Boussinesq Sloping-Substratum Fixed-Head 1D

Note

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Steady synthetic groundwater-flow case used to validate one sloping substratum profile beyond the trivial constant-thickness solution.

Case Setup

• quasi-1D strip (40 x 5, single layer for launcher-backed runs),

• linear topography from 25 m on the west side to 20 m on the east side,

• linear substratum from 5 m on the west side to 0 m on the east side,

• geometric aquifer thickness 20 m,

• west/east heads 15 m / 5 m,

• no recharge, no drainage, homogeneous K = 1e-4 m/s.

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

What It Shows

• keep the same simple sloping geometry as the constant-thickness case,

• remove recharge and drainage so the comparison isolates only the lateral Boussinesq balance on a sloping bottom,

• compare modflow_nwt, modflow6, modflow6_irregular_tri, and the local boussinesq backend against one semi-analytical no-recharge profile.

• Direct validation of sloping-bottom flow beyond the trivial constant-thickness branch

• 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: row-averaged head profile along x,

• reference: steady no-recharge Boussinesq relation on b(x) above a linear substratum, solved through its implicit integral law,

\[q=K\,b\left(S_0-\frac{\mathrm{d}b}{\mathrm{d}x}\right),\qquad S_0=-\frac{\mathrm{d}z_b}{\mathrm{d}x}\]

\[x-x_{\min}=\int_{b_w}^{b(x)} \frac{K\,\beta\,\eta-q}{K\,\beta^2}\,\mathrm{d}\eta,\qquad \beta=S_0,\qquad H(x)=z_b(x)+b(x)\]

Solver Coverage

• Default solver: MODFLOW-NWT

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

MODFLOW-NWT

Boussinesq Sloping-Substratum Fixed-Head 1D rendered with MODFLOW-NWT for the analytical gallery.

Metrics - Head-profile RMSE: 0.0146 m - Head-profile max abs error: 0.0215 m - Cross-row head spread: 4.67e-07 m - Reference discharge per width: 1.942e-05 m2/s

• Config file: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml

• Tolerances: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml

• Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_sloping_substratum_fixed_head_1d.run_case --no-show --solver modflow_nwt

MODFLOW 6

Boussinesq Sloping-Substratum Fixed-Head 1D rendered with MODFLOW 6 for the analytical gallery.

Metrics - Head-profile RMSE: 0.0006 m - Head-profile max abs error: 0.0008 m - Cross-row head spread: 4.05e-06 m - Reference discharge per width: 1.942e-05 m2/s

• Config file: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml

• Tolerances: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6.toml

• Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_sloping_substratum_fixed_head_1d.run_case --no-show --solver modflow6

MODFLOW 6 irregular triangles

Boussinesq Sloping-Substratum Fixed-Head 1D rendered with MODFLOW 6 irregular triangles for the analytical gallery.

Metrics - Head-profile RMSE: 0.0395 m - Head-profile max abs error: 0.0832 m - Cross-row head spread: 1.78e-15 m - Reference discharge per width: 1.942e-05 m2/s

• Config file: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml

• Tolerances: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6_irregular_tri.toml

• Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_sloping_substratum_fixed_head_1d.run_case --no-show --solver modflow6_irregular_tri

Boussinesq

Boussinesq Sloping-Substratum Fixed-Head 1D rendered with Boussinesq for the analytical gallery.

Metrics - Head-profile RMSE: 0.0516 m - Head-profile max abs error: 0.1259 m - Cross-row head spread: 1.78e-15 m - Reference discharge per width: 1.942e-05 m2/s

• Config file: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_boussinesq.toml

• Tolerances: validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml

• Expected shape: 5 x 40

python -m validation_cases.analytical.steady.boussinesq_sloping_substratum_fixed_head_1d.run_case --no-show --solver boussinesq

Reproduce

Run the underlying example or validation case with:

python -m validation_cases.analytical.steady.boussinesq_sloping_substratum_fixed_head_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_sloping_substratum_fixed_head_1d/metadata.toml
xmax	Maximum x coordinate of the analytical reference domain.	400	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
west_head	Fixed west-boundary hydraulic head used by the reference solution.	15 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
east_head	Fixed east-boundary hydraulic head used by the reference solution.	5 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
topography_base_elevation_m	Base elevation used by the synthetic reference topography.	20 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
topography_right_to_left_amplitude_m	Right-to-left topographic amplitude used by the hillslope reference.	5 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
bottom_base_elevation_m	Base elevation used by the synthetic reference substratum.	0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
bottom_right_to_left_amplitude_m	Right-to-left amplitude used by the synthetic reference substratum.	5 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
aquifer_thickness_m	Aquifer thickness used by the reference formulation.	20 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_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/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml
profile_axis	Axis along which the validation profile is extracted.	0	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_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_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.grid.length_x	Synthetic-domain length along the x axis.	400.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.grid.length_y	Synthetic-domain length along the y axis.	50.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.grid.nx	Grid cell count along the x axis.	40	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.grid.ny	Grid cell count along the y axis.	5	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.topography.kind	Synthetic topography shape used by the benchmark.	linear	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.topography.base_elevation	Base land-surface elevation of the synthetic topography.	20 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
geographic.synthetic.topography.right_to_left_amplitude	Right-to-left topographic amplitude used by the synthetic topography.	5	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
domain.depth_model.type	Depth model used to build the aquifer support.	constant_thickness	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
domain.depth_model.thickness	Aquifer or support thickness used by the benchmark.	20.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.flow_regime	Steady or transient flow regime used by the benchmark.	steady	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.active_sinks_sources	Sink and source families activated in the benchmark.	[]	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.active_bc	Boundary-condition families activated in the benchmark.	[west_side, east_side]	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.param_list	Hydraulic parameter families explicitly configured by the benchmark.	[K]	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.param.K.field.kind	Parameterization mode used for K.	homogeneous	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.param.K.field.value	Homogeneous K value used by the benchmark.	1e-4 m/s	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.ic.type	Initial-condition policy used to start the benchmark.	custom	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.ic.value	Scalar initial head or top offset used by the initial-condition policy.	10.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.bc.dirichlet.west_side.value	Fixed head applied on the west side boundary.	15.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml
flow.bc.dirichlet.east_side.value	Fixed head applied on the east side boundary.	5.0 m	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml

Solver-Specific Overrides

MODFLOW-NWT

Field	Meaning	Value	Source
modflownwt.sgrid.planar.mode	Planar support construction mode used by MODFLOW-NWT.	resample_to_shape	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml
modflownwt.sgrid.planar.nx	Planar support cell count along x used by MODFLOW-NWT.	40	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml
modflownwt.sgrid.planar.ny	Planar support cell count along y used by MODFLOW-NWT.	5	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml
modflownwt.sgrid.planar.resampling	Planar support resampling mode used by MODFLOW-NWT.	nearest	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml
modflownwt.sgrid.vertical.nlay	Number of vertical layers used by MODFLOW-NWT.	1	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml

MODFLOW 6

Field	Meaning	Value	Source
modflow6.runtime.mf6_ims_complexity	Linear-solver complexity preset used by MODFLOW 6.	SIMPLE	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.runtime.mf_verbose	Solver-specific override applied to MODFLOW 6.	false	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.process_specific.vka	Vertical anisotropy ratio passed to MODFLOW 6.	1	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.sgrid.planar.mode	Planar support construction mode used by MODFLOW 6.	resample_to_shape	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.sgrid.planar.nx	Planar support cell count along x used by MODFLOW 6.	40	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.sgrid.planar.ny	Planar support cell count along y used by MODFLOW 6.	5	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.sgrid.planar.resampling	Planar support resampling mode used by MODFLOW 6.	nearest	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml
modflow6.sgrid.vertical.nlay	Number of vertical layers used by MODFLOW 6.	1	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml

MODFLOW 6 irregular triangles

Field	Meaning	Value	Source
modflow6.runtime.mf6_ims_complexity	Linear-solver complexity preset used by MODFLOW 6.	COMPLEX	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml
modflow6.runtime.mf_verbose	Solver-specific override applied to MODFLOW 6.	false	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml
modflow6.process_specific.vka	Vertical anisotropy ratio passed to MODFLOW 6.	1	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml
modflow6.sgrid.vertical.nlay	Number of vertical layers used by MODFLOW 6.	1	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_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_sloping_substratum_fixed_head_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_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml

Boussinesq

Field	Meaning	Value	Source
flow.runtime_backend	Runtime backend selected for the in-house solver.	petsc	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_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_sloping_substratum_fixed_head_1d/metadata.toml
output.expected_shape	Expected spatial output shape checked by the validation helper.	[5, 40]	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml

Acceptance Criteria by Solver

MODFLOW-NWT

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_sloping_substratum_fixed_head_1d/metadata.toml
head_profile.rmse	Maximum accepted root-mean-square error for head profile.	0.08	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml
head_profile.max_abs_error	Maximum accepted absolute error for head profile.	0.14	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml
head_profile.row_spread	Maximum accepted cross-row spread for head profile.	2e-05	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml

MODFLOW 6

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_sloping_substratum_fixed_head_1d/metadata.toml
head_profile.rmse	Maximum accepted root-mean-square error for head profile.	0.03	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6.toml
head_profile.max_abs_error	Maximum accepted absolute error for head profile.	0.06	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6.toml
head_profile.row_spread	Maximum accepted cross-row spread for head profile.	1e-05	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6.toml

MODFLOW 6 irregular triangles

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_sloping_substratum_fixed_head_1d/metadata.toml
head_profile.rmse	Maximum accepted root-mean-square error for head profile.	0.08	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6_irregular_tri.toml
head_profile.max_abs_error	Maximum accepted absolute error for head profile.	0.18	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6_irregular_tri.toml
head_profile.row_spread	Maximum accepted cross-row spread for head profile.	1e-09	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6_irregular_tri.toml

Boussinesq

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_sloping_substratum_fixed_head_1d/metadata.toml
head_profile.rmse	Maximum accepted root-mean-square error for head profile.	0.08	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml
head_profile.max_abs_error	Maximum accepted absolute error for head profile.	0.14	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml
head_profile.row_spread	Maximum accepted cross-row spread for head profile.	2e-05	validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml

Source Pointers

• validation_cases/README.md

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/README.md

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/reference.py

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/comparison.py

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/plotting.py

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/run_case.py

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/metadata.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_common.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/runtime_boussinesq.py

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/tolerances_modflow6_irregular_tri.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflownwt.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_modflow6_irregular_tri.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum_fixed_head_1d/config_boussinesq.toml

• validation_cases/analytical/steady/boussinesq_sloping_substratum.py

• validation_cases/shared/boussinesq_uniform_strip.py

Artifacts

• docs/source/_static/capability_gallery/validation/boussinesq_sloping_substratum_fixed_head_1d__modflownwt.png

• docs/source/_static/capability_gallery/validation/boussinesq_sloping_substratum_fixed_head_1d__modflow6.png

• docs/source/_static/capability_gallery/validation/boussinesq_sloping_substratum_fixed_head_1d__modflow6_irregular_tri.png

• docs/source/_static/capability_gallery/validation/boussinesq_sloping_substratum_fixed_head_1d__boussinesq.png

• docs/source/_static/capability_gallery/validation/boussinesq_sloping_substratum_fixed_head_1d_summary.json stores the displayed metrics plus source hashes used by python -m tools.doc_gallery --check.