Fluid Dynamics of Viscous Buckling Applicable to Folding of Layered Rocks

Abstract

For slow creep under relatively small differential stress rocks may in a first approximation be treated as very viscous fluids. Under such conditions it is therefore possible to apply the theory of fluid dynamics to the evolution of strain in complexes consisting of coherent assemblages of rock bodies with unlike shape and unlike effective viscosities. In this paper fluid dynamics is applied to models simulating layered rocks exposed to compression parallel with layering. As basis for discussion and experimental tests presented in the paper the fluid dynamic theory of the following models is briefly reviewed. (1) Single layer enclosed in uniform surroundings whose dimension normal to the layer is large relative to the characteristic wavelength. (2) Multilayers consisting of alternating competent and incompetent layers; (a) unrestricted by surrounding media, and (b) restricted by surrounding media. In all cases the contacts between adjacent layers are considered welded to prevent free slip. Layer-parallel shear is consequently included in the models. This unfortunately makes the mathematics somewhat cumbersome but also makes the models more realistic representatives of rock structures than free-slip models. The welded-contact model of an embedded single layer gives a somewhat smaller wavelength/thickness ratio than the free-slip model of Biot and others. For multilayers (unrestricted, case b) consisting of many alternating competent and incompetent layers, the welded-contact model gives results very different from the free-slip model. If the spacing between the competent layers, i.e., the thickness of the incompetent layers, is greater than a certain limit, the multilayer will buckle in many folds, whereas at smaller spacing the multilayer will buckle to 1 or ½ wave. Many of the theoretical predictions of the behavior of viscous models apply to elastic models if viscosity coefficients are replaced by rigidity modula in the equations. All important predictions of the theory have been tested by rubber models. Excellent agreement between theory and experiment has been established. The multilayers which buckle to a single wave if unrestricted (by adjacent bodies) will buckle to many waves with a characteristic wavelength if surrounded by uniform viscous materials. Equations for enclosed multilayers of this kind are also developed. A theory is developed for viscous buckling of a thin competent layer embedded in an incompetent layer of limited thickness which in turn is sandwiched between thick competent layers. The latter layers remain straight during compression whereas the middle layer buckles. The wavelength of the buckles is shown to be a function of the spacing between the buckling layer and the straight walls. The case of a multilayer taking the place of the thin central layer in the foregoing case is discussed. The approach throughout the work has been to develop the stream function ^psgr and the derivative velocity components U and V which satisfy the particular boundary conditions of the models. When the correct stream function is found for a particular situation, the situation is completely known from a fluid dynamic point of view because the velocities, the rate of change of strain, the stress variations and the variations of pressure throughout the system all follow by way of the general fluid dynamic equations. The models have been chosen with application to common rock structures in mind.