# 2.4: Summary

- Page ID
- 3529

Let's review what we've learned about diffusion and Darcy's law so far.

## The Diffusion Equation

Diffusion is how a substance moves from a region of high concentration to a region of low concentration. It can be described using the diffusion equation

\[\frac{dT}{dt}=\kappa\frac{d^2T}{dz^2}\]

To solve this PDE so that we can apply it to real problems in geology, we take several steps.

- We first determine what is the initial condition for the variable we are given?
- What are the boundary conditions for the variable?
- What is η in the error function equation?
- Use the initial conditions and boundary conditions to determine the constants of integration in the general solution

## Geological Applications of the Diffusion Equation and Darcy's Law

We applied these PDE solving skills to several problems, the Cooling of a Lithospheric Plate, Fault Scarp Erosion, and Diffusion of a Chemical Species.

We also looked at Darcy's law and how it applies to flow in closed spaces such as pipes and groundwater. Darcy's law can be written as

\[q=-K\frac{dh}{dx}\;\;\; or\;\;\; q=\frac{-k}{\mu}\frac{dp}{dx}\]

where q is Darcy flux, \(k\) is the permeability in m^{2},^{ }K is the hydraulic conductivity, \(\mu\) is the fluid viscosity in Pa·s, and \(\frac{dh}{dx}\) is the hydraulic gradient. It describes how a fluid flows in a porous medium.

On a smaller scale, fluid flow can either be laminar or turbulent. This depends on the fluid's Reynold's number (Re).

\[Re=\frac{\rho u_oL}{\mu}\;\;\;Reynold's\;number\]

where \(u_o\) is flow velocity and L is fluid depth.

To have Darcy flow, the Re<1-10 (laminar) or Re>1-10 (non-linear/not-laminar). An Re>1-10 may not be laminar flow, but it is not turbulent either. To have turbulent flow, the Re>2000.