Residence Time

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Nov 24, 2013 (3 years and 9 months ago)

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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


Bedrock


Permeability


Fracturing


Slope


Elevation

0
40
80
120
160
0
10
20
30
40
50
60
70
80
Distance from divide (m)
Mean Residence time (days)
MRT = 1.9(Distance) + 19.0
r^2 = 0.88
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


Goal: Simulate the shape of a hydrograph
given a known or designed water input (rain
or snowmelt)

time

Precipitation

time

flow

Hydrologic
Model

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


Simulate what happened in the past


Can serve as a calibration data set


time

Precipitation

time

flow

Hydrologic
Model

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



Q

= f(P
,
landscape properties)

time

Precipitation

time

flow

Hydrologic
Model

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


12

Integrated Hydrologic Models Are Used to

Understand

and

Predict

(Quantify)
the Movement of Water

How
?

Formalization

of hydrologic process equations

Lumped Model

Distributed

Model

e.g: Stanford Watershed Model

e.g: ModHMS, PIHM, FIHM, InHM

Semi
-
Distributed

Model

e.g: HSPF, LASCAM

q
p
t





p

q

REW 1

REW 2

REW 3

REW 4

REW 5

REW 6

REW 7

Data Requirement:

Computational Requirement:

Small

Large

Process Representation:

Parametric

Physics
-
Based

Predicted States Resolution:

Coarser

Fine

ss
Q
U
t









)
.(
)
.(



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,
tan
β



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

q
total

= q
sub

+ q
overland

We need equations based on
topography to calculate q
sub

(9.6)
and q
overland

(9.5)


a

β

a
sat

q
overland

q
subsurface

c

Subsurface Flow in TOPMODEL


q
sub

= Tctan
β


What is the origin of this equation?


What are the assumptions?


How do we obtain tan
β


How do we obtain T?

a

β

a
sat

q
overland

q
subsurface

c


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

β

a
sat

q
overland

q
subsurface

c

z

Transmissivity of Saturated Zone


K at any depth



Transmissivity

of a saturated thickness z
-
D





D

a

β

a
sat

q
overland

q
subsurface

c

z

Equations


Subsurface

Surface

Assume Subsurface flow = recharge rate

Topographic Index

Saturation deficit for
similar topography
regions

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

-
Read 9.5.1

P

t

Q

t

Mathematical
Transfer Function

Transfer Functions


2 Basic steps to rainfall
-
runoff transfer functions

1. Estimate “losses”.


W minus losses = effective precipitation (W
eff
) (eqns 9
-
43, 9
-
44)


Determines the volume of streamflow response


2. Distribute W
eff

in

time


Gives shape to the hydrograph


Recall that Q
ef

= W
eff

Q

t

Base Flow

Event flow (W
eff
)

Transfer Functions


General Concept

W

Losses

W
eff

= Q
ef

Task


Draw a line through the
hyetograph separating loss and
W
eff
volumes (Figure 9
-
40)

t

W

?

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

W

Uniform: W
err
(t) = W(t)
-

constant

Exponential: W
eff
(t) = W
0
e
-
ct


c is unique to each site

Examples of Transfer Function Models


Rational Method (p443)


q
pk
=u
r
C
r
i
eff
A
d


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


V
i

= initial abstraction = storage that must be
satisfied before event flow can begin



V
r

= retention = W that falls after initial
abstraction is satisfied but that does not
contribute to event flow



Q
ef
= W
eff

= 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:


V
r
/ V
max.
= W
eff
/(W
-
V
i
)



where V
max

is the maximum storage capacity of the
watershed



If V
r

= W
-
V
i
-
W
eff
,


max
2
)
(
V
V
W
V
W
W
i
i
eff




SCS Concept


Assuming V
i

= 0.2V
max
(??)




V
max
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 W
eff

2. Estimate temporal distribution

Base flow

Q

t

Volume of effective
Precipitation or event
flow

-
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

T
b
=2.67T
r

Q

t

On top of base flow

T
w

= duration of effective P

T
c
= time concentration

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 (W
eff
=1)


A watershed has a “characteristic” response


This characteristic response is the model


Many methods to construct the shape

Q
ef

t

1

1

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








d
t
U
W
t
Q
t
eff



0
)
(
U(t
-

) = time distributed Unit Hydrograph


W
eff
(

)= 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)


𝑡

𝑈
𝑡
=

(
𝑡
)



𝜏
𝑈
𝑡

𝜏
𝑑𝜏





𝑡

𝑈
𝑡
=

(
𝑡
)



𝑡

𝜏
𝑈
𝜏
𝑑𝜏




x
(t) = input function

U(t) = system response function

τ = dummy variable of integration

Convolution


Watch these:
http://www.youtube.com/watch?v=SNdNf3m
prrU


http://www.youtube.com/watch?v=SNdNf3m
prrU


http://www.youtube.com/watch?v=PV93ueRg
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)


𝑡

𝑈
𝑡
=

(
𝑡
)



𝜏
𝑈
𝑡

𝜏
𝑑𝜏





𝑡

𝑈
𝑡
=

(
𝑡
)



𝑡

𝜏
𝑈
𝜏
𝑑𝜏




x
(t) = input function

U(t) = system response function

τ = dummy variable of integration


Unit Hydrograph Convolution integral in
discrete form



)
1
(
)
(
)
(
1





i
t
U
i
W
t
Q
t
i
j
t
t
t
U
W
U
W
U
W
U
W
t
Q
1
3
2
2
1
1
...
)
(







J=n
-
i+1


𝑡

𝑈
𝑡
=

(
𝑡
)



𝑡

𝜏
𝑈
𝜏
𝑑𝜏





(
𝑡
)



𝑡

𝜏
𝑈
(
𝜏
)

𝜏
=



For Unit Hydrograph (see
pdf

notes)

Catchment Scale Mean Residence Time: An
Example from Wimbachtal, Germany

Wimbach Watershed




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




Streamflow Gaging Station

Precipitation Station

Major Spring Discharge

Maloszewski et. al. (1992)

Geology of Wimbach

Fractured Triassic Limestone and Karstic Triassic Dolomite




300 meter thick Pleistocene glacial deposits with Holocene
alluvial fans above

Many springs discharge at the base of
the Limestone unit

Maloszewski, Rauert, Trimborn, Herrmann, Rau (1992)

3 aquifer types


Porous, Karstic, Fractured

d
18
O in Precipitation and Springflow


Seasonal variation of
18
O 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)


Filter/

Transfer

Function

Watershed/Aquifer Processes

Weight

Normalized Time

0

1

Modeling by

Convolution


A convolution is an integral which expresses the amount of overlap of
one function
g

as it is shifted over another function
C
in
. It therefore
"blends" one function with another


where


C(t) = output signature

C
in
(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







t
in
d
t
g
t
C
t
C
0
)
(
)
(
)
(


Convolution Illustration




t
in
d
t
g
t
C
t
C
0
)
(
)
(
)
(


C
in
(

)



g
(

⤠=
e
-
a


Folding

g
(
-

)



e
-
(
-
a


Displacement

g
(
t
-

)



e
-
a(t
-


t

Multiplication

C
in
(

)
g
(
t
-

)

t

Integration

C(t)

t

t

Shaded
area

1

2

3

4

Step

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.






0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
t/T
g(t)
PFM

PFM

Maloszewski and Zuber

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
2
4
6
8
10
12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
g(t)
t/T
Maloszewski and Zuber

EM

EM

EM

EPM

DM

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
4
6
8
10
12
0
0.05
0.1
0.15
0.2
t/T
g(t)
DM

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
2
4
6
8
10
0
0.002
0.004
0.006
0.008
0.01
t/T
g(t)
(White et al. 2004)

DM

Maloszewski and Zuber

Input Function


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





t
in
d
t
C
t
g
t
C
0
)
(
)
(
)
(




out
out
i
N
i
i
i
i
in
C
C
C
P
P
N
t
C





1
)
(


where

P
i

= the monthly depth of precipitation

N

= number of months with observations


= summer/winter infiltration coefficient

C
out

= mean output 18O composition (mean infiltration composition)



Infiltration Coefficient




was calculated using 18O data from precipitation and springflow
following Grabczak et al., 1984















Application of this equation yielded an


value of 0.2, which means that
winter infiltration exceeds summer infiltration by five times















]
)
(
)
(
/[
]
)
(
)
(
[







s
i
i
s
out
w
w
i
out
i
i
C
P
Pi
C
P
C
C
P

Grabczak, J., Maloszewski, P., Rozanski, K. ans Zuber, A., 1984. Estimation of the tritium input function with the aid of st
abl
e

isotopes. Catena, 11: 105
-
114

where


C
out
(1988
-
1990) =
-
12.82
o
/
oo
(spring water)


Mean Weighted Precipitation (1978
-
1990) =
-
8.90
o
/
oo

and
-
13.30
o
/
oo
, for summer and winter,
respectively

Input Function



out
out
i
N
i
i
i
i
in
C
C
C
P
P
N
t
C





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 M
ath
ematical
Models in Isotope Hydrogeology, IAEA:9
-
58

Convolution Summation in
EXcel


Work in progress


Your Task:


Evaluate my spreadsheet. Figure out if I’m doing it
right


Get
FlowPC

to work


Reproduce
Wimbachtal

results


Run
FlowPC

or Excel for Dry Creek.