## DESCRIPTION

This numerical program calculates implicit transient, confined and unconfined groundwater flow in two dimensions 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.

Workflow of r.gwflow

r.gwflow calculates the piezometric head and optionally the water budget and the filter velocity field, based on the hydraulic conductivity and the piezometric head. The vector components can be visualized with paraview if they are exported with r.out.vtk.
The groundwater flow will always be calculated transient. For stady state computation set the timestep to a large number (billions of seconds) or set the storativity/ effective porosity raster map to zero.
The water budget is calculated for each non inactive cell. The sum of the budget for each non inactive cell must be near zero. This is an indicator of the quality of the numerical result.

## NOTES

The groundwater flow calculation is based on Darcy's law and a numerical implicit finite volume discretization. The discretization results in a symmetric and positive definit linear equation system in form of Ax = b, which must be solved. The groundwater flow partial differential equation is of the following form:

(dh/dt)*S = div (K grad h) + q

In detail for 2 dimensions:

(dh/dt)*S = Kxx * (d^2h/dx^2) + Kyy * (d^2h/dy^2) + q

- h -- the piezometric head im [m]
- dt -- the time step for transient calculation in [s]
- S -- the specific storage [1/m]
- Kxx -- the hydraulic conductivity tensor part in x direction in [m/s]
- Kyy -- the hydraulic conductivity tensor part in y direction in [m/s]
- q - inner source/sink in meter per second [1/s]

Confined and unconfined groundwater flow is supported. Be aware that the storativity input parameter is handled differently in case of unconfined flow. Instead of the storativity, the effective porosity is expected.

To compute unconfined groundwater flow, a simple Picard based linearization scheme is used to solve the resulting non-linear equation system.

Two different boundary conditions are implemented, the Dirichlet and Neumann conditions. By default the calculation area is surrounded by homogeneous Neumann boundary conditions. The calculation and boundary status of single cells must be set with a status map, the following states are supportet:

- 0 == inactive - the cell with status 0 will not be calculated, active cells will have a no flow boundary to this cell
- 1 == active - this cell is used for groundwater floaw calculation, inner sources and recharge can be defined for those cells
- 2 == Dirichlet - cells of this type will have a fixed piezometric head value which do not change over the time

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. An iterative solvers with sparse and quadratic matrices support is implemented. The conjugate gradients method with (pcg) and without (cg) precondition. Aditionally a direct Cholesky solver is available. This direct solver only work with normal 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 script to create a working groundwater flow area and data. Make sure you are not in a lat/lon projection. It includes drainage and river input as well.

    # set the region accordingly
    g.region res=25 res3=25 t=100 b=0 n=1000 s=0 w=0 e=1000

    #now create the input raster maps for confined and unconfined aquifers
    r.mapcalc --o expression="phead=if(row() == 1 , 50, 40)"
    r.mapcalc --o expression="status=if(row() == 1 , 2, 1)"
    r.mapcalc --o expression="well=if(row() == 20 && col() == 20 , -0.01, 0)"
    r.mapcalc --o expression="hydcond=0.00025"
    r.mapcalc --o expression="recharge=0"
    r.mapcalc --o expression="top_conf=20.0"
    r.mapcalc --o expression="top_unconf=70.0"
    r.mapcalc --o expression="bottom=0.0"
    r.mapcalc --o expression="null=0.0"
    r.mapcalc --o expression="poros=0.15"
    r.mapcalc --o expression="s=0.0001"

    # The maps of the river
    r.mapcalc --o expression="river_bed=if(col() == 35 , 48, null())"
    r.mapcalc --o expression="river_head=if(col() == 35 , 49, null())"
    r.mapcalc --o expression="river_leak=if(col() == 35 , 0.0001, null())"

    # The maps of the drainage
    r.mapcalc --o expression="drain_bed=if(col() == 5 , 48, null())"
    r.mapcalc --o expression="drain_leak=if(col() == 5 , 0.01, null())"

    #confined groundwater flow with cg solver and sparse matrix, river and drain
    #do not work with this confined aquifer (top == 20m)
    r.gwflow --o solver=cg top=top_conf bottom=bottom phead=phead status=status \
    hc_x=hydcond hc_y=hydcond q=well s=s recharge=recharge output=gwresult_conf \
    dt=8640000 type=confined vx=gwresult_conf_velocity_x vy=gwresult_conf_velocity_y budget=budget_conf

    #unconfined groundwater flow with cg solver and sparse matrix, river and drain are enabled
    # We use the effective porosity as storativity parameter
    r.gwflow --o solver=cg top=top_unconf bottom=bottom phead=phead \
    status=status hc_x=hydcond hc_y=hydcond q=well s=poros recharge=recharge \
    river_bed=river_bed river_head=river_head river_leak=river_leak \
    drain_bed=drain_bed drain_leak=drain_leak \
    output=gwresult_unconf dt=8640000 type=unconfined vx=gwresult_unconf_velocity_x \
    budget=budget_unconf vy=gwresult_unconf_velocity_y

    # The data can be visulaized with paraview when exported with r.out.vtk
    r.out.vtk -p in=gwresult_conf,status vector=gwresult_conf_velocity_x,gwresult_conf_velocity_y,null out=/tmp/gwdata_conf2d.vtk
    r.out.vtk -p elevation=gwresult_unconf in=gwresult_unconf,status vector=gwresult_unconf_velocity_x,gwresult_unconf_velocity_y,null out=/tmp/gwdata_unconf2d.vtk

    #now load the data into paraview
    paraview --data=/tmp/gwdata_conf2d.vtk &
    paraview --data=/tmp/gwdata_unconf2d.vtk &

## SEE ALSO

r.solute.transport
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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