## NAME

r3.gwflow - Numerical calculation program for transient, confined groundwater flow in three dimensions.

## KEYWORDS

raster3d, groundwater flow, voxel, hydrology

## SYNOPSIS

r3.gwflow
r3.gwflow --help
r3.gwflow [-mf] phead=name status=name hc_x=name hc_y=name hc_z=name [sink=name] yield=name [recharge=name] output=name [velocity_x=name] [velocity_y=name] [velocity_z=name] [budget=name] dtime=float [maxit=integer] [error=float] [solver=name] [--overwrite] [--help] [--verbose] [--quiet] [--ui]

### Flags:

-m
  Use 3D raster mask (if exists)
-f
  Use a full filled quadratic linear equation system, default is a sparse linear equation system.
--overwrite
  Allow output files to overwrite existing files
--help
  Print usage summary
--verbose
  Verbose module output
--quiet
  Quiet module output
--ui
  Force launching GUI dialog

### Parameters:

phead=name [required]
  Input 3D raster map with initial piezometric heads in [m]
status=name [required]
  Input 3D raster map providing the status for each cell, = 0 - inactive, 1 - active, 2 - dirichlet
hc_x=name [required]
  Input 3D raster map with the x-part of the hydraulic conductivity tensor in [m/s]
hc_y=name [required]
  Input 3D raster map with the y-part of the hydraulic conductivity tensor in [m/s]
hc_z=name [required]
  Input 3D raster map with the z-part of the hydraulic conductivity tensor in [m/s]
sink=name
  Input 3D raster map with sources and sinks in [m^3/s]
yield=name [required]
  Specific yield [1/m] input 3D raster map
recharge=name
  Recharge input 3D raster map in m^3/s
output=name [required]
  Output 3D raster map storing the piezometric head result of the numerical calculation
velocity_x=name
  Output 3D raster map storing the groundwater filter velocity vector part in x direction [m/s]
velocity_y=name
  Output 3D raster map storing the groundwater filter velocity vector part in y direction [m/s]
velocity_z=name
  Output 3D raster map storing the groundwater filter velocity vector part in z direction [m/s]
budget=name
  Output 3D raster map storing the groundwater budget for each cell [m^3/s]
dtime=float [required]
  The calculation time in seconds
  Default: 86400
maxit=integer
  Maximum number of iteration used to solve the linear equation system
  Default: 10000
error=float
  Error break criteria for iterative solver
  Default: 0.000001
solver=name
  The type of solver which should solve the symmetric linear equation system
  Options: cg, pcg, cholesky
  Default: cg

## DESCRIPTION

This numerical module calculates implicit transient and steady state, confined groundwater flow in three dimensions based on volume maps and the current 3D region settings. All initial- and boundary-conditions must be provided as volume maps. The unit of the current coordinate reference system 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.

The module calculates the piezometric head and optionally the water balance for each cell and the groundwater velocity field in 3 dimensions. The vector components can be visualized with ParaView if they are exported with r3.out.vtk.

The groundwater flow will always be calculated transient. For steady state computation the user should set the timestep to a large number (billions of seconds) or set the specific yield raster map to zero.

## 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 definite 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 3 dimensions:

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

- h -- the piezometric head im meters [m]
- dt -- the time step for transient calculation in seconds [s]
- S -- the specific yield [1/m]
- b -- the bottom surface of the aquifer meters [m]
- Kxx -- the hydraulic conductivity tensor part in x direction in meter per second [m/s]
- Kyy -- the hydraulic conductivity tensor part in y direction in meter per seconds [m/s]
- Kzz -- the hydraulic conductivity tensor part in z direction in meter per seconds [m/s]
- q - inner source/sinc in [1/s]

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 can be set with the status map, the following cell states are supported:

- 0 == inactive - the cell with status 0 will not be calculated, active cells will have a no flow boundary to an inactive cell
- 1 == active - this cell is used for groundwater calculation, inner sources can be defined for those cells
- 2 == Dirichlet - cells of this type will have a fixed piezometric head value which do not change over 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. Additionally 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). The user should always prefer to use a sparse matrix solver.

## EXAMPLE 1

This small script creates a working groundwater flow area and data. It cannot be run in a lat/lon project.

    # set the region accordingly
    g.region res=25 res3=25 t=100 b=0 n=1000 s=0 w=0 e=1000 -p3

    #now create the input raster maps for a confined aquifer
    r3.mapcalc expression="phead = if(row() == 1 && depth() == 4, 50, 40)"
    r3.mapcalc expression="status = if(row() == 1 && depth() == 4, 2, 1)"
    r3.mapcalc expression="well = if(row() == 20 && col() == 20 && depth() == 2, -0.25, 0)"
    r3.mapcalc expression="hydcond = 0.00025"
    r3.mapcalc expression="syield = 0.0001"
    r.mapcalc  expression="recharge = 0.0"

    r3.gwflow solver=cg phead=phead statuyield=status hc_x=hydcond hc_y=hydcond  \
       hc_z=hydcond sink=well yield=syield r=recharge output=gwresult dt=8640000 vx=vx vy=vy vz=vz budget=budget

    # The data can be visualized with ParaView when exported with r3.out.vtk
    r3.out.vtk -p in=gwresult,status,budget vector=vx,vy,vz out=/tmp/gwdata3d.vtk

    #now load the data into ParaView
    paraview --data=/tmp/gwdata3d.vtk

## EXAMPLE 2

This will create a nice 3D model with geological layer with different hydraulic conductivities. Make sure you are not in a lat/lon projection.

    # set the region accordingly
    g.region res=15 res3=15 t=500 b=0 n=1000 s=0 w=0 e=1000

    #now create the input raster maps for a confined aquifer
    r3.mapcalc expression="phead = if(col() == 1 && depth() == 33, 50, 40)"
    r3.mapcalc expression="status = if(col() == 1 && depth() == 33, 2, 1)"
    r3.mapcalc expression="well = if(row() == 20 && col() == 20 && depth() == 3, -0.25, 0)"
    r3.mapcalc expression="well = if(row() == 50 && col() == 50 && depth() == 3, -0.25, well)"
    r3.mapcalc expression="hydcond = 0.0025"
    r3.mapcalc expression="hydcond = if(depth() < 30 && depth() > 23 && col() < 60, 0.000025, hydcond)"
    r3.mapcalc expression="hydcond = if(depth() < 20 && depth() > 13 && col() >  7, 0.000025, hydcond)"
    r3.mapcalc expression="hydcond = if(depth() < 10 && depth() >  7 && col() < 60, 0.000025, hydcond)"
    r3.mapcalc expression="syield = 0.0001"

    r3.gwflow solver=cg phead=phead statuyield=status hc_x=hydcond hc_y=hydcond  \
       hc_z=hydcond sink=well yield=syield output=gwresult dt=8640000 vx=vx vy=vy vz=vz budget=budget

    # The data can be visualized with paraview when exported with r3.out.vtk
    r3.out.vtk -p in=gwresult,status,budget,hydcond,well vector=vx,vy,vz out=/tmp/gwdata3d.vtk

    #now load the data into paraview
    paraview --data=/tmp/gwdata3d.vtk

## SEE ALSO

r.gwflow, r.solute.transport, r3.out.vtk

## AUTHOR

Sören Gebbert

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

## SOURCE CODE

Available at: r3.gwflow source code (history)

Accessed: Thursday Oct 30 22:17:45 2025

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