Question

A sedimentary basin has a thickness of \(7 \mathrm{~km}\). Assuming that the crustal stretching model is applicable and that \(h_{c c}=35 \mathrm{~km}, \rho_{m}=3300 \mathrm{~kg} \mathrm{~m}^{-3}\), \(\rho_{c c}=2700 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(\rho_{s}=2450 \mathrm{~kg} \mathrm{~m}^{3},\) determine the stretching factor.

Short answer

The stretching factor is approximately 1.82.

Step-by-step solution

Understanding the Crustal Stretching Model

In the crustal stretching model, we assume that the thinning of the lithosphere by stretching leads to a passive subsidence of the surface. The stretching factor, denoted as \( \beta \), is a measure of how much the crust has been stretched. It is calculated using the initial and final crustal thicknesses.

Applying Archimedean Principle to Compute Subsidence

The sediment thickness \( h_s = 7 \text{ km} \) is supported by the subsidence of the crust. From the isostatic balance, the addition of sediment to the crust causes the crust to subside more until buoyancy is achieved. We can model this using the density of the sediments (\( \rho_s \)), the density of the original crust (\( \rho_{cc} \)), and the density of the mantle (\( \rho_m \)).

Set Up and Solve for the Stretching Factor

Using the given densities and sediment thickness, we use the formula for isostatic equilibrium:\[ \beta = 1 + \frac{\rho_s}{\rho_m - \rho_{cc}} \times \frac{h_s}{h_{cc}} \]Substitute the known values into the equation:\[ \beta = 1 + \frac{2450}{3300 - 2700} \times \frac{7}{35} \]\[ \beta = 1 + \frac{2450}{600} \times \frac{7}{35} \]

Calculating the Stretching Factor

Simplify the expression step by step:\[ \frac{2450}{600} = 4.083 \]\[ \frac{7}{35} = 0.2 \]Therefore,\[ \beta = 1 + 4.083 \times 0.2 = 1 + 0.8166 \]\[ \beta = 1.8166 \]

Conclusion

The calculated stretching factor, \(\beta\), indicates the crust has been stretched by a factor of 1.8166, reflecting the level of thinning the crust has undergone to accommodate the sediment.

Key concepts

Sedimentary Basin Subsidence

Sedimentary basin subsidence occurs when the Earth's crust sinks under the weight of accumulating sediments. This process is essential in understanding how layers of rock and sediment form structures like valleys and oceans over geological time.
Subsidence can happen due to various factors, but the driving process here is the weight of the deposited sediments causing the lithosphere to sink.

As more sediment piles up, the crust bends and sinks more to maintain isostatic equilibrium, which we will discuss in the next section. This gradually increases the thickness of sedimentary layers over time, leading to formations that can be mined for resources or studied for Earth's history insights.

Subsidence is crucial in the crustal stretching model, as it affects the crust's ability to accommodate new sediments without causing crumbling or fracturing of the crustal layers.

Isostatic Equilibrium

Isostatic equilibrium is the state where the Earth's crust is floating in gravitational balance on the more fluid mantle beneath, much like an iceberg floats on water.
When weight, such as sediment, is added to the crust, it will sink to maintain this equilibrium state.

The concept of isostasy is integral to geology, as it explains why regions of the Earth's surface are stable despite varying geological pressures and loads.

In the context of sedimentary basins, as new sediments accumulate and increase the weight of the crust, isostatic adjustments lead to further subsidence until the weight is balanced by the buoyancy force due to the mantle below.

• Archimedean principle helps determine how much the crust will sink.
• Isostatic equilibrium ensures that the Earth's surface does not collapse but rather adjusts gently over time.


This mechanism helps explain why certain regions are geologically more resilient and why we don’t experience catastrophic collapses more frequently.

Lithosphere Thinning

Lithosphere thinning refers to the process where the solid outer layer of the Earth, known as the lithosphere, becomes thinner due to stretching forces.
This often occurs at tectonic plate boundaries or through processes like the formation of rift valleys.

As the lithosphere stretches, it creates gaps that can be filled by magma upwelling from below or by sediment accumulation from above, as seen in sedimentary basins.

These thinning processes are vital to understanding how sedimentary basins are formed, as they allow the crust to stretch and accommodate thick layers of sediment without immediate folding or fracturing.

• Thinning alters the landscape by creating depressions ideal for sediment deposit.
• It facilitates the subsidence required to form deep sedimentary basins.
• Geologists use the stretching factor \( \beta \) to quantify these changes.

This knowledge helps geologists predict resource locations and explain seismic activities induced by tectonic movements.

Density of Sediments

The density of sediments plays a crucial role in sedimentary basin dynamics.
It determines how much weight the sediments add to the crust and how the crust will respond in terms of subsidence.

Sediment density varies depending on the type of material, with denser materials contributing more significantly to crustal subsidence.
For example, heavier minerals like iron-rich rocks will have higher densities compared to lighter sediments such as sand or clay, leading to more pronounced subsidence.

• The density affects both the buoyancy forces acting on the crust and the pressure transmitted through the sediment layers.
• Understanding sediment density helps predict how quickly a basin might fill with sediment and potentially evolve over time.

By utilizing densities like \( \rho_{s} = 2450 \,\frac{\text{kg}}{\text{m}^3} \), geologists can apply complex models to simulate basin development, aiding in exploration and disaster preparedness.