Structural Geology

Structural geology is the study of rock deformation and the forces that create geological structures such as folds, faults, and fractures. Understanding these structures is crucial for interpreting Earth's tectonic history, locating natural resources, and assessing geological hazards.

Stress and Strain

Stress Definitions

Stress is the force applied per unit area:

Where $\sigma$ is stress, F is force, and A is area.

Types of Stress

• Compressional stress: Squeezing forces, $\sigma_1 > \sigma_2 > \sigma_3$
• Tensional stress: Pulling forces, $\sigma_1 > \sigma_2 > \sigma_3$ (but $\sigma_3$ is smallest)
• Shear stress: Forces acting parallel to a surface

Stress States

• Uniaxial stress: Stress in one direction only
• Biaxial stress: Stress in two directions
• Triaxial stress: Stress in three directions

Strain Analysis

Strain is the deformation in response to stress:

Where $\epsilon$ is strain, $\Delta L$ is change in length, and $L_0$ is original length.

Types of Strain

• Elastic strain: Reversible deformation (rocks return to original shape)
• Plastic strain: Permanent deformation (rocks do not return to original shape)
• Brittle strain: Fracture and faulting

Rock Deformation Behavior

Factors Controlling Deformation

Physical Conditions

1. Temperature: Higher temperature promotes ductile behavior

   • $T > \frac{2}{3}T_{\text{melting}}$: Typically ductile
   • $T < \frac{1}{3}T_{\text{melting}}$: Typically brittle

2. Pressure: Higher confining pressure strengthens rocks

   • Prevents microcrack propagation
   • Promotes ductile behavior

3. Strain rate: How fast deformation occurs

   • Fast rates favor brittle behavior
   • Slow rates favor ductile behavior

4. Fluid pressure: Pore fluids affect effective stress

   • Reduces effective normal stress on fractures
   • Can promote brittle failure

Rock Properties

• Composition: Micas and clays promote ductile behavior
• Grain size: Smaller grains often promote ductile behavior
• Layering: Competent vs. incompetent layers behave differently
• Pre-existing fabrics: Previous structures influence deformation

Folds

Fold Classification

Descriptive Elements

• Hinge zone: Region of maximum curvature
• Limb: Relatively flat portions of folded surface
• Axial surface: Surface dividing fold as symmetrically as possible
• Fold axis: Line of maximum curvature (theoretical)

Geometric Classification

Based on interlimb angle:

• Tight folds: 0° < interlimb angle < 10°
• Isoclinal folds: 0° < interlimb angle < 5° (limbs parallel)
• Open folds: 70° < interlimb angle < 120°
• Gentle folds: interlimb angle > 120°

Based on axial surface orientation:

• Upright folds: Axial surface vertical (90°)
• Inclined folds: Axial surface inclined
• Recumbent folds: Axial surface near horizontal
• Overturned folds: One limb overturned

Fold Mechanisms

Buckling

Most common folding mechanism for competent layers:

Where $\lambda$ is wavelength, $t$ is layer thickness, $E$ is elastic modulus, and $\nu$ is Poisson's ratio.

Passive Folding

Deformation by homogeneous strain without buckling forces.

Flexural Slip

Layer-parallel slip between beds during folding.

Flow Folds

Formed by plastic flow without layering constraints.

Faults

Fault Classification

Based on Motion

1. Dip-slip faults: Vertical displacement parallel to dip

   • Normal faults: Hanging wall moves down (extensional)
   • Reverse faults: Hanging wall moves up (compressional)
   • Thrust faults: Low-angle reverse faults

2. Strike-slip faults: Horizontal displacement parallel to strike

   • Right-lateral: Opposite block moves right
   • Left-lateral: Opposite block moves left

3. Oblique-slip faults: Combination of dip-slip and strike-slip

Recognition Criteria

• Slickensides: Polished, striated fault surfaces
• Fault breccia: Angular fragments in fault zone
• Fault gouge: Fine-grained, ground-up rock
• Drag folds: Folding along fault contact
• Fault scarps: Topographic expression of fault

Andersonian Fault Theory

Based on the orientation of principal stress axes:

Normal Faults

• $\sigma_1$ = vertical, $\sigma_2$ and $\sigma_3$ horizontal
• Extensional stress regime

Reverse/Thrust Faults

• $\sigma_3$ = vertical, $\sigma_1$ and $\sigma_2$ horizontal
• Compressional stress regime

Strike-Slip Faults

• $\sigma_2$ = vertical, $\sigma_1$ and $\sigma_3$ horizontal
• Transpressional or transtensional stress regime

Joints and Fractures

Joint Classification

Systematic Joints

• Regular, parallel sets with consistent spacing
• Often conjugate (paired) systems at ~60°

Desiccation Joints

• Form during volume reduction (drying)
• Polygonal patterns (mud cracks)

Columnar Joints

• Form during cooling and contraction
• Hexagonal columns (basalt flows)

Fracture Mechanics

Where $K_I$ is the stress intensity factor, $\sigma$ is applied stress, and $a$ is crack length.

Brittle-Ductile Transition

Depth-Dependent Behavior

With increasing depth:

• Temperature increases
• Confining pressure increases
• Deformation transitions from brittle to ductile

Typical transition occurs at:

• Continental crust: 10-15 km depth
• Oceanic crust: 5-8 km depth

Strength Envelope

Where $\tau$ is shear stress at failure, $C$ is cohesion, $\sigma_n$ is normal stress, and $\phi$ is internal friction angle.

Structural Analysis Techniques

Stereographic Projection

Used to analyze orientations of planes and lines:

Where $\theta$ is angular distance, $\delta$ is latitude, and $\lambda$ is longitude.

Strain Analysis

Strain Ellipse

Where $\epsilon_1$ and $\epsilon_3$ are principal strains.

Vorticity

Measures rotation during deformation:

Where $\Omega$ is rotation rate and $D$ is deformation rate.

Tectonic Settings and Structural Styles

Extensional Settings

Continental Rifting

• Half-grabens: Asymmetrical down-dropped blocks
• Detachment faults: Low-angle normal faults
• Metamorphic core complexes: Exhumed deep crust

Oceanic Rifting

• Normal faults: Create seafloor topography
• Oceanic core complexes: Large-offset detachment systems

Compressional Settings

Fold and Thrust Belts

• Thrust sheets: Laterally transported rock units
• Triangle zones: Tapered thrust slices
• Duplex structures: Multiple stacked thrust sheets

Orogenic Systems

• Nappes: Large-scale thrust sheets
• Klippes: Isolated window remnants of nappes

Transcurrent Settings

Strike-Slip Systems

• Pull-apart basins: Extensional releasing bends
• Transpressional ridges: Compressional restraining bends
• Flower structures: Characteristic fault patterns

Structural Geology Applications

Resource Exploration

Hydrocarbons

• Traps: Anticlines, fault closures
• Migration pathways: Fracture networks
• Seals: Shale layers, fault zones

Metal Deposits

• Structural controls: Fractures, shear zones
• Hydrothermal pathways: Dilational sites on faults
• Host rocks: Preferred lithologies for mineralization

Engineering Considerations

Rock Mass Properties

• Joint spacing: Affects rock mass strength
• Orientation: Controls failure mechanisms
• Persistence: Controls discontinuity effects

Slope Stability

• Planar failure: Along weak layers
• Wedge failure: Along intersecting discontinuities
• Toppling failure: Columnar structures

Real-World Application: Seismic Hazard Assessment

Structural geology is critical for understanding earthquake hazards and the deformation history of fault systems.

Fault Analysis Example

# Seismic fault analysis for hazard assessment
fault_data = {
    'dip_angle': 60,    # degrees
    'slip_rate': 8,     # mm/year (long-term slip rate)
    'rupture_length': 35,  # km (typical rupture length for M7.0)
    'maximum_displacement': 4.0,  # meters (for M7.0 earthquake)
    'recurrence_interval': 300,  # years (average time between large events)
    'last_major_event': 1857    # year of last large earthquake
}

# Calculate time since last major event
current_year = 2025
time_since_last_event = current_year - fault_data['last_major_event']
strain_accumulated = fault_data['slip_rate'] * time_since_last_event  # mm

# Calculate probability of next large event
probability_in_50_years = 1 - math.exp(-50 / fault_data['recurrence_interval'])

print(f"Fault dip angle: {fault_data['dip_angle']}°")
print(f"Slip rate: {fault_data['slip_rate']} mm/year")
print(f"Time since last major event: {time_since_last_event} years")
print(f"Strain accumulated: {strain_accumulated:.1f} mm")
print(f"Recurrence interval: {fault_data['recurrence_interval']} years")
print(f"Probability of large event in next 50 years: {probability_in_50_years:.2f}")

# Estimate moment magnitude from fault dimensions
# Wells & Coppersmith (1994) scaling relationship
fault_area = fault_data['rupture_length'] * (fault_data['rupture_length'] * math.sin(math.radians(fault_data['dip_angle'])))  # km²
estimated_magnitude = math.log10(fault_area) + 3.98  # Mw

print(f"Estimated earthquake magnitude: {estimated_magnitude:.1f}")
print("This analysis helps engineers design structures to withstand fault motion")

Stress Field Analysis

Understanding the regional stress field helps predict potential fault movements and orientations.

Your Challenge: Fault Analysis and Restoration

Analyze a geological cross-section with multiple faults and perform a structural restoration.

Goal: Determine fault types, calculate displacement, and restore the original undeformed state.

Geological Data

import math

# Cross-section data for structural analysis
stratigraphic_units = {
    'formation_A': {'thickness': 100, 'original_position': 0},  # meters
    'formation_B': {'thickness': 150, 'original_position': 100}, # meters
    'formation_C': {'thickness': 200, 'original_position': 250}, # meters
    'formation_D': {'thickness': 120, 'original_position': 450}  # meters
}

# Fault data
faults = {
    'main_fault': {
        'dip_direction': 'east',  # direction of dip
        'dip_angle': 45,         # degrees
        'hanging_wall_offset': 300,  # meters offset
        'type': 'normal'          # normal, reverse, or strike-slip
    },
    'synthetic_fault': {
        'dip_direction': 'west',
        'dip_angle': 60,
        'hanging_wall_offset': 80,
        'type': 'normal'
    }
}

# Calculate fault displacement components
main_fault = faults['main_fault']
vertical_displacement = main_fault['hanging_wall_offset'] * math.sin(math.radians(main_fault['dip_angle']))  # meters
horizontal_displacement = main_fault['hanging_wall_offset'] * math.cos(math.radians(main_fault['dip_angle']))  # meters

# Calculate extension (for normal fault)
original_crustal_width = 10000  # meters (before faulting)
extension_ratio = (horizontal_displacement / original_crustal_width) * 100  # percent extension

# Depth of fault origin (for normal fault)
depth_origin = vertical_displacement / math.tan(math.radians(main_fault['dip_angle']))  # meters

Analyze the fault system and restore the cross-section to its pre-faulted state.

Hint:

• Use the fault displacement to back-restore the section
• Consider the fault type and its kinematic implications
• Calculate the amount of displacement and strain
• Estimate the original thickness of units if they were thinned by extension

# TODO: Calculate fault parameters
net_vertical_displacement = 0  # meters (total vertical offset)
net_horizontal_displacement = 0  # meters (total horizontal offset)
extension_ratio = 0  # percent (amount of crustal extension)
original_strat_thickness = 0  # meters (estimated original thickness of sequence)
fault_type = ""  # Normal, reverse, or strike-slip based on displacement
deformation_style = ""  # Extensional, compressional, or transcurrent

# Calculate restoration parameters
restoration_amount = 0  # meters (amount to restore by)
original_crustal_width = 10000  # meters (before deformation)

# Print results
print(f"Net vertical displacement: {net_vertical_displacement:.1f} m")
print(f"Net horizontal displacement: {net_horizontal_displacement:.1f} m")
print(f"Extension ratio: {extension_ratio:.2f}%")
print(f"Deformation style: {deformation_style}")
print(f"Original stratigraphic thickness: {original_strat_thickness:.1f} m")
print(f"Fault type: {fault_type}")

# Assess seismic hazard based on fault parameters
if net_vertical_displacement > 200:
    seismic_hazard = "High - large displacement fault"
elif net_vertical_displacement > 50:
    seismic_hazard = "Moderate - moderate displacement fault"
else:
    seismic_hazard = "Low - small displacement fault"
    
print(f"Seismic hazard assessment: {seismic_hazard}")

How would the cross-section differ if this were a compressional (reverse/thrust) system instead of an extensional (normal) system?