This document summarizes lessons learned from physically-based hydrological models. It discusses how distributed hydrological models can be useful tools for understanding processes like streamflow generation, solute transport, and groundwater-surface water interactions through detailed numerical simulations. While complex models may not be suited for predictions, they can serve as virtual laboratories for testing hypotheses. 2D and 3D simulations discussed provide insights into mechanisms of streamflow generation, the old water contribution to streams, and the impact of heterogeneity on solute transport. Simpler models that are less realistic but more generalizable, like the Boussinesq model, can also provide useful understanding when calibrated against more complex simulations. The document evaluates the performance of lumped transport models representing

1. Lessons (I have) learned from
physically-based hydrological models
Aldo Fiori
Università di Roma Tre, Italy
Workshop on coupled hydrological modeling
University of Padova, 23‐24 September 2015.

2. Role of distributed hydrological models
• Many processes of interest need further
investigation, e.g. streamflow generation
(rainfall-runoff transformation) solute
transport and travel time distribution, SW-
GW interactions, etc.
• Detailed numerical laboratories are useful
tools for understanding: not much/only for
predictions (complexity,
overparametrization, etc.) but as:
• numerical (virtual) experiments for understanding
• help formulate simplified and parsimonious models
• cheking hypotheses and models performance
• Particularly useful for hillslope processes,
that have a central role in catchment
hydrology (limited size, complexity of the
system and processes)

3. 2D simulations: streamflow generation
• Focus on streamflow generation
and age of water; setup was
loosely based on CB1 catchment
• The leading mechanism for this
particular case was groundwater
ridging (steep hillslope)
• Hydrological response can vary
considerably with the parameters
and it strongly depends on the
overall condutivity and the
conductivity contrast
• The prediction of time-to-peak is
very robust: streamflow
generation cannot be directly
related to “event” water
Fiori et al, JoH 2007

4. 2D simulations: Age of water
• Method: Continuous injection of a tracer
• Stream water is mostly “old”
• Partitioning mainly depends on the hillslope geometry and
the soil/bedrock conductivity contrast

5. Storage-Discharge relation
• Example on how to use a distributed model to infer simple
and parsimonious rules, to be employed in lumped
hydrological models
Ali et al, HP 2013

6. Insight from simpler models
• Useful information can be obtained through simple
models, less «realistic» but more prone to generalization
(still physically based); Approximations are dictated by
evidence from more complex simulations.
• Boussinesq flow: Dupuit assumption and «complete
mixing» in the vertical.
Fiori, WRR 2012

7. Complete mixing??
• Complete (or perfect) mixing
seldom encountered, even in the
vertical
• Reformulation of 2D purely-
advective transport and
integration along the vertical
vertical (assuming Dupuit)
• The final ADE is identical to the
one assuming perfect mixing
• Reason: «vertical sampling»
replaces «complete mixing»
• Mixing within the entire system
(reactor) is harder to justify
(relation to StorAGE function)
Courtesy of John Selker

8. Again on old water contribution..
• The important role of old
water is confirmed
• It is ruled by two simple
dimensionless parameters:
• The ratio between rain water
and pre-event water is crucial
• The dynamic component is not
so important
• The vertically integrated
ADE can be used for more
complex problems involving
advective transport at the
hillslope scale (no mixing is
involved!)

9. StorAGE functions
• Simple, Boussinesq-like models can be very helpful to gain insight on
the StorAGE selection function through a fully hydrodinamic model
• Example for steady-flow (analytical solution; more work on the way…)

10. 3D simulations: Flow
• 3D, fully
saturated/unsaturated,
heterogeneous setup;
system is ergodic;
Soil+subsoil system with
different properties
• Groundwater ridging is the
leading mechanism
• Hortonian flow is absent;
Direct precipitation on
saturated zone near the
river is present; ponded
areas vary in time
Fiori and Russo, WRR 2007

11. 3D simulations: Solute transport
 Focus on the travel
time distribution
after a pulse
 Major aims:
 Impact of
heterogeneity,
injection period,
external forcings
(precipitation, ET,
etc.)
 Check the validity of
common
assumptions/concep
tual models
Fiori and Russo, WRR 2008

12. Solute flux and Travel time distribution
• Heterogeneity not much
important
• Solute flux is highly variable
and it reflects the temporal
variability of precipitation; it
strongly depends on the
injection period
• Travel time distribution is not
unique (time-variant)
• ESS may help in reducing to
time-invariance
• Important effect of ET
(selective solute uptake)

13. Equivalent Steady State (ESS)
• Work with cumulate discharge instead
of calendar time (see e.g. Niemi)
• Tested with several configurations

14. Why the Gamma distribution?
• Power law: mainly determined
by fast, unsaturated flow in the
upper soil; exponent related to
soil properties;
• Exponential decay:
groundwater contribution
• Heterogeneity not important;
source-zone dispersion
dominates (similar to Rinaldo,
Marani, Rigon, 1991)
• Conclusions different from
Kirchner et al. (2000; 2001)

15. Solute transport modelling is a
complex hydrological problem
(Complex subsurface physical
and geochemical processes)
A meaningful and relevant
approach to quantitatively
estimate the transport of solute
into hillslopes or small
catchments is through the
analysis of the travel time
distribution (TTD).
Despite the increasing use of the TTD-based models, their
performances as function of the system flow condition (e.g. steady or
unsteady flow) have not been much explored to date.
Performance of lumped transport models

16. Virtual (numerical) experiments can help evaluate the
performance of some travel-time based models (Time invariant
and time variant).
The models are tested against the results from detailed and
high resolution numerical experiments employing a three
dimensional (3D) dynamic model of a conceptual hillslope with
real hydrological input (i.e. rainfall).
Advantage: all system variables and input/output are perfectly
known
Disadvantage: it’s not a real experiment!
Performance of lumped transport models

17. Numerical modelling
Flow – Richards Equation
•Transport – Advection – Diffusion
Equation Analytical
Solute
Transport
ModelsData sets
 Rainfall and Evaporation Data
(Denno, northern Italy, which belongs
to the Mediterranean humid climate)
 The hydraulic properties of the
system are heterogeneous, i.e. spatially
distributed – Random space function
Evaluation
Mass recovery
and
Concentration
Performance of lumped transport models

18. Analytical Models
Time invariant model based on
concentration (TIC)
Time invariant model based on
Flux (TIF)
Equivalent steady state
approximation (ESS)
Time variant model based on
random sampling/Complete mixing
(TV)
Input
Output

19. Time invariant model
based on concentration
(TIC)
Time invariant model based
on Flux (TIF)
Equivalent steady state
approximation (ESS)
Time variant model based on
random sampling (TV)
A widely employed approach
assuming a time-invariant travel time distribution
strictly valid only when the subsurface flow is stationary
Does not generally fulfil the basic continuity mass requirement under unsteady
flow conditions
where C₀(t) is inflow concentration, C(t) is the cumulated
outflow volume ps(t) is the transit time distribution (i.e. gamma)



0
00 )()()(*)()(  dtpCtptCtC ss

20. Replaced solute concentration with mass flux in the convolution in TIC model
Always fulfills mass continuity, and the total mass is recovered from the system
Partition parameter is introduced in order to model in the presence of
evapotranspiration (Botter et al.,2010)
 Thus, the solute fluxes which exit the system through Q and ET are written as
Time invariant model based
on concentration (TIC)
Time invariant model
based on Flux (TIF)
Equivalent steady state
approximation (ESS)
Time variant model based on
random sampling (TV)
where Q₀(t) is inflow, Q(t) is the cumulated outflow volume
ET(t) is evapotranspiration
       


0
0  dtpFtF sQ
        


0
0 1  dtpFtF ETET
)()()( 000 tCtQtF 
)()()( tCtQtF QQ 

21. Time invariant model based
on concentration (TIC)
Time invariant model based
on Flux (TIF)
Equivalent steady
state approximation
(ESS)
Time variant model based on
random sampling (TV)
ESS model implies that the same convolution appearing in TIC
can be applied by a simple rescaling of calendar times.
(Niemi,1977)
It fully preserves mass continuity
Following the ESS approach, the injection time (τ) and exit
time (t) of the solute flux are expressed by the newly introduced
rescaled times as:
where V₀(t) is the cumulated rainfall volume injected to the
control volume and V(t) is the cumulated outflow volume
     
t
R dQ
QQ
tV
0
0
0 1

   
t
R dQ
QQ
tV
t
0
1

         


0
00 * RRRRRsRRQ tptCdtpCtC 

22. Time invariant model based
on concentration (TIC)
Time invariant model based
on Flux (TIF)
Equivalent steady state
approximation (ESS) Time variant model based
on random sampling (TV)
It is based on a time-variant formulation of TTD
A more consistent and robust approach to model solute transport
complete and instantaneous mixing between the injected solute
and the water stored in the system is often assumed
Requires the definition of travel time distributions conditioned at
both injection and exit times
where S(t) is the total water storage and M(t) is the total mass in the
system
       tETtQtQ
dt
tdS
 0
       tFtFtF
dt
tdM
ETQ  0
      tCtC
tS
tQ
dt
tdC
 0
0
)(
)(
     
 
 



t dx
xS
xQ
de
S
Q
CtC
t
0
)(0
0
0



 
     
 
 




t dx
xS
xETxQ
de
tS
Q
CtC
t
0
)(
)(
0
0 

 

23. Calibration is made in the first period (spring) injection,
while validation is performed over the other three periods
(summer, fall, winter)
Two scenarios are considered
•Rain only (RO) in which no ET is present
•Rain and ET case in which ET is considered
In the RET scenario, The partition parameter θ is
calibrated through two step iteration. Three parameter (θ ,β
and α) calibration with constant θ and then assuming the
water flow route is described by the same TTD, temporarilly
variable θ(t) can be obtained through
       dtQtp
tQ
t   ,
)(
1
0

24. Result and Discussion –numerical
results of study cases

25. TIC
Result : Raifall only scenario (RO)
Spring
Winter
Injectiontime

26. TIF
Result : Raifall only scenario (RO)
Spring
Winter
Injectiontime

27. Result : Raifall only scenario (RO)
ESS
Spring
Winter
Injectiontime
R2 = 0.828

28. Result : Raifall only scenario (RO)
TV
Spring
Winter
Injectiontime
Comment: zero parameters, but «active» storage needed to be fixed…

29. Rainfall and ET scenario (RET)
In some of the previous studies, the ET-related solute flux has
been neglected or taken as proportional to the streamflow
concentration (Rodhe et al.,1996; Benettin et al.,2013; Bertuzzo
et al.,2013).
 In fact, solute concentration through the plant roots is typically
much more difficult to measure than concentration streamflow
(Rodhe et al.,1996).
•The relatively poor behavior of all models highlights the
importance of ET when studying solute transport in areas in
which ET is relevant (Van der Velde et al HP2015)
•The total mass is fully recovered. However, only the total mass
is preserved, while the separate contributions MQ and MET may
different from the "real" ones

30. TV
Result: Raifall and ET scenario (RET)
Spring
Winter
Injectiontime

31. Conclusions
• Water flow and solute transport in hillslopes are
challenging areas of research
• Role of numerical models:
• Understanding of the principal physical processes
• Test common assumptions/models
• Help in developing and testing simplified models
• Much insight can be gained from models
• Numerical models should be as much realistic as
possible (3D, sat/unsat, SW/GW, uptake by roots,
heterogeneous, etc.)
• Simple, lumped models are necessary, but they need to
have strong physical foundations

32. References
• Ali, M., A. Fiori, G. Bellotti, Analysis of the nonlinear storage-discharge relation for hillslope
through 2D numerical modelling. HYDROLOGICAL PROCESSES, 27:2683-2690, DOI:
10.1002/hyp.9397, 2013
• Ali, M., A. Fiori, D. Russo, A comparison of travel-time based catchment transport models, with
application to numerical experiments, JOURNAL OF HYDROLOGY, 511, pg. 605-
618, http://dx.doi.org/10.1016/j.jhydrol.2014.02.010, 2014.
• Fiori, A. Old water contribution to streamflow: Insight from a linear Boussinesq model. WATER
RESOURCES RESEARCH, 48(6), W06601, DOI: 10.1029/2011WR011606, 2012
• Fiori, A., M. Romanelli, D.J. Cavalli, D.Russo, Numerical experiments of streamflow generation in
steep catchments, JOURNAL OF HYDROLOGY, 339, 183-192, 2007.
• Fiori, A., D. Russo, Numerical Analyses of Subsurface Flow in a Steep Hillslope under Rainfall:
The Role of the Spatial Heterogeneity of the Formation Hydraulic Properties, WATER RESOURCES
RESEARCH, 43, W07445, doi:10.1029/2006WR005365, 2007
• Russo, D., A. Fiori. Equivalent Vadose Zone Steady-State Flow: An Assessment of its Capability
to Predict Transport in a Realistic Combined Vadose Zone - Groundwater Flow System. WATER
RESOURCES RESEARCH, 44, W09436, doi:10.1029/ 2007WR006170, 2008.
• Fiori, A., D. Russo, Travel Time Distribution in a Hillslope: Insight from Numerical Simulations.
WATER RESOURCES RESEARCH, 44, W12426, doi:10.1029/2008WR007135, 2008.
• Russo, D., and A. Fiori, Stochastic analysis of transport in a combined heterogeneous vadose
zone–groundwater flow system. WATER RESOURCES RESEARCH, 45, W03426,
doi:10.1029/2008WR007157, 2009.
• Fiori, A., D. Russo, M. Di Lazzaro. Stochastic analysis of transport in hillslopes: Travel time
distribution and source zone dispersion. WATER RESOURCES RESEARCH, 45, W08435, 2009.