Solute residence time distributions in
stream networks and comparison with
a compartmental representation
Anders Wörman
Joakim Riml
The Royal Institute of Technology
Sweden
Compartmentalized watershed model
aQP,i
6
5
14
13 4
7
8
15
Surface water
12
3
11
1
2
10
9
Soil water
 Sub-catchments
 Landscape pixels
 Zones and processes
Discretization of processes
and sub-catchments
R
Day
Soil compartments
Surface water
compartments
Statistical correlation
and model calibration
(SUFI-2, GLUE, ParaSol)
Q
and
C
Physical model constraints?
Schematic of the HBV-model
Day
Tracer injection
Morsa Brook, Norway
Dispersion
Retention behaviour
for unit solute pulses
- tracer tests
Wörman and Wachniew,
2007, WRR
Retention due to
groundwater exchange
Concentration (dpm/g)
Reactive solutes strongly retained
3H O
2
51Cr(III)
pdf
32P
3H O
2
15N
Time (hours)
Time (days)
Slope differs 25%
Säva Brook
Ekeby wetland
Exchange mechanisms for
solutes in streams
Pumping due to bed roughness
Exchange caused by stream
curvature and topography
t = 12.3 hr
Salehin, et al., WRR, 2004
 Episodical flooding
 Stage variations
 Molecular diffusion
 Bioturbation
Courtesy of Bayani Cardenas
Differential model for solute
transport with retention
Bencala and Walters, 1983, WRR
C
C
 2C
u
K
 a C s  C 
2
t
x
x
Cs
1
 a Cs  C 
t

C2
Compartmentalized
path analysis
β
Xu et al., 2007, J. Hydrol.
β
q
Cs,2
C3
β
β
Cs,3

dCn
 qCn 1  Cn    Cs,n  Cn
dt
dCs, n
dt
 

C s , n  Cn 

1

Superpositioning over stream network in
sub-catchment
- one compartmentalized pathway
 
 Ct   L 
0
 mX,t f (X,  t)d  W (X)dx
0
- Expected effluent BTC <C(t)>
- Source strength m (kg m-4 s-1)
- Unit response function f (s-1)
- PDF for distances X from
source to outlet (m-1)
X
Spatial parameter variability along
pathways
Reach-by-reach spatial variability
M
C(x  X,t) 
f 01 * f12 *L * f (N 1)N
Q
Reach 1-2

Reach 0-1

M
M
i1
i1
X   x i  i X
Comparison of temporal moments with
instantaneous exchange
t,comp  t x  0 
N
1  
Variability effect
q
1/Pe
Network effect
t
M
dist
 t x  0  
i1
X i
1 i 
ui
2D
1
 a1
 a 2CV 2 (X)
N
X u
u /1 
2
N
2

  x  0  2 1  

q
M 
2D X i
2 
2
2
 t dist   t x  0  
1 i  
3
ui

i1 
2
t,comp
2
t
 1  2 
2
i




x 
2

ui



2D
q

1  a1a 3  a 2a 3 Pe CV 2 (X )

Network effect
term
Effect of changing water
velocity with discharge
Constant compartmental model parameters;
N = 5; q = 5 10-4 s-1; ε = 0.2
 Routing to determine change of mean velocity
with discharge
Need for model resolution to
capture unit pulse response
Xu et al., 2007, J. Hydrol.
Differential model
Compartmental model
Conclusions
 Compartmental model parameters may change
with flow conditions
 Model discretisation (resolution) should be
adapted to hydrological response behaviour
 The relative importance of physical constraints
depends on specific model application