Freshwater lens (circle)

This example illustrates how to setup a very simple unstructured groundwater transport model using the imod package and associated packages.

In overview, we’ll set the following steps:

• Setting up the flow model, just like in the circle.py example

• set up the transport model

• Run the simulation.

• Visualize the results.

We’ll start with the following imports:

import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import xarray as xr
import xugrid as xu

import imod

Parameters

porosity = 0.10
max_concentration = 35.0
min_concentration = 0.0
max_density = 1025.0
min_density = 1000.0
k_value = 10.0

Create a mesh

The first steps consists of generating a mesh. In this example, we’ll use data included with iMOD Python for a circular mesh. Note that this is a Ugrid2D object. For more information on working with unstructured grids see the Xugrid documentation

grid_triangles = imod.data.circle()

fig, ax = plt.subplots()
xu.plot.line(grid_triangles)
ax.set_aspect(1)

However a triangular grid has the issue that the direction of the fluxes between cell centres is not perpendicular to the cell vertices. The default formulation of Modflow 6 does not account for this, which causes mass balance errors. The XT3D formulation is able to account for this, but the last version of Modflow 6 (6.3 at time of writing) does not support this in combination with the Buoyancy package and using XT3D comes with an extra, significant computational burden. It is therefore easier to use a voronoi grid, for which Xugrid has a very convenient method.

grid = grid_triangles.tesselate_centroidal_voronoi()

fig, ax = plt.subplots()
xu.plot.line(grid)
ax.set_aspect(1)

Create arrays

nface = grid.n_face
nlayer = 15

layer = np.arange(nlayer, dtype=int) + 1

idomain = xu.UgridDataArray(
    xr.DataArray(
        np.ones((nlayer, nface), dtype=np.int32),
        coords={"layer": layer},
        dims=["layer", grid.face_dimension],
    ),
    grid=grid,
)
icelltype = xu.full_like(idomain, 0)
k = xu.full_like(idomain, k_value, dtype=float)
k33 = k.copy()

top = 0.0
bottom = xr.DataArray(top - (layer * 10.0), dims=["layer"])

Recharge

We need a recharge rate for the fluid and a recharge rate for the solute. The fluid recharge rate is volumetric and per unit area, so the unit is length/time. The solute recharge rate is the concentration of solute in the recharge, and has concentration units.

rch_rate = xu.full_like(idomain.sel(layer=1), 0.001, dtype=float)
rch_concentration = xu.full_like(rch_rate, min_concentration)
rch_concentration = rch_concentration.expand_dims(species=["salinity"])

Unlike a recharge boundary, with a prescribed head boundary we don’t know a priori whether water will flow in over the boundary or leave across the boundary. If water flows into the model over the boundary, it carries a prescribed solute concentration. If it leaves, it leaves with the concentration that was computed for the cell.

In this example we set the prescribed head value to 0.0 and the external concentration to 35.0 as well. The boundary only operates on the top layer.

chd_location = xu.zeros_like(idomain.sel(layer=1), dtype=bool).ugrid.binary_dilation(
    border_value=True
)
constant_head = xu.full_like(idomain, 0.0, dtype=float).where(chd_location)
# Approximate face area
face_area = (1000.0 / 6) ** 2 * 0.5

conductance = xu.full_like(idomain, face_area * k_value, dtype=float).where(
    chd_location
)

constant_concentration = xu.full_like(constant_head, max_concentration).where(
    chd_location
)
constant_concentration = constant_concentration.expand_dims(species=["salinity"])

Add flow model to simulation

See the circle.py example for more information.

gwf_model = imod.mf6.GroundwaterFlowModel()
gwf_model["disv"] = imod.mf6.VerticesDiscretization(
    top=top, bottom=bottom, idomain=idomain
)
gwf_model["ghb"] = imod.mf6.GeneralHeadBoundary(
    constant_head,
    conductance=conductance,
    concentration=constant_concentration,
    print_input=True,
    print_flows=True,
    save_flows=True,
)
gwf_model["ic"] = imod.mf6.InitialConditions(start=0.0)
gwf_model["npf"] = imod.mf6.NodePropertyFlow(
    icelltype=icelltype,
    k=k,
    k33=k33,
    save_flows=True,
)
gwf_model["sto"] = imod.mf6.SpecificStorage(
    specific_storage=1.0e-5,
    specific_yield=0.15,
    transient=False,
    convertible=0,
)
gwf_model["oc"] = imod.mf6.OutputControl(save_head="last", save_budget="last")
gwf_model["rch"] = imod.mf6.Recharge(
    rch_rate, concentration=rch_concentration, print_flows=True, save_flows=True
)

simulation = imod.mf6.Modflow6Simulation("circle")
simulation["flow"] = gwf_model
simulation["flow_solver"] = imod.mf6.Solution(
    modelnames=["flow"],
    print_option="summary",
    outer_dvclose=1.0e-4,
    outer_maximum=500,
    under_relaxation=None,
    inner_dvclose=1.0e-4,
    inner_rclose=0.001,
    inner_maximum=100,
    linear_acceleration="bicgstab",
    scaling_method=None,
    reordering_method=None,
    relaxation_factor=0.97,
)

Set the timesteps, we want output each year, so we specify stress periods which last 1 year. However, timesteps of 1 year yield unstable results, so we set n_timesteps to 10, which sets the amount of timesteps within a stress period.

simtimes = pd.date_range(start="2000-01-01", end="2030-01-01", freq="As")
simulation.create_time_discretization(additional_times=simtimes)
simulation["time_discretization"]["n_timesteps"] = 10

Buoyancy

Since we are solving a variable density problem, we need to add the buoyancy package. It will use the species “salinity” that we are simulating with a transport model defined below.

slope = (max_density - min_density) / (max_concentration - min_concentration)
gwf_model["buoyancy"] = imod.mf6.Buoyancy(
    reference_density=min_density,
    modelname=["transport"],
    reference_concentration=[min_concentration],
    density_concentration_slope=[slope],
    species=["salinity"],
)

Add transport model to simulation

The transport model requires the flow field inside the domain computed by the flow model. It also needs the fluxes over the boundary. It uses the same discretization as the flow model. Here we create a transport model for salinity, derive sources and sinks based from the flow model, and tell it to use the same discretization.

transport_model = imod.mf6.GroundwaterTransportModel()
transport_model["ssm"] = imod.mf6.SourceSinkMixing.from_flow_model(
    gwf_model, "salinity"
)
transport_model["disv"] = gwf_model["disv"]

Now we define some transport packages for simulating the physical processes of advection, mechanical dispersion, and molecular diffusion dispersion. This example is transient, and the volume available for storage is the porosity, in this case 0.10.

al = 0.001

transport_model["dsp"] = imod.mf6.Dispersion(
    diffusion_coefficient=1e-4,
    longitudinal_horizontal=al,
    transversal_horizontal1=al * 0.1,
    transversal_vertical=al * 0.01,
    xt3d_off=False,
    xt3d_rhs=False,
)
transport_model["adv"] = imod.mf6.AdvectionUpstream()
transport_model["mst"] = imod.mf6.MobileStorageTransfer(porosity)

Define the maximum concentration as the initial conditions, also output options for the transport model, and assign the transport model to the simulation as well.

transport_model["ic"] = imod.mf6.InitialConditions(start=max_concentration)
transport_model["oc"] = imod.mf6.OutputControl(
    save_concentration="last", save_budget="last"
)

simulation["transport"] = transport_model
simulation["transport_solver"] = imod.mf6.Solution(
    modelnames=["transport"],
    print_option="summary",
    outer_dvclose=1.0e-4,
    outer_maximum=500,
    under_relaxation=None,
    inner_dvclose=1.0e-4,
    inner_rclose=0.001,
    inner_maximum=100,
    linear_acceleration="bicgstab",
    scaling_method=None,
    reordering_method=None,
    relaxation_factor=0.97,
)

We’ll create a new directory in which we will write and run the model.

modeldir = imod.util.temporary_directory()
simulation.write(modeldir, binary=False)

Run the model

Note

The following lines assume the mf6 executable is available on your PATH. The examples introduction shortly describes how to add it to yours.

simulation.run()

Open the results

sim_concentration = simulation.open_concentration().compute()
sim_head = simulation.open_head().compute()

Assign coordinates to output

The model output does not feature very useful coordinate values for time and z, therefore it is best to assign these to the datasets for more understandable plots.

First we have to compute values for a z coordinate. The

interfaces = np.concatenate([[top], bottom.values])
z = (interfaces[:-1] + interfaces[1:]) / 2

z

array([  -5.,  -15.,  -25.,  -35.,  -45.,  -55.,  -65.,  -75.,  -85.,
        -95., -105., -115., -125., -135., -145.])

Assign these new coordinate values to the dataset

coords = {"time": simtimes[1:], "z": ("layer", z)}

sim_head = sim_head.assign_coords(**coords)
sim_concentration = sim_concentration.assign_coords(**coords)

Visualize the results

We can quickly and easily visualize the output with the plotting functions provided by xarray and xugrid:

fig, ax = plt.subplots()
sim_head.isel(time=-1, layer=0).ugrid.plot(ax=ax)
ax.set_aspect(1)

We can draw a crossection through the center by selecting y=0, for which we can plot the contours as follows:

fig, ax = plt.subplots()
sim_concentration.isel(time=-1).ugrid.sel(y=0).plot.contourf(
    ax=ax, x="mesh2d_x", y="z", cmap="RdYlBu_r"
)

<matplotlib.contour.QuadContourSet object at 0x00000204D7E4F8D0>