Conservation law
mass
Residual equation
Two otherwise identical simulations are compared: a control case and a perturbed case in which surface-water stage is increased.
The changes in directional groundwater–surface water exchange are
$$\Delta Q_{\mathrm{SW\rightarrow GW}}
=
Q_{\mathrm{SW\rightarrow GW}}^{\mathrm{high\ stage}}
-
Q_{\mathrm{SW\rightarrow GW}}^{\mathrm{control}}$$
and
$$\Delta Q_{\mathrm{GW\rightarrow SW}}
=
Q_{\mathrm{GW\rightarrow SW}}^{\mathrm{high\ stage}}
-
Q_{\mathrm{GW\rightarrow SW}}^{\mathrm{control}}.$$
For an actively connected groundwater–surface water system, increasing surface-water stage should shift the exchange toward groundwater. Therefore,
$$\Delta Q_{\mathrm{SW\rightarrow GW}} \ge 0$$
and
$$\Delta Q_{\mathrm{GW\rightarrow SW}} \le 0.$$
Equivalently, the net groundwater contribution to surface water should not increase:
$$\Delta Q_{\mathrm{GW,net}} \le 0.$$
All exchange fluxes have units of water volume per unit time ($L^3T^{-1}$), or equivalent depth per unit time ($LT^{-1}$).
Tolerance and its denominator
The primary criterion is the direction of the exchange response. Increasing surface-water stage must not produce a substantial increase in groundwater discharge to surface water or a substantial decrease in surface-water leakage to groundwater.
The tolerance is scaled by the gross groundwater–surface water exchange in the control simulation:
$$D =
Q_{\mathrm{GW\rightarrow SW}}^{\mathrm{control}}
+
Q_{\mathrm{SW\rightarrow GW}}^{\mathrm{control}}.$$
Changes smaller than 5% of this gross exchange are treated as numerical tolerance:
$$T = 0.05D.$$
A change larger than this tolerance in the physically wrong direction fails the probe. The synthetic perturbation is chosen to produce an exchange response substantially larger than this tolerance in the reference model.
If the control gross exchange approaches zero, an absolute flux tolerance scaled to the synthetic case is used instead.
How would an unphysical model fail this?
A deliberately unphysical model could conserve water and report internally consistent groundwater–surface water fluxes while failing to respond correctly to a change in the hydraulic forcing.
For example, suppose the control simulation reports 100 units of groundwater discharge to surface water and 30 units of surface-water leakage to groundwater, giving a net groundwater contribution of 70 units. A broken model could return exactly the same three fluxes after surface-water stage is substantially increased.
Such a model would satisfy
$$Q_{\mathrm{GW\rightarrow SW}}
-
Q_{\mathrm{SW\rightarrow GW}}
=
Q_{\mathrm{GW,net}}$$
in both simulations and could also close the overall water budget. However, it would show no groundwater–surface water response to the imposed change in surface-water stage.
A second broken model could respond in the wrong direction, increasing groundwater discharge to surface water or decreasing surface-water leakage to groundwater when surface-water stage is raised.
The directional-response criterion catches both cases. Thus, this probe tests behavior that water-budget closure and groundwater–surface water flux accounting consistency alone cannot guarantee.
How is the case generated?
A small synthetic coupled groundwater–surface water system is generated from a fixed random seed. Two otherwise identical simulations are run.
In the control case, groundwater head and surface-water stage are prescribed so that the groundwater and surface-water systems are hydraulically connected and measurable exchange occurs across their interface.
In the perturbed case, surface-water stage is increased by a prescribed amount while precipitation, evapotranspiration demand, groundwater boundary conditions, hydraulic properties, and initial conditions are kept unchanged.
The synthetic system is constructed away from dry, disconnected, or threshold conditions so that increasing surface-water stage unambiguously shifts the hydraulic gradient toward groundwater. The stage perturbation is also chosen to be large enough that the expected directional response is substantially greater than numerical tolerance.
No exact groundwater–surface water exchange magnitude is prescribed. The probe tests only whether the model responds to the controlled hydraulic perturbation in the physically required direction.
Will you build it?
Yes — assign it to me
Conservation law
mass
Residual equation
Two otherwise identical simulations are compared: a control case and a perturbed case in which surface-water stage is increased.
The changes in directional groundwater–surface water exchange are
and
For an actively connected groundwater–surface water system, increasing surface-water stage should shift the exchange toward groundwater. Therefore,
and
Equivalently, the net groundwater contribution to surface water should not increase:
All exchange fluxes have units of water volume per unit time ($L^3T^{-1}$ ), or equivalent depth per unit time ($LT^{-1}$ ).
Tolerance and its denominator
The primary criterion is the direction of the exchange response. Increasing surface-water stage must not produce a substantial increase in groundwater discharge to surface water or a substantial decrease in surface-water leakage to groundwater.
The tolerance is scaled by the gross groundwater–surface water exchange in the control simulation:
Changes smaller than 5% of this gross exchange are treated as numerical tolerance:
A change larger than this tolerance in the physically wrong direction fails the probe. The synthetic perturbation is chosen to produce an exchange response substantially larger than this tolerance in the reference model.
If the control gross exchange approaches zero, an absolute flux tolerance scaled to the synthetic case is used instead.
How would an unphysical model fail this?
A deliberately unphysical model could conserve water and report internally consistent groundwater–surface water fluxes while failing to respond correctly to a change in the hydraulic forcing.
For example, suppose the control simulation reports 100 units of groundwater discharge to surface water and 30 units of surface-water leakage to groundwater, giving a net groundwater contribution of 70 units. A broken model could return exactly the same three fluxes after surface-water stage is substantially increased.
Such a model would satisfy
in both simulations and could also close the overall water budget. However, it would show no groundwater–surface water response to the imposed change in surface-water stage.
A second broken model could respond in the wrong direction, increasing groundwater discharge to surface water or decreasing surface-water leakage to groundwater when surface-water stage is raised.
The directional-response criterion catches both cases. Thus, this probe tests behavior that water-budget closure and groundwater–surface water flux accounting consistency alone cannot guarantee.
How is the case generated?
A small synthetic coupled groundwater–surface water system is generated from a fixed random seed. Two otherwise identical simulations are run.
In the control case, groundwater head and surface-water stage are prescribed so that the groundwater and surface-water systems are hydraulically connected and measurable exchange occurs across their interface.
In the perturbed case, surface-water stage is increased by a prescribed amount while precipitation, evapotranspiration demand, groundwater boundary conditions, hydraulic properties, and initial conditions are kept unchanged.
The synthetic system is constructed away from dry, disconnected, or threshold conditions so that increasing surface-water stage unambiguously shifts the hydraulic gradient toward groundwater. The stage perturbation is also chosen to be large enough that the expected directional response is substantially greater than numerical tolerance.
No exact groundwater–surface water exchange magnitude is prescribed. The probe tests only whether the model responds to the controlled hydraulic perturbation in the physically required direction.
Will you build it?
Yes — assign it to me