Boussinesq Fixed-Head Piecewise-K 1D#

Note

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Steady synthetic groundwater-flow case used to validate the launcher workflow against the exact 1D Boussinesq solution written on U = h^2 for an unconfined aquifer with:

Case Setup#

  • Strip with piecewise-constant K, fixed west/east heads, no recharge

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

What It Shows#

  • validate heterogeneous K mapping on a simple 1D support,

  • check flux continuity across conductivity jumps,

  • keep a lightweight analytical benchmark for future heterogeneous cases.

  • Heterogeneous conductivity mapping and flux continuity across conductivity jumps

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

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

Analytical Reference#

  • compared quantity: domain-averaged head profile along x

  • reference: steady 1D Boussinesq profile with piecewise-constant K

\[U=h^2,\qquad \frac{\mathrm{d}}{\mathrm{d}x}\left(K(x)\,\frac{\mathrm{d}U}{\mathrm{d}x}\right)=0\]
\[U(x)=h_w^2+\left(h_e^2-h_w^2\right)\frac{\int_{x_{\min}}^{x}\frac{\mathrm{d}s}{K(s)}}{\int_{x_{\min}}^{x_{\max}}\frac{\mathrm{d}s}{K(s)}}\]

Solver Coverage#

  • Default solver: MODFLOW-NWT

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

Boussinesq Fixed-Head Piecewise-K 1D validation figure for MODFLOW-NWT
Boussinesq Fixed-Head Piecewise-K 1D rendered with MODFLOW-NWT for the analytical gallery.

Metrics - Head-profile RMSE: 0.0107 m - Head-profile max abs error: 0.0231 m - Cross-row head spread: 3.81e-07 m

  • Config file: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

  • Tolerances: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

  • Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_fixed_head_piecewise_k_1d.run_case --no-show --solver modflow_nwt
Boussinesq Fixed-Head Piecewise-K 1D validation figure for MODFLOW 6
Boussinesq Fixed-Head Piecewise-K 1D rendered with MODFLOW 6 for the analytical gallery.

Metrics - Head-profile RMSE: 0.0082 m - Head-profile max abs error: 0.0190 m - Cross-row head spread: 2.38e-06 m

  • Config file: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

  • Tolerances: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_modflow6.toml

  • Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_fixed_head_piecewise_k_1d.run_case --no-show --solver modflow6
Boussinesq Fixed-Head Piecewise-K 1D validation figure for MODFLOW 6 irregular triangles
Boussinesq Fixed-Head Piecewise-K 1D rendered with MODFLOW 6 irregular triangles for the analytical gallery.

Metrics - Head-profile RMSE: 0.0237 m - Head-profile max abs error: 0.0701 m - Cross-row head spread: 8.88e-16 m

  • Config file: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

  • Tolerances: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

  • Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_fixed_head_piecewise_k_1d.run_case --no-show --solver modflow6_irregular_tri
Boussinesq Fixed-Head Piecewise-K 1D validation figure for Boussinesq
Boussinesq Fixed-Head Piecewise-K 1D rendered with Boussinesq for the analytical gallery.

Metrics - Head-profile RMSE: 0.0079 m - Head-profile max abs error: 0.0207 m - Cross-row head spread: 5.30e-02 m

  • Config file: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_boussinesq.toml

  • Tolerances: validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_boussinesq.toml

  • Expected shape: 3 x 40

python -m validation_cases.analytical.steady.boussinesq_fixed_head_piecewise_k_1d.run_case --no-show --solver boussinesq

Reproduce#

Run the underlying example or validation case with:

python -m validation_cases.analytical.steady.boussinesq_fixed_head_piecewise_k_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/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

xmax

Maximum x coordinate of the analytical reference domain.

400

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

west_head

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

10 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

east_head

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

5 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

x_zone_breaks_m

x-coordinate breaks used to define conductivity zones in the reference solution.

[120 m, 280 m]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

hydraulic_conductivity_m_per_s_by_zone

Reference parameter hydraulic_conductivity_m_per_s_by_zone used by the analytical or benchmark solution.

[0.0002, 5e-05, 0.0001]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

aquifer_thickness_m

Aquifer thickness used by the reference formulation.

20 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

profile_axis

Axis along which the validation profile is extracted.

0

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_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/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.grid.length_x

Synthetic-domain length along the x axis.

400.0 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.grid.length_y

Synthetic-domain length along the y axis.

50.0 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.grid.nx

Grid cell count along the x axis.

40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.grid.ny

Grid cell count along the y axis.

5

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.topography.kind

Synthetic topography shape used by the benchmark.

flat

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

geographic.synthetic.topography.base_elevation

Base land-surface elevation of the synthetic topography.

20 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.supports.k_bands.provider

How the support k_bands is generated.

generated_bands

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.supports.k_bands.axis

Axis used to build support k_bands.

x

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.supports.k_bands.coordinate_mode

Coordinate interpretation used to define support k_bands.

relative

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.supports.k_bands.breaks

Break values used to split support k_bands.

[0.3, 0.7]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.supports.k_bands.labels

Labels assigned to the zones of support k_bands.

[west_zone, middle_zone, east_zone]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.depth_model.type

Depth model used to build the aquifer support.

constant_thickness

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

domain.depth_model.thickness

Aquifer or support thickness used by the benchmark.

20.0 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.flow_regime

Steady or transient flow regime used by the benchmark.

steady

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.active_sinks_sources

Sink and source families activated in the benchmark.

[]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.active_bc

Boundary-condition families activated in the benchmark.

[west_side, east_side]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param_list

Hydraulic parameter families explicitly configured by the benchmark.

[K]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.kind

Parameterization mode used for K.

heterogeneous

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.values_source

Value source used for the heterogeneous K field.

inline

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.values.west_zone

Heterogeneous K value applied on support zone west_zone.

2e-4 m/s

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.values.middle_zone

Heterogeneous K value applied on support zone middle_zone.

5e-5 m/s

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.values.east_zone

Heterogeneous K value applied on support zone east_zone.

1e-4 m/s

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.param.K.field.field_spatial_id

Support identifier used to distribute the heterogeneous K field.

k_bands

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.ic.type

Initial-condition policy used to start the benchmark.

custom

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.ic.value

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

7.5 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.bc.dirichlet.west_side.value

Fixed head applied on the west side boundary.

10.0 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

flow.bc.dirichlet.east_side.value

Fixed head applied on the east side boundary.

5.0 m

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_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/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

modflownwt.sgrid.planar.nx

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

40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

modflownwt.sgrid.planar.ny

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

5

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

modflownwt.sgrid.planar.resampling

Planar support resampling mode used by MODFLOW-NWT.

nearest

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

modflownwt.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW-NWT.

1

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

Field

Meaning

Value

Source

modflow6.runtime.mf_verbose

Solver-specific override applied to MODFLOW 6.

false

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.runtime.mf6_ims_complexity

Linear-solver complexity preset used by MODFLOW 6.

COMPLEX

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.sgrid.planar.mode

Planar support construction mode used by MODFLOW 6.

resample_to_shape

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.sgrid.planar.nx

Planar support cell count along x used by MODFLOW 6.

40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.sgrid.planar.ny

Planar support cell count along y used by MODFLOW 6.

5

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.sgrid.planar.resampling

Planar support resampling mode used by MODFLOW 6.

nearest

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

modflow6.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW 6.

1

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

Field

Meaning

Value

Source

modflow6.runtime.mf_verbose

Solver-specific override applied to MODFLOW 6.

false

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

modflow6.runtime.mf6_ims_complexity

Linear-solver complexity preset used by MODFLOW 6.

COMPLEX

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

modflow6.sgrid.vertical.nlay

Number of vertical layers used by MODFLOW 6.

1

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

mesh_input.mesh_path

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

../../../shared/mesh_bundles/sloping_substratum_irregular_tri_400x50/mesh_2d.msh

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_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/sloping_substratum_irregular_tri_400x50

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

Field

Meaning

Value

Source

flow.runtime_backend

Runtime backend selected for the in-house solver.

petsc

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_boussinesq.toml

Acceptance Criteria#

Field

Meaning

Value

Source

output.observable_name

Simulated observable compared against the reference solution.

watertable_elevation

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

output.expected_shape

Expected spatial output shape checked by the validation helper.

[5, 40]

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_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 40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

head_profile.rmse

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

0.05

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.1

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

2e-05

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

Field

Meaning

Value

Source

expected_output

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

Expected shape: 5 x 40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

head_profile.rmse

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

0.012

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_modflow6.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.025

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_modflow6.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

1e-05

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_modflow6.toml

Field

Meaning

Value

Source

expected_output

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

Expected shape: 5 x 40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

head_profile.rmse

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

0.05

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.1

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

2e-05

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

Field

Meaning

Value

Source

expected_output

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

Expected shape: 3 x 40

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

head_profile.rmse

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

0.2

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_boussinesq.toml

head_profile.max_abs_error

Maximum accepted absolute error for head profile.

0.35

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_boussinesq.toml

head_profile.row_spread

Maximum accepted cross-row spread for head profile.

0.08

validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_boussinesq.toml

Source Pointers#

  • validation_cases/README.md

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/README.md

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/reference.py

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/comparison.py

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/plotting.py

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/run_case.py

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/metadata.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflownwt.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/runtime_boussinesq.py

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_boussinesq.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/tolerances_modflow6.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_modflow6_irregular_tri.toml

  • validation_cases/analytical/steady/boussinesq_fixed_head_piecewise_k_1d/config_boussinesq.toml

  • validation_cases/analytical/steady/boussinesq_piecewise.py

Artifacts#

  • docs/source/_static/capability_gallery/validation/boussinesq_fixed_head_piecewise_k_1d__modflownwt.png

  • docs/source/_static/capability_gallery/validation/boussinesq_fixed_head_piecewise_k_1d__modflow6.png

  • docs/source/_static/capability_gallery/validation/boussinesq_fixed_head_piecewise_k_1d__modflow6_irregular_tri.png

  • docs/source/_static/capability_gallery/validation/boussinesq_fixed_head_piecewise_k_1d__boussinesq.png

  • docs/source/_static/capability_gallery/validation/boussinesq_fixed_head_piecewise_k_1d_summary.json stores the displayed metrics plus source hashes used by python -m tools.doc_gallery --check.