## NAME

r.gwflow - Numerical calculation program for transient, confined and unconfined groundwater flow in two dimensions.

## KEYWORDS

raster, groundwater flow, hydrology

## SYNOPSIS

r.gwflow
r.gwflow --help
r.gwflow [-f] phead=name status=name hc_x=name hc_y=name [q=name] s=name [recharge=name] top=name bottom=name output=name [vx=name] [vy=name] [budget=name] type=string [river_bed=name] [river_head=name] [river_leak=name] [drain_bed=name] [drain_leak=name] dtime=float [maxit=integer] [maxit=integer] [error=float] [solver=name] [--overwrite] [--help] [--verbose] [--quiet] [--ui]

### Flags:

-f
  Allocate a full 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]
  Name of input raster map with initial piezometric head in [m]
status=name [required]
  Name of input raster map providing boundary condition status: 0-inactive, 1-active, 2-dirichlet
hc_x=name [required]
  Name of input raster map with x-part of the hydraulic conductivity tensor in [m/s]
hc_y=name [required]
  Name of input raster map with y-part of the hydraulic conductivity tensor in [m/s]
q=name
  Name of input raster map with water sources and sinks in [m^3/s]
s=name [required]
  Name of input raster map with storativity for confined or effective porosity for unconfined groundwater flow booth in [-]
recharge=name
  Recharge input raster map e.g: 6*10^-9 per cell in [m^3/s*m^2]
top=name [required]
  Name of input raster map describing the top surface of the aquifer in [m]
bottom=name [required]
  Name of input raster map describing the bottom surface of the aquifer in [m]
output=name [required]
  Output raster map storing the numerical result [m]
vx=name
  Output raster map to store the groundwater filter velocity vector part in x direction [m/s]
vy=name
  Output raster map to store the groundwater filter velocity vector part in y direction [m/s]
budget=name
  Output raster map to store the groundwater budget for each cell [m^3/s]
type=string [required]
  The type of groundwater flow
  Options: confined, unconfined
  Default: confined
river_bed=name
  Name of input raster map providing the height of the river bed in [m]
river_head=name
  Name of input raster map providing the water level (head) of the river with leakage connection in [m]
river_leak=name
  Name of input raster map providing the leakage coefficient of the river bed in [1/s].
drain_bed=name
  Name of input raster map providing the height of the drainage bed in [m]
drain_leak=name
  Name of input raster map providing the leakage coefficient of the drainage bed in [1/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
maxit=integer
  The maximum number of iterations in the linearization approach
  Default: 25
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 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 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 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. 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). 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 -p3

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

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

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

## SOURCE CODE

Available at: r.gwflow source code (history)

Latest change: Thursday Feb 03 11:10:06 2022 in commit: 547ff44e6aecfb4c9cbf6a4717fc14e521bec0be

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