the catchment hydrology (cathy) model course on cat… · the catchment hydrology (cathy) model...

The CATchment HYdrology (CATHY) Model

Claudio Paniconi and Mauro Sulis INRS-ETE, University of Quebec, Canada Mario Putti and Matteo Camporese University of Padova, Italy Stefano Orlandini, Giovanni Moretti, Alice Cusi, and Maurizio Cingi University of Modena & Reggio Emilia, Italy

UNESCO-IHP Water Programme for Environmental Sustainability - WPA II

Groundwater Modelling as a Management Tool, Foz do Iguaçu, Brazil, Dec 6 – 7, 2011

Context

Surface and subsurface waters are not isolated components of the hydrologic system, but interact in response to topographic, soil, geologic, and climatic factors.

The interaction of groundwater and surface flow is a key focus of interest for: • Hydrology (e.g., runoff generation,

groundwater recharge, evaporation) • Geomorphology (e.g., soil erosion, action of

groundwater seepage on channel initiation) • Water resources management (under the

effects of land use, demography, and climate changes)

• Water quality (e.g., role hyporheic fluxes in aquatic habitats)


Context

In the context of the UNESCO WPA II project on climate change effects on groundwater, a detailed description of the surface-subsurface flow interaction was considered to provide reliable predictions of: • Variations in groundwater recharge across drainage basins • Variations in water table depth across drainage basins


These variable are determined by the partitioning of potential atmospheric fluxes (precipitation and evaporative demand) at the land surface. They are, therefore, affected by topography, soil moisture dynamics, and surface runoff dynamics.

Context


• 20 years of work • Subsurface flow model developed by Paniconi and Putti since 1991 • Surface flow model develped by Orlandini since 1994 • Surface-subsurface flow interaction developed since 1998 • Data assimilation and other extensions developed since 2000 • The model is permanently under revision, but it is always usable • Alice, Giovanni, Rafael, and Felipe used CATHY within the project

Model description: Mathematical formulation

2

2 ( , )k h k LQ Q Qc D c q ht s s

ψ∂ ∂ ∂+ = +

∂ ∂ ∂

[ ]( ) ( )( ) ( )w s rw w z sS K K S q htψσ ψ η∂

= ∇ ⋅ ∇ + +∂

σ general storage term [1/L]: σ = SwSs + φ(dSw/dψ) Sw water saturation = θ/θs [/] θ volumetric moisture content [L3/L3] θs saturated moisture content [L3/L3] Ss specific storage [1/L] φ porosity (= θs if no swelling/shrinking) ψ pressure head [L] t time [T] Ks saturated conductivity tensor [L/T] Krw relative hydraulic conductivity [/] ηz zero in x and y and 1 in z direction

z vertical coordinate +ve upward [L] qs subsurface equation coupling term (more generally, source/sink term) [L3/L3T] h ponding head (depth of water on surface of each cell) [L] s hillslope/channel link coordinate [L] Q discharge along s [L3/T] ck kinematic wave celerity [L/T] Dh hydraulic diffusivity [L2/T] qL surface equation coupling term (overland flow rate) [L3/LT]

(1) Paniconi & Wood, Water Resour. Res., 29(6), 1993 ; Paniconi & Putti, Water Resour. Res., 30(12), 1994 (2) Orlandini & Rosso, J. Hydrologic Engrg., ASCE, 1(3), 1996 ; Orlandini & Rosso, Water Resour. Res., 34(8), 1998 (1)+(2) Camporese et al., Water Resour. Res., 46(2), W02512, 2010.

(1)

(2)

Model description: Numerical discretization

Surface:

• PDE of the kinematic wave solved by a finite difference (FD) scheme

• Numerical dispersion arising from the truncation error of the scheme is used to simulate the physical dispersion

• Unconditional stability reached by matching numerical and physical diffusivities through the temporal weighting factor used to discretize the kinematic wave model


Model description: Numerical discretization

Subsurface:

• PDE solved by a three-dimensional finite element (FE) spatial integrator and by a weighted finite difference (FD), i.e. Euler or Crank-Nicolson, scheme in time

• Nonlinearity arising from the storage σ(Sw) and conductivity Krw(Sw) terms are handled via a Picard or Newton linearization scheme

• Time varying boundary conditions: prescribed head (Dirichlet type) or flux (Neumann type), atmospheric fluxes, source/sink terms, and seepage faces


Model description: Surface–subsurface interactions

The coupling between the land surface and the subsurface is handled by an automatic boundary condition (BC) switching algorithm acting on the source/sink terms qs(h) and qL(h,ψ).

The coupling term is computed as the balance between atmospheric forcing (rainfall and potential evaporation) and the amount of water that can actually infiltrate or exfiltrate the soil.

The switching check is done surface node by surface node in order to account for soil and topographic variability.

The switching check is done at each time the surface equation is solved (according to the values of ponding heads at the surface) and at each subsurface time or iteration.


Model description: Surface–subsurface interaction


Flow Direction Algorithm

The mathematical formulation implemented is based on the use of:

Triangular facets introduced by Tarboton (1997, WRR)

Path-based analysis introduced by Orlandini et al. (2003, WRR)

Plan curvature computation introduced by Zevenbergen and Thorne (1987, ESPL)

Flow Direction Algorithm

Orlandini and Moretti (2009, JGR)

Orlandini et al. (2003, WRR)

Relatore

Note di presentazione

Contour Elevation Data

Moretti and Orlandini (2008, WRR)

Prediction of the Drainage Network

Hydraulic Geometry (Leopold and Maddock, 1953)

bW aQ=

fmY cQ=

mU k Q=

zfS t Q=

ySk rQ=

Parameterization of Stream Channel Geometry (Channels and Hillslope Rivulets)

( )at-a-station relationshipbW a Q ′′=

( )downstream relationshipbfW a Q ′′′′=

( )given frequency discharge, bankfull dischargewfQ u A=

( ),1 bW W A Q ′=

( ) ( ) ( ) ( ) ( ),1 , b w b bs f f s sW A W A Q Q A A A

′ ′′ ′− −=

Orlandini and Rosso (1998, WRR)

Parameterization of Conductance Coefficients (Channels and Hillslope Rivulets)

( )at-a-station relationshipySk r Q ′′=

( )downstream relationshipyS fk r Q ′′′′=

( )given frequency discharge, bankfull dischargewfQ u A=

( ),1 yS Sk k A Q ′=

( ) ( ) ( ) ( ) ( ),1 , y w y yS S s f f s sk A k A Q Q A A A

′ ′′ ′− −=

Orlandini (2002, WRR)

Diffusion Wave Modeling: Mathematical Model

k k LQ Qc c qt s

∂ ∂+ =

∂ ∂

k

dQc

d=

Ω

2

2k h k LQ Q Qc D c qt s s

∂ ∂ ∂+ = +

∂ ∂ ∂

f fk

S Sc

Q∂ ∂

= −∂Ω ∂

1cos

fh

SWD

Qβ∂

=∂

Kinematic wave model Diffusion wave model

Diffusion Wave Modeling: Parameterization of the Drainage System

( )2 3 5 3 1 2 1 ,S f SQ k W S k n W P−= Ω = ≈

2

2 4 3 10 3fS

QSk W −=

Ω

( )

2

4 32 10 3f

S

QS

k W Q −=Ω

Gauckler-Manning-Strickler Equation

Incorporating the variation of stream channel geometry

Incorporating the variation of conductance coefficient and stream channel geometry

( ) ( )

2

2 4 3 10 3f

S

QS

k Q W Q −=Ω

Diffusion Wave Modeling: Constitutive Equations

( ) ( ) ( ) ( ) ( )5 11 2

3 1 2 3 2 1 2 31 2 3 3 1 2 3,1 ,1 y b y by b y bS fQ k A W A S− ′ ′ ′ ′− + − +′ ′ ′ ′− + − += Ω

( ) ( )( )( ) ( )

( )( ) ( )

( )( )

2 1 13 1 2 4 3 2 1 2 4 3 3 1 2 4 3 15 2 35 1 2 3 5 1 2 3 10 1 2 3

5 1 ,1 ,1 ,13 1 2 3

y b y b y b y by b y b y b

k S fc k A W A S A Qy b

′ ′ ′ ′ ′ ′− + − + − + ′ ′+ − − ′ ′ ′ ′ ′ ′− + − + − +=′ ′− +

( ) ( )

1 cos

2 1 2 3 ,1

b

h

f

QD

y b W A S

β′−

=′ ′− +

Flow Rating Curve

Kinematic Celerity

Hydraulic Diffusivity

Diffusion Wave Modeling: Muskingum-Cunge Method with Variable Parameters

( )( ) ( )1

22 1

k

k

c t s XC

X c t s∆ ∆ −

=− + ∆ ∆

( )( ) ( )2

22 1

k

k

c t s XC

X c t s∆ ∆ +

=− + ∆ ∆

( ) ( )( ) ( )3

2 12 1

k

k

X c t sC

X c t s− − ∆ ∆

=− + ∆ ∆

( ) ( )42

2 1k

k

c tCX c t s

∆=

− + ∆ ∆

1 1 11 1 2 3 1 4 1

j j j j ji i i i LiQ C Q C Q C Q C q+ + ++ + += + + +

Diffusion Wave Modeling: Muskingum-Cunge Method with Variable Parameters

( )1 2n kD c s X= ∆ − 1cos

fh

SWDQβ

∂ = ∂

12 cos

f

k

SWXc s Qβ

∂= −

∆ ∂

n hD D=

The Muskingum-Cunge method with variable parameters is: • Unconditionally stable (Dn = Dh). • Accurate for Courant numbers not too far from 1 (∆s ≈ ck ∆t). • Independent of structural parameters ∆s and ∆t. (Cunge, 1969, JHR; Ponce, 1986, JHE; Orlandini and Rosso, 1996, JHE)

Application to the Toledo Catchment

Model calibration

• Theoretically unnecessary. Practically needed with limited possible ranges.

• Relatively easy for non-interacting processes. Constrained for interacting processes. This is not a limitation of detailed models, but rather a strength.



Example

• Control variables: precipitation as observed and air temperature increased

• Response variable: water table depth decrease in upland areas



Example

• Adaptation measure: suitable culture selection in these areas

• Motivations: water use and soil conservation / land stability

• The obtained results identify a future research direction: to determine the role of vegetation in a changed water cycle


Concluding Remarks

Modeling. Why?

• A model is the synthesis of our understanding (ability to observe & predict).

• It is essential to make predictions.

• It is important to sharpen questions. What do we need to observe?

Detailed modeling. Why?

• Simple, conceptual models are designed to predict hydrologic processes singly (and under heavy limiting assumptions). Very useful in many technical problems.

• Detailed, fully dynamic models are designed to predict hydrologic processes in combinations (with the minimum of limiting assumptions). Essential in the prediction of future (unknown) scenarios.

• Example. Horton infiltration-excess runoff vs. Dunne, saturation-excess. What is the dominant process in a changing environment?


Concluding Remarks

Why detailed modeling in this UNESCO WPA II project?

• Because complex models are needed to cope with complex problems.

• Because we believe in unselfish cooperation in research.

• Because we do not underestimate the capability of young scientists.

Acknowledgments

• Brazilian colleagues at PTI: Cicero Bley, Rafael Hernando de Aguiar González, and Felipe Marques.

• Italian colleagues at UNIMORE: Alice Cusi, Giovanni Moretti, Maurizio Cingi.

• Colleagues at UNESCO: Salvatore D’Angelo.


the catchment hydrology (cathy) model course on cat… · the catchment hydrology (cathy) model...

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