Linearized Unconfined Hillslope Drainage 1D#

Note

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Steady synthetic groundwater-flow case used to validate one sloping-topography setup with distributed top drainage under the linearized unconfined model.

Case Setup#

  • geometry: quasi-1D strip (50 x 5, single layer for launcher-backed runs),

  • topography: linear, from 7.0 m on the west side to 5.0 m on the east side,

  • aquifer thickness: 30.0 m,

  • west/east heads: 7.25 m / 5.25 m,

  • drainage conductance: 1e-5 m2/s,

  • simulated observable: watertable_elevation.

  • Available solver variants: MODFLOW-NWT, MODFLOW 6, MODFLOW 6 irregular triangles.

What It Shows#

  • Sloping-topography drainage behavior without introducing a free seepage-face problem

  • Primary metrics: head-profile RMSE, max abs error, cross-row spread, minimum clearance above topography.

  • Solver-specific figures and metrics are shown side by side so the same benchmark can be read across backends.

Analytical Reference#

  • compared quantity: row-averaged head profile along x,

  • reference: steady linearized solution of T h’’ - c_a (h - z_top) = 0,

\[T_0\,\frac{\mathrm{d}^2u}{\mathrm{d}x^2}-\lambda_d\,u=0,\qquad u=h-z_{\mathrm{top}}(x)\]
\[z_{\mathrm{top}}(x)=z_0+A\left(1-\frac{x-x_{\min}}{L}\right),\qquad u(x)=u_w\frac{\sinh\left(\beta(L-x)\right)}{\sinh\left(\beta L\right)}+u_e\frac{\sinh\left(\beta x\right)}{\sinh\left(\beta L\right)}\]

Solver Coverage#

  • Default solver: MODFLOW-NWT

  • Available variants: MODFLOW-NWT, MODFLOW 6, MODFLOW 6 irregular triangles

Linearized Unconfined Hillslope Drainage 1D validation figure for MODFLOW-NWT
Linearized Unconfined Hillslope Drainage 1D rendered with MODFLOW-NWT for the analytical gallery.

Metrics - Head-profile RMSE: 0.0046 m - Head-profile max abs error: 0.0072 m - Cross-row head spread: 2.34e-07 m - Min numerical clearance above topography: 0.1139 m

  • Config file: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

  • Tolerances: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

  • Expected shape: 5 x 50

python -m validation_cases.analytical.steady.linearized_unconfined_hillslope_drainage_1d.run_case --no-show --solver modflow_nwt
Linearized Unconfined Hillslope Drainage 1D validation figure for MODFLOW 6
Linearized Unconfined Hillslope Drainage 1D rendered with MODFLOW 6 for the analytical gallery.

Metrics - Head-profile RMSE: 0.0047 m - Head-profile max abs error: 0.0072 m - Cross-row head spread: 1.45e-06 m - Min numerical clearance above topography: 0.1141 m

  • Config file: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

  • Tolerances: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

  • Expected shape: 5 x 50

python -m validation_cases.analytical.steady.linearized_unconfined_hillslope_drainage_1d.run_case --no-show --solver modflow6
Linearized Unconfined Hillslope Drainage 1D validation figure for MODFLOW 6 irregular triangles
Linearized Unconfined Hillslope Drainage 1D rendered with MODFLOW 6 irregular triangles for the analytical gallery.

Metrics - Head-profile RMSE: 0.0204 m - Head-profile max abs error: 0.0322 m - Cross-row head spread: 8.88e-16 m - Min numerical clearance above topography: 0.0800 m

  • Config file: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

  • Tolerances: validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

  • Expected shape: 5 x 50

python -m validation_cases.analytical.steady.linearized_unconfined_hillslope_drainage_1d.run_case --no-show --solver modflow6_irregular_tri

Reproduce#

Run the underlying example or validation case with:

python -m validation_cases.analytical.steady.linearized_unconfined_hillslope_drainage_1d.run_case --no-show

Refresh the committed gallery artifacts with:

python -m tools.doc_gallery

Case Parameters#

Reference Parameters#

Field

Meaning

Value

Source

xmin

Minimum x coordinate of the analytical reference domain.

0

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

xmax

Maximum x coordinate of the analytical reference domain.

100

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

length_y_m

Reference-domain length along the y axis.

10 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

west_head_m

Fixed west-boundary hydraulic head used by the reference solution.

7.25 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

east_head_m

Fixed east-boundary hydraulic head used by the reference solution.

5.25 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

topography_base_elevation_m

Base elevation used by the synthetic reference topography.

5 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

topography_right_to_left_amplitude_m

Right-to-left topographic amplitude used by the hillslope reference.

2 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

drainage_conductance_m2_per_s

Drainage conductance used by the analytical drainage boundary.

1e-05 m2/s

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

hydraulic_conductivity_m_per_s

Hydraulic conductivity used by the analytical or benchmark reference.

0.0001 m/s

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

reference_saturated_thickness_m

Reference saturated thickness used by the linearized formulation.

30 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

aquifer_thickness_m

Aquifer thickness used by the reference formulation.

30 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

profile_axis

Axis along which the validation profile is extracted.

0

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

Common Numerical Setup#

Field

Meaning

Value

Source

workflow

Case-specific configuration field workflow used by the validation benchmark.

simulation

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.grid.length_x

Synthetic-domain length along the x axis.

100.0 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.grid.length_y

Synthetic-domain length along the y axis.

10.0 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.grid.nx

Grid cell count along the x axis.

50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.grid.ny

Grid cell count along the y axis.

5

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.topography.kind

Synthetic topography shape used by the benchmark.

linear

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.topography.base_elevation

Base land-surface elevation of the synthetic topography.

5 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

geographic.synthetic.topography.right_to_left_amplitude

Right-to-left topographic amplitude used by the synthetic topography.

2

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

domain.depth_model.type

Depth model used to build the aquifer support.

constant_thickness

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

domain.depth_model.thickness

Aquifer or support thickness used by the benchmark.

30.0 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.flow_regime

Steady or transient flow regime used by the benchmark.

steady

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.active_sinks_sources

Sink and source families activated in the benchmark.

[]

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.active_bc

Boundary-condition families activated in the benchmark.

[west_side, east_side, drainage]

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.param_list

Hydraulic parameter families explicitly configured by the benchmark.

[K]

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.param.K.field.kind

Parameterization mode used for K.

homogeneous

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.param.K.field.value

Homogeneous K value used by the benchmark.

1e-4 m/s

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.ic.type

Initial-condition policy used to start the benchmark.

custom

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.ic.value

Scalar initial head or top offset used by the initial-condition policy.

6.25 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.bc.dirichlet.west_side.value

Fixed head applied on the west side boundary.

7.25 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.bc.dirichlet.east_side.value

Fixed head applied on the east side boundary.

5.25 m

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.bc.cauchy.drainage.application_domain

Part of the domain where the drainage Cauchy boundary is applied.

top

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

flow.bc.cauchy.drainage.value

Cauchy coefficient or conductance applied on the drainage boundary.

1e-5 m2/s

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

Solver-Specific Overrides#

Field

Meaning

Value

Source

modflownwt.sgrid.planar.mode

Planar support construction mode used by MODFLOW-NWT.

resample_to_shape

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

modflownwt.sgrid.planar.nx

Planar support cell count along x used by MODFLOW-NWT.

50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

modflownwt.sgrid.planar.ny

Planar support cell count along y used by MODFLOW-NWT.

5

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

modflownwt.sgrid.planar.resampling

Planar support resampling mode used by MODFLOW-NWT.

nearest

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

modflownwt.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW-NWT.

1

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

Field

Meaning

Value

Source

modflow6.runtime.mf_verbose

Solver-specific override applied to MODFLOW 6.

false

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.runtime.mf6_ims_complexity

Linear-solver complexity preset used by MODFLOW 6.

COMPLEX

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.sgrid.planar.mode

Planar support construction mode used by MODFLOW 6.

resample_to_shape

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.sgrid.planar.nx

Planar support cell count along x used by MODFLOW 6.

50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.sgrid.planar.ny

Planar support cell count along y used by MODFLOW 6.

5

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.sgrid.planar.resampling

Planar support resampling mode used by MODFLOW 6.

nearest

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

modflow6.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW 6.

1

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

Field

Meaning

Value

Source

modflow6.runtime.mf_verbose

Solver-specific override applied to MODFLOW 6.

false

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

modflow6.runtime.mf6_ims_complexity

Linear-solver complexity preset used by MODFLOW 6.

COMPLEX

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

modflow6.process_specific.vka

Vertical anisotropy ratio passed to MODFLOW 6.

1

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

modflow6.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW 6.

1

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

mesh_input.mesh_path

Committed unstructured mesh file used by the irregular-mesh solver variant.

../../../shared/mesh_bundles/linearized_unconfined_hillslope_drainage_irregular_tri_100x10/mesh_2d.msh

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

mesh_input.bundle_dir

Committed mesh-bundle directory used to recover support metadata for the irregular-mesh solver variant.

../../../shared/mesh_bundles/linearized_unconfined_hillslope_drainage_irregular_tri_100x10

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

Acceptance Criteria#

Field

Meaning

Value

Source

output.observable_name

Simulated observable compared against the reference solution.

watertable_elevation

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

output.expected_shape

Expected spatial output shape checked by the validation helper.

[5, 50]

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

Acceptance Criteria by Solver#

Field

Meaning

Value

Source

expected_output

Expected output shape or time-space layout checked for this solver.

Expected shape: 5 x 50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

head_profile.rmse

Maximum accepted root-mean-square error for head profile.

0.08

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.12

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

2e-05

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

Field

Meaning

Value

Source

expected_output

Expected output shape or time-space layout checked for this solver.

Expected shape: 5 x 50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

head_profile.rmse

Maximum accepted root-mean-square error for head profile.

0.08

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.12

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

2e-05

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

Field

Meaning

Value

Source

expected_output

Expected output shape or time-space layout checked for this solver.

Expected shape: 5 x 50

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

head_profile.rmse

Maximum accepted root-mean-square error for head profile.

0.08

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.12

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

2e-05

validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

Source Pointers#

  • validation_cases/README.md

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/README.md

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/reference.py

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/comparison.py

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/plotting.py

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/run_case.py

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/metadata.toml

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflownwt.toml

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances.toml

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/tolerances_boussinesq.toml

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6.toml

  • validation_cases/analytical/steady/linearized_unconfined_hillslope_drainage_1d/config_modflow6_irregular_tri.toml

  • validation_cases/analytical/transient/linearized_unconfined_1d.py

Artifacts#

  • docs/source/_static/capability_gallery/validation/linearized_unconfined_hillslope_drainage_1d__modflownwt.png

  • docs/source/_static/capability_gallery/validation/linearized_unconfined_hillslope_drainage_1d__modflow6.png

  • docs/source/_static/capability_gallery/validation/linearized_unconfined_hillslope_drainage_1d__modflow6_irregular_tri.png

  • docs/source/_static/capability_gallery/validation/linearized_unconfined_hillslope_drainage_1d_summary.json stores the displayed metrics plus source hashes used by python -m tools.doc_gallery --check.