Lagrangian and Eulerian coordinates
PorousFlow is formulated in Lagrangian coordinates. This page contains a description of Eulerian and Lagrangian coordinate systems and the continuity equation in each. This page is self contained and uses notation that might be different from other PorousFlow documentation.
Eulerian coordinates
Introduce the notion of the "spatial coordinate frame". It is the coordinate frame of a stationary observer who is looking at the deforming porous solid from the outside. Denote its coordinates by , and the derivatives with respect those coordinates by .
Let be a volume that is attached to particles of the porous-solid skeleton. As the porous solid deforms, so too will . Denote the velocity of the porous solid is , measured in the spatial coordinate frame: . Then the change of a small volume element is computed by calculating the Jacobian, and it is Remember the derivative is differentiating with respect to the coordinates of the spatial coordinate frame. This formula is easy to motivate for readers familiar with solid mechanics because , which is the time derivative of the volumetric strain.
Let represent a quantity that is attached to the porous-solid skeleton, for instance the mass density of the solid. Express in the spatial coordinate frame: . As the porous solid deforms The second term is easy to motivate by considering a constant velocity with spatially-dependent but temporally-constant .
The continuity equation is where is the flux of out of , is the outward unit normal, and is the area element on (which is the surface of ). Using the above expressions, and the divergence theorem, the continuity equation reads (1) Specialising and to fluids and heat gives the equations mentioned in the main text.
Lagrangian coordinates
Introduce the notion of the "material coordinate frame". It is the coordinate frame of an observer who is fixed to a certain point in the porous solid (eg, a particular finite-element node). Denote the coordinates in this frame by . This is the frame used by PorousFlow: the material coordinate frame can be considered to be the mesh. Fluid properties (pressures, mass fractions), the temperature, etc, are all stored at the finite-element nodes or the quadpoints, and move with the mesh. At the very least, an Eulerian description would be inconvenient when visualising with paraview.
Introduce the material derivative . It is the total time derivative as seen by an observer living in the Lagrangian frame. For instance, if the mesh isn't deforming, but is moving as a rigid body through space, then since the observer will see no change. If is any property that is expressed in terms of the "spatial coordinate frame" (Eulerian coordinates): (for some function ) then the material derivative is defined to be The continuity Eq. (1) can be re-written as However, if is expressed in terms of the Lagrangian coordinates: (generally will have a different functional form than , thus the tilde to emphasise the difference) then the material derivative is expressed by but the continuity equation has the identical form: