Rock strata - 2D cross-section and drill log

What is modelled

A sedimentary basin built layer by layer. A rising and falling sea decides what is deposited where: sand near the shore, mud on the shelf, lime mud in deep water, thin coal in swamps at the water line. The floor of the basin slowly sinks, buried layers are squeezed by the weight above, high ground erodes, and volcanic ash falls now and then and can be dated. You can fold, fault, uplift or tilt the whole stack and watch new layers settle on the damaged surface.

Equations

• Sea level: h(t) = A sin(2πt/P) + 0.45 A sin(2πt/(3.1 P) + 1.3), a main cycle plus a slower one. Water depth is d = h − zsurface; the facies belts are coal (−4 < d < 0 m), sandstone (0 to 15 m), shale (15 to 80 m) and limestone (deeper than 80 m). Above sea level nothing is deposited and the land erodes. Deposition rates (120, 60, 30 and 8 m per million years at supply 1) are illustrative.
• Compaction (Athy): porosity falls with burial depth, φ(z) = φ0 e−cz. The solid thickness S = ∫(1 − φ) dz of a layer never changes, so its thickness h follows from h − (φ0/c) e−c z1 (1 − e−ch) = S, solved by Newton iteration from the top down. Sand, shale and limestone use the Sclater and Christie (1980) constants; the values for coal, ash and alluvium are illustrative.
• Erosion: hillslope diffusion ∂z/∂t = κ ∂²z/∂x², applied only where the ground is above the sea; the eroded solids are re-deposited next to it as alluvium, so the volume of solids is conserved.
• Tectonics move only the basement (kinematic): a fold is a sine-wave shift, a fault is a step, uplift and tilt are rigid; layers keep their thickness. Uplift above sea level triggers erosion and leaves an unconformity, a gap in the age record.
• Dating: ash layers carry zircon crystals that start with uranium and no lead; the daughter-to-parent ratio is D/P = eλt − 1 with λ = ln 2 / 4.468 Gyr for U-238 to Pb-206, so the age is t = ln(1 + D/P)/λ. The page adds 0.3 % random scatter to each measured ratio.

Checks run in the page

• For every layer of a simulated stack, a numerical integral of (1 − porosity) over the computed thickness equals the stored solid thickness.
• A fourth-order Runge-Kutta decay of U-238 leaves exactly half the atoms after one half-life, and the age equation recovers six test ages from 12 to 541 Ma.
• After folding, faulting, uplift and tilting, the age of the layers never decreases upward in any column (law of superposition), the layer thicknesses are unchanged, and a marker bed is offset by exactly the imposed throw.
• A cosine relief eroded by the real diffusion routine decays to the analytic value A exp(−κk²t) and the volume of solids is conserved.

Honest simplifications

• The section is one-dimensional columns side by side: there is no lateral flow of sediment except erosion between neighbouring columns, no river system, no real fault plane (the fault is a steep step across one column) and no horizontal shortening (folds are vertical shifts, not true buckle folds).
• Deposition rates, facies depths, the 0.45 second sea-level cycle, the diffusion constant and the nominal start age of 300 Ma are illustrative. Real sea-level curves, climate and sediment supply are far less regular.
• Compaction ignores cementation, overpressure and temperature; limestone is treated like a simple porosity curve.
• Coal, ash and alluvium compaction constants are guesses of the right order, not measurements. The cross-section is stretched vertically (you choose how much) so thin layers can be seen; the 3D page shows the same section as a block diagram on a field bench.

What to try

• Choose Big sea-level swings and read the sand, shale and limestone cycles on the front face.
• Press Uplift, then keep running: the old stack is eroded and new layers start on a surface with a missing interval; check the age-depth graph.
• Click an ash layer and compare the U-Pb age with the true age.