This document discusses improvements made to the CATHY hydrological model to reduce mass balance errors in model coupling. The authors analyzed sources of subsurface, surface, and coupling errors. They proposed a modified time step control strategy and new node-to-cell interpolation algorithm. Case studies showed the approaches reduced surface errors by up to 97% and improved recession limb simulation. The conclusions recommend further analysis of mesh size effects and application to real basins.

1. CONTROL OF
COUPLING MASS BALANCE ERROR
IN THE CATHY MODEL
Department
of
Engineering
“Enzo
Ferrari,”
University
of
Modena
and
Reggio
Emilia
Marcello Fiorentini
Stefano Orlandini, Claudio Paniconi
Workshop on Coupled Hydrological Modeling
23-24 September 2015 (Padua)

2. Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -
CATchment Hydrology (CATHY) model
University of Modena and Reggio Emilia, University of Padua, University of Quebec
Subsurface water flow
• 3D solver of Richards equation for variably saturated porous media
• pressure head based formulation
Surface water flow propagation
• quasi-2D (path-based) surface routing
Coupling algorithm
• node-to-cell and cell-to-node
interpolation algorithm
• sequential iterative coupling
strategy (Dagès et al., 2012)

3. • Develop criteria to measure the
sources of error in the model
• Run systematic tests to assess
model error
• Propose improvements (e.g.,
information passing and time
stepping schemes)
Objective:
Explore the nature of mass balance errors in a coupled distributed hydrological
model (CATHY)
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

4. Mass balance errors
• Subsurface error ( )
• Surface error ( )
• Coupling error ( )
)1()1()1( +++
+Δ−= k
ssw
k
ssw
k
ssw VSε
)1()1()1()1( ++++
−+Δ−= k
out
k
sw
k
sw
k
sw VVSε
)1()1()1()1()1( +++++
Δ−Δ−−= k
ssw
k
sw
k
out
k
a
k
SSVQε
sswε
swε
cε
)1()1()1()1( ++++
−−= k
sw
k
ssw
kk
c εεεε
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

5. Dimensionless indices
• Saturation index (SI)
• Degree of coupling index (CI)
b
sat
A
A
SI =
exf
SA
exf
SA
VV
VV
CI
+
+
=
inf
inf
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

6. Case studies
• Sloping plane (Sulis et. al., 2010)
• Tilted v-catchment (Sulis et. al., 2010)
• Enza River drainage basin
• Atmospheric forcing (step functions)
• Hortonian effect
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

7. Case studies
• Sloping plane
• Tilted v-catchment
• Enza River drainage basin
• Atmospheric forcing (step
functions)
• Hortonian effect
• Dimensionless indices
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

8. Sloping plane
• Sensitivity of subsurface and
coupling error to time step size
• Sensitivity of surface error to
mesh size
Sensitivity of mass balance
errors to recession limb
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

9. ⎪
⎩
⎪
⎨
⎧
≥+Δ
<≤Δ
<−Δ
=Δ +
2
)(
21
)(
1
)(
)1(
,
,
,
ititatm
itititt
ititstm
t
k
m
k
k
r
k
( )
( )( )
( )⎪
⎩
⎪
⎨
⎧
≥+Δ−
<≤Δ−
<−Δ−
=Δ +
2
)()(
21
)()(
1
)()(
)1(
,
,,1min
,
ititatWm
ititittWm
ititstWm
t
kk
CIm
kk
CIm
kk
CIr
k
)1(
)(
)(
1 +
−= k
k
k
CI
CI
CI
W
Heuristic time step size control
Normalized variation of the
degree of coupling index
Modified time
step size
control
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

10. Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

11. • interpolation of exchange fluxes
• surface water propagation
• cell water balance
• reconstruction of ponding heads
)()( k
L
celltonodek
s qq ⎯⎯⎯⎯ →⎯ −−
)1()1()1( +−−++
⎯⎯⎯⎯ →⎯→ knodetocellkk
sw hhS
sqQI
dt
dS
LΔ+−=
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

12. Interpolation exchange fluxes
• geometric interpolation
• new interpolation algorithm
⎪
⎩
⎪
⎨
⎧
=
>
Δ= ∑=
0,0
0,
4
1
),(
),(
4
1
),(
)(),(
ik
ik
j
j
jk
swiik
L
S
SwQ
sq )(S
jjj NNw =
∑=Δ
=
4
1
),(
)(
),(
4
1
j
jk
swi
ik
L Q
s
q
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

13. CATHY
modifications:
• original code (O)
• modified time
step control
strategy (TC)
• new node-to-cell
interpolation
algorithm (IA)
• combination
between TC and
IA (IT)
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

14. CATHY modifications:
• original code (O)
• modified time step control
strategy (TC)
• new node-to-cell
interpolation algorithm (IA)
• combination between TC
and IA (IT)
Results:
• Reduction of surface error
(up to 97%)
• Faster recession limbs
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -

15. Conclusion
• Sources of mass balance errors in CATHY were analyzed
• Global indices were presented to describe the hydrologic response of water
basins
• Modified algorithms to control the time step size were presented
• A new interpolation algorithm to pass exchange fluxes from surface to subsurface
compartment was introduced
• Mesh size with local refinements will be investigated
• Application to real water basins
Workshop on Coupled Hydrological Modeling
- Control of Coupling Mass Balance Error in the CATHY model -