Predicting orogenic wedge styles as a function of analogue erosion law and material softening

Authors


Abstract

[1] The evolution of a compressive frictional wedge on a weak, frictional and planar décollement, subjected to frontal accretion, is predicted with a two step method called sequential limit analysis. The first step consists in finding, with the kinematic approach of limit analysis, the length of the active décollement and the dips of the emerging ramp and of the conjugate shear plane composing the emerging thrust fold. The second step leads to a modification of the geometry, first, because of the thrust fold development due to compression and, second, because of erosion. Erosion consists in removing periodically any material above a fictitious line at a selected slope, as done in analogue experiments. This application of sequential limit analysis generalizes the critical Coulomb wedge theory since it follows the internal deformation development. With constant frictional properties, the deformation is mostly diffuse, a succession of thrust folds being activated so that the topographic slope reaches exactly the theoretical, critical value. Frictional weakening on the ramps results in a deformation style composed of thrust sheets and horses. Applying an erosion slope at the critical topographic value leads to exhumation in the frontal, central, or rear region of the wedge depending on the erosion period and the weakening. Erosion at slopes slightly above or below the critical value results in exhumation toward the foreland or the hinterland, respectively, regardless of the erosion period. Exhumation is associated with duplexes, imbricate fans, antiformal stacks, and major backthrusting. Comparisons with sandbox experiments confirm that the thickness, dips, vergence, and exhumation of thrust sheets can be reproduced with friction and erosion parameters within realistic ranges of values.

1. Introduction

[2] A large part of terrestrial reliefs is formed by the horizontal contraction of shallow sedimentary materials above décollement surfaces in response to oceanic subduction (accretionary prisms) or continental collision (fold-and-thrust belts). The global cross-sectional shape of the resulting wedges has a topographic slope well captured by the critical-taper theory [Davis et al., 1983; Lehner, 1986] and function of the bulk and décollement friction coefficients, relevant to characterize rock strength for low temperatures. The purpose of this contribution is to propose a simple method that combines geometrical constructions [e.g., Suppe, 1983] and optimization techniques [Cubas et al., 2008; Mary et al., 2013] to produce the internal deformation leading to these critical topographies.

[3] Part of the progress in unraveling the complexity of the growth and the inner structure of frictional wedges has come from analogue experiments and numerical analyses based on mechanics. They have both revealed two key parameters: material weakening and erosion/sedimentation processes. Weakening material strength leads to the strain localization that mimics faulting within the frictional sediments (although it is not a necessary condition for faulting [Rudnicki and Rice, 1975]). This rheological feature, visible in the field [Wojtal and Mitra, 1986], is at the core of the reproduction of thrusting cycles during analogue wedge growth [Malavieille, 1984; Merle and Abidi, 1995; Mandl, 2000; Lohrmann et al., 2003; Hoth et al., 2007], and represents a challenge for numerical simulation techniques [Leroy and Ortiz, 1989; Buiter et al., 2006]. The second key parameter in shaping the structure of a wedge is surface transport (erosion/sedimentation), which has been invoked, for example, to explain the sustainability of relief (topography) at large scales [Avouac and Burov, 1996; Braun, 2010], the activation of major backthrusts [Silver and Reed, 1988], the location of internal deformation and exhumation in generic examples [Willett et al., 1993; Willett, 1999; Konstantinovskaya and Malavieille, 2005; Hoth et al., 2006], in real settings [Bonnet et al., 2008; Malavieille, 2010], and thrust vergence at the imbricate thrust scale [Adam et al., 2004].

[4] The geometry of internal deformation style has been described by structural geologists who introduced geometric constructions of folding [Rich, 1934; Suppe, 1983]. These concepts lead to quantitative techniques for extrapolation of surface observations, cross-section balancing [Dahlstrom, 1970], and restoration [Boyer and Elliott, 1982; Zoetemeijer and Sassi, 1992]. These geometrical constructions make no links with mechanical equilibrium or rheology. These also revealed, however, the essential kinematic features of thrust-related folding and proved to be extremely efficient to reproduce thrusting sequences over large shortenings.

[5] We use here a semianalytical mechanical method to predict the evolution of thrusting in frictional wedges. The evolution is based on a basic incremental kinematics inspired from the fault-bend fold geometry, where the values of the geometric parameters are obtained at each increment by optimization. This optimization makes use of the maximum strength theorem (the kinematic approach of limit analysis), which relies on a weak form of mechanical equilibrium and, for our specific applications, on the Coulomb criterion for the décollement and any fault [Salençon, 1974, 2002; Maillot and Leroy, 2006]. Applying the method sequentially to follow the collapse of a civil engineering structure was first proposed by Yang [1993] and was used apparently for the first time for accreting wedges by Cubas et al. [2008]. True displacement discontinuities with large offsets are inherent to the method, in contrast to more general and complex methods based for example on finite-element techniques. Such improvement in the chosen kinematics has allowed us to follow deformation for unlimited shortenings ([Mary et al., 2013], referred to as paper 1 in what follows). Distribution of slip on the décollement appears to be constant stepwise, decreasing toward the foreland. The steps mark preferential positions of the ramps along the décollement, marking the cores of thrust folds in the absence of weakening. It is this activation of thrust folds that leads to the evolution of the surface topography toward the critical taper value [Dahlen, 1984]. However, the convergence analysis of the numerical scheme reveals the essentially unpredictable character of an individual thrusting event. Numerical convergence is achieved by switching to a convergence criterion based on the statistics of the position of the thrusting events (paper 1).

[6] In the present contribution, we introduce erosion and accretion and focus on the reproduction of analogue experiments. In section 2, we describe the prototype of our frictional wedge, and the two step method called sequential limit analysis. In sections 3 and 4, we perform a parametric study to identify and illustrate the respective roles of the ramp friction weakening and erosion, keeping both basal friction and initial ramp friction values constant. We observe in particular that weakening confers a finite thickness to the thrust sheets [Cubas et al., 2008], and that forward thrusting sequences are interrupted by out-of-sequence episodes. Erosion if concentrated in the highest parts of the relief, promotes deformation throughout the wedge and exhumation by imbricate fans, antiformal stacks and backthrusting. In section 5, comparisons with high and low basal friction sandbox experiments [Konstantinovskaya and Malavieille, 2005] demonstrate the ability of our method to reproduce thrust vergence and exhumation patterns. The final discussion is centered on the implications of our parametric analysis for analogue experiments and geological structures interpretations.

2. The Prototype

[7] The geometry of our prototype, classical in analogue experiments, is presented in Figure 1. The structure is composed of a homogenenous, cohesionless, and frictional material, with a bulk friction angle math formula, a weakened friction angle on the ramps math formula, and a friction angle math formula on the horizontal décollement (AC). Although the theory of limit analysis accounts for cohesion, we disregard it here because of the large size of the simulated structures (several kilometers thick, tens of kilometers long) and because this assumption is most often applied to scaled analogue experiments with sand. The weight density is uniform and its value is irrelevant since all cohesions are set to zero. It only sets the magnitude of the compressive force that is not commented here. The initial topography is segmented in three parts, covering the internal, the central, and the external (or accreting) regions. The three initial topography slopes are αi, α, and zero, respectively. The thickness at the boundary between the internal and the central regions is H, and the external region thickness is h. The topography is discretized with an initial point spacing Δx. This value evolves during the simulation since new points are added where necessary such that others could be deleted (see paper 1 for further information). The shortening is applied in increments of Δs by the movement of the back wall from the left to the right.

Figure 1.

(a) The prototype is composed of three regions: the internal (between BI), the central (IJ), and the external (JD), with initial topographic slopes αi, α, 0. (b) Displacement of the back wall (AB) generates thrusting with two discontinuities, the ramp (GE, dipping at γ) and the shear plane (GF, dipping at θ). Displacement of the back wall generates internal deformation and surface uplift. The foot wall is immobile, and the sliding wall translates rigidly. (c) Erosion consists in removing periodically any material above the erosion segmented line (BMND). The segment (BM) has the slope αi, the segment (MN) is defined by the height He and the slope αe (both constant during a simulation), and the segment (ND) is horizontal.

[8] The Critical Coulomb Wedge theory defines the critical slope αc, a function of math formula and math formula [Dahlen, 1984]. Deformation occurs at the front if math formula and at the back of the wedge if math formula. We always set the internal topography math formula, so that the internal region acts as a buffer to eliminate the influence of the back wall. The deformation thus occurs in the central region or at its boundaries (points I and J on topography). For this reason the central region is referred to further on as the “wedge.” This type of geometry with an internal region is also common to many sandbox experiments. Note that, during accretion, the wedge grows at the expense of the internal and external regions, and the points I and J migrate to the left and to the right, respectively. Their exact position at any time (shortening) is the leftmost position reached by point F and the rightmost position reached by point E, respectively.

[9] The incremental construction of our deformed wedge is decomposed into two steps. The first step consists in selecting the optimum failure mechanism (Figures 1a and 1b) composed of three regions: the sliding wall (SW), the ramp hanging-wall (HW), and the ramp foot-wall (FW). These regions are separated by two discontinuities: a shear plane, or hinge, (GF) with dip θ, and a ramp (GE) with dip γ. They are rooting at the common point G on the décollement, at a distance LAG from the back wall, defining the active décollement (AG). Any such failure mechanism requires a force math formula applied at the back wall. The kinematic approach of limit analysis stipulates that it is an upper bound to the actual, unknown tectonic force [Salençon, 2002]. We then select the triplet ( math formula) which minimizes that bound [Cubas et al., 2008, electronic supplement]. The least upper bound for a new thrust fold makes use of math formula, in which SR is the accumulated slip on the ramp. The least upper bound if associated with the active thrust-fold relies on math formula, because activated thrust ramps are known to be weaker than pristine bulk material. Any deactivation of the ramp, due to the selection of a more optimum orientation γ or length LAG, erases this damage as if the ramp had healed instantaneously.

[10] The second step in our incremental scheme consists in deforming the wedge to accommodate shortening by thrust folding and in modifying the topography because of erosion. For the thrust folding, any point within the SW, including the discretized topography, is sliding parallel to the active décollement by the amount Δs, and points in the HW are sliding parallel to the ramp by the amount math formula such that volume is conserved when crossing the shear plane [Maillot and Leroy, 2003]. Points that are partly in the SW and partly in the HW during the same increment have a trajectory that exactly accounts for the partial motion within each region. These points could belong to the topography as mentioned above or be part of internal markers introduced to visualize the internal deformation (shades of gray in Figure 1). Some topographic points could result in surface slopes greater than the repose angle, or even overhanging above the FW in the vicinity of the ramp outcrop. We propose a geometrical scheme that differs from the one adopted by Cubas et al. [2008] and which is essential to account for thrusting over irregularly shaped topography: the unstable sections of the topography are replaced by segments with slope set to math formula, conserving the volume of material (see collapsed material in vicinity of point E, Figure 1b for an example). These topography modifications are inspired by the natural process of slope instability observed in a sandbox [Yamada et al., 2010]. Further technical details are found in Mary [2012] and in paper 1.

[11] Erosion is optional during the geometry modifications. We could erode any material found above a line that is fixed with respect to the moving back wall, a protocol proposed by Konstantinovskaya and Malavieille [2005] (Figure 1c). The new discretization points of the topography are obtained by vertical projection on the erosion line. Erosion is applied at every shortening intervals Δse which are multiples of Δs. Although erosion is not necessarily applied at each increment, the total input by frontal accretion and the total output by erosion are balanced for shortening much larger than ΔSe. We shall see that Δse as well as the slope αe, the only two parameters of our erosion law, have important consequences on the wedge deformation style.

[12] In the next two sections, deep dark blue, yellow, and blue markers indicate all material initially at the maximum height h above the décollement. Other markers correspond to material initially higher up, therefore lying in the central or internal regions. Results presented here are obtained with the sequential limit analysis code SLAMTec [SLAMTec, 2013].

3. Reference Results

[13] We first describe two simulations in details, the first without and the second with erosion, to provide a reference to which subsequent simulations are compared to in section 4. The various parameters common to all the simulations of sections 3 and 4 are summarized in Table 1. The topographic surface and the décollement are discretized with a step Δx = 50 m. The shortening increment Δs = 5 m is much smaller than Δx for numerical convergence reasons (paper 1). The central region is initially at or near the critical state, with a surface slope math formula. Friction and erosion parameters are given in Table 2, with values in boldface for the two reference simulations.

Table 1. Geometric Parameters
FiguresABHhαiααcSΔs
Parametric Study
 (km)(km)(km)(°)(°)(°)(km)(m)
2–93.420.5cαe3.437405
         
Comparison to Sandbox Experiments
 (cm)(cm)(cm)(°)(°)(°)(cm)(cm)
11a5.4h3.115n.a.4 (obs.)102n.a.
11b12h3.115n.a.5.81020.1
11c12h3.115n.a.4.81020.1
11d8.6h3.112n.a.2.71020.01
12a7h3.115n.a.4 (obs.)51n.a.
12bsee parametric study       
13a10.4h3.615n.a.10125n.a.
13(b)12h3.615n.a.8.81250.1
Table 2. Friction and Erosion Parameters
Figure math formula math formula math formulaHeαeΔse
Parametric Study
 (°)(°)(°)(km)(°)(km)
2–9103015, 25, 30H math formula1, Δs, 5
Comparison to Sandbox Experiments
 (°)(°)(°)(cm)(°)(cm)
11a1230 5.442
11b163026.58.46.82
11c13.530268.45.82
11d83024n.a.2.72
12a1230 762
12b1030252(km)3.437200(m)
13a2430 782
13b2230261492

3.1. Reference Simulation Without Erosion

[14] The first simulation considers weakening ( math formula) and there is no erosion. The initial geometry is presented in Figures 2a0/b0. Note that the initially horizontal markers do not reveal the internal deformations necessary to reach our initial state. The markers only shed light on the internal deformation occurring after the initial state. The internal cumulative deformation of the wedge is visualized after 20 and 40 km of shortening (Figures 2a1 and 2a2, respectively). Horizontal positions are measured with respect to the moving back wall, and the origin is chosen at the initial position of point I. After 40 km of shortening, the central region has grown by 5 km to the left and 9.5 km to the right, increasing its width to about 39 km. The frontal half of the wedge is exclusively composed of accreted material that is thickened by up to a factor of 3 in the central part (black, yellow, and blue markers) compared to the external region. The black and yellow layers at the very base of the accreting material demonstrate that the uplift is localized at the core of more or less regularly spaced anticlines found at positions 0, 6.5, 13.5, 18.5,…km. Deformation is less intense at the rear: the material which was occupying the whole initial wedge, has been shortened and deformed by two pop-up structures (centered at positions 0 and 6.5 km, Figure 2a2). These two structures were sufficient to accommodate shortening and to thicken the wedge at the rear. One should notice that the second pop-up from the rear has developed overturned markers between 20 and 40 km of shortening.

Figure 2.

Reference simulations for the parametric study. (a0/b0) Initial state; cross sections at (a1) 20 km and (a2) 40 km of shortening, without erosion. (b1, b2) Simulation with erosion.

[15] Layer thickening estimated by the increased distance between two markers is the result of two processes. First, the very irregular limits of the yellow layer shows the thickening due to the slip along the ramps, which tends to stack the units on top of each other. Second, there is thickening as material crosses the shear plane, because the optimal dips γ and θ are in the range math formula and math formula, respectively. Therefore, the shear plane dip θ is always much less than the value required to conserve thickness, i.e., the angle bisecting the décollement-ramp angle: math formula. The actual shear-plane dip range implies that the SW material is thickened by about 50% when entering the HW. This thickening is visible in the hanging walls of the most hinterland thrusts. It is observed in sandbox models and geological examples [Maillot and Koyi, 2006; Koyi and Maillot, 2007] and in all simulations presented here.

[16] The most significant parameter to describe the position of the activated thrust fold during shortening is the length of the active décollement (AG) (point G is the common root of the ramp and the shear plane). The plot of the shortening versus the distance AG is called a G-gram (Figures 3a and 3b). This visualization was first proposed by Strayer et al. [2001] and used by Smit et al. [2003] and Cruz et al. [2010]. At each shortening increment, the position of G is represented with a black dot, and this will be called a thrusting event. If the point G remains immobile for several thrusting events, the black dots will be aligned, forming an inclined segment (because of the shortening of the SW) that will be called a series of thrusting events (close-up frame in Figure 3b with the tip of a series made of 10 events). A single thrust fold develops during a series. Finally, the succession of series is called a sequence of thrustings. During a series, the light blue line to the front of the wedge, corresponding to point J, is parallel to the G segment, and the light blue line to the back, corresponding to point I, is vertical. The wedge is shortening in length (horizontal distance from I to J), due to internal deformation. The point I is displaced to the left if the deformation reaches the back (first time after about 1 km of shortening). In the same way, the point J jumps to the right at each new frontal accretion. These two lines motions are alternating throughout the simulation and illustrate the increase of the wedge width by several hundred meters (Figure 3a). We count about 30 frontal thrusting series over the first 20 km of shortening, corresponding to an average period of 6.7 ka assuming a shortening velocity of math formula cm/a. This time lapse is of the order proposed by Hoth et al. [2007] math formula ka, and within the time range of natural examples, if extrapolated to thin accreting layers [Hoth et al., 2007, Figure 10].

Figure 3.

G-grams, showing the position of point G during the entire shortening, of the reference simulation (a) without erosion and (b) with zoom. Position refers to the horizontal scale in Figure 2. At each shortening increment, a black dot indicates the current position of point G. The solid light blue curves indicate current positions of points I (left curve) and J (right curve), respectively, i.e., the limits of the central region. Red (or dark) curves show the theoretical limits of the central region calculated by redistributing at the critical slope all accreted material in the central region. (c) Average surface slope of the central region (blue curve), theoretical critical slope (solid black line), and critical slope accounting for the presence of a weak ramp (dashed line).

[17] The G-gram in Figure 3a demonstrates that the growth of the wedge is not due to thrusting that display regular, forward, or backward sequences. The irregular pattern was analyzed in details in paper 1. The mean surface slope of the central region is always oscillating between the theoretical critical slope αc (Table 1), and the slope determined by limit analysis taking into account the weakened friction angle (Figure 3c). When the activity is toward the back ( math formula10 km), there is a sharp increase in the mean slope at each thrusting series, and when it is at the front ( math formula 10 km), the surface mean slope decreases gradually (compare Figures 3a and 3c between 18 and 20 km of shortening for example).

[18] The thrusting series are far apart toward the back of the wedge and closer toward the front. This difference reflects the thinning of the wedge [Strayer et al., 2001; Panian and Wiltschko, 2007]. The series are also longer at the back (more events in each series) than at the front, and they are more seldom activated than at the front. In addition, thrusting series return rather regularly to the same region of the wedge. At the back this occurs every 7–9 km of shortening, and every few hundred meters toward the front. This suggests the existence of a period of reactivation of the main thrusts, particularly clear at the back, that is linked to the ramp weakening, the wedge size, and the accreting mass flux.

[19] The long-term growth of the wedge can be estimated by assuming a continuous accretion of material and spreading this volume at the surface, such that the central slope is constantly at the critical value. It is thus possible to estimate the migrations of points I and J on the basis of volume conservation. These estimates correspond to the red solid curves (Figures 3a and 3b), demonstrating that our simulations do indeed conserve volume.

3.2. Reference Simulation With Erosion

[20] Erosion is applied every math formula of shortening in the second simulation. The erosion slope math formula (Table 2), so that the erosion line BMND (Figure 1c) matches exactly the critical topographic slope. The wedge is free to grow in length and thickness only between the erosion events.

[21] The cross sections at s = 20 and 40 km (Figures 2b0 to 2b2) display large losses of material initially forming the wedge and replaced by accreted material. The displaced initial topography is marked by a thin gray line. The light gray material between this line and the current topography has been eroded. At 20 km, the material initially at depth h (top of blue marker) is now outcropping at the positions 10 and 12 km, corresponding to the boundaries of a syncline. At 40 km of shortening, the first outcropping position is at 6.5 km. Using the markers initially below h, we thus define an exhumation window in the center of the wedge, between 7 and 12 km, where the dark basal marker is lifted to the top. Exhumation requires erosion and uplifting of initially deep material. The G-gram in Figure 4b shows that the associated thrusting is still discontinuous in time (shortening) in view of the spatial distribution of the series (inclined segments corresponding to a collection of successive events). Consequently, exhumation is a discontinuous process both in space and time. Figure 4a presents erosion profiles along the wedge at three shortenings steps (S = 15, 30 and 45 km). Note that the exhumation window (corresponding to an erosion larger than 0.8 km) cannot be inferred from the G-gram. For example, erosion (and exhumation) are the least at the front while important activity is noticed there according to the G-gram. In summary, erosion along a line at the critical slope with an erosion period Δse higher than the shortening increment Δs produces exhumation in the central part of the wedge.

Figure 4.

(a) Erosion profiles at three shortenings steps of the reference simulation with erosion. (b) Corresponding G-gram. (c) Average surface slope of the central region (blue curve), theoretical critical slope (solid black line), and critical slope accounting for the presence of a weak ramp (dashed line).

[22] Comparing the frontal activity in the G-gram of the Figures 3 and 4, one sees that there are less events with erosion. The period of frontal accretion is disturbed by the period of erosion. Another remark on the G-gram concerns the motion of the I- and J-points bounding the central region. These points are not spreading away, as in the previous simulation, signaling that there is no wedge growth and thus a balance between accreted and eroded masses. Finally, the mean surface slope is also oscillating between αc and the value predicted by accounting for weakening, but with some intrusions below αc not seen before (Figure 4c).

4. Parametric Study

[23] We now perform a systematic exploration of the influence of three parameters on the evolution of the wedge: the erosion slope αe, the erosion period Δse, and the weakened friction on the ramp math formula. The latter parameter has a well-known importance in geological structures, while the two former may appear less related to geological processes. However, they are routinely used in sandbox experiments, and as such, deserve an analysis. This point will be further discussed in section 6. For each parameter, we examine several values, higher, equal, or lower than those of the reference simulation (Table 2), selecting a total of 27 simulations. Figures 5 (cross sections) and 6 (G-grams) illustrate the effects of math formula and αe, for math formula, i.e., a continuous erosion. Figure 7 illustrates the same effects, for Δse = 1000 m (for conciseness, we do not show the corresponding G-grams). Figures 8 and 9 illustrate the effects of math formula and Δse, for αec. In all the simulations without weakening ( math formula), the G-grams are continuous and the representation with black dots is not appropriate (paper 1). It is replaced by a density of dots (Figures 6c and 9c), expressing the local percentage of accomodated shortening. The color scale is consistent for all the G-grams, since those that are discontinuous only display white (0%) or black (100%).

Figure 5.

Influence of the erosion line slope αe on the wedge deformation style. Three values are considered: math formula, and math formula. The critical surface slope, math formula, is computed from math formula and math formula. Erosion is applied along the dotted line (MN) at every shortening increment ( math formula m). The thin gray lines show the non eroded topography, often cut by a horizontal gray segment for reason of space.

Figure 6.

G-grams of the simulations of Figure 5. Abscissas: position in the cross section (as in Figure 5). Ordinates: applied shortening. The left and right light blue curves mark the positions of points I and J, respectively. In (c), G points have a continuous distribution with shortening because there is no ramp weakening. Slip on the décollement is then visualized with a color code representing the local percentage of accommodated shortening. Each colored pixel represents a 250 × 250 m2 window in the G-gram. Above each G-gram, three curves separated by shades of blue indicate the cumulative eroded thickness (in km) after shortenings of s = 15 km, 30 km, and 45 km.

Figure 7.

Influence of the erosion line slope αe on the wedge deformation style. Three values are considered: math formula, and math formula. The critical surface slope, math formula, is computed from math formula and math formula. Erosion is applied every 1 km of shortening ( math formula km).

Figure 8.

Influence of the period of erosion, Δse, on the wedge deformation style. The erosion line slope αe is set to math formula, the critical surface slope, computed from math formula and math formula.

Figure 9.

G-grams of the simulations of Figure 8. Abscissas: position in the cross section (as in Figure 8). Ordinates: applied shortening. The left and right light blue curves indicate the points I and J, respectively. (c) G points have a continuous distribution so that slip on the décollement is indicated by a color code representing the percentage of shortening accommodated at each position. Each colored pixel represents a 250 × 250 m2 window in the G-gram. Above each G-gram, three curves separated by shades of blue indicate the eroded material (in km) after shortenings of s = 15 km, 30 km, and 45 km.

Figure 10.

The map of exhumation location in the space spanned by the normalized erosion period and the weakening on the ramps.

4.1. The Erosion Slope

[24] The angle αe is essential in controlling the location of exhumation windows, and therefore the distribution of the internal deformation. For math formula, the internal deformation of the wedge is restricted to the frontal part, either in a narrow region if Δses (Figures 5a1, 5b1, and 5c1) or in a wider region if Δse = 1 km (Figures 7a1, 7b1, and 7c1). It results from the piling up of new thrust sheets as duplexes that are progressively rotated to vertical and overturned dips, forming an imbricate fan as more thrust sheets are formed ahead (Figures 5 and 7, 7a1, 7b1). In the absence of weakening, the imbricate fans are however replaced by a simpler structure where the markers are not much less hacked (Figures 5 and 7, 7c1). The regions of internal deformation are delimited to the rear by a backthrusting kinematics (at 17 km, Figure 5a1; 19 km, Figure 5b1 and 21 km, Figure 5c1, and similarly for Figures 7a1, 7b1, and 7c1). The rest of the wedge is a stagnation zone, i.e., without any deformation or erosion (Figures 6a1, 6b1, and 6c1). Notice the peculiar G-gram patterns made of oblique discontinuous segments aligned vertically with weakening (Figure 6a1 and 6b1) and the continuous vertical line without weakening (Figure 6c1). These vertical alignments (locked pattern) mean that the active décollement is always of the same length at the initiation of a sequence of thrusts. It is striking to observe that this motion is associated with a backward vergence of the thrust folds or overturned markers (Figure 5c1 and 5a1, 5b1), despite the incremental forward kinematics (Figure 1b). This backward vergence is reminiscent of triangular zones found in the frontal parts of thrust belts [e.g., Price, 1986].

[25] For math formula, each erosion event brings the wedge into a subcritical state, so that the internal deformation tends to raise the internal region to regain the steeper critical slope. Therefore, erosion is more active in the rear part of the wedge. This exhumation kinematics is similar to the one observed for math formula but is located in the rear part of the wedge (Figures 5b3, 5c3 and 7b3, 7c3). Accordingly, the locking patterns of the G-grams are now in the rear part of the wedge, and they are interrupted by phases of activity in the central and frontal regions. The accreted material crosses the whole wedge to feed the region of erosion and exhumation (Figures 6a3, 6b3, and 6c3). Cases of Figures 5b3 and 7b3 are reminiscent of the stacking of nappes forming a broad dome anticline found in basal accretion models [e.g., Malavieille, 2010]. This deformation style does not develop in the case of strong weakening which deorganizes the locking patterns of the G-grams and precludes the backthrusting, although exhumation is still concentrated at the rear (Figures 5a3, 6a3, and 7a3). Simulations using math formula or math formula yield essentially the same deformation styles as discussed above.

[26] Finally, the results obtained for αec are close to the ones corresponding to αec + 0.5 or to math formula, depending on Δse and, less systematically, on the ramp weakening. The results do compare to the ones of the set αe = αc + 0.5 for Δses (compare all cases 1 and 2 of Figures 5 and 6) as well as for math formula and strong weakening (Figures 7, 7a1 and 7a2). Results compare more to the ones of the set math formula for little or no weakening and math formula (Figures 7, b2, b3, c2, c3). The position of the most intense deformation within the wedge therefore depends on the periodicity of erosion Δse if setting αec. This point is further discussed after the following comments on the effect of the weakening on the ramp.

4.2. The Weakened Friction on the Ramps

[27] The thrust sheets are thick, and the thrust ramps are far apart and accumulate large slips if there is large weakening on the ramps ( math formula, Figures 5a and 7a). For low weakening, the sheets are thinner and the ramps accumulate less slip (Figures 5b and 7b). At zero weakening, the deformation is not hacked and relatively smooth (Figures 5c and 7c) (in fact, because of discretization, the sheets have the length determined by Δx, and their lifetime is Δs). This difference in the lifetime and the length of the ramps are displayed by the G-grams with longest lifetimes and longest thrust sheets in the weakening cases (Figures 6a and 6b), and a continuous activation without weakening (Figure 6c).

[28] In addition, this weakening parameter offers a unifying view of the variation from imbricate fan to antiformal stack and to backthrusting, regardless of the period of erosion (cases a3, b3, c3, respectively, in Figures 5 and 7).

4.3. The Erosion Period

[29] The erosion period has a mild influence on the extent of the deformed region and none on its style nor its position for math formula. The erosion period is only important for αec considered here. Final cross sections and the associated G-grams are presented in Figures 8 and 9, in addition to the two values (0 and 1 km) of the preceding figures.

[30] For math formula, after an initial adjustment phase in the central and rear parts of the wedge, the deformation and the exhumation are locked at the front for Δse = 200 m (Figures 8a1 and 9a1). For Δse = 2 km, deformation extends from the front to the center of the wedge and exhumation is concentrated in the center (Figures 8a2 and 9a2). If Δse = 5 km the deformation is distributed throughout the wedge, with stacked sheets and several imbricate fans, and exhumation is mostly at the rear (Figures 8a3 and 9a3). For math formula, the same variation in deformation style occurs for Δse between 100 and 500 m, and is achieved completely at 2 km, with exhumation at the rear through antiformal stacks (Figures 8b and 9b). For math formula, the change in deformation style occurs around a period of 20 m since deformation is locked at the front for Δse = 10 m, distributed with maximum at the center for 20 m and more toward the back for 50 m (Figures 8c and 9c), and 1000 m (Figure 7c2). Accordingly, erosion is more evenly spread for increasing Δse.

[31] In summary, a continuous low-erosion period maintains the deformation and the exhumation in the frontal part of the wedge. Increasing the erosion period results in a more distributed deformation that affects all the wedge for large periods. The values of the period for which these transitions between frontal and distributed deformation styles occur increase with weakening. This dependence is illustrated in Figure 10, where a map of exhumation is presented in the space spanned by math formula and the weakening math formula. Three domains are observed corresponding to rear exhumation in the upper left of the plot, a transition zone where exhumation is essentially located at the center and a frontal exhumation in the lower triangular region of the plot. The boundaries between these regions were obtained by a least squares linear regression. The data points are found from a series of 540 simulations and based on a qualitative interpretation, the error bar signaling the difficulty of such a criterion.

[32] This study does not elucidate all behaviors of our prototype. At least two additional parameters should be discussed. First, the influence of the décollement friction was examined, and the broad results regarding exhumation agree with the present conclusions. In the next section, we present low and high basal friction examples. Second, the internal topographic slope αi influences the internal deformation, depending on the ratio αi∕αc. In particular, it affects the vergence of the thrust fold which is mostly to the rear. This has potential consequences on the interpretation of sandbox experiments commonly using a steep internal region to eliminate the effect of the rigid back wall.

5. Comparison With Sandbox Experiments

[33] Three sandbox experiments among those published by Konstantinovskaya and Malavieille [2005] were chosen for comparison with our predictions. They were chosen because they combine the effects of basal friction and erosion and display very different structural styles. The initial geometry of the sand pack, the applied shortening, the erosion period, and the colored markers are the same as in the experiments. Note that the initial geometry has only an internal region dipping at 15° and a flat external region.

[34] The value of the critical slope αc in the sandboxes was measured from the cross sections at the end of an experiment without erosion. The error on these measurements could be of the order of math formula. It should be mentioned that this estimated αc is certainly larger than the true theoretical value, since sand develops some weakening not accounted for in the critical taper theory. This theoretical error is of the order of math formula for math formula as seen in Figure 3 and increases with weakening ( math formula for math formula). Such overestimation of the theoretical αc should place any experiment in the supercritical cases of the parametric study ( math formula). We shall see that such minute changes in erosion slope could indeed drastically affect the final deformation styles.

[35] The erosion process was slightly different during the analogue experiments: two erosion steps were performed along erosion lines at low heights during the initial shortening before setting the erosion line at a constant height for the rest of the shortening. We did not perform these initial erosion steps in the simulations and directly set the erosion line at a constant height. The weakened friction values in the sandbox experiments were not given by Konstantinovskaya and Malavieille [2005]. The values chosen here ( math formula between 24 and math formula) are within the range measured in the laboratory for various sands [Lohrmann et al., 2003]. In the simulations, we slightly varied the basal friction value and the erosion line dip to better fit the experiments. One should indeed not expect a perfect fit of the parameters, owing to the inherent experimental uncertainties due, for example, to spurious friction of the sand body on the lateral glass walls [Souloumiac et al., 2012]. All comparisons are presented in Figures 11-13 and the corresponding parameters are found in Tables 1 and 2. Note that all lengths are now in centimeters instead of kilometers, but they can be compared directly since the experiments were performed, assuming a length scale ratio of math formula (i.e., 1 cm represents 1 km in the field).

5.1. Low-Friction Base Examples

[36] In Figure 11, we compare three simulations to the experiment LF4 of Konstantinovskaya and Malavieille [2005]. In this experiment, the basal friction is low (around math formula (coefficient 0.21)) and the erosion line dip is set at the observed experimental critical taper of math formula. In the simulation reported in Figure 11d only, we tested an early version of the erosion rule, where the erosion line is attached to the advancing front of the deformation rather than to the back wall (i.e., N=J in Figure 1c). This difference in erosion law explains the much lower relief observed in Figure 11d compared to Figures 11b and 11c.

Figure 11.

(a) Sandbox experiment LF4 of Konstantinovskaya and Malavieille [2005] after 102 cm of shortening. (b)–(d) Numerical simulations. In Figure 11d, the erosion line is attached to the current frontal part of the wedge rather than to the back wall, explaining the lower relief.

[37] Striking similarities are observed: the frontal thrusting produces successive thrust sheets that are progressively rotated and eroded to exhume along vertical corridors the deepest material originally in contact with the décollement. The thickness and number of thrust sheets in the lower basal friction simulation (Figure 11d) are very close to those in the experiment, while the anticline structure at the toe of the internal region is present only in the simulations using the same erosion rule (Figures 11b and 11c). Judging from the observed backthrusting in the parametric study, a more pronounced backthrust would have been obtained in these simulations if the two initial erosion steps had been applied, as in the experiment.

[38] In a second experiment, named LF6, using an overcritical erosion slope (Figure 12), Konstantinovskaya and Malavieille [2005] observe an initial stage of deformation throughout the wedge, and then a localization of the deformation and an exhumation toward the center or the front of the wedge, limited to the rear by a major backthrust zone, behind which the structures are stagnant. We observe very similar behaviors in our parametric study, using αe = αc (Figures 5a2, 5b2, 7a2, 8a1, and 8b1). The corresponding G-grams allow us to identify the stagnation area (Figures 6a2, 6b2, 9a1, and 9b1) at the rear of the exhumation area.

Figure 12.

Comparison between (a) the sandbox experiment LF6 after 51 cm of shortening [Konstantinovskaya and Malavieille, 2005, Fig.13b] and (b) the numerical simulation and its G-gram.

5.2. High-Friction Base Example

[39] Changing the basal friction to a high value (around 24°) while maintaining the erosion slope at the critical taper of math formula, and keeping all other parameters unchanged, produces an experimental cross section, with longer thrust sheets, less rotation, and hinterland thrust dips not exceeding around 45°. Exhumed material comes from slightly less deep regions, but outcrops more uniformly around the center of the wedge (Figure 13a). All these features are qualitatively reproduced with a numerical simulation using math formula, and math formula. The numerical horses are nevertheless segmented, whereas the experimental horses are continuous from the décollement to the surface (Figure 13b).

Figure 13.

Comparisons between (a) the sandbox experiment HF8 of Konstantinovskaya and Malavieille [2005] after 125 cm of shortening and (b) the numerical simulation.

[40] To illustrate the influence of minute changes in αe for this example, note that setting αec (error of math formula!) leads to a deformation style at the rear of the antiformal stack type instead of the imbricate fan style. The exhumation window would then be closer to the rear.

6. Summary and Discussion

[41] We have presented numerical simulations of a frictional wedge growing by frontal accretion. The material rests on a planar low friction surface (the décollement), and is shortened by the imposed movement parallel to the décollement, of a perpendicular, rigid back wall. The incremental deformation and evolution of the topography due to tectonic are consequences of the activation of a fault-bend fold described with a ramp and a shear plane rooting at the same place on the décollement. The length of active décollement and the dips of the ramp and shear plane are optimized at each increment by selecting those which yield the least upper bound to the compressive force at the back wall, according to the kinematic approach of limit analysis. The program SLAMTec repeats these calculations at every increment of shortening to obtain the complete evolution. It requires only a few mechanical parameters: the friction angles of the décollement and the pristine material through which cuts a new ramp, and the weakened friction angle on the ramp activated at the previous step. A simple erosion rule is accounted for, which periodically removes any material above a line of constant, imposed slope. The choice of this physically simplified approach has important numerical consequences: (i) we only need to discretize the topographic surface and not the interior of the wedge; (ii) optimization of the upper bound is semianalytical; and (iii) the décollement and ramps are treated as true discontinuities instead of shear zones of finite thickness. The method is very efficient, in memory, in amount of operations, and in coding. A typical simulation, with 45 km of shortening, applied in steps Δs = 5 m, with 3200 topographic and décollement points, requires approximately 350 Mo of memory. The computation time largely depends on the value of math formula, because for low values of math formula, ramps have a long lifetime, and the optimization performed at every time step is fast. For math formula, the simulation takes generally less than 40 mn on a cluster node (for example: Intel Xeon X5675, 3.1 GHz, 64 bits, 192 Go RAM), and takes more than 1 h on a laptop ThinkPad X220 (Core i7, 64 bits, 2.8 GHz, 4 Go RAM). If math formula (bulk friction), the simulations take less than 7 h on the Intel node.

[42] We show that for all amounts of shortening (up to around 300%) and without erosion, the mean slope of the wedge formed by frontal accretion matches the theoretical critical taper slope (αc), and its thickness is such that there is no global volume loss of material, i.e., the cross section is balanced. A parametric study of the effects of the ramp friction ( math formula), the erosion slope (αe), and the erosion period (Δse) reveals that these three parameters exert a deep influence on the internal deformation of the wedge. The erosion slope controls strongly the region of exhumation: at the front if math formula, distributed throughout if αec, depending on Δse with threshold values function of math formula, and at the back if math formula. Exhumation at the front is organized as an imbricate fan, possibly limited at the rear by a zone of backthrusting, the rest of the wedge being a stagnation zone. Exhumation, if in the center, is organized by duplexes with verticalized narrow thrust sheets exhuming the deepest material. Exhumation at the back or the front is made by imbricate fans, antiformal stacks, or backthrusting, depending on math formula. Independent of the erosion parameters, the thickness of the thrust sheets is controlled by the amount of weakening math formula. At zero weakening, the sheets are of infinitesimal thickness and lifetime, and the deformation appears to be diffuse. For increasing weakening, the thrust sheets grow in thickness to values of the order of the thickness of the sedimentary pile.

[43] To illustrate the qualitative and quantitative relevance of our parametric study, we have favorably compared our predictions with sandbox experiments combining erosion and low or high décollement friction [Konstantinovskaya and Malavieille, 2005]. The contrasting style of their internal structure and exhumation pattern are reproduced over very large shortenings ( math formula%), using parameter values that are compatible with the experimental data. Their deformation styles are typical of geological cases such as the contrasted Taiwan, Olympics, and New Zealand orogens as it is now further discussed.

[44] In the case of large weakening ( math formula) and shallow erosion slope ( math formula), the thrust sheets are thick, they become steeper and almost vertical toward the hinterland, and the exhumation reaches the deepest rocks, that were originally very close to the décollement (Figure 5a3). These are all features found in the Western foothills of Taiwan wedge and up to the Central range [Konstantinovskaya and Malavieille, 2005, Figure 11].

[45] We have mentioned above that exhumation in the central region and development of a stagnation zone could be obtained by setting αec and considering less weakening ( math formula). Reducing the erosion slope to math formula during the shortening would shift the exhumation to the rear of the wedge and reactivate the stagnant zone (Figures 5b3 or 7b3), and thus would produce the wide distributed area of exhumed rocks observed in the Olympic region of the Cascadia wedge [Konstantinovskaya and Malavieille, 2005].

[46] The large-scale backthrust structure observed without weakening and a shallow erosion slope (Figures 5c3 or 7c3) is strikingly similar to that deduced in the New Zealand Alps from the surface gradient in metamorphic grades [Batt and Brandon, 2002]. This comparison offers an explanation of the backthrusting that is in line with the explanation of Batt and Braun [1997, 1999] based on erosion, and also offers an alternative to the passive roof duplex model of Bonini [2001] based on ductility and sedimentation.

[47] It is somewhat satisfying to observe in these geological examples, that the shallower the décollement, the more weakening on the ramps is needed to fit the structures: Taiwan wedge was best reproduced with a large weakening ( math formula, décollement around 10 km max depth), while the Cascadia wedge ( math formula), and the New Zealand Alps without weakening ( math formula), both with décollements around 30 km max depth. Indeed, beyond 10 km depth, a constant friction is likely to better approximate the real rheology of the faults than the weakened friction found at shallow depths. Also, all these examples require erosion slopes that are equal to or, more often, less than the critical taper. A shallow erosion slope produces an erosion that is concentrated in the highest parts of the wedge, and therefore in agreement with the common idea that erosion is more active in the highest parts of the relief.

[48] The erosion along a straight line is certainly too simple to approximate the shape of a mountain belt. It is however the most widely used erosion law in analogue sandbox experiments [Graveleau et al., 2012, section 3.1.2], and it has been successful in reproducing some features of mountain belts like the Alps [Bonnet et al., 2007, 2008] and Taiwan [Malavieille, 2010]. In these analogue experiments, the slope αe is only set to a precision of math formula, and erosion is applied at certain shortening intervals not well constrained. Our parametric study could first help in choosing the value of Δse such that erosion can be considered as continuous (Figure 10) and second in revealing the consequences of an error on the slope of erosion. We do acknowledge that these parameters are not those required to describe surface processes accounting for fluvial incision and hillslope landslides, which yield curved mean 1-D cross-range topographies. Considering the high sensitivity of the tectonic structures to the value of αe, the incorporation of 1-D erosion curves based on more realistic processes [Lavé, 2005; Willett, 2010] is a necessary way forward for 2-D simulations.

7. Conclusion

[49] Our simulations rely on a simplified rheology based on Coulomb friction (no elasticity, no stress-strain relation), a simplified kinematics (incrementally rigid blocks), and a simplified erosion process (no material above an imposed line). In this respect, our sequential limit analysis (SLA) generalizes the Critical Coulomb Wedge theory from the onset to the development of fault-related folding. Our approach also permits to account for frontal accretion, material softening (by addition of a friction parameter), and erosion (by addition of two parameters). The parametric study based on SLA has revealed that a wide variety of deformation styles and histories can be described with these very few parameters. The comparison to sandbox experiments allows us to conclude that SLA is able to reproduce complex and fundamental features of analogue experiments with very large shortenings, using both low and high basal friction values. Some of the cross sections obtained in the parametric study are reminiscent of geological structures observed in Taiwan, the Cascadia wedge (Olympic region), and the New Zealand Alps. Our parametric study can be interpreted as an extension of the experiments of Konstantinovskaya and Malavieille [2005], either to cases that cannot be done experimentally (indeed, math formula cannot be chosen arbitrarily in experiments), or to interpolate results between experiments using markedly different parameters. This possibility to conduct numerical analysis with hypotheses close to the ones relevant for laboratory experiments constitutes certainly a new step toward the goal of matching physical and numerical experiments. The ease to conduct SLA analysis could also be beneficial for inversion techniques applied to analogue experiments [Maillot et al., 2007; Cubas et al., 2013].

[50] In the future, SLA should be complemented in various ways to better reproduce field observations at the crustal scale. Among the various possibilities, one could mention taking account of the pore pressure [Pons et al., 2011], the possibility of extension instead of or in complement of compression (as in many delta region), as well as the introduction of more realistic erosion laws. Also, the passive markers could be used to track down the evolution of material position and to determine the evolution of temperature or pore pressure with time in a staggered scheme.

Acknowledgments

[51] Baptiste Mary was supported by the French Ministry of Research during his doctoral thesis. Computer engineer Yann Costes is thanked for his technical support in running the programme SLAMTec on the cluster of the University of Cergy-Pontoise. We thank S. Schmalholz (University of Lausanne) and S. Willett (ETH) for their constructive reviews.

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