Scale a Geological Diffusion Model

5 questions0m 0s

Prerequisites

Calculus, second derivatives, dimensional analysis, and physical units.

Learn the skill

For u_t = D u_xx with constant positive D, set x = L xi and t = (L^2/D) tau. The equation becomes u_tau = u_xixi. The natural timescale L^2/D is not automatically the exact time to reach a specified tolerance.

Worked example

Tripling L at fixed D multiplies the characteristic time by nine. Boundary conditions and initial profiles still determine the actual solution and any numerical prefactor.

Model and Assumptions

Use a simplified one-dimensional diffusion model for a tracer in a rock. D has units length^2/time. Let L = 4 and D = 2 in compatible units. Assume constant D, no advection, and stated boundary conditions.

1. What is the characteristic time L^2/D? Answer

2. By what factor does the timescale change if L doubles? Answer

3. Which expression is dimensionless time? Dt/L^2 DLt tD/L

4. What is missing from a timescale alone? The definition of squaring All information about units Boundary and initial conditions

5. If advection becomes important, what should change? Nothing if the curve looks smooth The governing model and its dimensionless groups Only the name of the time unit

Further inquiry

Carry out both derivative substitutions explicitly. Add a constant advection speed v and derive the dimensionless equation. Identify the group vL/D and discuss when ignoring it is unjustified.

Review criteria

Show the chain-rule factors for time and space.
State a consistent sign convention for advection.
Connect the dimensionless group to competing transport processes.

Research notes (not saved or graded)