Lecture:Introduction to physical modeling

From GeoMod

In this discussion of physical modeling we will be dealing with computer models designed to capture physical processes by using the basic physics equations developed by physicists, engineers and geologists.

Governing Equations

To do this type of modeling you need the physical equations. Typically there is one or a small set of general equations that governing each system. Here are a few examples,

• Heat transport: Heat transport problems are common in geology.

• Examples:
  • If a magma chamber forms in the crust the zones of metamorphism surrounding the magma chamber, and the timing of metamorphism, depend on heat transport through the crust.
  • When the crust moves over a hotspot (or vice-versa) one of the effects is that the hotspot warms the crust and results in the crust expanding as its temperature rises. A certain region around the hotspot will also warm. Understanding the timing of these processes can be accomplished through numerical modeling.

• The differential equation shown here relates the change in temperature (T) over time (t) to the heat transported by fluids (first term on right side) and the diffusion of heat due to the temperature gradient over space (second term on right). This equation was first solved analytically by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822 (Wikipedia Heat_equation).

If you don't like differential equation, don't panic quite yet. We will approach these problems in a more intuitive manner.

• Groundwater flow: Most of my research has to do with groundwater flow and the transport of solutes. What you will notice is that the governing equations are remarkably similar to the temperature equation.

• Groundwater flow: This shows the change in hydraulic head (h) (which is related to water pressure) over time.

• Solute transport: This equation shows the change in concentration over time due to the advection of fluid (first term on right side) and diffusion of fluid (second term).

Erosion Model in Excel

We will look at the example of landscape erosion and derive the relavent equations and solve the problem numerically in one dimension using MS Excel.

• Media:Erosion model.xls - This is an example of a solution to the problem programmed in Visual Basic for Excel.

• What determines erosion rates

• Conservation of mass

• Assembling an equation

• Setting up the model

• Initial conditions
• Material properties
• Time

• Steady state
• Transient

• Transient solution (explicit)

• Testing the model
  • Internal consistency (eg. conservation of mass).
    • Numerical errors
  • Model calibration.
    • How would you do it?