Boussinesq Hillslope Recharge-Step Interception 1D

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Transient synthetic groundwater-flow case used to validate the dense in-house flow/boussinesq runtime on a sloping hillslope where a recharge step causes an interception zone to appear progressively from the outlet toward the divide.

Case Setup

• same sloping strip geometry as the steady interception benchmark (40 x 3),

• linear topography from 10 m to 5 m,

• flat bottom at 0 m,

• initial state: constant head 5 m,

• east fixed head 5 m,

• recharge step to 2 mm/day from the first transient period,

• 12 periods of 10 days.

• Available solver variants: Boussinesq.

What It Shows

• validate the transient onset of interception on a topographic slope,

• compare the time of first inland interception against a linearized approximation,

• compare the discrete interception trajectory x_int(t) on the early-moving front where the approximation remains informative.

• Transient appearance of seepage/interception on a hillslope with the PETSc TS VI Boussinesq runtime

• Primary metrics: onset-time error, interception-trajectory RMSE, interception-trajectory max abs error, cross-row spread.

Analytical Reference

• Linearized onset approximation for the moving interception front

\[S_y\frac{\partial \eta}{\partial t}=T_0\frac{\partial^2 \eta}{\partial x^2}+R_0\,H(t),\qquad \eta_{\mathrm{steady}}(x)=\frac{R_0}{2T_0}\left(L^2-x^2\right)\]

\[z_{\mathrm{top}}(x)=z_{\mathrm{toe}}+S\left(L-x\right),\qquad h(x,t)=z_{\mathrm{top}}(x)\ \text{defines the interception front}\]

Solver Coverage

• Default solver: Boussinesq

• Available variants: Boussinesq

Boussinesq Hillslope Recharge-Step Interception 1D rendered with Boussinesq for the analytical gallery.

Metrics - Analytical onset time: 60.0 d - Numerical onset time: 70.0 d - Onset-time error: 10.0 d - Interception-trajectory RMSE: 5.00 m - Interception-trajectory max abs error: 10.00 m - Trajectory reversal: 0.00e+00 m - Cross-row head spread: 8.88e-16 m

• Config file: validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/config_boussinesq.toml

• Tolerances: validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml

• Expected output: 12 periods, spatial shape 3 x 40

python -m validation_cases.analytical.transient.boussinesq_hillslope_recharge_step_interception_1d.run_case --no-show

Reproduce

Run the underlying example or validation case with:

python -m validation_cases.analytical.transient.boussinesq_hillslope_recharge_step_interception_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/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
xmax	Maximum x coordinate of the analytical reference domain.	400	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
base_head_m	Baseline hydraulic head around which the linearized reference is expressed.	5 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
toe_elevation_m	Toe elevation used by the hillslope reference geometry.	5 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
topography_slope_m_per_m	Topographic slope used by the hillslope reference.	0.0125 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
recharge_mm_day	Recharge rate used by the reference solution.	2 mm/day	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
hydraulic_conductivity_m_per_s	Hydraulic conductivity used by the analytical or benchmark reference.	0.0001 m/s	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
reference_saturated_thickness_m	Reference saturated thickness used by the linearized formulation.	5 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
specific_yield	Specific yield used by the transient reference formulation.	0.1 -	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
profile_axis	Axis along which the validation profile is extracted.	0	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
numerical_contact_tolerance_m	Tolerance used when comparing the numerical contact or interception position.	0.075 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
inland_contact_threshold_x_m	x threshold used to detect inland contact in the reference solution.	385 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
n_terms	Number of terms retained in the analytical series expansion.	400	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
dt_seconds	Reference time step used by the analytical evaluator.	864000 s	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml

Solver-Specific Overrides

Boussinesq

• No additional override beyond the common validation setup.

Acceptance Criteria

Field	Meaning	Value	Source
output.observable_name	Simulated observable compared against the reference solution.	watertable_elevation	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
output.expected_periods	Expected number of stored time periods checked by the validation helper.	12	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
output.expected_spatial_shape	Expected spatial shape for each stored time step.	[3, 40]	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml

Acceptance Criteria by Solver

Boussinesq

Field	Meaning	Value	Source
expected_output	Expected output shape or time-space layout checked for this solver.	Expected output: 12 periods, spatial shape 3 x 40	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml
onset.time_error_days	Acceptance threshold for onset.time_error_days.	31 days	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml
trajectory.rmse_m	Acceptance threshold for trajectory.rmse_m.	9 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml
trajectory.max_abs_error_m	Acceptance threshold for trajectory.max_abs_error_m.	21 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml
trajectory.reversal_m	Acceptance threshold for trajectory.reversal_m.	1e-06 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml
uniformity.row_spread	Maximum accepted cross-row spread for uniformity.	0.0001	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml
contact.max_positive_clearance_m	Acceptance threshold for contact.max_positive_clearance_m.	1e-06 m	validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml

Source Pointers

• validation_cases/README.md

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/README.md

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/reference.py

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/comparison.py

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/plotting.py

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/run_case.py

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/metadata.toml

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/runtime_boussinesq.py

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/tolerances.toml

• validation_cases/analytical/transient/boussinesq_hillslope_recharge_step_interception_1d/config_boussinesq.toml

• validation_cases/analytical/transient/linearized_unconfined_1d.py

Artifacts

• docs/source/_static/capability_gallery/validation/boussinesq_hillslope_recharge_step_interception_1d__boussinesq.png

• docs/source/_static/capability_gallery/validation/boussinesq_hillslope_recharge_step_interception_1d_summary.json stores the displayed metrics plus source hashes used by python -m tools.doc_gallery --check.