Additional Transport Calculations

In addition to coupling with diffusion or poroelasticy, NairnMPM can couple to several other transport equations. Most of these options are in development and therefore only available in OSParticulas. This information will be expanded when ported to NairnMPM.

General Transport Analysis

A generalized transport equation to be solved on the MPM grid for flow of some conserved content, [math]\displaystyle{ \tau }[/math], per unit volume can be cast as:

      [math]\displaystyle{ c_\theta {\partial \theta\over \partial t} = -\nabla \cdot \vec q(\vec x) + \dot q(\vec x) }[/math]

where [math]\displaystyle{ \theta }[/math] is transport "value," [math]\displaystyle{ c_\theta }[/math] is transport "capacity" (defining [math]\displaystyle{ \tau }[/math] per unit transport value per unit volume), [math]\displaystyle{ \vec q(\vec x) = -\kappa\nabla \theta }[/math] is flow of [math]\displaystyle{ \tau }[/math] per unit area with units Length-(units of [math]\displaystyle{ c_\theta }[/math])-(units of [math]\displaystyle{ \tau }[/math])/sec, and [math]\displaystyle{ \dot q(\vec x) }[/math] is a source term with units (units of [math]\displaystyle{ c_\theta }[/math])-(units of [math]\displaystyle{ \theta }[/math])/sec. In the flow term, [math]\displaystyle{ \kappa }[/math] is a ``conductivity (or ``diffusion) tensor with units of Length²-(units of [math]\displaystyle{ c_\theta }[/math])/sec.