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Peter J. Hudleston, Associate Fellow

Numerical Modeling of Geological Structures and Rock Rheology

Folds are good examples of ductile structures in which rock is deformed, often to a very large degree, while remaining cohesive and continuous. Faults are structures that involve loss of continuity and brittle behavior. Despite these differences, both folds and faults are intimately related in natural deformed rocks. However, the relationship between the two is not well understood. This is an important topic because fold-and-thrust belts, in which this association most commonly occurs, are found throughout the world at the borders of nearly all mountain belts and are one manifestation of the response of the crust to the collisional events of plate tectonics.

Most previous work on this problem has taken folds to be the geometric consequence of movement of layered rocks over a non-planar fault. The most usual configuration is to consider a fault parallel to the layering in places called flats, and which elsewhere, cuts obliquely across layering in places called ramps. Folds that form as a geometric consequence of movement of the hangingwall over the footwall are termed fault-bend folds. This work focuses on how folds form and how rheological and geometrical parameters influence the shape of folds as the fault hangingwall moves over a non-planar fault surface.

Kinematic models developed to explain these structures idealize the folds as being straight-limbed and sharp hinged. Although fold shape appears to often match the model predictions quite well, the evolution of the shapes does not. Thus, a better understanding of the phenomenon of folding associated with faulting must come from a mechanical analysis.

Early analytical work on the mechanics of the problem considered the ideal case of a rigid footwall with an isotropic Newtonian fluid representing the hanging wall, but such a model results in fold shapes unlike those seen in nature. Physical model experiments have produced structures similar to those seen in nature, but it is difficult to control physical properties and to evaluate the effects of varying model parameters. A number of numerical models of fault-bend folding have now been carried out using finite element or finite difference methods. These models employ homogeneous and isotropic media and involve moving the hangingwall over a preexisting discontinuity that represents the thrust fault and ramp. In all these models, fold shape produced in hangingwall is rounded and sharp hinges are lacking, quite unlike the straight-limbed flat-topped folds predicted by the geometric models and seen in many natural examples. The shape of fault-bend folds produced in the current computer simulations in anisotropic materials is very similar to that predicted by the geometric models. These researchers have found that sharp hinges of the folds can be developed in either anisotropic materials or nonlinear materials. This research is carrying out a series of numerical models of the fault-bend folds in nonlinear and anisotropic materials. Comparison of these results helps in obtaining information about fold-and-thrust dynamics.

These researchers are utilizing a two-dimensional, finite element method for slow (quasi-static) flow of an incompressible anisotropic nonlinear fluid. With appropriate choice of constitutive relationships, the code allows for the nodal positions to be updated after each time step, to simulate fold-and-thrust deformation and to compute cumulative strain in each element. The elements are linear convex quadrilaterals, and grids of 1200-1600 elements are used to represent a fold-and-thrust belt. Tens to a hundred or so iterations are required for convergence to a steady state velocity solution for non-linear rock materials. This is done by solving a series of steady-state problems. The velocities at all nodes are calculated for the initial mesh and boundary conditions. An increment of time is chosen and the distance that each node moves in this time is calculated. The nodes are then moved to their new positions, and a new steady-state (quasi-static) solution is obtained. Cumulative strain and strain rates are calculated at each increment. The units of length, velocity, and time are non-dimensionalized by choosing the unit of length of the layer as the layer thickness and the unit of time as the time increment used to update nodal coordinates.

The footwall of the thrust will be taken to be nearly rigid and represent the hanging wall by a layered anisotropic viscous medium with different viscosities. The discontinuity that is the fault in nature is represented by a layer made up of very soft viscous elements in which large deformation is allowed. To avoid extreme element distortion in the fault zone, the element grid has been updated as appropriate. Displacement on the fault due to forces or velocities applied to the boundaries of the model results in both a geometric change in the hanging wall and a dynamic instability due to the nonlinearity or anisotropy. The geometry of the fault (e.g., dip angle of ramp), displacement on the fault, and degree of non-linearity of the hanging wall can then be varied to examine how these parameters affect the resulting fold geometry and strain variation in the hanging walls. Focus is placed on the effect of variations of competition of rock nonlinearity or anisotropy on fault-related fold development in fold-and-thrust belts. The results of the models are compared with previous results and natural folds-in terms of the geometry of the folds, fold location with respect to the faults, and evolution of the structures with time.

This information is available in alternative formats upon request by individuals with disabilities. Please send email to alt-format@msi.umn.edu or call 612-624-0528.

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