Modelling Viscous Flow and Elastic Deformation in Fold-Thrust Belts and Magmatic Intrusions

Fluid dynamics governs many phenomena on the Earth's surface and interior, from the emplacement of fluid magma, to the viscous deformation of mountain ranges on the longest time scales. Understanding these processes presents a challenge to traditional modelling techniques. However, simplifying models of the leading-order features of the flow can give insight into the dominant physical balances at play. In this dissertation I use theoretical analysis, numerical simulations, and laboratory experiments to address two geophysical processes: the formation of fold-thrust belts and the dynamics of shallow magmatic intrusions. Although geophysically distinct, these two problems both involve the interplay between viscous flow and elastic deformation and so inform the modelling of one another. Fold-thrust belts are formed at convergent margins, where accretion of weak sediments to the front of the overriding plate results in continued flexural subsidence of the underthrusting plate. In this dissertation I build a new dynamic model to investigate both the role of the thickness and material properties of the incoming sediment, and the flexure in the underthrusting plate in controlling the behaviour and evolution of fold-thrust belts. The analysis shows that the evolution of fold-thrust belts can be dominated by either gravitational spreading or vertical thickening. I apply the model to the Makran accretionary prism and the Indo-Burman Ranges, and show that for the Makran flexure is important, while in the Indo-Burman Ranges the incoming sediment thickness has a first-order control on topography. The propagation of shallow magmatic intrusions is governed by the interplay between elastically deforming sedimentary layers, the viscous flow of magma beneath, and the requirement to fracture at the front. In this dissertation I describe this process by extending the model for elastic-plated gravity currents to an axisymmetric geometry and show that adhesion (or fracture toughness) gives rise to two dynamical regimes of spreading; viscosity dominant spreading and adhesion dominant spreading. Experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapour tip, and confirm the existence of spreading regimes controlled by viscosity and adhesion. I extend this laminar model of magma propagation to large mafic sills, which are thought to exhibit turbulent flow. Using a hybrid laminar-turbulent flow model I examine the transition to turbulence and show that volume fluxes several orders of magnitude above the average are required to reproduce the aspect ratios of large mafic sills measured in the field. Finally, I explore the role topographic gradients may play in driving magmatic intrusions by carrying out further experiments where the elastic sheets are inclined at an angle to the horizontal. Experimental observations show the formation of a transient head and a static tail structure with good first order comparisons to the deformation patterns of the Piton de la Fournaise flank sill intrusion.Natural Environment Research Council (NERC

Static and dynamic fluid-driven fracturing of adhered elastica

The transient spreading of a viscous fluid beneath an elastic sheet adhered to the substrate is controlled by the dynamics at the tip where the divergence of viscous stresses necessitates the formation of a vapor tip separating the fluid front and fracture front. The model for elastic-plated currents is extended for an axisymmetric geometry with analysis showing that adhesion gives rise to the possibility of static, elastic droplets and to two dynamical regimes of spreading; viscosity dominant spreading controlled by flow of viscous fluid into the vapor tip, and adhesion dominant spreading. Constant flux experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapor tip and confirm the existence of spreading regimes controlled by viscosity and adhesion. The theory and experiments thereby provide an important test coupling the dynamics of flow with elastic deformation and have implications in fluid-driven fracturing of elastic media more generally

Controls on the geometry and evolution of thin-skinned fold-thrust belts, and applications to the Makran accretionary prism and Indo-Burman Ranges

The formation of fold-thrust belts at convergent margins is a dynamic process. Accretion of weak sediments to the front of the overriding plate results in crustal thickening and continued flexural subsidence of the underthrusting plate. Fold-thrust belts are often treated as a Coulomb wedge having self-similar geometries with a critical taper, and either a rigid or isostatically compensated base. In this paper we build upon this work by developing a new dynamic model to investigate both the role of the thickness and material properties of the incoming sediment, and the flexure in the underthrusting plate in controlling the behaviour and evolution of fold-thrust belts. Our analysis shows that the evolution of fold-thrust belts can be dominated by either gravitational spreading or vertical thickening, depending on the relative importance of sediment flux, material properties and flexure. We apply our model to the Makran accretionary prism and the Indo-Burman Ranges, and show that for the Makran flexure must be considered in order to explain the dip of the sediment-basement interface from seismic reflection profiles. In the Indo-Burman Ranges, we show that incoming sediment thickness has a first-order control on the variations in the characteristics of the topography from north to south of the Shillong Plateau

Viscoplastic rimming flow instead a rotating cylinder

A theoretical analysis is presented for the flow of a Herschel-Bulkley fluid around the inside surface of a rotating cylinder, at rotation speeds for which the fluid largely collects in a prominent pool in the lower part of the cylinder. The analysis, based on lubrication theory, predicts the steady states typically reached after a small mumber of rotations. The analysis is also modified to consider the drainage of the film around a stationary cylinder, which allows an exploration of the dynamics when a rotating drum is suddenly stopped. The predictions of the theory are compared with experiments in which a Carbopol suspension is rotated inside an acrylic drum

Viscoplastic plates

An asymptotic model is constructed to describe the bending of thin sheets, or plates, of viscoplastic fluid described by the Herschel–Bulkley constitutive law, which incorporates the von Mises yield condition and a nonlinear viscous stress. The model reduces to a number of previous ones from plasticity theory and viscous fluid mechanics in various limits. It is characterized by a yield criterion proposed by Ilyushin which compactly combines the effect of the bending moment and in-plane stress tensors through three particular invariants. The model is used to explore the bending of loaded flat plates, the deflection of impulsively driven circular plates, and the tension-controlled deflection of loaded beams

Static and dynamic fluid-driven fracturing of adhered elastica

The geometry and propagation of fluid-driven fractures is determined by a competition between the flow of viscous fluid, the elastic deformation of the solid, and the energy required to create new surfaces through fracturing. To date, much research has focused on the formation of idealised penny-shaped cracks in elastic media [1]. However, the dynamics of fluid-driven fracturing of thin adhered elastica remain unexplored and unobserved, and provide an experimentally accessible and theoretically simpler setting in which to assess the underlying physical processes. We present a theoretical and experimental approach to model a â fractureâ produced when fluid is injected from a point source between a solid horizontal plane and an elastic sheet, which is adhered to the plane. Divergence of viscous stresses necessitates the formation of a vapour tip between the fluid front and fracture front. This results in two dynamical regimes of spreading: viscosity dominant spreading controlled by the flow of viscous fluid into the vapour tip, and adhesion dominant spreading controlled by the energy required to fracture the two layers. Constant flux experiments using clear elastic sheets (PDMS) enable new, direct measurements of the vapour tip and confirm the existence of spreading regimes controlled by viscosity and adhesion. We extend this work to consider the possibility of turbulent flow within the body of the fracture and assess the scale of the laminar tip at the fracture front. This analysis identifies the transition from turbulent to laminar control of the spreading, or equivalently the transition from bulk to tip control. These processes primarily feature industrially in the hydraulic fracturing of shale [2], but are also commonplace in nature, from magmatic intrusions in the Earthâ s crust [3, 4], to the propagation of cracks at the base of glaciers [5]. [1] D. I. Garagash and E. Detournay, â The Tip region of a Fluid-Driven fracture in an Elastic Medium,â J. Appl. Mech. 67, 183-192 (1999) [2] E. Detournay, â Mechanics of Hydraulic Fractures,â Annu. Rev. Fluid Mech. 48, 311-339 (2016) [3] C. Michaut, â Dynamics of Magmatic Intrusions in the Upper Crust: Theory and Applications to Laccoliths on Earth and the Moon,â J. Geophys. Res. 116, 1-19 (2011) [4] A. M. Rubin, â Propagation of Magma Filled Cracks,â Annu. Rev. Earth Planet. Sci. 23, 287-336 (1995) [5] V. C. Tsai and J. R. Rice, â A Model for Turbulent Hydraulic Fracture and Application to Crack Propagation at Glacier Beds,â J. Geophys. Res. Earth Surf. 115, 1-18 (2010)Non UBCUnreviewedAuthor affiliation: Cambridge UniversityGraduat

Viscoplastic Saffman-Taylor fingers with and without wall slip

The Saffman-Taylor instability for the flow of a Herschel-Bulkley fluid through a Hele-Shaw cell is explored theoretically and experimentally. The theoretical analysis adopts conventional Hele-Shaw approximations, but generalized to account for a Herschel-Bulkley rheology and to include a model for the effective slip of the fluid over smooth walls. A linear stability analysis is presented for fingering instabilities on both planar and axisymmetrical interfaces. The linear instability of a planar interface is continued numerically into the nonlinear regime. It is found that certain finger widths are selected and controlled by the yield stress. Stresses also fall sufficiently behind the fingertips to allow the yield stress to block the cell to either side. Experiments are conducted using aqueous suspensions of Carbopol pumped into a Hele-Shaw cell through a circular vent. Instabilities are created by first pumping a disk of Carbopol into the cell, then either pumping air into the fluid-filled cell or withdrawing the Carbopol through the vent. In both cases, the fingers forming on the retreating air-Carbopol interface are interrogated as a function of flux, gap size and the type of cell walls. The instability is very different for cells with either rough or smooth walls, an effect that we attribute to effective slip. The trends observed in the experiments are in broad agreement with theoretical predictions

Fracture patterns in viscoplastic gravity currents

Constant-flux gravity currents of viscoplastic fluid remain axisymmetric when extruded onto a dry horizontal plane. However, if the plane is coated with a shallow layer of water, the current suffers a dramatic non-axisymmetric instability in which localized v-shaped cuts appear in the outer edge where the viscoplastic fluid is in contact with water. These ‘fractures’ lengthen and guide the subsequent radial outflow, leading to distinctive flower-like patterns. This pattern formation process is illustrated for two viscoplastic materials, an aqueous suspension of Carbopol, and a mixture of water and joint compound (a kaolin-based, commercially available product). The fracturing spreads over the entire upper surface of the current when deeper water baths are used, complicating the extrusion patterns. The instability can be removed entirely when the ambient water layer is replaced by an immiscible liquid of comparable viscosity, indicating that the presence of water at the surface is key to the pattern formation process. We conjecture that the underlying mechanism is the fracture under tension of the viscoplastic material, exacerbated by the ambient water