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Strain and Competence Contrast Estimation from Fold Shape

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Strain and Competence Contrast Estimation from Fold Shape Tectonophysics 340 (2001) 195–213 www.elsevier.com/locate/tecto Strain and competence contrast estimation from fold shape Stefan M. Schmalholz*, Yuri Yu. Podladchikov Geologisches Institut, ETH Zentrum, 8092 Zu¨rich, Switzerland Received 14 February 2000; accepted 21 February 2001 Abstract A new method to estimate strain and competence contrast from natural fold shapes is developed and verified by analogue and numerical experiments. Strain is estimated relative to the nucleation amplitude, AN, which is the fold amplitude when the amplification velocities caused by kinematic layer thickening and dynamic folding are identical. AN is defined as the initial amplitude corresponding to zero strain because folding at amplitudes smaller than AN is dominantly by kinematic layer thickening. For amplitudes larger than AN, estimates of strain and competence contrast are contoured in thickness-to-wavelength (H/l) and amplitude-to-wavelength (A/l) space. These quantities can be measured for any observed fold shape. Contour maps are constructed using existing linear theories of folding, a new nonlinear theory of folding and numerical simulations, all for single-layer folding. The method represents a significant improvement to the arc length method. The strain estimation method is applied to folds in viscous (Newtonian), power-law (non-Newtonian) and viscoelastic layers. Also, strain partitioning in fold trains is investigated. Strain partitioning refers to the difference in strain accommodated by individual folds in the fold train and by the whole fold train. Fold trains within layers exhibiting viscous and viscoelastic rheology show different characteristic strain partitioning patterns. Strain partitioning patterns of natural fold trains can be used to assess the rheological behaviour during fold initiation. D 2001 Elsevier Science B.V. All rights reserved. Keywords: Strain estimation; Folding; Fold shape; Competence contrast; Strain partitioning; Fold trains 1. Introduction reconstructions of folded regions such as mountain belts. Although numerous studies have investigated A major geological question concerning folding is the mechanics of folding using analytical techniques how much strain (or shortening) is associated with (e.g., Biot, 1961; Johnson and Fletcher, 1994; Hunt et observed fold shapes. In this study, strain is defined as al., 1996), analogue experiments (e.g., Currie et al., the difference between the deformed and the initial 1962; Ramberg, 1963; Hudleston, 1973; Abbassi and length of a line element normalized by its initial Mancktelow, 1992) and numerical simulations (e.g., length. Information of strain is relevant to palinspastic Dieterich, 1970; Cobbold, 1977; Zhang et al., 1996; Mancktelow, 1999; Schmalholz et al., 2001), there is no satisfactory method to estimate strain from natural * Corresponding author. Now at Geomodelling Solutions fold shapes. One reason is that the existing analytical GmbH, Binzstrasse 18, 8045 Zu¨rich, Switzerland. Fax: +41-1- folding solutions are only valid for small limb dips 455-6390. E-mail address: [email protected] (10° to 15°, e.g., Fletcher and Sherwin, 1978), (S.M. Schmalholz). whereas most natural folds exhibit much larger limb 0040-1951/01/$ - see front matter D 2001 Elsevier Science B.V. All rights reserved. PII: S 0040-1951(01)00151-2 196 S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 dips. An intuitively attractive estimation method is the is accommodated while amplitudes are smaller than so-called arc length method (e.g., Dahlstrom, 1969). AN is ignored because this strain is not recorded This method assumes that the observed fold arc length through the fold shape. corresponds to the initial fold wavelength (Fig. 1). The variation of fold shape with strain depends on Hence, strain can be estimated from the difference material properties. This was shown by analogue and between the observed arc length and fold wavelength. numerical experiments where the variation of geo- However, the arc length method results in large errors metrical parameters has been recorded as a function of if it is applied to folds that developed in settings where strain. Manifestations of material dependence are the the competence contrast between the folded layer and alteration of the ratios of (i) amplitude to initial its matrix is small. These errors are a result of the amplitude (Chapple, 1968; Hudleston and Stephans- relatively large component of homogeneous layer son, 1973; Abbassi and Mancktelow, 1992), (ii) arc thickening (Sherwin and Chapple, 1968). This study length to initial arc length (Hudleston, 1973; Johnson applies a new analytical solution for folding valid up and Fletcher, 1994), (iii) amplitude to thickness to large limb dips (Schmalholz and Podladchikov, (Hudleston, 1973), (iv) amplitude to wavelength (Cur- 2000) and numerical simulations of folding to develop rie et al., 1962) and (v) wavelength to thickness (Lan a strain estimation method. This method reliably and Hudleston, 1995a). One specific fold shape, estimates strain and has the additional advantage that corresponding to different material properties, may it enables estimation of the competence contrast. theoretically correspond to different strains. Without The unknown material properties and initial geom- knowledge of the material properties, a unique strain etry of natural folds are the major obstacles to strain estimate is not possible from a single geometrical estimation. An essentially flat layer with infinitesimal parameter that quantifies the fold shape. To avoid this fold amplitudes may accommodate several hundreds obstacle, we employ two observable geometrical of percent strain by homogeneous layer thickening parameters simultaneously to characterize folds. Thus, without developing observable fold limb dips. A layer during progressive folding, strain is contoured in the that appears unfolded in the field may have accom- thickness-to-wavelength (H/l) and the amplitude-to- modated substantial layer-parallel strain. With increas- wavelength (A/l) space. These ratios can be measured ing limb dip the strain that is accommodated by layer for any observed fold shape. Strain contours as a thickening decreases, and that accommodated by function of A/l and H/l provide a ‘‘strain contour folding increases. In this study, we define a nucleation map’’, which uniquely defines strain for a continuous amplitude, AN (Table 1), as the initial (or reference) range of competence contrasts. The idea of this study amplitude for strain estimation in order to circumvent is to use two observable geometrical ratios to con- the obstacle of unknown initial geometry. Only the strain two unknowns: the competence contrast and the strain that is accommodated while amplitudes are strain that is accommodated by the folds after they larger than AN is, therefore, estimated. The strain that exceed the nucleation amplitude. Fig. 1. Major geometrical parameters of folds are the arc length (Larc), wavelength (l), amplitude (A) and thickness (H). In the initial folding stages, where amplitudes are very small, the arc length has approximately the same length than the wavelength. S.M. Schmalholz, Y.Yu. Podladchikov / Tectonophysics 340 (2001) 195–213 197 Table 1 relationships for the nonlinear folding stages (Schmal- List of symbols used in the text holz and Podladchikov, 2000) are used to choose the a general folding growth rate geometrical parameters for the strain estimation ad growth rate for ductile layers method and to evaluate its range of applicability. a dominant growth rate for ductile layers dd Next, numerical simulations are used to construct ae growth rate for elastic layers ade dominant growth rate for elastic layers the new method of strain and competence contrasts A fold amplitude estimates. All results are derived for ductile (viscous A0 initial fold amplitude pffiffiffiffiffiffi and power-law) as well as for viscoelastic layers to AC crossover amplitude: AC ¼ 1=ðp 2aÞ avoid a restriction of the strain estimation method to a AN nucleation amplitude: AN = 1/(2a) D flexural rigidity of the layer certain layer rheology. Otherwise, this rheology needs e engineering strain: e=(l0 À l)/l0 to be known by the geologist for the observed folds, en natural strain: en = ln(l0/l) which is unlikely in the majority of the cases. The new earc strain that is estimated by the method is then verified by numerical and analogue arc length method experiments and applied to constrain the strain parti- e˙ pure shear background strain rate G shear modulus of the layer tioning in natural and experimental fold trains. H thickness of the layer H0 initial thickness of the layer l fold wavelength 2. The nucleation amplitude for ductile and visco- l0 initial fold wavelength elastic layers ldd dominant wavelength for ductile layers lde dominant wavelength for elastic layers leff effective dominant wavelength A layer subjected to layer-parallel shortening Larc fold arc length accommodates strain by both layer thickening and Larc0 initial fold arc length rotation. The deformation process, during which con- ml viscosity of the layer siderable rotation of fold limbs takes place, is termed mm viscosity of the matrix n power-law exponent of the layer folding or buckling (e.g., Ramsay and Huber, 1987; P layer-parallel stress Price and Cosgrove, 1990). Layer thickening and R dominant wavelength ratio: folding can occur simultaneously, but it is possible R = ldd/lde or alternatively R = ade/add to discriminate between the velocity fields of the two
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