Sedimentary Processes

4.1 Stokes Settling Velocity

When a particle falls through a fluid, three forces act on it: gravity, buoyancy, and viscous drag. At terminal velocity, these forces balance.

Derivation of Stokes' Law

For a spherical particle of radius $r$ and density $\rho_s$ settling through a fluid of density $\rho_f$ and dynamic viscosity $\eta$, the forces are:

At terminal velocity, $F_{\text{gravity}} = F_{\text{buoyancy}} + F_{\text{drag}}$:

Solving for the settling velocity:

Key features of Stokes' law:

Reynolds Number and Turbulence

Stokes' law is valid only in the laminar regime. The particle Reynolds number is:

Stokes' law applies for $\text{Re}_p < 0.5$. For quartz grains ($\rho_s = 2650$ kg/m$^3$) settling in water ($\rho_f = 1000$ kg/m$^3$, $\eta = 10^{-3}$ Pa·s), this limit corresponds to grains smaller than about 0.1 mm (fine sand).

For larger particles ($\text{Re}_p > 0.5$), the drag coefficient must be modified. In the intermediate regime ($0.5 < \text{Re} < 1000$):

For very large particles ($\text{Re} > 1000$), the drag coefficient becomes nearly constant ($C_D \approx 0.44$), giving the impact law:

4.2 The Hjulstrom Diagram

Filip Hjulstrom (1935) empirically determined the relationship between flow velocity and grain size for erosion, transport, and deposition in rivers. The diagram defines three fields:

Critical Erosion Velocity

For coarse grains (sand and gravel, $d > 0.5$ mm), the erosion velocity increases approximately as:

This follows from the balance between fluid shear stress and particle weight. However, for fine grains (silt and clay, $d < 0.06$ mm), the erosion velocity increases as grain size decreases. This counterintuitive result arises because fine particles are cohesive — electrostatic and Van der Waals forces bind clay minerals together:

The deposition velocity is much simpler — it depends only on settling velocity and thus decreases monotonically with decreasing grain size:

The gap between erosion and deposition velocities is narrow for coarse grains but wide for fine grains. This explains why clay, once eroded, can be transported great distances before settling — it requires high velocity to erode but very low velocity to deposit.

4.3 Sediment Transport

Shields Criterion for Bedload Initiation

Albert Shields (1936) defined a dimensionless shear stress for the onset of sediment motion. The bed shear stress is:

where $h$ is flow depth and $S$ is the slope. The Shields parameter normalizes this by the submerged weight of a grain:

The critical Shields parameter $\theta_c$ for initiation of motion is a function of the particle Reynolds number $\text{Re}_* = u_* d / \nu$, where $u_* = \sqrt{\tau_b / \rho_f}$ is the shear velocity and $\nu$ is kinematic viscosity. Empirically:

For most natural sediments, $\theta_c \approx 0.045 - 0.06$ provides a reasonable estimate. The critical shear stress for quartz sand ($d = 1$ mm) in water is then:

Suspended Load: The Rouse Profile

Suspended sediment concentration varies with height above the bed according to the Rouse profile. The steady-state balance between upward turbulent diffusion and downward settling gives:

where $w_s$ is settling velocity, $C$ is sediment concentration, and $K_s$ is the sediment diffusivity. Using the parabolic eddy viscosity model $K_s = \kappa u_* z(1 - z/h)$where $\kappa = 0.41$ is von Karman's constant:

Integrating from reference height $a$ to height $z$:

where $P = w_s / (\kappa u_*)$ is the Rouse number. The Rouse number controls the vertical distribution:

4.4 Stratigraphic Principles

Walther's Law of Facies

Johannes Walther (1894) recognized that the vertical succession of facies in a conformable sequence reflects the lateral juxtaposition of depositional environments. Formally:

This means that a transgressive sequence (rising sea level) produces a vertical succession of: fluvial → deltaic → shallow marine → deep marine facies, mirroring the horizontal zonation of environments. Only facies that are laterally adjacent can appear in vertical contact (in a conformable sequence).

Sequence Stratigraphy

Sequence stratigraphy provides a framework for understanding sedimentary packages in terms of relative sea-level change. A depositional sequence is bounded by unconformities and their correlative conformities.

The key concept is accommodation space — the volume available for sediment accumulation, controlled by:

where $A$ is the rate of accommodation creation. A complete sequence contains:

Sediment Compaction

As sediment is buried, porosity decreases exponentially with depth:

where $\phi_0$ is the surface porosity and $\lambda$ is the compaction length scale. Typical values: shale ($\phi_0 = 0.63$, $\lambda = 2.0$ km), sandstone ($\phi_0 = 0.49$, $\lambda = 3.7$ km). The decompacted thickness of a layer originally deposited with thickness $h_0$ at depth $z$ is:

4.5 Diagenesis and Sedimentary Basins

Mechanical Compaction

As sediment is buried, the weight of the overburden compresses the grain framework. The effective stress on the grain framework is:

where $\sigma_{\text{total}} = \rho_{\text{bulk}} g z$ is the lithostatic stress and $P_f$is the pore fluid pressure. Under normal compaction, $P_f = \rho_w g z$ (hydrostatic), giving:

Overpressure occurs when fluid cannot escape fast enough ($P_f > \rho_w g z$), reducing the effective stress and retarding compaction. This is common in rapidly deposited shales and is critical for petroleum geology. The overpressure ratio is:

When $\lambda \to 1$, the effective stress approaches zero and hydraulic fracturing can occur.

Chemical Diagenesis

Chemical changes during burial include cementation, dissolution, replacement, and authigenic mineral growth. The solubility of quartz increases with temperature:

where $C$ is in ppm and $T$ in Kelvin. This drives silica redistribution: quartz dissolves at grain contacts (pressure solution) where stress concentrations are highest, and reprecipitates as overgrowths in adjacent pore space (Ostwald ripening at the grain scale).

Basin Subsidence

Sedimentary basins form by tectonic subsidence mechanisms. For a rifted basin, the McKenzie (1978) uniform stretching model gives the initial (fault-related) subsidence:

where $\beta$ is the stretching factor (ratio of extended to original crustal width). The subsequent thermal subsidence decays exponentially:

where $\tau = a^2/(\pi^2 \kappa) \approx 62.8$ Ma is the thermal time constant.

4.6 Carbonate Sedimentology

Carbonate sediments are produced primarily by biological processes and are fundamentally different from siliciclastic sediments. The key reaction governing carbonate precipitation and dissolution is:

The solubility product is:

The saturation state $\Omega$ determines whether precipitation ($\Omega > 1$) or dissolution ($\Omega < 1$) occurs:

Carbonate Compensation Depth (CCD)

Calcite solubility increases with pressure (depth) and decreasing temperature. The lysocline is the depth where dissolution begins to be significant, and the Carbonate Compensation Depth (CCD) is where the rate of supply equals the rate of dissolution — below this, no carbonate accumulates. The CCD is typically at ~4500 m in the modern ocean but varies with latitude, productivity, and CO$_2$ levels.

The pressure dependence of the calcite solubility product follows:

where $\Delta V_r$ is the molar volume change of the dissolution reaction and$\Delta \kappa_r$ is the compressibility change. Both favor increased dissolution at depth.

Computational Simulations