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Hydrological modeling of coupled surface-subsurface
flow and transport phenomena: the
CATchment-HYdrology Flow-Transport (C...
£
¢
 
¡INTRODUCTION CATHY_FT MODEL PERFORMANCE
Many challenges in improving and testing current state-of-the-art
models fo...
II. CATchment HYdrology Flow
and Transport model
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
CATchment HYdrology (CATHY) model



Sw Ss
∂ψ
∂t
+ φ∂Sw
∂t
= − · q + qss...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
CATHY Flow-Transport (CATHY_FT) model



Sw Ss
∂ψ
∂t
+ φ∂Sw
∂t
= − · q +...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
Richards’ equation (subsurface flow)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
Richards’ equation (subsurface flow)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
Richards’ equation (subsurface flow)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
Richards’ equation (subsurface flow)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
ADE equation (subsurface transport)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
ADE equation (subsurface transport)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
ADE equation (subsurface transport)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
ADE equation (subsurface transport)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
ADE equation (subsurface transport)



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Numerical model
Surface flow and transport equations



Sw Ss
∂ψ
∂t
+ φ∂S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Coupling in CATHY_FT
1 Surface flow 2 Surface transport 3 Subsurface flow 4 S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Coupling in CATHY_FT
1 Surface flow 2 Surface transport 3 Subsurface flow 4 S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Coupling in CATHY_FT
1 Surface flow 2 Surface transport 3 Subsurface flow 4 S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Coupling in CATHY_FT
1 Surface flow 2 Surface transport 3 Subsurface flow 4 S...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Coupling in CATHY_FT
1 Surface flow 2 Surface transport 3 Subsurface flow
Atm...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
C Scudeler Padua Works...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
Sw Ss
∂ψ
∂t
+ φ
∂Sw
∂t...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
∂θc
∂t
= · [−qc + D c]...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
∂θc
∂t
= · [−qc + D c]...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
∂θc
∂t
= · [−qc + D c]...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Model accuracy
Ability of the model to conserve mass
∂θc
∂t
= · [−qc + D c]...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Larson-Niklasson technique
Domain discretized by ne tetrahedral elements an...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Larson-Niklasson technique
Domain discretized by ne tetrahedral elements an...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Larson-Niklasson technique
Domain discretized by ne tetrahedral elements an...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
Larson-Niklasson technique
Domain discretized by ne tetrahedral elements an...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
INTRODUCTION
£
¢
 
¡CATHY_FT MODEL PERFORMANCE
LN velocity reconstruction results
1. Convergent streamlines towards an out...
III. Testing CATHY_FT at the
Landscape Evolution
Observatory (LEO)
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
The Landscape Evolution Observatory (LEO)
LEO, Biosphere 2, Oracle,
Arizona...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
The Landscape Evolution Observatory (LEO)
LEO, Biosphere 2, Oracle,
Arizona...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
The Landscape Evolution Observatory (LEO)
LEO, Biosphere 2, Oracle,
Arizona...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
The Landscape Evolution Observatory (LEO)
LEO, Biosphere 2, Oracle,
Arizona...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Computational domain
60 x 22 grid cells
30 layers; more refined cl...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Test case
Seepage Face
Outlet
Computational domain
60 x 22 grid cells
30 la...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Input
Water and solute mass inflow Cumulative volume and mass
0.005
0.01
0.0...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Input
Water and solute mass inflow Cumulative volume and mass
0.005
0.01
0.0...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Input
Water and solute mass inflow Cumulative volume and mass
0.005
0.01
0.0...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Water balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 1...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Water balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 1...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Water balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 1...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Water balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 1...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Water balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 1...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Mass balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 12...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Mass balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 12...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Mass balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 12...
INTRODUCTION CATHY_FT
£
¢
 
¡MODEL PERFORMANCE
Results
Mass balance
40
80
Vr
(%)
-40
0
40
80
∆Vst
(%)
40
80
Vsf
(%)
0 6 12...
INTRODUCTION CATHY_FT MODEL PERFORMANCE
Conclusions
1. P1 Galerkin solution is mass-conservative while the velocities are
...
INTRODUCTION CATHY_FT MODEL PERFORMANCE
Conclusions
1. P1 Galerkin solution is mass-conservative while the velocities are
...
INTRODUCTION CATHY_FT MODEL PERFORMANCE
Conclusions
1. P1 Galerkin solution is mass-conservative while the velocities are
...
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Carlotta Scudeler

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Hydrological modeling of coupled surface-subsurface flow and transport phenomena: the CAtchment-HYdrology Flow-Transport (CATHY_FT) model

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Carlotta Scudeler

  1. 1. Hydrological modeling of coupled surface-subsurface flow and transport phenomena: the CATchment-HYdrology Flow-Transport (CATHY_FT) model Workshop on coupled hydrological modeling Carlotta Scudeler, Claudio Paniconi, Mario Putti Padua, 23-09-2015
  2. 2. £ ¢   ¡INTRODUCTION CATHY_FT MODEL PERFORMANCE Many challenges in improving and testing current state-of-the-art models for integrated hydrological simulation Not so many models address both flow and transport interactions between the subsurface and surface I am presenting the CATchment-HYdrology Flow-Transport model and I am showing its performance under hillslope drainage, seepage face, and runoff generation C Scudeler Padua Workshop, Padua, 23-09-2015 2/17
  3. 3. II. CATchment HYdrology Flow and Transport model
  4. 4. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE CATchment HYdrology (CATHY) model    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts C Scudeler Padua Workshop, Padua, 23-09-2015 4/17
  5. 5. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE CATHY Flow-Transport (CATHY_FT) model    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts C Scudeler Padua Workshop, Padua, 23-09-2015 5/17
  6. 6. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model Richards’ equation (subsurface flow)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: P1 Galerkin finite element (FE) model in space and implicit finite difference model in time C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  7. 7. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model Richards’ equation (subsurface flow)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: P1 Galerkin finite element (FE) model in space and implicit finite difference model in time 1. Nodal solution for ψ → continuous and piecewise linear C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  8. 8. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model Richards’ equation (subsurface flow)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: P1 Galerkin finite element (FE) model in space and implicit finite difference model in time 1. Nodal solution for ψ → continuous and piecewise linear 2. Elementwise post-computation of the velocity field q from direct application of Darcy’s law → elementwise constant, normal flux discontinous and not mass-conservative across every face C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  9. 9. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model Richards’ equation (subsurface flow)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: P1 Galerkin finite element (FE) model in space and implicit finite difference model in time 1. Nodal solution for ψ → continuous and piecewise linear 2. Elementwise post-computation of the velocity field q from direct application of Darcy’s law → elementwise constant, normal flux discontinous and not mass-conservative across every face 3. Larson-Niklasson (LN) velocity field q reconstruction C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  10. 10. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model ADE equation (subsurface transport)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: High resolution finite volume (for - · qc advective step) and FE (for · (D c) dispersive step) combined with a time-splitting technique C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  11. 11. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model ADE equation (subsurface transport)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: High resolution finite volume (for - · qc advective step) and FE (for · (D c) dispersive step) combined with a time-splitting technique 1. Advective time-explicit step for the elementwise c C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  12. 12. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model ADE equation (subsurface transport)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: High resolution finite volume (for - · qc advective step) and FE (for · (D c) dispersive step) combined with a time-splitting technique 1. Advective time-explicit step for the elementwise c 2. Mass-conservative element→node c reconstruction C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  13. 13. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model ADE equation (subsurface transport)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: High resolution finite volume (for - · qc advective step) and FE (for · (D c) dispersive step) combined with a time-splitting technique 1. Advective time-explicit step for the elementwise c 2. Mass-conservative element→node c reconstruction 3. Dispersive time-implicit step for the nodal c C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  14. 14. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model ADE equation (subsurface transport)    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: High resolution finite volume (for - · qc advective step) and FE (for · (D c) dispersive step) combined with a time-splitting technique 1. Advective time-explicit step for the elementwise c 2. Mass-conservative element→node c reconstruction 3. Dispersive time-implicit step for the nodal c 4. Mass-conservative node→element c reconstruction C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  15. 15. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Numerical model Surface flow and transport equations    Sw Ss ∂ψ ∂t + φ∂Sw ∂t = − · q + qss ∂Q ∂t + ck ∂Q ∂s = Dh ∂2 Q ∂s2 + ck qs    ∂θc ∂t = · [−qc + D c] + qtss ∂Qm ∂t + ct ∂Qm ∂s = Dc ∂2 Qm ∂s2 + ct qts Numerics: Explicit finite difference scheme in space and time for both surface flow and transport solution C Scudeler Padua Workshop, Padua, 23-09-2015 6/17
  16. 16. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Coupling in CATHY_FT 1 Surface flow 2 Surface transport 3 Subsurface flow 4 Subsurface transport qs k qts k Qk+1 ,hk+1 Qm k+1 ,csurf k+1 ψk+1 ,qk+1 BC switching ck+1 BC switchingqss k+1 Atmospheric BCk+1 qss k+1 qtss k+1 qtss k+1 qs k+1 qts k+1 C Scudeler Padua Workshop, Padua, 23-09-2015 7/17
  17. 17. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Coupling in CATHY_FT 1 Surface flow 2 Surface transport 3 Subsurface flow 4 Subsurface transport qs k qts k Qk+1 ,hk+1 Qm k+1 ,csurf k+1 ψk+1 ,qk+1 BC switching Atmospheric BCk+1 ck+1 BC switchingqss k+1 qss k+1 qtss k+1 qtss k+1 qs k+1 qts k+1 C Scudeler Padua Workshop, Padua, 23-09-2015 7/17
  18. 18. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Coupling in CATHY_FT 1 Surface flow 2 Surface transport 3 Subsurface flow 4 Subsurface transport qs k qts k Qk+1 ,hk+1 Qm k+1 ,csurf k+1 Atmospheric BCk+1 ψk+1 ,qk+1 BC switching ck+1 BC switchingqss k+1 qss k+1 qtss k+1 qtss k+1 qs k+1 qts k+1 C Scudeler Padua Workshop, Padua, 23-09-2015 7/17
  19. 19. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Coupling in CATHY_FT 1 Surface flow 2 Surface transport 3 Subsurface flow 4 Subsurface transport qs k qts k Qk+1 ,hk+1 Atmospheric BCk+1 Qm k+1 ,csurf k+1 ψk+1 ,qk+1 BC switching ck+1 BC switchingqss k+1 qss k+1 qtss k+1 qtss k+1 qs k+1 qts k+1 C Scudeler Padua Workshop, Padua, 23-09-2015 7/17
  20. 20. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Coupling in CATHY_FT 1 Surface flow 2 Surface transport 3 Subsurface flow Atmospheric BCk+1 4 Subsurface transport qs k qts k Qk+1 ,hk+1 Qm k+1 ,csurf k+1 ψk+1 ,qk+1 BC switching ck+1 BC switchingqss k+1 qss k+1 qtss k+1 qtss k+1 qs k+1 qts k+1 C Scudeler Padua Workshop, Padua, 23-09-2015 7/17
  21. 21. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  22. 22. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass Sw Ss ∂ψ ∂t + φ ∂Sw ∂t = − · q + qss → Mass-conservative solution achieved solving the equation in its ψ − Sw mixed form [Celia et al., 1990] C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  23. 23. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass ∂θc ∂t = · [−qc + D c] + qtss → HRFV mass-conservative solution if q is mass-conservative. C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  24. 24. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass ∂θc ∂t = · [−qc + D c] + qtss → HRFV mass-conservative solution if q is mass-conservative. P1 Galerkin q is not mass-conservative C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  25. 25. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass ∂θc ∂t = · [−qc + D c] + qtss → HRFV mass-conservative solution if q is mass-conservative. P1 Galerkin q is not mass-conservative To make q mass-conservative: change the numerical scheme from FE =⇒ High computational cost to Mixed Hybrid Finite Element (MHFE) or add mass-conservative velocity field =⇒ Low computational cost reconstruction C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  26. 26. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Model accuracy Ability of the model to conserve mass ∂θc ∂t = · [−qc + D c] + qtss → HRFV mass-conservative solution if q is mass-conservative. P1 Galerkin q is not mass-conservative To make q mass-conservative: change the numerical scheme from FE =⇒ High computational cost to Mixed Hybrid Finite Element (MHFE) or add mass-conservative velocity field =⇒ Low computational cost reconstruction In CATHY_FT: FE =⇒ FE+Larson-Niklasson (LN) post-processing technique C Scudeler Padua Workshop, Padua, 23-09-2015 8/17
  27. 27. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Larson-Niklasson technique Domain discretized by ne tetrahedral elements and n nodes At each time step C Scudeler Padua Workshop, Padua, 23-09-2015 9/17
  28. 28. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Larson-Niklasson technique Domain discretized by ne tetrahedral elements and n nodes At each time step CATHY solution · ψ nodal solution · qe non mass-conservative where: qe is the non mass-conservative element velocity C Scudeler Padua Workshop, Padua, 23-09-2015 9/17
  29. 29. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Larson-Niklasson technique Domain discretized by ne tetrahedral elements and n nodes At each time step CATHY solution · ψ nodal solution · qe non mass-conservative · Re i · q·n where: qe is the non mass-conservative element velocity Re i is the element residual associated to each node i n is the vector normal to each element faces C Scudeler Padua Workshop, Padua, 23-09-2015 9/17
  30. 30. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE Larson-Niklasson technique Domain discretized by ne tetrahedral elements and n nodes At each time step CATHY solution · ψ nodal solution · qe non mass-conservative · Re i · q·n Larson-Niklasson · new qLN ·n · new mass-conservative qe LN where: qe is the non mass-conservative element velocity Re i is the element residual associated to each node i n is the vector normal to each element faces qe LN is the mass-conservative element velocity C Scudeler Padua Workshop, Padua, 23-09-2015 9/17
  31. 31. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 2. High streamline curvatures due to heterogeneity C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  32. 32. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet D=50 m D=0 m qN=0 m/s cin =1 C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  33. 33. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 0 1 2 3 4 Time (h) 25 50 75 100 Mass(%) Mst - P1 Mout - P1 Err - P1 Mst → mass stored Mout → cumulative mass flown out Min → mass initially in the system Err=Min − Mst − Mout C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  34. 34. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 0 1 2 3 4 Time (h) 25 50 75 100 Mass(%) Mst - P1 Mout - P1 Err - P1 Mst → mass stored Mout → cumulative mass flown out Min → mass initially in the system Err=Min − Mst − Mout At the end Mout = Min ⇒ P1 Galerkin q exits from the 0 flux boundary C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  35. 35. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 0 1 2 3 4 Time (h) 25 50 75 100 Mass(%) Mst - LN Mout - LN C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  36. 36. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 0 1 2 3 4 Time (h) 25 50 75 100 Mass(%) Mst - LN Mout - LN Velocities reconstructed with LN do not violate the 0 flux boundaries C Scudeler Padua Workshop, Padua, 23-09-2015 10/17
  37. 37. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 2. High streamline curvatures due to heterogeneity D=50 m D=0 m qN=0 m/s cin =1 Ks (m/s) 2x10-4 2x10-12 C Scudeler Padua Workshop, Padua, 23-09-2015 11/17
  38. 38. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 2. High streamline curvatures due to heterogeneity 0 2 4 6 8 10 Time (h) Mst - LN Mstf - LN 0 2 4 6 8 Time (h) 25 50 75 100 Mass(%) Mst - P1 Mstf - P1 Mstf → mass stored in the unpermeable soil Mst → mass stored C Scudeler Padua Workshop, Padua, 23-09-2015 11/17
  39. 39. INTRODUCTION £ ¢   ¡CATHY_FT MODEL PERFORMANCE LN velocity reconstruction results 1. Convergent streamlines towards an outlet 2. High streamline curvatures due to heterogeneity 0 2 4 6 8 10 Time (h) Mst - LN Mstf - LN 0 2 4 6 8 Time (h) 25 50 75 100 Mass(%) Mst - P1 Mstf - P1 Mstf → mass stored in the unpermeable soil Mst → mass stored At the end for P1 Mstf = Mst =0 ⇒ Solute mass get trapped in the unpermeable soil At the end for LN Mstf = Mst =0 ⇒ Solute mass slightly crosses the unpermeable soil C Scudeler Padua Workshop, Padua, 23-09-2015 11/17
  40. 40. III. Testing CATHY_FT at the Landscape Evolution Observatory (LEO)
  41. 41. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE The Landscape Evolution Observatory (LEO) LEO, Biosphere 2, Oracle, Arizona, U.S.A. 3 convergent landscapes 30 m long, 11.5 m wide dense sensor and sampler network rainfall simulator (3-45 mm/h) C Scudeler Padua Workshop, Padua, 23-09-2015 13/17
  42. 42. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE The Landscape Evolution Observatory (LEO) LEO, Biosphere 2, Oracle, Arizona, U.S.A. 3 convergent landscapes 30 m long, 11.5 m wide dense sensor and sampler network rainfall simulator (3-45 mm/h) In Figure: View of one of the three hillslopes from top C Scudeler Padua Workshop, Padua, 23-09-2015 13/17
  43. 43. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE The Landscape Evolution Observatory (LEO) LEO, Biosphere 2, Oracle, Arizona, U.S.A. 3 convergent landscapes 30 m long, 11.5 m wide dense sensor and sampler network rainfall simulator (3-45 mm/h) In Figure: View of one of the three hillslopes from top Tipping bucket for low seepage face flow C Scudeler Padua Workshop, Padua, 23-09-2015 13/17
  44. 44. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE The Landscape Evolution Observatory (LEO) LEO, Biosphere 2, Oracle, Arizona, U.S.A. 3 convergent landscapes 30 m long, 11.5 m wide dense sensor and sampler network rainfall simulator (3-45 mm/h) In Figure: View of one of the three hillslopes from top Tipping bucket for low seepage face flow Rainfall simulator C Scudeler Padua Workshop, Padua, 23-09-2015 13/17
  45. 45. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  46. 46. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  47. 47. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  48. 48. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport 1) Rainfall C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  49. 49. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport 1) Rainfall 2) Seepage face flow C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  50. 50. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport 1) Rainfall 2) Seepage face flow 3) Drainage under variably saturated conditions C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  51. 51. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport 1) Rainfall 2) Seepage face flow 3) Drainage under variably saturated conditions 4) Surface flow C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  52. 52. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Test case Seepage Face Outlet Computational domain 60 x 22 grid cells 30 layers; more refined close to the surface and at bottom Material model: homogeneity with Ks=1×10−4 m/s and φ=0.39 Van Genuchten parameters nVG=2.26, θres=0.002, ψsat =-0.6 m Model performance for Subsurface-Surface flow and transport 1) Rainfall 2) Seepage face flow 3) Drainage under variably saturated conditions 4) Surface flow C Scudeler Padua Workshop, Padua, 23-09-2015 14/17
  53. 53. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Input Water and solute mass inflow Cumulative volume and mass 0.005 0.01 0.015 Qr (m 3 /s) 0 6 12 18 24 30 36 42 48 Time (h) 0.005 0.01 0.015 Qm (mg/s) 15 30 45 60 Vr (m 3 ) 0 6 12 18 24 30 36 42 48 Time (h) 15 30 45 60 Min (mg) Initial conditions: 119 m3 of water initially present in the system (water table set at 0.4 m from bottom) and 0 solute mass C Scudeler Padua Workshop, Padua, 23-09-2015 15/17
  54. 54. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Input Water and solute mass inflow Cumulative volume and mass 0.005 0.01 0.015 Qr (m 3 /s) 0 6 12 18 24 30 36 42 48 Time (h) 0.005 0.01 0.015 Qm (mg/s) Qr=0.012 m3 /s 15 30 45 60 Vr (m 3 ) 0 6 12 18 24 30 36 42 48 Time (h) 15 30 45 60 Min (mg) Vr=40.4 m3 Initial conditions: 119 m3 of water initially present in the system (water table set at 0.4 m from bottom) and 0 solute mass Flow input: pulse of homogenous rain Qr =0.012 m3 /s for 1 h→ cumulative volume injected Vr =40.4 m3 C Scudeler Padua Workshop, Padua, 23-09-2015 15/17
  55. 55. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Input Water and solute mass inflow Cumulative volume and mass 0.005 0.01 0.015 Qr (m 3 /s) 0 6 12 18 24 30 36 42 48 Time (h) 0.005 0.01 0.015 Qm (mg/s) Qm=0.012 mg/s 15 30 45 60 Vr (m 3 ) 0 6 12 18 24 30 36 42 48 Time (h) 15 30 45 60 Min (mg) Min=40.4 mg Initial conditions: 119 m3 of water initially present in the system (water table set at 0.4 m from bottom) and 0 solute mass Flow input: pulse of homogenous rain Qr =0.012 m3 /s for 1 h→ cumulative volume injected Vr =40.4 m3 Transport input: solute injection with c=1 mg/m3 of rain pulse→ mass inflow Qm=0.012 mg/s and cumulative mass injected Min=40.4 mg C Scudeler Padua Workshop, Padua, 23-09-2015 15/17
  56. 56. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Water balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Vr − ∆Vst − Vsf − Vout = Flow Error Min − ∆Mst − Msf − Mout = Transport Error C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  57. 57. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Water balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) Vr=100% 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Vr − ∆Vst − Vsf − Vout ⇒100 Min − ∆Mst − Msf − Mout = Transport Error C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  58. 58. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Water balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) -48.17%∆Vst= 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Vr − ∆Vst − Vsf − Vout ⇒100+48.17 Min − ∆Mst − Msf − Mout = Transport Error C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  59. 59. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Water balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) Vsf=77.62% 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62 Min − ∆Mst − Msf − Mout = Transport Error C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  60. 60. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Water balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) Vout=70.58% 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62-70.58=o(0.01)% Min − ∆Mst − Msf − Mout = Transport Error C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  61. 61. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Mass balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Min=100% Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62-70.58=o(0.01)% Min − ∆Mst − Msf − Mout ⇒100 C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  62. 62. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Mass balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) ∆Mst=28.62% Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62-70.58=o(0.01)% Min − ∆Mst − Msf − Mout ⇒100-28.62 C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  63. 63. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Mass balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Msf=6.86% Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62-70.58=o(0.01)% Min − ∆Mst − Msf − Mout ⇒100-28.62-6.86 C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  64. 64. INTRODUCTION CATHY_FT £ ¢   ¡MODEL PERFORMANCE Results Mass balance 40 80 Vr (%) -40 0 40 80 ∆Vst (%) 40 80 Vsf (%) 0 6 12 18 24 30 36 42 48 Time (h) 40 80 Vout (%) 40 80 Min (%) 10 20 30 40 ∆Mst (%) 5 10 Msf (%) 0 6 12 18 24 30 36 42 48 Time (h) 30 60 90 Mout (%) Mout=64.42% Vr − ∆Vst − Vsf − Vout ⇒100+48.17-77.62-70.58=o(0.01)% Min − ∆Mst − Msf − Mout ⇒100-28.62-6.86-64.42=o(0.1)% C Scudeler Padua Workshop, Padua, 23-09-2015 16/17
  65. 65. INTRODUCTION CATHY_FT MODEL PERFORMANCE Conclusions 1. P1 Galerkin solution is mass-conservative while the velocities are not; this causes problems for transport simulations. This requires a post-processing technique to ensure mass-conservation C Scudeler Padua Workshop, Padua, 23-09-2015 17/17
  66. 66. INTRODUCTION CATHY_FT MODEL PERFORMANCE Conclusions 1. P1 Galerkin solution is mass-conservative while the velocities are not; this causes problems for transport simulations. This requires a post-processing technique to ensure mass-conservation 2. Results so far indicate that LN reconstructed velocities are as accurate as MHFE velocities and achieve much better computational efficiency C Scudeler Padua Workshop, Padua, 23-09-2015 17/17
  67. 67. INTRODUCTION CATHY_FT MODEL PERFORMANCE Conclusions 1. P1 Galerkin solution is mass-conservative while the velocities are not; this causes problems for transport simulations. This requires a post-processing technique to ensure mass-conservation 2. Results so far indicate that LN reconstructed velocities are as accurate as MHFE velocities and achieve much better computational efficiency 3. Exchange processes in integrated surface-subsurface models are highly complex and need to be carefully formulated and resolved C Scudeler Padua Workshop, Padua, 23-09-2015 17/17
  68. 68. Thanks for your attention

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