Trishear Model Research Articles

Propagation history of the Osaka-wan blind thrust, Japan, from trishear modeling

Mapping the nucleation and 3D fault tip growth of the active Osaka-wan blind thrust provides an opportunity to asses how reactivated thrusts build slip from preexisting faults and the threat they pose as sources of large earthquakes. Analysis of folded growth strata, based on 2D trishear inverse modeling allows a range of best-fit models of the evolution of slip and propagation of the fault to be defined. The depth of the fault tip at 1200 ka varies between ∼1.5–4.5 km, suggesting the fault grew upward from high in the crust, and that it is reactivated. From its onset at ∼1500 ka, the fault grew rapidly along strike in ∼300 ky, and upwards with a P/S ratio of 2.5–3.0, but variable fault slip in space and time. Shallower depths of the fault tip at initiation and thinner basin fill correlates with slower propagation with time, contradicting models that argue for sediments as inhibitors of fault growth. Results also suggest the displacement profile of the currently active thrust is offset from its predecessor, assuming shallower depths to the original fault correlate with greater displacement in its prior history. These results suggest reactivated faults may accrue slip differently than newly developed ones, based on the history of upward fault propagation.

Oligocene shortening in the Little Burro Mountains of southwest New Mexico

Laramide-style shortening structures in southwest New Mexico chiefly affect Cretaceous and older rocks, making the estimation of the cessation of shortening in the Paleogene problematic. The cessation of Laramide shortening is generally thought to be about 40 Ma in southwest New Mexico. However, in the Silver City Range, Grant County, New Mexico, shortening younger than 34.6 Ma was documented by Copeland et al. (2011). The Little Burro Mountains, 10–20 km south of the Silver City Range, contain a monocline trending northwest–southeast with beds dipping ∼12° in the backlimb and up to 30° in the forelimb. The youngest folded unit in the monocline is the tuff of Wind Mountain (Twt), which yielded a 206 Pb/ 238 U zircon age of 30.9 ± 0.5 Ma. Normal faults strike orthogonal to the axial trace of the fold with low displacement (10s of meters), which we interpret to have formed synchronous with the monocline to accommodate variations in shortening along strike. Field observations, trishear fault-propagationfold modeling, and the structural style and trends of the region are consistent with the development of the monocline related to a deeply rooted blind thrust or reverse fault. These results indicate that Laramide-style shortening in southwest New Mexico was active into the late Oligocene.

Do cataclastic deformation bands form parallel to lines of no finite elongation (LNFE) or zero extension directions?

Conjugate cataclastic deformation bands cut unconsolidated sand and gravel at McKinleyville, California, and dip shallowly towards the north-northeast and south-southwest. The acute dihedral angle between the two sets of deformation bands is 47° and is bisected by the sub-horizontal, north-northeast directed incremental and finite shortening directions. Trishear models of fault propagation folding above the McKinleyville fault predict two sets of LNFE (lines of no finite elongation) that plunge steeply and shallowly to the south and north. These predictions are inconsistent with deformation band orientations and suggest that deformation bands did not form parallel to these LNFE. During plane strain, zero extension directions with acute dihedral angles of 47° develop when the dilatancy rate (dV/dε1) is −4.3. Experimental dilatancy rates for Vosges sandstone (cohesion > 0) and unconsolidated Hostun sand suggest the deformation bands either developed parallel to zero extension directions or in accordance with the Mohr–Coulomb criterion, assuming initial porosities of 22% and 39%, respectively. An empirical relationship between dV/dε1, relative density and mean stress suggests that dilatancy rates for Vosges sandstone overestimate dV/dε1 at McKinleyville. Deformation bands at McKinleyville likely developed either in a Mohr–Coulomb orientation, or an intermediate orientation bounded by the Mohr–Coulomb (θC) and Roscoe (θR) angles.

A kinematic trishear model to predict deformation bands in a fault-propagation fold, East Kaibab monocline, Utah

Well core with numerous deformation bands in the reservoir interval may or may not be indicative of reduced effective reservoir permeability at the scale of well drainage areas. Many dense concentrations of deformation bands are related to the damage zone of a larger fault. However, some populations are more broadly distributed. We analyze one such population associated with the East Kaibab monocline in southern Utah. A kinematic trishear analysis is compared with field-based strain measurements. We find that the widespread dense deformation-band populations correspond to a broad zone of relatively high strain across the structure. Fault-damage zone models are inadequate to explain these occurrences. Our results show that, where deformation bands are known to occur from core in a folded reservoir, finite strains can be used to estimate their lateral and volumetric extents. However, we also find that the orientations of deformation bands predicted by our modeling are highly sensitive to the strain path. This indicates that path-independent methods for estimating strain such as curvature analysis are not fully appropriate for application to deformation bands. Ultimately, any such method requires information relating rock properties with propensity to form deformation bands to be predictive.

Determining the uniqueness of best-fit trishear models

We show the application of a simulated annealing algorithm to trishear inverse modeling. The algorithm traverses the parameter space in search for best-fit models without being trapped in local minima, and thus sampling for more possible solutions globally. Simulated annealing is a robust and efficient technique to determine the uniqueness of best-fit trishear models; the spread of possible trishear models that can fit a structure. We first apply the algorithm to a decameter-size, contractional fault-propagation fold in west-central Taiwan, for which there is an exceptional exposure of pre-growth and growth strata. Simulated annealing shows that even for this complete fold dataset with low uncertainties, there is a range of models and fault slip/uplift histories that can fit the data, with the consequent implications for the assessment of seismic hazard. We then apply the algorithm to a kilometer-size, extensional fault-propagation fold, the Hadahid monocline, Gulf of Suez Rift, Egypt. In this monocline there is only surface coverage in the footwall anticline areas and the algorithm was used to delimit the range of possible models that can fit the data and their uncertainties, thus avoiding biases in the interpretation. Simulated annealing suggests that the along-strike structural variability of the monocline can result from along-strike variability in fault slip, fault propagation to fault slip ratio and depth of fault nucleation. Both examples illustrate the benefits of searching for a possible range of models rather than a precise best-fit model when modeling fault-propagation folds. In an attempt to understand which parameters control fault development, and also how the spread of possible solutions varies with fold growth, we apply the algorithm to four sequential stages of a published, analog clay model of an extensional forced fold. The inversions of the natural examples and the analog model suggest that the spread of the possible models is a manifestation of the data uncertainties, the suitability of the trishear model, fold evolution, and rock mechanical properties.

The glaciotectonic deformation of Quaternary sediments by fault-propagation folding

Fault-propagation folding is an important yet seldom recognised structural style within sediments affected by glacier-induced deformation. Fault-propagation folds develop in the hanging wall of low angle thrust faults and compensate part of the slip along the fault. Field examples are recognised across northern Europe, in glaciotectonic complexes of north Germany, Wales and the Isle of Man. The recognition of the fault-propagation fold mechanism in glaciotectonic deformation is important because resultant structures are related to exactly the same phase of deformation (i.e. the same phase of ice advance), and thus play a critical role in analyses of the temporal and spatial evolution of glacier-induced deformation. Some field examples show monoclinal geometries that are in good agreement with predictions of trishear kinematic theory. The trishear approach is appropriate to model these structures because the structures analysed in the field and simulated below show characteristics that are compatible with fault-propagation folds that were produced by trishear kinematics. The curved forelimb and the monocline geometry of the fault-propagation folds fit to the trishear model. The occurrence of footwall synclines is also in good agreement with trishear kinematics. These synclines show the typical thickening of the strata in the hinge. With respect to the modelling output, most important factors for the structural evolution of the fault-propagation folds is the ramp angle of the thrust, the position of the tip line and the propagation-to-slip ratio along the fault. This fits to observations made by previous studies at large scale fault-propagation folds in fold-and-thrust belts.

Optimized trishear inverse modeling

We describe the application of optimization methods to 2D and 3D trishear inverse modeling. These optimization methods traverse the parameter space in search for the combination of trishear parameters that best restores a fold profile to a straight line in 2D, or a folded surface to a plane in 3D. Optimized trishear inverse modeling is significantly faster than the standard grid-search method. An optimized search of all six parameters of a 2D trishear model can be conducted in seconds. Gradient-based optimization methods converge to the solution much faster than the simplex optimization method, and therefore are the preferred choice for 2D or 3D inversions involving a large number of parameters. The optimization algorithms, however, are affected by local minima. This is particularly an issue in the inversion of actual fold data, where local minima can occur even in a parameter space of low dimensions. Local minima can be assessed by running several optimizations from different regions in the parameter space. Physically meaningless local minima can be avoided by constraining the optimization. Fast, optimized trishear inverse modeling makes possible within hours, heavy tasks such as estimating the uncertainties of the best-fit parameters by the inversion of hundreds of fold realizations. We illustrate this technique using synthetic fold data and an actual fault-propagation fold from the Los Angeles Basin, the Santa Fe Springs anticline.

A Quadshear kinematic model for folding and clay smear development in fault zones

A new kinematic model, based on the Trishear model, has been developed to simulate shearing and folding of a ductile interval sandwiched between two brittle layers, as faults propagate from the brittle layers into the ductile interval. This Quadshear model can be used to study fault zone evolution and the development of clay smears, which often form when a stratigraphic sequence including ductile clay-rich beds is sheared into a developing fault zone. The model also gives the critical clay smear factor (CCSF), the ratio of fault displacement to bed thickness at which a clay smear becomes discontinuous (and hence a potential leak point). The main control on clay smear development is the position of the clay-rich beds relative to brittle, fracture-prone layers: clay-rich beds near the centre of a ductile interval can shear to form much longer continuous clay smears than clay-rich beds adjacent to brittle layers. Fault propagation rates are also a strong control on clay smear development: slow propagating faults form longer continuous clay smears than fast propagating faults. However the width of the shear zone, the dip of the fault and the presence of releasing or restraining bends in the fault are minor controls on clay smear development.

Growth trishear model and its application to the Gilbertown graben system, southwest Alabama

Fault-propagation folding associated with an upward propagating fault in the Gilbertown graben system is revealed by well-based 3-D subsurface mapping and dipmeter analysis. The fold is developed in the Selma chalk, which is an oil reservoir along the southern margin of the graben. Area-depth-strain analysis suggests that the Cretaceous strata were growth units, the Jurassic strata were pregrowth units, and the graben system is detached in the Louann Salt. The growth trishear model has been applied in this paper to study the evolution and kinematics of extensional fault-propagation folding. Models indicate that the propagation to slip ( p/ s) ratio of the underlying fault plays an important role in governing the geometry of the resulting extensional fault-propagation fold. With a greater p/ s ratio, the fold is more localized in the vicinity of the propagating fault. The extensional fault-propagation fold in the Gilbertown graben is modeled by both a compactional and a non-compactional growth trishear model. Both models predict a similar geometry of the extensional fault-propagation fold. The trishear model with compaction best predicts the fold geometry.

A new active tectonic model for the construction of the Northern Apennines mountain front near Bologna (Italy)

We integrate existing and new geologic data [REtreating TRench, Extension, and Accretion Tectonics (RETREAT project)], particularly on the origin, growth, and activity of the mountain front at Bologna, Italy, into a new model that explains Apennine orogenesis in the context of a slab rollback ‐ upper plate retreat process. The Bologna mountain front is an actively growing structure driving rock uplift ∼1 mm/year, cored by a midcrustal flat‐ramp structure that accommodates ongoing shortening driven by Adria subduction at a rate of ∼2.5 mm/year. The data we use are assembled from river terraces and associated Pleistocene growth strata, geodesy including releveling surveys, reinterpretation of published reflection lines, and a new high‐resolution reflection line. These data constrain a simple trishear model that inverts for blind thrust ramp depth, dip, and slip. Apennine extension is recognized both in the foreland, as high‐angle normal faults and modest stretching in the carapace of the growing mountain front, and in the hinterland, with larger normal faults that accomplish some crustal thinning as the upper plate retreats. This coevolution of extension and shortening shares some notable characteristics with other basement‐involved collisional orogens including the early Tertiary Laramide orogeny in the American West and the Oligocene to Miocene evolution of the Alps. We propose a possible relationship between underplating and the development of the Po as a sag basin as a Quaternary phenomenon that may also apply to past periods of Apennine deformation (Tortonian). Continued shortening on the structure beneath the Bologna mountain front represents by far the most important and underappreciated seismogenic source in the front of the northern Apennines.

A methodology for efficiently populating faulted corner point grids with strain

This article describes an algorithm to compute finite strain in faulted corner point grids using the software Havana. The algorithm is based on a simple fault displacement formula, and a volumetric computation of strain in the grid's deformed configuration. The volumetric computation of strain is tested by comparing the finite strain of a 3D trishear model calculated by this method, with that calculated by the tetrahedrons method. The agreement between both methods confirms the validity of the volumetric strain computation. The algorithm is applied to synthetic models of one and three intersecting normal faults, and to a real model with seven faults, the Emerald Field. In all cases the computed finite strain is consistent with the fault network and with the variation of slip along the faults. There is one parameter that affects the computation significantly: the drag radius ( r d ) or extent of folding across a fault. Low r d models yield high finite strain and strain gradients but limited fault interaction, and vice versa. Using empirical relations between fault throw and damage zone width, r d can be narrowed down and further constrained by evaluating the quality of the grid's restoration. The strain algorithm can be integrated easily into a reservoir modelling workflow and in stochastic modelling. The algorithm provides criteria for conditioning the distribution of deformational features within the reservoir zones affected by faulting, based on the magnitude of finite strain.

Trishear in 3D. Algorithms, implementation, and limitations

The algorithms and implementation of pseudo-3D and true-3D trishear models are explained, including a strategy to model lateral fault propagation. I show that the pseudo-3D algorithm is adequate and sufficient to model trishear in three-dimensions. Although ad-hoc, the pseudo-3D algorithm preserves volume in simulations without and with lateral fault propagation. A disadvantage of the pseudo-3D algorithm is that it produces very simple, and perhaps not realistic hanging wall geometries, specially in simulations in which the fault slip varies along strike. The true-3D algorithm has a more elaborate and richer kinematics that produces more realistic hanging wall geometries. However, the true-3D algorithm contains mathematical inconsistencies that result in considerable volume changes when the slip gradients along the tip line are high and the tip line is highly oblique to the slip vector and/or the fault strike. The volume changes occur to a large extent in the hanging wall, and to a minor extent in the forelimb and footwall areas.

Inference of trishear-faulting processes from deformed pregrowth and growth strata

Near surface strata deformed by propagation of a fault situated underneath these strata hold explicit records of fault activities. The delicate deformed strata exposed in investigation trenches for paleoseismologic study conform to fault-propagation folds growing above a blind thrust. Growth strata deposited subsequently across the coseismic fold scarps are preserved by onlap and unconformities. Information from these sedimentary and folding processes was used in conjunction with inverse modeling using the trishear model technique proposed by Allmendinger (Allmendinger, R.W., 1998. Inverse and forward numerical modeling of trishear fault-propagation folds. Tectonics 17, 640–656) in estimating the times and amounts of slips. This methodology becomes necessary especially when the extent of a deformed profile is insufficiently exposed to observation or the location of the fault tip is invisible. The proposed method can deal with growth strata that can be selectively removed during fold restoration. The interpretations on the deformation history obtained by inverse analysis and trench investigation were then compared. The combination of these two approaches seemed to enhance and complement each other and allows a better and more complete understanding of the trishear-faulting processes and is useful in seismic hazard assessment.

Coseismic fold scarps and their kinematic behavior in the 1999 Chi‐Chi earthquake Taiwan

Discrete scarps that are created or reactivated during large earthquakes are a locus of concentrated hazard. A number of the coseismic scarps activated in the 1999 Chi‐Chi earthquake are actually fold scarps, which display several types of ground deformation characterized by localized folding and are distinct from classic fault scarps, which form by a fault cutting the surface. This paper documents and analyzes fold scarps that formed or reactivated in the 1999 Chi‐Chi Taiwan earthquake. Our results show the Chi‐Chi fold scarps can be generally divided into two types: (1) those associated with folding ahead of the tip of a blind thrust fault at shallow depths and (2) those associated with folding by kink band migration above fault bends at substantial depths ranging from ∼0.8 to 5 km). The previously published trishear model can be applied to model the former type, while a new curved hinge kink band migration model is provided to describe the behavior of the latter type. A key feature of fold scarps of the second type is that hinge zones are typically wide (25–100 m) relative to the displacement in a single earthquake (1–10 m), which exerts a significant control on fold scarp morphology and evolution. Because the coseismic strains of both types of fold scarps display relatively wide deformation zones (10–100 m) relative to fault scarps, wider set‐back zones might be appropriate from a public policy point of view to alleviate the risk to structural damage and collapse resulting from permanent ground deformation.

Application of trishear fault-propagation folding to active reverse faults: examples from the Dalong Fault, Gansu Province, NW China

Determining accurate fault slip rates at 1 ka to 1 Ma timescales requires well-constrained palinspastic reconstructions of dateable geomorphic and/or geologic markers. Although general kinematic models have been developed to simultaneously reconstruct both bedrock (e.g. bedding and fault attitudes) and neotectonic markers (e.g. strath terraces) along active strike-slip and thrust faults, it is not clear if these models can also account for deformation along steeply dipping (>45°) reverse faults. To address this problem, we have investigated the active, ∼50° dipping, Dalong reverse fault system. This ∼40-km-long fault system forms part of the Aksai restraining stepover along the active, left-slip Altyn Tagh Fault in northwestern China. Our geometric and kinematic analyses show that conventional fault-bend fold models cannot satisfy the steeply-dipping fault geometry we observe in the bedrock record. Likewise, standard fault-propagation fold models fail to match our measurements of a set of fluvial terraces. However, by expanding the trishear model of fault-propagation folding to track both bedrock and neotectonic markers, we are able to match both sets of records. In particular, we have developed trishear kinematic models for two sites (Liuchenzi and Qingyazi) using the numerical modeling program, Fault/Fold v.5.0. This work indicates that an important implication of active trishear fault-propagation folding is that terrace deformation extends for over 1 km on either side of the fault trace. Thus, to accurately measure the total magnitude of vertical separation between matching terraces in the hanging wall and footwall, terrace profiles across active reverse faults must extend 1–2 km on either side of this zone of deformation.

Trishear kinematic modeling of extensional fault-propagation folding

We present a new non-linear 2-D numerical trishear model that is capable of modeling both symmetrical and asymmetrical trishear zones and a strain prediction approach based on the velocity field within the trishear zone. A simple relationship exists between the shape factor ( r) of the velocity field, the hanging wall apical angle, and footwall apical angle. The implication of this relationship is that when dealing with a real deformed state cross-section, one can measure the hanging wall and footwall apical angles to determine the r value to specify the velocity field within the trishear zone for either the forward modeling or restoration. The new trishear model can accurately reproduce the geometry of two experimental clay models and an extensional fault-propagation fold from the Gulf of Suez. Predicted strain is greatest directly above the fault tip and decreases with distance from the fault tip, as in the experimental models. Greater strain occurs in the hanging wall because of the greater velocity gradient there, whereas the footwall is much less intensely deformed. Consequently, secondary faulting and fracturing, if present, should be better developed along the fault tip line or in the hanging wall. The orientation of fractures or minor faults can be predicted from the strain ellipses calculated from the trishear velocity field.

Influence of mechanical stratigraphy and initial stress state on the formation of two fault propagation folds

Kinematic and mechanical modeling of the Rip Van Winkle (SE New York, USA) and La Zeta (SW Mendoza, Argentina) anticlines illustrate the influence of mechanical stratigraphy and initial stress state on the kinematics of fault propagation folding. In both anticlines, faults nucleating at distinct stratigraphic levels open upward into triangular zones of folding. Folding intensity and finite strain attenuates with distance from the fault tip. Trishear reproduces the bulk geometry and finite strain of the relatively homogeneous limestone sequence of the Rip Van Winkle anticline and predicts an initial location of the fault tip consistent with the field observations. Folding of the heterogeneous sedimentary section of La Zeta anticline, however, cannot be simulated by the trishear model alone. An additional mode of deformation involving transport of material from the backlimb into the hinge area and extension parallel to the direction of fault propagation is necessary to reproduce the geometry and finite strain of the anticline. Mechanical, distinct element modeling (DEM) of the anticlines indicates that their contrasting kinematics could have resulted from differences in mechanical stratigraphy and initial stress state. Folding of a homogenous, normally consolidated assemblage (initial horizontal to vertical stress ratio, K o=1) is trishear like and resembles the Rip Van Winkle anticline. Folding of a heterogenous (layered such as La Zeta), over-consolidated assemblage ( K o>1) departs from the trishear model and resembles La Zeta anticline. Based on the DEM simulations, we postulate that the Rip Van Winkle anticline formed at high depths (high overburden loads and lithostatic stress conditions), and that La Zeta anticline formed at shallow depths, after substantial uplift and erosion of the Andean mountain front (which induced over-consolidation and high K o).

Trishear modeling of fold bedding data along a topographic profile

This paper describes an algorithm that extends the use of the trishear kinematic model to the commonly encountered case of a set of fold bedding observations along a profile. In a universe of trishear models that can possibly fit the structure, the algorithm searches for the model that minimizes the difference between observed and modeled beds (i.e. best fit model). The robustness and versatility of the algorithm are shown by applying it to a synthetic fold geometry, and to a transect of the Waterpocket monocline in southern Utah. In the first case, the best fit parameters are exactly those used in the synthetic model. In the second case, the best fit model reproduces fairly well the observable features of the monocline.

Reactivation, trishear modeling, and folded basement in Laramide uplifts: Implications for the origins of intra-continental faults

Reactivation, trishear modeling, and folded basement in Laramide uplifts: Implications for the origins of intra-continental faults

Extensional fault-related folding, northwestern Red Sea, Egypt

The Duwi and Hamadat areas on the northwestern margin of the Red Sea rift system exhibit superb outcrop examples of kilometric scale, extensional fault-related folds. In this area, the Nakheil and Hamadat faults of the Rift border fault system consist of a series of WNW- and NW-trending segments that link through breached relay ramps. The fault hanging walls are characterised by a series of offset synclines with non-linear axial traces sub-parallel to the faults. The synclines are doubly plunging and, in the immediate hanging wall to the normal faults, the strata dip sub-parallel to the fault. These folds result from along-strike displacement variations on the individual fault segments together with extension-related fault-propagation folding as the faults propagated upwards through a highly anisotropic pre-rift sedimentary section. In the Duwi area, the axial traces of the synclines are offset and bend in the regions of relay ramps. In the Hamadat area, the synclines associated with individual fault segments are separated by transverse hanging-wall anticlines that formed where segment linkage occurred. Numerical simulations of extension-related fault-propagation folding using a trishear model produce geometries that closely match those of the Duwi and Hamadat folds. The dimensions of these hanging-wall synclines and their relationships to fault displacement variations indicate that they are formed by extensional fault-propagation folding rather than by frictional drag along the fault surface.