Yes this is a bit broader question but I want to know which processes play their role in movement of magma. If one wants to model magma intrusion in rocks, which processes should one not miss at all and which equations discuss these processes?. As a starter I can that we can use Navier Stokes equations for velocity fields, continuity equation for pressure distribution etc.
I recommend reviewing the equations of the following topics:
a) A spherical expansion source at depth can simulate the change in storage of a magma chamber and account for deformation caused by magma intrusion events [Mogi, 1958].
b) Finite Element Models
c) Inverse analyses
d) 3-D visco-elasto-plastic numerical models of rock intrusion emplacement incorporating in particular the temperature-sensitive properties of both magma and country rocks. Thermomechanical modelling of magma intrusion involves simultaneous and intense deformation of materials with very contrasting rheological properties:
1.- the country
2.- crustal rocks
3.- visco-elasto-plastic while the intruding magma is a low viscosity
4.-complex fluid
Reference: Burg, J.P. & Gerya, T.V. (2008)
Hi Ather,
the problem can be classified as a 2-way fluid-structure interaction (FSI) where the fluid is represented by magma and the surrounding rocks are the structure. Magma is a multicomponent fluid with each components in gaseous and/or liquid state driven by the classical Navier-Stokes equations (NaS). The chemical properties of the magma depend on the local conditions of pressure, temperature,, velocity
and phase distribution under some different assumptions ofn the mixture, for ex. linear or magmatic mixture.
From a structural point of view rock is treated as a deformable elastic isotropic material. The numerical simulation of the coupled dynamics can be based on the Arbitrary Lagrangian Eulerian (ALE) mesh framework. Lagrangian for the rocks (i.e. each node of the mesh follows the associated material particle during motion), Eulerian for the magma flow (i.e., fixed mesh) .
The motion of rock is given by the elastodynamics equation based on the main quantities: deformation gradient, Piola-Kirchhoff stress tensor and Lamè coefficients, see for ex. the classical continuum mechanics book of M.Gurtin for this notions.
The general approach to solve numerically this type of problem can be summarized as follows:
1) Define domains for magma chamber and rock then discretize them by making
meshes.
2) Set initial conditions and solve NaS for magma flow. Then compute magma stress tensor and traction. The traction became the Neumann boundary conditions on interface.
3) Solve Elastodynamics equation with this interface conditions.
4) Pass structural deformation as Dirichlet boundary conditions on interface. Solve NaS for magma fluid motion.
5) Update the mesh.
6) And so on and on until convergence is reached.
Hope this help you a little bit,
Fabrizio