MATHEMATICAL PHOSPHORUS

REPRESENTATION VARIATIONS

IN A N O N - S T R A T I F I E D

SOUTHERN CONSTANTINE

E. M E R I C A S

OF S H O R T T E R M

LAKE and R O N A L D F. M A L O N E

Dept. of Civil Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

(Received February 17, 1983) Abstract. Total phosphorus concentrations in a small urban hypereutrophic lake undergoing restoration were continually modeled by a modified version of a Vollenweider model for the latter part of a three and one-quarter year restoration period. Results of the modeling effort on Crest Lake were analyzed to determine the compatibility of the basic Vollenweider model with short term applications to highly eutrophic southern lakes. Input parameters such as rainfall and wind speed were considered on a daily basis to see if the model could represent the short term variations observed in the lake. Model simulations were compared to total phosphorus observations collected on the lake at intervals as short as one day. Results indicated that the model can be used as a tool for analysis of system responses on a seasonal basis, but short term variations on the weekly or daily timeframe were poorly represented by the model. Intensive temporal and spatial sampling within the lake verified the validity of the completely mixed assumption for short term applications. Nine sampling locations within the lake, distributed with area and depth, displayed a mean coefficient of variation of only 10.9% when averaged daily for the 54 day intense sampling period. Short term variations were poorly correlated with meterotogical inputs, suggesting that variability be best represented by a stochastic approach if a practical management tool is desired for situations where variability is critical. The seasonal trends were well represented by the model, reflecting the rapid response ofhypereutrophic systems to alterations in phosphorus loadings.

1. Introduction The feasibility of restoring highly eutrophic lakes to more desirable trophic conditions has increased dramatically in recent years. This is the result of both increased understanding of the complex phenomena contributing to the problem, and experience gained from initial restoration efforts. The number of restoration projects actually undertaken is small when compared to the magnitude of the problem. This is due in part to the inability of the scientific and engineering community to model, in an inexpensive manner, various restoration options that may be available for a given eutrophic system. The high cost associated with modeling efforts seriously inhibits evaluation of restoration alternatives on all but the largest or most politically sensitive lake systems. Clearly, there is a need to refine our modeling capabilities and bring down the cost of assessing restoration options. A mass balance model originally proposed by Vollenweider (1969) and since widely applied, was selected for application to a small eutrophic lake undergoing restoration. This paper examines the ability of the Vollenweider model to represent short term fluctuations in total phosphorus concentrations in Crest Lake. During the prerestoration period, instability in the lake was believed to be a major factor in inducing fish kills. Following initial restoration activities, significant improvement in average concenEnvironmental Monitoring and Assessment 4 (1984) 85-98. 9 1984 by D. Reidel Publishing Company.

0167-6369/84/0041-0085502.10.

86

c.E. MERICASANDR. F. MALONE

trations was observed, but short term fluctuations continued. Consideration of the variability of the phosphorus concentrations was considered crucial if the modeling efforts were to realistically represent the potential for post restoration fish kills.

2. Background The University Lakes system (Figure 1) consists of six small urban lakes ranging in size from 3.0 to 220.0 acres. The lakes were formed in the 1930's when low lying cypress swamps were timbered and dammed. The expansion of the Louisiana State University campus to the west and rapid residential development to the east, led to the development of causeways and drainage systems which subdivided the original lake into its present configuration of six lakes. These lakes are representative of a large number of small urban lakes in the south that have been adversely impacted by intense development of

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MATHEMATICAL REPRESENTATION O F SHORT TERM P H O S P H O R U S

87

surrounding lands. Increased nutrient loading due to deterioration of runoff quality has led to highly eutrophic (or hypereutrophic) conditions. These conditions in turn led to frequent fish kills which substantially reduce the lakes' recreational value and periodically result in offensive odors. A lake restoration project sponsored by the United States Environmental Protection Agency and the City-Parish Government of East Baton Rouge was initiated in 1978 to deepen the lakes and correct runoff problems. It is anticipated that this restoration effort will be completed by the fall of 1983. The center of interest of this paper is Crest Lake, one of the smallest lakes in the system. Crest Lake is characterized by a surface area of 3.42 ha (8.45 acres), a mean depth of 1.6 m (5.25 feet) and a detention time of 561 days (1.54 yr). It is the deepest lake in the University Lakes system and possesses the smallest drainage basin to surface area ratio, 0.65. Benefits to be derived from dredging were judged to be minimal, so dredging of Crest Lake was not undertaken as part of the restoration project. External inputs of Crest Lake consist of runoff from the limited surrounding residential area and wind driven exchange through culverts connecting the lake with the adjacent University Lake. University Lake has a surface area of 89.2 ha (220 acres), a mean depth of 0.6 m (2.0 feet), and a drainage basin to surface area ratio of 3.72. University Lake has been historically heavily contaminated with raw sewage from leaks and designed overflows in an aging sewage collection system. The interconnection with University Lake thus contributed to the development of hypereutrophic conditions in Crest Lake. This observation led to the isolation of Crest Lake from the larger University Lake by construction of a sand bag and plywood barrier in the culvert connecting the two lakes. The goals of this temporary restoration effort were: (1) to protect Crest Lake from high sediment levels that would occur when University Lake was dredged; and (2) to improve the water quality so that desirable game fish could be stocked and later released into University Lake. Previous work (Mericas, 1982) has established that the fish kills observed in the University Lakes are associated with in-lake total phosphorus concentrations above 0.4 mg 1-1 p. A specific water quality objective of maintaining Crest Lake below this level was established.

3. Methodology Water quality data from Crest Lake has been collected on a regular basis since June, 1979. From June, 1979 to May, 1981, samples were taken monthly, with bi-weekly sampling during the highly productive months of June, July, and August. A weekly sampling program was initiated in June, 1981, and continued through December of 1982. Samples were taken from one foot below the surface and one foot above the bottom at a station roughly 15 feet from the shore. Between June 18 and August 10, 1982, an intense daily sampling program was conducted in addition to regular sampling. Samples were taken from three depths at three in-lake stations on a daily basis. Sampling depths were uniformly six inches, three feet and five feet from the surface. Samples were drawn through tygon tubes fixed at the three depths to a bouyed taut line

88

C. E. MERICAS A N D R. F. MALONE

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support designed to minimize disturbances of the water column during sampling. Figure 2 identifies the sampling locations used for this study. All samples were analyzed for total phosphorus using the persulfate digestion-ascorbic acid technique described in Standard Methods (1980). All determinations were done in triplicate and regularly subjected to quality control checks. 4. Model Description The classical Vollenweider model (VoUenweider, 1969) is summarized as follows V --dP = Q~Pt - a P V dt

- QoutP

(1)

M A T H E M A T I C A L REPRESENTATION OF SHORT TERM P H O S P H O R U S

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where V = Lake volume (m3). P = In-lake phosphorus concentration (g m - 3 ) . Q~ = Rate of water inflow (m 3 t - l). Pe = Inflow phosphorus concentration (g m - 3 ) . a = Sedimentation coefficient (t- 1). Qout = Rate of water outflow (m 3 t - 1). Application of the model to Crest Lake required modification of this basic relationship to account for inter-lake exchange as a loading source (Figure 3). The volume of water -

~ Q i Pi

UNIVERSITY CREST Fig. 3. A schematic representation of the phosphorus budget in Crest Lake.

that is exchanged between lakes, designated Qex, is the result of wind induced head changes. As wind blows along the axis of University Lake for a period of time, it induces a seiche. This results in an increase in lake level, or setup on the downwind side of the lake and corresponding decrease on the upwind side. When the wind shifts, the lake level returns to normal as water from the elevated side flows back towards the depressed side. Crest Lake responds to the head differential by either receiving or releasing water through its connections with University Lake. This exchange is on a short term basis. Qex was estimated by a function of the resultant daily wind component along the long axis of University Lake (320 o). B ased upon literature equations for wind setup (Linsley and Franzini, 1979), the relationship was established as O~x = ( f ) ( w ) 2

(2)

where Q~• = Volume of water exchanged (m3). f = Empirical constant (m 3 m p h - 2) w = Resultant wind component along the 320 ~ axis (mph). The exchange of volumes of water between the lakes assumes that water enters Crest at the mean University Lake concentration and exits at the mean Crest concentration. This simplifying assumption is consistent with the completely mixed assumption under

90

C. E. MERICASAND R. F. MALONE

which the Vollenweider model normally operates (Mericas and Malone, 1982). Thus, the total mass of phosphorus introduced to Crest by this mechanism is represented by (3)

Mex = Oex(Pu - P )

where M~x = Rate of mass input resulting from exchange processes (g day- 1). Q~x = Volume rate of water exchange (m 3 day- 1). Pu = In-lake phosphorus concentration in University (g m - 3). P = In-lake phosphorus concentration in Crest Lake (g m-3). The concentration of total phosphorus in University Lake at any point in time, P,, was defined by linear interpolation of the routine monitoring data collected on University

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where AP = Change in water column phosphorus (g m-3). At = Time step (days). L = Rainfall runoff loading (g day- 1). A hydraulic model was coupled to the phosphorus model to provide adjustments to the volume term. Daily wind and evaporation data were obtained from the Louisiana State Climatologist's office at LSU and were based on observations made at a station three miles from Crest Lake. Dally rainfall data was obtained from the LSU Forestry Department which has a recording station within one mile of the lake. Runoffflow input was calculated by coupling a runoff flow equation (Overton and Meadows, 1976) with an empirical phosphorus loading equation developed for the University Lakes drainage area. Details of these submodels, as well as other details of application not crucial to this paper, are available (Mericas and Malone, 1982; Mericas, 1982; Wegener, 1982). TABLE I Coefficients used during the restoration simulation Period

Sedimentation coefficient (day - 1)

Wind exchange constant (m 3 mph - 2)

Physical status

6/79-5/80 6/80-9/81 10/81-2/82 3/82-8/82

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5. Results

The Vollenweider model was used as a management tool for analysis of Crest Lake for the latter part of a three and one-quarter year period from June of 1979 through August of 1982 (Figure 5). During this period, the lake was actively responding to reclamation efforts. The model is very sensitive (Figures 6 and 7) to two coefficients: the sedimentation coefficient, a, and the empirical exchange constant, f. Optimum coefficients were selected by minimizing residual errors under the assumption of a constant sedimentation coefficient. The empirical exchange constant was varied to reflect the changing physical condition of the interconnections between the two lakes. Table I presents a summary of the coefficients used in the simulation of the restoration effort. A sedimentation coefficient of 0.006 day- i was found to accurately represent the system throughout the entire simulation period. The model was regularly compared with the routine monitoring data. Descrepancies between modeling projections and actual responses led to field investigations and model refinement. The four phases of simulation illustrated in Figure 5 resulted from this process. Alterations in the empirical wind constant, f, reflected physical alterations in the connections undertaken by the public works department between Crest and University Lakes.

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The first phase (June 1979-May 1980) was characterized by the virtually unrestricted flow ( f -- 350 m 3 mph- 2) that existed between the two lakes prior to the installation of a plywood and sand barrier in the box culvert under the causeway. In the second phase (June 1980-September 1981), the connection between the two lakes was assumed to be diminished but not eliminated because of leakage through the box culvert barrier ( f = 52.5m3mph-2). From October, 1981, it was assumed that the lakes were essentially isolated as the result of additional sand bagging efforts such that f = 0.0 m 3 mph- 2. However, additional field investigations prompted by discrepancies between model projections and monitoring data revealed the presence of a previously unknown 36 inch clay pipe beneath the causeway separating Crest Lake from University Lake. It is believed that this connection has historically been blocked by sedimentation. The artificial blockage of the larger box culvert under the causeway apparently created enough head differential between the two lakes to permit scour, thus reopening the pipe. Thus, the empirical constant, f, was modified to 5.25 m 3 mph - 2 to reflect partial leakage through the causeway for the period between February and August of 1982. As the quality of University Lake deteriorated (total phosphorus levels average 1.0 mg 1- 1 for the months of February through May, 1982) due to in-lake dredging activities, the model projected (Figure 8) Crest Lake's extreme sensitivity to even minor leakage through the causeway. Recognizing the drainage problems that would result from the permanent

94

C. E. MERICAS AND R. F. MALONE

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isolation of Crest and the ramifications of a fish kill of freshly stocked gamefish, the secondary objective of using Crest Lake for an early stocking program was abandoned. 6. Discussion

This modeling application must be considered a successful demonstration of the value of a simple model of the classical Vollenweider format as a tool to assist in the planning, implementation and evaluation of a restoration project. The principle objective of applying such a model is to provide reliable information that can be used as a basis for decisions. The application of the model in this case allowed the quantification of the major processes controlling the trophic level in Crest Lake. This information proved invaluable in assessing the effects of restoration efforts and in identifying necessary corrective actions. Management decisions were made with a clear understanding of their impact. The Vollenweider model represents the major processes, loading and sedimentation, that control the 'steady-state' trophic level in a lake. Typical applications of the Vollenweider Model (Reckhow, 1979a) usually involve the simulation of mean annual total phosphorus levels over years. However, management models must be able to simulate effectively over a short timeframe if they are to provide a basis for management decisions on small lakes. The monitoring results indicate that small hypereutrophic lakes do respond to alterations in loading regimes rapidly enough to allow use of a steady-state

MATHEMATICAL REPRESENTATION OF SHORT TERM PHOSPHORUS

95

analysis on a monthly timeframe. A comparison of the extensive calibration data (Figure 5) illustrate the ability of the model to represent the average conditions and transitional trends occurring during the different phases of the Crest restoration effort. This resolution was sufficient to contribute significantly to the implementation of the restoration plan. These observations are drawn in the context of hypereutrophic systems. The high phosphorus levels characteristic of these systems apparently cannot be maintained for any length of time without constant loading from external sources (in this case through exchange). The Vollenweider model is well designed to handle loading dominated systems. A major limitation of the model is its inability to represent the daily or even weekly variations that occur about the mean trend line. The daily or weekly variations of phosphorus in the water column were on occasion of a large enough magnitude to drive the system past the threshold level of 0.4 m g l - l p , thus exposing the system to conditions associated with fish kills. This problem was examined more closely through an intensive dally sampling of Crest. This intensive sampling regime allowed some definition of the dally variability of phosphorus levels within the lake. The spatial variations were found to be moderate, with the dally observations displaying coefficient of variations in the range of 1.4 to 20.8 ~o. The average coefficient of variation for the daily observations was 10.9~o, indicating that the completely mixed assumption under which the model operates was fundamentally valid. A comparison of the model's projections with observed average dally in-lake conditions derived from nine in-lake sampling points for the period of June 18 to August 10, 0.45 ZONE

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96

C. E. MERICAS AND R. F. MALONE

1982 is presented in Figure 9. The model was run through this period with the projected concentration for the first day reset to the observed mean value for that day. During this period, the model projected a slowly declining in-lake phosphorus concentration with a mean of 0.255 mg 1- ~ P while the actual average dally phosphorus concentrations had a mean value of 0.281 and varied widely from 0.213 to 0.353 mg 1- 1 p. The implication of these observations can be more clearly illustrated if it is assumed that both the projected and the observed values are representative of a constant (or steady-state) trophic state for the period of study. Under such an assumption, probability density distributions (Figure 10) can be developed to illustrate the relationship between the in-lake conditions and the fish kill threshold value. These continuous distributions were developed from the average total phosphorus concentrations derived from the intensive monitoring data or the mean in-lake concentrations projected by the model on a daily basis. Distribution types (normal for monitoring data and uniform for modeling projections) were determined by examination of discrete frequency distributions. The narrower more sharply peaked density function in Figure 10 represents the in-lake conditions as 'perceived' by the model. The distribution of projected values is characterized by a low level of variability. The more widely distributed function represents the distribution of mean daily in-lake phosphorus values observed. It can be seen that the observed distribution reflects the occurrence of phosphorus levels in excess of the critical 0.4 mg 1- ~ P as the area under the right hand tail of the curve. The projected distribution derived from model projections shows no occurrence beyond a level of about 0.280 mg 1- 1 p. Clearly the actual potential for fish kills was greater than strict interpretation of the model projections would have led one to believe. Examination of these daily variations has led to the suspicion that they are primarily

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MATHEMATICAL REPRESENTATION OF SHORT TERM PHOSPHORUS

97

the result of internal cycling of nutrients resulting from the cyclic sedimentation and release of phosphorus from the bottom. The importance of these processes in defining the in-lake phosphorus levels has been frequently discussed (Larsen and Malueg, 1980; Ryding and Forsberg, 1980), Variability becomes a very significant factor when dealing with lake systems that exhibit threshold responses such as fish kills. As it has been illustrated (Figure 10), interpretation of mean nutrient concentrations alone can be very deceptive in assessing the potential for threshold responses. Models in the format of the basic Vollenweider model are incapable of representing the wide range of total phosphorus values actually observed in the lake. Although variability induced by loading phenomena may be well represented, the significant variability induced by internal cycling phenomena is neglected. Even in this application to a lake whose seasonal trophic state seems to be dominated by loading phenomena, the variability induced by internal cycling remains too large to be safely neglected. The objective of developing a low cost modeling format for use in restoration efforts suggests that a stochastic enhancement of the Vollenweider model would be the most appropriate manner to deal with these variations. This approach would also be compatible with the observation that many of these cyclic processes appear to be functions of random factors such as temperature, wind mixing, and solar radiation (Ryding and Forsberg, 1980; Barica, 1978). The use of a stochastic format to represent the in-lake variability in highly eutrophic lakes would be a valuable asset to engineers and managers with a need to quantify the risks associated with various management options. The format of the basic Vollenweider model would seem compatible with this approach. The model's simplicity and minimum data requirements suggest its usefulness as a management tool. Within the context of ongoing monitoring programs, it is capable of generating seasonal means with a reasonable degree of accuracy. These observations provide a basis for directing further research towards the objective of developing a cost effective simulation approach for use in restoration efforts. 7. Conclusions

The modified VoUenweider model was capable of simulating monthly trends in total phosphorus levels in a small hypereutrophic urban lake dominated by exchange processes with an adjacent lake system. The model was a valuable tool in implementing restoration objectives. The model's inability to represent daily variations occurring about the trend lines will limit the value of the model in applications where such variations may lead to violation of threshold restoration constraints. This limitation could be addressed by application of a stochastic format model utilizing the basic Vollenweider structure.

98

r E. MERICASAND R. F. MALONE

Acknowledgements Funding for this research was derived in part from the United States Environmental Protection Agency, City/Parish Government of East Baton Rouge, and the State of Louisiana through a cooperative lake restoration effort under the Clean Lakes Program. This paper was not subject to review by the funding agencies. Findings of the paper reflect the opinions of the authors only. The data base used in this paper includes contributions by Mr Glenn McKenna, Mr Daniel Burden, and Mr Andrew Eversull. References Barica, J.: 1978, 'Collapse of Aphonizomenon Flos-Aquae Blooms Resulting in Massive Fish Kills in Eutrophic Lakes: Effect of Weather', Verh. Internat. Verein. Limnol. 20, 208-213. Larsen, D. P. and Malueg, K. W.: 1980, 'Whatever Became of Shagawa Lake?', Restoration of Lakes and Inland Waters, EPA 440/5-81-010, 67-72. Linsley, R. K. and Franzini, J. B.: 1979, Water Resources Engineering, McGraw-Hill, 165. Mericas, C. E.: 1982, 'Phosphorus Dynamics and the Control of Eutrophication in a Southern Urban Lake', Master's Thesis, Louisiana State University. Mericas, C. E. and Malone, R. F.: 1982, 'Short-term Application of a Modified Version of the Basic Vollenwider Nutrient Model to a Hypereutrophic Urban Lake', Developments in Environmental Modelling 5, Elsevier Scientific Publ., 609-613. Overton, D. E. and Meadows, M. E.: 1976, Stormwater Modelling, Academic Press, pp. 35-38. Reckhow, K. H.: 1979a, 'Uncertainty Analysis Applied to Vollenweider's Phosphorus Loading Criterion', Journal WPCF 51, 2123-2128. Reckhow, K. H.: 1979b, 'Quantitative Techniques for the Assessment of Lake Quality', EPA 440/5-79-015. Ryding, S. O. and Forsberg, C.: 1980, 'Short-term Load-Response Relationships in Shallow, Polluted Lakes', Developments in Hydrobiology 2, 95-103. Standard Methods for the Examination of Water and Wastewater: 14th ed., American Public Health Association, Inc., New York, 1980. Vollenweider, R. A.: 1969, 'Possibilities and Limits of Elementary Models Concerning the Budget of Substances in Lakes', Arch. Hydrobiology 66, 1-36. Wegener, K.: 1982, 'Selection of Parametric Stormwater Pollutant Loading Models Using Predictive Reliability Analysis', Master's Thesis, Louisiana State University.