# Residence Time

AI and Robotics

Nov 24, 2013 (4 years and 7 months ago)

91 views

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

-

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

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.

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

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:
prrU

prrU

iXE&feature=related

http://en.wikipedia.org/wiki/Convolution

Convolution

Convolution is a mathematical operation

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

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

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.