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Lateral variations of strain in experimental forced folds
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Tectonophysics 295 (1998) 79–91
Lateral variations of strain in experimental forced folds Gary D. Couples Ł , Helen Lewis 1 Department of Geology and Applied Geology, University of Glasgow, Glasgow G12 8QQ, UK Received 6 April 1997; accepted 28 May 1998
Abstract Packages of rock layers, deformed under confining pressure by the uplift and rotation of a steel-block forcing assembly, translate both towards, and away from, the margin of the principal uplifted block. The resulting asymmetric forced folds, and especially their long, planar limbs, exhibit along-layer variations in strain. Alternations of layer elongation and contraction occur along profiles extending away from the antiform=synform couplet. Layer-normal strains are mostly nil, so the longitudinal strains largely equate to volume strains in these plane-strain models. Spaced anomalies in outcrops, indicating either increased cementation, or erosional weakness, may suggest that similar processes operate in nature to produce variations in damage caused during the flexural-slip process. Two, non-exclusive explanations are offered to account for the patterns of strain observed in the experiments: (1) they may be caused by decaying wavetrains of small-scale flexural deflections (and their local strain patterns) related to the bending of the major forced folds; or (2) they may be caused by a ‘patchy’ development of layer-parallel slip, and the consequent spatial variability in displacements. 1998 Elsevier Science B.V. All rights reserved. Keywords: experimental models; strain analysis; flexural slip; bedding plane faults
1. Introduction 1.1. Previous work with rock-layer models Studies involving the experimental deformation of rock-layer models have been inspired by structural problems identified from nature. An important precursor to such work was the recognition that rock types can be chosen so as to provide contrasts in ductility and strength at convenient confining pressures and room temperature (Handin and Hager, 1957). Ł Corresponding
author. Present address: Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Tel.: C44-131-4513123; Fax: C44-1314513127; E-mail: [email protected] 1 Present address: Department of Petroleum Engineering, HeriotWatt University, Edinburgh, EH14 4AS, UK.
A further antecedent condition was the appreciation that the use of rock materials permitted petrofabric analysis of the induced deformation, such that stress and strain states could be determined following the experiments (Friedman, 1964; Friedman and Sowers, 1970). In the first of a series of rock-layer modelling studies, Handin et al. (1972) investigate the buckling of single-layer folds in which the layer consists of a machined prism of rock. They describe the resulting states of stress and strain, and subsequently extend this work to multi-layer models that illustrate the role of mechanical contrasts (Handin et al., 1976). The experimental and analysis techniques pioneered by Handin and his colleagues (including, of course, John Logan, to whom this volume is dedicated) have been employed throughout a series of investigations
0040-1951/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 1 9 5 1 ( 9 8 ) 0 0 1 1 6 - 4
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Fig. 1. Geometry of experimental models showing relationship of folded rock-layer package (layers of lead, dolostone, and limestone) to steel-block forcing assembly. See Couples et al. (1994) for further details of the model assembly. Note particularly the convergence of the dolostone and steel block on the right end of model. In high structural relief models, the layer package in this location translates closer to the adjacent forced fold, and steepens in dip, while the lead continues to withdraw into the fold area.
that use rock-layer models to examine the mechanical development of several structural types. Thrusting is another problem investigated by rock models in which layers are subjected to loading within their plane. Morse (1977), Serra (1977), and Chester et al. (1988, 1991) describe rock-layer models of thrust faults in which the ramp–flat–ramp geometry is essentially fixed, but where the translating sheet develops a range of strain states depending on the makeup of the layered package and the loading conditions. Bartlett et al. (1981) describe wrenchfault models in which single-layer rock sheets are deformed by the motion of rigid blocks that experience relative movements only in the plane of the layer — much as might be imagined for the initiation of an ideal natural wrench structure. A considerable contrast exists between the geometries and deformations of the models noted above, and another extensive set of experimental studies which emphasize the role of displacements imposed at high angles to the layering. This latter series of models was created to address a number of structural issues that had been raised in relation to the formation of basement-cored uplifts in Wyoming and adjacent areas (Stearns, 1971, 1978). Friedman et al. (1976a,b, 1980) and Logan et al. (1978) describe rock-layer models using a design in which the layers are subjected to rigid-block motions that are of a reverse sense, but without any rotation between
blocks. A further development in this series involves the design of a steel-block forcing assembly that allows both differential uplift (at right angles to the layering) and rotation to be imposed onto a layered package (Fig. 1; Stearns and Weinberg, 1975; Weinberg, 1979). Couples et al. (1994) and Couples and Lewis (1998) also use this forcing-block assembly, to quantify the kinematics of the folding, and address the stress-state effects of multiple layer interfaces, respectively. Still another set of rock-layer models is described by Patton et al. (1998). Their design again focuses on loading that is transverse to the layering, but these models cause the layers (both single- and multi-layer packages) to be extended by means of displacements that occur along a pre-machined ‘normal fault’. In this set of experiments, there is no rotation of the forcing blocks, so the results are most readily compared with the non-rotational, reverse-fault models of Friedman et al. (1976a, 1980). The transverse-load models described above can be grouped into: (1) those that impose shortening (Friedman et al., 1976a,b, 1980; Logan et al., 1978); (2) those that are neutral (Stearns and Weinberg, 1975; Weinberg, 1979; Couples et al., 1994; Couples and Lewis, 1998); and (3) those which extend the layering (Patton et al., 1998). In most of the ‘shortening’ and all of the ‘neutral’ models, the layered package is, to a greater or lesser extent, mechani-
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cally isolated from the underlying forcing blocks by means of lubrication, or by a very ductile basal layer. The models of Couples et al. (1994), Couples and Lewis (1998), and Weinberg (1979) all have a basal ductile unit (as do some of the models of Friedman et al., 1980), and the entire layered package in these models experiences considerable lateral motion (relative to the forcing assembly) during folding (see Haneberg, 1993, for another discussion about the role of the basal ductile unit). Couples et al. (1994) detail these lateral movements of the layered package and relate them to the growth history of the induced forced folds, and to major structural events that occur in the folding process. 1.2. This paper This paper extends the analysis of displacements in the forced-fold models of Couples et al. (1994) by determining layer-parallel motions along continuous profiles extending away from the induced forced folds. Within-layer variations in displacement indicate lateral variations in strains that occur during the folding. These laterally varying strains affect the nearly planar limbs of the experimental folds, and, if they have natural counterparts, these deformations could well have important petrophysical consequences. We suggest two potential mechanical explanations, and we note that folds observed in outcrop possess subtle features that may represent a natural counterpart to the patterns of strains observed in our models.
2. Experimental method All experiments are deformed under confining pressure (50 MPa), at room temperature, in the large-specimen apparatus described by Handin et al. (1972). An assembly of machined steel blocks is attached to the loading pistons, and layered packages are placed onto these blocks (Fig. 1) so that the blocks’ motions cause the packages of layers to experience differential transverse displacements, and rotations. The layered packages are held short of the pistons, with the resulting ‘space’ filled by weak plasticine that transmits only confining pressure to the ends of the layers. The layered package of each
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of the models is virtually identical in each starting configuration (see Couples et al., 1994, for details). This package consists of a basal unit of lead (2.0 mm thick), a layer of dolostone (1.55 mm thick), and a thicker layer of limestone (7.35 mm thick). The layers are 30 mm wide and 110 mm long. When the steel blocks of the forcing assembly move in response to the piston loads, the layered package is deformed into a series of forced folds (Stearns, 1978) whose geometry is primarily controlled by the imposed rotations and translations of the forcing assembly. In these models, the layer of lead flows in a very ductile fashion, the dolostone breaks into rigid segments, and the limestone experiences both fracture and flow (Fig. 2). In the crestal region of the antiform, a prominent graben forms in the limestone, and a reverse fault develops through the synform. These faults do not offset the dolostone=limestone interface. Twelve models (numbered 31 to 41, and 46) comprise this series, with the only difference between them being the amount of uplift imposed (see Couples et al., 1994). For our purposes, each model also has another crucial element — a side jacket of lead onto which a pre-deformation rectangular grid is inscribed. This jacket faithfully records the end-of-experiment geometry since the confining pressure completely welds it to the steel blocks or to the subjacent layered package (Couples et al., 1994). Distortions of the grid (see below) permit a determination of the displacements in this model set, and consequently, an analysis of strains. Following each experiment, the model is impregnated with epoxy and sectioned along its medial plane. The layer-scale and grain-scale structures which are revealed have been interpreted in terms of stress states, and related to the sequence of structural events, by Couples et al. (1994). Here we describe strains calculated from distortions of the side-jacket grids.
3. Analysis of displacements 3.1. Distortions of grids The distorted side-jacket grids (Fig. 3) reveal a displacement pattern with two important compo-
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Fig. 2. Deformed state of model 40. Short lines indicate microfractures visible on slabbed medial surface or in thin section. Dolostone layer is 1.55 mm thick. Note particularly the prominent graben in the antiform crest, the reverse fault through the synform hinge, the rigid-pieces of dolostone in the fold, and the striking thickness changes of the lead. Indiana D limestone; Blair D dolostone. Top of steel block forcing assembly shown by lowermost line. Large black areas are voids that opened following release of confining pressure.
Fig. 3. Photograph of gridded lead side jacket for model 40. Grid lines approximately 2.5 mm apart. Note layer-parallel translation apparent from offsets of the grid.
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nents. The most significant element of the displacement field is the rigid-body motion (both translation and rotation) of the forcing blocks and their overlying veneer of layers, and the second element is a localized shear which occurs at the boundary between the steel blocks and the layered package. A third, but more minor and local, component is due to the localized strains that are associated with the curved portions of the model where bending is concentrated. The displacements of the grid can be resolved into components parallel .u/ and perpendicular .v/ to the forcing-block=layered-package interface, and to facilitate the following analysis, we define a local coordinate axis ‘i’ parallel to the layering, with local axis ‘ j’ normal to this (Fig. 4B). Displacement data derived from these grids are used by Couples et al. (1994) to investigate if natu-
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ral forced folds should be subjected to interpretation techniques driven by balancing arguments. These authors select (arbitrary) reference points located 15 mm away from the uplift margin on both the upthrown and the downthrown blocks, and measure the line-lengths of the layers between these points, and the layer-parallel translations at those points (layerparallel translation is the relative motion of the layered package against a ‘fixed’ forcing-block below it, with clock-wise shear taken as positive). Based on these measurements, Couples et al. (1994) show that early in the uplift history, these layer-parallel translations represent motion away from the uplift, but that the layered packages subsequently move towards the uplift margin when the structural relief increases. They point out that, even accounting for these motions, the line-lengths of the layered packages do not balance, and the conclusion is drawn that finite strains are significant in these experimental folds. In these models, nearly all of the relative motion between the steel blocks and the layered package is localized at the lead=steel interface, rather than being distributed through the ductile basal unit of lead. The interiors of the side jackets preserve (in a mold=cast fashion) minor imperfections from the edges of the lead sheets which constitute the basal unit of the package, but the sharp steel=lead contact line (shear plane) on the interior of the jackets is represented by a broader shear zone on the jacket exterior (Fig. 4A). Couples et al. (1994) explain this relationship as being analogous to the upward widening of a wrench fault zone as observed in the models of Bartlett et al. (1981). The finite thickness of the lead side jackets, and the consequent de-focusing of the displacement field (from interior to exterior, as described), mean that it is difficult to relate any minor grid distortions to the specifics of the model underneath (the exception is the major layer-parallel translation, noted above, which is clearly localized to the steel=lead interface). 3.2. Variations in layer-parallel motion
Fig. 4. Schematic drawings illustrating method of determining layer-parallel translation from offset grid lines of lead side jackets. (A) Discrete slip zone on interior of side jacket becomes shear zone on its exterior surface. (B) Details of method for determining layer-parallel slip on lead=steel interface. Note local coordinate system i–j.
The layer-parallel translations used for the arguments of Couples et al. (1994) are located at two arbitrary reference points, but the side-jacket grids allow a similar determination of layer-parallel translations (by the method illustrated in Fig. 4) along
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traverses extending along the fold limbs away from the uplift margin, on both the upthrown and the downthrown blocks. Layer-parallel translation can also be determined along the top of the left-most steel block (as shown in Fig. 1). In all models, relative motion between the layered package and steel block is nil in this location, with layer-parallel slip becoming discernable only when the layer starts to incline up into the forced fold at the boundary between steel blocks 1 and 2. On the opposite (right) end of the models, layer-parallel translations cannot be readily determined above steel block 4; this is because the technique described in Fig. 4 relies on the parallelism of the layered package and the top of the underlying steel block. Above steel block 4, there is a divergence in orientation in all models because the lead withdraws from underneath the rock layers. This movement is similar to the flow of lead in the other fold locations (and discussed in detail by Couples et al., 1994), and here this motion is certainly enhanced by the relatively short distance between the location of this fold and the physical end of the model. (In higher-uplift models, the ends of the rock layers actually contact the right-most steel block as the lead withdraws into the adjacent forced fold.) When layer-parallel translation measurements are plotted against position (Fig. 5), we see that there has been relative motion between the steel blocks and the layered package at all locations above steel blocks 2 and 3 at some stage in the uplift history. Thus, there is no valid location along the planar limbs of the central forced folds for siting the notional ‘pin line’ from which most balancing methods begin. As noted above, relative translations are zero above steel block 1 (to the left in Fig. 5), and are unknown above steel block 4 (to the right in Fig. 5). A pin line could be sited above steel block 1 in this model set, but folds created with more complicated forcing-assembly geometries (Weinberg, 1979) have no location with zero relative translation. We suggest that the apparent presence of a pin line in our models is merely an artifact, and that there is therefore no firm basis for assuming where such a location might occur in an unknown setting. However, as determination of strain requires knowledge only of the relative motions, it can be calculated here in spite of the potential difficulty in siting pin lines. To calculate longitudinal strains
Fig. 5. Plot of layer-parallel translations vs. lateral position. Heavy lines join data points. Numbers adjacent to curves identify individual models. Amount of uplift (mm, in parentheses) increases from 33 (4.203), to 35 (4.399), to 40 (6.862), to 46 (7.597). Positive positions are on the upthrown side of the central uplift. Thin vertical lines denote size of steel blocks. These curves demonstrate significant layer-parallel translations (>2 mm) at large structural relief.
within the folded layered package, the translation data (Fig. 5) can be treated as a displacement function: u.i/. The longitudinal (along-layer) strain is simply the partial derivative of u with respect to i: @u @i Because the translation data are determined only where grid lines cross the basal slip surface, we only know the average longitudinal strain between measurement sites about 2.5 mm apart. In Fig. 6, these average longitudinal strains are plotted at the midpoints between measurement points. The variations in layer-parallel translation (relative displacement) along traverses extending away from the central fold couplet represent lateral variations in longitudinal (layer parallel) strain within the layered packages. The maximum slopes in the displacement plots (longitudinal strains: ei ) range from C0.389 to ei D
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Fig. 6. Plots of longitudinal (volume) strain (ordinate) vs. lateral position (in cm; abcissa). Data represent the slopes of the segments from displacement plots (as in Fig. 5), with the mid-points of those segments used as positions in this figure. Elongational longitudinal (dilational volumetric) strain is positive. Plots shown in groups (A–D) of models which have similar amount of uplift (in mm). (A) Model 36 (0.254), model 34 (1.016). (B) Model 38 (2.147), model 32 (2.540), model 37 (2.638). (C) Model 39 (3.411), model 41 (3.801), model 33 (4.203), model 35 (4.399). (D) Model 40 (6.862), model 46 (7.597).
0.433 (model 46), corresponding to lateral strains of 39% elongation and 43% contraction, respectively (Table 1). More typical maxima and minima are 20–25% elongation (models 32 and 33) and 16% contraction (models 33, 37, 39, 40 and 41). Smaller Table 1 Maximum slopes in the displacement plots Model No.
Maximum longitudinal strains (%) contractional
elongational
32 33 34 35 36 37 38 39 40 41 46
11.8 16.7 8.6 10.7 8.3 16.1 9.7 16.2 16.1 14.7 43.3
21.4 25 15 17.2 8.6 6.7 10.3 12.9 14.3 9.7 38.9
maximum strains characterize other models, but longitudinal strains of at least 8–10% are present at some point in all models. Note especially the alternation in longitudinal-strain ‘highs’ and ‘lows’ to suggest a resemblance to the stretching and bunching of an inchworm (cf. Price, 1988; Means, 1989). The accuracy of the displacement measurements is better than 0.01 mm. Based on an uncertainty assumed to be 0.01 mm, and given the nominal 2.5 mm spacing of grid lines, a 10% nominal longitudinal strain (in Fig. 6, derived from a slope of 0.1 in Fig. 5) has an error bar of š2% strain. The lateral position of the displacement measurement is taken as the projected location of the grid line which is attached to the steel block. The grid spacing limits the spatial resolution of the data (and the smearing effect illustrated in Fig. 4 precludes recognition of small strains other than those occurring along the basal shear surface), so no attempt is made here to calculate the areal distribution of strains within the layered package. And as in the approach of Cou-
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ples et al. (1994), the history of longitudinal strain can only be re-constructed from strain profiles determined from models that represent discrete points in structural-history space. The four panels in Fig. 6 are arranged in groups of models which have similar structural relief; the groups increase in relief from A to D. The longitudinal strains indicate variations in deformation at lateral positions considerably removed from the margin of the central uplift, and along planar limbs away from the prominent forced-fold couplet which is caused by the uplift. How are these strains expressed in the rock layers? The limestone is a porous (¾15%), granular rock composed of bioclasts and ooids ranging from about one to several mm in size. Significant fractures and microfaults are present, both in the region of the prominent graben at the uplift crest (Fig. 2), and in a similar, but smaller, graben located near the crest of the fold at the right side of the model (in the orientation illustrated in Fig. 1). Calcite twin gliding, along with microfractures, have been used by Couples et al. (1994) to infer deformation states in the immediate region of the major fold couplet, but calcite twinning is not observed in the parts of the limestone removed from that region — suggesting lower differential stresses. We infer that any strains in the limestone layer away from the main folds are accommodated by indistinct grain-boundary movements facilitated by the high porosity of this material (see comments below concerning thin sections), although we have not observed any clear indications of material damage in these sites. In the dolostone, there is no significant porosity, and the crystal size is very small (a few hundred microns). Microfracturing is the primary deformation mechanism for this rock under the extant experimental conditions, and the rigid blocks of dolostone which constitute the ‘fold’ over the uplift margin are bounded by such features (Fig. 2). Away from the major folds, the dolostone forms long, planar ‘limbs’. Within these long limbs, there are small clusters of (nominally) layer-normal microfractures (in groups of up to about six) that occur within an along-layer distance of 0.5 mm or less. Individual microfractures may cut across the dolostone layer, or they may terminate part way through it. Adjacent clusters are no closer than 4 or 5 mm apart, but
the spacing between visible clusters is sometimes greater. These results are derived from observations made by binocular microscope on the slabbed medial surfaces of the models, and from acetate peels. Quality thin sections proved impossible to make because of lead that smeared during cutting and polishing; this unfortunate event inhibited good epoxy penetration to some parts of the models. Because of these problems with observations of the deformation fabrics of the models, we are not able to make a one-to-one correlation between all of the strain peaks (as indicated in Fig. 6) and definitive fabrics revealing damage within the rock layers. However, in models where the clusters of dolostone microfractures are clearest (on the acetate peels), these clusters do correspond to the strain peaks plotted for that model. Therefore, we believe that the calculated strain curves represent localization of true deformation in the long, planar limbs of these forced folds. 3.3. Volumetric strains The displacement component perpendicular to layering, v. j/, is shown by Couples et al. (1994) to be generally nil except in the immediate vicinity of the macroscopic folds, or in small, localized areas which develop a distinct graben (located on the upthrown-block fold-crest of all models; Fig. 2). Thus, the strain component normal to layering is, except in local sites, nil: e j D¾ 0. If we assume plane strain deformation (appropriate to the medial plane, and, presumably, the remainder of the model), then the longitudinal (layer-parallel) strain is a uniaxial strain except (in small areas) where the layer-normal strain is non-zero. Therefore, in most of the model, the longitudinal strain approximates the volumetric strain: ev D ei C e j C ek D ¾ ei Given the sign convention selected (clockwise sense of shear is positive), positive slopes in Fig. 5 represent dilatant volume strains (volume increase), and negative slopes represent compactant volume strains (at all sites where e j is nil). The data plotted in Fig. 6, therefore, depict volumetric strains in all limb locations away from the graben. The volumetric strains created in these exper-
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imental asymmetric folds locally reach significant values (25C%; see Table 1 and Fig. 6), but they are nevertheless of little significance in terms of the arguments about the validity of balancing techniques. When integrated over distances of at least several grid squares (10C mm), the net volumetric strains average less than 3% dilation. Such low volumetric strain values, if typical of natural folds, would probably have only a very limited effect on the accuracy of re-constructing the geometry of the structural feature. However, the effect on line lengths may be important since line-length discrepancies of this magnitude are now being considered as significant (Groshong and Epard, 1994). As structural methods achieve greater precision, longitudinal and volumetric strains at sites away from ‘obvious’ deformation, and their potential effects on line lengths and=or areas, need to be given appropriate consideration.
4. Discussion 4.1. Petrophysics The prime issue with respect to volumetric strains of moderate-to-large magnitude (¾5–10%, or larger) may not lie in the geometric re-construction of cross sections, but, rather, in their impact on rock properties. Here we consider fracture processes that change the volume occupied by a constant mass of material, rather than pressure solution or other mass-transfer phenomena which could also affect a rock’s volume. Such ‘brittle’ deformation (i.e., characterized by fracturing) is often associated with dilatant strain, but both dilatant and compactant strains can be produced by fracture mechanisms (Nelson and Handin, 1977; Nelson, 1985; Antonellini et al., 1994). The porosity changes caused by deformation usually result in associated permeability changes (and not necessarily of the same sign; see D’Onfro et al., 1994). Examples of permeability alteration can be seen in our models: some compactant volumetric strain areas in the limestone layer have suffered considerable reduction in permeability, as deduced from the failure of the impregnating epoxy to reach regions isolated by zones of compactant strains. If the permeability effects we see associated with the volumetric strains in our experimental folds also
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apply to natural asymmetric folds, then there is considerable potential for significant alteration of the flow character of folded rock sequences (Edwards et al., 1993; Lewis and Couples, 1993; Gabrielsen et al., 1998; Gibson, 1998; Manzocchi et al., 1998). However, it is important to emphasize that we are not suggesting the existence of large (10%CC) volumetric strains in natural structures. As we have argued elsewhere (Couples et al., 1998), natural rock sequences, with their multitude of bedding interfaces, are likely to suffer considerably smaller ‘material’ strains (damage) because of the effects of bedding-plane slip. Nevertheless, the point we seek to make here is that the models have revealed a process (an ‘inchworm’ motion of the fold limbs, with along-limb longitudinal-strain variations) that should be considered in terms of its possible impact on fluid flow, even though the natural expression of the strains may be quite subtle. The opportunity to study quite a number of wellexposed, large-scale structures has emphasized to us the repeated occurrence of prominent, anomalously well-cemented ‘ribs’ of rock on the gentle limbs of many folds. We speculate as to whether these may be the result of enhanced fluid flow, localized to high-permeability pathways cutting across the strata (e.g., dilatant volumetric strains), and consequent diagenetic alterations. We also note that many folds in outcrop reveal a semi-regular pattern of erosional gullies (Fig. 7). Are these features suggesting the presence of (un-cemented) dilational zones which allow accelerated weathering? The recognition of lateral strain variations in our experiments suggests that there may be a non-random, process-based explanation for the ubiquitous occurrence of such spaced features within large-scale, flexural-slip folds. 4.2. Two possible explanations The strain variations (Fig. 6) of our models occur at an apparent half-wavelength of about 4–5 mm (about two times the sampling interval). They are associated with deformation (mostly fracturing, but possibly also including grain=grain motions) that is presumably caused when the imposed distortions raised stress levels to the point of failure. So, what is responsible for these (short-wavelength) distortions that are spaced along the layers?
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Fig. 7. Photograph of north wall of Shoshone Canyon, Rattlesnake Mountain, northwestern Wyoming showing the gentle limb of a major forced fold. Note semi-regular spacing to erosional weaknesses. Elevation difference from canyon floor to top of cliff is approximately 1 km.
4.2.1. Flexure In our models, the main, central fold couplet, and the folds overlying the flanking ‘faults’ in the forcing assembly, represent concentrated bending of the layered package (Fig. 1). The patterns of layer-elongation and layer-shortening associated with these major flexures (as revealed by both macroscopic examinations and petrographic study; see Friedman et al., 1976b; Couples et al., 1994) imply that significant bending moments are created. The inferred stress fields in these experiments are those predicted for such bending (Hafner, 1951; Couples, 1977; Gangi et al., 1977; Couples et al., 1998). We suggest that these localized, large bending moments, which are a result of the forced folding of the layered package over the uplifted and rotated steel blocks beneath, are responsible for propagating a decaying wavetrain of small-scale flexural deformation to locations away from the sites of sharp bending. It is possible that both the fracturing we noted above, and the associated strains we have calculated, are responses to these secondary and tertiary minor bending distortions of the almost-planar limbs of the major folds (Fig. 8). Positive and negative in-
Fig. 8. Illustration of supposed wavetrain of flexural deflections propagating away from sites of major bending. Top of steel block 3 shown along with geometry of layered package above (in grey).
terference of these decaying wavetrains (propagated away from the adjacent major flexures) may explain the occasional spikes in the data, as well as the ‘quiet’ zones observed in some plots. It is likely that the dolostone has a very high flexural rigidity (at least until it fractures), but the limestone will also contribute to the flexural stiffness of the layered package. The lead has, heretofore, been considered as a very weak material, but the possibility needs to be entertained that it also contributes some stiffness to this system. Mathematical models of our explanation have not been
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Fig. 9. Sketch showing spatial arrangement of fracture clusters associated with localized bedding-plane slip (after Cooke and Pollard, 1994, 1997).
constructed; they will, in any case, be considerably more complicated than the flexural models now popular for studying basin subsidence (cf. Beaumont et al., 1988). An interesting avenue to consider is the analytical approach described by Patton and Fletcher (1995); by means of appropriate series expansions, deformation-state predictions could be made for the inchworm motions implied by our data. 4.2.2. Irregular layer-parallel slip Cooke and Pollard (1994, 1997), and Cooke et al. (1998), consider the deformational consequences of localized bedding-plane slip during flexural-slip folding. They associate the creation of fracture clusters, at the ‘ends’ of patches of such slip surfaces, to strain compatibility, and to stress-state modifications (Fig. 9). We suggest that the volumetric strains we note in our models may be expressions of such ‘end-of-slip’ strains, where in this case, the patch of slip occurs at the base of the layered package. This notion is consistent with arguments we have made elsewhere (Couples et al., 1998) concerning the localization of bedding-parallel detachments in flexural-slip folds.
5. Conclusions Lateral variations in layer-parallel translation reflect alternation of elongation and contraction strains at locations away from the obvious folds (in otherwise virtually planar fold limbs). These longitudinal-strain anomalies can exceed 25%, but more typically are ¾10%, and have zones of lower strain lying between. Volume strains may approach or equal the magnitude of the longitudinal strains. The integrated longitudinal- (and volume-) strains (if calculated over large
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areas of the fold) are, however, considerably smaller (<5%), making it difficult to recognize these features in nature without the sort of detailed data coverage (of displacements) available in the rock-layer models. Nevertheless, natural folds exhibit a variety of spaced ‘anomalies’ which seem to be related to dilatancy; we suggest that the deformation processes in our models (which we do not yet fully understand) may apply to natural folds. The petrophysical effects of such alterations could be significant. The asymmetric forced folds created in these experiments, and the details of their deformation, reveal a considerable complexity of strain state resulting from the ‘simple’ process of bending. Because the layer-parallel translations we observe in the models are purely the result of the uplift and bending of the layered package, they are therefore processes inherent to the folding. Wavetrains of flexural deflections propagated away from the sites of major bending may be responsible for creating the along-layer strain zones. An alternate possibility is that the observed strains are reflecting spatial variations in slip on that basal shear surface (or within the layered package?). These two explanations are not mutually exclusive.
Acknowledgements The experimental work was undertaken at the Center for Tectonophysics, Texas A&M University, and we thank John Logan, the late Mel Friedman, and the late John Handin for their support. We also wish to thank a number of colleagues who have participated in friendly discussion of the mechanisms of forced folding: especially, of course, Dave Stearns and John Handin, and including John Logan, Mel Friedman, Dave Weinberg, Jamie Jamison, Jim Morse, and Mike Fahy. We thank Mohammed Ameen, Chris Banks, Ken McClay, Mark Verschuren, and several anonymous reviewers for commenting on earlier drafts of this paper.
References Antonellini, M.A., Aydin, A., Pollard, D.D., 1994. Microstructure of deformation bands in porous sandstones at Arches National Park, Utah. J. Struct. Geol. 16, 941–959.
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Bartlett, W.L., Friedman, M., Logan, J.M., 1981. Wrench faults in limestone layers. Tectonophysics 79, 255–277. Beaumont, C., Quinlan, G., Hamilton, J., 1988. Orogeny and stratigraphy: Numerical models of the Paleozoic in the eastern interior of North America. Tectonics 7, 389–416. Chester, J.S., Spang, J.H., Logan, J.M., 1988. Comparison of thrust fault rock models to basement-cored folds in the Rocky Mountains foreland. In: Schmidt, C.J., Perry, Jr., W.J., (Eds.), Interaction of the Rocky Mountain foreland and the Cordilleran thrust belt. Geol. Soc. Am. Mem. 171, 65–74. Chester, J.S., Logan, J.M., Spang, J.H., 1991. Influence of layering and boundary conditions on fault-bend and fault-propagation folding. Geol. Soc. Am. Bull. 103, 1059–1072. Cooke, M.L., Pollard, D.D., 1994. Development of bedding plane faults and fracture localization in a flexed multilayer: a numerical model. In: Nelson, P.P., Laubach, S.E. (Eds.), Rock Mechanics, Models and Measurements, Challenges from Industry. Proc. First North American Rock Mechanics Symposium. Balkema, Rotterdam, pp. 131–138. Cooke, M.L., Pollard, D.D., 1997. Bedding plane slip in initial stages of fault-related folding. J. Struct. Geol. 19, 567–581. Cooke, M., Mollema, P., Pollard, D.D., Aydin, A., 1998. Interlayer slip and joint localization in East Kaibab monocline, Utah: field evidence and results from numerical modeling. In: Cosgrove, J.H., Ameen, M.S. (Eds.), Forced (Drape) Folds and Associated Fractures. Geol. Soc. London Spec. Publ. (in press). Couples, G.D., 1977. Stress and shear fracture (fault) trajectories resulting from a suite of complicated boundary conditions with applications to the Wind River Mountains. Pure Appl. Geophys. 115, 113–133. Couples, G.D., Lewis, H., 1998. Effects of interlayer slip in model forced folds. In: Cosgrove, J.H., Ameen, M.S. (Eds.), Forced (Drape) Folds and Associated Fractures. Geol. Soc. London Spec. Publ. (in press). Couples, G.D., Stearns, D.W., Handin, J.W., 1994. Kinematics of experimental forced folds and their relevance to cross-section balancing. Tectonophysics 233, 193–213. Couples, G.D., Lewis, H., Tanner, P.W.G., 1998. Strain partitioning during flexural slip folding. In: Coward, M.P., Daltaban, T.S., Johnson, H. (Eds.), Structural Geology in Reservoir Characterization. Geol. Soc. London Spec. Publ. 127, 149– 165. D’Onfro, P., Fahy, M.F., Rizer, W., 1994. Geomechanical model for fault sealing in sandstone reservoirs. Am. Assoc. Pet. Geol. Annu. Conv. Official Progr. 3, 130–131. Edwards, H.E., Becker, A.D., Howell, J.A., 1993. Compartmentalisation of an aeolian sandstone by structural heterogeneities: Permo-Triassic Hopeman Sandstone, Moray Firth, Scotland. In: North, C.P., Prosser, D.J. (Eds.), Characterization of Fluvial and Aeolian Reservoirs. Geol. Soc. London Spec. Publ. 73, 339–365. Friedman, M., 1964. Petrofabric techniques for the determination of principal stress directions in rocks. In: Judd, W.R. (Ed.), State of Stress in the Earth’s Crust. Elsevier, New York, NY, pp. 451–552.
Friedman, M., Sowers, G.M., 1970. Petrofabrics: a critical review. Can. J. Earth Sci. 7, 477–497. Friedman, M., Handin, J., Logan, J.M., Min, K.D., Stearns, D.W., 1976a. Experimental folding of rocks under confining pressure: Part III. Faulted drape folds in multilithologic layered specimens. Geol. Soc. Am. Bull. 87, 1049–1066. Friedman, M., Teufel, L.W., Morse, J.D., 1976b. Strain and stress analyses from calcite twin lamellae in experimental buckles and faulted drape-folds. Philos. Trans. R. Soc. London 4283, 87–107. Friedman, M., Hugman, R.H.H., Handin, J., 1980. Experimental folding of rocks under confining pressure, part VIII — Forced folding of unconsolidated sand and of lubricated layers of limestone and sandstone. Geol. Soc. Am. Bull. 91, 307–312. Gabrielsen, R.H., Aarland, R.-K., Alsaker, E., 1998. Identification and spatial distribution of fractures in porous, siliciclastic sediments. In: Coward, M.P., Daltaban, T.S., Johnson, H. (Eds.), Structural Geology in Reservoir Characterization. Geol. Soc. London Spec. Publ. 127, 49–64. Gangi, A.F., Min, K.D., Logan, J.M., 1977. Experimental folding of rocks under confining pressure: Part IV. Theoretical analysis of faulted drape folds. Tectonophysics 42, 227–260. Gibson, R.G., 1998. Physical character and fluid-flow properties of sandstone-derived fault zones. In: Coward, M.P., Daltaban, T.S., Johnson, H. (Eds.), Structural Geology in Reservoir Characterization. Geol. Soc. London Spec. Publ. 127, 83–97. Groshong, R.H., Epard, J.-L., 1994. The role of strain in areaconstant detachment folding. J. Struct. Geol. 16, 613–618. Handin, J., Hager, R.V., 1957. Experimental deformation of sedimentary rocks under confining pressure: tests at room temperature on dry samples. Am. Assoc. Pet. Geol. Bull. 41, 1–50. Handin, J., Friedman, M., Logan, J.M., Pattison, L.J., Swolfs, H.S., 1972. Experimental folding of rocks under confining pressure: Buckling of single-layer rock beams. Am. Geophys. Union Monogr. 16, 1–28. Handin, J., Friedman, M., Min, K.D., Pattison, L.J., 1976. Experimental folding of rocks under confining pressure, Part II: buckling of multilayered rock beams. Geol. Soc. Am. Bull. 87, 1035–1048. Haneberg, W.C., 1993. Drape folding of compressible elastic layers — II. Matrix solution for two-layer folds. J. Struct. Geol. 15, 923–932. Hafner, W., 1951. Stress distribution and faulting. Geol. Soc. Am. Bull. 62, 373–398. Lewis, H., Couples, G.D., 1993. Production evidence for geological heterogeneities in the Anschutz Ranch East Field, western USA. In: North, C.P., Prosser, D.J. (Eds.), Characterization of Fluvial and Aeolian Reservoirs. Geol. Soc. London Spec. Publ. 73, 321–338. Logan, J.M., Friedman, M., Stearns, M.T., 1978. Experimental folding of rocks under confining pressure, Part VI — Further studies of faulted drape folds. In: Matthews, V. (Ed.), Laramide folding associated with basement block faulting in the western United States. Geol. Soc. Am. Mem. 151, 79–99. Manzocchi, T., Ringrose, P.S., Underhill, J.R., 1998. Flow through fault systems in high-porosity sandstones. In: Coward,
G.D. Couples, H. Lewis / Tectonophysics 295 (1998) 79–91 M.P., Daltaban, T.S., Johnson, H. (Eds.), Structural Geology in Reservoir Characterization. Geol. Soc. London Spec. Publ. 127, 65–82. Means, W.D., 1989. Stretching faults. Geology 17, 893–896. Morse, J.D., 1977. Deformation in ramp regions of overthrust faults: experiments with small scale rock models. Wyo. Geol. Assoc. 29th Annu. Field Conf. Guideb., pp. 457–470. Nelson, R.A., 1985. Geologic Analysis of Naturally Fractured Reservoirs. Gulf Publishing, Houston, 320 pp. Nelson, R.A., Handin, J.W., 1977. Experimental study of fracture permeability in porous rocks. Am. Assoc. Pet. Geol. Bull. 61, 227–236. Patton, T.L., Fletcher, R.C., 1995. Mathematical block-motion model for deformation of a layer above a buried fault of arbitrary dip and sense of slip. J. Struct. Geol. 17, 1455–1472. Patton, T.L., Logan, J.M., Friedman, M., 1998. Experimentally generated normal faults in single- and multilayer limestone beams at confining pressure, Tectonophysics 295, 53–77 (this issue).
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Price, R.A., 1988. The mechanical paradox of large overthrusts. Geol. Soc. Am. Bull. 100, 1898–1908. Serra, S., 1977. Styles of deformation in the ramp regions of overthrust faults. Wyo. Geol. Assoc. 29th Annu. Field Conf. Guideb., pp. 487–498. Stearns, D.W., 1971. Mechanisms of drape folding in the Wyoming province. Wyo. Geol. Assoc. 23rd Annu. Field Conf. Guideb., pp. 125–144. Stearns, D.W., 1978. Faulting and forced folding in the Rocky Mountains foreland. In: Matthews, V. (Ed.), Laramide Folding Associated with Basement Block Faulting in the Western United States. Geol. Soc. Am. Mem. 151, 1–37. Stearns, D.W., Weinberg, D.M., 1975. A comparison of experimentally created and naturally deformed drape folds. Wyo. Geol. Assoc. 27th Annu. Field Conf. Guideb., pp. 159–166. Weinberg, D.M., 1979. Experimental folding of rocks under confining pressure, Part VII — Partially scaled models of drape folds. Tectonophysics 54, 1–24.
Tectonophysics 295 (1998) 79–91
Lateral variations of strain in experimental forced folds Gary D. Couples Ł , Helen Lewis 1 Department of Geology and Applied Geology, University of Glasgow, Glasgow G12 8QQ, UK Received 6 April 1997; accepted 28 May 1998
Abstract Packages of rock layers, deformed under confining pressure by the uplift and rotation of a steel-block forcing assembly, translate both towards, and away from, the margin of the principal uplifted block. The resulting asymmetric forced folds, and especially their long, planar limbs, exhibit along-layer variations in strain. Alternations of layer elongation and contraction occur along profiles extending away from the antiform=synform couplet. Layer-normal strains are mostly nil, so the longitudinal strains largely equate to volume strains in these plane-strain models. Spaced anomalies in outcrops, indicating either increased cementation, or erosional weakness, may suggest that similar processes operate in nature to produce variations in damage caused during the flexural-slip process. Two, non-exclusive explanations are offered to account for the patterns of strain observed in the experiments: (1) they may be caused by decaying wavetrains of small-scale flexural deflections (and their local strain patterns) related to the bending of the major forced folds; or (2) they may be caused by a ‘patchy’ development of layer-parallel slip, and the consequent spatial variability in displacements. 1998 Elsevier Science B.V. All rights reserved. Keywords: experimental models; strain analysis; flexural slip; bedding plane faults
1. Introduction 1.1. Previous work with rock-layer models Studies involving the experimental deformation of rock-layer models have been inspired by structural problems identified from nature. An important precursor to such work was the recognition that rock types can be chosen so as to provide contrasts in ductility and strength at convenient confining pressures and room temperature (Handin and Hager, 1957). Ł Corresponding
author. Present address: Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, UK. Tel.: C44-131-4513123; Fax: C44-1314513127; E-mail: [email protected] 1 Present address: Department of Petroleum Engineering, HeriotWatt University, Edinburgh, EH14 4AS, UK.
A further antecedent condition was the appreciation that the use of rock materials permitted petrofabric analysis of the induced deformation, such that stress and strain states could be determined following the experiments (Friedman, 1964; Friedman and Sowers, 1970). In the first of a series of rock-layer modelling studies, Handin et al. (1972) investigate the buckling of single-layer folds in which the layer consists of a machined prism of rock. They describe the resulting states of stress and strain, and subsequently extend this work to multi-layer models that illustrate the role of mechanical contrasts (Handin et al., 1976). The experimental and analysis techniques pioneered by Handin and his colleagues (including, of course, John Logan, to whom this volume is dedicated) have been employed throughout a series of investigations
0040-1951/98/$19.00 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 1 9 5 1 ( 9 8 ) 0 0 1 1 6 - 4
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Fig. 1. Geometry of experimental models showing relationship of folded rock-layer package (layers of lead, dolostone, and limestone) to steel-block forcing assembly. See Couples et al. (1994) for further details of the model assembly. Note particularly the convergence of the dolostone and steel block on the right end of model. In high structural relief models, the layer package in this location translates closer to the adjacent forced fold, and steepens in dip, while the lead continues to withdraw into the fold area.
that use rock-layer models to examine the mechanical development of several structural types. Thrusting is another problem investigated by rock models in which layers are subjected to loading within their plane. Morse (1977), Serra (1977), and Chester et al. (1988, 1991) describe rock-layer models of thrust faults in which the ramp–flat–ramp geometry is essentially fixed, but where the translating sheet develops a range of strain states depending on the makeup of the layered package and the loading conditions. Bartlett et al. (1981) describe wrenchfault models in which single-layer rock sheets are deformed by the motion of rigid blocks that experience relative movements only in the plane of the layer — much as might be imagined for the initiation of an ideal natural wrench structure. A considerable contrast exists between the geometries and deformations of the models noted above, and another extensive set of experimental studies which emphasize the role of displacements imposed at high angles to the layering. This latter series of models was created to address a number of structural issues that had been raised in relation to the formation of basement-cored uplifts in Wyoming and adjacent areas (Stearns, 1971, 1978). Friedman et al. (1976a,b, 1980) and Logan et al. (1978) describe rock-layer models using a design in which the layers are subjected to rigid-block motions that are of a reverse sense, but without any rotation between
blocks. A further development in this series involves the design of a steel-block forcing assembly that allows both differential uplift (at right angles to the layering) and rotation to be imposed onto a layered package (Fig. 1; Stearns and Weinberg, 1975; Weinberg, 1979). Couples et al. (1994) and Couples and Lewis (1998) also use this forcing-block assembly, to quantify the kinematics of the folding, and address the stress-state effects of multiple layer interfaces, respectively. Still another set of rock-layer models is described by Patton et al. (1998). Their design again focuses on loading that is transverse to the layering, but these models cause the layers (both single- and multi-layer packages) to be extended by means of displacements that occur along a pre-machined ‘normal fault’. In this set of experiments, there is no rotation of the forcing blocks, so the results are most readily compared with the non-rotational, reverse-fault models of Friedman et al. (1976a, 1980). The transverse-load models described above can be grouped into: (1) those that impose shortening (Friedman et al., 1976a,b, 1980; Logan et al., 1978); (2) those that are neutral (Stearns and Weinberg, 1975; Weinberg, 1979; Couples et al., 1994; Couples and Lewis, 1998); and (3) those which extend the layering (Patton et al., 1998). In most of the ‘shortening’ and all of the ‘neutral’ models, the layered package is, to a greater or lesser extent, mechani-
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cally isolated from the underlying forcing blocks by means of lubrication, or by a very ductile basal layer. The models of Couples et al. (1994), Couples and Lewis (1998), and Weinberg (1979) all have a basal ductile unit (as do some of the models of Friedman et al., 1980), and the entire layered package in these models experiences considerable lateral motion (relative to the forcing assembly) during folding (see Haneberg, 1993, for another discussion about the role of the basal ductile unit). Couples et al. (1994) detail these lateral movements of the layered package and relate them to the growth history of the induced forced folds, and to major structural events that occur in the folding process. 1.2. This paper This paper extends the analysis of displacements in the forced-fold models of Couples et al. (1994) by determining layer-parallel motions along continuous profiles extending away from the induced forced folds. Within-layer variations in displacement indicate lateral variations in strains that occur during the folding. These laterally varying strains affect the nearly planar limbs of the experimental folds, and, if they have natural counterparts, these deformations could well have important petrophysical consequences. We suggest two potential mechanical explanations, and we note that folds observed in outcrop possess subtle features that may represent a natural counterpart to the patterns of strains observed in our models.
2. Experimental method All experiments are deformed under confining pressure (50 MPa), at room temperature, in the large-specimen apparatus described by Handin et al. (1972). An assembly of machined steel blocks is attached to the loading pistons, and layered packages are placed onto these blocks (Fig. 1) so that the blocks’ motions cause the packages of layers to experience differential transverse displacements, and rotations. The layered packages are held short of the pistons, with the resulting ‘space’ filled by weak plasticine that transmits only confining pressure to the ends of the layers. The layered package of each
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of the models is virtually identical in each starting configuration (see Couples et al., 1994, for details). This package consists of a basal unit of lead (2.0 mm thick), a layer of dolostone (1.55 mm thick), and a thicker layer of limestone (7.35 mm thick). The layers are 30 mm wide and 110 mm long. When the steel blocks of the forcing assembly move in response to the piston loads, the layered package is deformed into a series of forced folds (Stearns, 1978) whose geometry is primarily controlled by the imposed rotations and translations of the forcing assembly. In these models, the layer of lead flows in a very ductile fashion, the dolostone breaks into rigid segments, and the limestone experiences both fracture and flow (Fig. 2). In the crestal region of the antiform, a prominent graben forms in the limestone, and a reverse fault develops through the synform. These faults do not offset the dolostone=limestone interface. Twelve models (numbered 31 to 41, and 46) comprise this series, with the only difference between them being the amount of uplift imposed (see Couples et al., 1994). For our purposes, each model also has another crucial element — a side jacket of lead onto which a pre-deformation rectangular grid is inscribed. This jacket faithfully records the end-of-experiment geometry since the confining pressure completely welds it to the steel blocks or to the subjacent layered package (Couples et al., 1994). Distortions of the grid (see below) permit a determination of the displacements in this model set, and consequently, an analysis of strains. Following each experiment, the model is impregnated with epoxy and sectioned along its medial plane. The layer-scale and grain-scale structures which are revealed have been interpreted in terms of stress states, and related to the sequence of structural events, by Couples et al. (1994). Here we describe strains calculated from distortions of the side-jacket grids.
3. Analysis of displacements 3.1. Distortions of grids The distorted side-jacket grids (Fig. 3) reveal a displacement pattern with two important compo-
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Fig. 2. Deformed state of model 40. Short lines indicate microfractures visible on slabbed medial surface or in thin section. Dolostone layer is 1.55 mm thick. Note particularly the prominent graben in the antiform crest, the reverse fault through the synform hinge, the rigid-pieces of dolostone in the fold, and the striking thickness changes of the lead. Indiana D limestone; Blair D dolostone. Top of steel block forcing assembly shown by lowermost line. Large black areas are voids that opened following release of confining pressure.
Fig. 3. Photograph of gridded lead side jacket for model 40. Grid lines approximately 2.5 mm apart. Note layer-parallel translation apparent from offsets of the grid.
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nents. The most significant element of the displacement field is the rigid-body motion (both translation and rotation) of the forcing blocks and their overlying veneer of layers, and the second element is a localized shear which occurs at the boundary between the steel blocks and the layered package. A third, but more minor and local, component is due to the localized strains that are associated with the curved portions of the model where bending is concentrated. The displacements of the grid can be resolved into components parallel .u/ and perpendicular .v/ to the forcing-block=layered-package interface, and to facilitate the following analysis, we define a local coordinate axis ‘i’ parallel to the layering, with local axis ‘ j’ normal to this (Fig. 4B). Displacement data derived from these grids are used by Couples et al. (1994) to investigate if natu-
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ral forced folds should be subjected to interpretation techniques driven by balancing arguments. These authors select (arbitrary) reference points located 15 mm away from the uplift margin on both the upthrown and the downthrown blocks, and measure the line-lengths of the layers between these points, and the layer-parallel translations at those points (layerparallel translation is the relative motion of the layered package against a ‘fixed’ forcing-block below it, with clock-wise shear taken as positive). Based on these measurements, Couples et al. (1994) show that early in the uplift history, these layer-parallel translations represent motion away from the uplift, but that the layered packages subsequently move towards the uplift margin when the structural relief increases. They point out that, even accounting for these motions, the line-lengths of the layered packages do not balance, and the conclusion is drawn that finite strains are significant in these experimental folds. In these models, nearly all of the relative motion between the steel blocks and the layered package is localized at the lead=steel interface, rather than being distributed through the ductile basal unit of lead. The interiors of the side jackets preserve (in a mold=cast fashion) minor imperfections from the edges of the lead sheets which constitute the basal unit of the package, but the sharp steel=lead contact line (shear plane) on the interior of the jackets is represented by a broader shear zone on the jacket exterior (Fig. 4A). Couples et al. (1994) explain this relationship as being analogous to the upward widening of a wrench fault zone as observed in the models of Bartlett et al. (1981). The finite thickness of the lead side jackets, and the consequent de-focusing of the displacement field (from interior to exterior, as described), mean that it is difficult to relate any minor grid distortions to the specifics of the model underneath (the exception is the major layer-parallel translation, noted above, which is clearly localized to the steel=lead interface). 3.2. Variations in layer-parallel motion
Fig. 4. Schematic drawings illustrating method of determining layer-parallel translation from offset grid lines of lead side jackets. (A) Discrete slip zone on interior of side jacket becomes shear zone on its exterior surface. (B) Details of method for determining layer-parallel slip on lead=steel interface. Note local coordinate system i–j.
The layer-parallel translations used for the arguments of Couples et al. (1994) are located at two arbitrary reference points, but the side-jacket grids allow a similar determination of layer-parallel translations (by the method illustrated in Fig. 4) along
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traverses extending along the fold limbs away from the uplift margin, on both the upthrown and the downthrown blocks. Layer-parallel translation can also be determined along the top of the left-most steel block (as shown in Fig. 1). In all models, relative motion between the layered package and steel block is nil in this location, with layer-parallel slip becoming discernable only when the layer starts to incline up into the forced fold at the boundary between steel blocks 1 and 2. On the opposite (right) end of the models, layer-parallel translations cannot be readily determined above steel block 4; this is because the technique described in Fig. 4 relies on the parallelism of the layered package and the top of the underlying steel block. Above steel block 4, there is a divergence in orientation in all models because the lead withdraws from underneath the rock layers. This movement is similar to the flow of lead in the other fold locations (and discussed in detail by Couples et al., 1994), and here this motion is certainly enhanced by the relatively short distance between the location of this fold and the physical end of the model. (In higher-uplift models, the ends of the rock layers actually contact the right-most steel block as the lead withdraws into the adjacent forced fold.) When layer-parallel translation measurements are plotted against position (Fig. 5), we see that there has been relative motion between the steel blocks and the layered package at all locations above steel blocks 2 and 3 at some stage in the uplift history. Thus, there is no valid location along the planar limbs of the central forced folds for siting the notional ‘pin line’ from which most balancing methods begin. As noted above, relative translations are zero above steel block 1 (to the left in Fig. 5), and are unknown above steel block 4 (to the right in Fig. 5). A pin line could be sited above steel block 1 in this model set, but folds created with more complicated forcing-assembly geometries (Weinberg, 1979) have no location with zero relative translation. We suggest that the apparent presence of a pin line in our models is merely an artifact, and that there is therefore no firm basis for assuming where such a location might occur in an unknown setting. However, as determination of strain requires knowledge only of the relative motions, it can be calculated here in spite of the potential difficulty in siting pin lines. To calculate longitudinal strains
Fig. 5. Plot of layer-parallel translations vs. lateral position. Heavy lines join data points. Numbers adjacent to curves identify individual models. Amount of uplift (mm, in parentheses) increases from 33 (4.203), to 35 (4.399), to 40 (6.862), to 46 (7.597). Positive positions are on the upthrown side of the central uplift. Thin vertical lines denote size of steel blocks. These curves demonstrate significant layer-parallel translations (>2 mm) at large structural relief.
within the folded layered package, the translation data (Fig. 5) can be treated as a displacement function: u.i/. The longitudinal (along-layer) strain is simply the partial derivative of u with respect to i: @u @i Because the translation data are determined only where grid lines cross the basal slip surface, we only know the average longitudinal strain between measurement sites about 2.5 mm apart. In Fig. 6, these average longitudinal strains are plotted at the midpoints between measurement points. The variations in layer-parallel translation (relative displacement) along traverses extending away from the central fold couplet represent lateral variations in longitudinal (layer parallel) strain within the layered packages. The maximum slopes in the displacement plots (longitudinal strains: ei ) range from C0.389 to ei D
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Fig. 6. Plots of longitudinal (volume) strain (ordinate) vs. lateral position (in cm; abcissa). Data represent the slopes of the segments from displacement plots (as in Fig. 5), with the mid-points of those segments used as positions in this figure. Elongational longitudinal (dilational volumetric) strain is positive. Plots shown in groups (A–D) of models which have similar amount of uplift (in mm). (A) Model 36 (0.254), model 34 (1.016). (B) Model 38 (2.147), model 32 (2.540), model 37 (2.638). (C) Model 39 (3.411), model 41 (3.801), model 33 (4.203), model 35 (4.399). (D) Model 40 (6.862), model 46 (7.597).
0.433 (model 46), corresponding to lateral strains of 39% elongation and 43% contraction, respectively (Table 1). More typical maxima and minima are 20–25% elongation (models 32 and 33) and 16% contraction (models 33, 37, 39, 40 and 41). Smaller Table 1 Maximum slopes in the displacement plots Model No.
Maximum longitudinal strains (%) contractional
elongational
32 33 34 35 36 37 38 39 40 41 46
11.8 16.7 8.6 10.7 8.3 16.1 9.7 16.2 16.1 14.7 43.3
21.4 25 15 17.2 8.6 6.7 10.3 12.9 14.3 9.7 38.9
maximum strains characterize other models, but longitudinal strains of at least 8–10% are present at some point in all models. Note especially the alternation in longitudinal-strain ‘highs’ and ‘lows’ to suggest a resemblance to the stretching and bunching of an inchworm (cf. Price, 1988; Means, 1989). The accuracy of the displacement measurements is better than 0.01 mm. Based on an uncertainty assumed to be 0.01 mm, and given the nominal 2.5 mm spacing of grid lines, a 10% nominal longitudinal strain (in Fig. 6, derived from a slope of 0.1 in Fig. 5) has an error bar of š2% strain. The lateral position of the displacement measurement is taken as the projected location of the grid line which is attached to the steel block. The grid spacing limits the spatial resolution of the data (and the smearing effect illustrated in Fig. 4 precludes recognition of small strains other than those occurring along the basal shear surface), so no attempt is made here to calculate the areal distribution of strains within the layered package. And as in the approach of Cou-
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ples et al. (1994), the history of longitudinal strain can only be re-constructed from strain profiles determined from models that represent discrete points in structural-history space. The four panels in Fig. 6 are arranged in groups of models which have similar structural relief; the groups increase in relief from A to D. The longitudinal strains indicate variations in deformation at lateral positions considerably removed from the margin of the central uplift, and along planar limbs away from the prominent forced-fold couplet which is caused by the uplift. How are these strains expressed in the rock layers? The limestone is a porous (¾15%), granular rock composed of bioclasts and ooids ranging from about one to several mm in size. Significant fractures and microfaults are present, both in the region of the prominent graben at the uplift crest (Fig. 2), and in a similar, but smaller, graben located near the crest of the fold at the right side of the model (in the orientation illustrated in Fig. 1). Calcite twin gliding, along with microfractures, have been used by Couples et al. (1994) to infer deformation states in the immediate region of the major fold couplet, but calcite twinning is not observed in the parts of the limestone removed from that region — suggesting lower differential stresses. We infer that any strains in the limestone layer away from the main folds are accommodated by indistinct grain-boundary movements facilitated by the high porosity of this material (see comments below concerning thin sections), although we have not observed any clear indications of material damage in these sites. In the dolostone, there is no significant porosity, and the crystal size is very small (a few hundred microns). Microfracturing is the primary deformation mechanism for this rock under the extant experimental conditions, and the rigid blocks of dolostone which constitute the ‘fold’ over the uplift margin are bounded by such features (Fig. 2). Away from the major folds, the dolostone forms long, planar ‘limbs’. Within these long limbs, there are small clusters of (nominally) layer-normal microfractures (in groups of up to about six) that occur within an along-layer distance of 0.5 mm or less. Individual microfractures may cut across the dolostone layer, or they may terminate part way through it. Adjacent clusters are no closer than 4 or 5 mm apart, but
the spacing between visible clusters is sometimes greater. These results are derived from observations made by binocular microscope on the slabbed medial surfaces of the models, and from acetate peels. Quality thin sections proved impossible to make because of lead that smeared during cutting and polishing; this unfortunate event inhibited good epoxy penetration to some parts of the models. Because of these problems with observations of the deformation fabrics of the models, we are not able to make a one-to-one correlation between all of the strain peaks (as indicated in Fig. 6) and definitive fabrics revealing damage within the rock layers. However, in models where the clusters of dolostone microfractures are clearest (on the acetate peels), these clusters do correspond to the strain peaks plotted for that model. Therefore, we believe that the calculated strain curves represent localization of true deformation in the long, planar limbs of these forced folds. 3.3. Volumetric strains The displacement component perpendicular to layering, v. j/, is shown by Couples et al. (1994) to be generally nil except in the immediate vicinity of the macroscopic folds, or in small, localized areas which develop a distinct graben (located on the upthrown-block fold-crest of all models; Fig. 2). Thus, the strain component normal to layering is, except in local sites, nil: e j D¾ 0. If we assume plane strain deformation (appropriate to the medial plane, and, presumably, the remainder of the model), then the longitudinal (layer-parallel) strain is a uniaxial strain except (in small areas) where the layer-normal strain is non-zero. Therefore, in most of the model, the longitudinal strain approximates the volumetric strain: ev D ei C e j C ek D ¾ ei Given the sign convention selected (clockwise sense of shear is positive), positive slopes in Fig. 5 represent dilatant volume strains (volume increase), and negative slopes represent compactant volume strains (at all sites where e j is nil). The data plotted in Fig. 6, therefore, depict volumetric strains in all limb locations away from the graben. The volumetric strains created in these exper-
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imental asymmetric folds locally reach significant values (25C%; see Table 1 and Fig. 6), but they are nevertheless of little significance in terms of the arguments about the validity of balancing techniques. When integrated over distances of at least several grid squares (10C mm), the net volumetric strains average less than 3% dilation. Such low volumetric strain values, if typical of natural folds, would probably have only a very limited effect on the accuracy of re-constructing the geometry of the structural feature. However, the effect on line lengths may be important since line-length discrepancies of this magnitude are now being considered as significant (Groshong and Epard, 1994). As structural methods achieve greater precision, longitudinal and volumetric strains at sites away from ‘obvious’ deformation, and their potential effects on line lengths and=or areas, need to be given appropriate consideration.
4. Discussion 4.1. Petrophysics The prime issue with respect to volumetric strains of moderate-to-large magnitude (¾5–10%, or larger) may not lie in the geometric re-construction of cross sections, but, rather, in their impact on rock properties. Here we consider fracture processes that change the volume occupied by a constant mass of material, rather than pressure solution or other mass-transfer phenomena which could also affect a rock’s volume. Such ‘brittle’ deformation (i.e., characterized by fracturing) is often associated with dilatant strain, but both dilatant and compactant strains can be produced by fracture mechanisms (Nelson and Handin, 1977; Nelson, 1985; Antonellini et al., 1994). The porosity changes caused by deformation usually result in associated permeability changes (and not necessarily of the same sign; see D’Onfro et al., 1994). Examples of permeability alteration can be seen in our models: some compactant volumetric strain areas in the limestone layer have suffered considerable reduction in permeability, as deduced from the failure of the impregnating epoxy to reach regions isolated by zones of compactant strains. If the permeability effects we see associated with the volumetric strains in our experimental folds also
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apply to natural asymmetric folds, then there is considerable potential for significant alteration of the flow character of folded rock sequences (Edwards et al., 1993; Lewis and Couples, 1993; Gabrielsen et al., 1998; Gibson, 1998; Manzocchi et al., 1998). However, it is important to emphasize that we are not suggesting the existence of large (10%CC) volumetric strains in natural structures. As we have argued elsewhere (Couples et al., 1998), natural rock sequences, with their multitude of bedding interfaces, are likely to suffer considerably smaller ‘material’ strains (damage) because of the effects of bedding-plane slip. Nevertheless, the point we seek to make here is that the models have revealed a process (an ‘inchworm’ motion of the fold limbs, with along-limb longitudinal-strain variations) that should be considered in terms of its possible impact on fluid flow, even though the natural expression of the strains may be quite subtle. The opportunity to study quite a number of wellexposed, large-scale structures has emphasized to us the repeated occurrence of prominent, anomalously well-cemented ‘ribs’ of rock on the gentle limbs of many folds. We speculate as to whether these may be the result of enhanced fluid flow, localized to high-permeability pathways cutting across the strata (e.g., dilatant volumetric strains), and consequent diagenetic alterations. We also note that many folds in outcrop reveal a semi-regular pattern of erosional gullies (Fig. 7). Are these features suggesting the presence of (un-cemented) dilational zones which allow accelerated weathering? The recognition of lateral strain variations in our experiments suggests that there may be a non-random, process-based explanation for the ubiquitous occurrence of such spaced features within large-scale, flexural-slip folds. 4.2. Two possible explanations The strain variations (Fig. 6) of our models occur at an apparent half-wavelength of about 4–5 mm (about two times the sampling interval). They are associated with deformation (mostly fracturing, but possibly also including grain=grain motions) that is presumably caused when the imposed distortions raised stress levels to the point of failure. So, what is responsible for these (short-wavelength) distortions that are spaced along the layers?
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Fig. 7. Photograph of north wall of Shoshone Canyon, Rattlesnake Mountain, northwestern Wyoming showing the gentle limb of a major forced fold. Note semi-regular spacing to erosional weaknesses. Elevation difference from canyon floor to top of cliff is approximately 1 km.
4.2.1. Flexure In our models, the main, central fold couplet, and the folds overlying the flanking ‘faults’ in the forcing assembly, represent concentrated bending of the layered package (Fig. 1). The patterns of layer-elongation and layer-shortening associated with these major flexures (as revealed by both macroscopic examinations and petrographic study; see Friedman et al., 1976b; Couples et al., 1994) imply that significant bending moments are created. The inferred stress fields in these experiments are those predicted for such bending (Hafner, 1951; Couples, 1977; Gangi et al., 1977; Couples et al., 1998). We suggest that these localized, large bending moments, which are a result of the forced folding of the layered package over the uplifted and rotated steel blocks beneath, are responsible for propagating a decaying wavetrain of small-scale flexural deformation to locations away from the sites of sharp bending. It is possible that both the fracturing we noted above, and the associated strains we have calculated, are responses to these secondary and tertiary minor bending distortions of the almost-planar limbs of the major folds (Fig. 8). Positive and negative in-
Fig. 8. Illustration of supposed wavetrain of flexural deflections propagating away from sites of major bending. Top of steel block 3 shown along with geometry of layered package above (in grey).
terference of these decaying wavetrains (propagated away from the adjacent major flexures) may explain the occasional spikes in the data, as well as the ‘quiet’ zones observed in some plots. It is likely that the dolostone has a very high flexural rigidity (at least until it fractures), but the limestone will also contribute to the flexural stiffness of the layered package. The lead has, heretofore, been considered as a very weak material, but the possibility needs to be entertained that it also contributes some stiffness to this system. Mathematical models of our explanation have not been
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Fig. 9. Sketch showing spatial arrangement of fracture clusters associated with localized bedding-plane slip (after Cooke and Pollard, 1994, 1997).
constructed; they will, in any case, be considerably more complicated than the flexural models now popular for studying basin subsidence (cf. Beaumont et al., 1988). An interesting avenue to consider is the analytical approach described by Patton and Fletcher (1995); by means of appropriate series expansions, deformation-state predictions could be made for the inchworm motions implied by our data. 4.2.2. Irregular layer-parallel slip Cooke and Pollard (1994, 1997), and Cooke et al. (1998), consider the deformational consequences of localized bedding-plane slip during flexural-slip folding. They associate the creation of fracture clusters, at the ‘ends’ of patches of such slip surfaces, to strain compatibility, and to stress-state modifications (Fig. 9). We suggest that the volumetric strains we note in our models may be expressions of such ‘end-of-slip’ strains, where in this case, the patch of slip occurs at the base of the layered package. This notion is consistent with arguments we have made elsewhere (Couples et al., 1998) concerning the localization of bedding-parallel detachments in flexural-slip folds.
5. Conclusions Lateral variations in layer-parallel translation reflect alternation of elongation and contraction strains at locations away from the obvious folds (in otherwise virtually planar fold limbs). These longitudinal-strain anomalies can exceed 25%, but more typically are ¾10%, and have zones of lower strain lying between. Volume strains may approach or equal the magnitude of the longitudinal strains. The integrated longitudinal- (and volume-) strains (if calculated over large
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areas of the fold) are, however, considerably smaller (<5%), making it difficult to recognize these features in nature without the sort of detailed data coverage (of displacements) available in the rock-layer models. Nevertheless, natural folds exhibit a variety of spaced ‘anomalies’ which seem to be related to dilatancy; we suggest that the deformation processes in our models (which we do not yet fully understand) may apply to natural folds. The petrophysical effects of such alterations could be significant. The asymmetric forced folds created in these experiments, and the details of their deformation, reveal a considerable complexity of strain state resulting from the ‘simple’ process of bending. Because the layer-parallel translations we observe in the models are purely the result of the uplift and bending of the layered package, they are therefore processes inherent to the folding. Wavetrains of flexural deflections propagated away from the sites of major bending may be responsible for creating the along-layer strain zones. An alternate possibility is that the observed strains are reflecting spatial variations in slip on that basal shear surface (or within the layered package?). These two explanations are not mutually exclusive.
Acknowledgements The experimental work was undertaken at the Center for Tectonophysics, Texas A&M University, and we thank John Logan, the late Mel Friedman, and the late John Handin for their support. We also wish to thank a number of colleagues who have participated in friendly discussion of the mechanisms of forced folding: especially, of course, Dave Stearns and John Handin, and including John Logan, Mel Friedman, Dave Weinberg, Jamie Jamison, Jim Morse, and Mike Fahy. We thank Mohammed Ameen, Chris Banks, Ken McClay, Mark Verschuren, and several anonymous reviewers for commenting on earlier drafts of this paper.
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