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Lessons (I have) learned from physically-based hydrological models Aldo Fiori Università di Roma Tre, Italy Workshop on coupled hydrological modeling University of Padova, 2324 September 2015.

Aldo Fiori

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Page 1: Aldo Fiori

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.

Page 2: Aldo Fiori

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)

Page 3: Aldo Fiori

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

Page 4: Aldo Fiori

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

Page 5: Aldo Fiori

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

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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

Page 7: Aldo Fiori

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

Page 8: Aldo Fiori

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!)

Page 9: Aldo Fiori

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…)

Page 10: Aldo Fiori

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

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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

Page 12: Aldo Fiori

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)

Page 13: Aldo Fiori

Equivalent Steady State (ESS)• Work with cumulate discharge instead

of calendar time (see e.g. Niemi) • Tested with several configurations

Page 14: Aldo Fiori

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)

Page 15: Aldo Fiori

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

Page 16: Aldo Fiori

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

Page 17: Aldo Fiori

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

Page 18: Aldo Fiori

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

Page 19: Aldo Fiori

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

Page 20: Aldo Fiori

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

Page 21: Aldo Fiori

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

Page 22: Aldo Fiori

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

Page 23: Aldo Fiori

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

Page 24: Aldo Fiori

Result and Discussion –numerical results of study cases

Page 25: Aldo Fiori

TICResult : Raifall only scenario (RO)

Spring

Winter

Inje

ctio

n tim

e

Page 26: Aldo Fiori

TIFResult : Raifall only scenario (RO)

Spring

Winter

Inje

ctio

n tim

e

Page 27: Aldo Fiori

Result : Raifall only scenario (RO)

ESS

Spring

Winter

Inje

ctio

n tim

e

R2 = 0.828

Page 28: Aldo Fiori

Result : Raifall only scenario (RO)

TV

Spring

Winter

Inje

ctio

n tim

e

Comment: zero parameters, but «active» storage needed to be fixed…

Page 29: Aldo Fiori

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

Page 30: Aldo Fiori

TVResult: Raifall and ET scenario (RET)

Spring

Winter

Inje

ctio

n tim

e

Page 31: Aldo Fiori

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

Page 32: Aldo Fiori

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.