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ADZTOOL v 1.0 - User Manual

Matthew Lees, Luis Camacho Department of Civil and Environmental Engineering, Imperial College of

Science, Technology and Medicine, London, SW7 2BU, UK.

e-mail: [email protected]

tel: 0171 5946019

fax: 0171 5946124



Introduction

The Aggregated Dead Zone (ADZ) modelling technique is a relatively recent approach to

modelling dispersion processes that provides accurate predictions of the time of travel and

spread of a pollutant moving downstream in a natural stream. These predictions are useful in

applications involving real time ‘on-line’ management of water intakes during pollution

incidents or in applications involving the identification or confirmation of the sources of past

pollution events.

ADZ research (Beer and Young 1983, Wallis et al. 1989a,b, Young and Wallis 1993) has led

to the formulation of two methodologies for estimation of the advection and dispersion

parameters. The first approach is a simple subjective method that uses derived relationships

from observed concentration-time data measured at two downstream locations (Wallis et al.

1989b); while the second objective method is based on the Simplified Refined Instrumental

Variable (SRIV, Young 1984; 1992) method of system identification. Although the latter

method is superior, the former method is simpler and is relatively accurate if reliable tracerdata are available.

To demonstrate the applicability of the subjective methodology, and to help modellers

become familiarised with ADZ techniques, a simple tool (ADZTOOL v 1.0) has been

developed at Imperial College. The tool allows practising engineers or environmental

protection officers to develop simple yet reliable model to simulate the effects of conservative

solute transport in a river reach. It is intended mainly as a tool for teaching purposes and,

hopefully, its simplicity will provide the user with some insight into ADZ modelling

concepts, techniques and capabilities.

This brief user manual describes the ADZTOOL, the data requirements and the various model

outputs generated within each worksheet of the Excel workbook. The tool’s applicability,

scope and main features are also discussed.

Applicability and scope

Firstly, it should be noted that the simple deterministic methodology used here to calibrate the

ADZ model is only an approximate method. Its application is limited to cases in which

reliable tracer experiment data are available in at least two locations in the stream of interest.

Limitations in this simple methodology occur mainly for three reasons (Wallis et al. 1994):

firstly, the method does not involve any optimisation; secondly, the first order discrete-time

model implemented is only an approximation of the governing differential equation; and

thirdly, the estimates of the parameters derived from the experimental data will be subjected

to some error. Also the user must be aware that the model parameters may vary for different

flow regimes in a stream and calibration should ideally involve the analysis of tracer

experiments carried out under various flow conditions (Young and Wallis, 1993).

Despite these limitations, the tool can be used to estimate the parameters of a first order ADZ

model for conservative solute transport. Once calibrated, the model can also be used in a

simulation mode to compute the response to a user-defined pollution event and to different

user-specified model parameters. Also, as a by-product of the former calibration mode, the



tool can be used to compute dilution gauging flow discharge estimates or to analyse the

reliability of the tracer experiment data.

ADZ model calibration

The ADZ model parameters, shown graphically on Fig. 1., are calculated directly from the

tracer experiment data. For each distribution or concentration-time curve, the time of first

arrival and the location of the centroid are computed. Estimates of the mean travel time, time

delay and ADZ residence time are calculated for each reach between consecutive sampling

stations as follows,

12 t t t −= (1)

12 τ τ τ −= (2)

τ −= t T (3)

Fig. 1 – Graphical calculation of ADZ time delay and residence time

Using these parameters the dispersive fraction is computed as,

t T DF / = (4)

It is worth noting from these relationships that only two parameters, either ( DF , t ) or

( τ ,T ) are required to define the others.

After the parameters have been calculated, the response of a first order discrete ADZ model is

computed by means of the following difference equation,

δ −− ⋅+⋅−= k k k U bC aC 01 (5)

where k C is the output concentration in the reach at sampling interval (k ) [ML

-3∗]; 1−k

C is

the output concentration at the previous sampling interval [ML-3

];δ −k

U is the input

concentration at the upstream end of the reach at sampling interval δ −k [ML-3

]; δ is the

∗ ML-3 is used here as a generalisation for any mass and length units, for example kg/m3



advective time delay in sampling intervals or time steps of duration t ∆ , corresponding to the

nearest lower integer value of the ratio t ∆ / τ ; and a and 0b are the coefficients of the first

order discrete transfer function related to the residence time by,

) / exp( T t a ∆−−= (6a)

ab += 10 (6b)

ADZ model simulation

ADZTOOL has a simulation mode where the calibrated model (or specified model) can be

used to analyse the advection and dispersion processes of pollution event scenarios. In this

mode the difference equation form of the model (5) is used to simulate the downstream

response of a user-defined upstream pollution incident.

Tracer experiment data reliability

Accurate ADZ model calibration and simulation obviously depends upon the availability of reliable tracer experiment data. To assess data reliability, the present tool incorporates a mass

conservation test between sampling stations and the injection point, and a comparison

between measured discharges and estimated discharges from the tracer data. The latter

estimates are obtained using standard dilution gauging procedures (see, for example, Herschy,

1995). A mass conservation test between sampling stations can be performed by computing

the areas under the measured time-concentration curves. These areas will be identical if the

assumptions of conservative behaviour, steady flow conditions and no lateral inflow or

outflow to the reach are valid.

It is well known that data from conservative tracer experiments can be used for dilution

gauging purposes (Herschy 1995) if the injected mass of tracer is known; the tracer becomes

fully mixed in the reach; and if the concentration measurements are carried out until the tracer

has passed through completely at a downstream cross-section. For a gulp injection of tracer of

mass M a discharge estimate Q is given by,

∫ =

f T

t dt t C

M Q

0

)((7)

where )(t C is the measured concentration in time at the downstream location; 0t is the time

of injection; and f

T is the time taken for the tracer concentration to return to the background

concentration.

If an accurate discharge measurement is available from an adjacent gauging station or from a

velocity-area method measurement, then comparison provides a means of testing tracer

experiment reliability. The agreement between the computed value and the measured

discharge will serve to demonstrate that the tracer is conservative; that complete mixing was

achieved at the downstream location; that the experiment was carried under steady-state flow

conditions; and that lateral inflow did not entered the reach between the upstream and

downstream locations. In other words, if good agreement is obtained then the data can be used

with confidence for the calibration of a predictive transport model.



Model execution

ADZTOOL is a Microsoft Office (Office 97) Excel workbook (“ADZTOOL.xls”) that

consists of various worksheets. The provided workbook includes some sample data from a

tracer experiment carried out in the River Mimram in England to demonstrate the ADZ

modelling methodology. To apply the tool to a new data set open an original copy of the

ADZTOOL.xls workbook and save it with a different name.

The workbook structure is shown on the bottom toolbar (see the bottom of Fig. 2). As can be

seen, the workbook consists of the following worksheets: INTRO, Data, Data-Fig, Reliability,

ADZ Parameters, ADZ Results, Res-Fig, ADZ Simul, and Sim-Fig.

Apart from the sheets that contain figures, all other sheets have been protected in order to

avoid accidental changes to the structure and formulae. Only cells in purple are non-protected,

and correspond to data input cells where the user is asked to specify information.

Data

Figure 2 shows the section of the “Data” sheet where the user is asked to input the tracer

experiment data as described below.

Fig. 2 – The “Data” sheet

- Tracer mass injected [kg], and observed or measured discharge [m3s

-1]: This information

is not essential, however as described previously, it is required to test the reliability of the

tracer data. If this information is given, reliability analysis will be performed with results

given in the “Reliability” sheet.

- Time since injection [s]: Uniform-sampling intervals must be entered in the provided

purple area (a maximum of 218 data cells are provided, keep clear cells if fewer data are

given).

- Observed or measured concentration [mg/l]: Observed concentration time series can be

specified for up to four sites, cross sections or sampling stations. Note that the



background concentration should be removed from the original measured data (that is

why in this example the concentration at the initial injection time is zero, see row 12 in

Fig. 2). Data should be provided for all time intervals specified (note zero values at some

sites at the end of the table).

- Cross-sectional area [m2] and wetted perimeter [m] at each site together with a short

description: this information is not essential. However, if this information is provided,

cross-sectional velocities (Discharge/Area) can be calculated and compared against reach

velocity estimates (Reach Length/travel time) in the “ADZ Parameters” sheet. See Chapra

(1997, pp. 242) for an interesting discussion of reach velocity estimates.

- Reach length [m] and Manning-n estimates for each reach. Again, this information is not

essential. The former parameter is required to compute reach velocity estimates (see Fig.

5 at the bottom) while Manning-n is required for the theoretical calculation of cross-

sectional velocity.

Once these data have been specified, the whole workbook will be automatically recalculated

and the figures updated.

Data-Fig

This sheet contains a plot of the concentration data time-series specified in the “Data” sheet.

Note that adjustments to the figure data ranges may be required to obtain the “best” figure for

each particular user-application. If adjustment is required, click on the graph, select Chart

source from the Chart menu and modify the particular data-ranges from the Series options

(see Fig 3).

Fig. 3 – Adjustment of figure data ranges option

This procedure should be followed in the “Res-fig” and “Sim-fig” worksheets that containplots of the ADZ model calibration results and simulation mode results.



Reliability

Figure 4 shows the results of the reliability analysis based on the information specified in the

“Data” sheet. Results are only presented in the case that the injected mass of tracer and

observed discharge are available. In any case, if concentration-time curves are available at

more than one sampling station a mass conservation calculation, based on the area under the

distributions, is performed.

Initially, the agreement between tracer dilution gauging estimates computed by means of Eq.

(7), and the observed discharge is tested, and next the mass conservation test is performed.

The results are presented to the user in a self-explanatory way with messages depending upon

relative error and steady-state gain values (area under output concentration / area under input

concentration).

This sheet also contains intermediate calculations that the user can examine below row 27.

These correspond to the computation of the area and centroid of each distribution. The former

variable is calculated using a trapezoidal rule approximation and the latter parameter from,

I

ii

I A

At t

∑ ⋅= (8)

where, I A is the area of the concentration-time curve of site I , I t is the centroid of the

concentration-time curve of site I, it is the centroid of the i-th trapezoidal area, i A , defined

by the data at two consecutive sampling intervals. The summation is carried out for all

sampling intervals specified in the “Data” sheet. The computed values define the location of

the centroid indicated in the “ADZ Parameters” sheet (see Fig. 5, cell E12).

Fig. 4 – Reliability analysis of the data of the tracer experiment

ADZ parameters

Figure 5 presents the results of the parameters that result from the application of Eq. (1) to Eq.

(4). An important aspect though, must be considered in the identification of the time delayparameter. The time delay is defined as the time taken for the leading edge of the solute cloud



to be advected through the reach. In theory, this can be determined by inspection of the time

concentration data once the background concentration has been removed. However, in many

cases, due to measurement errors, small variations in concentration may be registered which

do not correspond to the first arrival of pollutant at the downstream location. To solve this

problem, the user is asked to define a threshold value as a percentage of the peak concentration of each distribution (say, up to 5%), which when crossed will define the time of

pollutant arrival at the downstream location. The user can vary this value in cell E4 of the

“ADZ Parameters” sheet (see Fig. 5). The particular threshold values that result are indicated

in row 9 and can be checked against the data in the “Data” sheet to confirm that they are

reasonable. However, ultimately the definite test of this subjective choice is provided by the

calibration results that are presented quantitatively in the “ADZ Results” sheet and

graphically in the “Res-Fig” sheet.

Fig. 5 – ADZ model parameters computations

ADZ calibration results

The “ADZ Results” sheet shows the discrete model (5) coefficients ( 0,ba ) as calculated from

the residence time parameter by means of Eq (6a), and (6b) (see rows 6 and 7 of Fig. 6). The

derived discrete model is then automatically applied recursively to generate ADZ results at

each cross-section (see columns D, G and J in Fig. 6). Note also that in Fig. 6 the input

concentrations at each reach correspond to the observed concentrations at the upstream cross-

section or site, but that the values have been automatically delayed by the time specified by

the time delay parameter (columns C, F and I). Also note that a column has been provided

with the model residuals defined as the difference between the modelled and the observed

concentration (columns E, H and K).

Importantly, note that the input concentrations to each reach do not correspond to the output

concentrations of the upstream reach, i.e. calibration is performed and tested independently

for each reach starting from the upstream-observed data.



The overall variance explained by the calibrated model is summarised in the coefficient of

determination2

T R . It can be interpreted as a measure of how well the ADZ model explains the

tracer experiment data. It is given by,

2

22

1 y

T

Rσ

σ −= (9a)

∑=

−−

=n

i

oimicc

n 1

22 )()1(

1σ (9b)

∑=

−−

=n

i

ooi y ccn 1

22 )()1(

1σ (9c)

where,2

σ is the sample variance of the model residuals;2

yσ is the sample variance of the

observed values about its mean value;mi

c is the computed ADZ model concentration at

sampling interval i; oic is the observed concentration at the same sampling interval; and oc is

the mean value of the observed concentration values at each sampling cross-section. The

summation is carried out for all specified sampling intervals.

Figure 6 shows the calibration fit obtained for the River Mimram data. It is worth noting the

particularly good fit obtained for the third reach with a coefficient of determination of 97.8%.

The fit for the second reach is not as good (92.1%), however, note in Fig. 7 (“Res-Fig” sheet)

how the time delay and the long recession curve characteristics of the observed time series are

reproduced reasonably well by the model.

Fig. 6 – ADZ Model calibration Results



Fig. 7 – Calibration for the river Mimram

ADZ simulation mode

Figure 8 shows part of the “ADZ simulation” sheet. The user can specified a set of model

parameters in cells C12 and C13 for the travel time and the dispersive fraction respectively,

and a pollution incident in the table at the bottom. The simulation results are computed

automatically in the right hand part of the table and they are shown graphically in the “Sim-

Fig” worksheet (see Fig. 9). The calibrated parameters are shown at the top of the sheet as a

reference for the user.

Fig. 8. – ADZ model simulation results

Future developments

ADZTOOL v. 1.0 is a first step towards a fully functional operational pollution incident

prediction tool. There are many possible improvements that could be made if user community

funding could be obtained. For most of the improvements suggested below, the underlying

research has been completed, so the main work requirement is related to coding Visual Basic

Macros in Excel. The wide spread availability and user expertise in Excel makes it an

attractive package for development of simple operational tools such as an extended



ADZTOOL. This development aims to increase the size of the ADZ user community, while

recognising that the fully featured ADZ software developed in the Environmental Science

Department of Lancaster University#

is most appropriate for large scale applications.

Fig. 9 – Example simulation results

Possible improvements to ADZTOOL can be categorised as follows.

Improved calibration

We are currently investigating the use of automatic optimisation to produce an improved first

order ADZ model calibration method. Alternatively, an objective system identification

method (Young, 1984, 1991) could be incorporated through the development of an ExcelMacro. However, as noted by Rutherford (1994) some experience in time-series analysis is

required to fit these potentially complex models to field data effectively, and especially to

interpret the physical significance of the estimated parameters. However, one major

advantage of the system identification approach is the estimation of parameter uncertainty

during the calibration procedure. This uncertainty estimate can be used in a Monte-Carlo

simulation procedure to calculate simulation results with associated confidence intervals as

demonstrated by Green and Beven (1993).

Unsteady flow conditions

As stated previously, one method of extending the ADZ simulation model to unsteady flowconditions is to carry out a number of tracer experiments at different discharges. Following

calibration, regression relationships can be derived for the ADZ parameters vs. discharge,

enabling simulations to be performed at any specified discharge.

An alternative method would be to utilise the procedure proposed by Whitehead et al. (1997).

Using the results of tracer experiments carried out at different discharges or from hydrological

records it is possible to estimate the velocity in a reach as a function of discharge using a

relationship of the form,

# Contact Professor Keith Beven for more details.



baQv = (10)

where, v is the velocity; Q is the discharge; and a , b are constant coefficients. The travel

time at some specified discharge can then be estimated as,

v

L

t = (11)

Since the dispersive fraction has been found to be approximately independent of discharge

(Wallis et al. 1989a,b), a single calibration value can be assumed to be valid for a range of

discharges. This means that by using Equation (11) an ADZ simulation model for any

specified discharge can be derived.

Non-conservative pollutants

It is relatively easy to incorporate a first order decay rate into the simulation mode of the tool.

This extension would allow non-conservative pollution prediction in a river reach that has

been characterised through a planned non-conservative tracer experiment.

River networks

Since the ADZTOOL is contained within an Excel spreadsheet, it is possible to specify a

multiple reach network for pollution incident protection simply by linking a number of

simulation worksheets together. However, consideration would have to be given to the

problem of within-reach pollution incidents and the effect of stream confluences.



References

Beer, T. and Young P.C., (1983) Longitudinal dispersion in natural streams, Journal of

Environmental Engineering., 109, No5, ASCE, 1049-1067.

Chapra, S.C., (1997) Surface Water-Quality Modelling, The McGraw-Hill Companies. Inc.,

New York.

Green, H. and Beven, K., (1993) Prediction of times of travel for a pollution incident on the

River Eden in March 1993, CRES Technical Report TR/99, Lancaster University.

Herschy, R.,W., (1995) Streamflow measurement, E and FN Spon, 2nd ed.

Lees, M.J., Camacho, L., Whitehead, P.G. (1998) Extension of the QUASAR water quality

model to incorporate dead-zone mixing, accepted for publication in Hydrology and Earth

System Sciences.

Rutherford, J.C. (1994) River mixing, J. Wiley, Chichester.

Wallis, S.G., Young P.C. and Beven K.J., (1989a) Experimental investigation of the

aggregated dead-zone model for longitudinal solute transport in stream channels, Proceedingsof the Institution of Civil Engineers, Part 2, 87, 1-22.

Wallis, S.G., Guymer I. and Bilgi A., (1989b) A practical engineering approach to modelling

longitudinal dispersion, Proc. of Int. Conf. on hydraulic and environmental modelling of

coastal, estuarine and river waters, Bradford, England 1-21 Sept.,291-300.

Wallis, S.G., (1994) Simulation of solute transport in open channel flow. In Mixing and

transport in the environment (Ed. K. Beven, P. Chatwin, J. Millbank). John Wiley & Sons,

Chichester, pp. 89-112.

Whitehead, P.G., Williams R.J. and Lewis D.R. (1997) Quality simulation along river

systems (QUASAR): model theory and development, The Science of the Total Environment ,

194/195, 447-456.Young, P.C., (1984) Recursive Estimation and Time-Series Analysis, Springer-Verlag,

Berlin.

Young P.C., (1991) Simplified Refined Instrumental Variable (SRIV) estimation and True

Digital Control (TDC): a tutorial introduction, Proc. First European Control Conference,

Grenoble, 1295-1306.

Young, P.C., (1992) Parallel processes in hydrology and water quality: a unified time-series

approach, J.IWEM , 6, 598-612.

Young, P.C., and Wallis S.G., (1993) Solute Transport and Dispersion in Channels. In

Channel Network Hydrology (Ed. K. Beven and M.J. Kirby). John Wiley & Sons, Chichester,

pp. 129-174.

Young, P.C., and Lees, M.J., (1993) The active mixing volume, a new concept in modelling

environmental systems. In statistics for the environment. (Ed. V. Barnett and K. Turkman).

John Wiley & Sons, Chichester, pp. 3-43.

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