Mechanical stability model of progradationalcarbonate platform margins under tectonicloads: Deformation of Cretaceous carbonateplatforms in the Sierra Madre Orientalfold-thrust belt (east central Mexico)Juan Contreras1 and Max Suter2

1Departamento de Geología, Centro de Investigación Científica y de Educación Superior de Ensenada, Ensenada, BajaCalifornia, Mexico, 2Instituto de Geología, Universidad Nacional Autónoma de México, Estación Regional del Noroeste,Hermosillo, Sonora, Mexico

Abstract Shortening in the Sierra Madre Oriental fold-thrust belt (east central Mexico) is localized along themargins of Cretaceous carbonate platforms and controlled by mechanical stratigraphy. The platform marginsare deformed by imbricate series of thrust ramps, whereas the coeval basins and platform interiors aredeformed by map-scale detachment folds. Here we present a finite element model to evaluate the influence ofthe boundary geometry and boundary conditions on the style of deformation observed at these basinwardprogradational platform margins. We calculate the stress distribution in a linearly elastic platform-basintransition zone under the action of horizontal tectonic stress, taking into account changes of rock mechanicalproperties across the platformmargin, as well as their dependence on direction, and infer the resulting fracturepatterns based on the Mohr-Coulomb failure criterion. Stress concentrations are predicted at the contactsbetween the massive rocks of the platformmargin and the well-layered rocks of both, the platform interior andthe adjacent basin. Brittle failure of the platform border can be mostly attributed to three effects: mechanicalcoupling between the carbonate platform and a substratum of moderate to low viscosity, variations in layeringand texture that governed themechanical properties of the involved carbonates as well as their dependence ondirection, and the development of sharp domain boundary corners associated with progradational facieschanges. In contrast, the dip of the basement and a possible taper of the overlying Upper Cretaceous shaletoward the basin appear to have little influence on the mechanical failure of the platform margin.

1. Introduction

Detailed geological field observations in the latest Cretaceous to early Tertiary Sierra Madre Oriental fold-thrustbelt (east central Mexico) indicate that shortening is localized at the margins of the Lower Cretaceous El Doctorand Valles-San Luis Potosí (VSP) carbonate platforms (Figure 1) and controlled by mechanical stratigraphy.Contrary to the platform margins, the platform interiors and basins are well stratified, and the platforms haveapproximately twice the thickness of the basins. The eastern platformmargins are deformed by imbricate seriesof thrust ramps, flats, duplexes, and fault-bend folds (Figures 1 and 2), whereas the coeval rocks in the basinand platform interior are mostly deformed by map-scale detachment folds [Suter, 1984, 1987; Suter et al., 1997;Carrillo-Martínez et al., 2001; Gray et al., 2001].

The localization of deformation at the platform margins suggests the presence of internal stress risers thatmay have weakened the material structure of the margin. To test this idea, we carry out an analysis similarto the one performed by Hafner [1951], in which tractions are imposed on the boundaries of a domain with alinear elastic rheology representing an undeformed carbonate platform. We then obtain the state of stress inthe simulated platform by the finite element method and subsequently use the Mohr-Coulomb failurecriterion to identify unstable areas and the orientation of potential shear fractures. With exception of thecentrifuge analog models by Dixon [2004] and Noble and Dixon [2011], discussed below, we are not aware ofany attempt to understand the tectonic deformation of carbonate platform margins within a rock mechanicsframework, whereas the synsedimentary deformation of carbonate platform margins has received moreattention [e.g., Resor and Flodin, 2010; Berra and Carminati, 2012; Boro, 2012].

CONTRERAS AND SUTER ©2015. American Geophysical Union. All Rights Reserved. 1

PUBLICATIONSJournal of Geophysical Research: Solid Earth

RESEARCH ARTICLE10.1002/2014JB011495

Key Points:• Mechanical stabilitymodel of carbonateplatform margins under tectonic loads

• Changes in rock properties across themargins cause spikes in Coulomb stress

• Realistic transverse tectonic loadsrender the platforms unstable

Correspondence to:M. Suter,SuterMax@alumnibasel.ch

Citation:Contreras, J., and M. Suter (2015),Mechanical stability model ofprogradational carbonate platformmargins under tectonic loads:Deformation of Cretaceous carbonateplatforms in the Sierra Madre Orientalfold-thrust belt (east central Mexico),J. Geophys. Res. Solid Earth, 120,doi:10.1002/2014JB011495.

Received 27 JUL 2014Accepted 30 DEC 2014Accepted article online 7 JAN 2015

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2. Structural Field Observations

Detailed structural field observations indicate that the eastern margins of the platforms are deformed bythrust faults that are typically of ramp-flat geometry with ramps (layer-oblique fault segments) in the LowerCretaceous platform margin carbonates and flats (layer-parallel fault segments) in the mechanically weak,clay-rich rocks of the Upper Cretaceous (Figures 1 and 2) [Suter, 1984, 1987; Suter et al., 1997]; north northwestof Laguna de Metztitlán (lower right part of Figure 1a), these structures form a major duplex. The thrustswere partly steepened by imbrication (Figure 3) indicating sequential southwest-to-northeast deformationtoward the foreland. The hanging wall carbonate strata of the imbricates along the eastern edge of the VSPplatform are cut under low angles of 6° to 10° (Figures 2 and 3). On an outcrop scale, the platform edgecarbonates are characterized by layer-oblique and layer-parallel shear fractures and tectonic stylolites; ooidsat the base of the Xilitla thrust (Figures 1 and 2) are unflattened but are marked by a closely spaced styloliticcleavage perpendicular to bedding [Suter, 1984].

According to our observations, deformation in the interior of the two platforms and in the coeval Zimapánand Tampico-Misantla shelf basins is dominantly by regional-scale detachment folds resulting fromdecoupling along several mechanically weak Jurassic stratigraphic units and, subordinately, by thrusts andthrust-related folds (Figure 1). On an outcrop scale, deformation is at some places additionally by buckle folds,dominantly of chevron style, and associated axial planar cleavage [Suter, 1987, 1990], which indicatessignificant variations in the amount of local shortening. Our observations differ from the regional structuralstyle adopted by Fitz-Díaz et al. [2011a, 2012], who did not identify on their structural maps the regional-scalefolds shown in Figure 1; instead, their sections show schematic mesoscopic buckle fold trains across theZimapán and Tampico-Misantla basins. What is more, Fitz-Díaz et al. [2011a, 2012] indicate on their structuralmaps numerous regional-scale cross faults, for which to our knowledge there is no evidence.

The angle between the shortening direction and the orientation of the platform margins also influencedthe structural style. A single thrust (e.g., El Doctor thrust) or a series of imbricates (e.g., La Misión,Lobo-Ciénega, and Agua Zarca thrusts on the eastern side of the VSP platform) exists along the platformwhere the margin is subparallel to the structural trend of the fold-thrust belt (Figure 1). On the other hand,an en échelon array of thrust faults can be observed where the platform edge is somewhat oblique tothe structural trend of the fold-thrust belt. Examples are the left-stepping array of the Jiliapan and ElVolantín thrusts on the western margin of the VSP and the right-stepping en échelon array on theeastern margin of the VSP platform in the northern part of Figure 1, which is composed of the La Misión,Lobo-Ciénega, Agua Zarca, and Xilitla thrusts and the frontal thrust to the north of Xilitla. Shortening of theindividual en échelon segments is a maximum at the platform edge and diminishes in the basin and

WSW ENE

Figure 2. Cross-sectional view, ~3500m wide, of the Xilitla thrust (Figure 1) from the south, across Highway 120 (whitearrows). The dotted lines indicate bedding traces. The hanging wall displays the platform edge (background) to basin(foreground) transition of the Lower Cretaceous Valles-San Luis Potosí platform and is thrust upon mechanically weak,clay-rich rocks of the Upper Cretaceous. The fault (red continuous line) cuts the strata of the hanging wall under low angles(8° in the eastern part and<2.5° in the western part) but is oriented parallel to the strata of the footwall. This configurationimplies the existence of a tectonic ramp in the subsurface, which can also be inferred from the fault-bend fold with asubvertical axial plane (dashed line) in the hanging wall.

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platform interior. The platform margin is the locus of a right-lateral strike-slip fault north of Xilitla (Figure 1),where the margin is subparallel to the direction of shortening.

Similar structural observations were also made along the northeastern edge of the VSP platform near Aramberri[Tardy et al., 1976], along the west central margin of the same platform [Aranda-Gómez et al., 2000], along thenortheastern edge of the Cretaceous Córdoba platform in southeastern Mexico [Mossman and Viniegra, 1976;Prost and Aranda, 2001; Roure et al., 2009], along thewesternmargin of the Cretaceous Guerrero-Morelos platformwhere the Teloloapan thrust system [Cerca et al., 2007] follows the platform margin outlined by de Cserna et al.[1978], and along themargins of carbonate platforms in the Alps [e.g.,Doglioni, 1985, 1988; Ford and Stahel, 1995].

Moreover, the structural style observed along the eastern margin of the VSP platform is an analog to thesetting of the hydrocarbon reservoirs of southeastern Mexico, which produce mostly from the fracture

Agua Fría Thrust Fault

UNDEFORMED CONFIGURATION OF THE VALLES-SAN LUIS POTOSÍ CARBONATE PLATFORM

Blind Thrust Fault

Lobo-Ciénega Thrust Fault

Agua Zarca Thrust Fault

Tamazunchale Thrust Fault

PRESENT CONFIGURATION

W E

tectonic transport ~33 km

platform interior facies

platform margin facies

basin facies Jurassic rocks

a)

b)

f )

e)

d)

c)

Pisaflores Anticlinorium

Figure 3. Kinematic forward model showing how shortening of the Valles-San Luis Potosí carbonate platform and the adjacent part of the Tampico-Misantla shelfbasin developed in a hinterland-to-foreland sequence. The system is open in the northeast. The trace of the lowermost section is marked in Figure 1.

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porosity of diagenetically dolomitized carbonate platform foreslope deposits in structural traps formed byfault-bend folds [e.g., Suter and Vargas, 1983; Marmisolle-Daguerre, 1984; García-Molina, 1994; Mitra et al.,2005; Contreras-Pérez, 2010]. Further understanding of this structural configuration by our study maytherefore be of economic relevance to the petroleum industry.

3. Model of Subsurface Structure and Kinematic Simulationof Map-Scale Deformation

A geomechanical model of the tectonic deformation at carbonate platform margins cannot rely on thelimited surficial structural observations but requires additional subsurface data for the lower parts of theplatforms and, where not available, a model of the subsurface structure. Thin-skinned deformation, with aslight (≤2°) inclination of the basal detachment toward the internal part of the belt, can be inferred fromthe calculation of the depth to detachment in area-balanced cross-sectional models of the Sierra MadreOriental fold-thrust belt, which are constrained by field observations and subsurface data from severalhydrocarbon exploration wells [Suter, 1987, 1990; Carrillo-Martínez et al., 2001] and based on the generalprinciple in continuum mechanics of the conservation of mass [Truesdell and Toupin, 1960; Malvern, 1969].Based on these sections, shortening measures approximately 10 km on the eastern edge of the El Doctor and10–12 km on either edge of the VSP carbonate platform.

We have incorporated these observations in the cross-sectional kinematic forward simulation in Figure 3,which illustrates how shortening of the VSP platform and the adjacent part of the Tampico-Misantla basindeveloped in time. The modeling is by a cellular automaton approach [Toffoli, 1984; Wolfram, 2002]and assumes that the hanging wall deforms by flexural slip parallel to the fault surface [Contreras and Suter,1990, 1997; Contreras, 1991, 2002; Contreras-Pérez, 2010]. The defined deformation corresponds to thesuperimposition of a vector displacement field and a strain field and does not cause a change in area, which istypical of deformation by simple shear [Truesdell and Toupin, 1960].

The resulting model of the deformed structural geometry (Figure 3) is in agreement with the availableobservational data. The system is open in the northeast. The unshortened part of the section is characterizedby a series of incipient thrust ramps. Spacing between the ramps varies between 10 and 25 km (Figure 3a)and is shortest between the two ramps that cut across the platform margin (Lobo-Ciénega and Agua Zarcathrust faults). It is likely that deformation proceeded from southwest to northeast (hinterland to foreland) inthe following sequence: (1) The Agua Fría thrust broke the platform into halves; its modeled steppedsubsurface geometry explains the two major folds within the thrust sheet near the western platform margin(Figures 1 and 3b). (2) The Lobo-Ciénega thrust fault, which cuts across the platform edge deposits, has a flatin the clay-rich Upper Cretaceous rocks (which are not shown in Figure 3). Motion along this steppedthrust fault is compatible with the structure observed within the thrust sheet and is likely to have passivelyrotated part of the Agua Fría thrust (Figure 3c). (3) Similarly, motion along the Agua Zarca thrust deformedpiggyback style parts of the overlying Lobo-Ciénega thrust (Figure 3d). (4) Area-balanced modeling of thesubsurface structure [Suter, 1987] suggests that the Tamazunchale thrust reaches, in the form of a ramp, thebasal detachment only below the western flank of the Pisaflores anticlinorium (Figure 3e). Motion alongthe Tamazunchale thrust further accumulated shortening at the platform margin and steepened the overlyingthrusts (Figure 3e). (5) Finally, motion along a blind thrust (Figure 3f) renders the model compatible with theavailable surficial structural observations [Suter, 1990]. Such a blind thrust, extending layer parallel into theChicontepec foredeep, requires detachment folding in the overlying basin fill, such as the Axtla synclineadjacent to the front of the fold-thrust belt (Figure 1). Overall shortening of the section is ~33 km or 39%.

South of our section, thrust sheets involving Middle Jurassic red beds as well as older sedimentary andvolcanic rocks have been documented [e.g., Burckhardt, 1930; Flores-Castro, 1993; Ochoa-Camarillo, 1996;Rosales-Lagarde et al., 2005]. In contrast, Fitz-Díaz et al. [2012] interpret the outcrops of Middle Jurassic redbeds in the Amajac River valley, which reach there elevations up to ~900m above sea level [Suter, 1990], to beautochthonous basement located beneath the basal detachment fault of the fold-thrust belt. However, onthe western slope of the nearby Claro River valley, in the projected continuation of the Tamazunchale thrust,the Naopa thrust (Figure 1) places the same Middle Jurassic red beds on top of Upper Jurassic rocksalong a subhorizontal thrust fault, 6 to 9 km wide [Ochoa-Camarillo, 1996], which causes us to questionthe interpretation by Fitz-Díaz et al. [2012]. Moreover, the interpretation by Fitz-Díaz et al. [2012] that the

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deformation in the Chicontepec basin is limited to gentle folds and meter-scale displacements on late thrustfaults (their D2 phase) is not corroborated by the observation of structures such as the Axtla syncline(Figure 1), where the rocks of the basin fill are vertical to overturned on a regional scale [Heim, 1940; Suter,1990; Santillán-Piña and Aguayo-Camargo, 2011] or the frontal thrust near Highway 85 (Figure 1), where theedge of the VSP platform was thrust on top of Chicontepec basin fill [Suter, 1980, 1990].

4. Mechanical Stratigraphy

Figure 4a synthesizes in a schematic section the mechanical stratigraphy and the undeformed boundarygeometry across the eastern margin of the Lower Cretaceous VSP platform. The platform and the coevalTampico-Misantla shelf basin (Figure 1) are of contrasting lithology and thickness. The platform edge andforeslope are composed of poorly stratified, massive carbonates, whereas the platform interior and the basin arewell stratified. The platform has approximately twice the thickness of the basins [Minero, 1991]. According toEnos and Stephens [1993], the depositional relief is about 1000m, and the foreslopes are as steep as 20–45°.Along the platform margins, the sediments of the platform interior are located vertically above the ones of theplatform edge and foreslope and the latter above the basin sediments due to progressive outbuilding of theplatform (Figure 4); the angle of progradation is 20–25° between the platform edge and the foreslope [Minero,1991; Enos and Stephens, 1993]. Vertically, the Lower Cretaceous carbonates are sandwiched by mechanicallyweaker Upper Jurassic and Upper Cretaceous stratigraphic units with a high shale content (Figure 4). No growthfaults have been observed along the platformmargins despite of claims of their existence [e.g.,Wilson andWard,1993; Carrillo-Martínez et al., 2001]; the faults at the platform margin are invariably thrust faults.

A conceptual model of the Lower Cretaceous carbonate platform edge environment in central Mexico is givenbyWilson [1975] andWilson and Ward [1993]. The platforms are rimmed by a framework of rudists, which weresessile bivalves (now extinct) having one valve attached to the substrate. These peculiar shallow-waterframebuilders were of large bulk and capable of rapid growth. Their buildups form mostly massive orthick-bedded limestone. The difference in thickness and the depositional relief between the platform andthe adjacent shelf basin (Figure 4a) resulted from the rapid production of carbonate by the rudist colonies duringa rise of sea level as opposed to the slow accumulation of fine-grained limemud in the basin, which derives fromthe skeletons of planktonic microorganisms. Since the rate of sea level rise was less than the vertical builduprate of the rudist colonies, these prograded basinward on top of their own debris (Figure 4a). The sediments on

a)

b)20 km 7.5 km 7.5 km

1.7 km750 m20o

Γv

ΓfΓt Γ0

Γf

15o

0.5 km200 m

1250 m

Figure 4. (a) Schematic section across the progradational eastern margin of the Valles-San Luis Potosí platform synthesizing the mechanical stratigraphy andthe undeformed boundary geometry of our model. (b) Sedimentary domains of our model and their boundary geometry, which is partly based on the betterconstrained Faja de Oro carbonate platform. The boundary conditions are described in the text and in Figure 6.

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the platform foreslope are mainly debris from upslope being distributed downslope by gravity, such as coarseblocks and sheets of forereef breccia, which merge toward the basin gradually with lime mud.

Since the reconstructed cross-sectional geometry of the Lower Cretaceous VSP carbonate platform marginshown in Figure 4a has significant uncertainties because of a shortage of subsurface information and dueto its subsequent incorporation into the Sierra Madre Oriental fold-thrust belt, we additionally takeobservations from the Faja de Oro carbonate platform into account. This platform is located to the east ofthe Sierra Madre Oriental fold-thrust belt, below the Chicontepec foreland basin (Figure 1b), and hasessentially remained undeformed. Moreover, its structure is well documented by seismic imaging and fromdeep boreholes [Enos, 1977, 1985; Loucks et al., 2011]. The relief between basin and platform is constrainedby wells as ~1000m and the thickness of the basin deposits as ~750m; the thickness of the platforminterior deposits can therefore be inferred as ~1750m (Figure 4b). The average inclination of the platformforeslope is ~15° [Coogan et al., 1972].

5. Geomechanical Model and Corresponding Boundary Value Problem

As outlined above, the major purpose of this study is an evaluation of the influence of the boundarygeometry, boundary conditions, and rock mechanical properties on the style of deformation observed atthese basinward progradational carbonate platform margins. We calculate for that purpose the stressdistribution in an undeformed, linearly elastic platform-basin transition zone under the action of horizontalcompressive tectonic stress, taking into account changes of rock mechanical properties across the platformborder as well as their transverse anisotropy in the stratified layers, and infer the resulting fracture patternsbased on the Mohr-Coulomb failure criterion. Since the pioneer work by Hafner [1951], numerous authorshave experimented with boundary conditions and rheologies to model natural fault patterns observedin rocks [e.g., Mandl, 1988; Gerbault et al., 1998; Panian and Wiltschko, 2007]. However, the effect ofboundary geometries more complicated than prisms, half spaces, and rectangular areas has not been givenmuch attention.

Our model does not deal with the development of the finite-strain, large-scale structures observed in theplatform interior and coeval basins (detachment folds in well-stratified carbonates) or at the platformmargins (ramps, flats, fault-bend folds, and duplexes in mostly poorly stratified, statistically homogeneouscarbonates). Instead, we focus on the question of where initial failure zones develop and how faultingin this initial stage is influenced by boundary conditions, pore pressure, lateral changes in materialstructure (i.e., changes from massive to well-stratified rocks), and external domain geometry[Contreras-Pérez, 1993].

5.1. Material Response

As outlined above, our goal is to find the state of stress prior to the linkage of fractures to large faults, i.e., thestress field associated with small displacements. At low temperature and confining pressure, most rocksdevelop shear fractures at strains ranging from 10�3 to 10�2 [Paterson and Wong, 2005]. It can be assumedthat these fractures will coalesce to large faults with finite displacements as deformation proceeds. For thatreason, we consider the rocks of our model to behave as linearly elastic solids. This approach is valid forsmall displacements, whereas for large displacements, the observed structural style is likely to have beeninfluenced by changes in rock strength across the platform margins (see discussion below).

The deformation of linearly elastic solids is described by the equation of conservation of momentum, whichcan be expressed as

∂σij∂xj

þ bi ¼ 0 (1)

[e.g.,Malvern, 1969], where σij is the stress tensor and bi the gravitational body force. Now Hooke’s law statesthat for small displacements stress and strain are linearly related

σij ¼ Cijklεkl; (2)

where εkl is the infinitesimal strain tensor, which is related to the displacement field, u, in the following way

εij ¼ 12

∂ui∂xj

þ ∂uj∂xi

� �: (3)

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In equation (2), Cijkl is the stiffness tensor,a fourth-rank tensor that incorporates thematerial response of the solid tomechanical loading. For isotropicmaterials, this equation simplifies to

σij ¼ E1þ ν

εij þ νE1þ νð Þ 1� 2νð Þ εkkδij;

(4)

where E is the Young’s modulus, ν is thePoisson’s ratio, and δij is the Kroneckerdelta. Layered, transverse anisotropic(orthotropic) materials, by contrast,require two pairs of Young’s moduli(E1 and E2) and Poisson’s ratios (ν12 andν21) to describe the material response

of the layers along the transverse and longitudinal directions. For such materials, the stress tensor isgiven by the following expressions

σ11 ¼ E11þ ν12ν21

ε11 þ ν12E11þ ν12ν21

ε22; (5)

σ22 ¼ E21þ ν12ν21

ε22 þ ν21E21þ ν12ν21

ε11; (6)

σ12 ¼ E21þ ν12

ε12: (7)

Rocks are semibrittle materials that yield at high deviatoric stress, at which they start to localize strain alongshear bands. Experimentally, it has been found that the peak stress satisfies the Mohr-Coulomb failurecriterion [e.g., Paterson and Wong, 2005; Hafner, 1951]

τj j � τ0 � μi σn � pwð Þ ≥ 0 (8)

(Figure 5), where τ0 is the cohesion and μi the coefficient of internal friction of the rock (intercept and slope,respectively, of the Mohr-Coulomb failure envelope on Figure 5); laboratory-derived coefficients of internalfriction range from about 0.5 to 2.0 with a mean value of ~1.2 [Zoback, 2007]. σn and τ are the normal andshear stresses acting on the potential fracture plane and are given by σn= σijnjni and τ = σij(δij� njni), where nis the unit vector normal to the fracture plane. The term (σn � pw) represents the effective normal stressacting on the potential fracture plane, which is the difference between the normal confining stress σn andthe pore pressure pw from intergranular fluids. The left-hand side of inequality (8) is known as the Coulombstress and is denoted by Δσc= ∣τ∣� τ0�μi(σn� pw). We present and discuss the results of our model interms of the Coulomb stress. Unlike the stress tensor σij, which has nine components and is difficult toconceptualize, the Coulomb stress captures the mechanical state of a volume of rock in a single scalarquantity that is straightforward to understand: changes in Δσc determine whether a volume of rock has beenbrought closer to, or further from, failure [King et al., 1994].

The orientations ϕ of the potential shear fractures, with respect to the orientation of the maximum principalstress, are given by the relation (Figure 5)

ϕ ¼ ± ½ arctan μið Þ (9)

Since an increase in pore pressure pw reduces the yield strength of rocks (equation (8)) [e.g., Engelder, 1993;Zoback, 2007], we will analyze two cases: (i) the case of no pore pressure (dry rock or no connectedporosity) and (ii) the case where the pores form an interconnected network, such that the fluids can flowfreely through the elastic rock matrix. In case (i) the pore pressure is simply pw = 0. In case (ii), the fluidsexert a hydrostatic pore pressure

pw ¼ ρwgx2; (10)

where ρw represents the density of water, g is the acceleration of gravity, and x2 represents depth.

Figure 5. Mohr circle and linearized Mohr-Coulomb failure envelopefor brittle and semibrittle materials. Instead of an exact value for thecohesion τ0, a tolerance Δτ0 of ± 5 MPa is used in our model to takeinto account the large variability in cohesion displayed by limestone.

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After the initiation of tectonic fracturing, the pore pressure within our system most likely approximated thehydrostatic pressure; pore pressure close to hydrostatic has been measured worldwide in the brittle crust ofintraplate regions by deep drilling [Townend and Zoback, 2000; Zoback et al., 2007].

5.2. Sedimentary Domains and Boundary Conditions

The sedimentary domains of our model and their boundary geometry are shown in Figure 4b. The transitionzone between the carbonate platform and the adjacent shelf basin is subdivided into four domains: platforminterior, platform edge, foreslope, and basin. Below, we will assign specific material properties to each ofthese domains. In our model, the platform interior measures 20 km, whereas the platform edge together withthe foreslope and the basin have a length of 7.5 km each (Figure 4b). For the upper part of the foreslope, weassume an inclination of 15°, whereas the surface of the lower part of the foreslope (Figure 4b) is approximatedby an error function, which lessens the stress concentration in the transition between the foreslope and thebasin (see below). For the angles of progradation between the platform edge and the foreslope and betweenthe foreslope and the basin, we assume 20°. For the thicknesses of the platform and the basin and thedepositional relief between them, we take the values from the Faja de Oro platform (1750m, 750m, and1000m, respectively).

We now describe the boundary conditions of our model. The VSP platform rests on organic-richcarbonaceous shale of the Upper Jurassic Pimienta Formation (Figure 4b), which can be assumed to have apressure-dependent rheology. Shale flows in a linear Newtonian fashion when subject to confiningpressure in excess of 30MPa [Chang and Zoback, 2009], which corresponds to the lithostatic pressure at adepth of ~1.5 km. Moreover, we assume this viscous layer to have a tapered geometry, with its thicknessdecreasing from ~500m below the platform interior (constrained by well data [Suter, 1987, 1990]) to athickness of ~200m below the adjacent basin (Figure 4b). The basement dips toward the hinterland at anangle of 0.5°.

Based on these considerations, we impose a fluid-solid boundary condition at the bottom of the platform.Applying the simple force balance analysis shown in Figure 6, the tangential (τ) and normal (σn) tractionsacting on the base of our model are

Figure 6. Sketch of the boundary conditions used in the model. The left vertical boundary Γt is subjected to a transversetectonic load σt. The bottom of the platform is coupled to the underlying shale of the Pimienta Formation through viscousstress. No loads are applied at the upper boundary, which can deform freely.

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τ ¼ �η ε̇ u1h; on Γv ; (11)

σn ¼ �η ε̇ u2 þ u1θh

þ Δρgu2; on Γv : (12)

where η is the viscosity of shale, ε̇ is the strain rate, h is the thickness of the Pimienta Formation, θ the dipof the structural basement, and Δρ is the density contrast between shale and limestone. Physically, theright-hand side of equation (11) represents viscous stress caused by simple shear deformation in the shale(τ12 in Figure 6). On the right-hand side of equation (12), the first term represents viscous stress related topure shear deformation in the shale (τ2 in Figure 6), whereas the second term represents an unbalancedpressure head due to the density contrast between shale and limestone. Notice that the latter equationincludes the term u1θ, which results from the upslope transport of the shale layer along the inclinedbasement [Reynolds, 1886].

Like other authors [e.g., Hafner, 1951; Liu and Ranalli, 1992; Gerbault et al., 1998], we impose on the left side ofthe domain region (boundary Γt in Figures 4b and 6) horizontal loads that increase linearly with depth

t ¼ 0; on Γt; (13)

σn ¼ �σtx2=H on Γt; (14)

where σt corresponds to the tectonic load applied at the base (H) of the carbonates.

The Lower Cretaceous platform and basin carbonates are covered by Upper Cretaceous rocks with a highshale content, which we consider a linearly elastic material. We further assume that the displacements arecontinuous across the contact between the Lower Cretaceous limestone and the Upper Cretaceous shale.By contrast, the top of the latter (the uppermost boundary of the model in Figures 4b and 6) can deformfreely, i.e.,

Sijnj ¼ 0; on Γf : (15)

Finally, the right boundary (Figure 4b) remains fixed

u1 ¼ u2 ¼ 0 on Γ0: (16)

5.3. Material Properties

The elastic properties of carbonates are scale dependent and display a wide range in values. Layering, clastsize, porosity, texture, and microfractures cause variations up to an order of magnitude [Zoback, 2007; Resorand Flodin, 2010, and references therein]. In our model, we assume layering and clast size to have themost important effect on stiffness. Well-layered rocks are compliant parallel to their stratification due to thinshale intercalations; as a result, the Young’s modulus measured parallel to bedding is a fraction, usuallyranging between one half and one fourth, of the Young’s modulus measured perpendicular to the beddingorientation [Miller et al., 2013]. Grain size exerts a primary control on the elastic parameters of low-porositycarbonates; coarse-grained carbonates have high Young’s moduli and Poisson ratios, whereas fine-grainedcarbonates are more compliant and compressible [Hatzor and Palchik, 1998, and references therein].

On the other hand, we ignore the effect of porosity on stiffness. Sedimentological studies indicate thematrix porosity in the platform margin rocks of our study area to be very low. Burial diagenesis, whichoccurred before thrust faulting, profoundly altered the composition and texture of these rocks bycompaction, cementation, and replacement by dolomite and anhydrite and reduced the initial porositysignificantly [Minero, 1991]. Similarly, laboratory measurements of the partly dolomitized limestone in theinterior of the El Doctor platform indicate only 1 to 5% matrix porosity [Palacios-Nieto, 1982] and so dogeophysical wireline measurements in the Cretaceous carbonate platform foreslope deposits in southeasternMexico [Marmisolle-Daguerre, 1984].

The values of the elastic properties assigned in our model (Table 1) are based on the lithology types in theplatform-basin transition described above (Figure 4a). The well-stratified carbonates of the platform interiorare represented by a stiff, moderately compressible, transverse anisotropic (orthotropic) medium with aPoisson’s ratio of 0.30 and Young’s moduli E1 of 15 GPa parallel to bedding and E2 of 70 GPa perpendicular tobedding (Table 1). Laboratory measurements of the partly dolomitized limestone in the interior of the El

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Doctor platform indicate a Poisson’s ratio of 0.29, whereas a Young’s modulus of 65 GPa was measured in thesame rocks in situ and moduli between 71GPa and 78GPa in the laboratory [Palacios-Nieto, 1982].

In contrast, the massive, poorly stratified carbonates of the platform edge are represented by a stiff,moderately compressible medium with a homogenous isotropic structure [Resor and Flodin, 2010], aPoisson’s ratio of 0.3, and a Young’s modulus of 70 GPa (Table 1). Similarly, the rocks of the foreslope arerepresented by a moderately compressible, isotropic but somewhat less stiff medium (ν: 0.25; E: 45 GPa;Table 1) due to grain size sorting downslope and incipient layering in the deposits of the outer foreslope. Thebasin carbonates, which have a relatively compliant material structure because of their fine grain size andthe intercalation of thin shale layers, are represented by a relatively low Poisson’s ratio of 0.2 and Young’smoduli E1 of 3.5 GPa parallel to bedding and E2 of 15 GPa perpendicular to bedding (Table 1).

Finally, the Upper Cretaceous organic matter-rich carbonaceous shale overlying the Lower Cretaceouscarbonates is represented by a weak, compressible anisotropic medium with E2 and E1 moduli of 15MPa and3.5MPa, respectively, and a Poisson ratio of 0.15. These are average values measured in samples of the UpperCretaceous Eagle Ford Formation of South Texas [Sone and Zoback, 2013a, 2013b], which is lithologicallysimilar to the Upper Cretaceous rocks of our study area.

The strain rate ε̇ in equations (11) and (12) was obtained by assuming linearly progressive deformation bysimple shear, ε̇ = tan (Δx/h)/Δt, of the basal shale layer (Pimienta Formation, Figures 4 and 6). A strain rateof 2.6 · 10�13 s�1 results from the linear shortening Δx of 33 km between the platform interior and theChicontepec foreland basin (Figures 1 and 3), an average thickness h of 300m of the shale layer, and a durationΔt of the deformation of 13.6Myr. The duration time is bracketed by the 62.2Ma age of posttectonic igneousintrusive rocks on the eastern platform edge [Suter, 1984] and the early Eocene age of the Chalma shale[Barker and Berggren, 1977], which is the youngest layer involved in the vertical to overturned western flank ofthe Axtla syncline in the foreland basin (Figure 1); it takes into account a 48.6Ma age for the top of the earlyEocene [Walker and Geissman, 2009]. A shortening rate, Δx/Δt, of 2.4mm/yr results from the same values.

Other parameters of the carbonates represented in the model are a constant coefficient of internal frictionof 0.75, a uniform bulk density of 2700 kg/m3, and a cohesion of 15MPa (Table 1). For the viscous layer at thebase (Pimienta Formation), we assume a density of 2100 kg/m3 and a viscosity of 5 × 1018 Pa s [Contrerasand Negrete-Aranda, 2014]. The viscosity of shale varies over 3 orders of magnitude; it falls between that ofhalite, which ranges between 1016 Pa s and 1018 Pa s [van Keken et al., 1993], and those of the much stifferlimestone and sandstone, which range between 1020 and 1022 Pa s [Nguyen et al., 2013].

The cohesion, τ0 (intercept of the Mohr-Coulomb failure envelope in Figure 5), is a critical parameter in ourmodel; it constrains the magnitude of the tectonic load that can be applied at the left boundary, which isthe driving force that causes the platform to fail. The internal state of stress cannot exceed locally the

Table 1. Material Properties Used in the Model

Symbol Platform Interior Platform Edge Foreslope BasinShale

OverburdenShale

Substratum Units

Materialstructure

- orthotropic(well stratified)

isotropic(poorly stratified)

isotropic(poorly stratified)

orthotropic(well stratified)

orthotropic(well stratified)

isotropicfluid

-

Young’smodulus

E 70a,c 70c 45c 15a 15a,d - GPa15b,c 3.5b 3.5b,d

Poissonratio

ν 0.3c 0.30c 0.25c 0.20c 0.15d - nondimensional

Viscosity η - - - - - 1019e Pa sCohesion τ0 15 15 15 15 - - MPaInternalfriction

μ 0.75 0.75 0.75 0.75 - - nondimensional

Density ρ 2700 2700 2700 2700 2100 2100 kg/m3

aDirection transverse to stratification.bDirection parallel to stratification.cBased on values in Resor and Flodin [2010].dBased on values in Sone and Zoback [2013a, 2013b].eBased on Contreras and Negrete-Aranda [2014].

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cohesive strength without the material deforming, e.g., along localized shear bands [e.g., Contreras, 2013]. Astate of stress exceeding the yield stress is for that reason physically unrealistic [Dunne and Petrinic, 2005]. Ourproblem, consequently, consists in finding the tectonic load σt (Figure 6) such that

τj j � τ0 þ μi σn � pwð Þ ≤ Δτ0: (17)

Expression (17) is a root-finding problem, in which Δτ0 (Figure 5) is, in principle, a small-order quantity knownas the tolerance. We solve problem (17) by means of a simple bisection method using a shear strength, τ0, of15MPa and a tolerance, Δτ0, of ±5MPa. Note that the value of the former parameter is up to an order ofmagnitude smaller than the ones obtained from laboratory samples [e.g., Palchik, 2006], which suffer fromscale effects that overestimate the strength of rocks at the regional scale [Scholz, 2002]. On the other hand,we provide a relatively large tolerance of 5MPa due to stress concentrations that appear where the systemboundaries form corners and reentrants (see below); without providing some leeway, the model would failat very low compressive loads.

5.4. Numerical Solution

We solve the equations of our model by the finite element method using the public domain partialdifferential equation solving program FreeFem++ [Hecht, 2012]. First, the modeled region is subdivided intotriangular elements: in the platform domain with a resolution of 80m, in the platform edge and foreslopedomains with a finer resolution of 30m, and near the contacts between the platform edge, foreslope, andbasin domains with a resolution of 20m. A piecewise continuous displacement field u is then calculated thatminimizes the work done by the applied tectonic load. Once u is known, we solve for the state of stress fromeither equation (4) or equations (5)–(7), depending on the material structure of the medium. Eventually, theunstable areas within the model are determined by the Mohr-Coulomb failure criterion from equation (8).

5.5. Results5.5.1. Case of No Pore PressureFirst, we review the results for a version of our model that does not take into account pore pressure, illustratedin Figure 7a, which requires a tectonic load σt of ~80MPa, applied at the base of the carbonates, to bringthe platform to failure. At this stress level, which is twice the frictional failure equilibrium of the uppermost2 km of the crust [Zoback and Healy, 1984], already existing, optimally oriented faults are likely to becomeactive before the platform fails. The distribution of the Coulomb stress (Δσc), resulting from equation (8), isshown in Figure 7a. The areas marked by cold colors are in a steady, clamped state where Δσc< 0; the areas inwhite are in a critical state of imminent failure; and the areas marked by warm colors are in an unclamped,unstable state where 0<Δσc ≤Δτ0. As expected, the platform interior is characterized by large Δσc values.This is a boundary effect produced by the horizontal load imposed on the boundary Γt and may havetriggered the formation of the Agua Fría thrust fault (Figure 3). The magnitude of Δσc decreases rapidlytoward the platform margin, but other Coulomb stress spikes Δσc result in the platform-basin transition zonenear sedimentary domain boundaries (Figure 7a). Notice that their magnitudes are partly higher than thosein the platform interior.

Figure 7b highlights the unstable regions (0<Δσc ≤Δτ0; regions in warm colors on Figure 7a), whichcover ~3% of the model, and additionally shows the orientations of the shear fractures predicted by theMohr-Coulomb failure criterion. Only the set of fractures compatible with the boundary conditions is shown.These are the fractures with the largest component along the tectonic displacement vector, which maximizethe dot product between the tectonic displacement vector and the unit vectors describing the orientationof the conjugate fractures. The arrows indicate the direction of material transport along these shear fractures.The model predicts fracturing in the platform interior only. It can be concluded that the case of no porepressure does not provide a satisfactory explanation of the deformation pattern observed in the VSPcarbonate platform. We will see below, however, that the stress spikes at the sedimentary domain boundariesare intensified by pore pressure, which could explain the nucleation of the Lobo-Ciénega and Agua Zarcathrust faults, structures that closely follow the contacts between the progradational domain boundaries ofthe platform-basin transition zone (Figure 3).

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5.5.2. Case of Hydrostatic Pore PressureAs mentioned previously, the pore pressure within the system most likely approximated the hydrostaticpressure after the initiation of tectonic fracturing and remained constant at tectonic timescales. In this case,the unstable area of the model increases to ~15% (Figure 7c), whereas the tectonic load σt on the base of theplatform, required to bring the platform to failure, is reduced to ~25MPa. This value is broadly equivalentto the frictional failure equilibrium of the uppermost crust. Thus, the boundary conditions for the case ofhydrostatic pore pressure are consistent with the state of stress prevailing in the crust. Another importantdifference brought about by the introduction of pore pressure into our model is a drastic change in thedistribution of unstable zones (Figure 7d): the Mohr-Coulomb failure criterion predicts most of the base of

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Figure 7. (a) Distribution of Coulomb stress resulting from the finite element calculations that do not take pore pressure into account. A transverse tectonic load of~80MPa at the left boundary of the model is required to bring the carbonate platform to failure. Stress concentrations in the platform-basin transition zone neardomain boundaries may explain the nucleation of the Lobo-Ciénega and Agua Zarca thrust faults. (b) Unstable areas and orientation of potential shear fracturespredicted by the Mohr-Coulomb failure criterion. Only the set of shear fractures compatible with the boundary conditions is shown. The arrows indicate the directionof potential material transport along these fractures. (c) Distribution of Coulomb stress for the calculations including hydrostatic pore pressure. A transverse tectonic loadof ~25MPa at the left boundary of the model is required to bring the carbonate platform to failure. (d) Unstable areas and orientation of potential shear fracturespredicted by the Mohr-Coulomb failure criterion. The unstable area of the model increases from ~3% in the case of no pore pressure to ~15%.

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the platform and the adjacent basin to be under failure. This effect was predicted by Hubbert and Rubey[1959] in their classical work, in which, given a sufficiently high pore pressure, longer fault blocks could bepushed over a nearly horizontal plane. This explains the formation of the thrust faults observed at theplatform edge (Figures 1 and 2) as well as the Tamazunchale thrust, which we assume to be rooted in thewestern part of the Tampico-Misantla basin (Figure 3). The distribution of unstable areas in Figure 7d alsosupports the existence of the detachment at the base of the shale layer (Pimienta Formation) beneaththe VSP carbonate platform assumed in our kinematic simulation (Figure 3). Therefore, the version of ourmodel incorporating hydrostatic pore pressure predicts the observed large-scale deformation pattern inthe VSP carbonate platform remarkably well.

The orientations of faults computed from the predicted fracture patterns are shown in Figure 8. These lines areconstructed such that the fractures are tangential to them; in a sense, they are similar to flow lines in fluiddynamics problems. Notice that they are directed toward the free surface of the model, where the confiningpressure decreases andwhere the rocks aremore prone to brittle failure. The faults indicatingmaterial transporttoward the tectonic basement have been omitted in Figure 8; they are unlikely to develop, since the basementis intrinsically more resistant and since the shear strength of the rocks increases at deeper levels. This leavesus with a set of potential faults confined to a narrow band at the base of the platform that probably decoupledthe carbonates from the underlying shale layer along a detachment surface. A second set of faults developswithin the platform-to-basin transition zone (Figure 8) and is oriented parallel to the progradational domainboundaries (Figure 4). These potential faults reach about halfway up the carbonate edifice.

A more complex fault pattern develops in the foreslope. Here the faults cut through the entire domain(Figure 8). They have a ramp-flat-ramp geometry that initiates at the base as a 30° dipping ramp, curves in themiddle part of the foreslope to a shallower dip, and cuts through the outermost foreslope, where theintegrated strength of the platform is weakest. This is a boundary effect resulting from the inclined geometryof the platform foreslope, since the principal stresses must be oriented perpendicular and parallel to thesurface of the domain region [Anderson, 1951]. This boundary effect causes a rotation of the principal stresstrajectories within the foreslope. It explains why the hanging wall carbonate strata of the imbricate thrustsdocumented along the eastern edge of the VSP platform are cut under low angles of 6° to 10° [Suter, 1984],which is half or less of the values commonly observed in map-scale thrusts [Suppe, 1985]. Similarly, asubhorizontal segment of the El Doctor thrust fault, 4.5 km long, is likely to exist in the subsurface and cut thethrust sheet under very low angles [Suter et al., 1997].

5.6. Sensitivity of the Model to Changes in Material Property Values and Boundary Geometry

We now perform a sensitivity analysis of the results (model version that incorporates hydrostatic porepressure) to test their robustness to changes in viscosity, material anisotropy, geometry of the overburden,and basement slope. We chose these parameters because they control the development of shear stress

20000 22500 25000 2700015000 17500

Distance (m)H

eigh

t (m

)

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Figure 8. Potential fault geometry (heavy lines) within the platform-basin transition of our model, computed from thepredicted fracture pattern for the case of a hydrostatic pore pressure. The arrows indicate the direction of potentialmaterial transport along the faults, which form a narrow band at the base of the platform and probably decoupled thecarbonates from the underlying shale layer along a detachment surface. The dashed line corresponds to the contactbetween carbonates and overburden. A set of faults develops within the platform-to-basin transition zone, is orientedparallel to the progradational domain boundaries, and reaches from the base about halfway up the carbonate edifice. Asimilar set cuts through the entire foreslope with a ramp-flat-ramp pattern that initiates at the base as a 30° dipping ramp,curves in the middle part of the foreslope to a shallower dip, and cuts through the outermost foreslope, where the strengthof the platform is least. This is a boundary effect resulting from the inclined geometry of the platform foreslope, whichcauses a rotation of the principal stress trajectories.

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along the base of the platform and the contrasts in elastic material properties across the platform border(Figure 6 and Table 1). The results of the analysis are summarized in Figure 9.

First, we evaluate how uncertainties in the viscosity of the underlying Pimienta Formation affect thedevelopment of shear instabilities in the model. Figure 9a illustrates the effect of decreasing the viscosity byan order of magnitude to a value of 5 × 1017 Pa s on Δσc. The model shows little sensitivity to viscosityreductions; Δσc is practically unaffected by the decrease in η, with exception of enhanced values in Δσc at thebase of the basin carbonates. In contrast, the model shows important changes when η is increased by anorder of magnitude to 5 × 1019 Pa s (Figure 9b). Most evident is the development of a lobe of positive Δσcvalues along the base of the platform, where the transverse load σt is a maximum (Figure 6). Anotherimportant outcome is a decrease of the stress concentration across the platform margin. Thus, a highly

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Figure 9. Sensitivity analysis for the model that incorporates hydrostatic pore pressure. The figures show how Coulomb stress is affected by changes in the followingparameters: (a) low-viscosity substratum (Pimienta Formation), (b) high-viscosity substratum, (c) isotropic material structure, (d) basement dipping at a steeper angle,and (d) tapered overburden. All other parameters were kept as indicated in Table 1.

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viscous substratum tends to minimize the influence of contrasting elastic material properties across theplatform border and to stabilize it.

A similar effect can be observed in Figure 9c, which shows the response to an isotropic material structure,i.e., E2 = E1. By eliminating the effect of layering, Δσc becomes strongly one-dimensional and acquires aquasi-linear distribution with depth. The model also exhibits little sensitivity to changes in basement oroverburden geometry. The resulting Coulomb stresses only change minimally by increasing the basementslope from 0.5° to 0.75° (Figure 9d) or decreasing the thickness of the shale overlying the Lower Cretaceousbasin carbonates to 500m (Figure 9e), which are both geologically realistic configurations.

In summary, the results of our sensitivity analysis reveal that the brittle failure of the platform border resultedfrom the mechanical coupling between the carbonate platform and a substratum of moderate to lowviscosity and variations in depositional structure and texture that governed the mechanical properties of theinvolved carbonates as well as their dependence on direction. In contrast, the dip of the basement and apossible taper of the overlying Upper Cretaceous shale toward the basin appear to have little influence on themechanical failure of the VSP platform border.

6. Discussion6.1. Influence of Bimaterial Effects and Sharp Corners on the Stability of the Platform Borders

Our model illustrates mechanisms that control the development of thrust faults in addition to the frictionalfailure criterion by Hubbert and Rubey [1959] and frictional reactivation [e.g., Sibson, 1985]. Our numericalsimulations suggest that the thrust faults observed along the margins of the VSP carbonate platformoriginated from a combination of localized phenomena and geometrical factors. The changes in materialproperties across the platform margins cause positive spikes in Coulomb stress that render the platformunstable under realistic tectonic loads. This phenomenon is frequently observed in the mechanical failure ofcomposite materials in engineering and material science experiments where the bonding joints betweencomponents of contrasting elastic properties are the locus of stress concentrations and residual stresses.Such highly stressed interfacial regions, in turn, are ideal for the activation of flaws and the propagation ofmicrocracks that weaken the joints, which ultimately leads to failure [e.g., Radaj et al., 2006, and referencestherein]. Moreover, the results of our model confirm the notion that the progradational domain boundaries ofthe platform-basin transition have the most favorable orientation to become yield surfaces.

Furthermore, experimental work in engineering andmaterial science indicates that shape also highly controlsthe location of fractures in brittle materials; for example, boundaries with a high curvature tend toconcentrate stress [Craig, 1996]. Stress singularities at sharp domain boundary corners of our model,dampened in the case of the foreslope-basin transition by an error function (Figure 4b), may have furtherpromoted the mechanical failure of the platform borders. Experiments with photoelastic materials[Miniatt et al., 1990] show that the local stress field at sharp corners in plates under remote transverse loadingfollows a relation of the form σ =A× r p�1, where A is a function of the radial direction, r is the radial distancemeasured from the corner, and p≈ 0.5 is a real exponent determined experimentally. This means that σ→∞ asr→0; since real materials cannot support infinite stresses, a region of plastic yielding, characterized byabundant cracking and reduction of grain size [e.g., Scholz, 2002], forms around these highly strained areasupon loading, which results in a loss of strength of the material.

6.2. Scenario for Sequence of Fault Initiation Across the VSP Platform Margin

In the light of the above discussion, we speculate that the VSP and El Doctor platforms first yielded at theirbase, not only at the contacts between the platform interior, platform edge, and foreslope domains butalso at the toe of the foreslope, in the transition to the basin carbonates. These locations acted as stress risersthat debilitated the material structure of the platform edge and foreslope carbonates, focused deformationalong their progradational contacts, and lead to a self-weakening feedback between cracking damageand strength reduction [Bercovici and Ricard, 2014]. Once these areas were yielding, the interdomain stressesindicated by our model (Figure 7), which are on the order of 10MPa, may have provided enough additionalenergy to form the observed thrust faults by the coalescence of cracks within their plastic zone. Wefurther hypothesize that slip then initiated along these large-scale shearing surfaces, with the related friction

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alleviating the stress at the corners of the domain boundaries. However, it should be borne in mind that wemade no effort to model such a scenario in the present investigation.

6.3. The Effect of Pore Pressure

Note that our model only becomes compatible with the crustal state of stress when the effect of porepressure is included. Otherwise, the magnitude of the tectonic load required to bring the platform margin tofailure is too large. Since the state of stress is controlled by the frictional strength of preexisting faults, thesewould become active before the platform yields. Bedding-parallel calcite veins within the carbonates ofour study area [Fitz-Díaz et al., 2011b] indicate that the pore fluid pressure exceeded the lithostatic loadduring their formation, most likely during diagenesis or the early stage of horizontal shortening, beforefracturing initiated. This suggests that the VSP and El Doctor platforms possibly yielded at even lower tectonicloads than the one obtained in our numerical model.

In contrast, we ignored the effects of matrix porosity and associated pore pressure in the unfracturedmedium. By the time the horizontal tectonic load initiated, the VSP carbonates likely had already lost most oftheir primary porosity (see above) and reached the currently observed 1–5% [Palacios-Nieto, 1982; Minero,1991]. Poroelasticity theory considers the macroscopic stress σ in a control volume to follow an effectivepressure law of the form σij= σij� α pw δij, where α is the effective pressure coefficient, a material parameterthat characterizes the coupling between the solid frame and fluid filling the pore space [Detournay andCheng, 1988]. For the matrix porosities observed in the carbonates of the VSP, the solid-fluid coupling wasnegligible, i.e., α~ 0.1 [Nur and Byerlee, 1971]. Furthermore, theoretical considerations and empirical datasuggest that the crust is a highly diffusive medium in which pore pressure rapidly equilibrates with long-termboundary loads, e.g., the free surface of the Earth. The equilibration time is in the order of 10 years [Townendand Zoback, 2000; Sarychikhina et al., 2009]; thus, pore pressure is independent of tectonic processes thattake place during much longer time periods.

6.4. The Effect of Dolomitization

An additional reason for the observed failure of the platform margins, not taken into account in our model,could be their partial dolomitization. According to Minero [1991], who studied the eastern edge of the VSPplatform along Highway 120 (Figure 1), dolomite comprises approximately 20% of the studied samples andbecomes more abundant from the platform margin (10%) toward the platform interior (50%). Dolostone ismore brittle than limestone; the degree of dolomitization correlates with fracture intensity, as documentedfrom geophysical wireline logs [Suter and Vargas, 1983] and in outcrops [Ortega et al., 2010]. Based oncompression tests of core samples, dolomitization decreases the compressive strength of limestone byapproximately 20% [Williams and McNamara, 1992].

6.5. Implications for the Coulomb Wedge Theory

Our study of the mechanical stability of carbonate platform margins undergoing fold-thrust deformation hassome affinity with the widely applied Coulomb wedge theory, which presents solutions for the state of stressin a fold-thrust belt of approximately wedge-shaped cross-sectional geometry, assuming the wedge to beat Coulomb yield stress [e.g., Dahlen, 1990; Buiter, 2012]. According to this paradigm, the wedge fails andincreases its taper up to a critical angle, at which the wedge is transported passively along the underlyingdetachment. However, the Coulomb wedge theory does not solve the problem of how critical wedgesdevelop from preexisting noncritical geometries such as carbonate platformmargins; critical wedge solutionsare of limited use for understanding the development of structures within tectonic wedges [Stockmal et al.,2007]. The fold-thrust shortening on the edges of the El Doctor and VSP carbonate platforms lowered thesurface slope angle of the platform-basin transition zone from ≤15° after deposition of the Upper Cretaceousrocks (Figure 4) to the ~3° surface slope toward the foreland typical of the Sierra Madre Oriental fold-thrustbelt wedge (Figure 3) [Eguiluz de Antuñano et al., 2000; Fitz-Díaz et al., 2011a], whereas the sole fault dips≤ 2° toward the hinterland [Suter, 1987; Carrillo-Martínez et al., 2001], which adds up to a critical taper angle of≤ 5°. The strain accumulation at the platform edge cannot be explained by the Coulomb wedge theory,according to which a wedge steeper than the critical taper, such as the ≤15° inclined undeformed platformmargin, is internally stable. Furthermore, in the Coulomb wedge theory the principal stresses are assumed tobe nearly vertical and horizontal, respectively, and constant throughout the wedge, whereas the shear

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fractures and faults initiating in our model (Figures 7 and 8) are not of constant dip as the ones predicted bythe Coulomb wedge theory.

6.6. Comparison With Analog Models

The scaled plasticine and silicone putty analog models of the deformation across carbonate platforms byDixon [2004] and Noble and Dixon [2011] have a certain resemblance to the structural style observed on theedges of the VSP platform. In Dixon’s model configuration where the basin is located on the hinterland sideof the platform, a fold tends to form along the platform edge at an early stage of shortening and evolvesinto a large, foreland-verging structure that carries the basin sediments over the platform margin. Thisexperimental result is similar to the observations on the western margin of the VSP platform, where the fill ofthe Zimapán basin was displaced along the El Volantín thrust onto the carbonate bank by forming a large,nearly recumbent fold nappe of regional extent (Bonanza anticline in Figure 1) [Carrillo-Martínez and Suter,1982; Carrillo-Martínez et al., 2001]. The preexisting anisotropy formed by the basinward facing foreslopesurface and layering had an ideal orientation for the El Volantín and Jiliapan thrusts to nucleate andpropagate [Suter, 1987]. In Dixon’s model configuration where the basin is located on the foreland side of theplatform, the results of both, his experiments and our model agree in that deformation initiated in theplatform-basin transition. In the experiments, a single buckle fold at the platform margin precedesdeformation within the basin and the platform [see Dixon, 2004, Figures 5a and 5b]. However, in Dixon’smodel, the basin is shortened by folding before the platform undergoes (out-of-sequence) deformation,which can be explained by the strong anisotropy and the relatively weak bedding planes of the deformedmaterial but is supported neither by our observations nor by our physical model.

The folds in the analog models by Dixon [2004] and Noble and Dixon [2011] have a remarkable similarity tothe mesoscopic buckle fold trains shown in the sections by Fitz-Díaz et al. [2011a, 2012] across the Zimapánand Tampico-Misantla basins. According to our observations, however, deformation in the basins isdominated by regional-scale detachment folds as well as thrusts and thrust-related folds (Figures 1 and 2),whereas the regional map by Fitz-Díaz et al. [2011a, 2012] does not indicate any regional-scale fold axes andappears to be schematic. As a result, the estimates by Fitz-Díaz et al. [2012] of the relative amount of shorteningaccommodated by mesoscopic deformation in the Zimapán and Tampico-Misantla basins, as compared toshortening by regional-scale thrusting and detachment folding, seem unlikely high and biased by the selectivestudy of sites with high local shortening. Consequently, the dichotomy in structural style assumed by Fitz-Díazet al. [2011a] between the Canadian Rockies (imbricate thrust sheets with relatively little internal deformation)and the Sierra Madre Oriental (deformation by meter-scale buckle folds) appears to be unwarranted.

7. Conclusions

We explored with a cross-sectional finite element model to what extent the material response to transversetectonic loads applied at progradational carbonate platform margins is controlled by their boundary geometryand material properties under the assumption of elastic and Mohr-Coulomb behavior. The motivation for ourstudy were geological field observations on the Valles-San Luis Potosí and El Doctor platforms (east centralMexico) and elsewhere within fold-thrust belts indicating an accumulation of shortening along the margins ofcarbonate platforms in the form of imbricate series of thrust ramps. Our simulations satisfactorily explain thefield observations. The changes in material properties across the platform margins cause positive spikes inCoulomb stress that render the platform unstable under realistic tectonic loads.

The results of our analysis reveal that the brittle failure of the platform border can be mostly attributed tothree effects: (i) mechanical coupling between the carbonate platform and a substratum of moderate to lowviscosity; (ii) variations in layering and texture that governed the mechanical properties of the involvedcarbonates as well as their dependence on direction; and (iii) the development of sharp domain boundarycorners associated with progradational facies changes. In contrast, the dip of the basement and a possibletaper of the overlying Upper Cretacous shale toward the basin appear to have little influence on themechanical failure of the VSP platform border.

Simulations that do not take the pore pressure into account require a tectonic load of ~80MPa to bring theplatform to failure. However, at this stress level, already existing, optimally oriented faults are likely to becomeactive before the platform yields. When a hydrostatic pore pressure is taken into account, the unstable area of

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the model increases to ~15% (Figure 7), whereas the tectonic load, required to bring the platform to failure,is reduced to ~25MPa. TheMohr-Coulomb failure criterion predicts most of the platform and the adjacent basinto be under failure at their base. A first set of faults computed from the predicted fracture patterns forms anarrow band at the base of the platform and probably decoupled the carbonates from the underlying shalelayer along a detachment surface (Figure 8). A second set of faults develops within the platform-to-basintransition zone, is oriented parallel to the progradational domain boundaries, and reaches from the baseabout halfway up the carbonate edifice. A similar third set of faults cuts through the entire foreslope with aramp-flat-ramp pattern that initiates at the base as a 30° dipping ramp, curves in themiddle part of the foreslopeto a shallower dip, and cuts through the outermost foreslope (Figure 8), where the integrated strength of theplatform is a minimum. This is a boundary effect resulting from the inclined geometry of the platform foreslope,which causes a rotation of the principal stress trajectories. It explains why the hanging wall carbonate strata ofthe imbricate thrusts along the eastern edge of the VSP platform are cut at low angles of 6° to 10°.

Our analysis is an alternative to the conventional critically tapered Coulomb wedge theory that treats thewedge as homogeneous and isotropic. The strain accumulation at the platform edge cannot be explained bythe Coulomb wedge theory, according to which a wedge steeper than the critical taper, such as the ≤ 15°inclined undeformed platform margin, is internally stable. Furthermore, in the Coulomb wedge theory theprincipal stresses are assumed to be nearly vertical and horizontal, respectively, and constant throughout thewedge, whereas the shear fractures and faults initiating in our model are not of constant dip as the onespredicted by the Coulomb wedge theory.

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AcknowledgmentsAll data necessary to understand,evaluate, replicate, and build upon thereported research are contained withinthe paper. The public domain finiteelement program FreeFem++ used inthe calculations is available from www.FreeFem.org/ff++/ (last accessed 13June 2014). We acknowledge thefinancial support by UNAM and CICESE(internal project 644143, to J.C.) and theGeological Society of America (researchgrant 4768–91, to J.C.). The research inthis paper was originally carried out bythe first author within his master thesisat UNAM; he thanks Gonzalo Alduncinand Ramón Zúñiga for their advice onnumerical modeling techniques. We arethankful for the prompt and detailedmanuscript evaluations by John M.Dixon, an anonymous reviewer, andAssociate Editor Isabelle Manighetti. Wededicate this paper to the memory ofDave Wiltschko, whose geomechanicalstudies of thrust-related structures havebeen inspirational to us.

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