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```Residence Time
Residence Time
•
Mean Water Residence Time (aka: turnover time, age of water leaving a system, exit age,
mean transit time, travel time, hydraulic age, flushing time, or kinematic age)
– T = V / Q = turnover time or age of water leaving a system
– For a 10 L capped bucket with a steady state flow through of 2 L/hr, T = 5 hours
•
•
Assumes all water is mobile
Assumes complete mixing
– For watersheds, we don’t know V or Q
•
Mean Tracer Residence Time (MRT) considers variations in flow path length and mobile and
immobile flow
Residence and Geomorphology
• Geomorphology controls fait of water molecule
• Soils
– Type
– Depth
– Permeability
– Fracturing
• Slope
• Elevation
Mean Residence time (days)
• Bedrock
160
MRT = 1.9(Distance) + 19.0
r^2 = 0.88
120
80
40
0
0
10
20
30
40
50
Distance from divide (m)
60
70
80
MRT estimated using Transfer Function
Models
Transfer Function Models
• Signal processing technique common in
– Electronics
– Seismology
– Anything with waves
– Hydrology
Transfer Function Models
• Brief reminder of transfer function
HYDROGRAPH model before returning to
Hydrograph Modeling
flow
Precipitation
• Goal: Simulate the shape of a hydrograph
given a known or designed water input (rain
or snowmelt)
time
Hydrologic
Model
time
Hydrograph Modeling:
The input signal
• Hyetograph can be
– A future “design” event
• What happens in response to a rainstorm of a
hypothetical magnitude and duration
– See http://hdsc.nws.noaa.gov/hdsc/pfds/
– A past storm
Hydrologic
Model
time
flow
Precipitation
• Simulate what happened in the past
• Can serve as a calibration data set
time
Hydrograph Modeling: The Model
• What do we do with the input signal?
– We mathematically manipulate the signal in a way
that represents how the watershed actually
manipulates the water
Hydrologic
Model
time
flow
Precipitation
• Q = f(P, landscape properties)
time
Hydrograph Modeling
• What is a model?
• What is the purpose of a model?
• Types of Models
– Physical
• http://uwrl.usu.edu/facilities/hydraulics/projects/projects.html
– Analog
• Ohm’s law analogous to Darcy’s law
– Mathematical
• Equations to represent hydrologic process
Types of Mathematical Models
• Process representation
– Physically Based
• Derived from equations representing actual physics of process
• i.e. energy balance snowmelt models
– Conceptual
• Short cuts full physics to capture essential processes
– Linear reservoir model
– Empirical/Regression
• i.e temperature index snowmelt model
– Stochastic
• Evaluates historical time series, based on probability
• Spatial representation
– Lumped
– Distributed
Integrated Hydrologic Models Are Used to Understand and Predict (Quantify)
the Movement of Water
How ? Formalization of hydrologic process equations
Lumped Model
Semi-Distributed Model
REW 2
p
t
 pq

REW 3
t
  .(  U )   .(    )  Q ss
REW 4
REW 1

Distributed Model
REW 5
REW 7
REW 6
q
e.g: Stanford Watershed Model
Process Representation:
e.g: HSPF, LASCAM
Parametric
Predicted States Resolution: Coarser
Data Requirement:
Small
e.g: ModHMS, PIHM, FIHM, InHM
Physics-Based
Fine
Large
Computational Requirement:
12
Hydrograph Modeling
• Physically Based, distributed
Physics-based equations for each process in
each grid cell
See dhsvm.pdf
Kelleners et al., 2009
Pros and cons?
Hydrologic Similarity Models
• Motivation: How can we retain the theory
behind the physically based model while
avoiding the computational difficulty? Identify
the most important driving features and
shortcut the rest.
TOPMODEL
•
•
Beven, K., R. Lamb, P. Quinn, R. Romanowicz and J. Freer, (1995), "TOPMODEL,"
Chapter 18 in Computer Models of Watershed Hydrology, Edited by V. P. Singh,
Water Resources Publications, Highlands Ranch, Colorado, p.627-668.
“TOPMODEL is not a hydrological modeling package. It is rather a set of
conceptual tools that can be used to reproduce the hydrological behaviour of
catchments in a distributed or semi-distributed way, in particular the dynamics of
surface or subsurface contributing areas.”
TOPMODEL
• Surface saturation and soil moisture deficits
based on topography
– Slope
– Specific Catchment Area
– Topographic Convergence
• Partial contributing area concept
• Saturation from below (Dunne) runoff
generation mechanism
Saturation in zones of convergent
topography
TOPMODEL
• Recognizes that topography is the dominant
control on water flow
• Predicts watershed streamflow by identifying
areas that are topographically similar,
computing the average subsurface and
overland flow for those regions, then adding it
all up. It is therefore a quasi-distributed
model.
Key Assumptions
from Beven, Rainfall-Runoff Modeling
• There is a saturated zone in equilibrium with a steady
recharge rate over an upslope contributing area a
• The water table is almost parallel to the surface such that the
effective hydraulic gradient is equal to the local surface slope,
tanβ
• The Transmissivity profile may be described by and
exponential function of storage deficit, with a value of To whe
the soil is just staurated to the surface (zero deficit
Hillslope Element
P
a
c
asat
qoverland
β
qsubsurface
qtotal = qsub + q overland
We need equations based on
topography to calculate qsub (9.6)
and qoverland (9.5)
Subsurface Flow in TOPMODEL
• qsub = Tctanβ
– What is the origin of this equation?
– What are the assumptions?
– How do we obtain tanβ
– How do we obtain T?
c
a
asat
qoverland
β
qsubsurface
•
•
•
Recall that one goal of TOPMODEL is to simplify the data required to run a watershed model.
We know that subsurface flow is highly dependent on the vertical distribution of K. We can
not easily measure K at depth, but we can measure or estimate K at the surface.
We can then incorporate some assumption about how K varies with depth (equation 9.7).
From equation 9.7 we can derive an expression for T based on surface K (9.9). Note that z is
now the depth to the water table.
a
z
c
asat
qoverland
β
qsubsurface
Transmissivity of Saturated Zone
• K at any depth
• Transmissivity of a saturated thickness z-D
a
z
c
qoverland
asat
D
β
qsubsurface
Equations
Subsurface
Surface
Assume Subsurface flow = recharge rate
Saturation deficit for
similar topography
regions
Topographic Index
Saturation Deficit
• Element as a function of local TI
• Catchment Average
• Element as a function of average
Hydrologic Modeling
Systems Approach
A transfer function represents the lumped processes operating in a watershed
-Transforms numerical inputs through simplified paramters that “lump”
processes to numerical outputs
-Modeled is calibrated to obtain proper parameters
-Predictions at outlet only
P
Mathematical
Transfer Function
t
Q
t
Transfer Functions
• 2 Basic steps to rainfall-runoff transfer functions
1. Estimate “losses”.
• W minus losses = effective precipitation (Weff) (eqns 9-43, 9-44)
• Determines the volume of streamflow response
2. Distribute Weff in time
• Gives shape to the hydrograph
Recall that Qef = Weff
Event flow (Weff)
Q
Base Flow
t
Transfer Functions
• General Concept
Draw a line through the
hyetograph separating loss and
Weff volumes (Figure 9-40)
W
Weff = Qef
W
?
Losses
t
Loss Methods
• Methods to estimate effective precipitation
– You have already done it one way…how?
• However, …
Q
t
Loss Methods
• Physically-based infiltration equations
• Chapter 6
– Green-ampt, Richards equation, Darcy…
• Kinematic approximations of infiltration and
storage
Exponential: Weff(t) = W0e-ct
c is unique to each site
W
Uniform: Werr(t) = W(t) - constant
Examples of Transfer Function Models
• Rational Method (p443)
•
•
•
•
No loss method
Duration of rainfall is the time of concentration
Flood peak only
Used for urban watersheds (see table 9-10)
• SCS Curve Number
– Estimates losses by surface properties
– Routes to stream with empirical equations
SCS Loss Method
• SCS curve # (page 445-447)
• Calculates the VOLUME of effective precipitation based
on watershed properties (soils)
• Assumes that this volume is “lost”
SCS Concepts
•
•
Precipitation (W) is partitioned into 3 fates
–
Vi = initial abstraction = storage that must be
satisfied before event flow can begin
–
Vr = retention = W that falls after initial
abstraction is satisfied but that does not
contribute to event flow
–
Qef = Weff = event flow
Method is based on an assumption that there
is a relationship between the runoff ratio and
the amount of storage that is filled:
–
Vr/ Vmax. = Weff/(W-Vi)
•
•
where Vmax is the maximum storage capacity of the
watershed
If Vr = W-Vi-Weff,
W eff 
(W  V i )
2
W  V i  V max
SCS Concept
• Assuming Vi = 0.2Vmax (??)
• Vmax is determined by a Curve Number
Curve Number
The SCS classified 8500 soils into four hydrologic groups according to
their infiltration characteristics
Curve Number
• Related to Land Use
Transfer Function
1. Estimate effective precipitation
– SCS method gives us Weff
2. Estimate temporal distribution
Volume of effective
Precipitation or event
flow
Q
Base flow
t
-What actually gives shape to the hydrograph?
Transfer Function
2. Estimate temporal distribution of effective precipitation
– Various methods “route” water to stream channel
• Many are based on a “time of concentration” and many other “rules”
– SCS method
• Assumes that the runoff hydrograph is a triangle
On top of base flow
Tw = duration of effective P
Tc= time concentration
Q
Tb=2.67Tr
t
How were these
equations developed?
Transfer Functions
•
Time of concentration equations attempt to relate residence time of water to watershed
properties
–
–
The time it takes water to travel from the hydraulically most distant part of the watershed to the
outlet
Empically derived, based on watershed properties
Once again, consider the assumptions…
Transfer Functions
2. Temporal distribution of effective
precipitation
– Unit Hydrograph
– An X (1,2,3,…) hour unit hydrograph is the
characteristic response (hydrograph) of a
watershed to a unit volume of effective water
input applied at a constant rate for x hours.
• 1 inch of effective rain in 6 hours produces a 6 hour unit
hydrograph
Unit Hydrograph
• The event hydrograph that would result from 1 unit
(cm, in,…) of effective precipitation (Weff=1)
– A watershed has a “characteristic” response
– This characteristic response is the model
– Many methods to construct the shape
1
Qef
1
t
Unit Hydrograph
1. How do we Develop the “characteristic response”
for the duration of interest – the transfer function ?
•
•
Empirical – page 451
Synthetic – page 453
2. How do we Apply the UH?:
•
For a storm of an appropriate duration, simply multiply
the y-axis of the unit hydrograph by the depth of the
actual storm (this is based convolution integral theory)
Unit Hydrograph
• Apply: For a storm of an appropriate duration, simply multiply
the y-axis of the unit hydrograph by the depth of the actual
storm.
– See spreadsheet example
– Assumes one burst of precipitation during the duration of the storm
In this picture, what duration
is 2.5 hours Referring to?
Where does 2.4 come from?
• What if storm comes in multiple bursts?
• Application of the Convolution Integral
– Convolves an input time series with a transfer
function to produce an output time series
t
Q ( t )   W eff  U t   d 
0
U(t-) = time distributed Unit Hydrograph
Weff()= effective precipitation
 =time lag between beginning time series of
rainfall excess and the UH
Convolution
• Convolution is a mathematical operation
– Addition, subtraction, multiplication, convolution…
• Whereas addition takes two numbers to make a third number,
convolution takes two functions to make a third function
∞
x(t)
∗   = () ≝
−
−∞
∞
U(t)
∗   = () ≝
−
−∞
y(t)
x(t) = input function
U(t) = system response function
τ = dummy variable of integration
Convolution
• Watch these:
prrU
prrU
iXE&feature=related
• http://en.wikipedia.org/wiki/Convolution
Convolution
• Convolution is a mathematical operation
– Addition, subtraction, multiplication, convolution…
• Whereas addition takes two numbers to make a third number,
convolution takes two functions to make a third function
∞
x(t)
∗   = () ≝
−
−∞
∞
U(t)
∗   = () ≝
−
−∞
y(t)
x(t) = input function
U(t) = system response function
τ = dummy variable of integration
• Unit Hydrograph Convolution integral in
discrete form
∞
∗   = () ≝
−
−∞
() ≝
∞
−  ()
=−∞
For Unit Hydrograph (see pdf notes)
Q (t ) 

t
i 1
W ( i )U ( t  i  1)
Q ( t )  W tU 1  W t 1U 2  W t  2U 3  ...  W 1U
j
J=n-i+1
Catchment Scale Mean Residence Time: An
Example from Wimbachtal, Germany
Wimbach Watershed
Streamflow Gaging Station
Major Spring Discharge
Precipitation Station
Maloszewski et. al. (1992)
•
Drainage area = 33.4 km2
•
Mean annual precipitation = 250 cm
•
Absent of streams in most areas
•
Mean annual runoff (subsurface
discharge to the topographic low) = 167
cm
Geology of Wimbach
Many springs discharge at the base of
the Limestone unit
Maloszewski, Rauert, Trimborn, Herrmann, Rau (1992)
3 aquifer types – Porous, Karstic, Fractured
300 meter thick Pleistocene glacial deposits with Holocene
alluvial fans above
Fractured Triassic Limestone and Karstic Triassic Dolomite
d18O in Precipitation and Springflow
•
•
•
Seasonal variation of 18O in precipitation and springflow
Variation becomes progressively more muted as residence time increases
These variations generally fit a model that incorporates assumptions about subsurface water flow
Modeling Approach
• Lumped-parameter models (black-box models):
– Origanilly adopted from linear systems and signal processing theory and involves a
convolution or filtering
– System is treated as a whole & flow pattern is assumed constant over the modeling
period (can have many system too)
Watershed/Aquifer Processes
Filter/
Transfer
Function
1
Weight
0
Normalized Time
Modeling by
Convolution
C (t ) 

t
0
C in ( t ) g ( t   ) d 
• A convolution is an integral which expresses the amount of overlap of
one function g as it is shifted over another function Cin. It therefore
"blends" one function with another
where
C(t) = output signature
Cin(t) = input signature
t = exit time from system
 = integration variable that describes the entry time into the system
g(t-) = travel time probability distribution for tracer molecules in the system
• It’s a frequency filter, i.e., it attenuates specific frequencies of the input
to produce the result
Convolution Illustration
C (t ) 

0
g() = e -a
Cin()
t
C in ( t ) g ( t   ) d 
Step

e
g(-)
-(-a
Folding
1

g(t-)
e -a(t-
2

t
Displacement
Cin()g(t-)
Multiplication
3
Integration
4
t
C(t)
t
area
t
Transfer Functions - Piston Flow (PFM)
•
Assumes all flow paths have same residence time
– All water moves with advection (no dispersion or diffusion)
Represented by a delta function
– This means the output signal at a given time is equal to the input concentration at
the mean residence time T earlier.
Maloszewski and Zuber
1
0.8
PFM
0.6
g(t)
•
0.4
0.2
0
0
1
2
t/T
3
4
PFM
Transfer Functions - Exponential (EM)
•
Assumes contribution from all flow paths lengths and heavy weighting of
young portion.
•
Similar to the concept of a “well-mixed” system in a linear reservoir model
0.16
DM
0.14
0.12
g(t)
0.1
0.08
EM EM EPM
0.06
0.04
0.02
0
0
2
4
6
8
10
12
t/T
Maloszewski and Zuber
EM
Exponential-piston Flow (EPM)
•
Combination of exponential and piston flow to allow for a delay of shortest
flow paths
•
This model is somewhat more realistic than the exponential model because it
allows for the existence of a delay
0.2
DM
g(t)
0.15
0.1
0.05
0
0
2
4
6
t/T
8
10
12
Maloszewski and Zuber
Dispersion (DM)
• Assumes that flow paths are effected by hydrodynamic dispersion or
geomorphological dispersion
– Geomorphological dispersion is a measure of the dispersion of a
disturbance by the drainage network structure
0.01
DM
0.008
g(t)
0.006
0.004
0.002
0
0
2
4
6
8
10
t/T
Maloszewski and Zuber
(White et al. 2004)
C (t ) 
Input Function

t
0
g ( t   ) C in ( t ) d 
• We must represent precipitation tracer flux to what actually goes
into the soil and groundwater
– Weighting functions are used to “amount-weight” the tracer values according
recharge: mass balance
C in ( t i ) 
N  Pi
N
 P
C
i

 C out  C out
i
i 1
where
Pi = the monthly depth of precipitation
N = number of months with observations
= summer/winter infiltration coefficient
Cout = mean output 18O composition (mean infiltration composition)
Infiltration Coefficient

 was calculated using 18O data from precipitation and springflow
following Grabczak et al., 1984
  [  ( P i C i )  C out  ( P i ) ] /[ C out  ( Pi )   ( P i C i ) ]
w
w
s
s
where
Cout (1988-1990) = -12.82o/oo (spring water)
Mean Weighted Precipitation (1978-1990) = -8.90o/oo and -13.30o/oo, for summer and winter,
respectively

Application of this equation yielded an  value of 0.2, which means that
winter infiltration exceeds summer infiltration by five times
Grabczak, J., Maloszewski, P., Rozanski, K. ans Zuber, A., 1984. Estimation of the tritium input function with the aid of stable
isotopes. Catena, 11: 105-114
Input Function
C in ( t i ) 
N  Pi
N
 P
C
i

 C out  C out
i
i 1
Convolution
using FLOWPC
Application of FLOWPC to estimate MRT for the
Wimbach Spring
Maloszewski, P., and Zuber, A., 1996. Lumped parameter models for interpretation of environmental tracer data. Manual on Mathematical
Models in Isotope Hydrogeology, IAEA:9-58
Convolution Summation in EXcel
• Work in progress