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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: [email protected]. 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 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. Fig. 1. Model setup. Shortening is applied only in the x-direction where the On one hand, this may be expected because slow viscous flow at shortening velocity vx is a function of the model width, X, and the constant strain low Reynolds numbers (described by the Stokes equations) is rate, e. vx is modified after each numerical time step with the new value of X to time-reversible (e.g., Bretherton, 1962), which was successfully maintain a constant strain rate. The top model surface is always free. S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 33 simulations are (Fig. 1): The bottom model side is kept planar (i.e. one shortening event with a constant direction is applied. The vz =0) with free slip whereas the top side is a free surface (i.e. flat layer is initially perturbed by elevating certain numerical normal stress and shear stresses on the surface are zero). The nodes within the layer by either 1/20th or 1/40th of the layer lateral sides orthogonal to the x-direction are always kept planar thickness in the z-direction (perpendicular to the layering). The with free slip. One lateral side orthogonal to the y-direction is initial perturbation exhibits either a point or line shape with always kept planar with free slip (i.e. vy =0). For the other side different orientations. The initial layer thickness, the Newtonian orthogonal to the y-direction two different conditions are applied: viscosity of the matrix and the shortening strain rate have all a thesideiskeptplanarwithfreeslip(i.e.vy =0, referred to as value of one and are used as characteristic values for length, boundary condition A) or is a free surface, so that the vis- viscosity and time, respectively. All other parameters and units cous material can extend in the y-direction during shortening in are scaled (or non-dimensionalized) by these three characteristic the x-direction (referred to as boundary condition B). Shortening values. The model width in the x- and y-directions is 80 times in the x-direction is performed under a horizontal constant strain the initial layer thickness and the model height is 16 times the rate which is achieved by modifying the horizontal velocity in the layer thickness. The competence contrast between layer and x-direction at one side after each numerical time step. At each time matrix is defined by the ratio of layer viscosity to matrix step, the horizontal velocities in the x-direction at all nodes of the viscosity. If the viscosity ratio is larger than one and an initial shortened side are identical. After each time step the calculated 3D perturbation geometry is present then a buckling instability (i.e. velocity field is used to advect the numerical grid and a new active folding) develops during layer-parallel shortening. velocity field is calculated for the new geometry (i.e. explicit time integration scheme). 3.2. Experiments

3. Forward modeling In the first series of experiments (Fig. 2) a point-shaped initial perturbation (1/20th of layer thickness) and boundary 3.1. Setting conditions A are applied (Fig. 2A) so that shortening of the incompressible material can generate extension in the vertical, In all simulations a layer, or plate, rests in the x–y plane on a z-direction only. The viscosity ratio between layer and matrix matrix that has a smaller viscosity than the layer (Fig. 1). Only is 75. After about 50% shortening parallel to the x-direction

Fig. 2. Results of numerical forward and reverse modeling for an initial geometrical point-shaped perturbation. Plots show the geometry of the layer and the layer surface is colored with values of the topography of the layer surface. The numbers in the legends are the vertical, z, values of the layer surface. Black color indicates high and white color low elevation. A) Initial and B) final layer geometry for a forward run with a viscosity ratio, R, of 75. C) to F) show results of the reverse models performed with different values of R. 34 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41

Fig. 3. Results of numerical forward and reverse modeling for an initial geometrical line perturbation in direction of the shortening. Plots show the geometry of the layer and the layer surface is colored with values of the topography of the layer surface. The numbers in the legends are the vertical, z, values of the layer surface. Black color indicates high and white color low elevation. A) Initial and B) final layer geometry for a forward run with a viscosity ratio, R, of 75. C) to F) show results of the reverse models performed with different values of R. three more or less cylindrical folds have developed (Fig. 2B). cylindrical folds with axes parallel and perpendicular to the For visualization, the folded layer is centered vertically around shortening direction. Folds at high angle to the shortening direction zero. The folds result from active folding caused by layer- are comparable to those obtained in the previous experiment. The parallel shortening of a stiff layer with an initial geometrical fold axis parallel to the shortening direction is anchored on the perturbation (i.e. buckling). The fold axes of the folds next to prescribed perturbation. the middle fold are slightly curved, showing minor culminations In the third series of experiments (Fig. 4) the line perturbation and depressions. The three more or less cylindrical folds (1/20th of layer thickness) is again parallel to the shortening developed from a point-shaped initial perturbation by fold direction (Figs. 3A and 4A) but boundary conditions B are propagation. This result could be predicted also based on applied. In that case, shortening in the x-direction can generate existing analytical solutions for 3D folding (Fletcher, 1991), but extension in both the y-andz-directions (Fig. 4B). The viscosity because the analytical solutions are only valid for small ratio between layer and matrix is 75. This different boundary amplitudes and limb slopes, the numerical results are essential condition has a strong influence on the fold pattern: indeed, there to provide the correct finite amplitude fold geometries. For is no fold parallel to the shortening direction along the prescribed example, the limb slopes at the center of the middle fold (at perturbation (compare Figs. 3B and 4B). Also, the fold y=40 and x=20, Fig. 2B) are nearly 90° whereas the limb amplitudes generated in the third series of experiments are slopes at the boundary of the model domain of the two other smaller than the amplitudes generated in the second series of folds (e.g. at y=0 and x=10, Fig. 2B) are around 25°, because experiments, which shows that the fold amplification rate also is the folds are propagating both in the y-direction, along axes, and strongly controlled by the boundary conditions (Figs. 3B and 4B). in the x-direction starting from the location of the initial point- In the fourth series of experiments (Fig. 5) the perturbation shaped perturbation. The analytical solution is only valid up to line (1/20th of layer thickness) trends 45° to the shortening 20° limb dip and it is therefore not possible to capture the high direction (Fig. 5A) and boundary conditions B are applied, as in amplitude fold propagation with the analytical solutions. the third series of experiments. The viscosity ratio between layer In the second series of experiments (Fig. 3) a line perturbation and matrix is 75. The resulting geometry shows a pattern with (1/20th of layer thickness) parallel to the shortening direction is folds being non-cylindrical and vanishing along-axis (Fig. 5B). introduced in one half of the plate (Fig. 3A) and boundary The highest fold amplitude developed at the corner of the model conditions A are applied. The viscosity ratio between layer and domain (x=40, y=0) where the initial perturbation line meets matrix is 75. The resulting fold geometry exhibits slightly non- the model boundaries. S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 35

Fig. 4. Results of numerical forward and reverse modeling for an initial geometrical line perturbation in direction of the shortening. The model side perpendicular to the y-direction at position y=80 is a free surface (compare A and B). The numbers in the legends are the vertical, z, values of the layer surface. Black color indicates high and white color low elevation. A) Initial and B) final layer geometry for a forward run with a viscosity ratio, R, of 75. C) to F) show results of the reverse models performed with different values of R.

Fig. 5. Results of numerical forward and reverse modeling for initial geometrical line perturbation in direction at 45° to the shortening. The model side perpendicular to the y-direction at position y=80 is a free surface (compare A and B). The numbers in the legends are the vertical, z, values of the layer surface. Black color indicates high and white color low elevation. A) Initial and B) final layer geometry for a forward run with a viscosity ratio, R, of 75. C) to F) show results of the reverse models performed with different values of R. 36 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41

Fig. 6. Results of numerical forward and reverse modeling for an initial geometrical line perturbation parallel to the shortening and a point-shaped perturbation. The numbers in the legends are the vertical, z, values of the layer surface. Black color indicates high and white color low elevation. A) Initial and B) final layer geometry for a forward run with a viscosity ratio, R, of 50. C) to F) show results of the reverse models performed with different values of R.

In the fifth series of experiments (Fig. 6) the perturbation kept the same as those of the forward models, but the velocity consists of a line that runs parallel to the x-direction throughout boundary condition is reversed to simulate extensional de- the model (y=60) and a point perturbation (x=40, y=20) both 1/ formation (i.e. sign of horizontal velocity in the x-direction is 40th of the layer thickness (Fig. 6A). Boundary conditions A are changed). Such well-defined reverse models are only possible applied and the viscosity ratio between layer and matrix is 50. with analytical or numerical models. The reverse models are run The resulting geometry shows synclines and anticlines with axes with the same number of time steps as the forward models. parallel to the x-direction that bound three folds with axes Ideally, after the last time step, the final geometry of the reverse parallel to the y-direction (Fig. 6B). The folds with axes parallel model is identical to the starting geometry of the forward model. to the y-direction originate from the point-shaped perturbation In the following experiments always four reverse simulations and amplify faster than folds with axes parallel to the x-direction have been performed: one with the same viscosity ratio as used and anchored on the x-parallel perturbation (Fig. 7A and B). in the forward model and three with different viscosity ratios to The results of the forward modeling show that 3D fold test the impact of uncertainties in the viscosity ratio on the shapes and patterns and their amplification rates are strongly reverse modeling. controlled by initial perturbation geometries and orientation with respect to shortening. 4.2. Experiments

4. Reverse modeling For the first series of experiments, the initial geometries of the reverse models utilize the final folded geometry of the forward A question relevant for structural geologists is whether the model shown in Fig. 2B with the same viscosity ratio of 75. The complex 3D fold shapes and patterns, which formed during a final step of the reverse model (Fig. 2C) reconstructs the point- single shortening event, can be restored to their initial, pre- shaped initial perturbation geometry shown in Fig. 2A, with only shortening geometry with a single extension event along the same some very low magnitude undulations (smaller than the magni- path as the shortening event. Reverse modeling is the useful test to tude of the point-shaped perturbation). Importantly, the magni- try reconstructing the deformation history of folded layers in 3D. tude of the point-shaped perturbation is close to the magnitude of the perturbation implemented for the forward modeling (values of 4.1. Setting grayscale colorbar in Fig. 2A and C). Three additional reverse models starting from folds of Fig. 2B were performed with The fold geometries obtained after forward modeling are viscosity ratios of 1, 10 and 500 instead of 75. The simulation with used as initial setting to reverse models. All the parameters are a viscosity ratio of 1 represents passive folding without buckling S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 37

Fig. 7. Evolution of fold interference pattern for the simulations of the fifth series of experiments (Fig. 6). The black area shows the folded layer and the white area shows the matrix cut by a x and y horizontal surface at the average vertical, z, position of the layer. A) to C) show the interference patterns for different deformation steps of the forward model. D) to F) show the interference patterns for the corresponding deformation steps of the reverse model with the same viscosity ratio as the forward model (i.e. 50). G) to I) show the interference patterns for the corresponding deformation steps of the reverse model with a viscosity ratio of 1. The reverse model with the same viscosity ratio as the forward model (D to F) correctly restores the fold evolution whereas the reverse model with a viscosity ratio of 1 (G to I) fails to restore the fold evolution. instability. The final layer geometries of the reverse models with For the fourth series of experiments (Fig. 5), four reverse viscosity ratios of 1 and 10 (Fig. 2D and E, respectively) are models were performed with viscosity ratio of 1, 50, 75, and significantly different from the initial geometry of the forward 250. The reverse model with the same viscosity ratio as the model. The final layer geometry of the reverse runs with viscosity forward model (Fig. 5C) reproduces rather accurately the initial ratio of 500 (Fig. 2F) is not as close as the reverse model with the geometry of the forward model, but undulations having a correct viscosity ratio, even if it would be an acceptable solution magnitude comparable to the initial perturbation of the forward in terms of error magnitude (b5%). model are present. The final geometry of the reverse model with For the second series of experiments (Fig. 3), four reverse a viscosity ratio of 250 (Fig. 5F) does not reach a satisfactory runs were performed with viscosity ratios of 1, 50, 75, and 250. solution since the initial perturbation of the forward model is The final geometry of the reverse model with the same viscosity hardly visible. There is again a strong boundary effect at the free ratio than the forward model (i.e. 75, Fig. 3C) is closest to the side where the perturbation is about one order of magnitude initial geometry of the forward model (Fig. 3A). The reverse larger than the initial perturbation of the forward model (about model with a viscosity ratio of 1 (Fig. 3D) shows a considerable 0.25 compared to 0.025). The reverse model exhibiting passive deviation from the initial geometry, whereas the other reverse folding (viscosity ratio equals 1, Fig. 5D) fails to restore the models with viscosity ratios of 50 (Fig. 3E) and 250 (Fig. 3F) initial perturbation geometry. yield acceptable results, relatively close to the correct solution. Forthefifthseriesofexperiments(Fig. 6), four reverse models For the third series of experiments (Fig. 4), four reverse models were performed with viscosity ratios of 1, 10, 50 and 250. The were performed with viscosity ratios of 50, 75, 100 and 250. The reverse model with the same viscosity ratio as the forward model reverse model with the same viscosity ratio as the forward run (Fig. 6C) accurately reproduces the initial geometry of the (Fig. 4C) best reconstructs the initial geometry of the corres- forward model but undulations having a magnitude comparable to ponding forward model, whereas the final geometry of the reverse the initial perturbation of the forward model are present. The final model with a viscosity ratio of 250 (Fig. 4F) does not reconstruct geometry of the reverse models with viscosity ratios of 1 (Fig. 6D) the initial geometry of the forward model. There is a clear and 10 (Fig. 6E) do not reach a satisfactory solution since the high boundary effect at the free side. The perturbation at this side is amplitude folds remain clearly visible. about one order of magnitude larger than the initial perturbation of Three deformation stages of the forward model from the fifth the forward model (around 0.2 compared to 0.025). series (Fig. 6) of experiments (Fig. 7A, B and C) are compared 38 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 to the three corresponding deformation stages of the reverse (Figs. 4 and 5). This is because with two free surfaces the flow model with the same viscosity ratio (Fig. 7D, E and F) and to of material is less confined and a larger number of flow paths are three deformation stages of the reverse model with a viscosity possible than in models with only one free surface. The ratio of 1 (Fig. 7G, H and I). For each deformation stage the fold uncertainties in viscosity ratio between the folded layer and the interference pattern of the layer (black) is shown. To visualize matrix may have minor or major impacts on the reverse the interference pattern, the model is cut along a horizontal modeling, depending on the boundary conditions of the forward plane that (i) is orthogonal to the z-direction and (ii) is at the runs. All reverse models with the same viscosity ratio as in the average vertical, z, position of the layer within the model forward model delivered a final layer and perturbation geometry domain. The areas within this plane which belong to the layer very close to the original geometry of corresponding forward are filled in black and the areas that belong to the matrix are models, which emphasises the validity of the method. filled in white. The evolution of the interference pattern shows Reverse models with a viscosity ratio significantly larger that the folds that originate from the point-shaped perturbation than the one of the corresponding forward model and boundary amplify and propagate faster than the folds that originate from conditions A provided relatively flat layer geometries after an the line-shaped perturbation (can be seen in Fig. 7A and B). extension that equaled the shortening in the forward model. This comparison shows that not only the initial layer geometry Therefore, the reverse modeling is not always able to estimate is accurately reconstructed by the reverse model, but also that an upper limit of the viscosity ratio. On the other hand, reverse the entire fold amplification and propagation is well recon- models with a viscosity ratio significantly smaller than the one structed as demonstrated by the similarity of interference of the corresponding forward model did not provide flat layer patterns between the corresponding stages of forward and geometries. In particular, the reverse models with a viscosity reverse models (Fig. 7C and F, B and E, A and D). The results ratio of 1 (passive folding) always exhibited significantly larger show that although folds grow asynchronous and propagate in amplitude folds after an extension equal to the shortening of the both x and y horizontal directions, the reverse model with the corresponding forward model. If the reverse models with a same viscosity ratio than the forward model successfully viscosity ratio significantly smaller than the one of the reconstructs the fold evolution, i.e. the deformation history. In corresponding forward are extended further until the layer is contrast, the reverse model with a viscosity ratio of 1 (Fig. 7Gto relatively flat, then the layer exhibits considerable thickness I) fails to reconstruct the fold evolution. variations showing the invalidity of the reverse model. The reverse modeling is therefore suitable to determine if natural 5. Discussion fold shapes have been generated by active or passive folding and may help to put a lower limit on the viscosity ratio of natural Complex fold patterns and non-cylindrical folds can be fold shapes. The upper limit for the viscosity ratio of natural generated either by two or more consecutive deformation events folds has to be estimated additionally with other parameters (i.e. truly superposed folds), or by a single constrictional such as the fold-span to layer thickness ratio which is controlled deformation event (Ramsay and Huber, 1983; Ramsay and by the effective viscosity ratio. Huber, 1987; Ghosh et al., 1995). The current numerical work Applications to natural folds require knowledge of many has demonstrated that non-cylindrical fold geometries (e.g., model parameters such as shortening direction, boundary folds with orthogonal fold axes) can be also formed by a single conditions or viscosity ratio which are usually only approxi- deformation event with a single shortening direction (Figs. 3 mately known. There is a need to perform reverse models with and 6). This is controlled by the initial geometry of the layers various viscosity ratios, in order to quantify the influence of the and the deformation boundary conditions. A detailed geometric uncertainty in the viscosity ratio, one of the biggest unknown in analysis of the fold shapes and interference patterns sometimes geological cases. allows determining, if folds have been generated by one or more The applied lateral boundary conditions, where one model deformation events (Ghosh et al., 1995). Reverse modeling is an side parallel to the shortening direction is either fixed or is a free additional method for determining the number of deformation surface, represent two end-member scenarios. In nature, the events that generated natural folds. actual boundary conditions may often be in between these two Folds with axes parallel to the shortening direction can end-member scenarios, but natural boundary conditions can be develop if the lateral sides parallel to the shortening direction are close to both of the two applied boundary conditions. fixed, i.e. not allowed to extend (Figs. 3B and 6B). Thereby, a In 2D, it is expected that the influence of the initial layer layer-parallel stress in the y-direction is generated, which is perturbation geometry on the final fold shape becomes smaller responsible for folding with axis orthogonal to this stress for larger viscosity ratios, because the wavelength selectivity direction (and this stress direction is orthogonal to the shortening increases with increasing viscosity ratio (Biot, 1961). In 3D, direction). larger viscosity ratios are not expected to reduce the influence of The accuracy of the reverse models depends on the the initial perturbation geometry on the final fold shape, because complexity of the deformation that generated the forward fold fold axes can develop parallel to the initial perturbation shapes. The results of the reverse models in the experiments orientations and larger viscosity ratios generate a faster growth with only one free surface (the top one, Figs. 2, 3 and 6) are of the folds with axes parallel to the initial perturbation. more accurate and less sensitive to uncertainties in viscosity For folding of a layer with viscous rheology, the shortening ratio than the results of reverse models with two free surfaces strain rate (which can vary during natural folding) needs not be S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 39

Fig. 8. A) Numerical folding of two viscous layers embedded in a matrix with 40 times smaller viscosity. Shortening was applied in both the x- and y-directions but the shortening rate in the x-direction was twice the rate in the y-direction. The initial perturbation geometry was random, where each numerical node of the layer was elevated randomly with a maximal value of 1/40th of the layer thickness. The initial model was 80 by 80 wide and 16 high with the layer thickness of one being the characteristic length scale. The displayed fold shapes were obtained after coeval 50% (x-direction) and 25% (y-direction) shortening. B) The corresponding fold interference pattern of the 3D fold shape shown in A) cut by a x and y horizontal surface at the average vertical, z, position of the two layers. The two layers are filled with black and the matrix is filled with white. known for the reverse modeling because the Newtonian Numerical reverse modeling has a high potential to restore viscosity is independent of the strain rate. This is different for rigorously the formation of 3D folds and may be applied to folding of a layer with power-law rheology where the effective natural fold shapes in order to quantify the folding history of an viscosity depends on the strain rate. Therefore, for correctly area, along with the physical parameters of rocks. reverse modeling power-law folding, the shortening strain rate that generated a natural fold has to be known. In this study, Acknowledgements reverse modeling is only applied for viscous folding and, therefore, any strain rate can be used for the reverse modeling. Thorough and constructive reviews by Yanhua Zhang and For folding of a layer with power-law rheology, the shortening Ray Fletcher, and comments by editor Mike Sandiford are strain rate that generated an observed fold shape has to be gratefully acknowledged. I thank Ray Fletcher for stimulating known to correctly reverse the folding, because the effective and helpful discussions. I thank Jean-Pierre Burg for detailed viscosity ratio depends on the shortening strain rate. The impact help and discussions during the preparation of the manuscript, of the strain rate on power-law folding becomes stronger if the Neil Mancktelow for stimulating discussions on 3D folding and difference in the power-law exponents of the layer and the Jaqueline Reber for collaboration during the development of the matrix becomes larger. However, if the shortening strain rate interference pattern visualization algorithm. can be estimated within an accuracy of one order of magnitude, the inaccuracy may be acceptable. Appendix A The initial geometries of the presented numerical forward models were intentionally kept simple for the current feasibility This appendix summarizes the self-developed finite element study on reverse modeling. However, with the current algorithm used in this study. It is a 3D version of the finite performance of computers, numerical modeling of 3D folding element algorithm described in Frehner and Schmalholz (2006). can also be applied for multilayers with an initial random The conservation equations for slow incompressible flow in the perturbation of the layer geometry (Fig. 8A). Such models can absence of body forces in three dimensions are (e.g., Bathe, be used in particular to study the formation of complex fold 1996; Haupt, 2002): interference patterns (Fig. 8B). 3D numerical folding/unfolding ∂r ∂r ∂r can become a suitable and standard tool for structural geologists xx þ xy þ xz ¼ 0 in the future to interpret natural fold shapes and patterns. ∂x ∂y ∂z ∂r ∂r ∂r xy þ yy þ yz ¼ 0 ðA1Þ 6. Conclusions ∂x ∂y ∂z ∂r ∂r ∂r xz þ yz þ zz ¼ 0 Complex 3D fold shapes and patterns exhibiting non- ∂x ∂y ∂z cylindrical fold axes and axes orientations spreading up to 90° can form during one shortening event with a single 1 ∂p ∂v ∂v ∂v ¼ x þ y þ z ðA2Þ shortening direction. The initial perturbation geometry of the K ∂t ∂x ∂y ∂z layer and the boundary conditions have a strong influence on the final fold shape. Therefore, different fold axis orientations where σxx, σyy and σzz are components of the total stress tensor and curved fold axes are not necessarily the result of more than in the x-, y- and z-directions, respectively, σxy, σxz and σyz are one deformation event. This conclusion impels some warning the shear stresses, p is the pressure, K is the compressibility points in interpreting dome-and-basin structures and other parameter and vx, vy and vz are the velocities in the x-, y- and z- interference patterns as the result of polyphase folding. directions, respectively. Eq. (A1) represents conservation of 40 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 linear momentum and Eq. (A2) represents conservation of mass. velocities times time step (i.e. explicit time integration). Then, Eq. (A2) deviates from the standard form for incompressible the new velocities are again calculated for the new grid. The ∂ ∂v ∂ vx þ y þ vz ¼ flow (i.e. ∂x ∂y ∂z 0), but is only applied for very large numerical code has been tested successfully with the analytical values of K, so that the resulting divergence of the velocity field solution for three-dimensional folding of an embedded viscous goes to zero, actually 10− 15 in this study. Application of Eq. layer presented in Fletcher (1991). (A2) is often referred to as the penalty approach for incompressible flow (Cuvelier et al., 1986; Hughes, 1987). References The constitutive equations for a linear viscous rheology are: 8 9 8 9 2 38 9 > r > > > > ∂ =∂ > > xx > > 1 > 4 2 2000> vx x > Aref, H., 1984. Stirring by chaotic advection. Journal of Fluid Mechanics 143, > r > > > 6 7> ∂ =∂ > <> yy => <> 1 => 6 24 20007<> vy y => 1–21 (JUN). r 6 7 ∂ =∂ zz ¼ 1 þ 1 l6 2 240007 vz z Bathe, K.-J., 1996. Finite Element Procedures. Prentice Hall, Upper Saddle > r > p> > 6 7> ∂ =∂ þ ∂ =∂ > > xz > > 0 > 3 6 0003007> vx z vz x > River, New Jersey. > r > > > 4 5> ∂ =∂ þ ∂ =∂ > :> xy ;> :> 0 ;> 000030:> vx y vy x ;> Biot, M.A., 1961. Theory of folding of stratified viscoelastic media and its r 0 000003 ∂v =∂y þ ∂v =∂z yz |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} z y implications in tectonics and orogenesis. Geological Society of America D Bulletin 72 (11), 1595–1620. ðA3Þ Bretherton, F.P., 1962. The motion of rigid particles in a shear flow at low Reynolds number. Journal of Fluid Mechanics 14 (2), 284–304. with μ the Newtonian viscosity. Discretization of the governing Cuvelier, C., Segal, A., van Steenhoven, A.A., 1986. Finite Element Methods equations and numerical integration is performed using an and the Navier–Stokes Equations. D. Reidel Publishing Company. 536 pp. isoparametric Q2-P1 element with 27 nodes for the biquadratic Fletcher, R.C., 1991. 3-Dimensional folding of an embedded viscous layer in pure shear. Journal of Structural Geology 13 (1), 87–96. continuous velocity degrees of freedom and 4 for the linear Frehner, M., Schmalholz, S.M., 2006. Numerical simulations of parasitic discontinuous pressure degrees of freedom (Hughes, 1987). folding in multilayers. Journal of Structural Geology 28 (9), 1647–1657. This element satisfies the inf–sup condition for incompressible Ghosh, S.K., Ramberg, H., 1968. Buckling experiments on intersecting fold flow (Hughes, 1987; Bathe, 1996). After discretization, the patterns. Tectonophysics 5 (2), 89–105. governing equations are given as (Hughes, 1987): Ghosh, S.K., Khan, D., Sengupta, S., 1995. Interfering folds in constrictional deformation. Journal of Structural Geology 17 (10), 1361–1373. "#() KQ Grujic, D., 1993. The influence of initial fold geometry on type-1 and type-2 v˜ 0 — M ¼ M ðA4Þ interference patterns an experimental approach. Journal of Structural T ˜new p˜old – – Q D p D Geology 15 (3 5), 293 307. K t K t Haupt, P., 2002. Continuum Mechanics and Theory of Materials. Springer, where swung dashes denote vectors containing nodal values of Berlin. the respective variables. The time derivative in Eq. (A2) has Hughes, T., 1987. The Finite Element Method. Dover Publications, Mineola, Δ New York. been replaced by a finite difference quotient with t being the Johns, M.K., Mosher, S., 1996. Physical models of regional fold superposition: the new old time increment (∂p/∂t≈(p −p )/Δt). The three matrices role of competence contrast. Journal of Structural Geology 18 (4), 475–492. K, Q and M are: Kaus, B.J.P., Podladchikov, Y.Y., 2001. Forward and reverse modeling of the three-dimensional viscous Rayleigh–Taylor instability. Geophysical Re- RRR RRR – K ¼ BT DBdxdydz; Q ¼ BT N dxdydz; ðA5Þ search Letters 28 (6), 1095 1098. RRR G p Kaus, B.J.P., Schmalholz, S.M., 2006. 3D finite amplitude folding: implications ¼ T M Np Npdxdydz for stress evolution during crustal and lithospheric deformation. Geophysical Research Letters 33 (14). Kocher, T., Mancktelow, N.S., 2005. Dynamic reverse modelling of flanking where vector NP contains the pressure shape functions and matrix B and vector B contain spatial derivatives of the structures: a source of quantitative kinematic information. Journal of G Structural Geology 27 (8), 1346–1354. velocity shape functions in a suitable organized way (e.g., Mancktelow, N.S., 2001. Single layer folds developed from initial random Zienkiewicz and Taylor, 1994). The integrations are performed perturbations: the effects of probability distribution, fractal dimension, phase numerically using 27 integration points per element. Using and amplitude. In: Koyi, H.A., Mancktelow, N.S. (Eds.), Tectonic Modeling: discontinuous pressure shape functions allows the elimination A Volume in Honor of Hans Ramberg. Geological Society of America, – of the pressure at the element level. This elimination leads to a Boulder, pp. 69 87. Odriscoll, E.S., 1962. Models for superposed laminar flow folding. Nature 196 system involving only unknown velocities: (4860), 1146–1148. ˜ ¼ ˜old ð Þ Pelletier, D., Fortin, A., Camarero, R., 1989. Are FEM solutions of Lv Qp A6 incompressible flows really incompressible? (or how simple flows can cause headaches!). International Journal for Numerical Methods in Fluids 9, where 99–112. Pine, D.J., Gollub, J.P., Brady, J.F., Leshansky, A.M., 2005. Chaos and threshold 1 T L ¼ K þ KDtQM Q : ðA7Þ for irreversibility in sheared suspensions. Nature 438 (7070), 997–1000. Ramsay, J.G., 1962. Interference patterns produced by the superposition of folds Values of p˜ new are restored during the Uzawa-type iteration of similar type. Journal of Geology 70 (4), 466–481. algorithm, during which Eq. (A6) is solved iteratively with Ramsay, J.G., Huber, M.I., 1983. The Techniques of Modern Structural updated values of p˜ old until the divergence of the velocity Geology. Volume 1: Strain analysis. Academic Press, London. Ramsay, J.G., Huber, M.I., 1987. The Techniques of Modern Structural converges towards zero (Pelletier et al., 1989). After every time Geology. Volume 2: Folds and fractures. Academic Press, London. step, the resulting velocities are used to move the nodes of each Schmalholz, S.M., 2006. Scaled amplification equation: a key to the folding element with the displacements resulting from the product of history of buckled viscous single-layers. Tectonophysics 419 (1–4), 41–53. S.M. Schmalholz / Tectonophysics 446 (2008) 31–41 41

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