update mrt model

docs/source/examples/running.rst

@@ -1,9 +1,10 @@
-============
-Running LBPM
-============
+==============
+LBPM examples 
+==============
 
 There are two main components to running LBPM simulators.
 First is understanding how to launch MPI tasks on your system,
 which depends on the particular implementation of MPI that you are using,
 as well as other details of the local configuration. The second component is
 understanding the LBPM input file structure. 
+

docs/source/userGuide/models/mrt/mrt.rst

@@ -1,6 +1,74 @@
-=============================================
+###############################################################################
 MRT model
-=============================================
+###############################################################################
+
+The LBPM single fluid model is implemented by combining a multi-relaxation time (MRT) D3Q19
+lattice Boltzmann equation (LBE) to solve for the momentum transport, recovering the Navier-Stokes
+equations to second order based on the Chapman-Enskog expansion. The MRT model is used to assess the
+permeability of digital rock images in either the Darcy or non-Darcy flow regimes. 
+
+A typical command to launch the LBPM color simulator is as follows
+
+```
+mpirun -np $NUMPROCS lbpm_permeability_simulator input.db
+```
+
+where ``$NUMPROCS`` is the number of MPI processors to be used and ``input.db`` is
+the name of the input database that provides the simulation parameters.
+Note that the specific syntax to launch MPI tasks may vary depending on your system.
+For additional details please refer to your local system documentation.
+
+***************************
+Model parameters
+***************************
+
+The essential model parameters for the color model are
+
+- ``tau`` -- control the fluid viscosity -- :math:`0.7 < \tau < 1.5`
+
+****************************
+Model Formulation
+****************************
+
+
+****************************
+Example Input File
+****************************
+
+
+
+  
+****************************
+Boundary Conditions
+****************************
+
+The following external boundary conditions are supported by ``lbpm_permeability_simulator``
+and can be set by setting the ``BC`` key values in the ``Domain`` section of the
+input file database
+
+- ``BC = 0`` -- fully periodic boundary conditions
+- ``BC = 3`` -- constant pressure boundary condition
+- ``BC = 4`` -- constant volumetric flux boundary condition
+
+For ``BC = 0`` any mass that exits on one side of the domain will re-enter at the other
+side. If the pore-structure for the image is tight, the mismatch between the inlet and
+outlet can artificially reduce the permeability of the sample due to the blockage of
+flow pathways at the boundary. LBPM includes an internal utility that will reduce the impact
+of the boundary mismatch by eroding the solid labels within the inlet and outlet layers
+(https://doi.org/10.1007/s10596-020-10028-9) to create a mixing layer.
+The number mixing layers to use can be set using the key values in the ``Domain`` section
+of the input database
+
+- ``InletLayers  = 5`` -- set the number of mixing layers to ``5`` voxels at the inlet
+- ``OUtletLayers  = 5`` -- set the number of mixing layers to ``5`` voxels at the outlet
+
+For the other boundary conditions a thin reservoir of fluid  (default ``3`` voxels)
+is established at either side of the domain. The inlet is defined as the boundary face
+where ``z = 0`` and the outlet is the boundary face where ``z = nprocz*nz``. By default a
+reservoir of fluid A is established at the inlet and a reservoir of fluid B is established at
+the outlet, each with a default thickness of three voxels. To over-ride the default label at
+the inlet or outlet, the ``Domain`` section of the database may specify the following key values
+
+- ``InletLayerPhase = 2`` -- establish a reservoir of component B at the inlet
+- ``OutletLayerPhase = 1`` -- establish a reservoir of component A at the outlet
 
-The multi-relaxation time (MRT) lattice Boltzmann model is constructed to approximate the
-solution of the Navier-Stokes equations.