3D numerical modeling of forward folding and reverse unfolding of a viscous single-layer: Implications for the formation of folds and fold patterns
Introduction
A method to reconstruct fold amplification and especially to estimate the amount of bulk shortening that generated a viscous (Newtonian) buckle-fold has been suggested for two dimensions (2D) (Schmalholz and Podladchikov, 2001, Schmalholz, 2006). A main difficulty, which was solved with this method, is to separate the amount of shortening by layer thickening from the amount of shortening by folding at constant layer thickness. This method can be applied in 3D for cylindrical fold shapes but not for more complex, non-cylindrical fold shapes. Additional difficulties arise in 3D fold reconstruction because the shortening direction may be unknown and also more than one shortening event with different shortening directions may have been active during the formation of natural fold shapes. In this study, 3D numerical reverse modeling is presented as a potential tool for 3D fold reconstruction. In this context, the impact of (i) the initial perturbation geometry of the layer, (ii) the boundary conditions and (iii) the viscosity ratio on 3D forward and reverse folding models is investigated. The results have important implications for the applicability of numerical reverse modeling to natural fold shapes and patterns.
In 3D, fold interference patterns have attracted considerable attention because such patterns may provide insight into the deformation history of the rock, such as the number of deformation phases, the shortening direction and the amount of bulk shortening. 3D fold interference patterns have been studied with two fundamentally different underlying assumptions: passive or active folding. In passive folding, the folded layers exhibit no competence contrast (e.g., the Newtonian viscosity of the layer and the matrix are identical) and, therefore, no mechanical instability is involved (Odriscoll, 1962, Ramsay, 1962, Thiessen and Means, 1980, Ramsay and Huber, 1987). Hence, no wavelength selection process is active and it is difficult to explain what mechanism has actually generated the often observed regularity and periodicity in natural folds. In active folding, the folded layers possess a higher competence than the embedding material and a buckling instability with its corresponding wavelength selection process is active (Ghosh and Ramberg, 1968, Skjernaa, 1975, Grujic, 1993, Johns and Mosher, 1996, Kaus and Schmalholz, 2006). In this study, only active folding generated by a mechanical instability is considered.
Important questions arising while studying 3D fold shapes are how many deformation events have generated the observed fold geometries and how much bulk shortening took place. Numerical simulations of single-layer folding performed in this study show that non-cylindrical 3D fold shapes, with curved fold axes and axes orientations varying up to 90°, can be generated during a single shortening event with one constant shortening direction. The simulations show that this is possible because the final fold shapes are strongly controlled by the initial perturbation geometry of the layer (e.g., Mancktelow, 2001) and the boundary conditions. Therefore, fold shapes with strongly varying fold axes can be generated by (i) a single, unidirectional shortening event, (ii) a single, multidirectional shortening event (e.g., constriction), or (iii) two or more shortening events (i.e. true superposed folds).
A suitable tool to investigate the shortening events generating 3D fold shapes and to reconstruct the evolution of 3D fold patterns is numerical modeling. Alternatively, existing analytical solutions for 3D single-layer folding could be used (e.g., Fletcher, 1991) but these analytical solutions are only valid for small amplitudes and limb dips. In this study numerical models are applied because (i) high amplitude folds are investigated and (ii) the numerical models can easily be used in the future for more complex scenarios such as multilayers or layers with strongly variable thickness. Usually, numerical 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 structures (Kocher and Mancktelow, 2005).
The reverse folding modeling performed in this study using linear viscous rheologies shows that the formation of the 3D folds can be accurately restored with a single extension event having a direction opposite to the original shortening direction. On one hand, this may be expected because slow viscous flow at low Reynolds numbers (described by the Stokes equations) is time-reversible (e.g., Bretherton, 1962), which was successfully demonstrated by an experiment of G.I. Taylor (see Taylor and Friedman, 1966). On the other hand, it was shown that low Reynolds number flows exhibiting contrasts in material properties (e.g., particles in shear flow) can exhibit chaotic advection and are not reversible (e.g., Aref, 1984, Yarin et al., 1997, Pine et al., 2005). For folding, an advection equation must be solved in addition to the Stokes equations to move the layer interfaces through the model domain during shortening. The reversibility is not obvious because the viscous flow moving the layer interfaces is unsteady (due to the moving boundaries and the buckling instability), and the viscous flow is sensitive to small changes in the shape of the layer interfaces. Therefore, the numerical simulations performed in this study have their justification in showing the numerical reversibility for high amplitude 3D viscous folding.
The applicability of the reverse folding modeling indicates that it is useful to apply numerical models to reverse the formation of 3D fold shapes that were generated by a mechanical instability. However, for practical fold reconstructions, the viscosity ratio between layer and matrix can only be roughly estimated for natural folds, and therefore the impact of uncertainties in viscosity ratios on the reverse models are quantified in this study. The results show that reverse modeling can be used to test whether fold shapes have been generated by either active or passive folding and may have a high potential to determine whether fold shapes have been generated by one or more deformation events.
The aims of the paper are (i) to show that 3D fold geometries with significantly varying fold axes can be generated by single, unidirectional shortening events, (ii) to test the feasibility of numerical reverse modeling for 3D fold reconstruction and (iii) to quantify the impact of different viscosity ratios on the retro-deformation of 3D fold shapes.
Section snippets
Methods
The numerical algorithm applied in this work for forward and reverse modeling is self-developed and the finite element method is employed to solve the continuum mechanics equations for slow viscous (Newtonian), incompressible flow in 3D in the absence of gravity (see Appendix). The boundary conditions applied for all simulations are (Fig. 1): The bottom model side is kept planar (i.e. vz = 0) with free slip whereas the top side is a free surface (i.e. normal stress and shear stresses on the
Setting
In all simulations a layer, or plate, rests in the x–y plane on a matrix that has a smaller viscosity than the layer (Fig. 1). Only one shortening event with a constant direction is applied. The flat layer is initially perturbed by elevating certain numerical nodes within the layer by either 1/20th or 1/40th of the layer thickness in the z-direction (perpendicular to the layering). The initial perturbation exhibits either a point or line shape with different orientations. The initial layer
Reverse modeling
A question relevant for structural geologists is whether the complex 3D fold shapes and patterns, which formed during a single shortening event, can be restored to their initial, pre-shortening geometry with a single extension event along the same path as the shortening event. Reverse modeling is the useful test to try reconstructing the deformation history of folded layers in 3D.
Discussion
Complex fold patterns and non-cylindrical folds can be generated either by two or more consecutive deformation events (i.e. truly superposed folds), or by a single constrictional deformation event (Ramsay and Huber, 1983, Ramsay and Huber, 1987, Ghosh et al., 1995). The current numerical work has demonstrated that non-cylindrical fold geometries (e.g., folds with orthogonal fold axes) can be also formed by a single deformation event with a single shortening direction (Fig. 3, Fig. 6). This is
Conclusions
Complex 3D fold shapes and patterns exhibiting non-cylindrical fold axes and axes orientations spreading up to 90° can form during one shortening event with a single shortening direction. The initial perturbation geometry of the layer and the boundary conditions have a strong influence on the final fold shape. Therefore, different fold axis orientations and curved fold axes are not necessarily the result of more than one deformation event. This conclusion impels some warning points in
Acknowledgements
Thorough and constructive reviews by Yanhua Zhang and Ray Fletcher, and comments by editor Mike Sandiford are gratefully acknowledged. I thank Ray Fletcher for stimulating and helpful discussions. I thank Jean-Pierre Burg for detailed help and discussions during the preparation of the manuscript, Neil Mancktelow for stimulating discussions on 3D folding and Jaqueline Reber for collaboration during the development of the interference pattern visualization algorithm.
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