch et al.,2003).

The final configuration of the model with a strongcover (breaking separation of 0.1R, Fig. 3b) issignificantly different: the fold in the cover is muchnarrower and steeper at the surface and individualbeds are approximately constant thickness around thefold displaying only minor synclinal thickening andanticlinal thinning. As before, the fold opens outwardsand upwards from the basement fault towards thesurface. However, the fold is now clearly breached bya fault which links to the basement fault at depth. This

Fig. 4. Results of the two-layer experiments after 15units of fault-parallel dispbreaking strain of 0.01 and (b) upper half of model has breaking strain of 0.

fault shallows upwards as it cuts through the fold andreaches the ground surface. As a result of the faultbreaching the fold, a prominent footwall syncline andhangingwall anticline are produced. Overall, the coverdeformation is more localised and is characterized byfaulting and localised folding rather than the distrib-uted folding and deformation observed in the weakcover model.

3.2. Two-layer cover sequences

In these two experiments the lower or upper sixlayers of the twelve layers of the cover sequence areeither strong (breaking separation of 0.1R) or weak(breaking separation of 0.01R), resulting in a simpletwo-layer cover with a marked strength contrast (strongoverlying weak or weak overlying strong). All othermodel parameters and boundary conditions are identicalto the homogeneous cover experiments discussedabove. The final configuration of the strong overlyingweak model is shown in Fig. 4a, and the weak overlyingstrong model in Fig. 4b. The experimental results showthat the two cover sequences behave independently butcompatibly, resulting in a close juxtaposition of verydifferent deformation styles and features, and one inwhich the surface fold (or upper cover) is little guide tothe deeper fold form.

In the case of strong overlying weak cover the finalmodel (Fig. 4a) shows a complexly deformed lower

lacement. (a) Upper half of model has breaking strain of 0.1, lower half01, lower half breaking strain of 0.1. See text for details.


(weak) sequence in which a tight, upright–overturnedsyncline has developed in the footwall of the basementfault which only shows minimal propagation into theweaker cover. While no major faulting is observed in thelower sequence, it is, however, markedly thickened inthe footwall syncline and markedly thinned immediatelyabove the basement fault tip — where the weak coversequence is highly attenuated between the rigidbasement and the strong upper sequence. The upper(strong) sequence is much more simply folded and isunfaulted, with individual units maintaining constantthickness across the monocline — which is a verysimple, narrow feature at the surface with a subtlesynclinal feature in the footwall block (Fig. 4a).

In the case of weak overlying strong cover the finalmodel (Fig. 4b) shows a simply deformed lower(strong) sequence in which all layers are breached bythe basement fault. Deformation in the lower sequenceis restricted to a narrow region adjacent to the faultleading to only a minor footwall syncline andhangingwall anticline. The upper (weak) sequence ismuch more strongly deformed, with a broad mono-cline having developed which exhibits marked thick-ening in the synclinal area and thinning (minorextensional faulting) in the anticlinal area. However,the fault in the lower sequence does not breach orpropagate into the upper sequence but rather termi-nates against it.

Fig. 5. Results of the three-layer experiments after 15units of fault-parallel diof 0.01, middle third a breaking strain of 0.1 and (b) upper and lower thirds o0.01. See text for details.

3.3. Three-layer cover sequences

In these experiments the lower and upper thirds of thecover sequence (bottom and top four layers) are eitherstrong or weak and the middle third (central four layers)is either weak or strong, resulting in a three-layer coversequence (sandwich) with a marked strength contrast(Fig. 5). All other model parameters are identical to theexperiments discussed previously. The final configura-tion of the model with a strong middle is shown in Fig.5a, and the model with a weak middle in Fig. 5b. Theexperimental results show that the three layers in thecover sequence behave almost independently butcompatibly — resulting in a cover sequence thatshows a close juxtaposition of brittle and macroscopi-cally “ductile” features.

In the case of a strong middle sequence embedded ina weaker cover sequence, the final model (Fig. 5a) hasdeveloped a gentle monocline at the surface and anupper (weak) sequence in which an open uprightsyncline has developed and which shows moderatethickening and thinning in synclinal and anticlinal areasrespectively. The lower (weak) sequence is thickenedand complexly deformed in the footwall and markedlythinned in the hangingwall. The basement fault cuts allof this lower sequence but has only partially propagatedinto (and offset) the middle (strong) sequence which ismuch more simply folded, and thus maintains

splacement. (a) Upper and lower thirds of model have a breaking strainf model have a breaking strain of 0.1, middle third a breaking strain of


approximately constant thickness across the monocline.The folding leads to a distinct synclinal feature in thefootwall block and a more subtle anticlinal feature in thehangingwall block.

In the case of a weak middle sequence embedded in astronger cover sequence, the final model (Fig. 5b)develops a steeper fold limb at the surface and an upper(strong) sequence which is simply folded leading to thedevelopment of a clear synclinal feature in the“footwall” block. The middle (weak) sequence is morestrongly deformed, but does not show such extremethickening and thinning as in previous experiments. Thelower (strong) sequence is more simply folded andfaulted within a short distance of the fault, with simplefootwall cutoffs and only a minor hangingwall anticlinedeveloping. The basement fault cuts all of the lowersequence and the lowest unit of the middle sequence butclearly terminates (or slip is accommodated in distrib-uted deformation) within the overlying weak layer.

4. Discussion and conclusions

We have investigated here, through discrete elementmodelling, the manner in which a heterogeneoussedimentary cover deforms in response to slip on adeeper basement fault or blind thrust. Our modelling isperhaps most appropriate to so-called ‘forced folds’ inwhich a weaker sedimentary cover overlies a stronger,rigid basement such as in Laramide structures orinverted normal faults. In this section we discuss ourmodel results, their implications and limitations.

We find that sedimentary cover sequences consistingof layers of very different strength materials deform ina complex manner controlled by the particulararrangement of materials with respect to the deeperfault. In general, homogeneous weak covers promotethe development of wide, gentle monoclines abovebasement faults (cf. Fig. 3a) while stronger coverspromote the development of narrower, steeper struc-tures (cf. Fig. 3b). Heterogeneous (layered) coversequences can present marked changes in deformationstyle as one moves upwards and outwards from thebasement fault, leading to rapid variations in forelimbdip, degree of bed thinning/thickening and faulting. Insimple two-layer models, strong basal sequences foldand fault in a localised, brittle manner and effectivelyact as the top of the mechanical basement withdeformation being restricted to a narrow regionadjacent to the fault (cf. Fig. 4b). Faulting terminates,or is transferred into more distributed deformation andfolding, in the overlying weak material. Weak basalsequences deform in a more “ductile” manner display-

ing marked thickness changes across the structure (cf.Fig. 4a), accommodating folding of the upper, strongersequence during slip on the basement fault. In three-layer models, stronger layers embedded within aweaker cover sequence simply fold during displace-ment on the basement fault (producing anticlines andsynclines) while the surrounding weaker units “flow”as a result of the relative motion of many elements.Weaker units embedded in a stronger cover sequencecan act as buffers to basement fault-propagation,effectively preventing upward fault-propagation fromlower, stronger, faulted sequences.

To illustrate the complex manner in which the coverdeforms we now show the density of broken bondsassociated with each element for a subset of our modelresults (Fig. 6). Elements in the models can vary fromhaving all bonds intact, to having 6 or more bondsbroken in the most intensely deformed locations. Thedensity of broken bonds gives an indication of therelative amount of “damage” or fracturing that aparticular part of a model has undergone. We illustratethe density of broken bonds for the homogeneous strongand weak cover models (Fig. 6a,b), as well as for the 2-layer model in which the upper half is weak (Fig. 6c)and the 3-layer model in which the middle sequence isweak (Fig. 6d). In the homogeneous models (Fig. 6a,b)the main difference between the strong and weak coverslies in the number and distribution of broken bonds. Inthe weak cover model, there is a widespread, broadlytriangular zone of deformation opening outwards andupwards from the basement fault tip, and intensefootwall deformation. This is similar to the triangularzone of penetrative shearing in the conceptual model oftrishear fault-propagation folding (cf. Fig. 1c). Incontrast, in the strong cover model the deformation ismuch more localised and is only associated with anarrow triangular region ahead of the fault tip, and thusmuch of the cover remains undeformed. Deformation inthe footwall is intense, but is restricted to a narrowregion adjacent to the basement fault. The 2-layer model(Fig. 6c) illustrates the combination of these twodeformation styles, leading to a narrow zone ofdeformation/fracturing in the lower, strong sequencethat passes abruptly (at the strength interface) into amuch wider, diffuse triangular zone of fracturing in theupper, weak sequence. Finally, the 3-layer modelillustrates well the manner in which a middle, weaklayer can buffer fault tip propagation and associateddeformation. In this model this leads to a weaklydeformed, and simply folded, upper sequence (Fig. 6d).The underlying weak layer is intensely deformed andfractured some distance into the footwall. The basal,

Fig. 6. Density of broken bonds in modelled assembly at end of model run, (a) homogeneous weak model, (b) homogeneous strong model, (c) 2-layermodel with weak upper sequence, and (d) 3-layer model with central weak sequence. Number of broken bonds at each element is shown, see key.


strong layer is deformed only in a narrow zoneimmediately adjacent to the basement fault.

It is interesting to consider the reasons that aparticular cover unit, without any obvious temperatureor pressure differences with respect to surroundingunits, would deform in a distributed, macroscopically“ductile” manner rather than a more localised brittlemanner. It may be a fundamental property of thematerial or lithology, controlled by initial beddingthickness, overpressuring, grain size, matrix composi-tion etc. However, it may also be the result of theavailability or easy creation of a network of fractures,joints and slip surfaces — the “block-controlledcatalclastic flow” of Ismat and Mitra (2005). Rheolog-ical models for fractured rock (Patton and Fletcher,1998), and studies of fractures/slip surfaces associatedwith folds (Bergbauer and Pollard, 2004; Ismat andMitra, 2005) have all emphasised the potential of afractured cover unit in the brittle regime to deform inbulk by folding by such a mechanism. It is important tonote that all fractures in our models occur as a result ofdeformation and we do not include prefolding joints orother weak surfaces which may be exploited in naturalexamples (cf. Bergbauer and Pollard, 2004). Whateverits cause, the observation that closely juxtaposed

lithologic units behave in mechanically such distinctmanners is important and this paper has shown that it is akey factor in controlling overall fold geometry.

This work has important implications for theconstruction of ‘balanced’ cross-sections across suchstructures where the deeper structure is often poorlyconstrained. Our model results suggest that a closejuxtaposition of macroscopically ductile deformation,with marked thinning and thickening of units, and moresimply folded or faulted units in which bed thickness ispreserved, is possible in such settings (cf. Fig. 4) andmust be borne in mind. The models also illustrate therapid vertical and lateral variations in fracturing/damagethat are possible in such settings. Simple geometricsection construction/restoration or kinematic modellingmay overlook such complex structural and stratigraphicrelationships. With specific reference to trishear fault-related folding, powerful forward and inverse methodshave been recently developed whereby the geometry ofa set of cover layers is used to predict overall fold andfault geometry, fault slip and propagation or vice versa(e.g. Allmendinger, 1998; Allmendinger and Shaw,2000; Bump, 2003). Our work suggests that wherecover stratigraphy is homogenous and deformationoccurs by penetrative shearing such approaches may


be appropriate and successful. However, where coverstratigraphy is more variable, with marked contrasts instrength and deformation style between distinct units,trishear modelling of such structures may not beappropriate.

A comparison of our model to a natural structure isillustrated in Fig. 7, where a cross-section through theHamilton Dome, a basement-involved fault-related foldfrom the Bighorn Basin is shown (Mitra and Mount,1998). This structure, where the lower part of the coversequence consists of Palaeozoic competent carbonatesand sandstones and the upper part of the sequenceconsists of incompetent shales, shows many of thefeatures predicted by our model. The structure ismoderately well-constrained by surface, well andseismic data, although, as the fault is poorly constrained,a range of steep basement fault dips could be used toconstruct the cross-section shown (Mitra and Mount,1998). We show a correctly scaled discrete elementmodel to compare with a published cross-section of thestructure (Fig. 7). In order to best reproduce thegeometries of the upper and lower cover sequences,given the displacement and cover thickness, we had to

Fig. 7. (a) Cross-section through the Hamilton dome fault-related fold (redrawof a two-layer experiment where upper two-thirds of model has a breaking strsequences indicated in different greyscales, and current erosion level by uppe

use breaking separations of 0.08 and 0.012 for the‘competent’ lower third (Palaeoizoic carbonates andsandstones) and ‘incompetent’ upper two thirds (shales)of the cover sequence respectively, homogeneous coversequences were found to be unable to reproduce theoverall geometry of the fold. This model reproduceswell the overall geometry of the structure includingextreme thinning of the upper sequence in the steepforelimb, thickening associated with the footwallsyncline and the amount of basement fault-propagationinto the cover. In addition, in both the model and naturalstructure, the lower sequence is only deformed in anarrow region immediately adjacent to the fault. Finally,we predict thinning (extensional faulting) associatedwith the upper parts of the hangingwall anticlinal regionwhich in the natural structure are now (possibly) lostthrough erosion. Clearly we do not reproduce the overalltilt of the structure, but the close similarities of thefeatures predicted by our model and those observed inthe natural example give us confidence in its applicationto other structures.

The model we have used here is only a simplificationof a complex basement-cover system but it produces

n from Mitra and Mount (1998)) showing well constraints. (b) Resultsain of 0.012, lower third has a breaking strain of 0.08. Lower and upperr surface in section and solid line in model. See text for further details.


many of the features observed in natural structures.However, the model itself is only a first step inmodelling such structures, and has certain limitationsand possible improvements. In using an elasto-plastic(frictionless) material we acknowledge that we havegreatly simplified rock behaviour. However, thisapproach has been previously shown to be a reasonablefirst approximation of brittle deformation in the upper-crust and to be appropriate for modelling fault-propagation (and in particular trishear) folds (Braunand Sambridge, 1994; Cardozo et al., 2003). Theaddition of frictional materials, with a pressure stressdependent yield, would be necessary to reproduce theincrease of yield stress with confining pressure. Thus wecannot directly compare our results to the Mohr–Coulomb model or to Byerlee's law. For a discussion ofthe effect of adding friction to the Lattice Solid Modelsee Place et al. (2002). In addition, we do not alsoconsider true ductile deformation in our models, as allbonds are strictly elastic and brittle. However, weakmodels in particular commonly display “ductile”geometries at the macroscopic scale, caused by therelative motion of many hundreds of elements. Strongmodels on the other hand display much more classically“brittle” geometries where deformation is much morelocalised and expressed by discrete faults. The inclusionof an additional contact law applicable to the ductilemode of deformation at greater temperatures andpressures, is in principal a straightforward task (Finch,1999).

In addition, many other parameters can influence theinitiation and growth of folds above blind basementfaults. In nature, folds are 3D entities that propagate andlink. Our model being two-dimensional it cannotinclude such effects. In addition, the role of bondannealing, fluid flow and other processes have not beenconsidered in the present model. Finally, erosion,transport and sedimentation are an integral part of folddevelopment. These issues can all be investigated withour technique, although, given the computationallyintensive nature of the approach they have not beeninvestigated in the present study.

Acknowledgements

This paper has benefited greatly from discussionswith many colleagues, in particular Eduard Roca andRick Allmendinger. This research has been supported byICREA (Institució Catalana de Recerca i EstudisAvançats), the Centre de Supercomputació de Catalunya(CESCA), the Geomod 3D project (CGL2004-05816-C02-01/BTE), the Geomodels programme, and GGAC

(Grup de Geodinàmica i Anàlisis de Conques) at theUniversity of Barcelona to SH, and by Norsk Hydro (toEF) whose support is gratefully acknowledged. Com-ments by two anonymous reviewers and Jean-PierreBurg have greatly improved this manuscript.

References

Allen, M.P., Tildesley, D.J., 1987. Computer Simulation of Liquids.Oxford Science Publications, Oxford.

Allmendinger, R.W., 1998. Inverse and forward numerical modelingof trishear fault-propagation folds. Tectonics 17, 640–656.

Allmendinger, R.W., Shaw, J., 2000. Estimation of fault propagationdistance from fold shape: implications for earthquake hazardassessment. Geology 28, 1099–1102.

Allmendinger, R.W., Zapata, T.R., Manceda, R., Dzelalija, F., 2004.Trishear kinematic modeling of structures with examples from theNeuquén Basin, Argentina. In: McClay, K.R. (Ed.), ThrustTectonics and Hydrocarbon Systems: AAPG Memoir. AmericanAssociation of Petroleum Geologists, Tulsa, vol. 82, pp. 356–371.

Antonellini, M.A., Pollard, D.D., 1995. Distinct element modeling ofdeformation bands in sandstone. Journal of Structural Geology 17,1165–1182.

Bardet, J.P., Proubet, J., 1992. The structure of shear bands in idealisedgranular materials. Applied Mechanics Reviews 45 (3), S118–S122.

Bergbauer, S., Pollard, D.D., 2004. A new conceptual fold-fracturemodel including pre-folding joints, based on the Emigrant Gapanticline, Wyoming. Geological Society of America Bulletin 116,294–307.

Bieniawski, Z.T., 1984. Rock Mechanics Design in Mining andTunnelling. A.A. Balkema, Netherlands. 272 pp.

Braun, J., Sambridge, M., 1994. Dynamical Lagrangian Remeshing(DLR): a new algorithm for solving large strain deformationproblems and its application to fault-propagation folding. Earthand Planetary Science Letters 124, 211–220.

Bump, A., 2003. Reactivation, trishear modeling and folded basementin Laramide uplifts: implications for the origins of intra-continentalfaults. GSA Today 13 (3), 4–10.

Camborde, F., Mariotti, C., Donzé, F.V., 2000. Numerical study ofrock and concrete behaviour by discrete element modelling.Computers and Geotechnics 27, 225–247.

Cardozo, N., Bawa-Bhalla, K., Zehnder, A., Allmendinger, R.W.,2003. Mechanical models of fault propagation folds andcomparison to the trishear kinematic model. Journal of StructuralGeology 25, 1–18.

Couples, G.D., Stearns, D.W., Handin, J.W., 1994. Kinematics ofexperimental forced folds and their relevance to cross-sectionbalancing. Tectonophysics 233, 193–213.

Cundall, P.A., 1971. Distinct element models of rock and soil structure.In: Brown, E.T. (Ed.), Analytical and Computational Methodsin Engineering Rock Mechanics. Unwin Publishers, London,pp. 129–163.

Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical models forgranular assemblies. Geotechnique 29 (1), 47–65.

Donzé, F., Mora, P., Magnier, S.A., 1994. Numerical simulation offaults and shear zones. Geophys. J. Int. 116, 46–52.

Donzé, F., Magnier, S.-A., Bouchez, J., 1996. Numerical modelling ofa highly explosive source in an elastic–brittle rock mass. Journal ofGeophysical Research 101 (2), 3103–3112.

Erslev, E.A., 1991. Trishear fault-propagation folding. Geology 19,617–620.


Erslev, E.A., Mayborn, K.R., 1997. Multiple geometries and modes offault-propagation folding in the Canadian thrust belt. Journal ofStructural Geology 19, 321–335.

Finch, E. 1999. A Lattice Solid Model for Crustal Extension, PhDThesis, University of Ulster, Coleraine.

Finch, E., Hardy, S., Gawthorpe, R.L., 2003. Discrete elementmodelling of contractional fault-propagation folding above rigidbasement fault blocks. Journal of Structural Geology 25, 515–528.

Finch, E., Hardy, S., Gawthorpe, R.L., 2004. Discrete elementmodelling of extensional fault-propagation folding above rigidbasement fault blocks. Basin Research 16, 489–506.

Ford, M., Williams, E.A., Artoni, A., Vergés, J., Hardy, S., 1997.Progressive evolution of a fault propagation fold pair as recordedby growth strata geometries, Sant Llorenç de Morunys, SEPyrenees. Journal of Structural Geology 19, 413–441.

Gawthorpe, R.L., Sharp, I., Underhill, J.R., Gupta, S., 1997. Linkedsequence stratigraphic and structural evolution of propagatingnormal faults. Geology 25, 795–798.

Gould, H., Tobochnik, J., 1988. Computer Simulation Methods: Part I.Addison Wesley, Reading, Mass.

Hardy, S., Ford, M., 1997. Numerical modeling of trishear fault-propagation folding. Tectonics 16, 841–854.

Hardy, S., McClay, K.R., 1999. Kinematic modeling of extensionalfault-propagation folding. Journal of Structural Geology 21,695–702.

Ismat, Z., Mitra, G., 2005. Folding by cataclastic flow: evolution ofcontrolling factors during deformation. Journal of StructuralGeology 27, 2181–2203.

Iwashita, K., Oda, M., 2000. Micro-deformation mechanism of shearbanding process based on modified distinct element method.Powder Technology 109, 192–205.

Johnson, K.M, Johnson, A.M., 2002a. Mechanical models of trishear-like folds. Journal of Structural Geology 24, 277–287.

Johnson, K.M., Johnson, A.M., 2002b. Mechanical analysis of thegeometry of forced-folds. Journal of Structural Geology 24,401–410.

Kuhn, M.R., 1999. Structured deformation in granular materials.Mechanics of Materials 31, 407–429.

McConnell, D.A., 1994. Fixed-hinge, basement-involved fault-prop-agation folds. Wyoming: GSA Bulletin, vol. 106, pp. 1583–1593.

Mitra, S., Islam, Q.T., 1994. Experimental (clay) models of inversionstructures. Tectonophysics 230, 211–222.

Mitra, S., Mount, V.S., 1998. Foreland basement-involved structures.American Association of Petroleum Geologists Bulletin 82,70–109.

Mora, P., Place, D., 1993. A lattice solid model for the non-lineardynamics of earthquakes. International Journal of Modern. PhysicsC 4 (6), 1059–1074.

Mora, P., Place, D., 1994. Simulation of the frictional stick-slipinstability. Pure and Applied Geophysics 143 (1–3), 61–87.

Narr, W., Suppe, J., 1993. Kinematics of basement-involvedcompressive structures. American Journal of Science 294,302–360.

Patton, T.L., Fletcher, R.C., 1995. Mathematical block-motion modelfor deformation of a layer above a buried fault of arbitrary dip andsense of slip. Journal of Structural Geology 17, 1455–1472.

Patton, T.L., Fletcher, R.C., 1998. A rheoloogical model for fracturedrock. Journal of Structural Geology 20, 491–502.

Place, D., Mora, P., 2001. A random lattice solid model for simulationof fault zone dynamics and fracture processes. In: Mulhaus, H.-B.,Dyskin, A.V., Pasternak, E. (Eds.), Bifurcation and LocalisationTheory for Soils and Rocks '99. A.A. Balkema, Rotterdam/Brookfield.

Place, D., Lombard, F., Mora, P., Abe, S., 2002. Simulation of themicro-physics of rocks using LSMEarth. Pure and AppliedGeophysics 159, 1911–1932.

Rutter, E.H., 1986. On the nomenclature of mode of failure transitionsin rocks. Tectonophysics 122, 381–387.

Saltzer, S.D., Pollard, D.D., 1992. Distinct element modeling ofstructures formed in sedimentary overburden by extensionalreactivation of basement normal faults. Tectonics 11, 165–174.

Scott, D.R., 1996. Seismicity and stress rotation in a granular model ofthe brittle crust. Nature 381 (6583), 592–595.

Sharp, I.R., Gawthorpe, R.L., Underhill, J.R., Gupta, S., 2000. Fault-propagation folding in extensional settings: examples of structuralstyle and synrift sedimentary response from the Suez rift, Sinai,Egypt. Geological Society of America Bulletin 112, 1877–1899.

Shaw, J.H., Shearer, P.M., 1999. An elusive blind-thrust fault beneathmetropolitan Los Angeles. Science 283, 1516–1518.

Stearns, D.W., 1978. Faulting and forced folding in the RockyMountains foreland. Geological Society of America Memoir 151,1–37.

Strayer, L.M., Huddleston, P.J., 1997. Numerical modelling of foldinitiation at thrust ramps. Journal of Structural Geology 19 (3–4),551–566.

Strayer, L.M., Suppe, J., 2002. Out-of-plane motion of a thrust sheetduring along-strike propagation of a thrust ramp: a distinct elementapproach. Journal of Structural Geology 24, 637–650.

Strayer, L.M., Erickson, S.G., Suppe, J., 2004. Influence of growthstrata on the evolution of fault-related folds—distinct elementmodels. In: McClay, K.R. (Ed.), Thrust Tectonics and Hydrocar-bon Systems. AAPG Memoir, vol. 82, pp. 413–437.

Toomey, A., Bean, C.J., 2000. Numerical simulation of seismic wavesusing a discrete particle scheme. Geophysical Journal International141, 595–604.

Wang, Y.-C., Yin, X.-C., Ke, F.-J., Xia, M.-F., Peng, K.-Y., 2000.Numerical simulation of rock failure and earthquake processes onmesoscopic scale. Pure and Applied Geophysics 157, 1905–1928.

Withjack, M.O., Olson, J., Peterson, E., 1990. Experimental models ofextensional forced folds. American Association of PetroleumGeologists Bulletin 74, 1038–1054.

Zehnder, A.T., Allmendinger, R.W., 2000. Velocity field for thetrishear model. Journal of Structural Geology 22, 1009–1014.

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