Description:

_Journal of Structural Geology 32 (2010) 818-831_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com/locate/jsg_ _Statistical analysis of an experimental compressional sand wedge_ _Nadaya Cubas a,1, Bertrand Maillot b,*, Christophe Barnesb_ _a École Normale Supérieure, CNRS, Laboratoire de Géologie, F-75005 Paris, France_ _b Université de Cergy-Pontoise, Département Géosciences et Environnement, F-95000 Cergy-Pontoise, France_ _article info_ _Article history: Received 9 October 2009; Received in revised form 19 May 2010; Accepted 21 May 2010; Available online 9 June 2010_ _Keywords: Analogue modeling Statistical modeling Accretionary wedge Experimental error bar_ _abstract_ _The quasi-static deformation of dry sand is widely used as an analogue to the brittle deformation of the upper crust. The quantitative comparison of analogue to natural tectonics, or to mechanical predictions, requires identifying sources of biases and estimating the intrinsic variability of the experimental results. We develop experimental and statistical methods that fulfill these requirements. We consider an initially perfect wedge resting on a flat layer, made of uniform dry sand in a rectangular glass box. The box is shortened lengthwise by translating one of its end walls towards the other. The lateral walls can remain fixed or be translated with the moving end wall. Upon shortening, the wedge is thrust above the flat layer forming classical fore-and back-thrusts, as essentially plane-strain structures. Lifetimes, locations, and dips of all thrusts constitute seven quantifiable output parameters (called observables), in addition to the shortening forces monitored at both end walls during shortening. Up to seventy measurements of each observable were performed in seven final-state cross-sections of ten experiments. A three-step statistical analysis allows us to prove that first, the observables vary independently, justifying their modeling with independent distributions. Second, the ergodic hypothesis holds, meaning that along-strike variations can be used to infer the intrinsic experimental variability. Measurements can thus be repeated on successive cross-sections in each experiment. Third, our data set is free from bias due to friction on the lateral walls or due to the finite length of the box. We then construct statistical models of each observable using either Gauss or Laplace distributions. For example, forethrusts dip at 38 ± 14° ± 3.2 ± 14°, and backthrusts, at 41 ± 14° ± 3.3 ± 14°. We finally show how to apply these statistical models to experiments using a different initial geometry. The statistical methods presented here are applicable to experiments with different setups, materials and observables, although the ergodic hypothesis is relevant only to plane-strain experiments._ _© 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _In the study of real tectonic structures, sandbox techniques are used to produce analog deformation histories from undisturbed sedimentary strata to final states that resemble actual structural geometries. Cadell (1888) provides one of the earliest examples of a mountain building due to horizontal shortening above a décollement, while the cross-section of the Northern Alps in granular materials made by Bonnet et al. (2007) is one of the latest and most impressive examples. Koyi (1997) provided a useful historical review of these experimental techniques._ _Comparisons of the results of sandbox models to real structures are often qualitative rather than quantitative, owing to structural complexity and to limitations concerning the quantitative description of the results of sandbox models. Attempts at developing quantitative interpretations started with King Hubert (1951), who showed that the quasi-static deformation of dry sand is consistent with the Coulomb failure criterion, opening the way for sandbox modeling of tectonic structures. Later, sandbox modeling of a wedge was used to validate theoretical predictions of critical wedge theory about the shape and evolution of an accretionary wedge (Davis et al., 1983). Further, the nature of accretion by underthrusting was successfully compared to work estimations using a boundary-element method (Del Castello and Cooke, 2007), and the dip of backthrusts in the wedge predicted from an analytical force balance with the principle of minimum dissipation was validated (Maillot and Koyi, 2006). During this time, both discrete element models (Saltzer, 1992; Seyferth and Henk, 2006; Egholm et al., 2007; Hardy et al., 2009) and finite-element models (Ellis et al., 2004; Crook et al., 2006) were used to conduct comparisons with sandbox experiments. Discrepancies between various numerical techniques and implementations have been demonstrated by benchmark tests (Buiter et al., 2006)._ _N. Cubas et al. Journal of Structural Geology 32 (2010) 818-831_ _819_ _In general, to have predictable quantitative experimental results, reproducibility must be prevalent. To our knowledge, the only attempt at quantitatively considering reproducibility is the benchmark tests by Schreurs et al. (2006). Ten different laboratories performed two plane-strain experiments, respectively in contraction and extension. Large discrepancies were observed in the positions, dips, and numbers of thrusts. These variations were attributed to differences in shear along the lateral walls of the boxes, in granular materials and in their handling methods. The problem of the relative movement of the lateral walls with respect to the moving end of the box was well described during a discussion between Koyi and Cotton (2004) and Costa and Vendeville (2004). Significant experimental artifacts have also been demonstrated when basal friction is very low compared with lateral friction (Vendeville, 2007). Here, we will explicitly demonstrate that our experimental data are free from such biases. Additionally, by using the same sand and the same handling protocol in all experiments, we eliminate the main biases identified in Schreurs et al. (2006) for our quantitative analysis of sandbox model behavior._ _The aim of this article is to quantify the experimental variability and to identify sources of experimental biases. We considered four different experimental prototypes or setups and repeated experiments for each setup. All setups consist of a pre-built sieved sand wedge occupying all the box width and approximately a third of its length. The rest of the box is filled with a flat sand layer. Differences in the setups concern only the relative slip sense of the lateral walls, the surface slope of the wedge and the dip of the base of the box (Table 1). We study the growth of the sand wedge by frontal accretion for an overall shortening of the box of about 8%, generating one or two forethrusts in a forward sequence. We then construct statistical distributions describing the variations of the observables (i.e., positions, lifetimes, dips of the thrusts, and the compressive forces at both ends of the box). Thus, the experimental results are described in a probabilistic framework, with the ultimate goal of inverting them to retrieve mechanical properties of the sand, as was done in a simpler experimental setup by Maillot et al. (2007) and for a case study of Nankai wedge (Cubas et al., 2008). These setups are very close to that proposed by Schreurs et al. (2006). In this respect the present study complements theirs, by repeating the experiments using the same protocol, sand, and box, instead of comparing the results of different laboratories, and by introducing a statistical treatment of the experimental results._ _2. Experiments_ _The experiments consist of shortening by a fixed amount of an initially perfect sand wedge resting on a flat sand layer (Fig. 1a). To accommodate the shortening, the sand is lifted in the flat layer at the front of the wedge, following a classical fore-backthrust geometry. Overall shortening magnitude was chosen to be sufficient to generate two fore-back thrust pairs (Fig. 1b). However, the intrinsic variability of the experiments yields the result that two out of ten experiments produce only one thrust pair (Fig. 1c)._ _2.1 The sandbox_ _The experimental box is rectangular, with inside dimensions of 280 mm (width), 370 mm (length) and 90 mm (height). It can be tilted lengthwise producing a dip of the basal plate in the direction of shortening, which is typical of accretionary wedges. All the box is built with 10 mm-thick glass with a precision of ±0.05 mm, which is an error smaller than our smallest grain sizes. In all experiments, shortening of the sand is imposed by translating the back wall (i.e., the wall on the wedge side) over 30 mm at a rate of 0.52 mm/s with an electric motor via a screw._ _This sandbox allows us to infer the effect of the horizontal shear stress due to sand friction on the lateral walls of the box by using two box configurations. In configuration A, the lateral walls are translated with the back wall, while in configuration B, the back wall moves alone (Fig. 2a,b). Any plane-strain sandbox experiment fits into one of these two categories. Sometimes in the literature, configuration A is referred to as a “pull” setup that is typical of boxes using a conveyor belt, and configuration B as a “push” setup. Note that in configuration B, the sand is overall lifted up by the movement of the back wall, while in A, the back wall remains at a constant altitude. Here, the box is horizontal (b = 0), so the only difference between the two configurations is the lateral friction. Basal and side glass walls were treated with a carbon-based product (“RainX”) that reduces friction with sand._ Ключевые слова: sand-box, schreurs, vendeville, eds analogue, thrust, rst forethrust, nite length, meas, measured, solid, central column, journal structural geology, independent, initial slope, society, wedge, error bar, graph measurement, slope, lateral wall, extension experiment, laplacian, comparison, bin, sand box, dispersion estimator, statistical model, surface, intrinsic variability, basal plate, model, crook, protocol, actual number, force sustained, dry sand, deg, mm df, strain, geology, experimental result, wall, structural, box width, curve, observables, statistical, experimental prototype, data population, cross, theoretical distribution, quasi-static deformation, observable, standard deviation, prototype data, sections, journal structural, tectonics, press, variability, forethrusts, lohrmann, tectonophysics, structural geology, maillot, shortening, fault dip, case, experimental apparatus, table, table column, prototype, analysis, initial, laplacian distribution, gaussian distribution, theoretical, standard, sand body, geophysical, lateral friction, prototype conguration, prototype experiment, cross-sections, sandbox, measurement observable, davis, granular material, friction coefcients, cross-sections separated, forethrust, laplacian case, setup, bias, gaussian case, number, geological, experimental insight, fault, experiment, independent measurement, nieuwland, strain gauge, test, cross sections, deviation, performed, cross-section, schematic illustration, backthrusts, mm mm, contribution, journal, force, koyi, geometric observables, mm, set, experiment note, wa, experimental setup, egholm, thrust dip, measurement, rejected, histogram, rst, gauge, length, sand, deformation, de?ne, congurations, journal geophysical, wygal, highest sample, angular observables, layer, result, fault location, column, elsevier, experiment prototype, fold-and-thrust belt, seventy measurement, sand-box modeling, lowest sample, geometric, change, nbr, negligible cohesion, numerical modelling, numerical, statistical analysis, repeated experiment, normalized force, fourth column, quantitative, con?guration, geometry, analogue, statistical modeling, sand distributor, king hubert, box, experimental, friction, cross-sections experiment, quantitative comparison, basal, distribution, force measurement, hardy, lubricant choice, data set, sample, krantz, general, shear stress, modeling, population, bin denition, graph, cubas, dip, ?rst, variation, gaussian, surface slope, doi, lubricant, sand wedge, cubas journal, saltzer, lateral, data, pair, central, measure