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)

Mea

n R

esid

ence

tim

e (d

ays)

MRT = 1.9(Distance) + 19.0r 2̂ = 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

Pre

cip

itat

ion

time

flo

w

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

Pre

cip

itat

ion

timefl

owHydrologic

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

Pre

cip

itat

ion

timefl

owHydrologic

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

qpt

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

ssQUt

).().(

Hydrograph Modeling

• Physically Based, distributedPhysics-based equations for each process in each grid cell

See dhsvm.pdfKelleners 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 Assumptionsfrom 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 ElementP

qtotal = qsub + q overland

We need equations based on topography to calculate qsub (9.6) and qoverland (9.5)

a

β

asat

qoverland

qsubsurface

c

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? a

β

asat

qoverland

qsubsurface

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

asat

qoverland

qsubsurface

c

z

Transmissivity of Saturated Zone

• K at any depth

• Transmissivity of a saturated thickness z-D

D

a

β

asat

qoverland

qsubsurface

c

z

Equations

Subsurface

Surface

Assume Subsurface flow = recharge rate

Topographic IndexSaturation deficit for similar topography regions

Saturation Deficit

• Element as a function of local TI

• Catchment Average

• Element as a function of average

Hydrologic ModelingSystems 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 functions1. 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

Q

t

Base Flow

Event flow (Weff)

Transfer Functions

• General Concept

W

Losses

Weff = Qef

TaskDraw a line through the hyetograph separating loss and Weff 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: Werr(t) = W(t) - constant

Exponential: Weff(t) = W0e-ct

c is unique to each site

Examples of Transfer Function Models

• Rational Method (p443)– qpk=urCrieffAd

• 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,

max

2)(

VVW

VWW

i

ieff

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

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

Tb=2.67Tr

Q

t

On top of base flow

Tw = duration of effective PTc= 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 (Weff=1)– A watershed has a “characteristic” response

– This characteristic response is the model– Many methods to construct the shape

Qef

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

dtUWtQt

eff 0

)(

U(t-t) = time distributed Unit Hydrograph

Weff(t)= effective precipitation t =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 )∗𝑈 (𝑡 )=𝑦 (𝑡)≝−∞

∞

𝑥 (𝜏 )𝑈 (𝑡−𝜏 )𝑑𝜏

𝑥 (𝑡 )∗𝑈 (𝑡 )=𝑦 (𝑡)≝−∞

∞

𝑥 (𝑡−𝜏 )𝑈 (𝜏 )𝑑𝜏

x(t) = input functionU(t) = system response functionτ = dummy variable of integration

Convolution

• Watch these: http://www.youtube.com/watch?v=SNdNf3mprrU

• http://www.youtube.com/watch?v=SNdNf3mprrU

• http://www.youtube.com/watch?v=PV93ueRgiXE&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 )∗𝑈 (𝑡 )=𝑦 (𝑡)≝−∞

∞

𝑥 (𝜏 )𝑈 (𝑡−𝜏 )𝑑𝜏

𝑥 (𝑡 )∗𝑈 (𝑡 )=𝑦 (𝑡)≝−∞

∞

𝑥 (𝑡−𝜏 )𝑈 (𝜏 )𝑑𝜏

x(t) = input functionU(t) = system response functionτ = dummy variable of integration

• Unit Hydrograph Convolution integral in discrete form

)1()()(1

itUiWtQ

t

i

jttt UWUWUWUWtQ 132211 ...)( J=n-i+1

𝑥 (𝑡 )∗𝑈 (𝑡 )=𝑦 (𝑡)≝−∞

∞

𝑥 (𝑡−𝜏 )𝑈 (𝜏 )𝑑𝜏

𝑦 (𝑡)≝ ∑𝜏=−∞

∞

𝑥 (𝑡−𝜏 )𝑈 (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

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)

Filter/Transfer Function

Watershed/Aquifer Processes

Weight

Normalized Time0

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 Cin. It therefore "blends" one function with another

where

C(t) = output signatureCin(t) = input signaturet = exit time from systemt = integration variable that describes the entry time into the systemg(t-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 dtgtCtC0

)()()(

Convolution Illustration t

in dtgtCtC0

)()()(

Cin(t)

t

g(t) = e -at

Folding

g(-t)

t

e -(-a )t

Displacement

g(t-t)

t

e -a(t- )t

t

Multiplication

Cin(t)g(t-t)

t

IntegrationC(t)

tt

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 40

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 10 120

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

g(t

)

t/T

Maloszewski and Zuber

EMEM

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 10 120

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 8 100

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 dtCtgtC0

)()()(

outoutiN

ii

iiin CCC

P

PNtC

1

)(

where

Pi = the monthly depth of precipitation

N = number of months with observations

a= 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

Application of this equation yielded an value of 0.2, which means that winter infiltration exceeds summer infiltration by five times

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

iis

outw w

ioutii CPPiCPCCP

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

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

Input Function outoutiN

ii

iiin CCC

P

PNtC

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• 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.