Unified notation

One reference list for every symbol used in the HydroModPy theory pages. SI units are the default. Per-page math notes link back here via the labels at the end of each row, so the same variable is never defined twice.

Hydraulic state and forcing

Symbol	Description	SI unit	Notes / cross-refs
h	Hydraulic head measured from a fixed datum	m	Field variable solved by every flow backend; see MODFLOW Governing Equation And CVFD Formulation.
z_top	Land surface elevation	m	Used in the Boussinesq surface-interaction closures.
z_bot	Aquifer bottom elevation	m	Domain depth declared via [domain.depth_model].
b = h - z_bot	Saturated thickness	m	Drives transmissivity in unconfined regimes.
q_x, q_y, q_z	Specific discharge along axis	m s-1	Darcy flux per unit area; see Solver Architecture.
Q	Volumetric discharge	m3 s-1	Streamflow, well rate, drainage outflow.
R	Recharge rate (positive into the aquifer)	m s-1	Forced via [flow.bc] recharge entries.
q_ex	Surface-interaction exchange flux	m s-1	Boussinesq closure between aquifer and ground surface.

Hydraulic parameters

Symbol	Description	SI unit	Notes / cross-refs
K	Saturated hydraulic conductivity	m s-1	Field declared in [flow.param.K]; supports homogeneous, heterogeneous, anisotropic.
T = K * b	Transmissivity	m2 s-1	Used in linearized Boussinesq variants.
S_s	Specific storage	m-1	Storage per unit volume.
S_y	Specific yield	dimensionless	Drainable porosity in unconfined regimes.
n	Porosity	dimensionless	Required for transport formulations.
alpha_L, alpha_T	Longitudinal and transverse dispersivities	m	Transport package input (MODFLOW 6 GWT, MT3DMS).

Geometry and discretisation

Symbol	Description	SI unit	Notes / cross-refs
L	Domain characteristic length (1D, 2D)	m	Used in analytical validation cases.
A	Catchment surface area	m2	Reported per project; gallery shows area_km2.
dx, dy	Cell size on a structured grid	m	Set via [<solver>.sgrid.planar].
nlay	Number of vertical layers	dimensionless	Set via [<solver>.sgrid.vertical].
dt	Solver internal timestep	s	Adaptive in PETSc backend; fixed in MODFLOW.
Pe	Mesh Peclet number for transport	dimensionless	Used to assess oscillations on coarse meshes.

Time and calibration

Symbol	Description	SI unit	Notes / cross-refs
t	Simulation time	s	Reported in catalog as ISO date when calendar metadata exists.
T_sim	Total simulated horizon	s	Set via [simulation.time].
RMSE	Root-mean-square error of an observable	same as observable	Default calibration objective.
NSE	Nash-Sutcliffe efficiency	dimensionless	Available via [calibration.objective].
KGE	Kling-Gupta efficiency	dimensionless	Optional alternative to NSE.

Conventions

• All times in TOML accept SI units ("3600 s", "1 h", "0.5 d"). The Pydantic Pint integration normalises them.

• Lengths default to metres; declarations may override with explicit units ("500 m", "2 km").

• Vector quantities follow the right-hand rule with z pointing up.

• Sign convention: Q > 0 and R > 0 represent flux into the aquifer.

For the in-scope backends and what each one represents, see Solver Capability Matrix.