## DESCRIPTION

This numerical program calculates numerical implicit transient and steady state solute transport in porous media in the saturated zone of an aquifer. The computation is based on raster maps and the current region settings. All initial- and boundary-conditions must be provided as raster maps. The unit in the location must be meters.

This module is sensitive to mask settings. All cells which are outside the mask are ignored and handled as no flow boundaries.
This module calculates the concentration of the solution and optional the velocity field, based on the hydraulic conductivity, the effective porosity and the initial piecometric heads. The vector components can be visualized with paraview if they are exported with r.out.vtk.
Use r.gwflow to compute the piezometric heights of the aquifer. The piezometric heights and the hydraulic conductivity are used to compute the flow direction and the mean velocity of the groundwater. This is the base of the solute transport computation.
The solute transport will always be calculated transient. For stady state computation set the timestep to a large number (billions of seconds).
To reduce the numerical dispersion, which is a consequence of the convection term and the finite volume discretization, you can use small time steps and choose between full and exponential upwinding.

## NOTES

The solute transport calculation is based on a diffusion/convection partial differential equation and a numerical implicit finite volume discretization. Specific for this kind of differential equation is the combination of a diffusion/dispersion term and a convection term. The discretization results in an unsymmetric linear equation system in form of Ax = b, which must be solved. The solute transport partial differential equation is of the following form:

(dc/dt)*R = div ( D grad c - uc) + cs -q/nf(c - c_in)

- c -- the concentration [kg/m^3]
- u -- vector of mean groundwater flow velocity
- dt -- the time step for transient calculation in seconds [s]
- R -- the linear retardation coefficient [-]
- D -- the diffusion and dispersion tensor [m^2/s]
- cs -- inner concentration sources/sinks [kg/m^3]
- c_in -- the solute concentration of influent water [kg/m^3]
- q -- inner well sources/sinks [m^3/s]
- nf -- the effective porosity [-]

Three different boundary conditions are implemented, the Dirichlet, Transmission and Neumann conditions. The calculation and boundary status of single cells can be set with the status map. The following states are supportet:

- 0 == inactive - the cell with status 0 will not be calulated, active cells will have a no flow boundary to an inactive cell
- 1 == active - this cell is used for sloute transport calculation, inner sources can be defined for those cells
- 2 == Dirichlet - cells of this type will have a fixed concentration value which do not change over time
- 3 == Transmission - cells of this type should be placed on out-flow boundaries to assure the flow of the solute stream out

Note that all required raster maps are read into main memory. Additionally the linear equation system will be allocated, so the memory consumption of this module rapidely grow with the size of the input maps.
The resulting linear equation system Ax = b can be solved with several solvers. Several iterative solvers with unsymmetric sparse and quadratic matrices support are implemented. The jacobi method, the Gauss-Seidel method and the biconjugate gradients-stabilized (bicgstab) method. Aditionally a direct Gauss solver and LU solver are available. Those direct solvers only work with quadratic matrices, so be careful using them with large maps (maps of size 10.000 cells will need more than one gigabyte of ram). Always prefer a sparse matrix solver.

## EXAMPLE

Use this small python script to create a working groundwater flow / solute transport area and data. Make sure you are not in a lat/lon projection.

    #!/usr/bin/env python
    # This is an example script how groundwater flow and solute transport are
    # computed within grass
    import sys
    import os
    import grass.script as grass

    # Overwrite existing maps
    grass.run_command("g.gisenv", set="OVERWRITE=1")

    grass.message(_("Set the region"))

    # The area is 200m x 100m with a cell size of 1m x 1m
    grass.run_command("g.region", res=1, res3=1, t=10, b=0, n=100, s=0, w=0, e=200)
    grass.run_command("r.mapcalc", expression="phead=if(col() == 1 , 50, 40)")
    grass.run_command("r.mapcalc", expression="phead=if(col() ==200  , 45 + row()/40, phead)")
    grass.run_command("r.mapcalc", expression="status=if(col() == 1 || col() == 200 , 2, 1)")
    grass.run_command("r.mapcalc", expression="well=if((row() == 50 && col() == 175) || (row() == 10 && col() == 135) , -0.001, 0)")
    grass.run_command("r.mapcalc", expression="hydcond=0.00005")
    grass.run_command("r.mapcalc", expression="recharge=0")
    grass.run_command("r.mapcalc", expression="top_conf=20")
    grass.run_command("r.mapcalc", expression="bottom=0")
    grass.run_command("r.mapcalc", expression="poros=0.17")
    grass.run_command("r.mapcalc", expression="syield=0.0001")
    grass.run_command("r.mapcalc", expression="null=0.0")

    grass.message(_("Compute a steady state groundwater flow"))

    grass.run_command("r.gwflow", solver="cg", top="top_conf", bottom="bottom", phead="phead",\
      status="status", hc_x="hydcond", hc_y="hydcond", q="well", s="syield",\
      recharge="recharge", output="gwresult_conf", dt=8640000000000, type="confined")

    grass.message(_("generate the transport data"))
    grass.run_command("r.mapcalc", expression="c=if(col() == 15 && row() == 75 , 500.0, 0.0)")
    grass.run_command("r.mapcalc", expression="cs=if(col() == 15 && row() == 75 , 0.0, 0.0)")
    grass.run_command("r.mapcalc", expression="tstatus=if(col() == 1 || col() == 200 , 3, 1)")
    grass.run_command("r.mapcalc", expression="diff=0.0000001")
    grass.run_command("r.mapcalc", expression="R=1.0")

    # Compute the initial state
    grass.run_command("r.solute.transport", solver="bicgstab", top="top_conf",\
      bottom="bottom", phead="gwresult_conf", status="tstatus", hc_x="hydcond", hc_y="hydcond",\
      rd="R", cs="cs", q="well", nf="poros", output="stresult_conf_0", dt=3600, diff_x="diff",\
      diff_y="diff", c="c", al=0.1, at=0.01)

    # Compute the solute transport for 300 days in 10 day steps
    for dt in range(30):
        grass.run_command("r.solute.transport", solver="bicgstab", top="top_conf",\
        bottom="bottom", phead="gwresult_conf", status="tstatus", hc_x="hydcond", hc_y="hydcond",\
        rd="R", cs="cs", q="well", nf="poros", output="stresult_conf_" + str(dt + 1), dt=864000, diff_x="diff",\
        diff_y="diff", c="stresult_conf_" + str(dt), al=0.1, at=0.01)

## SEE ALSO

r.gwflow
r3.gwflow
r.out.vtk

## AUTHOR

Sören Gebbert

This work is based on the Diploma Thesis of Sören Gebbert available here at Technical University Berlin in Germany.

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