R W Q M N .1

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Scientific and Technical Report No. 12

As background to the development of River Water Quality Model No. 1, the Task Group completed a critical evaluation of the current state of the practice in water quality modelling. A major limitation in present model formulations is the continued reliance on BOD as the primary state variable, despite the fact that BOD does not include all biodegradable matter. A related difficulty is the poor representation of benthic flux terms. As a result of these shortcomings, it is impossible to close mass balances completely in most existing models. These various limitations in current river water quality models impair their predictive ability in situations of marked changes in a river's pollutant load, streamflow, morphometry, or other basic characteristics. RWQM1 is intended to serve as a framework for river water quality models that overcome these deficiencies in traditional water quality models and most particularly the failure to close mass balances between the water column and sediment. In addition, the model is intended to be compatible with the existing IWA Activated Sludge Models (STR 9: Activated Sludge Models ASM1, ASM2, ASM2d and ASM3; ISBN: 1900222248) so that it can be straightforwardly linked to them. To these ends, the model incorporates fundamental water quality components and processes to characterise carbon, oxygen, nitrogen, and phosphorus (C, O, N, and P) cycling instead of biochemical oxygen demand as used in traditional models. The model is presented in terms of processes and components represented via a Petersen stoichiometry matrix, the same approach used for the IWA Activated Sludge Models. The full RWQM1 includes 24 components and 30 processes. The report provides detailed examples on reducing the numbers of components and processes to fit specific water quality problems. Thus, the model provides a framework for both complicated and simplified models. Detailed explanations of the model components, process equations, stoichiometric parameters, and kinetic parameters are provided, as are example parameter values and two case studies. The STR is intended to launch a participatory process of model development, application, and refinement. RWQM1 provides a framework for this process, but the goal of the Task Group is to involve water quality professionals worldwide in the continued work developing a new water quality modelling approach. This text will be an invaluable reference for researchers and graduate students specializing in water resources, hydrology, water quality, or environmental modelling in departments of environmental engineering, natural resources, civil engineering, chemical engineering, environmental sciences, and ecology. Water resources engineers, water quality engineers and technical specialists in environmental consultancy, government agencies or regulated industries will also value this critical assessment of the state of practice in water quality modelling. ISBN: 1 900222 82 5

ISSN: 1025-0913

Scientific and Technical Report No. 12 River Water Quality Model No.1

T

his Scientific and Technical Report (STR) presents the findings of the IWA Task Group on River Water Quality Modelling (RWQM). The task group was formed to create a scientific and technical base from which to formulate standardized, consistent river water quality models and guidelines for their implementation. This STR presents the first outcome in this effort: River Water Quality Model No. 1 (RWQM1).

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Scientific and Technical Report No. 12

River Water Quality Model No. 1 Edited by IWA Task Group on River Water Quality Modelling Peter Reichert Dietrich Borchardt Mogens Henze Wolfgang Rauch Peter Shanahan László Somlyódy Peter A. Vanrolleghem

Published by IWA Publishing, Alliance House, 12 Caxton Street, London SW1H 0QS, UK Telephone: +44 (0) 20 7654 5500; Fax: +44 (0) 20 7654 5555; Email: [email protected] Web: www.iwapublishing.com First published 2001 © 2001 IWA Publishing Edited and typeset by Jane Hammett, Leighton Buzzard, Bedfordshire, UK. Printed by TJ International (Ltd), Padstow, Cornwall, UK Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright, Designs and Patents Act (1998), no part of this publication may be reproduced, stored or transmitted in any form or by any means, without the prior permission in writing of the publisher, or, in the case of photographic reproduction, in accordance with the terms of licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licenses issued by the appropriate reproduction rights organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to IWA Publishing at the address printed above. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for errors or omissions that may be made. British Library Cataloguing in Publication Data A CIP catalogue record for this book is available from the British Library Library of Congress Cataloging- in-Publication Data A catalog record for this book is available from the Library of Congress

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Acknowledgements

This report was prepared by the IWA Task Group on River Water Quality Modelling. László Somlyódy conceived of the Task Group and served as its chair. Peter Shanahan served as editor of this report from the Task Group. Portions of this report have previously been published in modified form in Water Science and Technology by Rauch et al. (1998b)1, Shanahan et al. (1998), Somlyódy et al. (1998), Shanahan et al. (2001), Reichert et al. (2001a), Vanrolleghem et al. (2001), Reichert (2001), Borchardt and Reichert (2001), and Reichert and Vanrolleghem (2001). The Task Group greatly appreciates the financial support of IWA and the editorial assistance of Jane Hammett.

1

Dr László Koncsos contributed as an earlier member of the Task Group.

List of Task Group members

Peter Reichert Swiss Federal Institute of Environmental Science and Technology (EAWAG) and Department of Environmental Sciences, Swiss Federal Institute of Technology (ETH), Switzerland

Dietrich Borchardt Institute of Water Resources Research and Management, University of Kassel, Germany

Mogens Henze Department of Environment and Resources DTU, Technical University of Denmark, Denmark

Wolfgang Rauch Swiss Federal Institute of Environmental Science and Technology (EAWAG), Switzerland and Department of Environmental Engineering, University Innsbruck, Austria.

Peter Shanahan HydroAnalysis, Inc. and Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, USA

László Somlyódy Department of Sanitary and Environmental Engineering, Budapest University of Technology and Economics, Hungary

Peter A. Vanrolleghem BIOMATH Department, Ghent University, Belgium

Contents

Acknowledgements List of Task Group members

ix xi

1. Introduction 1.1 Background 1.2 Problem definition 1.3 Context 1.4 Objectives of RWQM1 1.5 Method of model presentation 1.5.1 Format and notation 1.5.2 Use in mass balances 1.5.3 Mass conservation check 1.5.4 Terminology 1.6 About the process of model development 2 Evaluation of current water quality modelling practice 2.1 Introduction 2.2 Water quality modelling and legislation 2.3 General concept of river water quality models 2.4 Hydrodynamics and hydraulics 2.5 Transport processes 2.6 Conversion processes 2.6.1 Introduction 2.6.2 Streeter–Phelps model 2.6.3 QUAL2 model family 2.6.4 Recent developments

1 1 3 3 4 5 5 7 8 8 8 10 10 11 12 14 15 17 17 17 19 22

vi

Contents

2.7 Software and computer programs 2.8 Model identification and testing 2.9 Shortcomings and needs 2.9.1 Problems in model application 2.9.2 Problems in model formulation 2.9.3 Problems in model calibration 2.9.4 Problems in data collection 2.9.5 Problems in predictive capability 2.10 Conclusions 3 Conversion model RWQM1 3.1 Introduction 3.2 Simplifying assumptions 3.3 Composition of organic compounds and organisms 3.4 Components used in the model 3.5 Measurability of model components 3.6 Biogeochemical conversion processes 3.7 Summary and conclusions 4 How to use the model 4.1 Introduction 4.2 Decision process 4.2.1 Step 1 – Define temporal representation 4.2.2 Step 2 – Define model spatial dimensions 4.2.3 Step 3 – Determine representation of mixing 4.2.4 Step 4 – Determine representation of advection 4.2.5 Step 5 – Determine reaction terms 4.2.6 Step 6 – Determine boundary conditions 4.3 Biogeochemical submodel selection 4.3.1 Biochemical model decision process 4.3.2 Compartments 4.3.3 Components and processes 4.4 Examples of biochemical submodel selection 4.5 Summary 5 Case studies 5.1 Introduction 5.2 River Glatt 5.2.1 Study site 5.2.2 Review of previous modelling studies 5.2.3 Goals of the present study 5.2.4 Submodel for oxygen, nitrogen and phosphorus conversion by constant benthic biomass 5.2.5 Extension of the submodel to the calculation of dissolved carbonate equilibria, calcite precipitation, and pH 5.2.6 Hypothetical simulation of the dynamics of benthic biomass 5.2.7 Summary and conclusions 5.3 River Lahn 5.3.1 Study site 5.3.2 Modelling approach 5.3.3 Model calibration 5.3.4 Modelling of oxygen time series in the surface flow and in the sediment

22 22 24 24 24 26 27 28 28 30 30 32 32 35 36 38 43 46 46 46 47 48 49 49 51 51 52 52 52 53 56 58 59 59 60 60 60 61 62 63 68 69 70 70 71 72 74

Contents

vii

5.3.5 System response to inputs of organic matter from a wastewater treatment plant and combined sewer outflows 5.3.6 Conclusions 6 Identifiability and uncertainty analysis 6.1 Introduction 6.2 Techniques 6.2.1 Selection of subsets of identifiable model parameters 6.2.2 Uncertainty analysis 6.3 Water quality submodel and measurement layouts 6.3.1 Identifiability analysis 6.3.2 Uncertainty analysis 6.4 Conclusions 7 Summary and future directions 7.1 Summary 7.2 Future directions 7.2.1 Case studies 7.2.2 Comprehensive sensitivity analysis 7.2.3 Integration with ASM 7.2.4 Enhanced or added model processes 7.2.5 Open-ended model development References Appendix 1: The river system concept Appendix 2: Formulas for stoichiometric coefficients Appendix 3: A numerical example Appendix 4: Bibliography for model enhancements Appendix 5: Definition of identifiability measures

76 78 80 80 81 81 82 82 85 86 87 89 89 91 91 91 91 92 94 95 102 105 117 120 123

Index

127

1 Introduction

1.1 BACKGROUND River water quality modelling has a long history that dates back to the pioneering work of Streeter and Phelps in 1925. Streeter and Phelps described the bacterial decomposition of organic carbon characterised by biochemical oxygen demand (BOD) and its impact on dissolved oxygen conditions. In the course of the next half-century, this simple, first-order kinetics approach was further developed in three major steps. The first was the refinement of the two-state-variable model by introducing the settling rate (of particulate matter) in addition to the decay rate (of dissolved matter) and the so-called sediment oxygen demand (as a parameter). The model was also improved by using research results on the surface reaeration rate. Finally, an extension was made by distinguishing between carbonaceous BOD (CBOD) and nitrogenous BOD (NBOD), which led to a third state variable. The second step was the incorporation of a simplified nitrogen cycle: ammonia, nitrate, and nitrite appeared as new components. This extension appears in QUAL1 (TWDB 1971), the first model of the QUAL family. Ten years later the third step further extended the approach by incorporating phosphorus cycling and algae, which resulted in organic nitrogen, organic phosphorus, dissolved phosphorus, and algae biomass (in terms of chlorophyll a) as additional state variables. This model is known today as QUAL2E and is widely used. It has also been adopted in a practically unchanged form in various simulation software and decision support systems (DSS).

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

2

River Water Quality Model No. 1

The above brief summary suggests a rather natural evolution. The three subsequent steps represent three different concepts (Masliev et al. 1995). The original Streeter–Phelps model is a phenomenological model, where BOD is not the concentration of a chemical substance, but the result of a bioassay test. The models of the second step have a typical chemical kinetic structure, where a group of first-order reactions represent in a cumulative manner the complex chain of processes related to electron transfer in aerobic conditions. Finally, in the third step, the algae model is an ecosystem dynamics model that accounts for non-linear growth and decay of phytoplankton and nutrient cycling. Since the procedure of model development was based mostly on the incorporation of “incremental” impacts by a more or less mechanical addition of model layers, these models contain inconsistencies. They also often lack clear operational definitions of the water quality variables involved. Bacterial decomposition also takes place under well-controlled conditions in biological wastewater treatment plants. Models have been applied here for several decades, as well. However, the development history is rather different than that for rivers. In 1987 the IAWPRC1 Activated Sludge Model (ASM) No. 1 was published (Henze et al. 1987). It was developed in one piece by a co-ordinated effort of professionals sharing a unified conceptual basis. The conceptual basis had developed as a standard model on the basis of experience gained with earlier activated sludge process models. The ASM1 effort led to a useful cross-fertilisation. Some of the researchers working in the field of water quality modelling (see for example Masliev et al. 1995; Maryns and Bauwens 1997) raised obvious questions about the similar processes in the two sets of models. While there are differences in the order of magnitude of biomass and concentrations between natural conditions in rivers and well-controlled conditions in reactors, it is nonetheless logical to ask, to what extent are QUAL2 and ASM1 similar to each other? Can one not draw conclusions from such a comparison for future model development? Could the two models be linked to each other to handle wastewater treatment plants (WWTPs) and rivers in an integrated fashion? And finally, would it not make sense to launch a systematic model development and harmonisation effort also for river water quality models? The implications of the above thoughts were strengthened by strategic needs in Europe. During the past few years significant efforts have been made to develop the EU–Water Framework Directive (approved October 2000; EU 2001a,b), which formed the basis of water policy for the European Union and associated countries. The basic goal of the EUWFD is to achieve “good status” of both surface and ground waters within given time periods with a strong focus on ecological criteria and by strategic integrated management. EU-initiated management and research activities clearly focus on integrated water resources management and the development of harmonised tools supporting it. Similar needs may show up in the future in other continents and large countries, for example, Australia. The establishment of the IWA Task Group on River Water Quality Modelling was an obvious consequence of all the above. The task group was formed to create a scientific and technical base from which to formulate standardised, consistent river water quality models and guidelines for their implementation. This effort was intended to lead to the development of river water quality models that are compatible with the existing IWA Activated Sludge Models (ASM1, ASM2 and ASM3; Henze et al. 2000) and can be straightforwardly linked to them. To this end, water quality components and model state variables characterising carbon, oxygen, nitrogen, and phosphorus cycling are a necessary part of the river model.

1

International Association on Water Pollution Research and Control. Later, IAWPRC was renamed to IAWQ and now IWA.

Introduction

3

1.2 PROBLEM DEFINITION Water quality models are used for many different problems and purposes. Existing models address some of these problems better than others. We have named the model presented in this report River Water Quality Model No. 1 (RWQM1). We have used a name similar to ASM1 to express the hope of initiating a similar development process as achieved for activated sludge models. Applications that RWQM1 is intended to address include: (1) dynamic problems of combined stormwater overflows and non-point source pollution; (2) impact of improved wastewater treatment plant operation and control; (3) extreme and surprising pollution events; (4) improved assessment of artificially influenced rivers (for example, by dams or renaturalisation); (5) data collection; (6) structured understanding, research, education, and improved communication (e.g. between wastewater engineers and receiving water quality experts); and, (7) regulatory applications including catchment planning. Chapter 7 provides additional discussion with respect to addressing these different types of applications.

1.3 CONTEXT Water quality changes in rivers are due to physical transport processes and biological, chemical, biochemical, and physical conversion processes. Physical transport includes advection and turbulent diffusion, which are separately described through hydraulic models. The above processes in the water phase are governed by a set of extended transport equations that can be represented conceptually as: Change in concentration with time

=

Change due to advection

+

Change due to diffusion or dispersion

+

Change due to conversion processes

(1.1)

To this conceptual equation, a similar mass conservation equation for the sediment should be added. Interface terms (e.g. sediment–water and water–air) appear as boundary conditions that are completed by specifying in- and outflows and boundary fluxes. Depending on integration, boundary conditions may enter the equation as sink or source terms (as a part of the aggregated conversion processes term). If we compare our knowledge on advection, dispersion, and their hydraulic backgrounds on the one side and conversion processes (mostly related to chemistry and biology) on the other side, clearly we are much weaker on the latter “soft” field. Here data collection, experimentation, and general empirical knowledge play a decisive role in identifying the model or submodel structure, and in performing model calibration and validation. This is particularly the case if we consider the uncertainties inherent in field data, model structure, and parameter values. Thus, methodologically three main components of the modelling context should be distinguished: (a) advection, dispersion, and hydraulics; (b) conversion

4

River Water Quality Model No. 1

processes; and (c) analyses related to identification, calibration, validation, and uncertainty. In this report we will deal primarily with the second element, conversion processes.

1.4 OBJECTIVES OF RWQM1 Our goal is to deal solely with the development of conversion submodels for traditional pollutants. This choice of focus is similar to that which led to the IWA Activated Sludge Models. Our choice recognises that there are well-developed models and tools to address the physical transport components of this problem. Particularly, one-dimensional, twodimensional, and, increasingly, three-dimensional hydrodynamic models are available to determine the velocity field and are becoming more practical with advancements in computer technology. We therefore see the future in water quality modelling as resting on the development of a well-structured description of conversion processes in Equation (1.1). On the basis of reviewing the state of the art and the problems with that state of the art, we define our detailed objectives as: (1) to develop a sequence of coherent and improved conversion submodels ranging from simple to complex; (2) to guide the selection of (a) hydraulic and physical transport model components, and (b) process submodels, and (c) to test the resulting water quality model; and, (3) to apply the submodels to real data from selected case studies. The first of these objectives entails the following subtasks: •

• • •

to re-evaluate models developed during the past three decades and to eliminate such inherent inconsistencies as the lack of closed mass balances (which mostly arise from an inadequate description of sediment-related processes and the use of BOD for the characterisation of organic matter); to guarantee compatibility with the IWA Activated Sludge Models to enable integrated analysis of wastewater treatment and receiving water quality impacts; to include and improve process descriptions such as nitrification, denitrification, and those related to sediment, benthic fluxes, attached bacteria and algae, and macrophytes; and, to formulate the resulting models in a way that is based on the innovative formalism used in the activated sludge models but at the same time considers the needs and practices of biologists, chemists, and other professionals dealing with rivers.

We expect these changes in model formulation to improve the predictive power of models to estimate multiple and non-linear effects from changes in emissions and other artificial alterations.

Introduction

5

1.5 METHOD OF MODEL PRESENTATION The following discussion is excerpted almost verbatim from Henze et al. (1987). We have adopted the identical model presentation method as employed by Henze et al. for the Activated Sludge Model No. 1, both to ensure compatibility with ASM1 and to take advantage of the intrinsic attractiveness of this simple and clear method, which has not previously been employed in river water quality modelling. Simulation of activated sludge (and river) system behaviour, incorporating phenomena such as carbon oxidation, nitrification, and denitrification, must necessarily account for a large number of reactions between a large number of components. To be mathematically tractable while providing realistic predictions, the reactions must be representative of the most important fundamental processes occurring within the system. In this context the term “process” is used to mean distinct mechanisms acting upon one or more system components. Furthermore, the model should quantify both the kinetics (rate-concentration dependence) and the stoichiometry (relationship that one component has to another in a reaction) of each process. Identification of the major processes and selection of the appropriate kinetic and stoichiometric expressions for each are the major conceptual tasks during development of a mathematical model. Consequently, most of this report will concern them.

1.5.1 Format and notation One problem often associated with papers presenting models describing complex systems is that it is difficult to follow the development of the author’s ideas. In particular, it is often difficult to trace all the interactions of the system components. Henze et al. (1987) concluded that a matrix format, based on the work of Petersen (1965), for presentation of the model offered the best opportunity for overcoming this problem while conveying the maximum amount of information. Furthermore, they felt that the notation recommended by a previous Task Group (Grau et al. 1982) should be used. An illustration will introduce the matrix format and the notation. Consider a situation in which heterotrophic bacteria are growing in an aerobic environment by utilising a soluble substrate for carbon and energy. In one simple conceptualisation of this situation, two fundamental processes occur: the biomass increases by cell growth and decreases by decay. Other events, such as oxygen utilisation and substrate removal, also occur, but these are not considered to be fundamental because they result from biomass growth and decay and are coupled to them through the system stoichiometry. The simplest model of this situation must consider the concentrations of three components: biomass, substrate, and dissolved oxygen. The matrix incorporating the fate of these three components in the two fundamental processes is shown in Table 1.1.

6

River Water Quality Model No. 1

Table 1.1: Process kinetics and stoichiometry for heterotrophic bacterial growth in an aerobic environment.

Continuity Component →

1

2

3

Process rate, ρj

Process ↓

XB

SB

S0

[ML–3T – 1]

1

Growth

1

1−Y Y

ˆS µ XB K S + SS

2

Decay

-1

-1

bXB

True Growth Yield: Y

1 Y

−

ri = ∑ rij = ∑ ν ij ρ j j

j

Oxygen (negative COD) [M(-COD)L-3]

Stoichiometric Parameters:

−

Substrate [M(COD)L-3]

Observed Conversion Rates [ML–3T–1]

Biomass [M(COD)L-3]

j

Mass Balance

i

Kinetic Parameters: Maximum specific ˆ growth rate: µ Half-velocity constant: KS Specific decay rate: b

The first step in setting up the matrix is to identify the components of relevance in the model. In this scenario these are biomass, substrate, and dissolved oxygen, which are listed across the top of Table 1.1 by symbol and across the bottom by name and units. To conform with IAWPRC nomenclature (Grau et al. 1982), insoluble constituents are given the symbol X and soluble components S. Subscripts are used to specify individual components: B for biomass, S for substrate, and O for oxygen. The index i is assigned to each component. In this case, i ranges from 1 to 3 for the three compounds in this simple model. The second step in developing the matrix is to identify the biological processes occurring in the system, i.e. the conversions or transformations that affect the components listed. Only two processes are included in this example: aerobic growth of biomass and its loss by decay. These processes are listed in the leftmost column of the matrix. The index j is assigned to each process: in this case, j = 1 or 2. The kinetic expressions or rate equations for each process are recorded in the rightmost column of the matrix in the appropriate row. Process rates are denoted by ρj where j corresponds to the process as numbered in the leftmost column. If we were to use the simple Monod–Herbert (Herbert 1958) model for this situation, the rate expressions would be those in Table 1.1. The Monod equation, ρ1, says that growth of biomass is proportional to biomass concentration in a first-order manner and to substrate concentration in a mixed-order manner. The Herbert expression, ρ2, states that biomass decay is first-order with respect to biomass concentration. The kinetic parameters used in the rate expressions are defined in the lower right corner of the table. The elements within the matrix comprise the stoichiometric coefficients, νij, which set out the mass relationships between the components in the individual processes. For example, growth of biomass (+1) occurs at the expense of soluble substrate (–1/Y); oxygen is utilised in the metabolic process [–(1–Y)/Y]. The coefficients, νij, are greatly simplified by working in consistent units. In this case, all organic constituents have been expressed as equivalent amounts of chemical oxygen demand (COD); likewise, oxygen is expressed as negative

Introduction

7

oxygen demand. The sign convention used in the matrix is negative for consumption and positive for production. All stoichiometric parameters are defined in the lower left corner of the table.

1.5.2 Use in mass balances Within a system, the concentration of a single component may be affected by a number of different processes. An important benefit of the matrix representation is that it allows rapid and easy recognition of the fate of each component, which aids in the preparation of mass balance equations. This may be seen by moving down the column representing a component, which is why the arrow marked “Mass Balance” is placed at the left-hand side. The basic equation for a mass balance within any defined system boundary is: Input – Output + Reaction = Accumulation

(1.2)

The input and output terms are transport terms and depend upon the physical characteristics of the system being modelled. The system reaction term, ri, is obtained by summing the products of the stoichiometric coefficients νij and the process rate expression ρj for the component i being considered in the mass balance: ri = ∑ ν ij ρ j

(1.3)

j

For example, the rate of reaction, r, for biomass, XB, at a point in the system would be: rX = B

ˆ SS µ X B − bX B K S + SS

(1.4)

ˆ SS 1 µ XB Y K S + SS

(1.5)

for soluble substrate, SS, it would be: rS = − S

for dissolved oxygen, SO, it would be: ˆ SS  1− Y  µ rS = − X B − bX B   Y  K S + SS 0

(1.6)

To create the mass balance for each component within a given system boundary (e.g. a completely mixed reactor), the conversion rate would be combined with the appropriate advective (flow) terms for the particular system. These terms have not been shown here because the purpose of the example is to demonstrate how the matrix is used to define the

8

River Water Quality Model No. 1

fundamental reactions regardless of the system configuration. It should be emphasised, however, that the modelling of a particular physical system requires definition of the system boundary with the associated advective terms.

1.5.3 Mass conservation check Another benefit of the matrix is that mass conservation can be checked by moving across the matrix; the sum of the stoichiometric coefficients multiplied by the mass fraction coefficients characterising the composition of the substances must be zero, if all substances affected by the process are considered in the matrix. This can be demonstrated by considering the decay process. Recalling that oxygen is negative COD so that its coefficient must be multiplied by –1, all COD lost from the biomass because of decay must be balanced by oxygen utilisation. Similarly, for the growth process, the substrate COD lost from solution due to growth minus the amount converted into new cells must equal the oxygen used for cell synthesis. Application of the mass conservation concept disqualifies the use of the traditional measure of organic matter in streams, BOD, in the process descriptions for RWQM1 since mass balances are not possible with BOD. The BOD test measures part of the material present, and measures varying parts depending upon the history of the organic material. The more oxidation the organic material has undergone, the smaller will the fraction be that is measured as BOD. For raw wastewater, typically 50% of the organic material is measured, while in biologically treated wastewater only 10–20% is measured. After degradation of part of the organic material in wastewater during its transport in a river, less than 5% of the organic material may be measured with the BOD test.

1.5.4 Terminology The Petersen matrix approach employs as terminology “components” to represent the chemical and biological species in the model and “processes” to represent the conversions or transformations that affect the components. In the context of environmental water quality modelling, we must add several additional terms. The term “compartment” is used to denote the conceptual subdivision of the system into different biochemical or physical environments. For example, it is typical to include separate compartments for the water column and the sediment. Appendix 1 presents an ecologically-based definition of the river system as a continuum of various zones and the compartments that define those zones. Yet another term can arise as part of the process of solving the model equations. Formulation of a finite-element, finite-difference, or otherwise spatially segmented model requires that the physical space represented in the model be subdivided. This creates spatial segments, which can go by a variety of names including spatial elements, finitedifference elements or grids, finite elements, and boxes. Although there is necessarily some relationship between model compartments and spatial segments, the latter arise from the method of solving the equations, whereas compartments are a more fundamental characteristic of the model. The term “state variable” indicates the model variables that must be solved for. The number of state variables is equal to the number of compartments multiplied by the number of components. Typical state variables are the concentrations of phosphorus in the water phase in the water column compartment, the interstitial water phase in the sediment compartment, and the adsorbed phase within the sediment compartment. In addition, the model includes a number of “parameters.” These are the various rate constants, proportionality coefficients, temperature-dependency coefficients, and other

Introduction

9

coefficients and constants that must be defined in the transport and process equations. Table 1.1, for example, defines several process and stoichiometric parameters in the model of heterotrophic bacterial growth. Model parameters are distinct from water quality parameters, in which case the term is used to represent some measurement of the state of the water quality in the environment. For example, the concentration of dissolved oxygen is a water quality parameter.

1.6 ABOUT THE PROCESS OF MODEL DEVELOPMENT We believe that achieving all of the objectives outlined above will require about a decadelong process of model development that includes such difficult tasks as improving the description of sediment processes. Our primary goal is only to launch this process, to define the framework, and to provide a first model version2 that we hope can be extended and further developed by a broad range of professionals dealing with water quality issues. Some readers of this report will seek a model “code”, that is, a computer program for application of the model. As with ASM1, our intention was not to create a code per se, but rather a conceptual model framework than can be implemented in a variety of codes. We have implemented the model in AQUASIM (Reichert 1994, 1995), a proprietary code available from EAWAG (http://www.aquasim.eawag.ch), and Meirlaen et al. (2001a,b) have implemented the model in the WEST modelling and simulation software (Vangheluwe et al., 1996, 1998, http://www.hemmiswest.com/). Meirlaen et al.'s application took particular advantage of RWQM1’s capability to integrate sewer and ASM-like models to simulate integrated real-time control of an urban wastewater system. Moreover, we encourage potential program users to create model versions in a variety of frameworks such as the US EPA WASP program (Ambrose et al. 1988), Matlab (http://www.mathworks.com/products/matlab/), Microsoft® Excel (http://www.microsoft.com/products/default.asp), and userdeveloped codes.

2

In the course of our work the conversion model obviously went through modifications. In relation to this, we note that the case studies in Chapter 5 demonstrate results obtained by the application of the “first” model version (Reichert et al. 2001a), while Chapter 3 presents a somewhat extended “second” version we felt to be the most relevant as we proceeded with our effort. Despite the creation of two versions of the model, we do not make a distinction such as RWQM1 and RWQM2 since the difference is not significant to the case studies because the “second”, extended version still contains the “first” version as a special case.

2 Evaluation of current water quality modelling practice

2.1 INTRODUCTION Water quality management of rivers requires mathematical models to, first, understand the cause-effect relation between emissions and water quality impacts and, second, to design control measures and to assess their effectiveness. Thus, to meet these research and policy goals, river water quality models seek to describe the spatial and temporal changes of constituents relevant to the state of the aquatic system. In 1925, Streeter and Phelps established the basic principles of such surface water quality modelling. In their groundbreaking work (Streeter and Phelps 1925), they found a relationship between constant pollution loading of the Ohio River and the resulting sag in the dissolved oxygen (DO) concentration in the downstream water. Since then, state variables have been gradually incorporated into the models following the evolution of water quality problems from simple dissolved oxygen household problems, to eutrophication, toxic materials, and so on. Model complexity has increased steadily over the years to address complex problems. Whereas the simple Streeter–Phelps model could describe the oxygen household sufficiently with only two state variables, models such as QUAL2 require about ten state variables to describe comprehensively O, C, N, and P cycling (Brown and Barnwell 1987). Even more complex are deterministic ecosystem models that may consider suspended solids, several classes of algae, zooplankton, invertebrates, plants, and fish (Boling et al. 1975; Wlosinski and Minshall 1983). Recent models have been © 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

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established with the aim of handling problems of non-point source pollution, interaction with wastewater treatment plant operation, sewerage and urban storm water overflows, and extreme pollution events. However, as pointed out by Rauch et al. (1998a, 2001), the lack of common water quality descriptors makes the combined use of such river water quality models together with models of the sewer system and the wastewater treatment plant a difficult task. In the end, the choice of the model applied for a particular task depends on many different factors such as the objectives of the analysis, and data and time availability. Stemming from our goals, we have limited our attention to deterministic models that handle the “traditional” components O, C, N, and P.

2.2 WATER QUALITY MODELLING AND LEGISLATION The current practice in water quality modelling is largely driven by legislation and regulation. Accordingly, the practice of water quality modelling varies from jurisdiction to jurisdiction with the regulatory framework. Despite its lack of recent developments, the most widely known and used computer program for river water quality modelling is the QUAL2E model developed by the US Environmental Protection Agency (EPA) (Brown and Barnwell 1987). QUAL2E has a long history, having been preceded by another version of QUAL2 (Roesner et al. 1981) and the similar but simpler QUAL1 model (TWDB 1971). As noted in Chapter 1, QUAL2E simulates dissolved oxygen and the many associated water quality parameters of the C, N, and P cycle in rivers and streams under conditions of steady streamflow and pollutant discharge. While QUAL2E has clear limitations—storm water flow events and other situations with unsteady hydraulics cannot be modelled—its formulation derives directly from the US regulatory framework for which it was developed and for which it is generally well suited. US federal laws establish a two-tier system in which effluent limitations are first set based on available technology and then, if this degree of restriction is insufficient to meet water quality standards for a particular river, a further reduction in emissions is determined through a wasteload allocation. A wasteload allocation is the computation of the maximum amount of waste that can be discharged to the river while still meeting water quality standards under the low-flow conditions specified as a part of the stream standards (usually 7Q10, the seven-consecutive-day low flow with a probability of occurring once in ten years). Wasteload allocations are performed for conditions of constant low flow and maximum permitted effluent discharge rate. QUAL2E is intended specifically for the steady-streamflow, steady-effluent-discharge conditions specified in the water quality regulations for wasteload allocation. The early and widespread use of QUAL2E and its predecessor models makes QUAL2E a standard against which other models are typically compared. The capabilities and state variables of the model are described further in Section 2.6.3 below. Few other countries have established water quality management laws in which water quality modelling is as integral a part of the process as it is in the US. Although water quality modelling is often used, it is typically as a planning tool in special studies of particular rivers. These models are frequently developed on a custom basis for each individual application. Thus, without the standardising requirements as created in the US wasteload allocation system, alternative modelling standards have yet to emerge in many other countries. It is this apparent lack of pressure towards the creation of individual

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River Water Quality Model No. 1

national operating standards for river water quality models that makes QUAL2E such a widespread tool. For example, until very recently in Europe, water quality modelling was far less prevalent in the regulatory process than in the US. Decision-makers typically employed dilution ratios (based on simple mass balances for the discharges to a river stretch) to assess expected water quality. Therefore, modelling the quantity of flow in a river was generally more important than modelling the quality. The recently released European Union Water Framework Directive (EU-WFD) has created new conditions of legislation (EU 2001a, b). The EU-WFD prescribes achieving the “good status” of surface (and subsurface) waters within 15 years. Good status considers both the ecological quality and chemical quality of the waters as a function of a number of physical, chemical, biological, hydrologic, morphologic and other parameters. EU-WFD should be applied together with a number of already existing directives including the Urban Wastewater Directive (EU 2001c). This rather complex and not yet well-defined system incorporates effluent standards and ambient criteria. Effluent standards for urban discharges include specifications for “normal” and “sensitive” areas that are defined based on eutrophication potential. Ambient criteria have yet to be defined in detail but will primarily serve to achieve the reduction of non-point sources if point source control alone will not lead to meeting the criteria. Overall, the two-tier system is likely to function similarly as in the US and thus it creates huge needs for water quality model development and applications over the forthcoming decade. In fact, this process has already been launched. Even prior to the EU-WFD, there has been a gradually increasing emphasis on water quality modelling, largely as the result of important developments in the UK, Germany, Denmark, and the Netherlands. UK environmental agencies use simple stochastic models (for example, SIMCAT (NRA 1990)) to summarise the two-week survey data typically collected by the agencies and to help to decide on future restoration activities or permits/consents for dischargers on the catchment scale. Monte Carlo simulation is incorporated in the procedure to compensate for the stochastic features of forcing functions and inherently large uncertainty in the sparse data set. Also in the UK, the Urban Pollution Management (UPM) procedure (FWR 1998) has been developed and relies upon a suite of water quality models of different levels of complexity, ranging from simple steady-state calculations implemented via a spreadsheet, to fully dynamic water quantity and quality models. The Danish Engineering Union was early to publish a detailed procedure for computing and assessing water pollution, focusing on intermittent oxygen depletion due to combined sewer overflow (Spildevandskomiteen 1985). In Germany, the ATV (Abwassertechnische Vereinigung – Association for Water Pollution Control) has developed a comprehensive water quality model (ATV 1996) that is increasingly applied. To summarise, in a number of countries, water quality models are increasingly being used on a case-by-case basis for specific environmental impact assessments or scenario analysis. Computer programs in use include QUAL2E, WASP (Ambrose et al. 1988), Mike-11 (DHI 1992), ISIS (Wallingford 1996, 1997), and DUFLOW (Aalderink et al. 1995). On the basis of the above, we anticipate significant further progress in the use of models in Europe.

2.3 GENERAL CONCEPT OF RIVER WATER QUALITY MODELS A river as a natural aquatic system consists not only of running water, but rather of three “compartments”: the gas phase, the water phase, and the sediment. The mathematical

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description of the system has to account for the processes within, and the interactions between, these compartments. As indicated in Chapter 1, the fate of substances in each compartment is due to (1) physical transport and exchange processes, and (2) biological, chemical, biochemical, and physical conversion processes. These processes are governed by a set of well-known extended transport equations (see for example Somlyódy and van Straten 1986):

∂c ∂c ∂c ∂c ∂  ∂c  ∂  ∂c  ∂  ∂c   + ε =−u −v − w + εx  + ε  + r( c , p ) (2.1) ∂t ∂x ∂y ∂z ∂x  ∂x  ∂y  y ∂y  ∂z  z ∂z  where c = n-dimensional mass concentration vector for the n state variables [ML–3]; t = time [T]; x, y, and z = spatial co-ordinates [L]; u, v, and w = corresponding velocity components [LT–1]; εx, εy, and εz = turbulent diffusion coefficients for the directions x, y and z, respectively [L2T–1]; r = n-dimensional vector of rates of change of state variables due to biological, chemical, and other conversion processes as a function of concentrations, c, and model parameters, p (subject to calibration) [ML–3T–1]. In principle, Equation (2.1) can be applied to describe the dynamics in all three compartments; however, the relative importance of the processes is dependent on the physical characteristics of the phase under consideration. For example, due to the high turbulence in the gas phase the constituents (most importantly oxygen) are typically fully mixed and the transport processes therefore no longer considered as relevant. Consequently, the gas phase is hardly ever looked upon in detail but considered via an interface term as a boundary condition of the water phase. To a lesser extent the same holds true for the sediment. In a number of river water quality models the sediment compartment is either neglected or taken into account as a boundary condition of the water phase. However, in contrast to the gas phase, the processes there cannot be generally neglected but are occasionally even decisive for the overall system dynamics. In that case Equation (2.1) must be applied for both the water phase and the sediment, but in the sediment phase convection and diffusion processes are slow and can be neglected. An interface term expresses the exchange of constituents between the water and sediment compartments. The description of physical transport and exchange processes (such as advection and diffusion/dispersion) is based upon detailed information of the flow field (velocity components u, v, and w, and turbulent diffusion coefficients εx, εy, and εz). Thus, Equation (2.1) in itself cannot be solved alone but requires the application of a hydraulic model as input. This stepwise, sequential solution procedure is correct as long as the hydraulic description is independent from the transport and exchange processes. If this condition is no longer guaranteed (for example, if sediments influenced bottom slope dynamically) Equation (2.1) would need to be directly linked with the hydraulic model equation. Equation (2.1) is a well-known partial differential equation (PDE) that can be solved either numerically (usually after averaging over the depth or the cross-sectional area which reduces the number of dimensions and leads to the introduction of the so-called dispersion coefficient), or by using a conceptual approach. In the latter case, it is assumed that the system consists of m interconnected (completely mixed and/or plug flow) tanks or segments, which leads to n by m ordinary differential equations. In both cases, some of the simplifying

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River Water Quality Model No. 1

assumptions enter the resulting equations as boundary conditions (basically as a part of the aggregated conversion submodel). Equation (2.1) offers not only the basic governing equation of water quality models, but it also specifies a useful framework and the main model elements. The components of this framework are the following (see also Chapter 1): • • • •

the hydrodynamic model for deriving velocity components u, v, and w, and turbulent diffusion coefficients εx, εy, and εz; the transport (or advection-diffusion) equation (describing the behaviour of so-called conservative substances) and its solution; the conversion submodel, r(c, p), which has much less solid theoretical grounds than hydrodynamics and, thus, must be developed using a combination of theoretical and empirical knowledge (Beck and van Straten 1983; Somlyódy 1982); and, methodologies such as calibration, validation, identification, sensitivity, and uncertainty analyses (Beck 1987) to aid model selection and testing.

All deterministic river water quality models follow the general concept outlined above. However, depending on the objective of a specific model, greater or fewer simplifying assumptions are used. A model designed on the basis of the above steps and elements is typically implemented on a computer, which raises additionally a number of software (and hardware) issues. In the remainder of Chapter 2, we discuss the state of the art of water quality models. We include but a short summary; for more details the reader is referred to, among others, Thomann (1972), Orlob (1982), Crabtree (1986), Thomann and Mueller (1987), McCutcheon (1989), and Somlyódy and Varis (1992).

2.4 HYDRODYNAMICS AND HYDRAULICS Flow of water in a river is described by the continuity and momentum equations. The latter is known as the Navier-Stokes or Reynolds equation. The actual form of a hydrodynamic model depends on assumptions made to characterise turbulence. Methods vary from the use of the eddy viscosity as a known parameter, to the application of the so-called k-ε theory (see Bedford et al. (1988) or Rodi (1993) for an overview of the state of the art of turbulence models). Complex hydrodynamic models are available (see for example Abbott (1979) and Naot and Rodi (1982)). Such models are useful in estimating local mixing processes that may be of interest at wastewater discharge sites. However, for estimating the effects of point sources, diffuse sources, and transformation processes over longer river reaches, crosssectionally averaged (one-dimensional) St. Venant equations or approximations to these equations are adequate hydrodynamic submodels. The St. Venant equations are shown in Equation (2.2) (Henderson 1966; French 1985; Yen 1973, 1979). Equation (2.2a) expresses continuity (mass conservation) and Equation (2.2b) momentum conservation. ∂A ∂Q + ∂t ∂x ∂Q ∂ Q2  + ∂t ∂ x  A

= q and

(2.2a)

 ∂y  + gA + gA ( S f − S 0 ) = 0 ∂x 

(2.2b)

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where A = cross-sectional area [L2]; Q = streamflow [L3T–1]; q = lateral inflow per unit length [L2T–1]; x = longitudinal co-ordinate [L]; S0 = bottom slope [–]; Sf = friction slope [–]; y = channel depth [L]; and g = gravitational acceleration [LT–2]. Many different forms and approximations to the St. Venant equations are known, depending upon whether the flow is steady or unsteady and the simplifications that are made. Thus, for water quality studies often the equation of steady, gradually varying flow is employed (which may be further simplified to the so-called Manning equation as done in QUAL2E). Unsteady models include the kinematic, diffusive, and dynamic wave approaches, which are all based on the continuity and momentum equations. The difference stems from simplifications of the latter: dynamic wave models solve the full equation, diffusive ones exclude the acceleration terms, while kinematic ones also disregard the pressure gradient term that is essential for the description of backwater effects. The hydrodynamic equations are generally solved by efficient finite difference methods (Abbott 1979; Amein and Fang 1970; Mahmood and Yevjevich 1975; Orlob 1982). For water quality issues, the acceleration terms in the momentum equation rarely play a significant role and conversion processes amplify the typical time scales. For these reasons, the diffusive wave approach is often a satisfactory approximation. An even greater simplification is employed in the so-called “hydrologic models”, which generally respect the continuity equation but replace the conservation of momentum with some conceptual relationship. The underlying concept is a cascade of reservoirs in series with the water being routed downstream. Hence this technique requires the river being discretised into a series of tanks. The system of equations for a single tank is written in its simplest form: ∂V V = Q i − Q e and Qe = K ∂t

(2.3)

where V = volume of tank [L3]; Qi = influent to the tank [L3T–1]; Qe = effluent from the tank [L3T–1] and K = storage constant [T]. The left equation expresses continuity and the right one is the conceptual equivalent for the conservation of momentum, which is needed for the calculation of water motion. This conceptual relation between outflow, inflow, and storage (V = f ( Qi,Qe)) might be formulated either linearly as in Equation (2.3) or non-linearly (Chow 1959). The critical parameter is the storage constant, K, which must be found by calibration. Hydrological methods have been applied frequently and with much success, the most important case study being the Muskingum River (Cunge 1969).

2.5 TRANSPORT PROCESSES As seen from Equation (2.1), the transport of dissolved substances in rivers is governed by advection and turbulent diffusion. The initial mixing in the “near field” zone may depend on momentum transport; this is a phenomenon that is important if the discharge flow is large (such as thermal discharges). The transport process is characterised by two mixing lengths, LV and LL. These are the distances of “complete” mixing along the depth and over the entire cross-section, respectively. For shallow rivers LV is short and thus a depth-integrated twodimensional form of Equation (2.1) can be applied. As a result of the integration, the impact of shear or spatial non-uniformity in the advective velocity appears in a term assumed to be

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River Water Quality Model No. 1

of Fickian type, and the “lumped” parameters of diffusion thus obtained are called dispersion coefficients. Their value strongly depends on the intensity of lateral mixing (dispersion coefficients increase with decreasing lateral mixing intensity). Lateral mixing is influenced by spatial non-uniformities of the river bed, such as non-uniformities in slope, morphology, roughness, or other stream characteristics. Dispersion coefficients can be estimated with semi-empirical formulas (Fischer et al. 1979) or with the aid of in situ tracer measurements. For water quality studies of rivers that are long in comparison to LL and dominated by longitudinal and temporal changes, further integration is possible along the width, which leads to the one-dimensional advection-dispersion equation (see for example Fischer et al. (1979); alternatively the dispersion effect can be approximately described by dividing the river cross-section into an advective and a stagnant zone, see Reichert and Wanner (1991) and references cited therein). In a vector form corresponding to Equation (2.1), this onedimensional equation can be written as (see e.g. Somlyódy and van Straten 1986): ∂ ( AC) ∂ (QC) ∂  ∂C  + =  AD L  + AR (C, P) ∂t ∂x ∂x  ∂x 

(2.4)

where A = cross-sectional area [L2]; Q = streamflow rate [L3T–1]; DL = longitudinal dispersion coefficient (see Fischer et al. 1979) [L2T–1]; C = vector of cross-sectionally averaged concentrations of various components [ML–3]; and R(C, P) = vector of rates [ML–3T–1] of change of cross-sectionally averaged concentrations due to conversion processes which are now a function of the concentrations, C, and of model parameters, P. It is stressed that due to integration, the R functions and P parameters differ from r and p in Equation (2.1). Initial and boundary conditions are also needed to solve Equation (2.4). The water quality model described by the function R can be developed stepwise and independently of the description of hydraulics. For steady problems, the dispersion term can often be neglected and the resulting ordinary differential equation (ODE) can be solved by introducing the travel time as an independent variable. Often another simplification is made: Equation (2.4) is integrated for subsequent river stretches within which complete mixing is assumed. This procedure leads to a sequence of interlinked reactors (and the application of the “reactor principle”) for which the mass balance is expressed by the vector equation:

d (VC* ) = QiCi − QC* + V dt

R* (C* , P* )

(2.5)

where C* = concentration vector [ML–3]; Ci = inflow concentration [ML–3]; Qi = inflow [L3T–1]; Q = outflow [L3T–1]; V = reactor or tank volume [L3]; and R* (C*, P*) = conversion rate vector [ML–3T–1] (asterisks denote that the interpretation of C, R, and P is again different than in the one-dimensional case). If the number of reactors is m, then n by m ordinary differential equations should be solved. Selection of the size of tanks should be done such that the introduced implicit or numerical dispersion is roughly equal to that of the actual system (Shanahan and Harleman 1984).

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Such box models are frequently used (e.g. Beck and Finney 1987; Rauch and Harremoes 1996; Meirlaen et al. 2001a), particularly due to the easier formulation of methods of calibration, validation, and parameter estimation in comparison to PDE model structures. However, note that the model suffers from a conceptual deficiency, as Equation (2.5) cannot take into account the different “travel velocities” of waves (wave celerity) and of substances in the water phase (transport velocity). Occasionally this phenomenon has a considerable effect on the overall systems dynamics (Krebs et al. 1999).

2.6 CONVERSION PROCESSES 2.6.1 Introduction As noted, conversion processes describe changes in the component concentrations that are due to biological, chemical, biochemical, and physical processes. The historical development of O, C, N, and P models shows step-by-step extensions and increasing complexity as follows (see also Chapter 1): (i) the starting point was the pioneering Streeter–Phelps model (Streeter and Phelps 1925) which describes the increase and following decrease of the oxygen deficit downstream of a source of organic material; (ii) the basic model was later enhanced to distinguish BOD removal by biodegradation and settling (Thomas 1948), to add the effects of dispersion (O’Connor 1961), sediment oxygen demand (SOD), and photosynthesis and respiration (P-R) (Dobbins 1964) and to distinguish nitrogenous (NBOD) and carbonaceous (CBOD) oxygen demand; (iii) it was later extended to include nitrogen processes (EPA’s detailed model with nitrification was called QUAL1 (Orlob 1982; TWDB 1971)); and (iv) finally phosphorus cycling and algae were added in creating the QUAL2 model family (Brown and Barnwell 1987; Roesner et al. 1981). Based on this general concept several newer models are available (which include both simplifications and extensions of the QUAL2 model) depending on the purpose of the use (research, regulation, etc.). An example of the latter is the recently released ATV river water quality model (ATV 1996). All biological reactions are strongly dependent on the conditions in their environment. Of all the different factors, temperature is undoubtedly the most significant one. The influence of temperature on parameters is most frequently expressed by the general function:

K (T) = K (20 ° ) e κ (T

− 20 )

(2.6)

where T = temperature [°C]; K = temperature-dependent parameter; and κ = temperature constant [–]. Equation (2.6) can be applied in the temperature range 0 to 30 °C, which encompasses most aquatic systems. The temperature constant κ has a value of typically 0.069°C–1, that is K(20°C) = 2·K(10°C).

2.6.2 Streeter–Phelps model This classical model for computing the detrimental effect of the discharge of organic waste on the oxygen concentration in a river is concerned with only two processes, that is, bacterial decomposition in the water phase and atmospheric reaeration. In spite of the fact that this is

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River Water Quality Model No. 1

the most primitive approach to water quality modelling, a discussion is worthwhile to outline the basic concepts of water quality modelling. The underlying idea of the bacterial decomposition process is that organic matter is degraded by pelagic (suspended) heterotrophic bacteria in the water phase. The degradation rate, K1, is first-order, and oxygen is consumed from the water phase equivalent to the quantity of the organic matter converted. The first major shortcoming of this model approach is that only the growth of pelagic bacteria is taken into account, but not the complete metabolism. That is, the bacteria groups are not state variables of the model and the effect of changes in mass and character is neglected altogether. This is a basic problem that also appears in the QUAL model family. Second, the model assumes that all degradation effects appear in the water phase only and completely disregards the equivalent process in the sediment compartment. Thus, degradation is potentially underestimated and the model must be fitted by means of an adaptation of the coefficient K1 in the first-order biodegradation process. Neglecting the sediment compartment can cause gross model errors, especially if spatial and temporal dynamics appear in sediment degradation processes. The introduction of the sediment oxygen demand (SOD) in later, refined models does not solve this general problem but is to be seen as only a boundary condition for the water-phase compartment. Reaeration is the term commonly used in river quality modelling for describing the diffusion of the gaseous oxygen into the liquid through the air–water interface. The description of the phenomenon is essentially similar to the one mentioned above: as with the SOD, atmospheric reaeration is introduced as a boundary condition for the water phase compartment. The detailed dynamics of oxygen in the atmosphere (gas phase) are not considered to be relevant for the modelling purpose. The formulation of this boundary condition assumes that the absorption of oxygen is directly proportional to the dissolved oxygen deficit, that is the deviation from the oxygen saturation concentration. The mass transfer rate of oxygen is hence usually expressed as:

aer RDO = K 2 .( DOsat − DO )

(2.7)

where RaerDO = contribution of reaeration to the conversion rate of dissolved oxygen, K2 = reaeration coefficient [T – 1]: DO = dissolved oxygen concentration in the water body [ML–3]; DOsat = oxygen saturation concentration [ML–3], where DOsat= 14.652 – 0.41022*T + 0.007991*T 2 – 0.000077774*T 3 with T = temperature [°C] (Elmore and Hayes 1960). Due to the importance of the reaeration coefficient for water quality modelling, numerous predictive models have been presented over the last decades (reviewed in Bowie et al. 1985; Gromiec et al. 1982; McCutcheon 1989). The typical approach is to express the reaeration coefficient as a function of simple hydraulic parameters, e.g. velocity and depth. The Streeter–Phelps model is shown in Table 2.1 in the Petersen matrix notation introduced in Section 1.5. The simplicity of the matrix reflects the simplicity of the model. Note that according to the general concept, reaeration is not a conversion process but a boundary condition.

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Table 2.1: Conversion processes of Streeter–Phelps model in matrix formulation. Component Process name Reaeration Biodegradation

1 DO 1 –1

2 BOD –1

Process rate ML–3T – 1 K2.(DOsat – DO) K1.BOD

Specification Boundary Condition Conversion Process

In the original form, this biokinetic model was applied to a steady, uniform flow description of the water motion mechanism, which made it possible to find an analytical solution (see examples given by Thomann and Mueller 1987). The result is denoted also as the dissolved oxygen sag curve and some effort has been made to prepare solutions based on these analytical considerations.

2.6.3 QUAL2 model family QUAL2 is the most comprehensive river water quality model developed over the years by the US EPA. It includes the following phenomena: degradation of organic material; growth and respiration of algae; nitrification (considering nitrite as an intermediate product); hydrolysis of organic nitrogen and phosphorus; reaeration; sedimentation of algae, organic phosphorus, and organic nitrogen; sediment uptake of oxygen; and sediment release of nitrogen and phosphorus (Brown and Barnwell 1987). All these processes consider the effect on the oxygen, nitrogen, and phosphorus cycles. Even though the model has not seen substantial improvement since 1987, QUAL2 is still seen as a standard for river water quality modelling (Section 2.2). Figure 2.1 shows schematically the processes included in this model.

ORG-N

AtR

NH4 NO2

SOD

BOD DO ORG-P

NO3

DIS-P Chla

Figure 2.1: Schematic description of the water quality model QUAL2 (Brown and Barnwell 1987). AtR = atmospheric reaeration; DO = dissolved oxygen; BOD = biological oxygen demand; SOD = sediment oxygen demand; NH4 = ammonia and ammonium; NO2 = nitrite; NO3 = nitrate; ORG-N = organic nitrogen; Chla = chlorophyll a (algae); ORG-P = organic phosphorus; DIS-P = dissolved Phosphorus.

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In spite of the fact that the model is significantly more complex, the underlying assumptions of the primitive Streeter–Phelps model are still apparent in QUAL2. As in the Streeter–Phelps formulation, the QUAL model fails to regard the bacterial biomass as a state variable, which makes the dynamic consideration of population changes in the bacterial composition impossible. In contrast, the algae biomass is a dynamic variable considered in terms of chlorophyll a. Also, the model focuses only on the water phase, thus neglecting the detailed dynamics in the other two compartments, the gas phase and the sediment. The interactions with both are taken into account in the form of boundary conditions as discussed above for the Streeter– Phelps model. But whereas the gas–water interface remains unchanged (reaeration process), the interaction between the water phase and the sediment contains now boundary conditions for several model components: oxygen is removed from the water phase due to the sediment oxygen demand, while nitrogen and phosphorus are released from the sediment into the water. All three water-sediment boundary conditions are formulated as zero-order processes, thus ignoring completely the temporal and spatial dynamics in the sediment. Table 2.2: Biochemical and source and sink terms of exchange processes with the sediment and atmosphere of the river water quality model QUAL2 in matrix notation. Component Process

1 2 3 4 DO BOD ABM ORG-N

1 Reaeration

1

2 Biodegradation

–1

3 BOD sedimentation 4 Sediment DO demand

–1

5 Photosynthesis

α3

5 NH4

6 NO2

7 NO3

8 ORG-P

9 DIS-P

K2·(DOsat-DO) –1

K1·BOD

–1

K3·BOD K4/d 1

–0.07·FNH4

–0.07·

–0.01

(1–FNH4) 6 Respiration

–α4

7 Algae sedimentation

–1

0.07

µmax·ABM ·f(L,N,P) ρ·ABM

0.01

σ1/d·ABM

–1

8 Nitrogen Hydrolysis 9 Nitrification 1st step

Process rate [ML–3T–1]

–1 –

β3·ORG-N

1 –1

β1·NH4·f(DO)

1

3.43 10 Nitrification 2nd step

–

–1

β2·NO2·f(DO)

1

1.14 11 N sedimentation

σ4·ORG-N

–1

12 N sediment release

σ3/d

1

13 P hydrolysis

–1

14 P sedimentation

–1

1

σ5·ORG-P

15 P sediment release

1 –3

β4·ORG-P

–3

σ2/d

where: DO = dissolved oxygen [ML ]; DOsat = DO saturation concentration [ML ]; BOD = biochemical –3 –3 –3 oxygen demand of organic material [ML ]; ABM = algal biomass [ML ]; ORG-N = organic nitrogen [ML ]; –3 –3 –3 NH4 = ammonia-N [ML ]; NO2 = nitrite-N [ML ]; NO3 = nitrate-N [ML ]; ORG-P = organic phosphorus –3 –3 –1 [ML ]; DIS-P = dissolved phosphorus [ML ]; K 1 = deoxygenation coefficient [T ]; K 2 = reaeration –1 –1 –2 –1 coefficient [T ]; K 3 = BOD settling rate [T ]; K 4 = sediment oxygen demand rate [ML T ]; d = mean –1 –1 stream depth [L]; µmax = maximum algal growth rate [T ]; ρ = algal respiration rate [T ]; σ 1 = algal –1 –2 –1 –2 –1 settling rate [LT ]; σ 2 = benthos source rate for P [ML T ]; σ 3 = benthos source rate for N [ML T ]; –1 –1 –1 σ 4 = N settling rate [T ]; σ5 = P settling rate [T ]; β1 = ammonia oxidation rate [T ]; β2 = nitrite oxidation –1 –1 –1 rate [T ]; β3 = N hydrolysis rate [T ]; β4 = P hydrolysis rate [T ]; α3 = oxygen production per unit of algal growth, gO/g ABM [-]; α4 = oxygen uptake per unit of algal respiration, gO/g ABM [-]; f ( L,N,P) = algal growth limitation factor; f ( D O ) = nitrification limitation factor; FNH4 = ammonia preference factor.

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The model formulation is given in Table 2.2 in Petersen matrix notation. Note that reaeration (1), sediment oxygen demand (4), nitrogen sediment release (12), and phosphorus sediment release (15) are not conversion processes but boundary conditions of the model. The main difference between QUAL2 and earlier model approaches is the consideration of the eutrophication phenomenon and its implications for the dissolved oxygen concentration and to the nutrient cycle. However, there are two dominant photoautotrophic constituents that contribute to eutrophication: (1) phytoplankton, which denotes all different forms of algae with the common feature of being distributed in the water phase, and (2) aquatic macrophytes, which denotes rooted green plants. Although both constituents are essentially subject to the same mechanism, QUAL2 (as many other models) neglects the latter. Primary production of photo-autotrophs (photosynthesis) is limited when light and/or nutrients are not available in a sufficient extent. The resulting growth limitation factor f tot has to take into account the individual influences. QUAL2 offers several possibilities about how to model the total limitation, the most important of which is the multiplicative method (f tot = f 1*f 2*..) and the minimum factor approach, the latter being based on Liebig’s law of the minimum (f tot = Min[f 1, f 2, ..]). For specifying nutrient limitations to photosynthesis, QUAL2 assumes that the stoichiometry of the algae biomass is constant and thus disregards the luxury uptake mechanism (Nyholm 1977). Due to this simplification, the limitation factors are only dependent on the nutrient concentration in the water phase and can be conveniently expressed by means of Monod-type equations:

f(N )=

NH 4 + NO3 DIS − P and f ( P ) = KN + ( NH 4 + NO3 ) KP + DIS − P

(2.8)

where KN and KP = half saturation constants for nitrogen and phosphorus [ML–3]. Note that here no preference is stated in the nitrogen uptake between ammonia and nitrate as the effect on the growth limitation is marginal. In order to predict the limiting effect of light to algae growth, both the light attenuation with depth into the water and the relation between actual light level and photosynthesis rate are considered. Light attenuation with depth is computed in QUAL2 and essentially all models by means of Beer’s law, assuming an exponential decrease of light over depth:

I ( z ) = I 0e −γ z

(2.9)

where I(z) = light intensity at depth z [EL–2T-1]; z = distance from surface [L]; I0 = surface light intensity [EL–2T–1]; and γ = light extinction coefficient [L–1]. The light extinction coefficient contains both a component expressing the constant base light extinction for water and a second component that takes into account the algae self-shading effect. Note that I0 varies with time of day, time of year, and location. Furthermore, meteorological conditions and shading may be of influence. In most models I0 is an input variable but there are also some equations available to compute the temporal variation.

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River Water Quality Model No. 1

QUAL2 expresses the actual relation between light and photosynthesis with one of three different options: saturation type, Smith (1936) formula, and Steele (1965) formula. All exhibit similar characteristics for lower and medium light intensities. The growth rate increases linearly with light at low intensities and reaches the maximum at an optimum light intensity (saturation). However, research has revealed that extreme light intensities cause a photoinhibition, resulting in a decrease of the growth rate when light exceeds saturation intensity. This effect is only taken into account by Steele’s formulation.

2.6.4 Recent developments Research in “traditional” water quality pollutants has waned over the last two decades, resulting in relatively few new developments. Recent model developments contain several extensions of the general concept developed for the QUAL model family. For example, Cerco and Cole (1995) include the description of the silica cycle and of sediment. Similar processes are also implemented in special model versions of MIKE11 (DHI 1992) and in the river water quality programme designed by ATV in Germany (ATV 1996). The latter includes silica, several classes of algae, consumers, suspended solids, pH, metals, and a description of the sediment in addition to the state variables used in QUAL2. Higher trophic states are also included in older stream ecosystem models (Boling et al. 1975; Smith 1978; Wlosinski and Minshall 1983). In addition to using temperature-dependent rates, most advanced water quality models including QUAL2 contain submodels that calculate water temperature. These submodels use short and long wave solar and atmospheric radiation, evaporation and sensible heat fluxes, and long wave emission in order to calculate river water temperature from a heat balance.

2.7 SOFTWARE AND COMPUTER PROGRAMS Other than the simplest approaches, all mathematical models for the prediction of water quality in rivers require the use of a computer. Due to the considerable effort needed to develop and implement a site-specific model, the use of existing computer programs is preferred whenever possible. The following classification aims to give only an overview of the most important computer programs and is by no means meant to be exhaustive. Relevant features for classification are the description of hydrodynamics and transport, model structure (important variables, processes and submodels), software structure (open/closed, meaning that the user can change the model structure), and systems analytic features supported by the program. Table 2.3 gives an overview of some important software products for river water quality modelling.

2.8 MODEL IDENTIFICATION AND TESTING Calibration, validation, and model structure identification have become increasingly important and difficult. Thus, although our primary aim is the development of standardised conversion submodels, this cannot be done without developing a framework for the entire modelling process (Chapter 1). The framework should incorporate methods that can and should be employed for the purposes of identification, calibration, validation, and the analysis of uncertainties of differing origins. Moreover, as shown by Equation (1.1), the framework also should include hydraulic and transport modules to be able to perform calculations for “real” systems. There is no unique methodology in this respect: many different but basically equivalent methods are known and frequently it is desired to switch

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from one approximation to the other (e.g. when moving from the so-called near field to the far field, or from the two-dimensional plume reach to the “completely” mixed onedimensional river reach). Table 2.3: Important software products (after Ambrose et al., 1996, with extensions). Hydrodynamics Transport Sediment Water quality

Systems analysis

Program: External Input Simulated Control structure Advection Dispersion Quality models Temperature Bacteria DO-BOD Nitrogen Phosphorus Silicon Phytoplankton Zooplankton Benthic algae Parameter estimation Sensitivity, uncertainty analysis

1 Y N N Y Y N Y N Y Y Y N Y N N N

2 Y Y N Y Y Y N N Y Y Y N Y N N N

3 N Y Y Y Y Y Y Y Y Y Y Y 3 Y N N

4 N Y Y Y Y N Y Y Y Y Y N Y N N N

5 Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N

6 N Y Y Y Y Y Y Y Y Y Y Y Y Y Y N

7 N Y Y Y Y Y Y Y Y Y Y Y Y N Y N

8 N Y Y Y Y N

9 N Y Y Y Y O

OS

OS

N

Y

10 Y Y Y Y Y Y N N Y Y Y N Y N N Y

Y

N

N

N

N

N

N

N

Y

Y

11 Y N N Y N O

OS

Y Y

O = open; OS = open structure. 1 = QUAL2 (US EPA; Brown and Barnwell 1987); 2 = WASP5 (US EPA; Ambrose et al. 1988); 3 = CE-QUAL-ICM (US Army Engineer Waterways Experiment Station; Cerco and Cole 1995); 4 = HEC5Q (US Army Engineer Hydrologic Engineering Center; HEC 1986); 5 = MIKE11 (Danish Hydraulic Institute; DHI 1992); 6 = ATV Model (ATV, Germany; ATV 1996); 7 = ISIS (HR Wallingford, UK; Wallingford 1996, 1997); 8 = DUFLOW (University of Wageningen, The Netherlands; Aalderink et al. 1995); 9 = AQUASIM (EAWAG, Switzerland; Reichert 1994); 10 = DESERT (IIASA; DeMarchi et al. 1999; Ivanov et al. 1996); 11 = WEST (Hemmis, Belgium; Vangheluwe et al. 1996, 1998).

Experiments and data collection are crucial for testing and selecting conversion submodels, but are often inadequate. Provided adequate data are collected, the values of model parameters must be assessed from the data. Typically one starts from the default values that are reported with the model description. For complex models often only finetuning is applied. More can be achieved by also “isolating” certain sets of conversion processes, performing experiments accordingly and estimating parameters from the data gathered (Somlyódy and van Straten 1986). In many practical applications of water quality models, heuristic approaches are followed to decide which parameters to adjust to obtain a “good” fit, choosing parameter values manually rather than by automated numerical techniques. Automated methods, on the other hand, depend on the model structure (PDE, ODE, linear, non-linear), noise assumed, loss function defined (for example, minimisation of the sum of least square deviations), optimisation method (batch or recursive estimation), number of variables and parameters, type of measurements, and so on (see among others Beck 1987; Beck and van Straten 1983; Brun et al. 2001; Dochain and Vanrolleghem 2001; van Straten and Somlyódy 1986). For smaller models – if a white noise assumption holds and regular observations are available – the Kalman filter or the extended Kalman filter can be applied. For more complex or non-linear models, the Hornberger, Spear, and Young (HSY) method offers attractive features (Beck 1987; Hornberger and Spear 1981, 1983; Young et al. 1978) and accounts for

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the fact that in water quality studies generally insufficient data are available to calibrate a model. Further useful techniques of calibration, validation, uncertainty, and sensitivity analyses are discussed by Beck (1987).

2.9 SHORTCOMINGS AND NEEDS 2.9.1 Problems in model application The limitations of the QUAL2 formulation become apparent when attempting to simulate conditions other than the steady-streamflow, constant-emission conditions for which it is intended. QUAL2 is best suited for point sources of pollutants. For even these problems, however, the model is unsuitable for rivers that experience temporal variation in streamflow or where the major discharges fluctuate significantly over a diurnal or shorter time period (other available models are able to simulate transient conditions – see Section 2.7). More significant are the limitations of the model when examining the contribution of nonpoint sources of pollutants to river water quality degradation. Non-point sources have assumed greater relative importance in water quality management as point sources have come under increasingly stringent control. Unfortunately, non-point source loads are often driven by rainfall events and thus both the wasteload and streamflow vary significantly over time. Both types of variation may deviate significantly from the assumptions of QUAL2. These limitations for point and non-point sources compromise the ability to model such problems as rivers regulated by hydropower or other dams that cause significant diurnal fluctuations in streamflow; combined sewer overflows and urban storm water effects; the effects of diurnal variation in the flow of municipal effluents; and the effects of industrial effluents discharged on a batch basis or with significant variation in flow during different working shifts. For many of these problems, there is currently no readily available, widely accepted water quality model. The limitations of the model are compounded in many applications by inexperience or insufficient knowledge on the part of the model user. The QUAL2E user’s manual (Brown and Barnwell 1987) is basically a description of the model formulation and input formats for a model user who is implicitly assumed to be experienced and knowledgeable. However, the widespread dissemination and ready availability of QUAL2 encourages use that sometimes falls short of this implicit expectation. In some cases, the model user simply lacks the depth of understanding needed to evaluate the applicability of the model to the problem at hand.

2.9.2 Problems in model formulation As with all models, QUAL2 incorporates certain simplifying assumptions and approximations. These pose specific limitations for some applications but more generally reduce the robustness of the model representation of basic water quality processes. The problems also impinge on the intended goal of this Task Group to integrate river water quality models with the more fundamentally based ASM models in order to develop control strategies that integrate river water quality with sewer overflows and wastewater treatment. The following lists several problems in the basic representation of water quality processes. QUAL2E and most river water quality models treat the river as a one-dimensional system. Implicit in this formulation is the assumption that any emissions to the river are instantaneously mixed across the full cross-section of the river. Actual experience is that discharged effluents, particularly if released from a shoreline outfall, may be distinguishable within the river flow for considerable distances downstream, and that transverse mixing is

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often a slow process. Transverse mixing distances are roughly proportional to the square of the width, and become very large in larger rivers. For example, the mixing distance on the Danube River at Budapest is about 200 times the width (i.e. on the order of one hundred kilometres). In terms of water quality, the result is the prediction of average river concentrations that may be much less than those observed in the field in the core of the effluent plume. A basic principle of stream water quality models is the conservation of mass. Thus, a very fundamental concern with the existing approach is the fact that using BOD as a state variable intrinsically means that mass balances cannot be closed because BOD is ill-defined and does not account for all biodegradable organic matter. Rather than being a unique material, BOD is the result of a bioassay measurement (Chapter 1), the yield of which changes with the type of substrate consumed. Hence, the amount of substrate consumed and biomass produced, and the rates of those processes, can vary significantly. Existing models, with a single BOD substance and decay rate, cannot account for these variations. An extreme example occurs with the highly refractory waste generated by paper mills. The BOD of paper-mill wastewater is sufficiently different from municipal wastewater that practitioners such as Whittemore (1983) have experimented with two-stage decay and similar artifices to properly simulate degradation. Concerns over BOD aside, virtually all models attempt to observe mass-balance principles within the water column, but often fail to close mass balances involving interaction with the sediment. Thus, for example, oxygen-demanding materials that settle to the streambed are lost from the model mass balance, yet their effect continues to be modelled through a sediment oxygen demand (SOD) flux term in the model equations that is unrelated to BOD sedimentation. In QUAL2, the following components are lost from the mass balance upon settling to the bottom: phytoplankton, organic nitrogen, organic phosphorus, and BOD. A more fundamental formulation would consider a complete mass balance and track mass in both the water column and sediment. A related issue is the treatment of benthic demands (flux terms). Most river water quality models employ user-specified fluxes such as SOD to model the effect of the benthos. However, if the complete mass balance approach were used, benthic fluxes could be determined based on the mass settled to the stream bottom and the population of mediating microbiota. Three mechanisms contribute to oxygen uptake from the sediments: (1) Sedimenting solids form sludge banks that usually undergo both anaerobic and aerobic decomposition. Aerobic decomposition on the bank surface consumes oxygen from the overlying water column. (2) Sessile bacteria degrade organic matter and consume dissolved oxygen from the water. (3) Microbial decomposition and endogenous respiration lead to a decay of biomass under oxygen consumption. A modelling approach that represents these basic phenomena could provide a more robust model formulation able to evaluate such changes as the transition from a discharge of untreated wastewater (and creation of a sludge bank) to secondarily treated wastewater (and eventual depletion of benthic sludge). Similar approaches are also possible for simulating the contribution of macrophytes to the stream-dissolved oxygen concentration. Mass balance approaches are also lacking for the pelagic bacteria that mediate biodegradation in the river water. These bacteria are typically not treated as state variables in the model but their effect is encapsulated within a single first-order degradation coefficient. As a result, changes in their population and character are not considered. Sedimentation,

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growth, and death affect bacterial population and are influenced by changes in the environment and emissions. The resulting dynamic changes in the degradation coefficient cannot be explained by the standard model approach. QUAL2 and most other existing river water quality models also lack a phenomenological submodel of the sessile microbiota that mediate nitrification and some of the benthic demands discussed above. Biological degradation of organic compounds and biochemical transformations of inorganic compounds in rivers are affected by bacteria or algae attached to substrates at the riverbed as well as by those suspended in the water column. Since small rivers have a much larger ratio of wetted surface area to volume of water, the contribution of attached bacteria to the total transformation in small rivers may be particularly significant. If conversion process rates are dominated by the activity of attached bacteria, a river water quality model must be able to describe the population dynamics of these bacteria in order to be able to predict changes in conversion rates that follow from changes in pollution loads. The processes governing such population dynamics are attachment and detachment of bacteria, and substrate limitations in the depth of the developing biofilms or in deeper interstitial zones. The following example from the river Necker in Switzerland (Uehlinger et al. 1996) illustrates these processes. The Necker has a gravel bed that creates a substrate for sessile algae. By comparing the results of a series of empirical models with measurements of algal surface densities over a measurement period of 18 months, the most important factors influencing the dynamics of algal surface densities in this river were identified. These factors were growth limitation with increasing biomass density, a slow increase in the algal detachment rate with increasing discharge, and nearly complete elimination of the algal population during bed-moving floods. Proper modelling of such a river’s water quality requires representation of these significant time-varying effects.

2.9.3 Problems in model calibration Regardless of the formulation chosen for the river water quality model, calibration of the model to the specific river is a key step in model use. Several aspects of stream water quality may complicate or impede water quality model calibration. Ironically, one is the improvement of stream water quality over the years in many countries. Many rivers now experience such good water quality (at least for conventional water quality parameters) that there is no clear response to pollutant loading (that is, no depletion of dissolved oxygen or “DO sag”). This leaves no clear “signal” against which to calibrate the water quality model. These streams may still experience water quality problems on an occasional basis such as extreme low flow or severe rainstorm events. However, the normal absence of a DO sag creates the need to gather data for model calibration during unusual or extreme conditions. Thus, a complete calibration process must include both dynamic loading conditions as well as loads that do not cause major changes in DO or other parameters. A key parameter in dissolved oxygen models for rivers is the reaeration coefficient, K2, a parameter to which model predictions are highly sensitive. Typically, K2 is taken to be a function of temperature and simple hydraulic parameters such as the stream depth and velocity (Bowie et al. 1985; McCutcheon 1989). In typical stream DO model applications, K2 is specified as a constant determined by calibration to each data set. However, intermittent discharges such as those associated with urban drainage, combined sewer overflows, or rainfall-derived non-point sources cause changes in streamflow and thus also in the reaeration coefficient. The implication of this type of variation is that determination of a constant reaeration coefficient by calibration is likely to result in a value that is not

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transferable to other conditions. This difficulty is compounded in small rivers, where determination of the reaeration coefficient is generally problematic. The uniqueness of the model calibration is also an issue, given that there are parameters that can counteract this such that several sets of parameter values lead to the same modelling results. For example, the BOD decay coefficient, K1, and the reaeration coefficient, K2, can be adjusted to compensate for each other such that multiple acceptable calibration combinations typically are possible. Similarly, Li (1962) has shown that a distributed inflow from non-point sources may mimic the effect of altered reaeration and degradation rates. His analytical solutions for a uniformly distributed inflow of constant BOD showed this inflow term to be mathematically indistinguishable from changes in the reaeration and BOD decay rates.

2.9.4 Problems in data collection A basic impediment to successful water quality modelling is the lack of adequate data for model calibration and verification. Thomann (1982) discusses issues of model calibration and verification, and stresses the importance of using independent field data sets, preferably reflecting different field conditions, for calibration and verification. The need for independent data places a burden upon the model user to conduct multiple field measurement programs over a variety of streamflow and meteorological conditions. As discussed above, the data sets most useful for calibration and verification may be achievable only during extreme or unusual conditions when there is a clearly measurable pollutant stress exerted on the river. Field data collection is often limited by such practical considerations as the financial and personnel resources available for data collection. Usually, data collection focuses on an intensive field survey of short duration – typically one, two, or three days. Protocols for surveys intended to support wasteload allocation studies with QUAL2 are given by Mills et al. (1986). These established procedures need to be modified for studies directed toward nonpoint sources or rainfall-driven events. Timing data collection to occur during such special events (either high flow or low flow) is difficult unless a dedicated field crew is available on a stand-by basis waiting for the desired event to occur, or advanced automatic monitoring stations are available and their limitations overcome (Van Griensven et al. 2000). Despite attempts to create protocols for the design of data collection campaigns for model calibration and testing, these procedures are hardly ever followed. Rather, ad hoc designs are made for a particular study. The experimental designs are typically characterised by a too low measuring frequency, both in time and in space, to accommodate calibration of complex dynamic conversion models. Usually samples are taken, say, weekly at different locations along the river. In other approaches a plug of liquid is followed (by labelling it with some inert dye) and samples are taken at regular times. Rarely, dispersion effects are determined by measuring the behaviour of the tracer. Increasingly, river quality modellers are aware of the lack of data and propose procedures such as (i) model-based design of the experimental set-up, (ii) high-frequency sampling at sufficient locations including the assessment of mixing and dispersion effects, and (iii) microcosm studies that follow the conversions taking place in a representative sample of river water. A different problem is raised by the aims of this Task Group, namely to develop models based upon state variables similar to those used in the ASM models. The change to state variables such as bacterial biomass implies that model state variables may not be measurable in the field. However, this problem also arises with ASM as well as models in other

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River Water Quality Model No. 1

disciplines. The solution has been to focus calibration and verification on those variables that can be measured and, when feasible, to employ surrogate measurements for variables that are otherwise not directly measurable. Moreover, the inclusion of a closed mass balance provides an additional data constraint that can compensate when certain state-variable fractions cannot be measured directly (although the presence of large counteracting fluxes will diminish the accuracy with which smaller fractions can be discriminated).

2.9.5 Problems in predictive capability The problems identified in the foregoing sections diminish the capability of water quality models to be truly predictive, particularly when significant changes alter the river’s pollutant load, streamflow, morphometry, or other basic characteristics. These problems arise when the changes to the river affect model parameters that are fixed based on an observed river condition that no longer controls the parameter. If, like most models at present, the model lacks the phenomenological structure to change model parameters to reflect changes in the river, then there is no predictive ability for at least those particular kinds of changes. A classic example of this situation is when a large-volume discharge that was untreated begins to receive secondary treatment. Standard water quality models cannot predict the resulting changes in light penetration due to a clearer effluent or sediment oxygen demand due to the depletion of sludge banks, nor can they predict the transition of the river from anaerobic to aerobic conditions or the accompanying cessation of denitrification. Denitrification occurs primarily in anoxic water zones and sediments and thus denitrification rates are higher in highly polluted rivers with major mud deposits (Billen et al. 1985; Chesterikoff et al. 1992). For this reason, it has been argued that a reduction in the organic load from sewage treatment plants without an accompanying increase in plant denitrification capacity may actually increase the nitrogen load passed downstream in the river. This happens because the in-river denitrification rate is reduced after restoration (Billen et al. 1985, 1986; Chesterikoff et al. 1992). It is impossible to quantify these processes without a model for the river sediment. Similar difficulties are faced by those trying to predict the effects of physical stream changes such as adding or removing a dam or the restoration of natural stream conditions, as advocated in recent US initiatives (US EPA 1995).

2.10

CONCLUSIONS

The current standard for river water quality modelling is the QUAL2 model developed by the US EPA. QUAL2 was specifically designed to conduct wasteload allocations – the determination of allowable maximum effluent loads under steady low streamflow to satisfy needs of legislation in the US. While QUAL2 and similar models are adequate for the specific regulatory situations for which they were developed, there is a need for a more comprehensive modelling framework for non-regulatory problems (for example, research and teaching) and for those water quality management problems not addressed by QUAL2 (such as storm water flow events, non-point sources, and transient streamflow), that is, for future regulatory conditions. Problems with current modelling practice include those of model application, model formulation, model calibration, data collection, and predictive ability. Problems in model application include the aforementioned management problems not addressed by QUAL2 as well as potential misuse by inexperienced users. Limitations in model formulation are continued reliance on BOD as the primary state variable, despite the fact that BOD does not include all biodegradable matter, and poor representation of benthic flux terms. As a result of

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these limitations, it is impossible to completely close mass balances in most existing models. Model calibration is hampered by the need for river characterisation under unusual or infrequently occurring conditions, a problem that is compounded by the general inadequacy of field data collection frequency in time. On top of all of this, pronounced concentration gradients are often missing for many restored rivers, making it extremely difficult to estimate parameters of counteracting fluxes. These various limitations in current river water quality models impair their predictive ability in situations of marked changes in the river’s pollutant load, streamflow, morphometry, or other basic characteristics. These limitations and the lack of consistency motivated the initial effort by the IWA Task Group on River Water Quality Modelling. More recently, it is further supported by emerging regulatory needs in Europe and many promising developments of monitoring.

3 Conversion model RWQM1

3.1 INTRODUCTION This chapter presents mathematical equations that represent the biological and chemical processes that comprise River Water Quality Model No. 1. The general formalism applied (Petersen matrix with process stoichiometry and kinetics) is identical to the formalism used in activated sludge models. However, in order to account for the habits and needs of water chemists and biologists, the description of the composition of organic material and of process stoichiometry is based on elemental mass fractions of organic compounds instead of chemical oxygen demand (COD) and content of nitrogen and phosphorus per unit of COD. Note that both formalisms are equivalent and, assuming the relevant parameters are known, parameters from one formalism can be converted to the other formalism and vice versa. These conversion formulas are also given in this chapter and in the numerical example given in Appendix 3 in units familiar to wastewater engineers. This demonstration of how to switch between measurement units familiar to wastewater engineers and those familiar to water chemists and biologists, and which parameters need to be known in order to make such a conversion unique, will aid communication between scientists and engineers working on models integrating wastewater and receiving water systems. The description of organic substances in wastewater engineering is based on COD and on fractions iC, iN, and iP of carbon, nitrogen, and phosphorus per unit of COD. For any measured quantity, the concept of what should be measured and the analytical procedure leading to a numeric result should ideally be distinguished (although the analytic procedure should be designed to give results very close to the “true” values expected for the concept). © 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

Conversion model RWQM1

31

In wastewater engineering, COD is interpreted as the result of a given measurement technique rather than as the concept for a characterisation of organic material. For this reason, the notion of theoretical oxygen demand (ThOD) has been introduced to characterise the corresponding concept, and in order to generalise the meaning also to inorganic compounds. In wastewater engineering the modelling approach is designed to model the results of the measurement procedures (for example, COD). A “composition matrix” is then used to convert these results into estimates for the corresponding conceptual characteristics (note that due to lack of better quantitative knowledge, typically conversion factors equal to unity are applied) (Henze et al. 2000). In contrast, chemists or limnologists generally prefer a description of organic substances by moles of chemical elements such as C106H263O110N16P for algal biomass (Stumm and Morgan 1981), by organic carbon content, or by dry or wet mass. These scientists usually interpret COD as the theoretical need of oxygen to reach the given reference state of mineralisation. There are various analytical techniques designed to give reasonable approximations to the theoretical value. Consequently, conceptual values are modelled, and an “observation equation” is used to convert modelled results to measured values. Similarly to the composition matrix mentioned above, factors not equal to unity in this observation equation can be used to account for systematic measurement errors (such as incomplete oxidation in COD measurements). In addition, the observation equation is used to derive parameters that summarise classes of state variables with similar properties, such as COD, total mass of particles, etc. In integrated modelling of engineered systems (such as sewers and wastewater treatment plants) and natural systems (such as rivers), these different practices make it necessary to convert measurement units between these two approaches. For the following reasons, we decided to base the main development on the elemental composition approach and to derive COD-based variables in a second step: • • •

•

•

There is increasing realisation of the importance of elemental composition and of the stoichiometry of biochemical conversion processes in ecology (Elser and Urabe 1999). Elemental analyses of organisms are increasingly applied in the environmental sciences (Elser et al. 1995; Fagerbakke et al. 1996; Norland et al. 1995). Other measurement units of biomass than COD, such as cell counts, organic carbon content, dry weight, or elemental mass fractions, are widely used to quantify measurements in natural systems. It seems unreasonable to propose replacing such measurements with COD measurements only. The use of elemental mass fractions builds a rigorous theoretical base for biogeochemical conversion processes that allows for the derivation of most other commonly used quantification measures. Although the actual elemental mass fractions will not be known in any application, their use at least makes the underlying assumptions of the model explicit. Sensitivity analysis can then be used to distinguish between more and less important assumptions. This report is aimed at engineers and natural scientists investigating surface water systems and to motivate them to use river water quality models similar to activated sludge sewage treatment plant models. The elemental mass balance approach facilitates this process.

It should be noted, however, that the two approaches are equivalent and that the elemental mass balance approach has the disadvantage of introducing more model parameters. The approach described in this report is an extension of the usual biogeochemical approach that is

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River Water Quality Model No. 1

based on given atomic ratios of the various elements in all compounds to mass fractions of elements in the compounds that are explicitly declared as model parameters. The first version of this approach described in Reichert et al. (2001a) considered the elements C, O, H, N, and P to be generalised by the inclusion of an additional mass fraction “X” that is used to summarise the total mass fraction of all elements other than C, O, H, N, and P.

3.2 SIMPLIFYING ASSUMPTIONS The equations of the conversion or biochemical model are based on the following simplifying assumptions: (1) The elemental composition of all compounds and organisms, as well as the stoichiometry of all processes, is assumed to be constant in time for each model application (but they may be different for different model applications). The elemental composition of compounds in the model explicitly considers the elements carbon (C), hydrogen (H), oxygen (O), nitrogen (N), and phosphorus (P). All the other elements are summarised in one additional mass fraction, X. As a special case of this simplification, growth limitation of diatoms by silicate is not considered in the model. (2) No adaptation of specific organisms takes place and changes in the composition within organism classes are neglected. This means that the time dependence of kinetic expressions and parameters is only via the dependence on other model constituents or environmental conditions, such as temperature, light, or pH. Among the consequences of this assumption, luxury uptake of nutrients by algae cannot be modelled. (3) It is assumed that oxygen and/or nitrate is always available. If anaerobic processes in the water column or the river sediment are of importance for the turnover of the compounds considered in the model, the model must be extended to account for these processes (for example, sulphur is not explicitly considered, nor is methane production).

3.3 COMPOSITION OF ORGANIC COMPOUNDS AND ORGANISMS The limitations to constant elemental composition of compounds and organisms and to a given set of elements to be explicitly considered make it possible to use the mass fractions of these elements as model parameters. The formulation of stoichiometric coefficients of conversion processes as functions of these parameters simplifies the adaptation of the model to cases where different composition of organic substances seems appropriate. Because different units are used to characterise organic substances, conversion formulas are given between mass of organic substances (OM; this is dry mass for particulate substances), organic carbon (orgC), and chemical oxygen demand (COD). The last unit is natural in the case of oxygen depletion simulations and it is of special importance for linkage to sewage treatment simulations (where organic substances are usually characterised by COD and ratios of N and P to COD). The composition of organic matter is approximated by mass fractions of

Conversion model RWQM1

33

the elements C, H, O, N, and P and the “compound” X that summarises all other elements. For this reason, the composition of organic substances can uniquely be described by the mass fractions αC, αH, αO, αN, αP, αX of the elements C, H, O, N, and P, and of the “compound” X describing the total mass fraction of all other elements. These mass fractions fulfil the constraint (see simplifying assumption 1 above) αC + αH + αO + α N + αP + α X = 1

(3.1)

The key process leading to conversion formulas between the elemental approach and the conceptual COD approach is the mineralisation process of organic compounds. In order to uniquely define this process, and the conceptual meaning of COD, the reference states of the mineralisation products must be defined. This is done by selecting the compounds HCO3–, H+, H2O, NH4+, HPO42– to be the reference compounds corresponding to the elements C, H, O, N, and P, respectively. Note that the reference compound for C has been changed from CO2, as used in Reichert et al. (2001a) to HCO3–. This compound is more convenient to assess expected pH changes using the stoichiometric coefficient of H+. Due to the consideration of the chemical equilibria of the inorganic carbon species, this change does not affect any of the calculations and both notations are correct. The reference compound for the “compound” X that summarises all elements other than C, H, O, N, and P is described by the following three parameters β+, βH, βO. These three parameters characterise the units of charge, the mass of hydrogen, and the mass of oxygen per unit mass of X in the reference compound for X. If X represents a single chemical element, the chemical formula of the reference compound of X can be written in the form X HMβH OMβO/16Mβ+ where M is the molar mass of X. In this chemical formula of the reference compound of X, the subscript MβH represents the number of hydrogen atoms; the subscript MβO/16 represents the number of oxygen atoms; and the superscript Mβ+ represents the net charge of the reference compound. The parameters β are defined on a mass basis of X instead of a molar basis in order to account for the possibility of describing a mixture of several elements with X. Note that the definitions of reference states given above are motivated by experimental procedures used to estimate COD, but that the following definition of COD still represents a conceptual definition and not an operative definition of a measurement procedure. Based on the definitions given above and using conservation principles of charge and of mass of the elements C, H, O, N, P, and the compound X, the following equation for the mineralisation process of organic compounds is obtained:

34

River Water Quality Model No. 1 α α β  3α 5α  1g OM + 8  C + α H − O − N + P + α X  β + − β H + O   gO O 2 8 14 31 8    3 3α 3α 1  αC  − α H + N + P + α X (β H − β + ) mole H 2 O →  2 6 14 31  α 2α P α  − α C gC HCO 3 +  C − N + − α X β +  gH H + 31  12 14  +

α N gN NH 4 + α P gP HPO 4

2+

+ α X gX XH Mβ O Mβ H

+

(3.2)

+ Mβ + O

/ 16

The conversion factor, γ (gCOD/gOM), from organic mass to chemical oxygen demand can be easily extracted as the factor for oxygen consumption in this mineralisation formula:

α α β  3α N 5α P  γ = 8 C + αH − O − + + α X  β + − β H + O    3 8 14 31 8    

(3.3)

This equation can then be used to derive the conversion equations for the parameters iN, iP, and iC:

iN =

iP =

iC =

αN

,

α α β  3α 5α  8 C + α H − O − N + P + α X  β + − β H + O   3 8 14 31 8   

αP  αC α O 3α N 5α P β   + αH − − + + α X  β + − β H + O   8 8 14 31 8    3

αC α α 3α β  5α  8 C + α H − O − N + P + α X  β + − β H + O   3 8 14 31 8   

=

1 iThOD

(3.4a)

,

(3.4b)

.

(3.4c)

The presence of the element(s) X in the organic compound leads to an equivalent COD of γX (gCOD/gX): γ X = 8 (β + − β H ) + β O

(3.5)

Conversion model RWQM1

35

If X describes a single element, this factor can be transformed to the units of gCOD/moleX (as used in Henze et al. 2000) by multiplication with the molar mass M of the element. Note that the parameters β only enter into the COD calculation in the form of the linear combination Equation (3.5). This means that for COD calculations with an unknown combination of elements in X, only this combination of parameters has to be estimated and not each of the parameters separately. Table 3.1 gives an overview of values of the parameters β and γX for important elements. In the table, the first five rows describe the elements that are already explicitly considered in the formalism for illustrative purposes. The equivalent COD provided in the last column demonstrates the equivalence of the formalism with Henze et al. (2000). Table 3.1: Examples of elements that could be contained in the mass fraction αv. Mass

Element(s)

fraction α

Reference

Composition parameters of

compound(s)

reference compound

X

β+

βH

βO

ch. un. gX

gH gX

gO gX

Equivalent COD γX

gCOD gX

gCOD moleX

αC

C

HCO3–

–1/12

1/12

48/12

+32/12

αH

H

H+

1

1

0

0

0

αO

O

H2O

0

2/16

16/16

0

0

αN

N

NH4+

1/14

4/14

0

–24/14

–24

αP

P

HPO42–

–2/31

1/31

64/31

+40/31

+40

αS

S

SO42–

–2/32

0

64/32

+48/32

+48

αCa

Ca

Ca2+

2/40

0

0

+16/40

+16

αK

K

K+

1/39

0

0

+ 8/39

+ 8

αMn

Mn

MnO2

0

0

32/55

+32/55

+32

αFe

Fe

FeOOH

0

1/56

32/56

+24/56

+24

αAl

Al

Al2O3

0

0

24/27

αS + αCa

αSS+αCaCa

SO42-, Ca2+

α α β X = S β S + Ca β Ca αX αX

+24/27 αS 48 α Ca 16 + α X 32 α X 40

+32

+24

–

3.4 COMPONENTS USED IN THE MODEL The following components are distinguished in the model: • • • • • •

SS: Dissolved organic substances, assumed to be available for rapid biodegradation by heterotrophic organisms. Composition is characterised by αC,SS, αH,SS, αO,SS, αN,SS, αP,SS, αX,SS. SI: Inert dissolved organic substances. These substances are assumed not to be biodegradable within the time frame of relevance. Composition is characterised by αC,SI, αH,SI, αO,SI, αN,SI, αP,SI, αX,SI. SNH4: Ammonium: NH4+. It is usually characterised by nitrogen mass. SNH3: Ammonia: NH3. It is usually characterised by nitrogen mass. SNO2: Nitrite: NO2–. It is usually characterised by nitrogen mass. SNO3: Nitrate: NO3–. It is usually characterised by nitrogen mass.

36 • • • • • • • • • • • • • •

•

• • •

River Water Quality Model No. 1 SHPO4: Part of inorganic dissolved phosphorus: HPO42–. It is usually characterised by phosphorus mass. (Total inorganic dissolved phosphorus consists of SHPO4 + SH2PO4 with the distribution depending on pH.) – SH2PO4: Part of inorganic dissolved phosphorus: H2PO4 . It is usually characterised by phosphorus mass. SO2: Dissolved oxygen: O2. SCO2: Sum of dissolved carbon dioxide (CO2) and H2CO3. It is usually measured as carbon mass. SHCO3: Bicarbonate: HCO3–. It is usually measured as carbon mass. SCO3: Dissolved carbonate: CO32–. It is usually measured as carbon mass. SH: Hydrogen ions: H+. pH can then be calculated as –log10(SH/1gH/L). SOH: Hydroxyl ions: OH–. It is usually measured as hydrogen mass (or moles). SCa: Dissolved calcium ions: Ca2+. XH: Heterotrophic organisms that are assumed to be able to grow aerobically as well as anoxically (at a slightly slower rate). Composition is characterised by αC,H, αH,H, αO,H, αN,H, αP,H, αX,H. XN1: Organisms oxidising ammonia to nitrite. Composition is characterised by αC,N1, αH,N1, αO,N1, αN,N1, αP,N1, αX,N1. XN2: Organisms oxidising nitrite to nitrate. Composition is characterised by αC,N2, αH,N2, αO,N2, αN,N2, αP,N2, αX,N2. XALG: Algae and macrophytes. In the model only one class of algae and macrophytes is introduced. A model extension to more classes can be made easily if this seems to be appropriate. Composition is characterised by αC,ALG, αH,ALG, αO,ALG, αN,ALG, αP,ALG, αX,ALG. XCON: Consumers. In the model only one class of consumers is introduced, which feeds on algae, heterotrophic and autotrophic organisms and biodegradable particulate organic matter. A model extension to more consumer classes can easily be made. Composition is characterised by αC,CON, αH,CON, αO,CON, αN,CON, αP,CON, αX,CON. XS: Particulate organic material, assumed to be available for biodegradation after hydrolysis. These substances must undergo hydrolysis catalysed by heterotrophic organisms before being directly degradable. Composition is characterised by αC,XS, αH,XS, αO,XS, αN,XS, αP,XS, αX,XS. XI: Inert particulate organic material. These substances are assumed to be not biodegradable within the time frame of relevance. Composition is characterised by αC,XI, αH,XI, αO,XI, αN,XI, αP,XI, αX,XI. XP: Phosphate adsorbed to particles. Usually measured as phosphorus mass. XII: Particulate inorganic material. In the basic model, particulate inorganic material is summarised in one class. However, an extension to classes of different size or composition can easily be made.

3.5 MEASURABILITY OF MODEL COMPONENTS One of the issues in any modelling exercise is the necessity to provide data for the model variables. Standard laboratory methods should preferably be applied. In this section, we relate the different state variables of the proposed model to measurable quantities such as COD, BOD, TOC, suspended solids, Kjeldahl nitrogen, dry mass, and element analyses. Both total and filtered samples are presumed to be available. Straightforward direct analysis can be performed for the following variables: SNH3 + SNH4, SNO2, SNO3, SHPO4 + SH2PO4, SO2, SCa, and SH, the latter being 10–pH. The sum of SCO2 and SHCO3 can be obtained from either a TIC (total inorganic carbon) analysis or an advanced alkalinity

Conversion model RWQM1

37

titration. Using the pH value subsequently allows differentiation between SCO2 and SHCO3, SNH4 and SNH3, and SH2PO4 and SHPO4. More problems exist with the determination of the multitude of organic substances of the model. However, overall measurement is an important starting point for the assessment of the different fractions. First, an overall COD analysis of the total and filtered samples allows differentiation between dissolved and particulate fractions, that is, CODtot = COD of (SS + SI + XH + XN1 + XN2 + XALG + XCON + XS + XI)

CODdiss = COD of (SS + SI)

,

CODpart = CODtot - CODdiss

(3.6)

(3.7)

Differentiating between SS and SI can proceed via the analysis of the biodegradable part of the dissolved organic fraction, that is, via a type of BOD analysis of the filtered sample (at the appropriate time scale). A similar experiment can be performed on the complete sample. By combining both results, the inert particulate fraction XI can be assessed (Lesouef et al. 1992). Organic substances can be analysed for their C, N, P, O, and H content. This leads to TOC (total organic carbon), DOC (dissolved organic carbon), TON, DON, TOP, and DOP: TOC = α C,SS SS + α C,SI S I + α C,XH S H + α C, N1 X N1 + α C, N2 X N2 + α C,ALG X ALG + α C,CON X CON + α C,XS X S + α C,XI X I

DOC = α C,SS S S + α C,SI S I

(3.8)

(3.9)

where equivalent expressions apply to N, P, O, and H. Note that total and dissolved Kjeldahlnitrogen are KNtot = TON + SNH4 + SNH3

,

KNdiss = DON + SNH4 + SNH3

(3.10)

and total phosphorus is

TP = TOP + SH2PO4 + SHPO4 + XP

(3.11)

which is useful to check mass balances (Nowak et al. 1999). Volatile suspended solids determination gives the sum of all organic particulate fractions, summing their carbon, nitrogen, oxygen, and hydrogen content

38

River Water Quality Model No. 1 VSS =

∑ (α

X + α H, Xi X i + α O, Xi X i + α N, Xi X i )

C, Xi i i = H, N1, N2, ALG,CON,S, I

(3.12)

and dry mass of organic particles is

OM =

∑X

i i = H, N1, N2,ALG,CON,S, I

(3.13)

Organism counts are often converted to wet mass units. These can then be converted to dry mass with the aid of an empirical conversion factor (between 0.1 and 0.4 depending on species). To determine the fractions of particulate organic material in samples is quite problematic. The particulate biodegradable material XS could be determined via respirometric analysis and an assumed yield coefficient (Vanrolleghem et al. 1999b). However, interference with endogenous respiration may occur. Determination of the organism fractions (XH, XN1, XN2, XALG, XCON) can be based either on activity measurement and a specific activity factor or a specific analysis of some property of the group of organisms (e.g. chlorophyll determination for algae). The activity measurements are COD-oxidation rates, nitrification rates, photosynthesis rates, and reduction of respiration rates in the presence of inhibitors that specifically inhibit certain groups such as nitrifiers or consumers. An alternative method, which has been quite successfully used in wastewater treatment fractionation of biomass, consists of calculating the amount of biomass grown under certain loading conditions using typical yield values and retention times (Vanrolleghem et al. 1999b). It can be expected that this approach also works for certain riverine situations.

3.6 BIOGEOCHEMICAL CONVERSION PROCESSES This section gives a complete description of the biogeochemical process equations. Note that recommendations for model simplifications are given in Chapter 4. In order to apply the model, these biochemical process equations must be supplemented by transport equations, equations for substance transfer between adjacent river compartments and to the atmosphere, geometrical conversions between concentrations in the water column and surface densities of sessile organisms, etc. The qualitative stoichiometric matrix of the model is given in Table 3.2; the stoichiometric parameters required to make all stoichiometric coefficients unique are listed in Table 3.3; the kinetic parameters are defined in Table 3.4; and the formulations of the process rates are given in Table 3.5. These definitions make the model stoichiometry unique up to the numerical values of the parameters. It is some work to calculate the stoichiometric coefficients from the stoichiometric parameters and from the composition parameters of organic compounds (αC, αH, αO, αN, αP, αX, β+, βH, and βO for all organic compounds) using conservation principles for the elements and for charge. In order to provide as detailed information as possible, these equations for the stoichiometric coefficients are given in Appendix 2 (two additional state variables, SN2 and SH2O, for molecular nitrogen gas and for water, respectively, are introduced in the appendix in order to make mass balance checks possible). Furthermore, these formulas are also implemented in a Microsoft® Excel

Conversion model RWQM1

39

spreadsheet that can be obtained from the authors (http://www.eawag.ch/~reichert). A numerical example is given in Appendix 3. In Table 3.2, the signs of all non-zero stoichiometric coefficients are given: “+” indicates a positive stoichiometric coefficient, “–” a negative coefficient, “?” indicates a coefficient the sign of which depends on the composition of the organic substances involved in the process and on the stoichiometric parameters, and “(+)” is the same as “?”, but in this case, the composition of compounds and the stoichiometric parameters should be chosen in a way that guarantees that this coefficient is non-negative (because there is no limiting factor to the corresponding compound in the process rate). In Table 3.5, limiting terms in square brackets can be omitted if the chosen stoichiometry is such that the corresponding component is not consumed. Stoichiometry and kinetics of processes are briefly discussed in the following paragraph. Table 3.2: Qualitative stoichiometric matrix of the complete River Water Quality Model No. 1 (see Chapter 4 for hints for model simplification). Component → Process ↓ j (1a) (1b) (2) (3a) (3b) (4) (5) (6) (7) (8) (9a) (9b) (10) (11) (12a) (12b) (12c) (12d) (12e) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)

i

Aerobic Growth of Heterotrophs with NH4 Aerobic Growth of Heterotrophs with NO3 Aerobic Respiration of Heterotrophs Anoxic Growth of Heterotrophs with NO3 Anoxic Growth of Heterotrophs with NO2 Anoxic Respiration of Heterotrophs Growth of 1st-stage Nitrifiers Aerobic Respiration of 1st-stage Nitrifiers Growth of 2nd-stage Nitrifiers Aerobic Respiration of 2nd-stage Nitrifiers Growth of Algae with NH4 Growth of Algae with NO3 Aerobic Respiration of Algae Death of Algae Growth of Consumers on XALG Growth of Consumers on XS Growth of Consumers on XH Growth of Consumers on XN1 Growth of Consumers on XN2 Aerobic Respiration of Consumers Death of Consumers Hydrolysis Equilibrium CO2 ↔ HCO3 Equilibrium HCO3 ↔ CO3 Equilibrium H2O ↔ H + OH Equilibrium NH4 ↔ NH3 Equilibrium H2PO4 ↔ HPO4 Equilibrium Ca ↔ CO3 Adsorption of Phosphate Desorption of Phosphate

(1) (2)

(3)

SS

SNH4 SNH3 SNO2 SNO3 SHPO4 SH2PO4 SO2 SCO2 SHCO3 SCO3

-

SI

(4)

(5)

(6)

?

-

+

-

+

-

-

+

-

-

+

+ -

+

+ -

+

(7)

(8)

(9)

(10)

(11)

(12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) SH SOH SCa XH XN1 XN2 XALG XCON XS

?

-

+

?

?

-

+

?

1

+

-

+

-

-1

?

+

?

1

?

+

?

1

+

+

-

-1

-

+

1

+

-

+

-

-1

+

-

-

-

-

1

+

-

+

-

-1

-

+ +

-

-

+ 1 1

+

+

-

+

-

-1

(+)

(+)

(+)

?

?

-1

(+)

(+)

-

?

?

-

(+)

(+)

-

?

?

(+)

(+)

-

?

?

(+)

(+)

-

?

?

(+)

(+)

-

?

?

+

+

-

+

-

-1

(+) (+)

(+) (+)

(+) (+)

? ? -1

? ? + +

-1

1 -1

1

XII

+

-

1

XP

+

-

1 -1

XI

1

-

+ + 1

+

1

-

+

1 -

1 -

1 + + -1

+

1

+ 1

-1

+ +

-1 1

1 1 -1

The following processes are considered in the model (numbers correspond to rows in Tables 3.2 and 3.5): (1) Aerobic Growth of Heterotrophs: Growth of heterotrophic organisms using dissolved organic substrate, dissolved oxygen, and nutrients. If the organic substrate contains enough phosphorus (αP,XH < YH,aer αP,SS), no phosphate uptake from the surrounding water is necessary and the limiting term with respect to phosphate can be neglected. If there is not

40

River Water Quality Model No. 1

enough nitrogen in the substrate (αN,XH > YH,aer αN,SS), ammonia is consumed by process (1a). If ammonia concentrations become very low, there is a switch to the nitrate uptake process (1b). The ammonia limitation term in process (1a) and the whole process (1b) can be omitted if there is enough nitrogen in the substrate (αN,XH < YH,aer αN,SS). In this case, the excess nitrogen is released as ammonia by process (1a). (2,6,8,10,13) Aerobic Endogenous Respiration: Loss of biomass by aerobic endogenous respiration. (3) Anoxic Growth of Heterotrophs: Growth of heterotrophic organisms with oxygen gained by reducing nitrate to nitrite or nitrite to molecular nitrogen (denitrification; processes 3a and 3b, respectively). If αP,XH < YH,aer αP,SS, SHPO4 must be available for growth. In the process rate, the phosphate limitation term (square brackets in Table 3.5) is only present if this condition is fulfilled. This process is inhibited by the presence of dissolved oxygen. (4) Anoxic Endogenous Respiration of Heterotrophic Organisms: Loss of heterotrophic biomass in the absence of dissolved oxygen by endogenous respiration with nitrate (for simplicity this process is formulated as a one-step reduction of nitrate to molecular nitrogen in contrast to anoxic growth). (5) Growth of 1st Stage Nitrifiers: Growth of organisms that oxidise ammonia to nitrite. (7) Growth of 2nd Stage Nitrifiers: Growth of organisms that oxidise nitrite to nitrate. In order to avoid problems in the absence of ammonia, it is assumed that the nitrogen source for build up of biomass is also nitrite (due to the small contribution to nitrite consumption this assumption is not important). Table 3.3: Stoichiometric parameters. Symbol Y Y Y f

H,aer

Description Yield for aerobic heterotrophic growth

gXH/gSS

H,anox,NO3

Yield for anoxic heterotrophic growth with nitrate

gXH/gSS

H,anox,NO2

Yield for anoxic heterotrophic growth with nitrite

gXH/gSS

Fraction of respired heterotrophic and autotrophic biomass that becomes inert Yield for growth of 1st step nitrifiers

gXI/gXH

I,BAC

Y Y

N1

nd

N2

fI,ALG Y Y

ALG,death

CON

F

e

F

I,CON

Y Y

CON,death

HYD

Yield for growth of 2

step nitrifiers

Unit

gXN1/gSNH4-N gXN2/gSNO2-N

Fraction of particulate organic matter that becomes inert during death of algae Yield for death of algae (set to a value that avoids consumption of nutrients and oxygen) Yield for grazing (set to a value that avoids consumption of nutrients and oxygen) Fraction of incorporated biomass that is excreted as faecal pellets

gXI/g(XS+XI)

Fraction of particulate organic matter that becomes inert during death of consumers Yield for death of consumers (set to a value that avoids consumption of nutrients and oxygen) Yield for hydrolysis (set to a value that avoids consumption of nutrients and oxygen)

gXI/g(XS+XI)

g(XS+XI)/gXALG gXCON/gXALG gXS/gXCON

g(XS+XI)/gXCON gSS/gXS

Conversion model RWQM1

41

Table 3.4: Kinetic parameters. Symbol

kdeath,ALG,To

Description Specific death rate for algae

T

–1

Units

kdeath,CON,To

Specific death rate for consumers

T

–1

kgro,ALG,To

Maximum specific growth rate for algae

T

–1

kgro,CON,To

L3M–1 T

kgro,H,aer,To

Maximum specific growth rate for consumers per mass unit of grazed organisms Maximum aerobic specific growth rate of heterotrophs

T

–1

kgro,H,anox,To

Maximum anoxic specific growth rate of heterotrophs

T

–1

kgro,N1,To

Maximum specific growth rate of 1st stage nitrifiers

T

–1

kgro,N2,To

Maximum specific growth rate of 2nd stage nitrifiers

T

–1

khyd,To

Hydrolysis rate constant

T

–1

kresp,ALG,To

Maximum specific respiration rate of algae

T

–1

kresp,CON,To

Maximum specific respiration rate of consumers

T

–1

kresp,H,aer,To

Maximum aerobic specific respiration rate of heterotrophs

T

–1

kresp,H,anox,To Maximum anoxic specific respiration rate of heteroptrophs

T

–1

kresp,N1,To

Maximum specific respiration rate of 1st stage nitrifiers

T

–1

kresp,N2,To

Maximum specific respiration rate of 2nd stage nitrifiers

T

–1

keq,1

Rate constant for CO2-HCO3– equilibrium*

T

–1

keq,2

Rate constant for HCO3–-CO32– equilibrium*

T

–1

keq,w

Rate constant for H+-OH– equilibrium*

ML–3 T

keq,N

Rate constant for NH4+-NH3 equilibrium*

T

–1

keq,P

Rate constant for H2PO4–-HPO42- equilibrium*

T

–1

keq,so

Rate constant for calcium carbonate equilibrium†

ML–3 T

kads

Phosphate adsorption rate constant

T

–1

kdes

Phosphate desorption rate constant

T

–1

KHPO4,ALG

Saturation coefficient for growth of algae on phosphate

ML–3

KHPO4,H,aer

ML

KN,ALG

Saturation coefficient for aerobic growth of heterotrophs on phosphate Saturation coefficient for anoxic growth of heterotrophs on phosphate Saturation coefficient for growth of 1st stage nitrifiers on phosphate Saturation coefficient for growth of 2nd stage nitrifiers on phosphate Saturation coefficient for growth of algae on nitrogen

KNH4,ALG

Saturation coefficient for growth of algae on ammonia

ML–3

KN,H,aer

Saturation coefficient for aerobic growth of heterotrophs on nitrogen Saturation coefficient for growth of 1st stage nitrifiers on ammonia

ML–3

KHPO4,H,anox KHPO4,N1 KHPO4,N2

KNH4,N1

–3

ML–3 ML–3 ML–3 ML–3

ML–3

–1

–1

–1

42

River Water Quality Model No. 1

Table 3.4: Kinetic parameters (continued). Symbol

KI

Description Saturation coefficient for growth of algae on light

Units EL–2

KNO3,H,anox

Saturation coefficient for anoxic growth of heterotrophs on nitrate

ML–3

KNO2,H,anox

Saturation coefficient for anoxic growth of heterotrophs on nitrite

ML–3

KNO2,H,anox

Saturation coefficient for anoxic growth of heterotrophs on nitrite

ML–3

KNO2,N2

Saturation coefficient for growth of 2nd stage nitrifiers on nitrite

ML–3

KO2,ALG

Saturation/inhibition coefficient for endogenous respiration of algae

ML–3

KO2,CON

ML–3

βALG

Saturation/inhibition coefficient for endogenous respiration of consumers Saturation/inhibition coefficient for aerobic endogenous respiration of heterotrophs Saturation/inhibition coefficient for aerobic endogenous respiration of 1st stage nitrifiers Saturation/inhibition coefficient for aerobic endogenous respiration of 2nd stage nitrifiers Saturation coefficient for aerobic growth of heterotrophs on dissolved organic substrate Saturation coefficient for anoxic growth of heterotrophs on dissolved organic substrate Temperature correction factor for algae growth rate

βCON

Temperature correction factor for consumer growth rate

ºC–1

βH

Temperature correction factor for heterotroph growth rate

ºC–1

βhyd

Temperature correction factor for hydrolysis

ºC–1

βN1

Temperature correction factor for 1st stage nitrifier growth rate

ºC–1

βN2

Temperature correction factor for 2nd stage nitrifier growth rate

ºC–1

KO2,H,aer KO2,N1 KO2,N2 KS,H,aer KS,H,anox

–

–

2–

–

+

–

ML–3 ML–3 ML–3 ML–3 ML–3 ºC–1

2–

* For the equilibria for CO2-HCO3 , HCO3 -CO3 , H2O-OH , NH4 -NH3, and H2PO4 -HPO4 , rate constants need not have a value realistic for the chemical processes; the value simply must be large enough to guarantee that the concentrations are always very close to their equilibrium values. † For calcium carbonate equilibrium, kinetic effects due to slow calcite precipitation or dissolution are – – – 2– + – typical in contrast to the equilibria for CO2-HCO3 , HCO3 -CO3 , H2O-OH , NH4 -NH3, and H2PO4 2– HPO4 .

(9) Growth of Algae: Growth of algae by primary production. This process is divided into two subprocesses describing growth with ammonia (preferred) or nitrate as the nitrogen source. The Steele (1965) function is used to describe light limitation and light inhibition. (11,14) Death of Algae or Consumers: Conversion of algae or consumers to slowly degradable and inert organic matter by death, lysis, etc. With the degree of simplification in this model, which uses a constant composition of organic substances for each class, death of algae and consumers is difficult to describe. This is because dead organic material may have a composition other than algae or consumers. This problem is solved with the introduction of a yield coefficient for the death process that is used to make mass conservation of all elements possible without requiring an uptake of oxygen, nitrogen, phosphorus, or carbon during the death process. The disadvantage of this concept is that, depending on differences in the composition of algae and particulate organic matter, the process may release oxygen,

Conversion model RWQM1

43

ammonia, phosphate, and carbon dioxide. If there is not strong evidence for different composition of different classes of organic material, this problem can be solved by using the same composition for algae, consumers, and dead organic substances and setting these yield coefficients to unity. (12) Growth of Consumers: Growth of consumers by grazing on algae, on particulate organic matter, and on heterotrophic and autotrophic organisms (subprocesses 12a and 12e, respectively) with production of faecal pellets in the form of slowly biodegradable particulate organic matter. It is assumed that organic matter is homogeneously distributed. Note that this assumption may be violated for sessile organisms. A simple way to consider this fact is discussed in the case study of the River Glatt in Chapter 5. The yield coefficient must be small enough to guarantee the availability of enough nitrogen and phosphorus in the food for building consumer biomass. A very simple process rate proportional to the product of food and consumer concentrations was chosen. In some cases limiting terms with respect to food or consumers may be necessary. (15) Hydrolysis: Dissolution of slowly biodegradable particulate organic matter to dissolved organic matter catalysed by heterotrophic biomass. Similarly to the death processes, a yield coefficient is introduced to guarantee that no oxygen, ammonia, or phosphate be consumed during the hydrolysis process. If there is not strong evidence that the composition of particulate and dissolved organic matter is different, the same composition should be used and the yield coefficient set equal to unity. (16-21) Chemical Equilibria: Chemical equilibria between CO2 and HCO3–, between HCO3– and CO32–, between H2O and H+ and OH–, between NH4+ and NH3, between H2PO4– and HPO42–, and between Ca2+ and CO32– and CaCO3(s). (22) Adsorption of Phosphate: Any type of binding of phosphate on particulate matter. (23) Desorption of Phosphate: Release of phosphate previously bound on particulate matter. Note that all process formulations given above are based on in situ concentrations of substrates and in situ light conditions. If a biofilm of sessile organisms is modelled without explicit consideration of substrate gradients and light availability, additional limiting factors must be formulated on an empirical basis (for an example of this see the case study of the River Glatt in Section 5.2).

3.7 SUMMARY AND CONCLUSIONS The biogeochemical conversion model for river water quality modelling presented in this section is rather complex. Special emphasis is given to a rigorous formulation of the mass balances of the elements. Although this approach introduces parameters that may not be identifiable in typical applications, it clarifies model assumptions and can be the basis for a thorough identifiability analysis. Two important reasons for the complexity of the model are the consideration of a rather complete set of processes that may be important under aerobic and anoxic conditions and the inclusion of inorganic carbon compounds for the calculation of pH. In specific applications, however, it may be possible to omit many of these processes. For this reason, recommendations are given in Chapter 4 for the selection of adequate submodels for specific applications. It is important to study the possibilities for such model simplifications very carefully before an unnecessarily complicated and non-identifiable model is applied. Nevertheless, because different applications entail different components and processes, it is useful to document the full model as it is done in this chapter.

Table 3.5: Process rates (terms in square brackets are omitted under certain circumstances, see text). No.

Process

Rate

(1a)

Aerobic Growth of Heterotrophs with NH4

(1b)

Aerobic Growth of Heterotrophs with NO3

    K N, H,aer SS SO2 S NO3 S HPO4 + S H2PO4 kgro, H,aer,To e β H (T −T0 )  XH KS, H,aer + SS K O2, H,aer + SO2 K N,H,aer + S NH 4 + S NH3 K N,H,aer + S NO3  K HPO4,H,aer + SHPO4 + S H2PO4   

(2)

Aerobic Endogenous Respiration of Heterotrophs

kresp, H,aer,To e β H (T −T0 )

SO2 XH K O2,H,aer + SO2

(3a)

Anoxic Growth of Heterotrophs with NO3

kgro, H,anox,To e β H (T −T0 )

  K O2,H,aer SS S NO3 S HPO4 + S H2PO4 XH  KS, H,anox + SS K O2,H,aer + SO2 K NO3,H,anox + S NO3  K HPO4,H,anox + S HPO4 + SH2PO4 

(3b)

Anoxic Growth of Heterotrophs with NO2

kgro, H,anox,To e β H (T −T0 )

K O2,H,aer SS S NO2 KS, H,anox + SS K O2,H,aer + SO2 K NO2,H,anox + S NO2

(4)

Anoxic Endogenous Respiration of Heterotrophs

kresp, H,anox,To e β H (T −T0 )

(5)

Growth of 1st-stage Nitrifiers

(6)

Aerobic Endogenous st Respiration of 1 -stage Nitrifiers

(7)

Growth of 2nd-stage nitrifiers

(8)

Aerobic Endogenous nd Respiration of 2 -stage Nitrifiers

(9a)

Growth of Algae with NH4

SS SO2 KS, H,aer + SS K O2,H,aer + SO2

kgro, H,aer,To e β H (T −T0 )

kgro, N1,To e β N1 (T −T0 )

   S NH4 + S NH3 S HPO4 + S H2PO4   XH  K N, H,aer + S NH4 + S NH3   K HPO4,H,aer + S HPO4 + S H2PO4 

  S HPO4 + S H2PO4 XH  K S S + + HPO4 H2PO4   HPO4,H,anox

K O2,H,aer S NO3 XH K O2, H,aer + SO2 K NO3,H,anox + S NO3

SO2 S NH4 + S NH3 S HPO4 + S H2PO4 X N1 K O2, N1 + SO2 K NH4, N1 + S NH4 + S NH3 K HPO4, N1 + S HPO4 + S H2PO4

kresp, N1,To e β N1 (T −T0 )

SO2 X N1 K O2, N1 + SO2

kgro, N2,To e β N2 (T −T0 )

SO2 S NO2 S HPO4 + S H2PO4 X N2 K O2, N2 + SO2 K NO2, N2 + S NO2 K HPO4, N2 + S HPO4 + S H2PO4

kresp, N2,To e β N2 (T −T0 )

SO2 X N2 K O2, N2 + SO2

kgro, ALG,To e β ALG (T −T0 )

 S NH4 + S NH3 + S NO3 S NH4 + S NH3 S HPO4 + SH2PO4 I I   X ALG exp1 −  K N, ALG + S NH4 + S NH3 + S NO3 K NH4,ALG + S NH4 + S NH3 K HPO4,ALG + S HPO4 + S H2PO4 K I K I 

(9b)

Growth of Algae with NO3

(10)

Aerobic Endogenous Respiration of Algae

k resp, ALG,To e β ALG (T −T0 )

(11)

Death of Algae

kdeath,ALG,To e β ALG (T −T0 ) X ALG

kgro, ALG,To e β ALG (T −T0 )

K NH4,ALG  S NH4 + S NH3 + S NO3 S HPO4 + SH2PO4 I I   X ALG exp1 − K N, ALG + S NH4 + S NH3 + S NO3 K NH4,ALG + S NH4 + S NH3 K HPO4,ALG + S HPO4 + S H2PO4 K I K I   SO2 X ALG K O2, ALG + SO2

(12a-e) Growth of Consumers on XI

kgro,CON,To e β CON (T −T0 )

SO2 X i X CON , K O2,CON + SO2

(13)

Aerobic Endogenous Respiration of Consumers

kresp,CON,To e β CON (T −T0 )

SO2 X CON K O2,CON + SO2

(14)

Death of Consumers

kdeath,CON,To e β CON (T −T0 ) X CON

(15)

Hydrolysis

k hyd,To e

(16)

Equilibrium CO2 ↔ HCO3– –

(17)

Equilibrium HCO3 ↔ CO3

(18)

Equilibrium H+ ↔ OH–

(19) (20)

keq, w (1 − S H SOH / K eq, w ) keq,2 ( S HCO3 − S Η SCO3 / K eq,2 )

Equilibrium NH4 ↔ NH3 –

Equilibrium H2PO4 ↔ HPO4 ↔ CO3

XS

keq,1( SCO2 − S H S HCO3 / K eq,1 ) 2–

+

2+

β hyd (T −T0 )

2–

keq, N ( S NH4 − S H S NH3 / K eq, N ) 2–

keq, P ( S H2PO4 − S H S HPO4 / K eq, P )

(21)

Equilibrium Ca

(22)

Adsorption of Phosphate

k ads S HPO4

(23)

Desorption of Phosphate

k des X P

keq,s0 (1 − SCa SCO3 / K eq,s0 )

i = ALG, S, H, N1, N2

4 How to use the model

4.1 INTRODUCTION Translating the formulations presented in Chapter 3 into a model of an actual river system requires the identification of the components and processes to be included as well as the specification of the spatial discretisation of the model domain and definition of model parameter values. As indicated in Chapter 3, the structure of the RWQM1 biochemical model is intended to be sufficiently broad to embrace a wide variety of potential water quality modelling situations. However, few actual models will require all of the components and processes identified in Table 3.2, since the process of applying RWQM1 will usually entail simplifying the broad structure to fit the specific problem. As a general principle, the modeller should seek the simplest model adequate for the problem at hand – spurious detail and complexity should be avoided. This chapter presents an approach to seeking an appropriate level of model complexity for a specific problem.

4.2 DECISION PROCESS The decision process is provided here as a series of six steps in which specific aspects of the model are determined.

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

How to use the model

47

4.2.1 Step 1 – Define temporal representation Different water-quality processes are exerted and manifested over different length scales, and with proportionally different time constants (i.e., the time frame within which changes are expected to occur). The length scale, L, is related to the time constant, τ, as:

τ

=

L u

(4.1)

where u = the average velocity over the length scale L. In nearly stagnant water bodies, such as lakes, L is often large while u is small, thus leading to long time constants. Water bodies with long time constants typically require a much less detailed representation than those that respond quickly. For example, lake eutrophication is a response to annual nutrient loading transpiring over many years such that many short-duration phenomena may be overlooked. In contrast, combined sewer overflows affect a river’s water quality within minutes or hours and must be modelled in considerable detail. Such disparate time constants necessitate different water-quality model formulations and, accordingly, the length scale and time constant are key factors in the model decision process. For this reason, we make it Step 1 of the decision process. Step 1 requires τ1 and τ2 to be defined, the lower and upper bounds of the characteristic time constant based on L1 and L2, the corresponding length scales. These bounds depend upon the problem being modelled and the processes that dominate that problem. As such, they depend upon both Equation (2.2) and its boundary conditions. Examples of time constants potentially involved include: rainstorm duration and watershed time of concentration (as used in hydrology; see Maidment 1993, page 9.15) for non-point source pollution problems, a day for photosynthesis, a day or week for variations in domestic wastewater flow, travel time for pollutant advection, operational periods for reservoirs, seasons for population dynamics, and longer periods for accumulative pollutants. Once τ1 and τ2 have been defined, the representation of processes over time – either dynamic or steady-state – can be selected. If the process of interest proceeds at some rate constant k (in units of inverse time), then a process time constant of τc = 1/k may be defined. If τ1 < τc < τ2, then a dynamic model is required. If τc >> τ2 then the process may be omitted. If τc << τ1 then a steady-state model will suffice. Yet another factor to be considered is the influence of sediment processes. A sediment time constant can be defined as τs which is equal to the time between floods that precipitate sediment resuspension and sedimentation or between similar relevant events, where typically, τs >> τ2. Processes in the sediment can be neglected only if τc >> τs. Processes in the sediment should be described dynamically if τ1 < τs < τc. This is typically the case in shallow rivers where benthic activity contributes significantly to conversion processes within the stream. In the typical case where sediment processes occur slowly, τs >> τc, then sediment processes can be captured as a time-invariant parameter, such as the sediment oxygen demand (SOD) term in traditional stream dissolved oxygen models.

48

River Water Quality Model No. 1

4.2.2 Step 2 – Determine model spatial dimensions Choosing the spatial dimensions to be represented in the water-quality model has significant implications for the model formulation. Particularly, the choice of dimensionality alters the representation of boundary conditions and model parameters. In this decision step, we consider the choices regarding the vertical resolution of the model as well as model dimensionality per se. Selection of model dimension is associated with the spatial resolution of the water body. A fully continuous three-dimensional description of the water column and sediment can often be reasonably approximated by a vertically mixed water column and one or two sediment compartments. Such a compartmentalisation of the water body leads to a formal reduction in dimensionality of the differential equations without the loss of all of the spatial resolution. This reduction is associated with a change of flux terms (across the new compartment boundaries) to source terms in the differential equations. Similarly, the boundary condition for gas exchange with the atmosphere changes to a source term. Deep sediment can be captured simply as a “burial” flux, which represents matter lost from the active sediment to deeper layers in which transformation rates are much slower. There can also be an opposite flux due to diffusion from deeper layers to the active layer. In traditional stream dissolved oxygen modelling, even the active sediment is not represented, but is instead captured through a SOD term. The choice of vertical resolution ultimately depends upon the problem being considered. Sediment is almost universally included in models of toxic substances that adsorb to sediment particles, but only rarely in dissolved oxygen models. In Section 2.9 we indicate the importance of representing sediment oxygen demand and how it may change over time; thus, we recommend that sediment be represented in river dissolved oxygen problems in which sediment demands are significant. How specifically to treat the sediment must also be decided. In the IWA Activated Sludge Models, there is no formal distinction between the water column and the sediment (sludge). The sediment (comprised of sludge fractions such as heterotrophic bacteria, XH, and others) is considered to be mixed into the water column. In that type of formulation, the vertical movement of particulates is represented by a vertical velocity that is the net of water advection, particulate settling, and particle buoyancy. Transport between the water column and sediment is determined by this continuous velocity function. Such a formulation is also possible for river water quality models, however a more practical alternative is to model physically distinct vertical zones, for example for the water column and sediment. In this case, transport between the sediment and the water column is represented through explicit transfer terms, such as a settling flux and resuspension flux. These terms capture both advective and diffusive transport. An excellent example of this approach is the sediment oxygen demand model by DiToro et al. (1990). Vertical resolution also needs to be determined for the water column. In shallow waters, no differentiation within the water column may be needed. In deeper waters, there may be significant differences in water quality with depth, requiring greater vertical resolution to properly represent the deeper water column. Once the resolution of the model is decided, its dimensionality can be chosen. Models can be three-dimensional, two-dimensional horizontally (xy-plane), two-dimensional vertically (xz-plane), one-dimensional in the x-direction, and finally zero-dimensional. A zerodimensional formulation is a so-called “box model” in which the water body is represented as one or more fully mixed tanks. Dimensionality may be selected by consideration of length scales. The length scales for lateral and vertical mixing, LL and LV, in a river may be estimated as:

How to use the model LL ≈

W2 u 2K y

LV ≈

49

h2 u 2K z

(4.2)

where W and h = river width and depth, Ky ≈ 0.6 u*h is the lateral dispersion coefficient, Kz ≈ 0.067 u*h is the vertical dispersion coefficient, u* =

ghS 0 is the friction velocity, g =

gravitational acceleration, and S0 = stream bottom slope (formulae for Ky and Kz are taken from pages 112 and 106, respectively, of Fischer et al. (1979), who offer further details on deriving dispersion coefficients). Criteria for dimensionality based on length scales are: if L1 >> LL, then a one-dimensional model will suffice; if L1 >> LV, but not L1 >> LL, then a two-dimensional model is needed; and if not L1 >> LV, then a three-dimensional model is needed.

4.2.3 Step 3 – Determine representation of mixing The representation of mixing is also dependent on the choice made in Step 2 as to dimensionality. If three dimensions are modelled, then turbulent diffusion is the operative mixing mechanism in the water column. However, if fewer dimensions are represented, then the resulting averaging of advection gives rise to the artificial mixing represented as dispersion. Whether modelled as dispersion or diffusion, the need to represent mixing processes varies from problem to problem. In traditional one-dimensional dissolved oxygen models, dispersion is often considered negligible when flow and sources are all steady. Dispersion may be neglected if:

τc

>>

2 DL u2

(4.3)

and:

τe

>>

2 DL L2 u3

(4.4)

where DL ≈ 0.011W 2 u 2 / u * h is the longitudinal dispersion coefficient (see again Fischer et al. 1979) and τe is the time constant associated with variations in concentrations induced by external processes or boundary conditions.

4.2.4 Step 4 – Determine representation of advection Advection affects the movement of water, sediment, dissolved substances, and particulates. The advection of the water phase can in almost all instances be calculated independently of the water-quality model. The exceptions are pollutants which alter hydrodynamics, such as high-concentration solids or high-volume thermal discharges, each of which may create a significantly non-uniform density distribution within the water column. These exceptions aside, a wide variety of methods to calculate flow are available based on the degree to which the St. Venant equations are or are not simplified. In many instances, hydrodynamic

50

River Water Quality Model No. 1

calculations are dispensed with in preference to simple measures of stream velocity such as time-of-travel studies (Kilpatrick and Wilson 1989). In situations in which streamflow varies significantly over time, such as non-point source pollution problems or where reservoir operations control flow, more complex hydrodynamics need to be considered. In terms of the time period of interest discussed above, this may be stated as a requirement to consider hydrodynamics when τ1 < τflow < τ2, where τflow is a representative time constant for flow variation. In those instances when field data are inadequate to define advection, it is necessary to employ a modelling approach. Consistent with the typical geometry of rivers, this review considers only one-dimensional models of stream hydraulics. Two- and three-dimensional representations are also possible; these have historically carried a considerable computational burden, which has decreased dramatically with improved computer technology. A comprehensive review is provided by Bedford et al. (1988). Mahmood and Yevjevich (1975) and Cunge et al. (1980) provide lengthy treatises on the one-dimensional solution for unsteady flow in open channels, including discussions of the various degrees to which the St. Venant equations may be simplified. The St. Venant equations represent the conservation of mass and momentum, as shown in Equations (4.5) and (4.6) for one dimension: ∂h 1 ∂Q q = − + ∂t W ∂x W

∂Q ∂  Q2 = −  ∂x  A ∂t

 ∂h  − gA + gA (S o − S f )  ∂x 

(4.5)

(4.6)

where Q is the streamflow, q is the lateral inflow per unit length of the river, Sf is the friction slope, and all other variables are as defined above. The full St. Venant equations were historically only rarely solved – the adequate handling of water quality problems did not require all the details and the computational burden of the full solution was substantial – and simplifications were typically employed. One particularly effective simplification, the kinematic wave model, recognises that the slope terms, So and Sf, dominate Equation (4.6). Thus, in the kinematic approach, Equation (4.5) is fully represented, but the differential terms of Equation (4.6) are neglected, resulting in the approximation that So = Sf. The resulting equations are considerably simpler to solve, but result in a reasonable representation of advection affected by an increase in streamflow (“flood wave”). In some systems, storage zones along the river or a mild channel slope tend to disperse a flood wave, or dams create backwater effects, none of which are well represented by the kinematic approach. For these situations, Equation (4.6) may be solved along with Equation (4.5), but with only the So, Sf, and gA ∂h/∂x terms retained in Equation (4.6). This results in the diffusive wave approximation, so-named because the equation derived takes the form of the diffusion equation (Equation (2.2)) with a “diffusion” coefficient that captures the dispersion of the flood wave. In practice, this coefficient is empirically increased in order to represent the storage effects caused by channel irregularities and off-channel storage. The diffusive wave approach represents a good compromise between computational complexity and accuracy required for most river-water quality problems.

How to use the model

51

There may also be advection of solid phases either through water-column sediment transport or via bed movement. Generally, the predominant movement of particulates in the water column is vertically downwards as represented through the settling velocity. However, during periods of high flow, or in some parts of the river system where streamflow velocities are high, resuspension and/or horizontal sediment and bedload transport may be significant (Vanoni 1975). These episodic events and other advective fluxes across the water-sediment boundary are not included in Equation (2.2) but are considered boundary conditions. The time constant for solid-phase advection may be represented as τs = h/ws, where ws is the settling velocity, which is a function of particle size and density (Thomann and Mueller 1987, p. 545). If τs << τ1 then sedimentation is effectively instantaneous and the watercolumn sediment phase need not be modelled. If τs >> τ2 then sedimentation occurs sufficiently slowly that the settling process may be neglected. In the situation that τ1 < τs < τ2 , then the sediment phase and sedimentation process must be considered. A similar analysis can be applied to the resuspension process by considering a resuspension velocity (i.e. ws < 0) that is a function of the bottom shear stress exerted by the stream velocity (Chapra 1997, p. 312). The potentially episodic nature of this process should also be considered in developing a modelling approach.

4.2.5 Step 5 – Determine reaction terms Step 5 is a fundamental part of the decision process: the determination of which components and processes to include in the model and which to omit. In terms of Equation (2.2), this step determines the elements in the concentration vector, c, and the expressions to be included in the reaction vector, r(c,p). We propose that this step be completed within the framework of the Petersen stoichiometry matrix as presented in Chapter 1. This step is sufficiently complicated to merit a separate discussion, which appears in Section 4.3.

4.2.6 Step 6 – Determine boundary conditions Once the model variables and reactions have been determined, it is possible to complete the final decision, the determination of model boundary conditions. The specification of boundary conditions is intrinsically related to the choice of model dimensionality. Boundary conditions of the fully three-dimensional model (Equation (2.2)) become source terms in equations in which the corresponding dimension is integrated or averaged over. For example, in a three-dimensional representation of stream dissolved oxygen, oxygen transfer across the water surface is a boundary condition at z = zo:

εz

∂c ∂z

z = zo

 c = K L  air − c  = K L (cs − c )   H

(4.7)

where KL is the interfacial transfer velocity, cair is the concentration of oxygen in the air, H is Henry’s Law constant for oxygen, c is the concentration of oxygen in the water, and cs is the saturation concentration of dissolved oxygen. In contrast, in a traditional one-dimensional model of stream dissolved oxygen the vertical dimension is integrated out, and oxygen transfer across the water surface becomes the familiar source term based on a reaeration rate (Thomann and Mueller 1987):

52

River Water Quality Model No. 1 rreaeration = K 2 (cs − c )

(4.8)

where rreaeration is the reaeration component of the r(c,p) term in Equation (2.2), c is the concentration of dissolved oxygen, and K2 is the reaeration coefficient = KL/h, where h is the water depth. Other fluxes at the vertical boundaries include oxygen and COD flux across the bottom boundary (corresponding to SOD in a traditional one-dimensional model), water flux at the bottom due to seepage loss or ground-water inflow, and other gas transfers at the surface. Similarly, boundary conditions for the flux of pollutants at lateral inflows become sources (pollutant loads) in models in which lateral distance (the y-co-ordinate) is averaged out. In this step of the model decision process, the modeller must identify these and other boundary conditions that affect the processes of interest and, depending upon the model dimensionality, formulate these as true boundary conditions or equivalent source terms. An example of rigorous treatment of model boundary conditions is the sediment oxygen demand model by DiToro et al. (1990). In addition, at the boundaries of the primary directional coordinate (usually the longitudinal distance x in a river model), the modeller must assign appropriate boundary conditions. In a typical river model, these will be a specified inflow concentration at the model headwater (c = c0 at x = 0), or flux ( F = Qc − K x ∂c / ∂x at x = 0), and zero change in concentration at the model terminus (∂c/∂x = 0 at x = L).

4.3 BIOCHEMICAL SUBMODEL SELECTION 4.3.1 Biochemical model decision process The purpose of this section is to help users of the River Water Quality Model No. 1 to decide on the adequate selection among the multitude of conversion terms, similar to the process in the preceding section for the adequate choice of the transport terms. Unfortunately, the quantitative measures that could be provided in the decision process for the transport equations are not as abundantly available for the conversion model. Hence, more qualitative reasoning needs to be conducted in the decision process. In the overall decision process of a water quality modelling exercise summarised in Section 4.2 above, Step 5 forms a fundamental part. Indeed, in this step it is determined which (a) compartments, (b) components, and (c) processes are to be included in the model where these individual decisions certainly influence each other. In terms of Equation (2.1), this step determines the elements in the concentration vector, c, and the expressions to be included in the reaction vector, r(c,p). We propose that this step be completed within the framework of the Petersen stoichiometry matrix as presented in Table 3.2. Below, we offer a not necessarily systematic discussion on the main parameters and features that influence the procedure in the practice.

4.3.2 Compartments One of the most important decisions in terms of submodel selection is whether it is necessary to consider one or more compartments in which the processes summarised in the Petersen matrix are occurring. As defined in Section 1.5.4, a compartment is a conceptual or physical subdivision of the modelling domain in which particular biochemical or physical conditions prevail. In case one decides for more compartments, the number of state variables in the model is increased, leading to potentially considerably longer calculation times.

How to use the model

53

The most complete model would contain all components in the water column compartment, particulate components attached to the surface of the river bed (interacting with dissolved compounds in the water column), all components in the sediment pore volume, and, finally, particulate components attached to sediment particles. In case the sediment is modelled as a biofilm, then the number of components is increased even more. Also in the case of the selection of several compartments, simplifications to such a complicated model will often be appropriate. In the following, we discuss adequate models for typical situations: •

•

•

•

Large river: In a very large river, bacteria and algae suspended in the water column may dominate conversion rates. For such a river, a one-compartment model, extended by sediment source and sink terms, may be sufficient for the description of nutrient dynamics in the water column (similar to QUAL2E). However, for the investigation of environmental conditions in the sediment, an additional sediment compartment is required. Small river: The large ratio of wetted surface to volume in a small river makes attached bacteria and algae much more important in comparison to the situation in a large river. In order to calculate nutrient or DO dynamics in a small river, a one-compartment model is a good modelling option. The model would include dissolved components in the water column, algae and bacteria attached to the riverbed, and dead organic particles in the water column and at the riverbed (see the River Glatt case study in Section 5.2). However, this option requires the consideration of additional processes. Due to the absence of light, nutrient, and substrate limitation for sessile algae and bacteria exposed to water column concentrations and light, the model equations presented in Section 3.4 lead to Lotka–Volterra type oscillations and even to unlimited growth in the absence of consumers. The case study in Section 5.2 shows that this problem can be solved by an empirical saturation factor with respect to algae or bacteria concentration (as a simple measure of biofilm thickness). In addition, deposition and detachment processes must be considered in such a model. Such an extended model can lead to good results in the water column. However, as shown in the case study in Section 5.3, water column concentrations may not be representative for the sediment. Then, a sediment compartment is required in order to allow the modeller to estimate the environmental conditions in the pore water of the sediment. River with significant conversion rates in the water column as well as in the sediment: A combination of the models described above must be applied. However, this leads to a very large number of unknown parameters and, therefore, to a very demanding model calibration. River sediment: In order to explore environmental conditions in the sediment pore water, in many situations it may be appropriate to decouple the water column and sediment models. This may be possible due to the small and slowly changing effect of sediment processes on the water column. In this approach, using typical river water concentrations as boundary conditions, the conditions in the sediment and the exchange fluxes between sediment and pore water can be calculated. In a second step, the exchange fluxes can be used as input to a simpler model for the water column of the river.

4.3.3 Components and processes In this section we turn our attention to the rules of conversion model selection, which are related to components and processes, and the conditions influencing them.

54

River Water Quality Model No. 1

Replacing concentrations as state variables by constants. A number of decisions (see below) may lead to elimination of the dynamics of certain component concentrations. However, this does not mean that these component concentrations are completely eliminated from the process descriptions. Rather, the kinetic expressions in which these component concentrations are present are modified. For instance, in case the dynamics of biomass concentrations are not considered, the X variable is replaced with a constant to be chosen by the user or estimated from data. Similarly, if the concentration of one or the other substrate of a reaction is assumed to be independent of time, the saturation terms involving this variable can be replaced by constants, in this way simplifying the overall model considerably. Such a simplification of the model is of special interest for sessile algae and bacteria, when not enough information for dynamic modelling of their population is available. Such a simplified model at least allows modelling of the short-term dynamics of dissolved components using sessile algae and bacteria as model parameters. However, it cannot account for longer-term changes in the populations of algae and bacteria (e.g. Section 5.2). Nitrite. Components (columns) 5 and 6 of the RWQM1 process matrix (Table 3.2) show reactions for the nitrite and nitrate species of nitrogen. Nitrite, however, is typically shortlived in rivers and stays low in concentration. For many rivers, it may be omitted from the river water quality model without loss of predictive power. In some rivers, it reaches higher concentrations and can become toxic to aquatic organisms. The decision to include or exclude nitrite thus depends upon observed conditions in the river being modelled, the quality criteria for the river, and the goals of the modelling exercise. If there is a significant ammonia input to the river, nitrite should be considered as a component, because there can be a significant nitrite build-up due to nitrification, especially during the summer months (Londong et al. 1994). If nitrite is concluded to be unimportant, components 5 and 6 of the matrix may be consolidated into a single column for oxidised nitrogen species, SNO (as in the Activated Sludge Models). Similarly, the first- and second-step nitrifying bacteria, XN1 and XN2, may also be consolidated in a single nitrifying bacteria population, XN. The corresponding processes 5 and 7, and 6 and 8 can also be consolidated into a single process for nitrifier growth and a single process for aerobic respiration of nitrifiers, respectively. Finally, the growth of consumers on the two nitrifier populations (processes 12d and 12e) can be combined into a single process. Anoxic conditions. In general, the modeller should assume that anoxic conditions (the absence of oxygen and simultaneous presence of nitrate/nitrite) may occur. However, field data may indicate that such conditions are virtually impossible. A typical example would be a highly aerated stream with a small organic load. In those cases that the modeller can assume anoxic conditions will not occur, processes 3a, 3b, and 4 for anoxic growth and anoxic respiration can be eliminated from the Petersen matrix (Table 3.2). Algae. In some riverine situations, algal activity may contribute insignificantly to the dissolved oxygen budget. For instance, if the hydraulic residence time is less than 4 to 7 days, then wash-out will limit the suspended algal population (Kimmel et al. 1990; Thomann and Mueller 1989). Otherwise, a rule of thumb used in traditional dissolved oxygen modelling is that algal influence can be ignored if the concentration of chlorophyll a in the water column is less than 10 µg/L. This rule of thumb is consistent with the approximate formulae given by Thomann and Mueller (1989) that P = 0.25 Chl-a and R = 0.025 Chl-a, where P is the maximum photosynthetic oxygen production rate in mg/L/day; R is the rate of oxygen consumption by algal respiration in mg/L/day; and Chl-a is the concentration of chlorophyll a in µg/L. Different considerations govern sessile algae. These algae may influence dissolved oxygen levels under conditions in which planktonic algae are unimportant. However, sessile

How to use the model

55

algae require a rocky substrate and will not be a factor in a river with a muddy bottom. They also require good light conditions and small water depth (note that macrophytes can be considered to be equivalent to sessile algal activity and may significantly contribute to the oxygen balance). If algae are not influential, then component 19 may be removed from the table, as well as processes 9a, 9b, 10, and 11. Predation of algae by consumers is then no longer possible and process 12a can be removed as well. Consumers. In a great many water quality and aquatic ecosystem models, consumers are not explicitly modelled. Not modelling consumers is tantamount to an implicit assumption that consumers exist at a constant concentration. If this simplification is chosen, the death rate for the different populations must be proportionally adjusted to account for the effects of consumers. If consumers are not modelled, then processes 12a through 14 and component 20 can be eliminated from the process matrix. The effect of consumers is then lumped into expressions for death or respiration of the organisms, leading to apparently higher rates. The decision to model consumers or not is largely driven by data availability: in many rivers, there will be insufficient data to evaluate the accuracy of the model of consumers and thus justification will be limited for inclusion of such a model. Heterotrophs. As with the population of consumers, heterotrophic bacteria need not be explicitly modelled, but may be implicitly treated as a constant concentration in the model (if this is representative for the system under study). In this instance, XH (component 16) may be eliminated. Model kinetic coefficients would require adjustment to capture the effect of the unmodelled, constant population. The assumption of constant heterotrophic population is, for instance, implicit in the QUAL2E model and other traditional approaches to water quality modelling. In general, we do not recommend this approach with RWQM1 inasmuch as a primary goal of the model is to capture heterotrophic population dynamics. Nitrifiers. Only under certain conditions will the size of a nitrifier population be of such significance that the composition of the river water is influenced by its activity. Such conditions are, for instance, a sufficient retention time or the possibility to form biofilms that are not overgrown by heterotrophs, that is, under not too organically polluted conditions. When nitrifiers (and nitrification) can be neglected, components 17 and 18 can be eliminated, as can the processes 5 to 8. Note that their absence will also allow the elimination of processes 12d and 12e from the process matrix. Another condition for elimination of an equation for the nitrifier concentration occurs when their concentration is not varying significantly over time, as when the load is constant or another constant limitation is affecting their growth. This corresponds to an exact compensation of the processes affecting the nitrifier concentration (and a vanishing of the stoichiometric coefficient when these processes are summed, for example for XN1 processes 5, 6, and 12d). Under these conditions, components 17 and 18 can be eliminated in the process matrix, but one should not forget that the constant value of the biomass concentration must be introduced in the corresponding combined process rate. Chemical equilibria. The decision of whether or not to model the chemical equilibria involving ammonium (SNH3 and SNH4), the carbonate system (SH2CO3, SHCO3, SCO3, and SCa), phosphate (SH2PO4 and SHPO4), and hydroxyl and proton concentrations (SOH and SH) depends upon the river characteristics and the modelling goals. The equilibria can often be eliminated as extraneous to the goals of modelling. Exceptions are those situations in which it is important to understand pH dynamics, where field measurements of total inorganic carbon indicate possible limitation of the growth of nitrifiers (XN1 and XN2) and algae (XALG), or where large pH variations have an important effect on the rates of processes included in the model. Such large pH variations can be caused, for instance, by algae growth, nitrification, or

56

River Water Quality Model No. 1

external disturbances such as acid or alkaline discharges. Other useful applications of these equilibria are those riverine situations where it is important to detect potential ammonia (SNH3) toxicity towards fish. If the chemical equilibria are not modelled, components 4, 8, and 10 through 15 and processes 16 to 21 may be removed from the Petersen matrix. General rules for submodel selection. Several model components are usually essential. Obviously, rapidly biodegradable organic matter, SS, and dissolved oxygen, SO2, correspond to the fundamental parameters BOD and DO in traditional water quality models and must usually be retained. Ammonium is similarly fundamental. Slowly biodegradable particulate organic matter (XS) arises, according to the matrix, from the death of consumers or algae. However, it is commonly introduced into a river by sources (that is, point- and non-point source loads). Hence, even if algae and consumers are not modelled, XS and hydrolysis must remain a part of the model. Similarly, SS, SNH4, and SHPO4 commonly derive from source loads. As indicated in the text above, elimination of some species or processes may precipitate the elimination of others. In general, it can be stated that, for any component with a negative relation in a row, the reaction cannot occur if that component is assumed to have a zero concentration, and the row can consequently be eliminated. The same effect could be achieved in the full model by programming a switching function in the process rate equation to yield a zero process rate when the component concentration is zero. However, this causes superfluous calculations to be made, increasing computation time. For example, if there were no XP, then there could be no phosphorus adsorption, and process 22 could be deleted altogether. A simple rule does not hold for elimination of components (columns) from the process matrix, however. If there are no terms in any particular component (that is, all boxes in the column are empty), then the component might not necessarily drop out of the process matrix. Indeed, if the column is dropped out of the Petersen matrix, it means that the component is completely eliminated from the river model (it is no longer in the component vector, c). The component may, however, still be needed in the model, for instance because it is present in one or the other kinetic relation, or it may be necessary to allow calculation of a variable that can be related to measurements, e.g. the total suspended solids. Moreover, the fact that all elements in a column are empty does not necessarily mean that the concentration of the corresponding component is not time-varying, because boundary conditions or sources and transport processes may also affect its concentrations. Some examples above indicate that the net conversion rates for components can become zero if different processes exactly compensate for each other (or are assumed to compensate). Essentially, this means that the summation of the different process rates involving the component that could potentially be eliminated should make the stoichiometric coefficient vanish. Only then is the elimination of the component allowed.

4.4 EXAMPLES OF BIOCHEMICAL SUBMODEL SELECTION In the following, some examples are presented that illustrate how simplifications of the basic River Water Quality Model No. 1 can be obtained for adequate description of particular situations in rivers. In Table 4.1, a simplified model is introduced, in which the influences of consumers, pH variations, and phosphorus adsorption/desorption on other variables in the system can be assumed to be negligible and their variation itself is of no interest to the model builder. This model may be selected in cases when pH measurements indicate only slight variations thereof, when phosphate is not the limiting nutrient, and when measurements indicating the

How to use the model

57

activity of consumers are not available or not sufficiently convincing to extend the model with this state variable and the corresponding processes. Table 4.1: Simplified River Water Quality Model No. 1 without consumers, pH variation, or phosphorus sorption. i → Component j Proc ess ↓ Aerobic Growth of (1a) Heterotrophs with NH4 Aerobic Growth of (1b) Heterotrophs with NO3 Aerobic Respiration of (2) Heterotrophs Anoxic Growth of (3a) Heterotrophs with NO3 Anoxic Growth of (3b) Heterotrophs with NO2 Anoxic Respiration of (4) Heterotrophs Growth of 1st-stage (5) Nitrifiers Aerobic Respiration of (6) 1st-stage Nitrifiers Growth of 2nd-stage (7) Nitrifiers Aerobic Respiration of (8) 2nd-stage Nitrifiers (9a) Growth of Algae with NH4 (9b) Growth of Algae with NO3 Aerobic Respiration of (10) Algae (11) Death of Algae (15) Hy droly sis

(1)

(2)

(3)

(5)

(6)

(7)

(9)

SS

SI

S NH4

S NO2

S NO3

S HPO4

S O2

XH

?

-

1

?

-

1

+

-

-1

-

?

-

+

-

+

-

-

-

+ -

+

+ -

+

+

?

1

?

1

+

-1

X N1

X N2

1

+

-

-1

-

-

1 -1

+

-

-

+ +

X A LG

(21) (22) XS

XI

+

-

+

(19)

+

-

-

(16) (17) (18)

+

+ 1 1

+

+

-

-1

(+) (+)

(+) (+)

(+) (+)

-1

+ + -1

+

Table 4.2 is a model extending the simplifications made in Table 4.1, with the additional assumptions that: • •

bacterial growth is compensated by respiration (leading to constant heterotrophic and nitrifying populations), and the rate-limiting function of hydrolysis is incorporated into the degradation rate. This formulation is conceptually similar to the QUAL2E model (Brown and Barnwell 1987).

Table 4.2: Simplified River Water Quality Model No. 1 similar to QUAL2E. Component → Process ↓ j

i

Aerobic Degradation of (1+2+15) Organic Material Anoxic Degradation of (3+4+15) Organic Material Growth and Respiration of (5+6) 1st-stage Nitrifiers Growth and Respiration of (7+8) 2nd-stage Nitrifiers Growth of Algae with NO3 (9b)

(3)

(5)

(6)

(7)

(9)

(19) (21)

S NH4

S NO2

S NO3

S HPO4

S O2

X ALG

XS

+

-

-

-

-

-

+ + -

-

+

+ -

+ -

-

+

+

58

River Water Quality Model No. 1

As a next example, Table 4.3 illustrates to what extent the model can eventually be simplified when the additional assumption is made that nitrification is absent (for example, because the organic load is too high, which leads to too strong competition with the heterotrophs) and anoxic degradation can be omitted from the overall description of the riverine situation under study (for example, because nitrate is absent or because aeration is intensive and no considerable biofilm is present). Here, even in this very simple model – and in contrast to many state-of-the-art models – a description of biomass accumulation (of algae) is still essential to describe the observed oxygen dynamics. This model is essentially the Streeter–Phelps model extended to include a photosynthesis-respiration term. Table 4.3: Simplified River Water Quality Model No. 1 similar to extended Streeter–Phelps model. → Component Process ↓ j

i

Aerobic Degradation of (1+2) Organic Material (9b) Growth of Algae with NO3

(3)

(6)

(7)

(9)

(19) (21)

SNH4

SNO3

SHPO4

SO2

XA LG XS

+

-

-

-

+

+

+ -

-

Finally, in the very simple model presented in Table 4.4, the assumption of constant heterotrophic population (hidden in the kinetic coefficient) is sufficient to describe the dissolved oxygen dynamics induced by organic material biodegradation. Additional assumptions here are that algae activity can be neglected and that ammonium and phosphate are not to be modelled, for example because they are not limiting the conversion processes. This is the reduction of the River Water Quality Model No. 1 into the classic Streeter–Phelps model. Table 4.4: Minimal River Water Quality Model No. 1 similar to the Streeter–Phelps model. → Component Process ↓ j

i

Aerobic Degradation of (1+2) Organic Material

(7)

(16)

SO2

XS

-

-

4.5 SUMMARY This chapter has shown how to simplify the River Water Quality Model No.1 under various circumstances. Guidelines on the choice of different submodels that can be selected from the multitude of biochemical process equations presented in Chapter 3 have been given. There are no clear-cut decision criteria for the conversion part of the model, but guidelines have been presented and some general rules for model selection specified.

5 Case studies

5.1 INTRODUCTION This chapter presents the results of two research-oriented case studies in which River Water Quality Model No. 1 was tested. The case studies include two small rivers in Germany and Switzerland. These case studies illustrate the capabilities of RWQM1 on rivers with some unusual features, and demonstrate the robust capabilities of the model for non-traditional situations. We hope that the publication of RWQM1 will encourage the completion of additional case studies as well as the identification of additional field data sets. These would aid in the further development of RWQM1. We particularly encourage the identification of a riverine water quality data set that includes both BOD and COD measurements that would enable a field-based comparison of traditional approaches and RWQM1 applied to a prototypical DO sag problem. The focus of the first case study, of the River Glatt in Switzerland, is the biochemical submodel of RWQM1; the focus of the second case study, of the River Lahn in Germany, is the compartmentalisation concept of the model. Consequently, in the first case study several submodels of the full biochemical model presented in Chapter 3 are selected as recommended in Chapter 4 and applied to the River Glatt. The goal of the second case study is the extension of the model description to the pore water in the sediment. From a biochemical point of view, this case study is kept very simple and is limited to the description of dissolved oxygen only. The two case studies are based on prior publications by Reichert (2001) and Borchardt and Reichert (2001). © 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

60

River Water Quality Model No. 1

5.2 RIVER GLATT 5.2.1 Study site The River Glatt is a small Swiss river of about 35 km length flowing from Lake Greifensee to the Rhine River. The characteristics of the river change from an unpolluted lake outlet to a slowly flowing, heavily polluted river with a small slope north of Zürich, and finally, to a relatively steep, highly aerated river with small drops every 50 m and some cascades. This study focuses on the last 10 km of the river. This reach can best be used for the investigation of conversion processes of substances transported with the river water because there are no significant dry weather tributaries. For the river stretch, upstream and downstream online measurements of temperature, pH, ammonia, nitrite, and oxygen concentrations are available from six measurement campaigns with duration between a few days and a few weeks (Berg 1991; Kändler unpublished). In addition, online measurements of temperature, pH, and oxygen concentration, and cumulative samples of many other substances are available at the downstream site from the Swiss National River Survey Program (NADUF).

5.2.2 Review of previous modelling studies Because it has been shown experimentally that conversion processes in samples from the water column are much slower than those observed in the river, all modelling approaches concentrate on conversion of substances dissolved in the water column caused by sessile bacteria and algae attached to the river bed. The goals of the first study (Reichert et al. 2001b) were to determine nitrification, production, and respiration rates in the river and to find a correlation between environmental factors and oxygen and nitrogen conversion rates. Stoichiometric coefficients of the simple model applied in this study were determined with the aid of a mass composition for organic substances that considered only the elements C, H, O, and N. For the dissolved substances ammonia (SNH4), nitrite (SNO2), and oxygen (SO2), conversion rates were formulated with a linear dependence of algae growth on light intensity, a constant community respiration rate, and with Monod-type limitation factors for oxygen and ammonia or nitrite for first- and second-stage nitrification, respectively. The parameters of these transformation rates were estimated using upstream and downstream online measurements for ammonia, nitrite, and oxygen from all six measurement campaigns mentioned above (Berg 1991; Kändler unpublished). The single exception was the gas exchange coefficient for oxygen, which was determined independently with SF6 volatilisation experiments. The most important results of the study were the following: • •

For each of the six measurement campaigns, a good fit was possible after adjusting the six model parameters. However, the values of the parameters differed from one measurement campaign to the other. No significant correlation between environmental parameters and conversion rate parameters was found. This led to the following conclusions: • the model is of an adequate complexity for the description of short-term dynamics of ammonia, nitrite, and oxygen concentrations (daily variations within periods of a few days); • over the time-scale of a few days to a few weeks, nitrogen and oxygen conversion rate parameters do not change significantly. Since these rates are caused by sessile algae and bacteria, this implies the presence of a limiting factor for the accumulation of active algae and bacteria such as detachment,

Case studies

•

61

grazing, or a limiting effect of active biomass in biofilms due to diffusion or shading; and due to the absence of a mechanistic description of the limiting factor for the accumulation of active biomass, the model cannot be used for long-term prediction or for the prediction of oxygen and nitrogen dynamics under changed external driving conditions (e.g. improved wastewater treatment).

The most important problem in the investigation described above was the inability to find a significant correlation between conversion rate parameters and environmental factors. One reason for this result was the small number of measurement campaigns that could not be increased significantly. In order to obtain a better statistical foundation at least for production and respiration rate parameters, the model was simplified to a model only for dissolved oxygen. This model could be calibrated with the downstream online measurements of oxygen alone (Uehlinger et al. 2000). These measurements are available over many years from the Swiss National River Survey Program (NADUF). With this model, 123 dry weather periods of 3 days length within a period of 6 years could be evaluated. One respiration parameter (total community respiration including nitrification) and two production parameters were fitted for all 123 investigation periods and a correlation analysis with discharge, time, temperature, seasonal temperature gradient, time since last flood, and global radiation was performed. This evaluation led to the following results: • •

The environmental parameters with most significant influence were temperature, seasonal temperature gradient, time, and global radiation (used for production parameters only). There was no significant effect of discharge or time since last flood.

The conclusions were that, in contrast to gravel-bed rivers (Uehlinger et al. 1996), floods have no significant effect on production and respiration in the River Glatt with its more stable river bed and that production and respiration activity is dominated by the seasonal effect.

5.2.3 Goals of the present study The focus of the previous research studies summarised above was to quantify nitrogen and oxygen conversion rates. This was done with parsimonious models that use coefficients in empirically formulated conversion rates as model parameters. These models do not describe changes in algal and bacterial densities mechanistically. Therefore, they are not able to predict conversion rates under changed external driving conditions (e.g. improved wastewater treatment). The goal of the present study is to perform speculative applications of adequate submodels of River Water Quality Model No. 1 in order to demonstrate how an increase in the predictive capability of river water quality models could be achieved. This goal is approached by discussing three submodels of increasing complexity. The three submodels of the River Water Quality Model No. 1 applied in these three steps, as well as the results of their application to the River Glatt, are described in the following three sections. All results shown in the following sections were gained by integrating the partial differential equations for river hydraulics, substance transport and substance conversion

62

River Water Quality Model No. 1

with the simulation and parameter identification program AQUASIM (Reichert 1994, 1995; http://www.aquasim.eawag.ch).

5.2.4 Submodel for oxygen, nitrogen and phosphorus conversion by constant benthic biomass The first submodel of the River Water Quality Model No. 1 is described qualitatively in Table 5.1. This model is a simplified version of the submodel described in Table 4.2 (constant organism densities, oxic conditions). In addition to the processes shown in this matrix, gas exchange of dissolved oxygen at the cascades and along the river is considered. Bacterial densities are assumed to be constant and are used as model parameters. The submodel shown in Table 5.1 is similar to the model used in the first Glatt study mentioned above (Reichert et al. 2001b). The major difference is that here the values of growth parameters and yields of algae and bacteria are assumed and that surface densities of sessile algae and bacteria are used as model parameters instead of more direct conversion rate parameters for nitrification, production, and respiration. This leads to more speculative results but it is a good first step towards a model with better predictive capabilities that tries to simulate the dynamics of algae and bacteria. A minor difference is the consideration of nitrate, SNO3, dissolved organic substrate, SS, and phosphate, SHPO4, as additional state variables. Anoxic processes, processes, and state variables for dissolved inorganic carbon compounds and all particulate state variables are omitted. Instead, algal and bacterial surface densities are used as model parameters. Table 5.1: First submodel of the River Water Quality Model No. 1 as it is used for the present investigation. Component → j Process ↓ (1a) (1b) (2) (5) (6) (7) (8) (9a) (9b) (10) (11) (15)

i

Aerobic Growth of Heterotrophs with NH4 Aerobic Growth of Heterotrophs with NO3 Aerobic Respiration of Heterotrophs Growth of 1st-stage Nitrifiers Aerobic Respiration of 1st-stage Nitrifiers Growth of 2nd-stage Nitrifiers Aerobic Respiration of 2nd-stage Nitrifiers Growth of Algae with NH4 Growth of Algae with NO3 Aerobic Respiration of Algae Death of Algae Hydrolysis

(1) (3+4)

(5)

SS SNH4 SNO2 -

(6)

?

-

+ -

+

+ -

+

+ -

+

+ (+) (+)

(7+8)

(9)

SNO3 SHPO4 SO2 ?

-

?

-

+

-

-

-

+

-

-

-

+

-

+ (+) (+)

+ + (+) (+)

Table 5.2 summarises the input data employed for the simulations with the model shown in Table 5.1. In addition to these data, light intensity and atmospheric pressure were used from a nearby measurement site of the Swiss Meteorological Institute and water temperature from Berg (1991).

Case studies

63

All parameter values of the model were taken from the numerical example given in Reichert et al. (2001b) with the exception of KI, the value of which was increased to 2000 W/m2 in order to account for the use of light intensity at the water surface instead of light intensity at the river bed (there was no significant light limitation or inhibition observed). The bacterial densities that are model parameters and not state variables in this simplified model were used to adapt the model results to the measurements. Table 5.3 gives the values of the model parameters for two measurement campaigns in July and November 1990 (the width of the river is about 17 m; biomass surface densities can be obtained by a division by 17 m). Figure 5.1 shows measured and calculated upstream and downstream concentrations of oxygen, ammonia and nitrate. Table 5.2: Input concentrations used for the model described in Table 5.1. Variable SS

July 90 7.7

Input concentration Nov. 90 Units 6.6 gCOD/m3 gN/m3

SNO2 SNO3

Measured time series of NH3+NH4 Measured time series 4.5 4.5

gN/m3 gN/m3

SHPO4

0.28

gP/m3

SO2

Measured time series

SNH4

0.21

gO/m3

Comment based on cumulative sample of river survey programme see Figure 5.1 see Figure 5.1 based on cumulative sample of river survey programme based on cumulative sample of river survey programme see Figure 5.1

Table 5.3: Parameter vales adjusted for adapting the simulated results to the downstream measurements. Parameter

XALGs XHs XSs XN1s XN2s

July 1990 650 350 250 20 3.5

Nov. 1990 800 0 0 50 15

Units gCOD/m gCOD/m gCOD/m gCOD/m gCOD/m

As already stated in Reichert et al. (2001b), the close agreement of calculated with measured concentrations for this simple model indicates that there is no significant change of active biomass in the river during the measurement periods.

5.2.5 Extension of the submodel to the calculation of dissolved carbonate equilibria, calcite precipitation, and pH Although the first submodel shown in Table 5.1 could describe the behaviour of the most crucial dissolved compounds in the river, it was felt important to extend the dissolved variables to the calculation of the pH value. This would demonstrate the capability of the model to perform pH calculations, which makes it possible to calculate the concentration of NH3 (toxic for fish) and it is a prerequisite for the use of pH-dependent rate expressions. As pointed out by Quinlan (1984) and Boon and Laudelout (1962), pH dependence is of special importance for nitrification; however, pH dependence is not actually implemented in this case study. The second submodel of the River Water Quality Model No. 1 was extended by adding inorganic carbon compounds and the calcium ion for pH calculation, and by the

64

River Water Quality Model No. 1

corresponding chemical equilibrium processes. This submodel is shown in Table 5.4. Gas exchange of carbon dioxide is taken into account in addition to gas exchange of oxygen. pH, July 1990

Conductivity, July 1990 600

9

uS/cm

log10(mol/l)

9.5

8.5 8 7.5 7/17/90

7/18/90

7/19/90

500

400 7/17/90

7/20/90

7/18/90

7/19/90

60 40 20 0 7/17/90

7/20/90

7/18/90

7/19/90

7/20/90

CO2, July 1990

HCO3, July 1990 1.5

45

gC/m3

gC/m3

7/20/90

Calcium, July 1990

50

40 35 30 7/17/90

7/19/90

80 gCa/m3

mol/m3

Alkalinity, July 1990 4.2 4 3.8 3.6 3.4 3.2 3 7/17/90

7/18/90

7/18/90

7/19/90

7/20/90

1 0.5 0 7/17/90

7/18/90

7/19/90

7/20/90

Figure 5.1: Measured (markers) and calculated upstream (thin lines) and downstream (thick lines) concentrations of oxygen (top), ammonia (middle), and nitrite (bottom).

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65

Table 5.4: Second submodel of the River Water Quality Model No. 1 for the River Glatt case study. (1) (3) (4) (5) Component i → SS SNH4 SNH3 SNO2 j Process ↓ Aerobic Growth of Heterotrophs (1a) ? with NH4 Aerobic Growth of Heterotrophs (1b) with NO3 Aerobic Respiration of + (2) Heterotrophs + (5) Growth of 1st-stage Nitrifiers Aerobic Respiration of 1st-stage (6) + Nitrifiers (7) Growth of 2nd-stage Nitrifiers Aerobic Respiration of 2nd-stage (8) + Nitrifiers (9a) Growth of Algae with NH4 (9b) Growth of Algae with NO3 + (10) Aerobic Respiration of Algae (+) (11) Death of Algae (15) Hydrolysis + (+) (16) Equilibrium CO2 ↔ HCO3 (17) Equilibrium HCO3 ↔ CO3 (18) Equilibrium H2O ↔ H + OH -1 1 (19) Equilibrium NH4 ↔ NH3 (20) Equilibrium H2PO4 ↔ HPO4 (21) Equilibrium Ca ↔ CO3

(6)

(7)

(8)

(9)

(10)

(11)

(12)

SNO3 SHPO4 SH2PO4 SO2 SCO2 SHCO3 SCO3

-

+

-

(13) (14) (15) SH

?

-

+

?

?

-

+

?

+

-

+

-

-

-

-

+

+

-

+

-

-

-

-

-

+

-

+

-

+ (+) (+)

+ + (+) (+)

+ ? ? -1

? ? + + 1 + +

1

1 -1

1

-1 +

SOH SCa

1

1

Table 5.5 shows the parameter values used in this second submodel. In addition, constant values, SK, SMg, SNa, SCl, and SSO4, for the concentration of potassium, magnesium, sodium, chlorine, and sulphate, were assumed in order to make a calculation of electrical conductivity possible (SK = 4.2 and 3.9 gK/m3, SMg = 16 and 16 gMg/m3, SNa = 17 and 16 gNa/m3, SCl = 23 and 24 gCl/m3, and SSO4 = 22 and 23 gSO4/m3 for the two simulation runs for July and November 1990). Note that only the most essential variables for the calculation of pH were introduced. More complete pH models can be found in Chapman (1982), Yeh and Tripathi (1991), and Runkel et al. (1996a,b).

66

River Water Quality Model No. 1

Table 5.5: Input concentrations used for the model described in Table 5.4. Variable

July 90

SS

Input concentration Nov. 90

7.7

SNH4

6.6

SNH / (1+Keq,N / SH )

Data source

Units gCOD/m3 gN/m3

based on cumulative sample of river survey programme SNH is the measured time series of NH4+NH3 (Figure 5.1) SNH is the measured time

SNH3

SNH / (1+SH /Keq,N )

gN/m3

SNO2

measured time series

gN/m3

see Figure 5.1

SNO3

4.5

gN/m3

based on cumulative sample

gN/m3

cumulative sample for

SHPO4

4.5

SPO / (1+SH / Keq,P) SPO / (1+Keq,P / SH)

gN/m3

SO2 SCO2

measured time series

SH SHCO3 / Keq,1

gO/m3 gC/m3

SHCO3

See Equation (5.1) below

gC/m3

SCO3

Keq,2 SHCO3 / SH

gC/m3

SH

1000 10-pH

gH/m3

SOH

Keq,w / SH

gH/m3

SH2PO4

SCa

65

73

gCa/m3

series of NH4+NH3 (Figure 5.1)

of river survey programme SPO is the measured phosphate (Table 5.4) SPO is the measured cumulative sample for phosphate (Table 5.4) see Figure 5.1 calculated from a charge balance based on measured time series of pH based on cumulative sample of river survey programme

Because pH dependence of process rates is neglected in the model, this second submodel leads to the same results for nutrients and oxygen as the first submodel (see Figure 5.1). However, dissolved inorganic carbon compounds, protons (and pH), hydroxyl ions, and calcium ions can also be calculated. This leads also to an increased number of inflow concentrations to be provided at the upstream end of the river section. Table 5.5 gives an overview on the source of the data used for the input. Because there were no quasicontinuous measurements available for some of the modelled substances, the concentrations of some substances were assumed to be constant at a value measured in cumulative samples from the river survey programme. The inflow concentration of bicarbonate ions was then calculated from a charge balance according to

S HCO3 =

2S S S S S 12   S H − S OH + NH4 − NO2 − NO3 − H2PO4 − HPO4 − 1 + 2 K eq,2 / S H  14 14 14 31 31 2 S SO4 2 S Ca 2 S Mg S K S Na S  + + + + − Cl  96 40 24.3 39 23 35.5 

(5.1)

Case studies pH, July 1990

Conductivity, July 1990 600

9

uS/cm

log10(mol/L )

9.5 8.5 8

7.5 7/17/90

67

7/18/90

7/19/90

500 400 7/17/90

7/20/90

7/18/90

7/19/90

60 40 20 0 7/17/90

7/20/90

7/18/90

7/19/90

7/20/90

CO2, July 1990

HCO3, July 1990 1.5

45

gC/m3

gC/m3

7/20/90

Calcium, July 1990

50 40 35 30 7/17/90

7/19/90

80 gCa/m3

mol/m3

Alkalinity, July 1990 4.2 4 3.8 3.6 3.4 3.2 3 7/17/90

7/18/90

7/18/90

7/19/90

7/20/90

1 0.5 0 7/17/90

7/18/90

7/19/90

7/20/90

Figure 5.2: Measured (markers) and calculated upstream (thin line) and downstream (thick line) pH value, electrical conductivity, alkalinity, and concentration of calcium, bicarbonate and CO2.

Figure 5.2 shows the results for some of the additional state variables and derived variables for the second submodel. For these simulations, the gas exchange efficiency for CO2 at the cascades and the kinetic coefficient for calcium dissolution/precipitation have been adjusted. Calculated values of pH (top left) are qualitatively correct, however, there is a time shift in comparison to the measurements. Electrical conductivity is in the correct order of magnitude (top right). The differences in the daily variations are probably due to unknown variations in some ion concentrations at the upstream end of the river reach (for the calculation, due to lack of data, most upstream ion concentrations were set to constant values; see Table 5.5). The plots for calcium and CO2 in Figure 5.2 show that the river is significantly supersaturated with both components. The dashed lines show the equilibrium levels for calcium based on the current CO3 concentration and for CO2 based on equilibrium with the atmosphere. Assumption of equilibration of calcium with calcite through precipitation (by increasing the value of keq,s0) would significantly diminish the amplitude of pH variations.

68

River Water Quality Model No. 1

5.2.6 Hypothetical simulation of the dynamics of benthic biomass The third submodel of the River Water Quality Model No. 1 used for a more speculative simulation is shown in Table 5.6. In addition to the dissolved state variables of the first model, it contains particulate state variables for sessile algae and bacteria (XH,s, XN1,s, XN2,s, XALG,s), for benthic consumers (XCON,s), for sedimented organic material (XS,s, XI,s), and for suspended organic material (XS, XI) (algae, bacteria, and consumers are assumed to be effective only at the sediment surface). Modelling sessile or benthic organisms requires some additional considerations. In order to make it easy to fulfil mass balances even in the case of fluctuating water levels, it is advantageous to introduce sessile or benthic biomass per unit river length instead of per unit river bed surface. In Table 5.6 such masses per unit length are indicated with an index “s” (for sessile) (e.g. XH,s denotes the mass of heterotrophic bacteria per unit river length). The coefficients in the stoichiometric matrix that convert from substances dissolved or suspended in the water column to substances attached to the river bed must then be multiplied with the wetted cross-sectional area A of the river (this is the water volume per unit river length). As additional physical processes, detachment of attached biomass to the water column and sedimentation of suspended organic material must be taken into account. The process rates in Reichert et al. (2001b) are formulated with in situ nutrient and oxygen concentrations and with in situ light conditions. If the benthic biofilm in the river is modelled on the basis of surface densities or masses per unit river length only, limiting effects due to self-shading and diffusion of nutrients into the biofilm are not considered. An empirical consideration of this effect is possible with a limiting factor for growth of algae and bacteria with increasing algal or bacterial biomass. This can be implemented by a factor

1+

1 kgro rmax

X

(5.2)

in the growth rate of algae and bacteria. A similar factor was also introduced for consumers. A final extension of the model concerns the assumption of a homogeneous mixture of the food of the consumers. This may not be realistic for sessile algae and bacteria. In contrast to algae, bacteria can grow without light, and therefore they can grow on places that may be difficult for consumers to access. This can be considered by multiplying all bacterial concentrations in the rate expression for consumer growth by an empirical factor αBAC. All these additional processes introduce a number of additional model parameters that cannot be identified. Figure 5.3 shows a simulation of the benthic population based on average nutrient concentration in the water and on actual light, discharge, and temperature data in the River Glatt. This figure demonstrates that simulations can lead to reasonable results (the dots in the right-hand plot show the seasonal pattern of respiration calculated from oxygen data of the years 1992–1997 by Uehlinger et al. (2000); some of the scatter of this data is due to plotting all six years). However, many more data on the algal and bacterial population at the river bed would be necessary in order to identify the parameters of such a river model.

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69

Table 5.6: Third submodel of the River Water Quality Model No. 1. Component → j Process ↓ (1a) (1b) (2) (5) (6) (7) (8) (9a) (9b) (10) (11) (12a)

(1) (3+4)

i

(5)

SS SNH4 SNO2

Aerobic Growth of Heterotrophs with NH4 Aerobic Growth of Heterotrophs with NO3 Aerobic Respiration of Heterotrophs Growth of 1st-stage Nitrifiers Aerobic Respiration of 1st-stage Nitrifiers Growth of 2nd-stage Nitrifiers Aerobic Respiration of 2nd-stage Nitrifiers Growth of Algae with NH4 Growth of Algae with NO3 Aerobic Respiration of Algae Death of Algae Growth of Consumers on XALG

Growth of Consumers on XS Growth of Consumers on XH Growth of Consumers on XN1 Growth of Consumers on XN2 Aerobic Respiration of (13) Consumers (14) Death of Consumers (15) Hydrolysis

-

(6)

(7+8)

?

-

+ -

+

+ -

+

-

A

?

-

A

+

-

-A

-

-

A

+

-

-A

-

-

A -A

-

+ (+)

+ + (+)

A A -A -A -

(+)

-

(+) (+) (+) (+)

-

+

+

-

-A

(+) (+)

(+) (+)

(+) (+)

-A

-

+ -

+ + -A

+

40

XH XS

500

1/1/92

gO/m2/d

gCOD/m

1000

A A A A A

+ +

Respiration Rates 1990/91 XALG

1/1/91

+

(+)

XCON

0 1/1/90

+

(+) (+) (+) (+)

Biomass per unit river length 1990/91

XI

+

+

1500

XS

+

+ -

+

?

+ (+)

(12b) (12c) (12d) (12e)

(9) (16s) (17s) (18s) (19s) (20s) (21s) (22s) (21) (22)

SNO3 SHPO4 SO2 XH,s XN1,s XN2,s XALG,s XCON,s XS,s XI,s

30 20 10 0 1/1/90

1/1/91

1/1/92

Figure 5.3: Calculated biomass densities (biomass per river length) and respiration rates (oxygen consumption per surface area) for the third submodel described in Table 5.6.

5.2.7 Summary and conclusions This case study demonstrates the applicability of the River Water Quality Model No. 1 to improve our understanding in three steps: • •

The application of the first and simplest submodel demonstrates its ability to reproduce previous results obtained with even simpler models. The second application shows the usefulness of the inorganic carbon submodel for modelling pH. With the consideration of pH-dependent rates this could lead to an interesting model extension.

70 •

River Water Quality Model No. 1 The third speculative application demonstrates that the model has the potential to be used for predicting algal and bacterial densities, as well as nutrient and oxygen conversion rates for changed external driving conditions. However, such a prediction needs much more experience with model parameters and formulations that must be gained by data evaluations for a wide spectrum of river types.

As shown with this case study, the prediction of nutrient and oxygen conversion rates in rivers is a difficult task that requires a lot of experience with model formulations and model parameters for a given type of river. It is hoped that River Water Quality Model No. 1 can help facilitate the exchange of experience among river water quality modellers and thereby can accelerate improvements in the predictive power of river water quality models.

5.3 RIVER LAHN 5.3.1 Study site The River Lahn is a right-sided tributary in the middle reach of the River Rhine with a total length of 245 km. The study site is located in the upstream part at Sarnau, Germany, 53 km from the source, with a drainage area of 453 km2 and a medium gradient of 2.36 m/km. The average discharge amounts to 7.3 m3/s with a base-flow of 0.57 m3/s (annual precipitation: 810 mm). We selected a reach 450 metres in length with two regular pool-riffle-sequences (10 to 15 m in width) and seven sampling transects (I through VII in Figure 5.4). At baseflow, the mean flow velocity approximates 0.3 m/s. A wastewater treatment plant for 15,500 population equivalent discharges to the river directly downstream of transect III. The upstream human population approximates 100,000 inhabitants with their wastewater being discharged from seven additional treatment plants. The proportion of treated wastewater in the river base flow is calculated to be 26% from upstream and 5% from the Sarnau plant. The temporal and spatial dynamic of flow and physicochemical parameters was studied by employing 50 stainless steel sampling pipes distributed at seven transects (Figure 5.4). For details of the methodology see Borchardt and Fischer (2000).

III

VI

V

IV

VII

Figure 5.4. Longitudinal profile of the sediment surface with sampling transects.

II I

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71

Besides the basic formulation of biochemical conversion processes, a focus of the RWQM1 approach is the compartment structure of running water ecosystems including their longitudinal, vertical, and lateral zonation patterns (Appendix 1). This aspect of the river has not so far gained much attention in river water quality modelling. Nevertheless, this may be an important aspect to be considered in order to achieve an ecologically appropriate choice of model compartments and state variables. Running waters are linked elements within the hydrological continuum. As a consequence, the hydrological interactions between surface and subsurface flows are important for system functions in rivers, as they influence the transport and storage of water, chemical compounds, and nutrients. Furthermore, the hyporheic zone of running waters has been recognised as an ecologically essential compartment. The upper sediment layers act both as an important habitat for the benthic community (every lotic species has life stages linked to the hyporheic zone) and as a reactor with an intense metabolism. As a consequence, water quality constituents may be related not only to the surface flow but also to the upper layers of the river sediment. This is especially important for impact assessments of point and diffusive pollution on benthic macroinvertebrates and gravel-spawning fish, and for an ecologically meaningful application of water quality standards. Quantifying the significance of exchange processes is complicated due to the high spatial and temporal variation of the hydrological system, which depends on a set of factors of which river morphology, sediment structure, and hydraulic gradients are the most important. Moreover, metabolic dynamics of rivers are shaped by a complex temporal pattern due to diurnal and seasonal changes in autotrophic and respiration activities. We narrowed this complexity by a systematic procedure and analysed the relevance of temporal dynamics of water constituents in a eutrophic shallow river. The goals of this case study are: (1) to outline the compartmentalisation approach of RWQM1; (2) to analyse the connectivity between surface and subsurface flows in running waters with emphasis on oxygen fluxes and sediment oxygen demand; and, (3) to discuss implications for ecologically sound river water quality management.

5.3.2 Modelling approach The study reach was modelled using measured cross-sectional profiles within the sampling reach and simplified profiles, a constant slope, and an effective friction coefficient upstream of the sampling reach. River hydraulics was calculated with the diffusive wave approximation of the full one-dimensional St. Venant hydrodynamic equations (Section 2.4). The Manning–Strickler equation for the friction slope, Sf,

Sf =

1 1 u2 K st2 R 4 / 3

(5.3)

was used to calculate the friction slope. In this equation Kst = friction coefficient according to Strickler (1923) [TL–1/3], R = hydraulic radius of the river [L], and u = the cross-sectionally averaged flow velocity [LT–1]. The longitudinal dispersion coefficient, DL, was estimated according to Fischer et al. (1979)

D L = cf

w2u 2 u* h

(5.4)

72

River Water Quality Model No. 1

where cf = non-dimensional dispersion coefficient [L2T–1], w = the surface width of the river [L], h = the mean river depth [L], and

u* = ghS f –1

(5.5)

2 –1

is the friction velocity [LT ]. DL is in units [L T ]. In addition to the vertically and laterally mixed water column, a sediment pore water compartment describing a vertically mixed sediment layer of 40 cm depth and a porosity of 0.23 was introduced. Diffusive as well as advective exchange (within infiltration and exfiltration zones) between these two compartments was considered. The parameters of these exchange processes were derived from uranine dye tracer experiments, continuous temperature records in different sediment layers, and freeze cores (Lenk and Saenger 2000; Saenger and Lenk 2000). All simulations were performed with an extended version of the simulation and data analysis tool AQUASIM (Reichert 1994, 1995; http://www.aquasim.eawag.ch.)

5.3.3 Model calibration We calibrated uranine tracer transients in the water column as well as in the sediment to two experimental series in August and October 1997. Because of the small effect of sediment exchange on the tracer concentrations in the water column, calibration was done in two steps: Step 1:

Step 2:

We calibrated the tracer mass (reduction factors η1 and η2 for loss of tracer in experiment 1 and 2, respectively), the effective Strickler friction coefficient in the reach with simplified geometry upstream of the investigation reach (Kst,eff,1 and Kst,eff,2 for experiment 1 and 2, respectively), and the non-dimensional dispersion coefficient (cf with the same value for both experiments) based on the tracer transients in the water column (using both tracer experiments). The values of the estimated parameters are given in Table 5.7. Figure 5.5 shows the water level profile in the investigation reach. Hydraulic head gradients between riffles and pools can be clearly seen with the calculated longitudinal water surface profile being in acceptable agreement with measured data. We calibrated the discharge in the sediment (Qsed,1 and Qsed,2 for riffle 1 and 2, respectively) and for the diffusive exchange velocity (vex) based on the transients in the sediment. The positions of infiltration and exfiltration reaches were chosen based on results from tracer experiments. Measurements were crosssectionally and depth averaged (averages of measurements taken at depths of 15 cm and 25 cm) for comparison with calculated concentrations in the sediment. Both tracer experiments were used simultaneously for the fit. The values of the estimated parameters are given in Table 5.8. The results shown here were first

Case studies

73

published in Borchardt and Reichert (2001). Later investigations led to the conclusion that advective processes in the pore water of the sediment are less important than as modelled in this study. A revised version of the simulations will be published in Ingendahl et al. (2002). Table 5.7: Parameter estimates from tracer transients in the water column. Parameter

Unit

η1 η2 Kst,eff,1 Kst,eff,2 cf

m1/3s–1 m1/3s–1

Value 0.2 0.76 3.9 8.7 0.006

Standard error 0.1 0.05 1.9 0.6 0.009

Table 5.8: Parameter estimates from tracer transients in the sediment. Parameter

Unit m3s–1 m3s–1 md–1

Qsed,1 Qsed,2 vex

Value 0.0031 0.0018 0.061

Water Level Profile

192.5 Bed and water level [m]

Standard error 0.0006 0.0004 0.018

192 191.5 191 190.5 0

200

400

600

distance [m]

Figure 5.5: Level of the river bed (black line) and measured (markers) and calculated (grey line) water level in the study reach.

There is a complex pattern of flow and tracer transport in the riffle and pool (Figure 5.6). Mass transport in the surface flow is characterised by distinct and symmetric breakthrough curves. For the riffle section, significant exchange processes can be identified from the delayed and asymmetric temporal concentration profiles. In contrast, as indicated by the low concentrations of tracer in the sediments of the pool, there are much smaller exchange rates with the surface flow in these river sections.

74

River Water Quality Model No. 1 Uranin Transients Riffle 1, Aug. 20/21 1997

Uranin Transients Riffle 1, Oct. 7/8 1997

calc, wat col meas, sed, I meas, sed, II calc, sed, I+II meas, sed, III calc, sed, III

15 10 5

100

60 40 20

0

0

9:00

15:00

21:00

3:00

12:00

9:00

10 5

6:00

meas, wat col calc, wat col meas, sed calc, sed

80 60 40 20 0

0 9:00

0:00

100 Uranine [ug/l]

meas, wat col calc, wat col meas, sed calc, sed

15

18:00

Uranin Transients Pool, Oct. 7/8 1997

Uranin Transients Pool, Aug. 20/21 1997

20 Uranine [ug/l]

meas, wat col calc, wat col meas, sed, I meas, sed, II calc, sed, I+II meas, sed, III calc, sed, III

80

Uranine [ug/l]

Uranine [ug/l]

20

15:00

21:00

3:00

12:00

9:00

18:00

0:00

6:00

Figure 5.6: Measured (markers) and calculated (lines) tracer transients in riffle 1 (top) and in the pool (bottom) for the tracer experiment at August 20/21, 1997 (left) and at October 7/8, 1997 (right).

5.3.4 Modelling of oxygen time series in the surface flow and in the sediment The submodel of River Water Quality Model No. 1 used for the River Lahn case study is described qualitatively in Petersen matrix format in Table 5.9. In addition to the processes shown in this matrix, constant gas exchange of dissolved oxygen at a spatially averaged scale is included in the model. Table 5.9: Qualitative Petersen matrix for model of River Lahn. Component

→

i

j (1a) + (1b) (2)+(5)+(6)+(7)+(8)+(10)+(12)+(13) (9a) + (9b)

Process ↓ Aerobic Growth of Heterotrophs Sum of Respiration and Nitrification Processes Growth of Algae

(1)

(9)

SS -

S O2 +

The oxygen balance of the River Lahn in the surface flow is characterised by high daily and seasonal temporal dynamics triggered by radiation and intense production-respiration processes (top, Figure 5.7). Despite enhanced gas exchange with the atmosphere due to low depth and turbulent flow, daily oxygen amplitudes exceeded saturation by up to 8 mg/L in August 1997 (top-left, Figure 5.7), while oxygen deficits were less pronounced. These patterns persisted in October but in a more narrow range of oxygen concentrations (top-right, Figure 5.7).

Case studies

75

The daily amplitude of the oxygen concentration in the uppermost sediment layers is an important consideration in eutrophic rivers with well-developed hyporheic zones. In particular, the uppermost sediment layers function as important habitat for the benthic community. Therefore, we directed specific attention to modelling this aspect of the oxygen balance. As a first step, we modelled the oxygen balance with a simple version of RWQM1 using Equation (5.6): r = K 2 (Csat − CO 2 ) +

PI R − h h

(5.6)

where r = net oxygen production rate [ML–3T–1], K2 = reaeration rate constant [T–1], Csat = oxygen saturation concentration [ML–3], CO2 = oxygen concentration in the water column [ML–3], I = light intensity [EL–2], h = mean river depth [L], and P = production parameter [ME–1T–1], and R = respiration parameters [ML–2T–1]. Estimates for physical reaeration were based on empirical assessments (Owens et al. 1964; Wolf 1974) with consideration of given flow velocities and water depth for the investigation periods. Empirical formulas resulted in K2-values ranging from 17.0 to 29 d–1. For model calculations we selected a value for K2 = 20 d–1. The fit results shown in Tables 5.9 and 5.10 below are conditional on this value. Based on these boundary conditions, production and respiration rate parameters were estimated from continuously measured oxygen time series (Table 5.10 and 5.11). With this approach, a meaningful agreement between modelled and calculated oxygen concentrations could be achieved (top, Figure 5.7). The model was then used to calculate oxygen time series in the sediment layers in the riffle and pool sections of the investigation site (middle and bottom, Figure 5.7). A sediment respiration rate coefficient of 2.5 d–1 was used to adjust calculated oxygen concentrations to some measured values available only at specific dates and for specific locations. Table 5.10: Parameter estimates from oxygen time series in the water column (August 1997). Parameter PI R

Unit g/(Wd) g/(m2d)

Value 0.0781 7.1

Standard error 0.0008 0.2

Table 5.11: Parameter estimates from oxygen time series in the water column (October 1997). Parameter PI R

Unit g/(Wd) g/(m2d)

Value 0.1472 16.04

Standard error 0.0008 0.06

Addition of sediment respiration results in a significant drop of the oxygen concentrations down to concentrations of 2–3 mg/L with an attenuation of the daily amplitude to 0.5–1 mg/L O2 in the pool sediments (bottom, Figure 5.7). A completely different pattern can be identified for the three measurement cross-sections in the upstream riffle (middle, Figure 5.7). At the infiltration area and transition zone (cross-section I, Figure 5.4), highly fluctuating oxygen concentrations are present. While infiltrated water travels through the riffle to the exfiltration zone (cross-section III, Figure 5.4) a significant depletion of oxygen concentrations occurs in a pattern that is comparable to the pool conditions. The difference of median concentrations in a range of 3.5–5 mg/L is close to levels reported for oxygen mass balances from in situ samples (Borchardt and Fischer 2000). We therefore conclude that our modelling approach may be sufficient to describe meaningfully surface-subsurface connectivity and oxygen concentration patterns in a eutrophic fluvial system in a spatially

76

River Water Quality Model No. 1

averaged way. It is clear that the approach is insufficient for a description of the spatial inhomogeneity being documented in the actual river by Borchardt and Fischer (2000). Oxygen in the Water Column 16

14

14

12 meas calc sat

10 8 6 4 8/13/97

8/17/97

8/21/97

Oxygen [mg/l]

Oxygen [mg/l]

Oxygen in the Water Column 16

12 8 6 4 10/1/97 10/3/97 10/5/97 10/7/97 10/9/97

8/25/97

Oxygen in the Riffle 1 Sediment 12

10

10

calc, I calc, III sat

6 4

Oxygen [mg/l]

Oxygen [mg/l]

Oxygen in the Riffle 1 Sediment 12

8

8

4

0

0 8/17/97

8/21/97

10/1/97 10/3/97 10/5/97 10/7/97 10/9/97

8/25/97

Oxygen in the Pool Sediment 12

10

10

8

calc sat

6 4 2

Oxygen [mg/l]

Oxygen [mg/l]

Oxygen in the Pool Sediment 12

8 6

calc sat

4 2

0 8/13/97

calc, I calc, III sat

6

2

2 8/13/97

meas calc sat

10

0 8/17/97

8/21/97

8/25/97

10/1/97 10/3/97 10/5/97 10/7/97 10/9/97

Figure 5.7: Oxygen time series in the water column (top), in the sediment of the pool (middle), and in the sediment of riffle 1 (bottom) for the August 1997 (left) and the October 1997 (right) measuring campaign. “meas” = measured, “calc” = calculated, “sat” = saturation concentration in the water column, and “I” and “III” refer to the measurement cross-section locations mentioned in the text (and shown in Figure 5.4).

5.3.5 System response to inputs of organic matter from a wastewater treatment plant and combined sewer overflows In a third step we applied our modelling approach to questions of system analysis and impact assessment of external inputs of organic matter from sewerage systems. In case of the River Lahn and the boundary conditions of wastewater load (see Section 5.3.1), an important aspect is how the infiltration of biologically treated wastewater and combined sewer overflows into the riffles affects sediment oxygen concentrations and sediment oxygen demand.

Case studies

77

A sewage treatment plant located at x = 350 m has a mean dry weather discharge of 0.04 m3s–1and a mean COD of 40 mg/L. In order to compare oxygen concentrations calculated with this actual load to a situation with stronger pollution, we calculated oxygen concentrations in riffle 2 below the sewage treatment plant effluent under the assumption of a constant COD concentration of 90 mg/L (upper legislative limit) and for a combined sewer overflow with a duration of 2.4 h, a discharge of 2 m3/s, and a COD concentration of 100 mg/L (approximately yearly average event). The effect of these effluents on oxygen concentrations in the river is based on a degradation rate coefficient of 1 d–1 in the water column and 10 d–1 in the sediment pore water. Figure 5.8 shows that the first case of an increased dry weather COD effluent concentration leads to a significant decrease in dissolved oxygen concentrations in the sediment. The decrease is between 1–2 mg/L while diurnal fluctuations do not significantly change. Oxygen in the Riffle 2 Sediment

Oxygen in the Riffle 2 Sediment

12

12

calc, V calc, VI calc, VII sat

8 6 4 2

calc, V calc, VI calc, VII sat

8 6 4 2

0

8/13/97

10

Oxygen [mg/l]

Oxygen [mg/l]

10

0

8/17/97

8/21/97

8/25/97

8/13/97

8/17/97

8/21/97

8/25/97

Figure 5.8: Sediment oxygen time series in riffle 2 (below the wastewater treatment plant effluent) for the August 1997 measuring campaign (left) and for a hypothetical situation with a wastewater treatment plant effluent with a COD of 90 mg/L instead of 40 mg/L (right).

The effect of the second case of a combined sewer overflow is seen in the last day of the simulation shown in the right-hand plot of Figure 5.8 and in more detail in Figure 5.9. The simulation is in agreement with documented effects of immediate and delayed oxygen depletion in the surface flow of running waters (Harremoes 1982). Krejci et al. (1994) showed that combined sewer overflows may result in very low oxygen concentrations in the hyporheic zone (<1 mg/L O2) of a small urban running water, while the surface flow was almost saturated at the same time. This result is of special importance because existing receiving water protection criteria are typically based on measurements of oxygen concentration in the water column only. The results in Figure 5.9 show the time series for COD and oxygen in riffle 2. The combined sewer overflow increased COD concentrations in the sediment at cross-section V significantly for a time period of more than 6 hours. Almost parallel to this increase in COD, oxygen concentrations decreased to nearly zero levels for several hours followed by a slow recovery. The oxygen depletion is even longer at cross-section VII where the exchange processes are slower. At the same time, water column oxygen concentrations do not fall below 4 mg/L. It may be concluded that for the boundary conditions given for the River Lahn combined sewer overflows have the potential to disrupt sediment oxygen balances in riffle sections for extended time periods. Those single events would mean a substantial endangering for populations of sediment dwelling benthic macro-organisms and gravelspawning fishes.

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River Water Quality Model No. 1 Oxygen and COD in the Riffle 2 Sediment

COD [mg/l]

25

6

20

4

15 10

2

5 0 12:00

Oxygen [mg/l]

8

30

COD, V COD, VII O2, V O2, VII

0 18:00

00:00

06:00

12:00

Figure. 5.9: Sediment time series for COD and oxygen in riffle 2 for a hypothetical situation with a combined sewer overflow effluent (discharge of 2 m3/s with a duration of 2.4 h and a COD of 100 mg/L).

5.3.6 Conclusions This case study addressed the application of the compartmentalisation approach of River Water Quality Model No. 1 (RWQM1). It was performed in order to illustrate the importance of modelling a sediment compartment for an ecologically meaningful assessment of the impact of wastewater effluents and combined sewer overflows. The focus of this case study is on the compartmentalisation approach of the RWQM1 that makes such a description possible. In contrast to this, a strongly simplified biochemical submodel is used that considers only oxygen and dissolved substrate. Starting with the implementation of a strongly simplified version of the biochemical part of the RWQM1, but with the consideration of a sediment pore water compartment in addition to the water column compartment, the model was calibrated to tracer data from the water column and the sediment. The calibrated model was then used to study the system response to wastewater treatment plant effluent and combined sewer overflow emissions. The modelling approach makes it possible to quantify the sediment oxygen demand; the spatial and temporal extent of sediment zones with oxygen depletion; and the temporal dynamic of oxygen in surface flow and sediment zones for spatially averaged scales. The modelling approach also considers the distinct patterns in pool and riffle zones. However, the spatially averaged approach does not account for inhomogeneities in the sediment. This potential is demonstrated by results showing the effect of emissions from wastewater treatment plants operating according to emissions standards infiltrating the riffles, which may be high enough to drop sediment oxygen concentrations to low levels while those in the surface flow remain elevated at corresponding times. It is shown that for this river with its coarse alluvial sediments, even moderate emissions from sewerage systems may be high enough to reduce sediment oxygen concentrations to low levels while those in the surface flow remain close to saturation. Similarly, it is demonstrated that combined sewer overflows may cause anoxic sediment oxygen conditions for extended time periods. At the same level of specific domestic wastewater pollution, combined sewer overflows may cause anoxic sediment oxygen conditions for extended time periods. Therefore, oxygen concentrations and sediment oxygen demand in the hyporheic flow of rivers with alluvial coarse sediments are a potential limiting factor for benthic macroinvertebrate communities

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79

and gravel-spawning fish species. This may be the case even under conditions of moderate wastewater loading that may not be captured by water quality modelling restricted to the surface flow. Therefore, the compartmentalisation approach opens new perspectives to study ecologically relevant system responses to a wide range of impact factors.

6 Identifiability and uncertainty analysis

6.1 INTRODUCTION The parameters of water quality models such as RWQM1 are not universal enough to make it possible to describe different systems with the same values of all model parameters. Hence, site-specific model parameters must be obtained by calibration to experimental data. Obviously, trying to estimate all model parameters from experimental data is utopian. Rather, a subset of parameters must be selected that, with the proper data, can yield a wellcalibrated model for a given application of the model to a real system. However, this raises the question of how to select such a subset of parameters to be adjusted. Important aspects to be considered for answering this question are: (1) prior knowledge on parameter values, universality, and uncertainty; (2) the experimental and initial conditions and the measurement layout used for data collection; (3) the measured data if already available; and, (4) information on the identifiability of model parameters for the given measurement layout (the model structure is assumed to be given; the aspect of structural uncertainty is not addressed here). It is obvious that the RWQM1 belongs to the category of models that are too large for unique parameter identification in a typical field application. Because this model may be applied to various field studies, it is important to know the parameters for which information can be gained under different measurement layouts and the expected magnitude of the uncertainties in the model predictions. It is the goal of this chapter to give answers to these © 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

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questions for a typical river and for typical measurement layouts and to demonstrate methodically how such answers could be found for other situations. We concentrate this analysis on the kinetic part of the model because the uncertainties are larger than in the stoichiometric part. Nevertheless, an analysis of the stoichiometric part is also of interest. This chapter is based on the analysis of Reichert and Vanrolleghem (2001). It is structured as follows. First, the techniques used for selecting identifiable parameter subsets and for uncertainty analysis are briefly reviewed (more detail is given in Appendix 5). Then the submodel of the RWQM1 used for a hypothetical case study analysis is introduced and the measurement layouts are discussed. Then the results of identifiability and uncertainty analyses are presented and discussed. Finally, the key results are summarised and conclusions are drawn.

6.2 TECHNIQUES 6.2.1 Selection of subsets of identifiable model parameters Identifiability analysis is performed using three measures related to identifiability. In order to calculate these measures, uncertainty ranges of the parameters, ∆θ j , and scale factors of the model results, sci, must be specified. In addition, the experimental layout (which variables are measured at which locations and at which points in time) must be given. The three measures used for identifiability analysis are defined in Appendix 5. Here we give a brief qualitative description of these three measures. •

•

•

Sensitivity measures δmsqr: This measure quantifies the sensitivity of the model results to a parameter. For each parameter, a value of δmsqr can be calculated. Larger sensitivity of model results to a parameter leads to a larger value of δmsqr. A low sensitivity indicates a poor identifiability of the corresponding parameter. Collinearity index γ: For a given set of model parameters, the collinearity index describes the degree to which the effect on the model results of a change in one parameter can be compensated by appropriate changes in the other parameters of the set (in linear approximation to the model equations). A collinearity index of 1 indicates that such a compensation is not possible, an index of 10 indicates that 90% of the effect can be compensated, and the limiting case of a full “compensability” would lead to a collinearity index of infinity. A collinearity index exceeding a critical value in the order of 10–15 indicates that the parameter set is very poorly identifiable from data even if the model results are sensitive to all parameters of the set. Measure of the extension of the confidence region ρ: ρ is a measure of the (linear) extension of the confidence region if the parameters would be estimated by the method of least squares. The values of ρ decrease with increasing size of the confidence region. This extension of the confidence region combines information on sensitivity (as it can be quantified with δmsqr) with information on collinearity or “compensability” (as it can be quantified with γ). Large values of ρ indicate good identifiability, low values poor identifiability.

Subsets of best identifiable parameters can be selected as follows. First a ranking of sensitivities δmsqr leads to a first overview of important parameters. At this step, parameters with a very low sensitivity can be eliminated in order to reduce the number of parameter combinations for the subsequent steps. For all subsets of size 2, 3, 4, etc. of the remaining parameters, the collinearity and confidence measures γ and ρ are calculated. The subsets of

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each size are then ordered according to decreasing values of ρ. This ranking then reflects decreasing degrees of identifiability of the parameter sets. The values of γ for these sets can then be used to define the threshold for identifiability. Empirical experience indicates that values for this threshold are in the range of 10–15.

6.2.2 Uncertainty analysis Uncertainty analysis is most often done by linear error propagation or by Monte Carlo simulation (Hammersley and Handscomb 1964; Rubinstein 1981). The advantage of linear error propagation is its computational efficiency. If the sensitivity functions have already been calculated for identifiability analysis, no further simulations are required to get an error estimate (see Appendix 5). However, if model non-linearities are significant within the uncertainty range of the parameters, the results of linear error propagation are inaccurate. Monte Carlo simulation is a simple technique to consider non-linearity. However, due to the very large number of simulations required, this technique is computationally very demanding. Convergence can be improved by using Latin hypercube sampling (McKay et al. 1979).

6.3 WATER QUALITY SUBMODEL AND MEASUREMENT LAYOUTS A hypothetical case study for identifiability and uncertainty analysis was drawn from our case study of the River Glatt (Section 5.2). As in the case study of the River Glatt, we used a simplified version of the RWQM1 that is based on constant benthic densities of bacteria, algae, and consumers. Similarly to Omlin et al. (2001b) and Brun et al. (2002), we divided the uncertainty ranges of the parameters into three classes. Accurately known parameters (class 1), ∆θ j =5%; very poorly known parameters (class 3), ∆θ j =50%; and an intermediate class 2 with ∆θ j =20%. All external and input parameters are assigned to class 1, growth rates and temperature dependence coefficients were assigned to class 2, and all remaining kinetic parameters were assigned to class 3. The class 3 parameters are halfsaturation concentrations and specific death and respiration rates that are very poorly known. Uncertainty in stoichiometric coefficients is not considered in the present analysis. Table 6.1 gives an overview of model parameters (from Appendix 3, with the exception of KI, which was increased in order to account for the use of water surface instead of in situ light intensities). Table 6.2 summarises input and system parameters. Measurement layouts considered in this analysis and scale factors are summarised in Table 6.3. Upstream inflows are assumed to be constant at the values given in Table 6.2. Downstream measurements according to the experimental layouts listed in Table 6.3 are assumed to be used for parameter estimation.

Identifiability and uncertainty analysis

83

Table 6.1: Values, uncertainty classes and uncertainty ranges of the kinetic conversion model parameters (for an explanation of these model parameters, see Chapter 3). Name

Value Uncertainty class range 3 0.050 kdeath,ALG,20 0.1 kdeath,CON,20 0.05 3 0.025 keq,1 100000 3 50000 keq,2 10000 3 5000 keq,N 10000 3 5000 keq,P 10000 3 5000 keq,s0 2 3 1 keq,w 10000 3 5000 kgro,ALG,20 2 2 0.4 kgro,CON,20 0.0002 2 0.00004 kgro,H,aer,20 2 2 0.4 kgro,H,anox,20 1.6 2 0.32 kgro,N1,20 0.8 2 0.16 kgro,N2,20 1.1 2 0.22 KHPO4,ALG 0.02 3 0.01 3 0.01 KHPO4,H,aer 0.02 3 0.01 KHPO4,H,anox 0.02

Unit

Name

Value

d–1 d–1 d–1 d–1 d–1 d–1 gCa/m3/d gH/m3/d d–1 m3/gCOD/d d–1 d–1 d–1 d–1 gP/m3 gP/m3

KO2,ALG KO2,CON KO2,H,aer KO2,N1 KO2,N2 kresp,ALG,20 kresp,CON,20 kresp,H,aer,20 kresp,H,anox,20 kresp,N1,20 kresp,N2,20 KS,H,aer KS,H,anox XALG XCON XH

0.2 0.5 0.2 0.5 0.5 0.1 0.05 0.2 0.1 0.05 0.05 2 2 500 100 200

Uncertainty class range 3 0.1 3 0.25 3 0.1 3 0.25 3 0.25 3 0.05 3 0.025 3 0.1 3 0.05 3 0.025 3 0.025 3 1 3 1 3 250 3 50 3 100

gP/m3

10

3

5

gCOD/m

KHPO4,N1 KHPO4,N2 khyd,20 KI KN,ALG KNH4,ALG KN,H,aer KNH4,N1 KNO2,H,anox KNO2,N2 KNO3,H,anox

gP/m gP/m3 d–1 W/m2 gN/m3 gN/m3 gN/m3 gN/m3 gN/m3 gN/m3 gN/m3

XN1 XH XN1 XN2 XS

200 10 5 100 0.046 0.08 0.07 0.07 0.098 0.069

3 3 3 3 2 2 2 2 2 2

100 5 2.5 50 0.0092 0.016 0.014 0.014 0.0196 0.0138

gCOD/m gCOD/m gCOD/m gCOD/m

0.02 0.02 3 2000 0.1 0.1 0.2 0.5 0.2 0.5 0.5

3 3 3 3 3 3 3 3 3 3 3

0.01 0.01 1.5 1000 0.05 0.05 0.1 0.25 0.1 0.25 0.25

3

βALG βCON βH βhyd βN1 βN2

Unit gO/m3 gO/m3 gO/m3 gO/m3 gO/m3 d–1 d–1 d–1 d–1 d–1 d–1

3

gCOD/m 3 gCOD/m gCOD/m gCOD/m gCOD/m

o

C–1 C–1 o –1 C o –1 C o –1 C o –1 C o

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River Water Quality Model No. 1

Table 6.2: Values, uncertainty classes and uncertainty ranges of input and system parameters. Light intensity was modelled as I = Imax sin(π(0.5+(t mod 1d – 0.5d)/tlight)) for 0.5(1d – tlight) < t mod 1d < 0.5(1d + tlight) and otherwise 0. Name

Value

Unit

Explanation

900 25

Uncertainty class range 1 45 1 1.25

Imax Kst

W/m2 m1/3/s

K2,CO2,20

16

1

0.8

d–1

K2,O2,20

20

1

1

d–1

p

101325

1

5066

Pa

pHin Qin S0 SCa,in

8 5 0.001 60

1 1 1 1

0.4 0.25 0.00005 3

SCl

20

1

1

gCl/m3

SHPO4+H2PO4,in

0.5

1

0.025

gP/m

SK

4

1

0.2

gK/m3

SMg

15

1

0.75

gMg/m3

SNa

15

1

0.75

gNa/m3

SNH4+NH3,in

1

1

0.05

gN/m3

SNO2,in SNO3,in

0.5 5

1 1

0.025 0.25

gN/m3 gN/m3

SO2,in

10

1

0.5

gO/m3

SS,in

10

1

0.5

gCOD/m3

SSO4

20

1

1

gSO4/m3

tlight

0.6

1

0.03

d

Tmax

20

1

1

o

C

Tmin

18

1

0.9

o

C

Maximum light intensity at noon Friction coefficient according to Strickler Gas exchange coefficient for carbon dioxide Gas exchange coefficient for oxygen Atmospheric pressure (used for O2 saturation) pH in inflow River discharge Slope of river bed Calcium concentration in the inflow Chloride concentration (used for conductivity) Phosphate concentration in the inflow Potassium concentration (used for conductivity) Magnesium concentration (used for conductivity) Sodium concentration (used for conductivity) Ammonia concentration in the inflow Nitrite concentration in the inflow Nitrate concentration in the inflow Oxygen concentration in the inflow Dissolved substrate concentration in the inflow Sulphate concentration (used for conductivity) Length of day (sin2 shape of light intensity) Maximum temperature during the day (sin shape) Minimum temperature during the day (sin shape)

m3/s gCa/m3

3

Identifiability and uncertainty analysis

85

Table 6.3: Experimental layouts 1 through 7. The list of variables is assumed to be measured at the downstream site and the values of the scale factors (sci) are used to make the outputs non-dimensional. The scale factors were chosen to be equal to the input concentration at the upstream end of the river reach. Layout 1 SO2

Layout 2 Layout 3 SO2 SO2 SNH4+SNH3 SNH4+SNH3 SNO2

Layout 4 SO2 SNH4+SNH3 SNO2 SNO3

Layout 5 SO2 SNH4+SNH3 SNO2 SNO3 SHPO4+ SH2PO4

Layout 6 SO2 SNH4+SNH3 SNO2 SNO3 SHPO4+ SH2PO4 pH

Layout 7 SO2 SNH4+SNH3 SNO2 SNO3 SHPO4+ SH2PO4 pH Scond

sci 10 mgO/l 1 mgN/l 0.5 mg N/l 5 mgN/l 0.5 mgP/l 8 500 µS/cm

6.3.1 Identifiability analysis Table 6.4 shows the results of sensitivity analysis for the experimental layouts 1–4. The table shows the parameters in decreasing order of δmsqr. Only those parameters with a δmsqr of at least 10% of the maximum of δmsqr are included in the list. Table 6.4: Results for the parameter sensitivities δmsqr for the experimental layouts 1–4. Layout 1 Parameter δmsqr

XALG KI kgro,ALG,20 XH XS XN1 kgro,H,aer,20 khyd,20

0.124 0.081 0.052 0.048 0.017 0.015 0.014 0.014

Layout 2 Parameter δmsqr

XN1 XALG KI kgro,ALG,20 kgro,N1,20 KNH4,N1 XH XCON kresp,ALG,20 kresp,H,aer,20 KNH4,ALG XS

0.119 0.109 0.079 0.049 0.048 0.047 0.038 0.023 0.019 0.018 0.016 0.012

Layout 3 Parameter δmsqr

XN2 XN1 KNO2,N2 XALG KI kgro,N2,20 KNH4,N1 kgro,N1,20 kgro,ALG,20 XH XCON kresp,ALG,20 kresp,H,aer,20 KNH4,ALG

0.126 0.116 0.092 0.091 0.066 0.053 0.047 0.047 0.041 0.031 0.019 0.016 0.015 0.013

Layout 4 Parameter δmsqr

XN2 XN1 KNO2,N2 XALG KI kgro,N2,20 KNH4,N1 kgro,N1,20 kgro,ALG,20 XH XCON kresp,ALG,20 kresp,H,aer,20 KNH4,ALG

0.110 0.101 0.080 0.080 0.058 0.046 0.041 0.041 0.036 0.027 0.017 0.014 0.013 0.012

In Table 6.5 the values of ρ and γ are indicated for the subset with the largest value of ρ for each subset size. In all cases we obtained parameter sets of given size that contained the parameters selected for the smaller subset sizes. For this reason, only the additional parameter is listed in each case in Table 6.5. Horizontal lines divide values for γ below 10 (interpreted as identifiable), and below 15 (interpreted as possibly identifiable) from larger values of γ (interpreted as not identifiable).

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River Water Quality Model No. 1

Table 6.5: Parameter sets (each set contains the parameter in the current row and all parameters in the rows above) with the largest value of ρ of all parameter subsets of given size together with the values of ρ and γ for layouts 1 to 4. The layouts 5-7 essentially lead to the same results as those for layout 4. Layout 1 Layout 2 new Set new ρ γ ρ param. size param. 1 XALG 0.57 1.0 XN1 0.77 2 XH 0.31 1.7 XALG 0.74 3 KI 0.15 14.0 XH 0.47 4 XN1 0.079 15.2 KNH4,ALG 0.30 5 6

0.050 18.2 KI khyd,20 KNH4,ALG 0.029 26.8 KNH4,N1

7 8 9

Layout 3 Layout 4 new new γ ρ γ ρ γ param. param. 1.0 XN2 1.00 1.0 XN2 1.01 1.0 1.0 XN1 1.6 XALG

0.88 0.82

2.2 XH 0.57 0.21 13.7 KNH4,ALG 0.38 0.13 27.6 KI

KNH4,N1 KNO2,N2

1.5 XN1 1.5 XALG

0.90

1.4

0.83

1.5

1.8 XH 0.58 2.5 KNH4,ALG 0.39

1.8 2.4

0.28 13.0 KI

0.30 10.3

0.21 15.2 KNH4,N1 0.14 94.1 XCON

0.22 13.4 0.17 14.8

KNO2,N2

0.13 93.5

The results summarised in Table 6.5 show how the size of the identifiable parameter subset increases with additional measurements for layouts 1 through 3 (from 2–3 to 5–6). However, the addition of four more measured variables in the layouts 4–7 only increases the size of the identifiable parameter subset to a very small degree (to 5–8 already identifiable for layout 4). The essential parameters identifiable from the data are bacteria and algae concentrations responsible for primary production, nitrification (two steps), and respiration. In addition, depending on the parameter values and the measurement accuracy, some halfsaturation concentrations may be identifiable. These poor identifiability results are consistent with the fact that the dominating processes can be described by 5 to 8 parameters and are identifiable from oxygen, ammonia, and nitrite measurements. The dominating processes in the river are primary production, respiration, and two nitrification steps. The additional measurements give information on process stoichiometry, which was assumed to be known in the present application. The results from the identifiability analysis imply that a large effort is necessary to collect highly informative dynamic data in order to make a calibration of RWQM1 possible. Experimental design techniques (Vanrolleghem et al. 1999a) may support the design of adequate measuring campaigns.

6.3.2 Uncertainty analysis Figure 6.1 shows model predictions and uncertainty estimates using prior uncertainties from Tables 6.1 and 6.2 and, for comparative purposes, assuming zero uncertainty for the five identifiable parameters. With the exception of nitrite, the uncertainty bounds using linear error propagation are not very different from those based on Monte Carlo simulation. Estimation of the five identifiable parameters leads to a significant reduction in prediction uncertainty. However, prediction uncertainty still remains very large (with the exception of nitrate, which is not significantly affected by the conversion processes).

Identifiability and uncertainty analysis Ammonia

Oxygen 20

NH4 [mgN/l]

O2 [mgO/l]

25

15 10 5 0 0

0.2

0.4

0.6

0.8

1.4 1.2 1 0.8 0.6 0.4 0.2 0

1

0

0.2

time [d]

NO3 [mgN/l]

NO2 [mgN/l]

0.8 0.6 0.4 0.2 0 0.4

0.8

1

0.6

0.8

0.8

1

0.8

1

7 6 5 4 3 2 1 0 0

1

0.2

0.4

0.6

time [d]

time [d]

pH

Phosphate 10

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

9.5 9 pH

HPO4 [mgP/l]

0.6

Nitrate

Nitrite

0.2

0.4

time [d]

1

0

87

8.5 8 7.5 7

0

0.2

0.4

0.6

time [d]

0.8

1

0

0.2

0.4

0.6

time [d]

Figure 6.1: Downstream (after 10 km) model results (thick solid line) and uncertainty ranges: 95% prior uncertainty estimates with linear error propagation (outer dashed lines), 95% prior uncertainty estimates for lognormal distributions calculated by Monte Carlo simulation (thin solid lines), and 95% uncertainty estimates (inner dashed lines) calculated with linear error propagation assuming no uncertainty for the 5 identifiable parameters.

6.4 CONCLUSIONS The identifiability analysis applied in this chapter and described in more detail in Appendix 5, Brun et al. (2001 and 2002) can readily be applied to the RWQM1. With a relatively small computational expense (calculation of linear sensitivity functions) it provides a significant amount of information for the selection of identifiable parameter subsets, for improving the understanding of model mechanisms and for linear uncertainty analyses. A crucial point of the analysis is the choice of parameter uncertainty ranges and model result scaling factors that are necessary in order to make parameters and results non-dimensional.

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River Water Quality Model No. 1

We analysed a rather steady-state river system (a hypothetical case study) where the only dynamics are induced by the diurnal variation in light intensity. Measurements of oxygen, ammonia, and nitrite over a few days led to the identifiability of only about 5 or 6 kinetic parameters of a simplified version of River Water Quality Model No. 1. The parameters that could be identified are the biomass densities (assumed to be constant) and some saturation coefficients. The availability of data on nitrate, phosphate, pH, dissolved ions, and conductivity does not significantly increase the number of identifiable kinetic parameters (5 to 8). Nevertheless, such measurements are useful in order to identify stoichiometric parameters and to test the model structure (which was assumed to be “correct” in this analysis). More information can only be gained by improved experimental designs, for instance, measurements under more dynamic conditions and/or more difficult measurements, such as the quantification of benthic biomass. Also, microcosm experiments in the laboratory might yield additional information to support the calibration of the RWQM1. An analysis of model results reveals a very large uncertainty in model predictions, which could only partially be decreased with the measurements described above. The result obtained in this simple, hypothetical case study challenges the use of complicated river water quality models as quantitative predictive tools without comprehensive and well-designed measurement campaigns. In particular, more knowledge on the dynamics of benthic biomass is necessary; an aspect that was not even considered in this study. Despite these problems, the models are still useful as knowledge archives, for didactical purposes, and for qualitative or short-term predictions.

7 Summary and future directions

7.1 SUMMARY The presentation of River Water Quality Model No. 1 (RWQM1) in this report is intended to initiate a development process of river water quality models that will create a knowledge base of relevant processes and improve integrated water management that considers wastewater and receiving water systems. The most important points addressed in this report are the following: •

Application of the Petersen matrix to describe the biochemical process model (Chapters 1, 2, 3, 4 and 5) Following its introduction into the wastewater engineering community with the Activated Sludge Model No. 1 (ASM1) (Henze et al. 1987), this form of presentation of biochemical process models became a standard technique in wastewater engineering. Unfortunately this is not the case in river water quality modelling. With the use of this formalism for the RWQM1, we seek to promote the application of this unified and consistent methodology to a wider community of scientists. This methodology significantly improves exchange of information and experience about biochemical models. It also improves communication between wastewater engineers and receiving water chemists and biologists, which is of primary importance from the viewpoint of integrated management

•

Use of elemental mass fractions (Chapter 3) Unlike current practice in wastewater engineering, we decided to use a description of the composition of organic material by elemental mass fractions instead of COD with

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

90

River Water Quality Model No. 1 nitrogen and phosphorus per unit of COD. This choice was made in order to accommodate the practices and needs of receiving water professionals. Note that the two formalisms are equivalent and, assuming the relevant parameters are known, parameters from one formalism can be converted to the other formalism and vice versa. Conversion formulas are given in Chapter 3, while in the numerical example in Appendix 3 the units familiar to wastewater engineers are used. This demonstration of how to switch between measurement units used in the two associated fields and of the parameters that need to be known in order to make such a conversion should improve communication between wastewater and river specialists as indicated above.

•

Presentation of a complex model (Chapter 3) and discussion of how to select a submodel for a specific application (Chapter 4) In order to account for different problems in different rivers, we propose a comprehensive model but also give advice on how to select an adequate submodel for a specific application. Due to the huge variety in river water quality problems to be solved and the variety in the types of rivers, there are still many more model extensions needed (see below). With RWQM1 we have used a similar name as ASM1 as an expression of our hope of initiating a similar development process as achieved for activated sludge models, in which many practitioners have adopted the framework and developed extensions, submodels, and alternative models.

•

Case studies to demonstrate the applicability of the approach (Chapter 5) The application of the model to two case studies shows that the approach is workable and that the selected parameter values lead to reasonable results. Unfortunately, the two case studies are not fully independent tests, as some model parameters are calibrated in one case study and used in the other. Moreover, there is a need for more case studies and testing of RWQM1. Here, as well, we hope to foster a user community that contributes to model development and testing.

•

Sensitivity, identifiability, and uncertainty analysis (Chapter 6) Chapter 6 illustrate three main findings: first, the number of identifiable parameters of RWQM1 or RWQM1 submodels is relatively small and it depends upon the experimental layout; second, the model predictions are highly uncertain in the absence of specific information from a river; and third, a methodology was identified and tested that addresses these questions in different contexts (river characteristics, dominant processes, experimental layouts, etc.).

In spite of this long list of issues addressed in this report, there is still considerable need for further work. We divide these needs into four areas as follows: (1) (2) (3) (4)

additional case studies (see Section 7.2.1); more sensitivity analyses (addressed in Section 7.2.2); better integration with wastewater treatment models (see Section 7.2.3); and model extensions (addressed in Section 7.2.4).

We end the report with a general comment on the model development process (Section 7.2.5).

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7.2 FUTURE DIRECTIONS 7.2.1 Case studies Case studies are the key means for testing models, creating ideas for model improvements and extensions, and for gaining experience with values of model parameters and their dependence on river morphology and external influence factors. If used for a significant number of studies, the framework provided by the RWQM1 can help to facilitate the exchange of information. At the same time, deficiencies of model structures can be found, experience on parameter values can be gained, and model extensions can be tested. Chapter 1 identifies the following list of applications that RWQM1 was intended to be capable of addressing: (1) (2) (3) (4)

dynamic problems of combined stormwater overflows and non-point source pollution; impact of improved wastewater treatment plant operation and control; extreme and surprising pollution events; improved assessment of artificially influenced rivers (for example, by dams or renaturalisation); (5) data collection; (6) structured understanding, research, education, and improved communication (e.g. between wastewater engineers and receiving water quality experts); and, (7) regulatory applications including catchment planning.

To date, very few of these applications have been tested in case studies. Thus, there is a pressing need to evaluate RWQM1 in a wide range of problems from this list and beyond.

7.2.2 Comprehensive sensitivity analyses In Chapter 6, a sensitivity and identifiability analysis is presented for a submodel of the RWQM1 for a river similar to the River Glatt described in the case study in Section 5.2. Such analyses can contribute considerably to the understanding of the behaviour of the model and to knowledge on reasonable parameter values. However, due to the large variety and variability of (a) meaningful submodels (see Chapter 4), (b) river morphology, hydrology, hydraulics, and compartmental structure, (c) type and pattern of emissions, and (d) other characteristics of the problem, more systematic analyses are required. Systematic studies based on idealised hypothetical problems, such as the comparison of QUAL2E and ASM1 by Masliev et al. (1995), are also very useful to learn about the behaviour of models and to evaluate changes or extensions to the model structure. For such analyses, different types of rivers can be created (large, small, shallow, headwater, or midwater streams), preferably starting from a case study with real data (see Chapter 5). Scales, dynamics, and likely dominating components and processes can be changed. To each of these situations a sequence of Petersen matrices can be ordered, from the complex to the simple. Although the procedure can be tedious, the comparison of numerical results usually leads to useful conclusions for the selection of the simplest meaningful conversion model.

7.2.3 Integration with ASM Section 1.4 describes one of the objectives of the RWQM1 model development as “to guarantee compatibility with the IWA Activated Sludge Models to enable integrated analysis

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of wastewater treatment and receiving water quality impacts.” While this objective is achieved in theory by virtue of the model construction, it is in principle possible as shown by the work of Meirlaen et al. (2001a) in which an ASM2d model of a wastewater treatment plant and a sewer model were linked to RWQM1, albeit with some major assumptions. There are no principal differences in the biological and chemical conversion processes between the ASMs and the RWQM1. The basic RWQM1 is far more complex, having more components and processes than the ASM1. The RWQM1 is thus a large model, which can be reduced depending on the actual problem in question, whereas the ASM1 was a minimum model for carbon decomposition and nitrification-denitrification, which could be expanded if needed (as was done with the ASM2 and ASM2d.) The main problem of linking ASM1 with RWQM1 is the possibility for the same state variables in the sewage treatment plant and in the river to have different compositions. If organic material in the river is not dominated by sewage treatment plant effluent, then there may be the need for a different composition, say of degradable organic matter XS in the river and in the sewage treatment plant. In such a case either additional state variables with different composition must be introduced or a conversion process is required to convert the concentrations of the sewage treatment plant effluent to concentrations in the river. This conversion process should be as simple as possible and must account for elemental mass conservation. Thus, the linking of ASM (and appropriate diffuse pollution or stormwater overflow models) with RWQM1 can be performed (although the work needed in order to do this is significant). This creates opportunities for research and legislation alike. In terms of research, the challenging question is, how can we improve stormwater management and treatment plant operation by focusing on ambient water quality impacts? In terms of regulation, the improved cause-effect relations as expressed by the conversion model creates new opportunities in Europe in light of the EU Water Framework Directive and elsewhere.

7.2.4 Enhanced or added model processes While there is a virtual infinity of possible model extensions and enhnacements, the Task Group has identified the following high-priority items.

7.2.4.1 Benthic processes A major difficulty for the application of the RWQM1 model is a lack of recommendations on how to quantify the contribution of the benthic community to changes in the concentration of dissolved and particulate substances in the water column as well as in the sediment pore water. While the equations described in this report can be applied to the benthic community as well as to suspended organisms, the additional processes of detachment and resuspension of sessile organisms or benthic biofilms and of sedimentation and attachment of suspended organisms are crucial for modelling the benthic community (and its effect on substance concentrations). Additionaly, unlike for ASMs, a portion of suspended solids is of watershed origin which is subject to sedimentation and bottom-shear-dependent resuspension. This fraction is mostly inert, but it influences light and sorption conditions. These processes depend much more strongly on the hydromorphology of the river bed than the biochemical processes described in this report. This makes it much more difficult to give universal recommendations. Only an analysis of a large number of case studies can lead to insight about the many-sided dependencies. Due to the introduction of a common notation, the River Water Quality Model No. 1 can contribute to simplifying such an analysis from the viewpoint of water quality impacts.

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We think that an additional module with formulations of detachment, resuspension, sedimentation and attachment processes, and with recommendations of how to estimate parameters for these processes using information on river bed morphology is a high priority for extending the River Water Quality Model No. 1. A bibliography of pertinent references for this task can be found in Appendix 4.

7.2.4.2 Dependence of rate coefficients on environmental factors The temperature dependence of rate coefficients (e.g. growth, respiration and death rates) is very simple in RWQM1, and other dependencies are neglected. The dependence of halfsaturation concentrations or inhibition concentrations on temperature, and the dependence of any of the model parameters on pH (and other external or internal variables) may become important for extended models (e.g. nitrification rates and half-saturation concentrations can vary significantly as a function of pH).

7.2.4.3 Variable mass composition For modelling of conversion processes it is normally assumed that the elemental composition of substrates, products, and organisms is constant. The formalism of the RWQM1 supports the use of a composition of organic compounds that can differ from one case study to the other. However, under some circumstances, a varying composition may be required even within a single case study. Examples for such a need are the process of “luxury” uptake of nutrients under non-limiting conditions and the growth of organisms with a smaller mass fraction of an element under strongly limiting conditions. The luxury uptake process (Nyholm 1977) is a phenomenon wherein algae consume greater quantities of nutrients than needed to support their current rate of growth and store nutrients for future needs. This is well documented in literature of lake eutrophication modeling (for example, Somlyódy and van Straten 1986). The result of luxury uptake is a change in algae biomass composition over time. Variable algae biomass compositions are also well known for different taxonomical groups that may follow distinct succession processes. Neither process is currently accounted for in RWQM1. Changing composition of materials and organisms can be modelled by a structured approach, as done in ΑSΜ2 (Henze et al. 1995) where Phosphate Accumulating Organisms (PAOs) are modelled as a shell of a given organic and nutrient composition. Inside the shell, fractions are present with an elemental composition different from that of the PAO shell itself. The two examples from ΑSΜ2 are the organic storage component, XPHA, representing glycogen and poly-β-hydrοxybutyrate, and the inorganic storage component, XPP, representing polyphosphate. A similar approach was applied to a biogeochemical lake model (Omlin et al. 2001a) in order to model growth of algae with lower phosphorus content in a lake under strongly phosphate-limiting conditions. The separation of organic substrate in RWQM1 into the fractions SS and XS is similar to this, although it is made from a reaction point of view and not specifically intended to model luxury uptake. The two organic fractions allow for varying the nitrogen and phosphorus content in the organic fractions and thus the changing character of the organic matter present in the process. These algorithms could be easily adapted to model luxury uptake in RWQM1.

7.2.4.4 Other processes In RWQM1, it is assumed that oxygen and/or nitrate is always available. If anaerobic processes in the water column or the river sediment are of importance for the turnover of the

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compounds considered in the model, the model must be extended to account for these effects. This would entail modelling the cycling of iron, manganese, and sulphur and production of methane. These processes would be needed only in highly polluted streams in which wastewater degradation has overwhelmed the assimilative capacity. Another extension in state variables may be required to model the limitation of growth of diatoms due to the availability of silica. The effect of macrophytes on the dissolved oxygen may be substantial in some streams. Previous models have incorporated representations of macrophytes, typically by analogy to algae and using similar process descriptions. Examples include Wright and McDonnell (1986), Cerco and Meyers (2000), and Cosgrove and Obropta (1995). Other pertinent references are provided in Appendix 4.

7.2.5 Open-ended model development As we note elsewhere in this report, the Task Group did not have the objective of producing a RWQM1 software. We solely wanted to develop the conversion model and to make it available (see http://www.eawag.ch/~reichert) to interested users who can implement it in their preferred software platform. It should be recognised that simulation of highly non-linear nutrient and oxygen related processes in rivers is an extremely difficult task, requiring considerable experience with model formulation and parameters for different types of rivers. It is hoped that the IWA River Water Quality Model No. 1, together with the open-ended software platform approach and today’s advanced communication tools, will help facilitate the exchange of experience among river water quality professionals worldwide. This process can lead to significant improvements in river water quality models.

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Appendix 1 The river system concept

Figure A1.1 presents a conceptual model of running water ecosystems consisting of abiotic and biotic elements linked within a hydrological continuum. Processes within and between elements are complex and can be described by a series of physico-chemical, hydromorphological, and biological parameters. Hydrology

Morphology

Biological Communities

Land-Water Interface

Physico-Chemistry

Figure A1.1. Structure of running water ecosystems.

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

Appendix 1: the river system concept

103

The abiotic and biotic structures of running waters are characterised by longitudinal, vertical, lateral, and temporal gradients (Table A1.1). This is especially important for the connectivity between surface and subsurface compartments, particularly the hyporheic zone (the zone in fluvial sediments in which the chemistry of the pore fluids is influenced by both ground water and surface water). The upper layers of the hyporheic zone act both as an important habitat for the benthic community (Schwoerbel 1961) and as a reactor with intense metabolism (Fisher and Likens 1973). River water quality modelling approaches must address the ecological characteristics of the river or stream by appropriate consideration of compartments and processes on representative spatial and temporal scales. For example, for headwater streams and midstream regions with coarse substrates, it is necessary to consider both suspended and benthic processes including the hyporheic zone. In contrast, large rivers are more likely to be dominated by transport and conversion processes in the surface flow. Furthermore, there are basic ecological relations of autotrophic and heterotrophic processes with characteristic ratios of Production/Respiration (P/R). Table A1.1: Conceptual description of physico-chemical characteristics and important system compartments of unpolluted running waters (modified from River Continuum Concept; Vannote et al. 1980). Zone

Processes and physico-chemical characteristics

Important compartments

Temperature regime dependent on spring type Temperature fluctuations low Headwater stream

Nutrient import from geogenous sources and bank/riparian vegetation

(Krenal, Epirhithral)

Ratio of Production (P) vs. Respiration (R) P/R < 1 (without allochthonous sources) Most important nutrient source for aquatic communities is allochthonous detritus; carbon import from groundwater

Pelagial

River bed

Hyporheic compartment (groundwater)

Elevated temperature amplitudes. Mid-reach stream

Primary production (PP) from sessile algae, beginning PP in surface flow/channel water

Metarhithral

Primary production from submersed macrophytes (P/R > 1)

Hyporhithral

Elevated nutrient concentrations; Increased variation of oxygen concentrations

Epipotamal

Most important nutrient source: allochthonous and autochthonous detritus

Pelagial

River bed

Hyporheic compartment

Temperature amplitudes balanced Downstream regions

Primary production dominated by phytoplankton with limitation by turbidity/suspended solids

Growth of herbivorous zooplankton MetaP/R <1 /Hypopotamal Oxygen variations smaller; Increased nutrient concentrations Most important nutrient source: autochthonous detritus

Pelagial

Riparian compartment

Flood plain

104 Definitions:

River Water Quality Model No. 1 allochthonous – originating from external sources; autochthonous – originating from internal sources epipotamal – ecological term (based on zoological criteria) for the upstream zones of rivers epirhithral – ecological term (based on zoological criteria) for the upstream zones of streams geogenous – originating from geological sources hypopotamal – ecological term (based on zoological criteria) for the downstream zones of rivers hyporheic – ecological term for the transition zone between surface water and ground water hyporhithral – ecological term (based on zoological criteria) for downstream zones of streams krenal – ecological term (based on zoological criteria) for the spring zones of running waters (creeks) metapotamal – ecological term (based on zoological criteria) for the midstream zones of rivers metarhithral – ecological term (based on zoological criteria) for the midstream zones of streams pelagial – ecological term for the open water compartment of surface waters riparian – ecological term (based on botanical criteria) for the bank compartment of running waters with vegetation dependent on fluctuating water tables

Appendix 2 Formulas for stoichiometric coefficients

Table A2.1 contains the formulas for calculating the stoichiometric coefficients from mass fractions of the organic compounds and stoichiometric parameters as described in Chapter 3. The stoichiometric coefficients have been calculated from mass balances of C, H, O, N, P, X and charge. In order to make a check of mass balances possible, the additional state variables SH2O (water), SN2 (nitrogen gas) and XCaCO3 (calcite) are introduced. The formulas given here are generalisations of the formulas presented in Reichert et al. (2001a). They include the extension introduced in Chapter 3. The formulas presented in Reichert et al. (2001a) can be reproduced from those given here by setting the mass fractions α X to zero (note that SCO2 has been replaced by SHCO3 and that some units are different; this has a minor effect on the formulas, but both formalisms are correct because the equilibrium between SCO2 an SHCO3 is included as process no. 16). The formulas given in Table A2.1 are also implemented in a Microsoft® Excel spreadsheet that can be obtained from the authors (http://www.eawag.ch/~reichert). This facilitates the calculation of stoichiometric coefficients from mass compositions and stoichiometric parameters considerably.

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

106

River Water Quality Model No. 1

Table A2.1: Stoichiometric coefficients of the river model.

Subst. value

Unit +

(1a) Aerobic growth of heterotrophs with NH4 : SS

−

1 YH,a er

gSS/gXH

SNH4

 1  α N,SS − α N, XH    14  YH,a er 

molesNH4+/gXH

SHPO4

 1  α P ,SS − α P , XH   31  YH,a er 

molesHPO42 /gXH

SO2

 1  α H,SS  1  α C,SS  1  α O,SS − α O,XH  −  − α H,XH  −  − α C,XH        32  YH,a er  4  YH,a er  12  YH,a er 

molesO2/gXH

+

–

  3  α N,SS 5  α P,SS − α N,XH  − − α P, XH      56  YH,a er  124  YH,a er 

 β  α X ,SS 1 − α X , XH  −  β + − β H + O   4 8  YH,a er  –

SHCO3

 1  α C,SS − α C, XH    12  YH,a er 

molesHCO3 /gXH

SH

 1  α N,SS  2  α P,SS  1  α C,SS − α C, XH  −  − α N,XH  +  − α P, XH        12  YH,a er  14  YH,a er  31  YH,a er 

molesH+/gXH

 α X ,SS  − β +  − α X , XH   YH,a er 

SH2O

 1  α C,SS  3  α N,SS  1  α H,SS − α H,XH  −  − α C,XH  −  − α N,XH   2  YH,a er  12  YH,a er  28  YH,a er  −

molesH2O/gXH

 1  α X ,SS  3  α P,SS − α P, XH  + (β + − β H ) − α X , XH      62  YH,a er  2  YH,a er 

XH

1

gXH/gXH

X

α X ,SS YH,a er

− α X , XH

gX/gXH

(1b) Aerobic growth of heterotrophs with NO3: SS SNO3

−

1

gSS/gXH

YH,a er

 1  α N,SS − α N, XH   14  YH,a er 

–

molesNO3 /gXH

Appendix 2: formulas for stoichiometric coefficients

107 –

SHPO4

 1  α P ,SS − α P , XH    31  YH,a er 

molesHPO42 /gXH

SO2

 1  α H,SS  1  α C,SS  1  α O,SS − α O,XH  −  − α H,XH  −  − α C,XH   32  YH,a er  4  YH,a er  12  YH,a er 

molesO2/gXH

−

  5  α N,SS 5  α P,SS − α N, XH  − − α P, XH    56  YH,a er  124  YH,a er 

 β  α X ,SS 1 −  β + − β H + O  − α X , XH  4 8  YH,a er  –

SHCO3

 1  α C,SS − α C, XH    12  YH,a er 

molesHCO3 /gXH

SH

 1  α N,SS  2  α P,SS  1  α C,SS − α C, XH  +  − α N,XH  +  − α P, XH   12  YH,a er  14  YH,a er  31  YH,a er 

molesH+/gXH

 α X ,SS  − β+  − α X , XH  Y   H,a er 

SH2O

 1  α C,SS  1  α N,SS  1  α H,SS − α H,XH  −  − α C,XH  −  − α N,XH   12  Y  28  Y  2  YH,a er   H,a er   H,a er  −

molesH2O/gXH

 1  α X ,SS  3  α P,SS − α P, XH  + (β + − β H ) − α X , XH  62  YH,a er 2 Y   H,a er 

XH

1

X

α X ,SS YH,a er

gXH/gXH − α X , XH

gX/gXH

(2,6,8,10,13) Aerobic endogenous respiration of Xi (i=H,N1,N2,ALG,CON): SNH4

1 (α N,Xi − f I,iα N,XI ) 14

molesNH4+/gXi

SHPO4

1 (α P,Xi − f I,iα P,XI ) 31

molesHPO42 /gXi

SO2

1 (α O,Xi − f I,iα O,XI ) − 1 (α H,Xi − f I,iα H,XI ) − 1 (α C,Xi − f I,iα C,XI ) 32 4 12 3 5 (α P,Xi − f I,iα P,XI ) + (α N,Xi − f I,iα N,XI ) − 56 124 β  1 −  β + − β H + O (α X,Xi − f I,iα X,XI ) 4 8 

molesO2/gXi

SHCO3

1 (α C,Xi − f ,iα C,XI ) 12

molesHCO3 /gXi

–

–

108

River Water Quality Model No. 1

SH

1 (α C,Xi − f I,iα C,XI ) − 1 (α N,Xi − f I,iα N,XI ) + 2 (α P,Xi − f I,iα P,XI ) 12 14 31 − β + (α X,Xi − f I,iα X,XI )

molesH+/gXi

SH2O

1 (α H,Xi − f I,iα H,XI ) − 1 (α C,Xi − f I,iα C,XI ) − 3 (α N,Xi − f I,iα N,XI ) 2 12 28 3 1 − (α P,Xi − f I,iα P, XI ) + (β + − β H )(α X,Xi − f I,iα X,XI ) 62 2

molesH2O/gXi

Xi

-1

gXH/gXi

XI

f I,i

X

α X,Xi − f I,iα X,XI

gXI/gXi gX/gXi

(3a) Anoxic growth of heterotrophs with NO3: SS SNO2

−

1 YH,a nox

gSS/gXH

−

 1 α  1 α  1  α O,SS − α O,XH  +  H,SS − α H,XH  +  C,SS − α C,XH   16  YH,a nox  2  YH,a nox  6  YH,a nox 

molesNO2 /gXH

+

  5  α 5  α N,SS − α N,XH  +  P,SS − α P, XH   28  YH,a nox   62  YH,a nox

–

 β  α 1 +  β + − β H + O  X ,SS − α X ,XH  2 8  YH,a nox 

SNO3

  1 α  1 α 1  α O,SS − α O,XH  −  H,SS − α H,XH  −  C,SS − α C,XH   16  YH,a nox   6  YH,a nox  2  YH,a nox −

–

molesNO3 /gXH

  5  α 3  α N ,SS − α N ,XH  −  P,SS − α P, XH  Y 28  YH,a nox 62   H,a nox 

 β  α 1 −  β + − β H + O  X ,SS − α X ,XH  2 8  YH,a nox  –

SHPO4

 1  α P,SS − α P, XH   31  YH,a nox 

molesHPO42 /gXH

SHCO3

 1  α C,SS − α C, XH    12  YH,a nox 

molesHCO3 /gXH

SH

  2 α  1α 1  α C,SS − α C,XH  +  N ,SS − α N ,XH  +  P,SS − α P, XH   12  YH,a nox   31  YH,a nox  14  YH,a nox

molesH+/gXH

α  − β +  X ,SS − α X ,XH  Y  H,a nox 

–

Appendix 2: formulas for stoichiometric coefficients SH2O

 1  α C,SS  1  α N ,SS  1  α H,SS −  −   − − − α α α H, XH C, XH N , XH  12  Y  28  Y  2  YH,a nox   H,a nox   H,a nox  −

XH X

109

molesH2O/gXH

  1 α 3  α P,SS − α P, XH  + (β + − β H ) X ,SS − α X ,XH  62  YH,a nox 2 Y    H,a nox

1

gXH/gXH

α X ,SS YH,a nox

− α X ,XH

gX/gXH

(3b) Anoxic growth of heterotrophs with NO2: SS SNO2

−

1 YH,a nox

 1 α  1 α  1  α O,SS − α O,XH  −  H,SS − α H,XH  −  C,SS − α C,XH   24  YH,a nox  3  YH,a nox  9  YH,a nox  −

gSS/gXH –

molesNO2 /gXH

 1  β  α 5  α P,SS − α P, XH  −  β + − β H + O  X ,SS − α X ,XH   93  YH,a nox 8  YH,a nox  3  –

SHPO4

 1  α P,SS − α P, XH    31  YH,a nox 

molesHPO42 /gXH

SHCO3

 1  α C,SS − α C, XH   12  YH,a nox 

molesHCO3 /gXH

SH

  1 α  1 α 1  α O,SS − α O,XH  −  H,SS − α H,XH  −  C,SS − α C,XH   24  YH,a nox   36  YH,a nox  3  YH,a nox

molesH+/gXH

SH2O

SN2

+

  1 β  α 1  α P,SS − α P, XH  −  4 β + − β H + O  X ,SS − α X ,XH   93  YH,a nox 8  YH,a nox   3

−

  1 α  2 α 1  α O,SS − α O,XH  +  H,SS − α H,XH  −  C,SS − α C,XH   48  YH,a nox   36  YH,a nox  3  YH,a nox

−

  1 β  α 2  α P,SS − α P, XH  +  4 β + − 4 β H + O  X ,SS − α X ,XH   93  YH,a nox 8  YH,a nox   6

−

  1α  1 α 1  α O,SS − α O,XH  +  H,SS − α H,XH  +  C,SS − α C,XH   48  YH,a nox   18  YH,a nox  6  YH,a nox

+

  5  α P,SS 1  α N,SS  − α P, XH  − α N,XH  +   28  YH,a nox   186  YH,a nox

–

molesH2O/gXH

molesN2/gXH

 β  α 1 +  β + − β H + O  X ,SS − α X ,XH  6 8  YH,a nox 

XH

1

gXH/gXH

110 X

River Water Quality Model No. 1 α X ,SS YH,a nox

− α X ,XH

gX/gXH

(4) Anoxic endogenous respiration of heterotrophs: SNH4

1 (α N,X H − f I α N,XI ) 14

molesNH4+/gXH

SNO3

1 (α O,X H − f I α O,XI ) − 1 (α H,X H − f I α H,XI ) − 1 (α C,X H − f I α C,XI ) 40 5 15 3 1 + (α N,X H − f I α N,XI ) − (α P,X H − f I α P,XI ) 70 31 βO  1 −  β+ − βH + (α X ,X H − f I α X ,XI ) 5 8 

molesNO3 /gXH

SHPO4

1 (α P,X H − f I α P,XI ) 31

molesHPO42 /gXH

SHCO3

1 (α C,X H − f I α C,XI ) 12

molesHCO3 /gXH

SH

1 (α O,X H − f I α O,XI ) − 1 (α H,X H − f I α H,XI ) + 1 (α C,X H − f I α C,XI ) 40 5 60 1 1 − (α N,X H − f I α N,XI ) + (α P,X H − f I α P,XI ) 35 31 βO  1 −  6β + − β H + (α X ,X H − f I α X ,XI ) 5 8 

molesH+/gXH

SH2O

1 (α O,X H − f I α O,XI ) + 3 (α H,X H − f I α H,XI ) − 1 (α C,X H − f I α C,XI ) 80 5 20 9 1 − (α N,X H − f I α N,XI ) − (α P, X H − f I α P,XI ) 70 31 β  1 +  6 β + − 6 β H + O (α X ,X H − f I α X ,XI ) 10  8 

molesH2O/gXH

1 (α O,X H − f I α O,XI ) + 1 (α H,X H − f I α H,XI ) + 1 (α C,X H − f I α C,XI ) 80 10 30 3 1 (α N,X H − f I α N,XI ) + (α P,X H − f I α P,XI ) − 140 62 βO  1 +  β+ − βH + (α X ,X H − f I α X ,XI ) 10  8 

molesN2/gXH

XH

-1

gXH/gXH

XI

fI

gXI/gXH

X

α X ,X H − f I α X ,XI

gX/gXH

SN2

−

−

–

–

–

(5) Growth of 1st stage nitrifiers: SNH4 SNO2

−

1 1 14 YN 1

1 1 1 − α N, N 1 14 YN 1 14

molesNH4+/gXN1 –

molesNO2 /gXN1

Appendix 2: formulas for stoichiometric coefficients SHPO4 SO2

SHCO3

111 –

−

1 α P, N 1 31

molesHPO42 /gXN1

−

3 1 1 1 1 3 5 + α C, N 1 + α H, N 1 − α O, N 1 + α N, N 1 + α P, N 1 28 YN 1 12 4 32 56 124

molesO2/gXN1

+

β  1  β + − β H + O α X , N 1 4 8 

−

1 α C, N 1 12

–

molesHCO3 /gXN1

SH

α N, N 1 2α P, N 1 1 1 1 − α C, N 1 − − + β +α X , N 1 7 YN 1 12 14 31

molesH+/gXN1

SH2O

α H, N 1 α N, N 1 3α P, N 1 1 1 1 1 + α C, N 1 − + + − (β + − β H )α X , N 1 14 YN 1 12 2 28 62 2

molesH2O/gXN1

XN1

1

gXN1/gXN1

X

− α X,N 1

gX/gXN1 nd

(7) Growth of 2 stage nitrifiers: SNO2 SNO3 SHPO4 SO2

SHCO3 SH SH2O

−

–

1 1 14 YN 2

molesNO2 /gXN2

1 1 1 − α N, N 2 14 YN 2 14

–

molesNO3 /gXN2 –

−

1 α P, N 2 31

molesHPO42 /gXN2

−

1 1 1 1 1 5 5 + α C, N 2 + α H, N 2 − α O, N 2 + α N, N 2 + α P, N 2 28 YN 2 12 4 32 56 124

molesO2/gXN2

+

β  1  β + − β H + O α X , N 2 4 8 

−

1 α C, N 2 12

molesHCO3 /gXN2

−

α N, N 2 2α P, N 2 1 − + β +α X , N 2 α C, N 2 − 12 14 31

molesH+/gXN2

α H, N 2

molesH2O/gXN2

−

2

+

–

α N, N 2 3α P, N 2 1 1 + − (β + − β H )α X , N 2 α C, N 2 + 12 28 62 2

XN2

1

gXN2/gXN2

X

− α X,N 2

gX/gXN2

(9a) Growth of algae with NH4: SNH4 SHPO4

−

1 α N, ALG 14

molesNH4+/gXALG

−

1 α P, ALG 31

molesHPO42 /gXALG

–

112 SO2

SHCO3 SH SH2O

River Water Quality Model No. 1 1 1 1 3 5 α C, ALG + α H,ALG − α O, ALG − α N,ALG + α P, ALG 12 4 32 56 124 β  1 +  β + − β H + O α X , ALG 4 8 

–

−

1 α C, ALG 12

molesHCO3 /gXALG

−

α N, ALG 2α P, ALG 1 − + β +α X ,ALG α C, ALG + 12 14 31

molesH+/gXALG

α H,ALG

molesH2O/gXALG

−

2

+

3α N,ALG 3α P, ALG 1 1 + − (β + − β H )α X ,ALG α C,ALG + 12 28 62 2

XALG 1 X

molesO2/gXALG

gXALG/gXALG

− α X , ALG

gX/gXALG

(9b) Growth of algae with NO3: SNO3 SHPO4 SO2

SHCO3 SH SH2O

–

−

1 α N, ALG 14

molesNO3 /gXALG

−

1 α P, ALG 31

molesHPO42 /gXALG

–

1 1 1 5 5 α C, ALG + α H,ALG − α O, ALG + α N,ALG + α P, ALG 12 4 32 56 124 β  1 +  β + − β H + O α X , ALG 4 8 

–

−

1 α C, ALG 12

molesHCO3 /gXALG

−

α N,ALG 2α P, ALG 1 − + β +α X ,ALG α C, ALG − 12 14 31

molesH+/gXALG

α H,ALG

molesH2O/gXALG

−

2

+

α N, ALG 3α P, ALG 1 1 + − (β + − β H )α X , ALG α C,ALG + 12 28 62 2

XALG 1 X

molesO2/gXALG

− α X , ALG

gXALG/gXALG gX/gXALG

(11,14) Death of Xi (i= ALG,CON): SNH4

1 (α N,i − (1 − f I ,i )Yi,deathα N,XS − f I ,i Yi,deathα N,XI ) 14

molesNH4+/gXi

SHPO4

1 (α P,i − (1 − f I ,i )Yi,deathα P,XS − f I ,iYi ,deathα P,XI ) 31

molesHPO42 /gXi

–

Appendix 2: formulas for stoichiometric coefficients

113

SO2

1 (α O,i − (1 − f I ,i )Yi,deathα O,XS − f I ,i Yi,deathα O,XI ) 32 1 − (α H,i − (1 − f I ,i )Yi ,death α H, XS − f I ,i Yi ,death α H, XI ) 4 1 − (α C,i − (1 − f I ,i )Yi ,death α C, XS − f I ,i Yi ,death α C, XI ) 12 3 + (α N,i − (1 − f I ,i )Yi,deathα N,XS − f I ,iYi,deathα N,XI ) 56 5 − (α P,i − (1 − f I ,i )Yi,deathα P,XS − f I ,iYi,deathα P,XI ) 124 β  1 −  β + − β H + O (α X ,i − (1 − f I ,i )Yi ,death α X ,XS − f I ,i Yi ,death α X ,XI ) 4 8 

molesO2/gXi

SHCO3

1 (α C,i − (1 − f I ,i )Yi,deathα C,XS − f I ,iYi,deathα C,XI ) 12

molesHCO3 /gXi

SH

1 (α C,i − (1 − f I ,i )Yi ,deathα C,XS − f I ,i Yi ,deathα C,XI ) 12 1 − (α N,i − (1 − f I ,i )Yi ,death α N, XS − f I ,i Yi ,death α N, XI ) 14 2 + (α P,i − (1 − f I ,i )Yi ,death α P, XS − f I ,i Yi ,death α P, XI ) 31 − β + (α X ,i − (1 − f I ,i )Yi ,death α X ,XS − f I ,i Yi ,death α X ,XI )

molesH/gXi

SH2O

1 (α H,i − (1 − f I,i )Yi,deathα H,XS − f I,iYi,deathα H,XI ) 2 1 − (α C,i − (1 − f I,i )Yi,deathα C,XS − f I,iYi,deathα C,XI ) 12 3 − (α N,i − (1 − f I,i )Yi,deathα N,XS − f I,iYi,deathα N,XI ) 28 3 (α P,i − (1 − f I,i )Yi,deathα P,XS − f I,iYi,deathα P,XI ) − 62 1 + (β + − β H )(α X,i − (1 − f I,i )Yi,deathα X,XS − f I,iYi,deathα X,XI ) 2

molesH2O/gXi

Xi

–1

gXi/gXi

XS

( 1 − f I,i )Yi, death

gXS/gXi

XI

f I,iYi,death

gXI/gXi

X

α X,i − (1 − f I,i )Yi,deathα X,XS − f I,iYi,deathα X,XI

gX/gXi

–

(12) Growth of consumers on Xi (i= ALG,S,H,N1,N2): SNH4

fα  1  α N,i  − e N,XS − α N,CON  14  YCON YCON 

molesNH4+/gXCON

SHPO4

fα  1  α P,i  − e P, XS − α P,CON  YCON 31  YCON 

molesHPO42 /gXCON

–

114 SO2

River Water Quality Model No. 1 fα fα  1 α  1  α O,i  − e O,XS − α O,CON  −  H,i − e H,XS − α H,CON  32  YCON YCON YCON  4  YCON  −

fα fα  3 α  1  α C,i  − e C,XS − α C,CON  +  N,i − e N,XS − α N,CON  12  YCON YCON YCON  56  YCON 

−

fα  5  α P,i  − e P, XS − α P,CON  124  YCON YCON 

molesO2/gXCON

fα  β  α 1 −  β + − β H + O  X ,i − e X ,XS − α X ,CON  4 8  YCON YCON  –

SHCO3

fα  1  α C,i  − e C,XS − α C,CON  12  YCON YCON 

molesHCO3 /gXCON

SH

fα fα  1 α  1  α C,i  − e C,XS − α C,CON  −  N,i − e N,XS − α N,CON  12  YCON YCON 14 Y Y CON   CON 

molesH+/gXCON

+

SH2O

Xi

fα fα  α  2  α P,i  − e P, XS − α P,CON  − β +  X ,i − e X ,XS − α X ,CON  31  YCON YCON Y Y CON   CON 

fα fα   1α 1  α H,i  − e H,XS − α H,CON  −  C,i − e C,XS − α C,CON  2  YCON YCON YCON   12  YCON −

fα fα  3 α  3  α N,i  − e N,XS − α N,CON  −  P,i − e P, XS − α P,CON  28  YCON YCON 62 Y Y CON   CON 

+

fα α  1 (β + − β H ) X ,i − e X ,XS − α X ,CON  2 Y Y CON  CON 

−

1 YCON

molesH2O/gXCON

gXi/gXCON

XCON 1

gXCON/gXCON

XS

fe YCON

gXS/gXCON

X

α X ,i YCON

−

f eα X ,XS YCON

− α X ,CON

gX/gXCON

(15) Hydrolysis: SS

YHYD

gSS/gXS

SNH4

1 (α N,XS − YHYDα N,SS ) 14

molesNH4+/gXS

SHPO4

1 (α P,XS − YHYDα P,SS ) 31

molesHPO42 /gXS

–

Appendix 2: formulas for stoichiometric coefficients

115

SO2

1 (α O,XS − YHYDα O,SS ) − 1 (α H,XS − YHYDα H,SS ) − 1 (α C,XS − YHYDα C,SS ) molesO2/gXS 32 4 12 3 5 (α P,XS − YHYDα P,SS ) + (α N,XS − YHYDα N,SS ) − 56 124 β  1 −  β + − β H + O (α X ,XS − YHYDα X ,SS ) 4 8 

SHCO3

1 (α C,XS − YHYDα C,SS ) 12

SH

+ 1 (α C,XS − YHYDα C,SS ) − 1 (α N,XS − YHYDα N,SS ) + 2 (α P,XS − YHYDα P,SS )molesH /gXS 12 14 31 − β + (α X ,XS − YHYDα X ,SS )

SH2O

1 (α H,XS − YHYDα H,SS ) − 1 (α C,XS − YHYDα C,SS ) − 3 (α N,XS − YHYDα N,SS ) molesH2O/gXS 2 12 28 3 1 − (α P, XS − YHYDα P,SS ) + (β + − β H )(α X ,XS − YHYDα X ,SS ) 62 2

XS

-1

gXS/gXS

X

α X ,XS − YHYDα X ,SS

gX/gXS

–

molesHCO3 /gXS

(16) Equilibrium SCO2 – SHCO3: SCO2

–1

–

molesCO2/molesHCO3 –

–

SHCO3 1

molesHCO3 /molesHCO3

SH

1

molesH+/molesHCO3

SH2O

–1

molesH2O/ molesHCO3

– –

(17) Equilibrium SHCO3 – SCO3: –

–

SHCO3 –1

molesHCO3 /molesHCO3

SCO3

1

molesCO32 /molesHCO3

SH

1

molesH+/molesHCO3

–

–

–

(18) Equilibrium SH2O – SH+SOH: SH

1

molesH+/molesH+

SOH

1

molesOH /molesH+

SH2O

–1

molesH2O/molesH+

–

(19) Equilibrium SNH4 – SNH3: SNH4

–1

molesNH4+/molesNH4+

SNH3

1

molesNH3/molesNH4+

SH

1

molesH+/molesNH4+

(20) Equilibrium SH2PO4 – SHPO4: SHPO4 1

–

–

molesHPO42 /molesHPO42

116

River Water Quality Model No. 1 –

SH2PO4 –1

molesH2PO42 – /molesHPO42

SH

molesH+/molesHPO42

1

–

(21) Equilibrium SCa+SCO3 – SCaCO3: SCa

1

molesCa2+/molesCa2+

SCO3

1

molesCO32 /molesCa2+

XCaCO3 –1

–

molesCaCO3/molesCa2+

Appendix 3 A numerical example

The following tables give examples for numerical values of the stoichiometric parameters, the kinetic parameters, and for the resulting stoichiometric coefficients. The numerical values given in these tables are not part of the River Water Quality Model No. 1. Reasonable values were estimated based on literature on the composition of organic material, on the activated sludge models, on existing river water quality models, on the case studies following this paper, and on the experience of the authors. The example is from Reichert et al. (2001a) with minor modifications to fit with the extended model presented in Chapter 3 (the extended model makes compositions with any number of elements possible; in the numerical example, however, only the most important element C, H, O, N, and P are considered). Table A3.1: Mass fractions of elements on organic compounds. SS 0.57

SI 0.61

XH 0.52

XN1 0.52

XN2 0.52

XALG

XCON 0.36

XS 0.57

XI 0.61

Units

0.36

αH

0.08

0.07

0.08

0.08

0.08

0.07

0.07

0.08

0.07

gH/gOM

αO

0.28

0.28

0.25

0.25

0.25

0.50

0.50

0.28

0.28

gO/gOM

αN

0.06

0.03

0.12

0.12

0.12

0.06

0.06

0.06

0.03

gN/gOM

αP

0.01

0.01

0.03

0.03

0.03

0.01

0.01

0.01

0.01

gP/gOM

αX

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

gX/gOM

αC

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

gC/gOM

118

River Water Quality Model No. 1

Table A3.2: Stoichiometric parameters. Symbol YH,aer

Value 0.60

Unit gXH/gSS

Symbol YALG,death

Value 0.62

Unit g(XS+XI)/gXALG

YH,anox,NO3 0.50

gXH/gSS

YCON

0.20

gXCON/gXALG

YH,anox,NO2 0.30

gXH/gSS

fe

0.40

gXS/gXCON

gXI/gXH;N1;N2

fI,CON

0.20

gXI/g(XS+XI)

YCON,death

0.62

g(XS+XI)/gXCON

YHYD

1.00

gSS/gXS

fI,BAC

0.20

YN1

0.13

YN2 fI,ALG

0.03 0.20

gXN1/gSNH4–N gXN2/gSNO2–N gXI/g(XS+XI)

Table A3.3: Chemical equilibria (Stumm and Morgan 1981; Sigg and Stumm 1994; modified, T in ºC). Symbol

Keq,w Keq,1 Keq,2 Keq,N Keq,P Keq,s0

Value

10

Unit

-4470.99/( 273.15 +T ) +12.0875-0.01706(27 3.15+T )

gH2/m6

17.843-3404.71/(273.15+T )-0.032786(273.15+T )

gH/m3

10 10 9.494-2902.39/(273.15+T )-0.02379(273.15+T ) 10 2.891-2727/(273.15+T ) 10 -3.46-219.4/(273.15+T ) 12 ⋅ 40 ⋅ 1019.87-3059/(273.15+T )-0.04035(273.15+T )

gH/m3 gH/m3 gH/m3 gCagC/m6

Table A3.4: Kinetic parameters (T0 is equal to 20 ºC). Symbol

Value

Unit

Symbol

Value

Unit

kdeath,ALG,To kdeath,CON,To kgro,ALG,To kgro,CON,To kgro,H,aer,To kgro,H,anox,To kgro,N1,To kgro,N2,To khyd,To kresp,ALG,To kresp,CON,To kresp,H,aer,To kresp,H,anox,To kresp,N1,To kresp,N2,To keq,1 keq,2 keq,w keq,N keq,P keq,so kads kdes KHPO4,ALG KHPO4,H,aer

0.1 0.05 2.0 0.0002 2.0 1.6 0.8 1.1 3.0 0.1 0.05 0.2 0.1 0.05 0.05 100000 10000 10000 10000 10000 2 – – 0.02 0.02

d–1 d–1 d–1 m3/gCOD/d d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 d–1 gH/m3/d d–1 d–1 gCa/m3/d d–1 d–1 gP/m3 gP/m3

KHPO4,H,anox KHPO4,N1 KHPO4,N2 KN,ALG KNH4,ALG KN,H,aer KNH4,N1 KI KNO3,H,anox KNO2,H,anox KNO2,N2 KO2,ALG KO2,CON KO2,H,aer KO2,N1 KO2,N2 KS,H,aer KS,H,anox βALG βCON βH βhyd βN1 βN2

0.02 0.02 0.02 0.1 0.1 0.2 0.5 500 0.5 0.2 0.5 0.2 0.5 0.2 0.5 0.5 2.0 2.0 0.046 0.08 0.07 0.07 0.098 0.069

gP/m3 gP/m3 gP/m3 gN/m3 gN/m3 gN/m3 gN/m3 W/m2 gN/m3 gN/m3 gN/m3 gO/m3 gO/m3 gO/m3 gO/m3 gO/m3 gCOD/m3 gCOD/m3 ºC–1 ºC–1 ºC–1 ºC–1 ºC–1 ºC–1

Table A3.5: Stoichiometric coefficients based on the parameters given in Tables A3.1 and A3.2. Component → i j Process ↓ 1a Aerobic Growth of Heterotrophs with NH4 1b Aerobic Growth of Heterotrophs with NO3 Aerobic Endogenous Respiration 2 of Heterotrophs 3a Anoxic Growth of Heterotrophs with NO3 3b Anoxic Growth of Heterotrophs with NO2 Anoxic Endogenous Respiration of 4 Heterotrophs st 5 Growth of 1 –stage Nitrifiers st Aerobic Endogenous Respiration of 1 – 6 stage Nitrifiers nd 7 Growth of 2 –stage nitrifiers nd Aerobic Endogenous Respiration of 2 – 8 stage Nitrifiers 9a Growth of Algae with NH4 9b Growth of Algae with NO3 10 Aerobic Endogenous Respiration of Algae 11 Death of Algae 12a Growth of Consumers on XALG 12b Growth of Consumers on XS 12c Growth of Consumers on XH 12d Growth of Consumers on XN1 12e Growth of Consumers on XN2 Aerobic Endogenous Respiration 13 of Consumers 14 Death of Consumers 15 Hydrolysis – 16 Equilibrium CO2 ↔ HCO3 + – 17 Equilibrium H ↔ OH – 2– 18 Equilibrium HCO3 ↔ CO3 + 19 Equilibrium NH4 ↔ NH3 – 2– 20 Equilibrium H2PO4 ↔ HPO4 2+ 2– 21 Equilibrium Ca ↔ CO3 22 Adsorption of Phosphate 23 Desorption of Phosphate Units

1 SS –1.9 –1.9

2 SI

3 SNH4 –0.012

4 SNH3

5 SNO2

6 SNO3

7 8 SHPO4 SH2PO4 –0.0083 –0.012 –0.0083

9 SO2 –0.85 –0.80

0.017

–0.77

0.071 –2.2 –3.7

1.1 –1.6 0.071

–1.1 –0.27

–4.8

4.7

0.071

–0.0062 0.0021 0.017 –0.019

–15

11 SHCO3 0.27 0.27

12 SCO3

13 SH 0.023 0.021

14 SOH

17 XN1

0.25

0.017

–1.0

0.39 0.86

0.032 –0.045

1.0 1.0

0.25

0.0025

–1.0

–0.32

0.65

1.0 –1.0

18 XN2

19 XALG

20 XCON

0.25

0.017

–0.32

–0.033

1.0

0.071

0.017

–0.77

0.25

0.017

–1.0

–0.065 0.058 0.029 0.13 0.13 0.45 0.45 0.45

–0.011 –0.065 –0.011 0.0086 0.0041 0.022 0.022 0.13 0.13 0.13

1.0 1.3 –0.60 0.20 –0.15 –4.8 –3.8 –3.8 –3.8

–0.39 –0.39 0.26 0.0018 0.32 1.5 1.2 1.2 1.2

–0.028 –0.038 0.018 0.0016 0.019 0.11 0.075 0.075 0.075

0.058

0.0086

–0.60

0.26

0.018

–1.0

0.029 0.0

0.0041 0.0

0.20 0.0

0.0018 0.0 1.0 –1.0

0.0016 0.0 0.083 0.083 1.0 0.071 0.032

–1.0

1.0 1.0

–1.0

gN

gN

gN

gN

gP

gO

gC

gC

gC

23 XP

0.23

0.23 1.0 1.0 –1.0 –1.0 –5.0

–8.7 –8.7 –8.7

1.0 1.0 1.0 1.0 1.0

0.95 3.8 –5.8 3.8 3.8 3.8

0.40 0.25

0.40 0.95 –1.0

0.25

1.0

0.30 –1.0 1.0 gP

22 XI

0.23

–22

1.0

21 XS

0.23

–0.77

–1.0

gCOD

16 XH 1.0 1.0

0.017 21

–1.0

gCOD

15 SCa

–0.019

–21

1.0

10 SCO2

1.0

gH

gH

gCa

gCOD

gCOD

gCOD

gCOD

gCOD

gCOD

gCOD

1.0 –1.0 gP

Appendix 4 Bibliography for model enhancements

A4.1 REFERENCES WITH RESPECT TO THE ROLE OF BENTHIC ALGAE Biggs, B.J.F. (1995) The contribution of flood disturbance, catchment geology and land-use to the habitat template of periphyton in stream ecosystems. Freshwater Biol. 33(3), 419–438. Biggs, B.J.F. and Hickey, C.W. (1994) Periphyton responses to a hydraulic-gradient in a regulated river in New-Zealand. Freshwater Biol. 32(1), 49. Biggs, B.J.F., Kilroy, C. and Lowe, R.L. (1998) Periphyton development in three valley segments of a New Zealand grassland river: test of a habitat matrix conceptual model within a catchment. Arch. Hydrobiol. 143(2), 147–177. Biggs, B.J.F., Smith, R.A. and Duncan, M.J. (1999) Velocity and sediment disturbance of periphyton in headwater streams: biomass and metabolism. J. N. Am. Benthol. Soc. 18(2), 222–241. Cerco, C.F. and Seitzinger, S.P. (1997) Measured and modeled effects of benthic algae on eutrophication in Indian River Rehoboth Bay, Delaware. Estuaries, 20(1), 231–248. Francoeur, S.N., Biggs, B.J.F., Smith, R.A. and Lowe, R.L. (1999) Nutrient limitation of algal biomass accrual in streams: seasonal patterns and a comparison of methods. J. N. Am. Benthol. Soc. 18(2), 242–260. Gustina, G.W. and Hoffmann, J.P. (2000) Periphyton dynamics in a sub-alpine mountain stream during winter. Arct. Antarct. Alp. Res. 32(2), 127–134. Peterson, C.G., Weibel, A.C., Grimm, N.B. and Fisher, S.G. (1994) Mechanisms of benthic algal recovery following spates – comparison of simulated and natural events. Oecologia 98(3-4), 280–290. Pfister, P. (1993) Seasonality of macroalgal distribution patterns within the reach of a gravel stream (Isar, Tyrol, Austria). Arch. Hydrobiol. 129(1), 89–107. Uehlinger, U. (1991), Spatial and temporal variability of the periphyton biomass in a pre-alpine river (Necker, Switzerland). Arch. Hydrobiol. 123(2), 219–237.

© 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

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Uehlinger, U. (2000) Resistance and resilience of ecosystem metabolism in a flood-prone river system. Freshwater Biol. 45, 319–332. Uehlinger, U., König, Ch. and Reichert, P. (2000) Variability of photosynthesis-irradiance curves and ecosystem respiration in a small river. Freshwater Biol. 44, 493–507.

A4.2 REFERENCES WITH RESPECT TO THE ROLE OF BENTHIC BACTERIA Battin, T.J. and Sengschmitt, D. (1999) Linking sediment biofilms, hydrodynamics, and river bed clogging: evidence from a large river. Microb. Ecol. 37(3), 185–196. Cabrita, M.T. and Brotas, V. (2000) Seasonal variation in denitrification and dissolved nitrogen fluxes in intertidal sediments of the Tagus estuary. Mar. Ecol. Prog. Ser. 202, 51–65. Romani, A.M. and Sabater, S. (1999) Effect of primary producers on the heterotrophic metabolism of a stream biofilm. Freshwater Biol. 41(4), 729–736. Romani, A.M. and Sabater, S. (2000) Variability of heterotrophic activity in Mediterranean stream biofilms: a multivariate analysis of physical-chemical and biological factors. Aquat. Sci. 62(3), 205–215. Romani, A.M., Butturini, A., Sabater, F. and Sabater, S. (1998) Heterotrophic metabolism in a forest stream sediment: surface versus subsurface zones. Aquat. Microb. Ecol. 16(2), 143–151. Uehlinger, U. and Naegeli, M.W. (1998) Ecosystem metabolism, disturbance, and stability in a prealpine gravel bed river. J. N. Am. Benth. Soc. 17(2), 165–178.

A4.3 REFERENCES WITH RESPECT TO THE ROLE OF BENTHIC MACRO-INVERTEBRATES Bachmann, V. and Usseglio-Polatera, P. (1999) Contribution of the macrobenthic compartment to the oxygen budget of a large regulated river: the Mosel. Hydrobiol. 410, 39–46. Biggs, B.J.F., Kilroy, C. and Lowe, R.L. (1998) Periphyton development in three valley segments of a New Zealand grassland river: test of a habitat matrix conceptual model within a catchment. Arch. Hydrobiol. 143(2), 147–177. Lamberti, G.A., Gregory, S.V., Askhenhas, L.R., Steinman, A.D. and McIntire, C.D. (1989) Productive capacity of algae as determinant of plant-herbivore interactions in streams. Ecol. 70, 1840–1856. Stevenson, R.J., Bothwell, M.L. and Lowe, R.L. (1996) Algal Ecology: Freshwater Benthic Ecosystems, Academic Press, San Diego, CA.

A4.4 REFERENCES WITH RESPECT TO THE ROLE OF BENTHIC COMMUNITIES Biggs, B.J.F., Kilroy, C. and Lowe, R.L. (1998) Periphyton development in three valley segments of a New Zealand grassland river: test of a habitat matrix conceptual model within a catchment. Arch. Hydrobiol. 143(2), 147–177. Cerco, C.F. and Seitzinger, S.P. (1997) Measured and modeled effects of benthic algae on eutrophication in Indian River Rehoboth Bay, Delaware. Estuaries 20(1), 231–248. McIntire, C.D. (1973), Periphyton dynamics in laboratory streams: A simulation model and its implications. Ecol. Monog. 43(3), 399–420. Rauch, W. and Vanrolleghem, P.A. (1998) Modelling benthic activity in shallow eutrophic rivers. Wat. Sci. Tech. 37(3), 1129–137. Rutherford, J.C., Scarbrook, M.R. and Beoekhuizen, N. (2000), Grazer control of stream algae: Modeling temperature and flood effects. J. Environ. Eng., ASCE 126, 331–339. Uehlinger, U., Bührer, H. and Reichert, P. (1996), Periphyton dynamics in a flood-prone pre-alpine river: Evaluation of significant processes by modelling. Freshwater Biol. 36, 249–263.

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A4.5 REFERENCES WITH RESPECT TO THE EFFECT OF SUSPENDED SOLIDS ON LIGHT PENETRATION AND ADSORPTION Goldsborough, W.J. and Kemp, W.M. (1988) Light responses of a submersed macrophyte: Implications for survival in turbid waters. Ecology 69, 1775–1786. Huisman, J. and Weissing, F.J. (1994) Light-limited growth and competition for light in well-mixed aquatic environments: An elementary model. Ecology 75, 507–520.

A4.6 REFERENCES WITH RESPECT TO MODELLING AQUATIC MACROPHYTES Cerco, C.F. and Meyers, M. (2000) Tributary refinements to Chesapeake Bay model. J. Environ. Eng. ASCE 124(2), 164–174. Collins, C. and Wlosinski, J.H. (1989) A macrophyte submodel for aquatic ecosystems. Aquat. Bot. 33, 191–206. Cosby, B.J. and Hornberger, G.M. (1984) Identification of photosynthesis-light model for aquatic systems. I. Theory and simulations. Ecol. Model. 22, 1–24. Cosby, B.J., Hornberger, G.M. and Kelly, M.G. (1984) Identification of photosynthesis-light model for aquatic systems. II. Application to a macrophyte dominated stream. Ecol. Model. 22, 25–51. Cosgrove, J.F. and Obropta, C.C. (1995) Modeling the impact of macrophytes on instream dissolved oxygen dynamics in the Pennsauken watershed. In Integrated Water Resources Planning for the 21st Century, Proceedings of the 22nd Annual Conference, Cambridge, MA, 7–11 May, pp. 504– 507, American Society of Civil Engineers, New York. Kemp, W.M., Boynton, W.R. and Hermann, A.J. (1995) Simulation models of an estuarine macrophyte ecosystem. In Complex Ecology (ed. B.C. Patten), pp. 262–277, Prentice Hall, Englewood Cliffs, NJ. Madden, C.J. and Kemp, W.M. (1996) Ecosystem model of an estuarine submersed plant community: calibration and simulation of eutrophication responses. Estuaries 19(2B), 457–474. Muhammetoğlu, A. and Soyupak, S. (2000) A three-dimensional water quality-macrophyte interaction model for shallow lakes. Ecol. Model. 133, 161–180. Wiland, B.L. and LeBlanc, K. (2000) LA-QUAL for Windows, User’s Manual, Model Version 3.02, Manual Rev. B. Louisiana Department of Environmental Quality, Baton Rouge, LA. Wright, R.M. and McDonnell, A.J. (1986) Macrophyte growth in shallow streams: field investigations. J. Environ. Eng. ASCE 112(5), 953–966. Wright, R.M. and McDonnell, A.J. (1986) Macrophyte growth in shallow streams: biomass model. J. Environ. Eng. ASCE 112(5), 967–982.

Appendix 5 Definition of identifiability measures

In this appendix, we briefly review a recently proposed technique for identifiable parameter subset selection for large environmental simulation models. This technique can be viewed as a quantification of the visual assessment of sensitivity functions (see Holmberg 1982, Reichert et al. 1995, and Dochain and Vanrolleghem 2001, among others). The technique combines quantitative identifiability measures derived from the Fisher Information Matrix (Hidalgo and Ayesa 2001; Söderström and Stoica 1989; Walter and Pronzato 1990; Weijers and Vanrolleghem 1997) with measures used to quantify approximate linear dependence of sensitivity functions (Belsley 1991; Brun et al. 2001). Some of these identifiability measures have been calculated for identifiable parameter subset selection for ASM1 (Henze et al. 2000) by Weijers and Vanrolleghem (1997) and for a biogeochemical model of a lake by Omlin et al. (2001a,b). The technique as described here follows the presentations by Brun et al. (2001 and 2002); the application given in Chapter 6 follows Reichert and Vanrolleghem (2001). We assume the outputs of a deterministic model to be described by the function

y (θ)

(A5.1)

where y = ( y1 ,..., y n ) T is the vector of model outcomes that correspond to the measured output variables and θ = (θ1,…, θm)T is the vector of model parameters. Note that for state– space models this equation combines the state equations with the observation equations. In addition to the deterministic model, this equation also contains the experimental layout © 2001 IWA Publishing. River Water Quality Model No. 1. Edited by IWA Task Group on River Water Quality Modelling. ISBN: 1 900222 82 5.

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specifying which variables are measured at which points in space and time. The observation vector y contains all observed variables at all observation locations in space and time. The mathematical form of the model is not relevant. It can consist of algebraic equations, ordinary or partial differential equations, and/or integral equations. The technique for the selection of identifiable model parameter subsets is based on sensitivity functions. It is useful to define the following two non-dimensional sensitivity functions:

s i, j =

∆θ j ∂y i sc i ∂θ j

~ si , j =

,

si , j

(A5.2)

n

∑s k =1

2 k, j

~

si , j ) , and the column vectors We also define the matrices S = ( si , j ) and S = ( ~ s j = ( s1, j ,..., s n , j ) T and

~s = ( ~ s1, j ,..., ~ sn, j )T . Note that the vectors j

s j by normalisation: ~s j = s j

~s are constructed from j

s j . The dimensional scaling parameters in this equation,

∆θ j and sci , are very important for the analysis. ∆θ j represents the uncertainty range of the parameter θ j according to prior knowledge and sci is a characteristic scale of the variable yi . Note that the specification of parameter uncertainties is essential for estimating the range of variation of the model results and the scaling of model results is important to make different model outputs numerically comparable with each other. If the number of measurements for different variables differs significantly, this could also be considered by an appropriate choice of the scaling factor. Note that the normalised sensitivity functions, ~ sj , are independent of ∆θ j and of a common multiplicative factor in all sci . Poor identifiability of model parameters can be caused by a small sensitivity of the model results to a parameter, or by an approximate linear dependence of sensitivity functions of the results with respect to the parameters. The technique proposed by Brun et al. (2001) is based on two different measures of sensitivity and “compensability” to quantify the two problems separately. First a sensitivity ranking of parameters is done using

(θ) = δ msqr j

1 n 1 2 sj ∑ si, j = n i =1 n

(A5.3)

Appendix 5: definition of identifiability measures

125

Then the “compensability” is quantified for interesting parameter subsets by the “collinearity index”

γ (θ) =

min

β =1

1 1 ~s β + ... + ~s β = ~ = min β =1 Sβ m m 1 1

1

( [ ])

~ ~ min EV S T S

(A5.4)

(EV[.] represents the eigenvalues of the argument) which quantifies the minimum achievable norm of a linear combination of the normalised sensitivity functions with normalised coefficients (γ is unity for linearly independent sensitivity functions and approaches infinity with increasing degree of linear dependence). Note that each δmsqr characterises a single parameter whereas the value of γ gives information on problems due to "compensability" within the subset of parameters. As a third identifiability measure we use the 2mth root of the determinant of the Fisher information matrix (FIM)

(

ρ (θ) = det[S T S]

)

1 / 2m

(A5.5)

(we calculate the Fisher information matrix using the same scaling as used above). ρ is a measure of the (linear) extension of the confidence region if the parameters would be estimated by the method of least squares. The values of ρ decrease with increasing size of the confidence region. This extension of the confidence region combines information on sensitivity (as it can be quantified with δmsqr) with information on “compensability” (as it can be quantified with γ). Subsets of best identifiable parameters can be selected as follows. First a ranking of sensitivities δmsqr leads to a first insight in important parameters. At this step, parameters with a very low sensitivity can be eliminated in order to reduce the number of parameter combinations for the subsequent steps. For all subsets of size 2, 3, 4, etc. of the remaining parameters, the identifiability measures γ and ρ are calculated. The subsets for each size are then ordered according to decreasing values of ρ. This ranking then reflects the decreasing degree of identifiability of the parameter sets. The values of γ for these sets can then be used to define the threshold for identifiability. Empirical experience indicates that this threshold is in the range of values of 10 to 15.

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Scientific and Technical Report No. 12

As background to the development of River Water Quality Model No. 1, the Task Group completed a critical evaluation of the current state of the practice in water quality modelling. A major limitation in present model formulations is the continued reliance on BOD as the primary state variable, despite the fact that BOD does not include all biodegradable matter. A related difficulty is the poor representation of benthic flux terms. As a result of these shortcomings, it is impossible to close mass balances completely in most existing models. These various limitations in current river water quality models impair their predictive ability in situations of marked changes in a river's pollutant load, streamflow, morphometry, or other basic characteristics. RWQM1 is intended to serve as a framework for river water quality models that overcome these deficiencies in traditional water quality models and most particularly the failure to close mass balances between the water column and sediment. In addition, the model is intended to be compatible with the existing IWA Activated Sludge Models (STR 9: Activated Sludge Models ASM1, ASM2, ASM2d and ASM3; ISBN: 1900222248) so that it can be straightforwardly linked to them. To these ends, the model incorporates fundamental water quality components and processes to characterise carbon, oxygen, nitrogen, and phosphorus (C, O, N, and P) cycling instead of biochemical oxygen demand as used in traditional models. The model is presented in terms of processes and components represented via a Petersen stoichiometry matrix, the same approach used for the IWA Activated Sludge Models. The full RWQM1 includes 24 components and 30 processes. The report provides detailed examples on reducing the numbers of components and processes to fit specific water quality problems. Thus, the model provides a framework for both complicated and simplified models. Detailed explanations of the model components, process equations, stoichiometric parameters, and kinetic parameters are provided, as are example parameter values and two case studies. The STR is intended to launch a participatory process of model development, application, and refinement. RWQM1 provides a framework for this process, but the goal of the Task Group is to involve water quality professionals worldwide in the continued work developing a new water quality modelling approach. This text will be an invaluable reference for researchers and graduate students specializing in water resources, hydrology, water quality, or environmental modelling in departments of environmental engineering, natural resources, civil engineering, chemical engineering, environmental sciences, and ecology. Water resources engineers, water quality engineers and technical specialists in environmental consultancy, government agencies or regulated industries will also value this critical assessment of the state of practice in water quality modelling. ISBN: 1 900222 82 5

ISSN: 1025-0913

Scientific and Technical Report No. 12 River Water Quality Model No.1

T

his Scientific and Technical Report (STR) presents the findings of the IWA Task Group on River Water Quality Modelling (RWQM). The task group was formed to create a scientific and technical base from which to formulate standardized, consistent river water quality models and guidelines for their implementation. This STR presents the first outcome in this effort: River Water Quality Model No. 1 (RWQM1).

RIVER WATER QUALITY MODEL NO.1 BY BY

IWA IWA T TASK ASK G GROUP ROUP ON ON R RIVER IVER W WATER ATER Q QUALITY UALITY M MODELLING ODELLING