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A mathematical deterministic river-quality model. Part 1: Formulation and description
Wttter Re.~earch Vol. 12. pp. 1149 to 1153 Pergamon Press Ltd. 1978. Printed in Great Britain
A MATHEMATICAL DETERMINISTIC RIVER-QUALITY MODEL. PART 1: F O R M U L A T I O N A N D DESCRIPTION G. KNOWLES and A. C. WAKEFORD Water Research Centre, Stevenage Laboratory, Elder Way, Stevenage, Herts, SGI ITH (Received 10 September 1977; in revised form 14 June 1978)
Abstract--A comprehensive river water-quality model, deterministic in type, is described in Part 1 and its application to a river catchment is given in Part 2, for which there is a separate abstract. The eleven processes represented in the model include nitrification, denitrification, photosynthesis, BOD decay, re-aeration and others, the rate equations for each being detailed, as well as the numerical integration employed to solve these. A choice of FORTRAN computer programs is available, each embodying the model--one being for "interactive" use from a computer terminal, one for conventional "batch" use at any computer centre, while a third form is envisaged for incorporation in a minicomputer or microprocessor controlling remote automatic water-quality monitoring stations. This third form would allow predictions of downstream quality immediately data are received from each station, so enabling early warning of impending substandard conditions.
INTRODUCTION River-quality models are perhaps most usefully applied in long-term planning to maintain or improve the quality of rivers, and also in the daily management of river quality. The model described here was tested by Casapieri et al. (1978), against data recorded for the Blackwater river catchment and, following good agreement, used to predict the effects of possible changes in population and sewage-treatment practices. Their work is the subject of Part 2 of this paper. Although the application of such models to day-today management needs investigating, their use may possibly allow the amount of manual sampling to be reduced; and where automatic monitoring stations are concerned, a reduction in the number of stations is possible because a model could be applied to the virtually continuous stretch downstream of a selected point. Predictions can include concentrations of dissolved oxygen, BOD, ammonia, and nitrate for the whole length of a river if so desired, and the computer program representing the model can be applied to any river because the user is able to define a particular river by means of the data he supplies to the program for a "run".
GENERAL DESCRIPTION The model simulates a body of water travelling downstream and sums the effects of eleven river processes on its quality, usually for each hour of travel. The rate equations used for these river processes are detailed in the Appendix and include photosynthesis, plant respiration, BOD, river-bed respiration, nitrification, denitrification, re-aeration through the surface, dissolved-oxygen changes at weirs, and settling and resuspension of particulate BOD, any or all of which can be selected for a particular "run". Numerical in-
tegration of the rate equations using the Runge-Kutta method (Kuo, 1965) is necessary since they form an interlinked set too complex to be integrated by simple methods, the only action required of the user being the specification of the time step for the numerical integration. Sufficient accuracy is obtained using a typical value of 1 h with a short computer run, but a smaller time step is appropriate if it is thought that any of the processes may be unusually rapid at any point along the river. The model is thus largely "deterministic" in type, though statistical in the sense that the separate rate equations are based on field work described by the authors quoted in the Appendix. It is essentially a steady-state model with the refinement that varying light intensity is taken into account because the model, in effect, follows the movement of a body of water along a river. It seems likely that such stochastic models would be less reliable than deterministic ones for predicting conditions substantially different from those during which the values and loads were measured. Usually too the stochastic approach will be less convenient because the relationships between loads and qualities have to be established by data collection and statistical analysis for the particular rivers, while deterministic models could be applied to any river without collection of quality data except a small amount for checking purposes, such as from one or two 24-h sampiing surveys. A simplified flow chart is given in Fig. 1 for the computer program representing the model described in this paper. This model does not include longitudinal mixing or the effect of hydraulic changes such as fluctuations in depth caused by variable flow: however, the authors have also developed models which include those processes, but the use of such models requires computer run times which are far longer than
1149
1150
G. KNOWLESand A. C. WAKEFORD
//o'Sf;o°',,%7=:o;,°0.// I
"
C o n c e n t r a t i o n s calculated after mixing of t r i b u t a r y
or effluent with river
=='°°1
NO
Fig. 1. River simulation. Flow chart of simple "'batch" computer program. those required by the present model, which appears to be adequate for most purposes. Only in one special (but perhaps important) case is longitudinal dispersion likely to be very significant, which is the prediction of concentrations of toxic chemicals downstream of accidental spillages, and it would probably be worthwhile to keep a special computer program of a mathematical model including longitudinal dispersion. in readiness to handle predictions of this kind. Such a model, described by Owens. Knowles & Clark (1969), was found to give good agreement with the moving and flattening concentration/time curves observed in a river after addition of a dose of radioactive tracer. Modelling of longitudinal dispersion makes the differential equations "'partial" ones (i.e. more than one independent variable), which can be solved--though often at the expense of much computer time--by methods given in texts by Ames (1969) and Mitchell 0969). FORMS OF T H E RIVER-QUALITY M O D E L
Two forms of the computer program representing the river-quality model are available, while a third, for working in "real time", is envisaged. The first available form of the computer program is a "batch" version in FORTRAN usable in nearly every computer centre; the second is an "interactive" version, again in FORTRAN, suitable for nearly every time-sharing system--for example, it can be operated from a teletype linked by a telephone to a computer bureau. The envisaged third form would be for small computers or microprocessors that control automatic river-quality monitors. This "monitor" version would
make use of the quality data as they were acquired and make immediate predictions of downstream conditions which would be valid for the following few hours or even days, in some instances, the period depending on the accuracy with which downstream effluents can be estimated in advance. M E T H O D OF USING THE RIVER-QUALITY MODEL, "BATCH" OR "INTERACTIVE" VERSION
The user divides the river into stretches, a new stretch commencing wherever a tributary of effluent enters or wherever there is a weir or significant change in the depth of the river, its cross-section, plant density, or the constants in rate equations. In the model it is supposed that a body of water is moving down river and the user has to supply the time (GMT) at which the body of water starts its journey. In addition, depending on which river processes the user wishes to include, he must supply some or all of the following: volume rate of flow, starting concentrations of dissolved oxygen, BOD and other variables, rate-equation constants (values for which are suggested in the Appendix) maximum light intensity, duration of daylight, and the time step--see above-for the integration of the rate equations. As the progress of the theoretical body of water down a river is simulated in the program, the values predicted for dissolved oxygen, BOD, ammonia, and nitrate for each hour of travel, together with the corresponding distance down the river, and the time (GMT) are printed out. Flow rates and polluting loads of tributaries and effluents are also required, and when using the "batch" version the user has to supply these for each of the 24 h of the day to enable the program, which keeps account of time, to interpolate a value for each item whenever the supposed body of water encounters a tributary or effluent. When using the "interactive" version, however, the program itself asks for the flow and polluting load for each tributary and effluent, in a message on the teletype or visual display unit, immediately after it has printed or displayed the predictions for the preceding stretch, the requested information being typed in by the user, or "read" in from prepunched paper tape, unless he decides to discontinue the "run". Again, as the program follows the progress of the supposed body of water, calculating its changes in quality for each time step--usually 1 h--of travel, account has to be taken of the entries of tributaries and effluents. In order to add the contributions of these to the quality of the body of water, the time at which the beginning of each stretch is reached has to be available to the simulation since it is here that a tributary or effluent enters. In the WRC programs, the time (GMT) at the beginning of each stretch is worked out by the programs from the flows and the length of each stretch taken in conjunction with their average depth and width, all these being supplied as data for the "run" by the user: how-
A mathematical deterministic river-quality--I ever, the authors of Part 2 (Casapieri et al.) of this paper modified the program so that depths and widths were no longer required, times at the beginning of each stretch now being found by the program from the user-supplied "time-of-travel" along each stretch, these times-of-travel already being known from tracer studies of the river. The "batch" version of the program has the additional feature of being able to print out the dissolved oxygen (mg 1-1) added to removed by the relevant processes, viz BOD oxidation, photosynthesis, reaeration, plant respiration, bed respiration, ammonia oxidation, and the milligrams of nitrogen per litre lost by denitrification for each hour of travel. This gives a clear picture of the comparative magnitude of these processes, and this information as well as the values of dissolved oxygen and other qualities can also be obtained as a graph produced on the conventional computer-printing devices; special graph-plotting features are not required. The features mentioned in the last sentence are not provided by the "interactive" version, when the usual narrowwidth teletype is used, but are available if a full-width (130 characters) printing device is employed. The rate equations and the associated suggested values for their constants are from published work, this being comprehensive for some of the processes, though further experimental and field work seem desirable, to confirm or improve the representations of settling and resuspension. In testing the model by seeing how well it would have predicted observed data, there can be individual values or period averages. If period averages are used, then the values supplied to the model for flows, loads, depths, temperatures, and other qualities can likewise be period averages.
1151
along the whole length of any river, by taking into account the results of a wide range of river processes, including photosynthesis, bed respiration, and the effects of weirs and entry of effluents and tributaries. This model, which is in the form of a computer program, was checked by the Thames Water Authority against past results for a river catchment (Casapieri et al.) and used for their long-term planning. It should also be useful for day-to-day management of river quality. An "interactive" version, in FORTRAN, allows predictions to be made at a few minutes' notice using time-shared computers via the ordinary G P O STD telephone system, while a "batch" version, also in FORTRAN, is suitable for more conventional computer processing. A "monitor" version could quickly be created for placing in any type of processor used for controlling automatic river waterquality monitors, to allow continuous automatic prediction of conditions in the stretches of rivers between monitors. In the mathematical model, the effects of eleven major river processes on the qualities of a supposed body of water are repeatedly summed as the body of water moves down the river, numerical integration being employed to obtain the actual concentration changes from the rate equations for the river processes. Predictions for dissolved oxygen, BOD, ammonia, and nitrate are printed for the end of each stretch of the river, together with a detailed breakdown of the dissolved oxygen 1 # contributed or removed by each process. Acknowledgement--This paper is published by permission of the Director of the Water Research Centre.
REFERENCES
SENSITIVITY ANALYSIS Though sensitivity analysis is desirable, the processes are in this case so interlinked that a long discussion would be required if generalizations were not to be misleading. However, it is of interest to note that nitrification and plant respiration can sometimes have a major effect on the dissolved-oxygen concentration in smaller rivers; this is seen in results from the model, the output from which includes the addition of or consumption of oxygen for each relevant process for each stretch of river. Likewise the interaction between equations makes it difficult to estimate errors resulting from changes in rate constants in the equations in the Appendix, but it is hoped later to quantify effects on water-quality predictions caused by these changes, in typical cases.
Ames W. F. (1969) Numerical Methods for Partial Differential Equations. Nelson, London. Casapieri P., Fox T. M., Owers P. & Thomson G. D. (1978) A mathematical deterministic river-quality model. Part 2: Use in evaluating the water quality of the Blackwater catchment. Wat. Res. 12, 1155-1161. Kuo S. S. (1965) Numerical Methods and Computers. Addison-Wesley, Reading, Mass., USA. Mitchell A. R. (1969) Computational Methods in Partial Differential Equations. Wiley, New York. Owens M., Knowles G. & Clark A. (1969) The prediction of the distribution of dissolved oxygen in rivers. Proc. 4th Int. Conf. War. Pollut. Res., pp. 125-137 and 139-147. Pergamon Press, Oxford. APPENDIX: PROCESSES MODELLED AND EQUATIONS USED
1. Rate of change in 5-day BOD
(a) Rate of change due to oxidation (1)
CONCLUSIONS
= - 1 . 0 4 7 ( r - 2 ° ~ ( K B ) ( B O D ) m g l - 1 m i n -1
A deterministic river model developed at the Water Research Centre can be used to predict concentration of dissolved oxygen, BOD, ammonia and nitrate
where BOD is concentration in mg 1-t, K B is BOD decay constant (approximately 0.000139 for equation 1), and T is temperature in °C.
A MATHEMATICAL DETERMINISTIC RIVER-QUALITY MODEL. PART 1: F O R M U L A T I O N A N D DESCRIPTION G. KNOWLES and A. C. WAKEFORD Water Research Centre, Stevenage Laboratory, Elder Way, Stevenage, Herts, SGI ITH (Received 10 September 1977; in revised form 14 June 1978)
Abstract--A comprehensive river water-quality model, deterministic in type, is described in Part 1 and its application to a river catchment is given in Part 2, for which there is a separate abstract. The eleven processes represented in the model include nitrification, denitrification, photosynthesis, BOD decay, re-aeration and others, the rate equations for each being detailed, as well as the numerical integration employed to solve these. A choice of FORTRAN computer programs is available, each embodying the model--one being for "interactive" use from a computer terminal, one for conventional "batch" use at any computer centre, while a third form is envisaged for incorporation in a minicomputer or microprocessor controlling remote automatic water-quality monitoring stations. This third form would allow predictions of downstream quality immediately data are received from each station, so enabling early warning of impending substandard conditions.
INTRODUCTION River-quality models are perhaps most usefully applied in long-term planning to maintain or improve the quality of rivers, and also in the daily management of river quality. The model described here was tested by Casapieri et al. (1978), against data recorded for the Blackwater river catchment and, following good agreement, used to predict the effects of possible changes in population and sewage-treatment practices. Their work is the subject of Part 2 of this paper. Although the application of such models to day-today management needs investigating, their use may possibly allow the amount of manual sampling to be reduced; and where automatic monitoring stations are concerned, a reduction in the number of stations is possible because a model could be applied to the virtually continuous stretch downstream of a selected point. Predictions can include concentrations of dissolved oxygen, BOD, ammonia, and nitrate for the whole length of a river if so desired, and the computer program representing the model can be applied to any river because the user is able to define a particular river by means of the data he supplies to the program for a "run".
GENERAL DESCRIPTION The model simulates a body of water travelling downstream and sums the effects of eleven river processes on its quality, usually for each hour of travel. The rate equations used for these river processes are detailed in the Appendix and include photosynthesis, plant respiration, BOD, river-bed respiration, nitrification, denitrification, re-aeration through the surface, dissolved-oxygen changes at weirs, and settling and resuspension of particulate BOD, any or all of which can be selected for a particular "run". Numerical in-
tegration of the rate equations using the Runge-Kutta method (Kuo, 1965) is necessary since they form an interlinked set too complex to be integrated by simple methods, the only action required of the user being the specification of the time step for the numerical integration. Sufficient accuracy is obtained using a typical value of 1 h with a short computer run, but a smaller time step is appropriate if it is thought that any of the processes may be unusually rapid at any point along the river. The model is thus largely "deterministic" in type, though statistical in the sense that the separate rate equations are based on field work described by the authors quoted in the Appendix. It is essentially a steady-state model with the refinement that varying light intensity is taken into account because the model, in effect, follows the movement of a body of water along a river. It seems likely that such stochastic models would be less reliable than deterministic ones for predicting conditions substantially different from those during which the values and loads were measured. Usually too the stochastic approach will be less convenient because the relationships between loads and qualities have to be established by data collection and statistical analysis for the particular rivers, while deterministic models could be applied to any river without collection of quality data except a small amount for checking purposes, such as from one or two 24-h sampiing surveys. A simplified flow chart is given in Fig. 1 for the computer program representing the model described in this paper. This model does not include longitudinal mixing or the effect of hydraulic changes such as fluctuations in depth caused by variable flow: however, the authors have also developed models which include those processes, but the use of such models requires computer run times which are far longer than
1149
1150
G. KNOWLESand A. C. WAKEFORD
//o'Sf;o°',,%7=:o;,°0.// I
"
C o n c e n t r a t i o n s calculated after mixing of t r i b u t a r y
or effluent with river
=='°°1
NO
Fig. 1. River simulation. Flow chart of simple "'batch" computer program. those required by the present model, which appears to be adequate for most purposes. Only in one special (but perhaps important) case is longitudinal dispersion likely to be very significant, which is the prediction of concentrations of toxic chemicals downstream of accidental spillages, and it would probably be worthwhile to keep a special computer program of a mathematical model including longitudinal dispersion. in readiness to handle predictions of this kind. Such a model, described by Owens. Knowles & Clark (1969), was found to give good agreement with the moving and flattening concentration/time curves observed in a river after addition of a dose of radioactive tracer. Modelling of longitudinal dispersion makes the differential equations "'partial" ones (i.e. more than one independent variable), which can be solved--though often at the expense of much computer time--by methods given in texts by Ames (1969) and Mitchell 0969). FORMS OF T H E RIVER-QUALITY M O D E L
Two forms of the computer program representing the river-quality model are available, while a third, for working in "real time", is envisaged. The first available form of the computer program is a "batch" version in FORTRAN usable in nearly every computer centre; the second is an "interactive" version, again in FORTRAN, suitable for nearly every time-sharing system--for example, it can be operated from a teletype linked by a telephone to a computer bureau. The envisaged third form would be for small computers or microprocessors that control automatic river-quality monitors. This "monitor" version would
make use of the quality data as they were acquired and make immediate predictions of downstream conditions which would be valid for the following few hours or even days, in some instances, the period depending on the accuracy with which downstream effluents can be estimated in advance. M E T H O D OF USING THE RIVER-QUALITY MODEL, "BATCH" OR "INTERACTIVE" VERSION
The user divides the river into stretches, a new stretch commencing wherever a tributary of effluent enters or wherever there is a weir or significant change in the depth of the river, its cross-section, plant density, or the constants in rate equations. In the model it is supposed that a body of water is moving down river and the user has to supply the time (GMT) at which the body of water starts its journey. In addition, depending on which river processes the user wishes to include, he must supply some or all of the following: volume rate of flow, starting concentrations of dissolved oxygen, BOD and other variables, rate-equation constants (values for which are suggested in the Appendix) maximum light intensity, duration of daylight, and the time step--see above-for the integration of the rate equations. As the progress of the theoretical body of water down a river is simulated in the program, the values predicted for dissolved oxygen, BOD, ammonia, and nitrate for each hour of travel, together with the corresponding distance down the river, and the time (GMT) are printed out. Flow rates and polluting loads of tributaries and effluents are also required, and when using the "batch" version the user has to supply these for each of the 24 h of the day to enable the program, which keeps account of time, to interpolate a value for each item whenever the supposed body of water encounters a tributary or effluent. When using the "interactive" version, however, the program itself asks for the flow and polluting load for each tributary and effluent, in a message on the teletype or visual display unit, immediately after it has printed or displayed the predictions for the preceding stretch, the requested information being typed in by the user, or "read" in from prepunched paper tape, unless he decides to discontinue the "run". Again, as the program follows the progress of the supposed body of water, calculating its changes in quality for each time step--usually 1 h--of travel, account has to be taken of the entries of tributaries and effluents. In order to add the contributions of these to the quality of the body of water, the time at which the beginning of each stretch is reached has to be available to the simulation since it is here that a tributary or effluent enters. In the WRC programs, the time (GMT) at the beginning of each stretch is worked out by the programs from the flows and the length of each stretch taken in conjunction with their average depth and width, all these being supplied as data for the "run" by the user: how-
A mathematical deterministic river-quality--I ever, the authors of Part 2 (Casapieri et al.) of this paper modified the program so that depths and widths were no longer required, times at the beginning of each stretch now being found by the program from the user-supplied "time-of-travel" along each stretch, these times-of-travel already being known from tracer studies of the river. The "batch" version of the program has the additional feature of being able to print out the dissolved oxygen (mg 1-1) added to removed by the relevant processes, viz BOD oxidation, photosynthesis, reaeration, plant respiration, bed respiration, ammonia oxidation, and the milligrams of nitrogen per litre lost by denitrification for each hour of travel. This gives a clear picture of the comparative magnitude of these processes, and this information as well as the values of dissolved oxygen and other qualities can also be obtained as a graph produced on the conventional computer-printing devices; special graph-plotting features are not required. The features mentioned in the last sentence are not provided by the "interactive" version, when the usual narrowwidth teletype is used, but are available if a full-width (130 characters) printing device is employed. The rate equations and the associated suggested values for their constants are from published work, this being comprehensive for some of the processes, though further experimental and field work seem desirable, to confirm or improve the representations of settling and resuspension. In testing the model by seeing how well it would have predicted observed data, there can be individual values or period averages. If period averages are used, then the values supplied to the model for flows, loads, depths, temperatures, and other qualities can likewise be period averages.
1151
along the whole length of any river, by taking into account the results of a wide range of river processes, including photosynthesis, bed respiration, and the effects of weirs and entry of effluents and tributaries. This model, which is in the form of a computer program, was checked by the Thames Water Authority against past results for a river catchment (Casapieri et al.) and used for their long-term planning. It should also be useful for day-to-day management of river quality. An "interactive" version, in FORTRAN, allows predictions to be made at a few minutes' notice using time-shared computers via the ordinary G P O STD telephone system, while a "batch" version, also in FORTRAN, is suitable for more conventional computer processing. A "monitor" version could quickly be created for placing in any type of processor used for controlling automatic river waterquality monitors, to allow continuous automatic prediction of conditions in the stretches of rivers between monitors. In the mathematical model, the effects of eleven major river processes on the qualities of a supposed body of water are repeatedly summed as the body of water moves down the river, numerical integration being employed to obtain the actual concentration changes from the rate equations for the river processes. Predictions for dissolved oxygen, BOD, ammonia, and nitrate are printed for the end of each stretch of the river, together with a detailed breakdown of the dissolved oxygen 1 # contributed or removed by each process. Acknowledgement--This paper is published by permission of the Director of the Water Research Centre.
REFERENCES
SENSITIVITY ANALYSIS Though sensitivity analysis is desirable, the processes are in this case so interlinked that a long discussion would be required if generalizations were not to be misleading. However, it is of interest to note that nitrification and plant respiration can sometimes have a major effect on the dissolved-oxygen concentration in smaller rivers; this is seen in results from the model, the output from which includes the addition of or consumption of oxygen for each relevant process for each stretch of river. Likewise the interaction between equations makes it difficult to estimate errors resulting from changes in rate constants in the equations in the Appendix, but it is hoped later to quantify effects on water-quality predictions caused by these changes, in typical cases.
Ames W. F. (1969) Numerical Methods for Partial Differential Equations. Nelson, London. Casapieri P., Fox T. M., Owers P. & Thomson G. D. (1978) A mathematical deterministic river-quality model. Part 2: Use in evaluating the water quality of the Blackwater catchment. Wat. Res. 12, 1155-1161. Kuo S. S. (1965) Numerical Methods and Computers. Addison-Wesley, Reading, Mass., USA. Mitchell A. R. (1969) Computational Methods in Partial Differential Equations. Wiley, New York. Owens M., Knowles G. & Clark A. (1969) The prediction of the distribution of dissolved oxygen in rivers. Proc. 4th Int. Conf. War. Pollut. Res., pp. 125-137 and 139-147. Pergamon Press, Oxford. APPENDIX: PROCESSES MODELLED AND EQUATIONS USED
1. Rate of change in 5-day BOD
(a) Rate of change due to oxidation (1)
CONCLUSIONS
= - 1 . 0 4 7 ( r - 2 ° ~ ( K B ) ( B O D ) m g l - 1 m i n -1
A deterministic river model developed at the Water Research Centre can be used to predict concentration of dissolved oxygen, BOD, ammonia and nitrate
where BOD is concentration in mg 1-t, K B is BOD decay constant (approximately 0.000139 for equation 1), and T is temperature in °C.