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## Milestone 4: Results and Analysis

Discussion of the system being modeled, and details of how to run the model with different initial conditions in 2 and 3D.

### Problem Description

This milestone solves a porosity pressure system which involves the coupling of a Darcy flow to describe the compressibility of the permeable solid matrix and a time dependent advection equation for the porosity field. Together these equations allow for non-linear dispersive porosity waves. Given an initial porosity distribution which is itself a porosity wave, this wave should advect at a speed determined by the amplitude and the power used to determine the permeability from the porosity (as well as the velocity of the background solid), without any diffusion. If the initial porosity distribution is not itself a solitary porosity wave, with time these should emerge from the porosity field.

### Running the Simulations

In order to run the solitary waves model, (int 2D) first:

  cd Magma/Models/Milestone4/SolitaryWaves2D


Then make a symbolic link to the executable binary as:

  ln -s ../../../../build/bin/StGermain .}}}


The simulation may then be run in parallel as:

  mpirun -np <# of procs> ./StGermain SolitaryWaves.xml


This will run the code with the default initial porosity distribution of a solitary wave with a wave speed of 7 and a porosity exponent of 3. The background solid velocity has been set in the file VelocityField.xml as -2, such that the wave should rise with a speed of 5. In order to verify that the wave is advecting at the correct speed, this may be changed to -7, which should then show the wave to be stationary.

An alternative initial porosity distribution of a vertically changing noisy 1D solitary wave may be set from the file sWaveSetup.xml by modifying the referenceSolutionFileName parameter to ./input/solitaryWaves1DGlobal.dat. This should then show a set of 2D solitary porosity waves emerging from the 1D distribution with time.

A 3D model may also be run by changing directories to:

  cd Magma/Models/Milestone4/SolitaryWaves3D


and then repeating the procedures detailed above for the 2D system. The initial condition, read in from the file ./input/solitaryWaves3DGlobal.dat, is that of a single 1D solitary wave in the vertical direction set against a noisy background distribution which evolves with time into a set of 3D solitary waves.

 2D Solitary Wave. A 2D solitary wave with a wave speed of 7 rising through a solid with a constant speed of -2. The wave shows no visible diffusive behavior. Noisy 1D Solitary Wave Initial Condition. Initial condition of a vertically changing 1D solitary wave with a certain amount of introduced noise, which allows 2D solitary waves to emerge over time. Emerging 2D Solitary Waves. Solitary waves emerging from a noisy 1D solitary wave initial condition. Emergent 2D Solitary Waves. Solitary waves having emerged from a noisy 1D solitary wave initial distribution.

Emergent 3D Solitary Waves. 3D Solitary Waves emerging from a noisy 1D Solitary Wave initial distribution