3d numerical modeling of forward folding and reverse ... · unidirectional shortening event and...

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Tectonophysics 446

3D numerical modeling of forward folding and reverse unfolding of a viscoussingle-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 2007Available 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 3Dbuckle-folds. 3D viscous (Newtonian) single-layer folding is numerically simulated with the finite element method to investigate the formation offold shapes during one shortening event. An initially flat layer rests on a matrix with smaller viscosity and is shortened in one direction parallel tothe layering. Forward modeling with different initial geometrical perturbations on the flat layer and different lateral boundary conditions generatesnon-cylindrical 3D fold shapes and patterns. The simulations show that, in reality, the initial layer geometry and the boundary conditions stronglycontrol the final fold geometry. Fold geometries produced from the forward folding models are used as initial setting in numerical reverse foldingmodels with parameters identical to those of forward models. These reverse models accurately reconstruct the initial geometry of forward modelswith also only one extension event parallel to the previous shortening direction. The starting geometry of the forward models is inaccuratelyreconstructed by the reverse models if a significantly different viscosity ratio than in the forward models is used. This work demonstrates thatreverse modeling has a high potential for reconstructing the deformation history of folded regions and rheological constraints such as viscosityratio. Reverse models may be applied to natural 3D fold shapes and patterns in order to determine if they formed (i) during a single or multipledeformation events and (ii) as active buckle-folds with a viscosity ratio ≫ 1 or as passive, kinematic folds without buckling. This approach mayfind 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

A method to reconstruct fold amplification and especially toestimate 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 toseparate the amount of shortening by layer thickening from theamount of shortening by folding at constant layer thickness. Thismethod can be applied in 3D for cylindrical fold shapes but not formore complex, non-cylindrical fold shapes. Additional difficul-ties arise in 3D fold reconstruction because the shortening direc-

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tion may be unknown and also more than one shortening eventwith different shortening directions may have been active duringthe formation of natural fold shapes. In this study, 3D numericalreverse modeling is presented as a potential tool for 3D foldreconstruction. In this context, the impact of (i) the initialperturbation geometry of the layer, (ii) the boundary conditionsand (iii) the viscosity ratio on 3D forward and reverse foldingmodels is investigated. The results have important implicationsfor the applicability of numerical reverse modeling to natural foldshapes and patterns.

In 3D, fold interference patterns have attracted considerableattention because such patterns may provide insight into thedeformation history of the rock, such as the number ofdeformation phases, the shortening direction and the amount ofbulk shortening. 3D fold interference patterns have been studied

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http://dx.doi.org/10.1016/j.tecto.2007.09.005

Fig. 1. Model setup. Shortening is applied only in the x-direction where theshortening velocity vx is a function of the model width, X, and the constant strainrate, e. vx is modified after each numerical time step with the new value of X tomaintain a constant strain rate. The top model surface is always free.

32 S.M. Schmalholz / Tectonophysics 446 (2008) 31–41

with two fundamentally different underlying assumptions:passive or active folding. In passive folding, the folded layersexhibit no competence contrast (e.g., the Newtonian viscosity ofthe layer and the matrix are identical) and, therefore, nomechanical 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 difficultto explain what mechanism has actually generated the oftenobserved regularity and periodicity in natural folds. In activefolding, the folded layers possess a higher competence than theembedding material and a buckling instability with its corres-ponding wavelength selection process is active (Ghosh andRamberg, 1968; Skjernaa, 1975; Grujic, 1993; Johns andMosher,1996; Kaus and Schmalholz, 2006). In this study, only activefolding generated by a mechanical instability is considered.

Important questions arising while studying 3D fold shapesare how many deformation events have generated the observedfold geometries and how much bulk shortening took place.Numerical simulations of single-layer folding performed in thisstudy show that non-cylindrical 3D fold shapes, with curvedfold axes and axes orientations varying up to 90°, can begenerated during a single shortening event with one constantshortening direction. The simulations show that this is possiblebecause the final fold shapes are strongly controlled by theinitial perturbation geometry of the layer (e.g., Mancktelow,2001) and the boundary conditions. Therefore, fold shapes withstrongly varying fold axes can be generated by (i) a single,unidirectional shortening event, (ii) a single, multidirectionalshortening event (e.g., constriction), or (iii) two or moreshortening events (i.e. true superposed folds).

A suitable tool to investigate the shortening eventsgenerating 3D fold shapes and to reconstruct the evolution of3D fold patterns is numerical modeling. Alternatively, existinganalytical solutions for 3D single-layer folding could be used(e.g., Fletcher, 1991) but these analytical solutions are onlyvalid for small amplitudes and limb dips. In this study numericalmodels are applied because (i) high amplitude folds areinvestigated and (ii) the numerical models can easily be usedin the future for more complex scenarios such as multilayers orlayers with strongly variable thickness. Usually, numericalmodels are used to simulate the formation of buckle-foldsduring the shortening of stiff layers (Kaus and Schmalholz,2006); they are referred to as forward models. On the otherhand, it is also possible to use the fold geometries as initialsetting of a numerical model and extend the model in a directionopposite to the shortening direction used to produce the folds.Such models are referred to as reverse models. Reversemodeling has been for example applied to Rayleigh–Taylorinstabilities (Kaus and Podladchikov, 2001) and flankingstructures (Kocher and Mancktelow, 2005).

The reverse folding modeling performed in this study usinglinear viscous rheologies shows that the formation of the 3Dfolds can be accurately restored with a single extension eventhaving a direction opposite to the original shortening direction.On one hand, this may be expected because slow viscous flow atlow Reynolds numbers (described by the Stokes equations) istime-reversible (e.g., Bretherton, 1962), which was successfully

demonstrated by an experiment of G.I. Taylor (see Taylor andFriedman, 1966). On the other hand, it was shown that lowReynolds number flows exhibiting contrasts in materialproperties (e.g., particles in shear flow) can exhibit chaoticadvection and are not reversible (e.g., Aref, 1984; Yarin et al.,1997; Pine et al., 2005). For folding, an advection equation mustbe solved in addition to the Stokes equations to move the layerinterfaces through the model domain during shortening. Thereversibility is not obvious because the viscous flow moving thelayer interfaces is unsteady (due to the moving boundaries andthe buckling instability), and the viscous flow is sensitive tosmall changes in the shape of the layer interfaces. Therefore, thenumerical simulations performed in this study have theirjustification in showing the numerical reversibility for highamplitude 3D viscous folding.

The applicability of the reverse foldingmodeling indicates thatit is useful to apply numerical models to reverse the formation of3D fold shapes that were generated by a mechanical instability.However, for practical fold reconstructions, the viscosity ratiobetween layer and matrix can only be roughly estimated fornatural folds, and therefore the impact of uncertainties in viscosityratios on the reverse models are quantified in this study. Theresults show that reverse modeling can be used to test whetherfold shapes have been generated by either active or passivefolding and may have a high potential to determine whether foldshapes have been generated by one or more deformation events.

The aims of the paper are (i) to show that 3D fold geometrieswith significantly varying fold axes can be generated by single,unidirectional shortening events, (ii) to test the feasibility ofnumerical reverse modeling for 3D fold reconstruction and (iii)to quantify the impact of different viscosity ratios on the retro-deformation of 3D fold shapes.

2. Methods

The numerical algorithm applied in this work for forward andreverse modeling is self-developed and the finite element methodis employed to solve the continuummechanics equations for slowviscous (Newtonian), incompressible flow in 3D in the absence ofgravity (see Appendix). The boundary conditions applied for all

33S.M. Schmalholz / Tectonophysics 446 (2008) 31–41

simulations are (Fig. 1): The bottommodel 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 surface are zero). Thelateral sides orthogonal to the x-direction are always kept planarwith free slip. One lateral side orthogonal to the y-direction isalways kept planar with free slip (i.e. vy=0). For the other sideorthogonal to the y-direction two different conditions are applied:the side is kept planar with free slip (i.e. vy=0, referred to asboundary condition A) or is a free surface, so that the vis-cous material can extend in the y-direction during shortening inthe x-direction (referred to as boundary condition B). Shorteningin the x-direction is performed under a horizontal constant strainrate which is achieved bymodifying the horizontal velocity in thex-direction at one side after each numerical time step. At each timestep, the horizontal velocities in the x-direction at all nodes of theshortened side are identical. After each time step the calculated 3Dvelocity field is used to advect the numerical grid and a newvelocity field is calculated for the new geometry (i.e. explicit timeintegration scheme).

3. Forward modeling

3.1. Setting

In all simulations a layer, or plate, rests in the x–y plane on amatrix that has a smaller viscosity than the layer (Fig. 1). Only

Fig. 2. Results of numerical forward and reverse modeling for an initial geometricasurface is colored with values of the topography of the layer surface. The numbers inhigh and white color low elevation. A) Initial and B) final layer geometry for a forwaperformed with different values of R.

one shortening event with a constant direction is applied. Theflat layer is initially perturbed by elevating certain numericalnodes within the layer by either 1/20th or 1/40th of the layerthickness in the z-direction (perpendicular to the layering). Theinitial perturbation exhibits either a point or line shape withdifferent orientations. The initial layer thickness, the Newtonianviscosity of the matrix and the shortening strain rate have all avalue of one and are used as characteristic values for length,viscosity and time, respectively. All other parameters and unitsare scaled (or non-dimensionalized) by these three characteristicvalues. The model width in the x- and y-directions is 80 timesthe initial layer thickness and the model height is 16 times thelayer thickness. The competence contrast between layer andmatrix is defined by the ratio of layer viscosity to matrixviscosity. If the viscosity ratio is larger than one and an initialperturbation geometry is present then a buckling instability (i.e.active folding) develops during layer-parallel shortening.

3.2. Experiments

In the first series of experiments (Fig. 2) a point-shapedinitial perturbation (1/20th of layer thickness) and boundaryconditions A are applied (Fig. 2A) so that shortening of theincompressible material can generate extension in the vertical,z-direction only. The viscosity ratio between layer and matrixis 75. After about 50% shortening parallel to the x-direction

l point-shaped perturbation. Plots show the geometry of the layer and the layerthe legends are the vertical, z, values of the layer surface. Black color indicates

rd run with a viscosity ratio, R, of 75. C) to F) show results of the reverse models

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 thelayer 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. Blackcolor 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 thereverse models performed with different values of R.


three more or less cylindrical folds have developed (Fig. 2B).For visualization, the folded layer is centered vertically aroundzero. The folds result from active folding caused by layer-parallel shortening of a stiff layer with an initial geometricalperturbation (i.e. buckling). The fold axes of the folds next tothe middle fold are slightly curved, showing minor culminationsand depressions. The three more or less cylindrical foldsdeveloped from a point-shaped initial perturbation by foldpropagation. This result could be predicted also based onexisting analytical solutions for 3D folding (Fletcher, 1991), butbecause the analytical solutions are only valid for smallamplitudes and limb slopes, the numerical results are essentialto provide the correct finite amplitude fold geometries. Forexample, the limb slopes at the center of the middle fold (aty=40 and x=20, Fig. 2B) are nearly 90° whereas the limbslopes at the boundary of the model domain of the two otherfolds (e.g. at y=0 and x=10, Fig. 2B) are around 25°, becausethe folds are propagating both in the y-direction, along axes, andin the x-direction starting from the location of the initial point-shaped perturbation. The analytical solution is only valid up to20° limb dip and it is therefore not possible to capture the highamplitude fold propagation with the analytical solutions.

In the second series of experiments (Fig. 3) a line perturbation(1/20th of layer thickness) parallel to the shortening direction isintroduced in one half of the plate (Fig. 3A) and boundaryconditions A are applied. The viscosity ratio between layer andmatrix is 75. The resulting fold geometry exhibits slightly non-

cylindrical folds with axes parallel and perpendicular to theshortening direction. Folds at high angle to the shortening directionare comparable to those obtained in the previous experiment. Thefold axis parallel to the shortening direction is anchored on theprescribed perturbation.

In the third series of experiments (Fig. 4) the line perturbation(1/20th of layer thickness) is again parallel to the shorteningdirection (Figs. 3A and 4A) but boundary conditions B areapplied. In that case, shortening in the x-direction can generateextension in both the y- and z-directions (Fig. 4B). The viscosityratio between layer and matrix is 75. This different boundarycondition has a strong influence on the fold pattern: indeed, thereis no fold parallel to the shortening direction along the prescribedperturbation (compare Figs. 3B and 4B). Also, the foldamplitudes generated in the third series of experiments aresmaller than the amplitudes generated in the second series ofexperiments, which shows that the fold amplification rate also isstrongly controlled by the boundary conditions (Figs. 3B and 4B).

In the fourth series of experiments (Fig. 5) the perturbationline (1/20th of layer thickness) trends 45° to the shorteningdirection (Fig. 5A) and boundary conditions B are applied, as inthe third series of experiments. The viscosity ratio between layerand matrix is 75. The resulting geometry shows a pattern withfolds being non-cylindrical and vanishing along-axis (Fig. 5B).The highest fold amplitude developed at the corner of the modeldomain (x=40, y=0) where the initial perturbation line meetsthe model boundaries.

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 tothe 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 indicateshigh 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 modelsperformed with different values of R.

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 they-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 highand 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 modelsperformed with different values of R.


Fig. 6. Results of numerical forward and reverse modeling for an initial geometrical line perturbation parallel to the shortening and a point-shaped perturbation. Thenumbers 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 fora 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 perturbationconsists of a line that runs parallel to the x-direction throughoutthe model (y=60) and a point perturbation (x=40, y=20) both 1/40th of the layer thickness (Fig. 6A). Boundary conditions A areapplied and the viscosity ratio between layer and matrix is 50.The resulting geometry shows synclines and anticlines with axesparallel to the x-direction that bound three folds with axesparallel to the y-direction (Fig. 6B). The folds with axes parallelto the y-direction originate from the point-shaped perturbationand amplify faster than folds with axes parallel to the x-directionand anchored on the x-parallel perturbation (Fig. 7A and B).

The results of the forward modeling show that 3D foldshapes and patterns and their amplification rates are stronglycontrolled by initial perturbation geometries and orientationwith respect to shortening.

4. Reverse modeling

A question relevant for structural geologists is whether thecomplex 3D fold shapes and patterns, which formed during asingle shortening event, can be restored to their initial, pre-shortening geometry with a single extension event along the samepath as the shortening event. Reversemodeling is the useful test totry reconstructing the deformation history of folded layers in 3D.

4.1. Setting

The fold geometries obtained after forward modeling areused as initial setting to reverse models. All the parameters are

kept the same as those of the forward models, but the velocityboundary condition is reversed to simulate extensional de-formation (i.e. sign of horizontal velocity in the x-direction ischanged). Such well-defined reverse models are only possiblewith analytical or numerical models. The reverse models are runwith the same number of time steps as the forward models.Ideally, after the last time step, the final geometry of the reversemodel is identical to the starting geometry of the forward model.In the following experiments always four reverse simulationshave been performed: one with the same viscosity ratio as usedin the forward model and three with different viscosity ratios totest the impact of uncertainties in the viscosity ratio on thereverse modeling.

4.2. Experiments

For the first series of experiments, the initial geometries of thereverse models utilize the final folded geometry of the forwardmodel shown in Fig. 2B with the same viscosity ratio of 75. Thefinal step of the reverse model (Fig. 2C) reconstructs the point-shaped initial perturbation geometry shown in Fig. 2A, with onlysome very low magnitude undulations (smaller than the magni-tude of the point-shaped perturbation). Importantly, the magni-tude of the point-shaped perturbation is close to the magnitude ofthe perturbation implemented for the forwardmodeling (values ofgrayscale colorbar in Fig. 2A and C). Three additional reversemodels starting from folds of Fig. 2B were performed withviscosity ratios of 1, 10 and 500 instead of 75. The simulationwitha viscosity ratio of 1 represents passive folding without buckling

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 areashows 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 deformationsteps 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 theforward 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 reversemodel 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) failsto restore the fold evolution.


instability. The final layer geometries of the reverse models withviscosity ratios of 1 and 10 (Fig. 2D and E, respectively) aresignificantly different from the initial geometry of the forwardmodel. The final layer geometry of the reverse runs with viscosityratio of 500 (Fig. 2F) is not as close as the reverse model with thecorrect viscosity ratio, even if it would be an acceptable solutionin terms of error magnitude (b5%).

For the second series of experiments (Fig. 3), four reverseruns were performed with viscosity ratios of 1, 50, 75, and 250.The final geometry of the reverse model with the same viscosityratio than the forward model (i.e. 75, Fig. 3C) is closest to theinitial geometry of the forward model (Fig. 3A). The reversemodel with a viscosity ratio of 1 (Fig. 3D) shows a considerabledeviation from the initial geometry, whereas the other reversemodels with viscosity ratios of 50 (Fig. 3E) and 250 (Fig. 3F)yield acceptable results, relatively close to the correct solution.

For the third series of experiments (Fig. 4), four reversemodelswere performed with viscosity ratios of 50, 75, 100 and 250. Thereverse model with the same viscosity ratio as the forward run(Fig. 4C) best reconstructs the initial geometry of the corres-ponding forwardmodel, whereas the final geometry of the reversemodel with a viscosity ratio of 250 (Fig. 4F) does not reconstructthe initial geometry of the forward model. There is a clearboundary effect at the free side. The perturbation at this side isabout one order ofmagnitude larger than the initial perturbation ofthe forward model (around 0.2 compared to 0.025).

For the fourth series of experiments (Fig. 5), four reversemodels were performed with viscosity ratio of 1, 50, 75, and250. The reverse model with the same viscosity ratio as theforward model (Fig. 5C) reproduces rather accurately the initialgeometry of the forward model, but undulations having amagnitude comparable to the initial perturbation of the forwardmodel are present. The final geometry of the reverse model witha viscosity ratio of 250 (Fig. 5F) does not reach a satisfactorysolution since the initial perturbation of the forward model ishardly visible. There is again a strong boundary effect at the freeside where the perturbation is about one order of magnitudelarger than the initial perturbation of the forward model (about0.25 compared to 0.025). The reverse model exhibiting passivefolding (viscosity ratio equals 1, Fig. 5D) fails to restore theinitial perturbation geometry.

For the fifth series of experiments (Fig. 6), four reverse modelswere performed with viscosity ratios of 1, 10, 50 and 250. Thereverse model with the same viscosity ratio as the forward model(Fig. 6C) accurately reproduces the initial geometry of theforwardmodel but undulations having amagnitude comparable tothe initial perturbation of the forward model are present. The finalgeometry of the reversemodelswith viscosity ratios of 1 (Fig. 6D)and 10 (Fig. 6E) do not reach a satisfactory solution since the highamplitude folds remain clearly visible.

Three deformation stages of the forward model from the fifthseries (Fig. 6) of experiments (Fig. 7A, B and C) are compared


to the three corresponding deformation stages of the reversemodel with the same viscosity ratio (Fig. 7D, E and F) and tothree deformation stages of the reverse model with a viscosityratio of 1 (Fig. 7G, H and I). For each deformation stage the foldinterference pattern of the layer (black) is shown. To visualizethe interference pattern, the model is cut along a horizontalplane that (i) is orthogonal to the z-direction and (ii) is at theaverage vertical, z, position of the layer within the modeldomain. The areas within this plane which belong to the layerare filled in black and the areas that belong to the matrix arefilled in white. The evolution of the interference pattern showsthat the folds that originate from the point-shaped perturbationamplify and propagate faster than the folds that originate fromthe line-shaped perturbation (can be seen in Fig. 7A and B).This comparison shows that not only the initial layer geometryis accurately reconstructed by the reverse model, but also thatthe entire fold amplification and propagation is well recon-structed as demonstrated by the similarity of interferencepatterns between the corresponding stages of forward andreverse models (Fig. 7C and F, B and E, A and D). The resultsshow that although folds grow asynchronous and propagate inboth x and y horizontal directions, the reverse model with thesame viscosity ratio than the forward model successfullyreconstructs the fold evolution, i.e. the deformation history. Incontrast, the reverse model with a viscosity ratio of 1 (Fig. 7G toI) fails to reconstruct the fold evolution.

5. Discussion

Complex fold patterns and non-cylindrical folds can begenerated either by two or more consecutive deformation events(i.e. truly superposed folds), or by a single constrictionaldeformation event (Ramsay and Huber, 1983; Ramsay andHuber, 1987; Ghosh et al., 1995). The current numerical workhas demonstrated that non-cylindrical fold geometries (e.g.,folds with orthogonal fold axes) can be also formed by a singledeformation event with a single shortening direction (Figs. 3and 6). This is controlled by the initial geometry of the layersand the deformation boundary conditions. A detailed geometricanalysis of the fold shapes and interference patterns sometimesallows determining, if folds have been generated by one or moredeformation events (Ghosh et al., 1995). Reverse modeling is anadditional method for determining the number of deformationevents that generated natural folds.

Folds with axes parallel to the shortening direction candevelop if the lateral sides parallel to the shortening direction arefixed, i.e. not allowed to extend (Figs. 3B and 6B). Thereby, alayer-parallel stress in the y-direction is generated, which isresponsible for folding with axis orthogonal to this stressdirection (and this stress direction is orthogonal to the shorteningdirection).

The accuracy of the reverse models depends on thecomplexity of the deformation that generated the forward foldshapes. The results of the reverse models in the experimentswith only one free surface (the top one, Figs. 2, 3 and 6) aremore accurate and less sensitive to uncertainties in viscosityratio than the results of reverse models with two free surfaces

(Figs. 4 and 5). This is because with two free surfaces the flowof material is less confined and a larger number of flow paths arepossible than in models with only one free surface. Theuncertainties in viscosity ratio between the folded layer and thematrix may have minor or major impacts on the reversemodeling, depending on the boundary conditions of the forwardruns. All reverse models with the same viscosity ratio as in theforward model delivered a final layer and perturbation geometryvery close to the original geometry of corresponding forwardmodels, which emphasises the validity of the method.

Reverse models with a viscosity ratio significantly largerthan the one of the corresponding forward model and boundaryconditions A provided relatively flat layer geometries after anextension that equaled the shortening in the forward model.Therefore, the reverse modeling is not always able to estimatean upper limit of the viscosity ratio. On the other hand, reversemodels with a viscosity ratio significantly smaller than the oneof the corresponding forward model did not provide flat layergeometries. In particular, the reverse models with a viscosityratio of 1 (passive folding) always exhibited significantly largeramplitude folds after an extension equal to the shortening of thecorresponding forward model. If the reverse models with aviscosity ratio significantly smaller than the one of thecorresponding forward are extended further until the layer isrelatively flat, then the layer exhibits considerable thicknessvariations showing the invalidity of the reverse model. Thereverse modeling is therefore suitable to determine if naturalfold shapes have been generated by active or passive foldingand may help to put a lower limit on the viscosity ratio of naturalfold shapes. The upper limit for the viscosity ratio of naturalfolds has to be estimated additionally with other parameterssuch as the fold-span to layer thickness ratio which is controlledby the effective viscosity ratio.

Applications to natural folds require knowledge of manymodel parameters such as shortening direction, boundaryconditions or viscosity ratio which are usually only approxi-mately known. There is a need to perform reverse models withvarious viscosity ratios, in order to quantify the influence of theuncertainty in the viscosity ratio, one of the biggest unknown ingeological cases.

The applied lateral boundary conditions, where one modelside parallel to the shortening direction is either fixed or is a freesurface, represent two end-member scenarios. In nature, theactual boundary conditions may often be in between these twoend-member scenarios, but natural boundary conditions can beclose to both of the two applied boundary conditions.

In 2D, it is expected that the influence of the initial layerperturbation geometry on the final fold shape becomes smallerfor larger viscosity ratios, because the wavelength selectivityincreases with increasing viscosity ratio (Biot, 1961). In 3D,larger viscosity ratios are not expected to reduce the influence ofthe initial perturbation geometry on the final fold shape, becausefold axes can develop parallel to the initial perturbationorientations and larger viscosity ratios generate a faster growthof the folds with axes parallel to the initial perturbation.

For folding of a layer with viscous rheology, the shorteningstrain rate (which can vary during natural folding) needs not be

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 theshortening 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 waselevated 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 thecharacteristic length scale. The displayed fold shapes were obtained after coeval 50% (x-direction) and 25% (y-direction) shortening. B) The corresponding foldinterference 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 filledwith black and the matrix is filled with white.


known for the reverse modeling because the Newtonianviscosity is independent of the strain rate. This is different forfolding of a layer with power-law rheology where the effectiveviscosity depends on the strain rate. Therefore, for correctlyreverse modeling power-law folding, the shortening strain ratethat generated a natural fold has to be known. In this study,reverse modeling is only applied for viscous folding and,therefore, any strain rate can be used for the reverse modeling.For folding of a layer with power-law rheology, the shorteningstrain rate that generated an observed fold shape has to beknown to correctly reverse the folding, because the effectiveviscosity ratio depends on the shortening strain rate. The impactof the strain rate on power-law folding becomes stronger if thedifference in the power-law exponents of the layer and thematrix becomes larger. However, if the shortening strain ratecan be estimated within an accuracy of one order of magnitude,the inaccuracy may be acceptable.

The initial geometries of the presented numerical forwardmodels were intentionally kept simple for the current feasibilitystudy on reverse modeling. However, with the currentperformance of computers, numerical modeling of 3D foldingcan also be applied for multilayers with an initial randomperturbation of the layer geometry (Fig. 8A). Such models canbe used in particular to study the formation of complex foldinterference patterns (Fig. 8B). 3D numerical folding/unfoldingcan become a suitable and standard tool for structural geologistsin the future to interpret natural fold shapes and patterns.

6. Conclusions

Complex 3D fold shapes and patterns exhibiting non-cylindrical fold axes and axes orientations spreading up to90° can form during one shortening event with a singleshortening direction. The initial perturbation geometry of thelayer and the boundary conditions have a strong influence onthe final fold shape. Therefore, different fold axis orientationsand curved fold axes are not necessarily the result of more thanone deformation event. This conclusion impels some warningpoints in interpreting dome-and-basin structures and otherinterference patterns as the result of polyphase folding.

Numerical reverse modeling has a high potential to restorerigorously the formation of 3D folds and may be applied tonatural fold shapes in order to quantify the folding history of anarea, along with the physical parameters of rocks.

Acknowledgements

Thorough and constructive reviews by Yanhua Zhang andRay Fletcher, and comments by editor Mike Sandiford aregratefully acknowledged. I thank Ray Fletcher for stimulatingand helpful discussions. I thank Jean-Pierre Burg for detailedhelp and discussions during the preparation of the manuscript,Neil Mancktelow for stimulating discussions on 3D folding andJaqueline Reber for collaboration during the development of theinterference pattern visualization algorithm.

Appendix A

This appendix summarizes the self-developed finite elementalgorithm used in this study. It is a 3D version of the finiteelement algorithm described in Frehner and Schmalholz (2006).The conservation equations for slow incompressible flow in theabsence of body forces in three dimensions are (e.g., Bathe,1996; Haupt, 2002):

∂rxx∂x

þ ∂rxy∂y

þ ∂rxz∂z

¼ 0

∂rxy∂x

þ ∂ryy∂y

þ ∂ryz∂z

¼ 0

∂rxz∂x

þ ∂ryz∂y

þ ∂rzz∂z

¼ 0

ðA1Þ

1K∂p∂t

¼ � ∂vx∂x

þ ∂vy∂y

þ ∂vz∂z

� �ðA2Þ

where σxx, σyy and σzz are components of the total stress tensorin the x-, y- and z-directions, respectively, σxy, σxz and σyz arethe shear stresses, p is the pressure, K is the compressibilityparameter and vx, vy and vz are the velocities in the x-, y- and z-directions, respectively. Eq. (A1) represents conservation of


linear momentum and Eq. (A2) represents conservation of mass.Eq. (A2) deviates from the standard form for incompressibleflow (i.e. ∂vx∂x þ ∂vy

∂y þ ∂vz∂z ¼ 0), but is only applied for very large

values of K, so that the resulting divergence of the velocity fieldgoes to zero, actually 10−15 in this study. Application of Eq.(A2) is often referred to as the penalty approach forincompressible flow (Cuvelier et al., 1986; Hughes, 1987).The constitutive equations for a linear viscous rheology are:

rxxryyrzzrxzrxyryz

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼ �p

111000

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ 1

3l

4 �2 �2 0 0 0�2 4 �2 0 0 0�2 �2 4 0 0 00 0 0 3 0 00 0 0 0 3 00 0 0 0 0 3

26666664

37777775

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}D

∂vx=∂x∂vy=∂y∂vz=∂z

∂vx=∂zþ ∂vz=∂x∂vx=∂yþ ∂vy=∂x∂vz=∂yþ ∂vy=∂z

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ðA3Þwith μ the Newtonian viscosity. Discretization of the governingequations and numerical integration is performed using anisoparametric Q2-P1 element with 27 nodes for the biquadraticcontinuous velocity degrees of freedom and 4 for the lineardiscontinuous pressure degrees of freedom (Hughes, 1987).This element satisfies the inf–sup condition for incompressibleflow (Hughes, 1987; Bathe, 1996). After discretization, thegoverning equations are given as (Hughes, 1987):

K Q

QT � MKDt

" #v

pnew

� �¼

0

� MKDt

pold

( )ðA4Þ

where swung dashes denote vectors containing nodal values ofthe respective variables. The time derivative in Eq. (A2) hasbeen replaced by a finite difference quotient with Δt being thetime increment (∂p /∂t≈ (pnew−pold) / Δt). The three matricesK, Q and M are:

K ¼ R R RBTDBdxdydz; Q ¼ �R R R

BTGNpdxdydz;

M ¼ R R RNT

pNpdxdydz

ðA5Þ

where vector NP contains the pressure shape functions andmatrix B and vector BG contain spatial derivatives of thevelocity shape functions in a suitable organized way (e.g.,Zienkiewicz and Taylor, 1994). The integrations are performednumerically using 27 integration points per element. Usingdiscontinuous pressure shape functions allows the eliminationof the pressure at the element level. This elimination leads to asystem involving only unknown velocities:

L v ¼ �Q pold ðA6Þwhere

L ¼ K þ KDtQM�1QT : ðA7ÞValues of pnew are restored during the Uzawa-type iteration

algorithm, during which Eq. (A6) is solved iteratively withupdated values of pold until the divergence of the velocityconverges towards zero (Pelletier et al., 1989). After every timestep, the resulting velocities are used to move the nodes of eachelement with the displacements resulting from the product of

velocities times time step (i.e. explicit time integration). Then,the new velocities are again calculated for the new grid. Thenumerical code has been tested successfully with the analyticalsolution for three-dimensional folding of an embedded viscouslayer presented in Fletcher (1991).

References

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