Available online at www.sciencedirect.com Tectonophysics 446 (2008) 31–41 www.elsevier.com/locate/tecto 3D numerical modeling of forward folding and reverse unfolding of a viscous single-layer: Implications for the formation of folds and fold patterns ⁎ Stefan M. Schmalholz Geological Institute, ETH Zurich, Switzerland Received 13 April 2007; received in revised form 24 September 2007; accepted 27 September 2007 Available online 6 October 2007 Abstract The main aims of this study are to show (i) that non-cylindrical three-dimensional (3D) fold shapes and patterns can form during a single, unidirectional shortening event and (ii) that numerical reverse modeling of 3D folding is a feasible method to reconstruct the formation of 3D buckle-folds. 3D viscous (Newtonian) single-layer folding is numerically simulated with the finite element method to investigate the formation of fold shapes during one shortening event. An initially flat layer rests on a matrix with smaller viscosity and is shortened in one direction parallel to the layering. Forward modeling with different initial geometrical perturbations on the flat layer and different lateral boundary conditions generates non-cylindrical 3D fold shapes and patterns. The simulations show that, in reality, the initial layer geometry and the boundary conditions strongly control the final fold geometry. Fold geometries produced from the forward folding models are used as initial setting in numerical reverse folding models with parameters identical to those of forward models. These reverse models accurately reconstruct the initial geometry of forward models with also only one extension event parallel to the previous shortening direction. The starting geometry of the forward models is inaccurately reconstructed by the reverse models if a significantly different viscosity ratio than in the forward models is used. This work demonstrates that reverse modeling has a high potential for reconstructing the deformation history of folded regions and rheological constraints such as viscosity ratio. Reverse models may be applied to natural 3D fold shapes and patterns in order to determine if they formed (i) during a single or multiple deformation events and (ii) as active buckle-folds with a viscosity ratio ≫ 1 or as passive, kinematic folds without buckling. This approach may find much application to fold interference patterns, in particular. © 2007 Elsevier B.V. All rights reserved. Keywords: Folding; Buckling; 3D; Reverse modeling; Interference patterns 1. Introduction tion may be unknown and also more than one shortening event with different shortening directions may have been active during A method to reconstruct fold amplification and especially to the formation of natural fold shapes. In this study, 3D numerical estimate the amount of bulk shortening that generated a viscous reverse modeling is presented as a potential tool for 3D fold (Newtonian) buckle-fold has been suggested for two dimensions reconstruction. In this context, the impact of (i) the initial (2D) (Schmalholz and Podladchikov, 2001; Schmalholz, 2006). perturbation geometry of the layer, (ii) the boundary conditions A main difficulty, which was solved with this method, is to and (iii) the viscosity ratio on 3D forward and reverse folding separate the amount of shortening by layer thickening from the models is investigated. The results have important implications amount of shortening by folding at constant layer thickness. This for the applicability of numerical reverse modeling to natural fold method can be applied in 3D for cylindrical fold shapes but not for shapes and patterns. more complex, non-cylindrical fold shapes. Additional difficul- In 3D, fold interference patterns have attracted considerable ties arise in 3D fold reconstruction because the shortening direc- attention because such patterns may provide insight into the deformation history of the rock, such as the number of ⁎ Tel.: +41 44 632 8167; fax: +41 44 632 1030. deformation phases, the shortening direction and the amount of E-mail address: schmalholz@erdw.ethz.ch. bulk shortening. 3D fold interference patterns have been studied 0040-1951/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2007.09.005 32 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 with two fundamentally different underlying assumptions: demonstrated by an experiment of G.I. Taylor (see Taylor and passive or active folding. In passive folding, the folded layers Friedman, 1966). On the other hand, it was shown that low exhibit no competence contrast (e.g., the Newtonian viscosity of Reynolds number flows exhibiting contrasts in material the layer and the matrix are identical) and, therefore, no properties (e.g., particles in shear flow) can exhibit chaotic mechanical instability is involved (Odriscoll, 1962; Ramsay, advection and are not reversible (e.g., Aref, 1984; Yarin et al., 1962; Thiessen and Means, 1980; Ramsay and Huber, 1987). 1997; Pine et al., 2005). For folding, an advection equation must Hence, no wavelength selection process is active and it is difficult be solved in addition to the Stokes equations to move the layer to explain what mechanism has actually generated the often interfaces through the model domain during shortening. The observed regularity and periodicity in natural folds. In active reversibility is not obvious because the viscous flow moving the folding, the folded layers possess a higher competence than the layer interfaces is unsteady (due to the moving boundaries and embedding material and a buckling instability with its corres- the buckling instability), and the viscous flow is sensitive to ponding wavelength selection process is active (Ghosh and small changes in the shape of the layer interfaces. Therefore, the Ramberg, 1968; Skjernaa, 1975; Grujic, 1993; Johns and Mosher, numerical simulations performed in this study have their 1996; Kaus and Schmalholz, 2006). In this study, only active justification in showing the numerical reversibility for high folding generated by a mechanical instability is considered. amplitude 3D viscous folding. Important questions arising while studying 3D fold shapes The applicability of the reverse folding modeling indicates that are how many deformation events have generated the observed it is useful to apply numerical models to reverse the formation of fold geometries and how much bulk shortening took place. 3D fold shapes that were generated by a mechanical instability. Numerical simulations of single-layer folding performed in this However, for practical fold reconstructions, the viscosity ratio study show that non-cylindrical 3D fold shapes, with curved between layer and matrix can only be roughly estimated for fold axes and axes orientations varying up to 90°, can be natural folds, and therefore the impact of uncertainties in viscosity generated during a single shortening event with one constant ratios on the reverse models are quantified in this study. The shortening direction. The simulations show that this is possible results show that reverse modeling can be used to test whether because the final fold shapes are strongly controlled by the fold shapes have been generated by either active or passive initial perturbation geometry of the layer (e.g., Mancktelow, folding and may have a high potential to determine whether fold 2001) and the boundary conditions. Therefore, fold shapes with shapes have been generated by one or more deformation events. strongly varying fold axes can be generated by (i) a single, The aims of the paper are (i) to show that 3D fold geometries unidirectional shortening event, (ii) a single, multidirectional with significantly varying fold axes can be generated by single, shortening event (e.g., constriction), or (iii) two or more unidirectional shortening events, (ii) to test the feasibility of shortening events (i.e. true superposed folds). numerical reverse modeling for 3D fold reconstruction and (iii) A suitable tool to investigate the shortening events to quantify the impact of different viscosity ratios on the retro- generating 3D fold shapes and to reconstruct the evolution of deformation of 3D fold shapes. 3D fold patterns is numerical modeling. Alternatively, existing analytical solutions for 3D single-layer folding could be used 2. Methods (e.g., Fletcher, 1991) but these analytical solutions are only valid for small amplitudes and limb dips. In this study numerical The numerical algorithm applied in this work for forward and models are applied because (i) high amplitude folds are reverse modeling is self-developed and the finite element method investigated and (ii) the numerical models can easily be used is employed to solve the continuum mechanics equations for slow in the future for more complex scenarios such as multilayers or viscous (Newtonian), incompressible flow in 3D in the absence of layers with strongly variable thickness. Usually, numerical gravity (see Appendix). The boundary conditions applied for all models are used to simulate the formation of buckle-folds during the shortening of stiff layers (Kaus and Schmalholz, 2006); they are referred to as forward models. On the other hand, it is also possible to use the fold geometries as initial setting of a numerical model and extend the model in a direction opposite to the shortening direction used to produce the folds. Such models are referred to as reverse models. Reverse modeling has been for example applied to Rayleigh–Taylor instabilities (Kaus and Podladchikov, 2001) and flanking