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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

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Residence time

Residence Time


Residence time1

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

Residence and Geomorphology

  • Geomorphology controls fait of water molecule

  • Soils

    • Type

    • Depth

  • Bedrock

    • Permeability

    • Fracturing

  • Slope

  • Elevation


Mrt estimated using transfer function models

MRT estimated using Transfer Function Models


Transfer function models

Transfer Function Models

  • Signal processing technique common in

    • Electronics

    • Seismology

    • Anything with waves

    • Hydrology


Transfer function models1

Transfer Function Models

  • Brief reminder of transfer function HYDROGRAPH model before returning to


Hydrograph modeling

flow

Precipitation

Hydrologic Model

time

time

Hydrograph Modeling

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


Hydrograph modeling the input signal

flow

Precipitation

Hydrologic Model

time

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

      • Simulate what happened in the past

      • Can serve as a calibration data set


Hydrograph modeling the model

flow

Precipitation

Hydrologic Model

time

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

      • Q= f(P, landscape properties)


Hydrograph modeling1

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

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


Residence time

Integrated Hydrologic Models Are Used toUnderstandandPredict(Quantify) the Movement of Water

REW 2

REW 3

REW 4

REW 1

REW 5

p

REW 7

REW 6

q

Small

Large

Coarser

Parametric

Physics-Based

Fine

How ?Formalizationof hydrologic process equations

Semi-DistributedModel

DistributedModel

Lumped Model

e.g: Stanford Watershed Model

e.g: HSPF, LASCAM

e.g: ModHMS, PIHM, FIHM, InHM

Process Representation:

Predicted States Resolution:

Data Requirement:

Computational Requirement:


Hydrograph modeling2

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

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

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.”


Topmodel1

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


Residence time

Saturation in zones of convergent topography


Topmodel2

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

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 element

Hillslope Element

P

a

c

asat

qoverland

β

qsubsurface

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

qtotal = qsub + q overland


Subsurface flow in topmodel

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

c

asat

qoverland

β

qsubsurface


Residence time

  • 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

c

asat

qoverland

z

β

qsubsurface


Transmissivity of saturated zone

Transmissivity of Saturated Zone

  • K at any depth

  • Transmissivity of a saturated thickness z-D

a

c

asat

D

qoverland

z

β

qsubsurface


Equations

Equations

Subsurface

Assume Subsurface flow = recharge rate

Saturation deficit for similar topography regions

Surface

Topographic Index


Saturation deficit

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

Hydrologic ModelingSystems Approach

P

Mathematical Transfer Function

Q

t

t


Transfer functions

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)

Base Flow


Transfer functions1

Transfer Functions

  • General Concept

Task

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

W

Weff = Qef

W

?

Losses

t


Loss methods

Q

t

Loss Methods

  • Methods to estimate effective precipitation

    • You have already done it one way…how?

      • However, …


Loss methods1

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

    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 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

    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,


    Scs concept

    SCS Concept

    • Assuming Vi = 0.2Vmax (??)

    • Vmax is determined by a Curve Number


    Curve number

    Curve Number

    The SCS classified 8500 soils into four hydrologic groups according to their infiltration characteristics


    Curve number1

    Curve Number

    • Related to Land Use


    Transfer function

    Q

    Base flow

    t

    Transfer Function

    1. Estimate effective precipitation

    • SCS method gives us Weff

      2. Estimate temporal distribution

    Volume of effective Precipitation or event flow

    -What actually gives shape to the hydrograph?


    Transfer function1

    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

    How were these equations developed?

    Tb=2.67Tr

    t


    Transfer functions2

    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 functions3

    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

    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 hydrograph1

    Unit Hydrograph

    • How do we Develop the “characteristic response” for the duration of interest – the transfer function ?

      • Empirical – page 451

      • Synthetic – page 453

    • 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 hydrograph2

    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?


    Residence time

    • 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

    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

    • 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


    Convolution1

    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


    Convolution2

    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


    Residence time

    • Unit Hydrograph Convolution integral in discrete form

    For Unit Hydrograph (see pdf notes)

    J=n-i+1


    Catchment scale mean residence time an example from wimbachtal germany

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


    Wimbach watershed

    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

    Major Spring Discharge

    Precipitation Station

    Maloszewski et. al. (1992)


    Geology of wimbach

    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


    D 18 o in precipitation and springflow

    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

    Watershed/Aquifer Processes

    Filter/

    Transfer

    Function

    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)

    1

    Weight

    0

    Normalized Time


    Modeling by convolution

    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 signature

      Cin(t) = input signature

      t = exit time from system

      t = integration variable that describes the entry time into the system

      g(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


    Convolution illustration

    Cin(t)

    g(t) = e -at

    t

    g(-t)

    e -(-at)

    t

    e -a(t-t)

    g(t-t)

    t

    Cin(t)g(t-t)

    t

    C(t)

    Shaded area

    t

    t

    t

    Folding

    Multiplication

    Displacement

    Integration

    Convolution Illustration

    Step

    1

    2

    3

    4


    Transfer functions piston flow pfm

    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

    PFM

    PFM


    Transfer functions exponential em

    DM

    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

    EM

    EM

    EPM

    EM

    Maloszewski and Zuber


    Exponential piston flow 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

    Maloszewski and Zuber


    Dispersion dm

    DM

    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

    Maloszewski and Zuber

    (White et al. 2004)


    Input function

    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

    • 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

    Infiltration Coefficient

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

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

    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

    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 function1

    Input Function

    Convolution using FLOWPC


    Application of flowpc to estimate mrt for the wimbach spring

    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

    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 FlowPCor Excel for Dry Creek.


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