Analytical and Experimental Study of Simple Geologic Structures

Abstract

An analytical method is derived which gives the elastic response of a homogeneous rock layer to two-dimensional distributions of vertical displacement applied along its lower boundary. Displacement field, stress distribution, and distortional strain-energy density diagrams are presented for three types of displacement applied at the lower boundary of 5-km thick layers possessing average sedimentary rock properties. These three types of displacement are: (1) sinusoidal vertical displacement and no horizontal displacement; (2) an approximate step in vertical displacement and no horizontal displacement; and (3) sinusoidal vertical and horizontal displacement (horizontal displacement 90° out of phase with the vertical). Displacement fields and stress distributions for each type of applied displacement are nearly independent of the elastic characteristics of the layers. The magnitudes of displacement necessary to initiate fracture at some point in the layer are small (3.7–8.2 m) for the three types of applied displacement. For applied displacement (1) and (3), the initial fracture is a vertical tensile crack at the crest of the fold. For applied displacement (2), the initial fracture is either a vertical crack at the upper surface or a shear fracture at the lower surface. Displacement fields and fracture patterns for scale-model experiments of two problems similar to the analytical examples are presented. For a sinusoidal vertical displacement, the fracture pattern is a complex zone of normal faults which taper inward toward the axis of the fold and die out at depth. For a step in vertical displacement, the fracture pattern is (1) a series of reverse faults which start vertically at the base of the layer, curve, and intersect the upper surface at low angles, and (2) a series of normal faults dipping toward the convex side of the reverse faults. Of particular interest are the reverse faults which show that vertical movement at depth can generate low-angle faulting at the surface. Displacement fields found in the elastic analyses are good first-order approximations of displacement fields in the scale-model experiments. Points of initial fracture observed in the model experiments agree closely with those computed in the elastic analyses. The line of fracture for the curved reverse faults in the model experiments can be predicted on the basis of the Mohr fracture criterion and the stress distribution from an elastic analysis.

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