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_Journal of Structural Geology 32 (2010) 566e579_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg A model for low amplitude detachment folding and syntectonic stratigraphy based on the conservation of mass equation J. Contreras* Departamento de Geolog?a, Centro de Investigaci?n Cient??ca y de Educaci?n Superior de Ensenda (CICESE), km 107 carretera Tijuana-Ensenada, Ensenada, BC 22860, Mexico Article info Article history: Received 10 June 2009 Received in revised form 8 March 2010 Accepted 11 March 2010 Available online 20 March 2010 Keywords: Stratigraphic modeling Detachment folding Growth strata Self-af?ne deformation Abstract This paper presents a model for the structural evolution of low strain detachment folds in which rock is treated as an incompressible deformable material and its kinematics is governed by the continuity equation. The model also considers the following boundary conditions: (i) the vertical ?ux across the core of the fold is described by a cosine function, (ii) the horizontal ?ux due to transport along a basal detachment remains constant with depth, and (iii) the depth of the detachment remains ?xed. Upon finding analytical expressions for the velocity ?eld of this deformation process, a model for the accumulation of growth strata and the degradation of the topography created by folding is then derived. These processes are approximated by means of the transport-diffusion equation. Solutions of these two models are in excellent agreement with geometric, and stratigraphic relations documented in fold-and-thrust belts, and analog experiments. Moreover, the model indicates that fold growth in this class of structures is a self-af?ne process. A compilation of detachment folds around the world indicates that these structures share a common pro?le. Moreover, fold amplitude and wavelength of these folds are scaled by different amounts, con?rming this result. The stratigraphy obtained with the growth strata model exhibits the typical thinning, and truncation of timelines toward the core of the anticline, cross-cutting relations often observed in seismic cross-sections and ?eld data. ? 2010 Elsevier Ltd. All rights reserved. 1. Introduction Detachment folds (Fig. 1) are one of the three architectural elements of thrustbelts; the other two being fault-bend folds and fault-propagation folds (Suppe, 1985; Jamison, 1987; Mitra, 1990; Suppe and Medwedeff, 1990; Nem x14cok et al., 2005). Despite their importance as the building blocks of contractional terrains, little progress has been made toward finding analytical solutions describing the evolution of these structures. Analytical solutions provide with the exact behavior of the system being modeled in terms of well-known elementary functions and material parameters. Moreover, characteristic times and lengths can be readily identi?ed that may reveal fundamental relationships that can be tested against datasets. Of course only the simplest cases have closed-form solutions, and even those are highly idealized. This work is also motivated by the shortcomings in many of the existing kinematic models of fault-related folding that assume bed length is conserved, and a stylized geometry in which folds are straight-limbed and sharp hinged (Suppe, 1983; Hardy, 1995; Hardy and Poblet, 1995; Contreras and Suter, 1990, 1997; Zehnder and', '* corresponding author E-mail address: juanc@cicese.mx 0191-8141 $ e see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016 j.jsg.2010.03.006 Allmendinger, 2000; Suppe et al., 2004; Hardy and Connors, 2006). A further criticism is that these models consider fault-related folding as a steady state process (Poblet et al., 2004). Some of these assumptions are not physically realistic (e.g., Kwon et al., 2005) and appear to be unwarranted in the light of ?eld data (Wiltschko and Chapple, 1977; Verg?s et al., 1996; Poblet et al., 2004), experimental results (Biot, 1961; Storti et al., 1997; Da?ron et al., 2007), and other stratigraphic relationships discussed below. Within this context kinematic models for low amplitude detachment folds and associated growth strata are illustrated here. I selected low amplitude detachment folds because, as it is shown in Fig. 1, they have simple geometries and growth strata. In this ?gure the pre-growth sequences display a smooth symmetrical shape. Sharp hinges, an idealization often made in the literature (e.g., Poblet and Hardy, 1995; Poblet et al., 1997), are not evident. Instead, fold curvature seems to change evenly from a maximum value at the crest of the structure to a minimum value toward the ?anking synclines. Syntectonic strata also display a strong symmetry indicating that these structures grow by acquiring amplitude as shortening is accommodated by folding. All these features suggest that strain is accommodated in a continuous and smooth fashion and that periodic functions are best suited to model the evolution of this class of folds. J. Contreras Journal of Structural Geology 32 (2010) 566e579 567 Fig. 1. Example of a detachment fold from the Campos and Santos basin, offshore Brazil (Demercian et al., 1993). These structures form by ?ow of ductile rocks and by parallel folding of more competent rocks. The stratigraphy of these structures consists of two successions: the pre-growth strata, deposited previous to folding, with a homogenous thickness, and growth strata synchronous with folding. Existing kinematic models of detachment folding often treat independently axial surface activity, limb rotation, limb lengthening, uplift rate, as well as the accumulation of syntectonic sediments (e.g., Hardy and Poblet, 1994; Poblet and McClay, 1996; Poblet et al., 1997; Wilkerson et al., 2004; Da?ron et al., 2007). Even if mass conservation is imposed, additional geometrical constrains such as self-similarity (or the lack of) are required to bring the number of degrees of freedom to a few manageable parameters. Moreover, velocity ?elds in these works are derived in a heuristic manner, not ab initio. The model presented here is of extreme simplicity. It only considers that mass is conserved, a stationary Eulerian velocity ?eld, and a constant shortening rate applied on the limbs of the structure. These conditions severely restrict the kinematics and time-evolution of folding in the model and reduce the number of free variables to the rate of tectonic uplift, the coef?cient of mass diffusion, and the initial dimensions of the fold. In spite of its simplicity, the model reproduces accurately the shape of these folds, their kinematics, and stratigraphic relations observed in seismic lines and analog experiments. Following the principles of ?uid dynamics and previous work by Waltham and Hardy (1995), Hardy and Poblet (1995), and Zehnder and Allmendinger (2000), I start by posing the boundary value problem describing the kinematics of detachment folding and growth strata, as well as their solutions, in an Eulerian reference frame that describes the motion of material points passing through ?xed positions in space. Next, the pathlines of the deformation process are obtained; results are then expressed in a more natural Lagrangian reference frame that follows the motion of parcels through time and space. This shift in reference frame is necessary to compare the model results with measurements in analog experiments that track the evolution of material markers through time as deformation progresses. I will refer to positions and velocity components in the Eulerian reference system using capital letters, i.e., X, Y, and VX, VY; for the Lagrangian reference frame I will use lower case letters, i.e., x, y, and vx, vy. Also following the approach of Hardy and Poblet (1994), Hardy et al. (1996) and den Bezemer et al. (1999) the accumulation of growth strata is modeled by means of a boundary value problem in which the topography generated by detachment folding degrades by the combined effects of erosion and sedimentation. These processes are approximated by the transport-diffusion equation whose expression can also be derived from the principle of mass conservation. Once the solution of the boundary value problem is found, the stratigraphic timelines can be obtained by evaluating the solution for past times; bundles of those lines can be compared with sedimentary sequences, which are strata bounded by surfaces that are assumed to represent time lines (e.g., Mial, 1997). 2. The mass conservation equation and other fundamental relations The mass conservation equation states that the mass change inside an arbitrary volume v ?xed in space is equal to the mass ?ux crossing the bounding surface G of the volume, plus the mass added by sources inside the volume. The equation in its integral form is expressed as follows: Z x18v vt x19 ? V,V r dv ? Z Z q,n dG? 4_ dv; (1) v G v where t is the time, r is the mass density, V is the Eulerian velocity, q is the mass ?ux, n is the unit vector normal to the bounding surface G, and 4_ represents the mass sources. The differential operator ?v_vt ? V,V? is the Lagrangian derivative; the first term is the Eulerian (spatial) derivative whereas the second term represents advection or simply the mass transported by the deforming medium. Specialized forms of this general equation can be derived to characterize the kinematics of deformation as well as the redistribution of mass associated with erosion and sedimentation and are discussed next. Two assumptions can be made to derive a simpler equation than that of expression (1) governing the kinematics of deformable bodies. The first one is to assume that rocks do not undergo chemical reactions or phase changes. A second assumption is to consider that no expulsion of intra-granular ?uids takes place during the burial of sediments. Under such conditions the system remains closed (i.e., there are no sources or sinks of mass), and r is constant. Expression (1) then simplifies._ Ключевые слова: spatial position, erosion, surface, groshong, velocity eld, central asia, model derived, deformation progress, position, mass conservation, mial, function, detachment fold, vx vy, detachment, thinning-upward pattern, epard, author, pclet number, younger sediment, reciprocal relation, material position, source, excess area, rst term, geometrical, incoming, small, bounding surface, poblet, geometry, sandbox experiment, jamison storti, st, limiting case, fold axis, tectonophysics, growth stratum, process, suter, shear, text, folding, central anticline, kinematic modeling, fold limb, jamison, american association, case, fault, aapg bulletin, expression, component, term, central, strain, cosine function, hardy hardy, shortening, self-afne process, differential, scharer, topography, characteristic time, kinematic, covered anticline, fold shape, accommodated continuously, doi, wavelength, paper, fold growth, ?eld, condition, ductile layer, suppe, geology, closed-form solution, thrust, detachment folding, model, mitra, model reproduces, equation, sedimentary sequence, connors, structural geology, ?ux, hardy poblet, uplift, syntectonic, exponential term, kinematic model, mcclay, initial conguration, transport, l? ?, approximate solution, deformation, analog experiment, problem, deformation function, journal structural, boundary condition, velocity component, result, hardy, sedimentation, evolution, structure, simple shear, change, basin, kinematics, appendix, depth, derived, seismic cross-sections, thrust tectonics, bulletin, common prole, applied, vu, ratio, vy, stratigraphic, diffusion, verg?s, stratum, parameter, limb, vx, vertical component, pattern, stratigraphic horizon, topographic gradient, fold-and-thrust belt, time, normalized change, journal structural geology, deformation process, seismic cross-section, simple, local sedimentation, tectonics, biot, rock, amplitude, characteristic, detachment surface, plot, aspect ratio, rectangular region, seismic, eq, velocity description, growth, fold amplitude, demercian, york, distant source, relief, discussion, contreras journal, linear, geological, fault-related folding, material, velocity, structural, tulsa, model result, aapg, scaling factor, cos?px, vertical direction, mass leaving, chapple epard, region, sediment, contreras, development, erosion sedimentation, presented, mass diffusion, tectonic uplift, journal, height, blackwell, bed, model presented, syntectonic sediment, experiment, initial, form, fold, mass, tectonic, burbank, solution, relation, basal detachment, kink band, thickness, vertical, constant, topographic relief, spatial term, pre-growth stratum, non-dimensional quantity, transport-diffusion equation, boundary, material point, structural relief, rate, area, cloetingh