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.
Lessons (I have) learned from physically-based hydrological models
Aldo FioriUniversità di Roma Tre, Italy
Workshop on coupled hydrological modelingUniversity of Padova, 23‐24 September 2015.
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 usefultools 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 catchmenthydrology (limited size, complexity of the system and processes)
2D simulations: streamflow generation• Focus on streamflow generation
and age of water; setup wasloosely based on CB1 catchment
• The leading mechanism for thisparticular case was groundwaterridging (steep hillslope)
• Hydrological response can varyconsiderably with the parametersand it strongly depends on the overall condutivity and the conductivity contrast
• The prediction of time-to-peak isvery robust: streamflowgeneration cannot be directlyrelated to “event” water
Fiori et al, JoH 2007
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
Storage-Discharge relation• Example on how to use a distributed model to infer simple
and parsimonious rules, to be employed in lumpedhydrological models
Ali et al, HP 2013
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
Complete mixing??• Complete (or perfect) mixing
seldom encountered, even in the vertical
• Reformulation of 2D purely-advective transport and integration along the verticalvertical (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
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!)
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…)
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; pondedareas vary in time
Fiori and Russo, WRR 2007
3D simulations: Solute transport Focus on the travel
time distributionafter a pulse
Major aims: Impact of
heterogeneity, injection period, external forcings(precipitation, ET, etc.)
Check the validity ofcommon assumptions/conceptual models
Fiori and Russo, WRR 2008
Solute flux and Travel time distribution• Heterogeneity not much
important• Solute flux is highly variable
and it reflects the temporalvariability of precipitation; itstrongly depends on the injection period
• Travel time distribution is notunique (time-variant)
• ESS may help in reducing to time-invariance
• Important effect of ET (selective solute uptake)
Equivalent Steady State (ESS)• Work with cumulate discharge instead
of calendar time (see e.g. Niemi) • Tested with several configurations
Why the Gamma distribution?
• Power law: mainly determinedby fast, unsaturated flow in the upper soil; exponent related to soil properties;
• Exponential decay: groundwater contribution
• Heterogeneity not important; source-zone dispersiondominates (similar to Rinaldo, Marani, Rigon, 1991)
• Conclusions different from Kirchner et al. (2000; 2001)
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, theirperformances as function of the system flow condition (e.g. steady orunsteady flow) have not been much explored to date.
Performance of lumped transport models
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 knownDisadvantage: it’s not a real experiment!
Performance of lumped transport models
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
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
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
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
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 continuityFollowing 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 dQQQ
tV
00
0 1
t
R dQQQ
tVt0
1
0
00 * RRRRRsRRQ tptCdtpCtC
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 TTDA more consistent and robust approach to model solute transportcomplete and instantaneous mixing between the injected solute and the water stored in the system is often assumedRequires 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
tETtQtQdt
tdS 0
tFtFtFdt
tdMETQ 0
tCtCtStQ
dttdC
00
)()(
t dx
xSxQ
deS
QCtC
t
0
)(00
0
t dxxS
xETxQ
detS
QCtC
t
0
)()(
00
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, temporarillyvariable θ(t) can be obtained through
dtQtptQ
t ,)(
1
0
Result and Discussion –numerical results of study cases
TICResult : Raifall only scenario (RO)
Spring
Winter
Inje
ctio
n tim
e
TIFResult : Raifall only scenario (RO)
Spring
Winter
Inje
ctio
n tim
e
Result : Raifall only scenario (RO)
ESS
Spring
Winter
Inje
ctio
n tim
e
R2 = 0.828
Result : Raifall only scenario (RO)
TV
Spring
Winter
Inje
ctio
n tim
e
Comment: zero parameters, but «active» storage needed to be fixed…
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; Bertuzzoet 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 theimportance of ET when studying solute transport in areas inwhich ET is relevant (Van der Velde et al HP2015)•The total mass is fully recovered. However, only the total massis preserved, while the separate contributions MQ and MET maydifferent from the "real" ones
TVResult: Raifall and ET scenario (RET)
Spring
Winter
Inje
ctio
n tim
e
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
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. WATERRESOURCES 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 RESOURCESRESEARCH, 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. WATERRESOURCES 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 vadosezone–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.