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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

- Geomorphology controls fait of water molecule
- Soils
- Type
- Depth

- Bedrock
- Permeability
- Fracturing

- Slope
- Elevation

- Signal processing technique common in
- Electronics
- Seismology
- Anything with waves
- Hydrology

- Brief reminder of transfer function HYDROGRAPH model before returning to

flow

Precipitation

Hydrologic Model

time

time

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

flow

Precipitation

Hydrologic Model

time

time

- 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/

- What happens in response to a rainstorm of a hypothetical magnitude and duration
- A past storm
- Simulate what happened in the past
- Can serve as a calibration data set

- A future “design” event

flow

Precipitation

Hydrologic Model

time

time

- 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)

- We mathematically manipulate the signal in a way that represents how the watershed actually manipulates the water

- 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

- Physical

- 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

- Short cuts full physics to capture essential processes
- Empirical/Regression
- i.e temperature index snowmelt model

- Stochastic
- Evaluates historical time series, based on probability

- Physically Based
- Spatial representation
- Lumped
- Distributed

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:

- Physically Based, distributed

Physics-based equations for each process in each grid cell

See dhsvm.pdf

Kelleners et al., 2009

Pros and cons?

- 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.

- 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.”

- 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

- 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.

- 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

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

- 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

- 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

- K at any depth
- Transmissivity of a saturated thickness z-D

a

c

asat

D

qoverland

z

β

qsubsurface

Subsurface

Assume Subsurface flow = recharge rate

Saturation deficit for similar topography regions

Surface

Topographic Index

- Element as a function of local TI
- Catchment Average
- Element as a function of average

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

Q

t

- 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

- General Concept

Task

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

W

Weff = Qef

W

?

Losses

t

Q

t

- Methods to estimate effective precipitation
- You have already done it one way…how?
- However, …

- You have already done it one way…how?

- Physically-based infiltration equations
- Chapter 6
- Green-ampt, Richards equation, Darcy…

- Chapter 6

Exponential: Weff(t) = W0e-ct

c is unique to each site

W

Uniform: Werr(t) = W(t) - constant

- 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)

- qpk=urCrieffAd
- SCS Curve Number
- Estimates losses by surface properties
- Routes to stream with empirical equations

- SCS curve # (page 445-447)
- Calculates the VOLUME of effective precipitation based on watershed properties (soils)
- Assumes that this volume is “lost”

- 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

- Vr/ Vmax. = Weff/(W-Vi)
- If Vr = W-Vi-Weff,

- Assuming Vi = 0.2Vmax (??)
- Vmax is determined by a Curve Number

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

- Related to Land Use

Q

Base flow

t

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?

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

- 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…

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

- 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

- A watershed has a “characteristic” response

1

Qef

1

t

- 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)

- 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

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 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

- Addition, subtraction, multiplication, convolution…

x(t)

U(t)

y(t)

x(t) = input function

U(t) = system response function

τ = dummy variable of integration

- 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 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

- Addition, subtraction, multiplication, convolution…

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)

J=n-i+1

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

- 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)

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

- 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

Watershed/Aquifer Processes

Filter/

Transfer

Function

- 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

- 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

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

Step

1

2

3

4

- 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

DM

- 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

DM

- 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

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)

- 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)

- 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

Convolution using FLOWPC

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

- 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.