Tectonic and gravity extensional collapses in over-pressured cohesive and frictional wedges2

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Geology of extensional collapses

Natural examples of gravity and tectonic extensional modes

In this manuscript, we will study two kinds of extensional collapse modes above a weak, fric-tional detachment in the upper crust. One extensional collapse is gravity-driven failure due to an inclined topography above a weak detachment. This inclination can come from excessive sedimentation as seen from regional gravity shale tectonics system in SW Niger Delta [Damuth, 1994; Corredor et al., 2005; Mourgues et al., 2009], Figure 1.1a. Deformation in the overpres-sured delta results from gravitational thin-skinned instabilities that formed various structural zones in this region. The gravitational collapse where the slip on the detachment is toward the wedge tip is the first mechanism to be studied in the context of fluid-saturated wedges, Figure 1.1a, b. This collapse mode is also found in small-scale landslides such as the Storegga slide in Norway [Kvalstad et al., 2005].
The second extensional collapse mode is the slip of wedge on a low-angle detachment fault driven by lithospheric stresses (or a regional extensional tectonic event) which is opposite to the wedge tip, Figure 1.1a, c. This mode is referred to as a tectonic extensional collapse for many crustal-scale problems including the late phase deformation in the northern part of the Basin and Range province, Nevada and Utah [Anderson et al., 1983], and the extensional province above a gently seaward dipped detachment in NE Niger Delta, Figure 1.1a and 1.1c. Sorel [2000] proposed four steps of fault sequences to illustrate the evolution process of the Corinthe rift system on a weak detachment based on seismic section of Rigo et al. [1996] (Figure 1.2). More impressively, the structural style in the Albuquerque Basin, New Mexico, also presents various deformation patterns resulting from normal faulting and sedimentation above a low-angle detachment due to the Rio Grande Rift [Russell and Snelson, 1994], as seen from cross-sections in Figure 1.3. These observations call for new theoretical developments to capture the failure onset and evolution process resulting from normal faulting and sedimentation through time.
The gravity and tectonic extensional collapses in upper crust are also found in hyper-extended, magma-poor rifted margins [Nirrengarten et al., 2016]. The wedges in the upper and lower plate margins correspond respectively to the hanging wall and footwall of the de-tachment system, as shown in the cross-section of Porcupine basin (Figure 1.4a) and the same cross-section corrected for post-rift sediment loading and thermal subsidence (Figure 1.4b).
The upper plate wedge corresponds to a tectonic extensional wedge, the one in the lower plate matches that of a gravity extensional wedge, as illustrated in Figure 1.4c.

Kinematic model in extension

The kinematic model in extension is referred to as an half-graben. To capture the evolution process of normal faulting, Groshong [1989] developed the basic balanced geometrical model for extensional faulting and related bending of the half-graben model, as shown in Figure 1.5. For example, this kinematic model implies the deformation of gravity and tectonic extensional collapses in offshore Niger Delta (Figure 1.1a) and the Rio Grande Rift, New Mexico (Figure 1.3), respectively. In this half-graben kinematics of Figure 1.5, the footwall does not deform or rotate during deformation. It is based on the geometry of a planar normal fault that joins a planar detachment at depth (Figure 1.5a). The hanging wall slides down along the normal fault and is sheared through the conjugated axial surface. As a result of extension, an half-graben develops, bounded on one side by the normal fault and on the other by the active axial surface, causing the total structure to be asymmetric. As displacement increases on the normal fault (Figure 1.5b and 1.5c), the flat bottom of the half-graben gradually disappears as it drops down and shifts laterally into the domain of antithetic dip. The antithetic dip domain is bounded by two parallel axial surfaces that dip at an angle equal and opposite to that of the normal fault.
The half-graben kinematics in Groshong [1989] has considered so far the models involve only preexisting or pregrowth strata. To account for more natural field examples (e.g. Gulf Coast rollovers), Xiao and Suppe [1992] updated the half-graben kinematics to account for the bed deposition during deformation, that is, growth strata. Figure 1.6 shows the effect of syntectonic sedimentation on the rollover geometry for a constant fault shape [Xiao and Suppe, 1992]. The new features are the deposited beds passing through the active axial surface. The surface that connects the active and inactive axial surfaces is called the growth axial surface in their model. This growth axial surface links the top points of active and inactive axial surfaces, Figure 1.6, and records at each bed the location of the active axial surface on the sea bottom at the time of deposition.
For the structural evolution in extension, the kinematics of half-graben proposed by Groshong [1989] and Xiao and Suppe [1992] resulting from normal faulting and the sedimentation is elegant, but is not mechanically proven. For example, the dips of the normal fault and active axial surface are not constrained. Additionally, the footwall slope in Figure 1.5 might be not sustainable due to the depression of topography during extension. Our mechanical method used in this manuscript requires firstly to choose a failure kinematics, that is the half-graben geometrical model of Groshong [1989] in Figure 1.5. The parameters (dips of the normal fault and the axial surface and length of active detachment) will be determined by mechanical optimization.

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Analogue modeling

To study the extensional deformations, wet clay is a widely used analogue material in experi-ments [Dula, 1991; Withjack et al., 1995; Bose and Mitra, 2009] because of its strong cohesion which easily allows to conduct the extensional tests. The property of volume change and creep of wet clay is often captured by the Cam-Clay model [Roscoe et al., 1958; Roscoe and Burland , 1968] which is distinct from Mohr-Coulomb criterion for dry sand materials. Thus, the use of wet clay in experiments to characterize deformation in the brittle crust is controversial. With jack and Schlische [2006] reported a significant difference of extensional deformation between the materials of dry sand and of wet clay; sand analogue characterises a planar fault whereas clay materials in extension generates listric fault. Additionally, in the clay experiments, the ma-terials sit on a metal plate which is forced to slip as a detachment fault, thus it is questionable whether these simulations can capture the sequences of normal faulting in natural field exam-ples of upper crust. In contrast, the sand analogue experiments have been proposed to study the tectonic extensional collapse at the subduction scale [Xiao et al., 1991] which captures the typical features of migration of half-graben geometries during extension. Thus, the proposed mechanical methodology will use Mohr-Coulomb model which is more suitable to characterise brittle extensional failure of upper crust of interest in this thesis.

Table of contents :

1 General Introduction 
1.1 Geology of extensional collapses
1.1.1 Natural examples of gravity and tectonic extensional modes
1.1.2 Kinematic model in extension
1.1.3 Analogue modeling
1.2 Mechanics of extensional collapses
1.2.1 Fluid Pressure
1.2.2 The Critical Coulomb Wedge (CCW) theory
1.2.3 Numerical modeling
1.3 Methodology used here : Limit Analysis and Sequential Limit Analysis
1.4 Manuscript content
2 Tectonic and gravity extensional collapses in over-pressured cohesive and frictional wedges2 
2.1 Introduction
2.2 Limit analysis for extension
2.2.1 Prototype and collapse mechanisms
2.2.2 Theorem of effective virtual powers
2.2.3 Maximum strength theorem (MST)
2.3 Gravitational collapse
2.3.1 General stability conditions based on the MST
2.3.2 Comparison with CCW theory
2.3.3 The role of cohesion in a triangular wedge
2.3.4 Experimental validation
2.4 Tectonic extensional collapse
2.4.1 Mechanism (1): decollement fully activated
2.4.2 Mechanism (2): a normal fault rooting at the back-wall
2.4.3 Mechanism (3): a normal fault and a shear plane rooting on the decollement
2.4.4 Comparison with CCW theory for extensional collapse
2.4.5 Experimental validation
2.5 Application to North Chile
2.6 Conclusion
3 Deformation pattern during normal faulting: a sequential limit analysis 
3.1 Introduction
3.2 Sequential Limit Analysis of a Homogenous Wedge under Extension
3.3 Simulation Results and Interpretation by the CCW Theory
3.3.1 Numerical Results for the Case of a Dry Wedge
3.3.2 Internal Deformation Pattern for the Case of a Dry Wedge
3.3.3 Numerical Results for the Case of an Overpressured Wedge
3.4 Roles of Fault Softening and of Sedimentation
3.4.1 Reference Simulations
3.4.2 Fault Softening
3.4.3 Sedimentation and No Fault Softening
3.4.4 Combined Effects of Sedimentation and Fault Softening
3.5 Application to Jeanne d’Arc Basin, Grand Banks, Newfoundland
3.6 Conclusions
4 Reappraisal of gravity instability conditions for offshore wedges: consequences for overpressures in the Niger Delta 
4.1 Introduction
4.2 Gravity instabilities with Limit Analysis
4.2.1 General prototype
4.2.2 Application of Limit Analysis
4.3 Validation for an inclined layer
4.3.1 Comparison with other analytical results
4.3.2 Validation with sandbox experiments in fluid overpressured conditions
4.4 Application to the offshore Niger Delta
4.4.1 Stability conditions
4.4.2 Stability Analysis of Niger Delta
4.5 Concluding discussions
5 Role of fluid overpressures on the shape of normal faults in brittle, upper crust 
5.1 Introduction
5.2 The prototype
5.2.1 The velocity field
5.2.2 The bounding of the tectonic force
5.3 Convergence analysis and validation with slip-line theory
5.3.1 Convergence analysis
5.3.2 Validation with the slip-line theory
5.4 Results of normal faulting
5.4.1 Influence of geometries of the prototype
5.4.2 Influence of fluid-retention depth ZFRD and material cohesion
5.5 Applications to sedimentary upper crust
5.5.1 Faulting in NW Gulf of Mexico
5.5.2 Faulting in offshore Niger Delta
5.6 Concluding discussions
6 Conclusions 
6.1 Limit Analysis
6.1.1 Gravity and tectonic extensional collapses
6.1.2 Gravity instability with a resistive toe
6.1.3 Formation of low-angle and listric normal fault
6.2 Sequential Limit Analysis
6.3 Perspectives
A Appendix for Chapter 2
1 Different fluid pressure parametrizations
2 Exact critical Coulomb wedge theory (ECCW)
B Electronic Supplement to Chapter 2
1 Theorem of virtual powers with acceleration contribution
2 A weak expression of Archimedes theorem
3 Derivation of three upper bounds
C Electronic Supplement to Chapter 3
1 Preliminary
2 The upper bound forces for three mechanisms in Section 3.3
3 Upper bound for a normal fault piercing the cover, Sections 3.4 and 3.5
4 Kinematics of hanging-wall deformation
5 Cross-sections of Jeanne d’Arc Basin, Section 3.5
6 Some results of inverse analysis for Section 3.5
D Appendix for Chapter 4 173
1 General solution of the kinematic approach of LA
1.1 General prototype
1.2 Inclined layer
1.3 Triangular Wedge
E Electronic Supplement to Chapter 4
1 Different fluid pressure parametrizations
2 Analytical collapse length and fault dips for an inclined layer
F Appendix for Chapter 5
1 Geometric relations
2 Velocity relations and Limit Analysis

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