and Water Lab
A mechanistic, physically based model for pollutant release from a surface source, such as field-spread manure, was hypothesized and laboratory and field-tested. Stable sources and a conservative "pollutant" (KCl) were used in the laboratory investigation so that the dynamic effects of source dissolution and chemical transformations could be ignored and transport processes isolated. The field investigation utilized runoff and soluble reactive phosphorus data collected from a dairy-manure-spread field in the Cannonsville watershed in the Catskills region of New York State. The model predictions corroborated well with observations of runoff and pollutant delivery in both the laboratory and the field. "Pollutant" release from surface sources was generally predicted within 11% of laboratory KCl measurements and field P observations. Laboratory flume runoff predictions with 15% and 26% errors, for 2.5 cm/hr and 1.5 cm/hr simulated rainfall intensity experiments respectively, represented root mean square errors of less than 0.2 ml/s. A 4% error was calculated for overland flow predictions in the field, which translated into approximately a 0.07 l/s error. Results suggest that the hypothesized model satisfactorily represents the primary mechanisms in pollutant release from surface sources.
Though runoff from manure spread fields is identified as a source of nonpoint source pollution, primarily nutrients, sediment, oxygen demanding compounds, and pathogens (Young, 1976; USEPA, 1990), there are no models which mechanistically describe transport from a field-spread-manure-type source. The objective of this project was to create and test a simple pollutant release model which describes mechanisms for pollutant release from manure-like sources. Furthermore, in lieu of good understanding of the environmental longevity of important pollutants, particularly Cryptosporidium parvum, the theoretical focus was limited to conservative pollutants. A conservative assumption provides a worst-case scenario. The model was field tested using phosphorus, which is generally not considered to be a conservative (non-reactive) substance, to demonstrate that the "conservative" assumption just mentioned doesn’t seriously impair the utility of this model to substances receiving current attention in environmental studies.
Agricultural runoff water quality research has been primarily through field experiments (e.g.: Young and Mutchler, 1976; Khaleel et al., 1980; Westerman and Overcash, 1980; Edwards and Daniel, 1993). The modeling efforts aimed at predicting pollutant release from manured fields have been, by-and-large, empirical and primarily focused on particulate transport (Wischmeier and Smith, 1978; Khaleel et al., 1979). Wang et al. (1994) developed a model to simulate runoff transport of land-applied manure constituents which utilized the convective-dispersion equation and the Green and Ampt (1911) equation (Mein and Larson, 1973). While this model is largely mechanistic (Ibrahim and Scott, 1990), it ignores or overly simplifies the processes around the pollutant source, assuming that rainfall leaches all soluble constituents from the manure.
To justify limiting the scope of this work to conservative contaminants, consider one pollutant associated with field-spread manure which is currently receiving acute attention is the pathogen, Cyptosporidium parvum (C. Parvum) (Moore and Zeman, 1991; Anderson and Hall, 1982; Leek and Fayer, 1984; Ongerth and Stibbs, 1989; Garber et al., 1994; Mawdsley et al., 1995), which has been identified as the cause of major waterborne outbreaks over the past decade (Gallaher, 1989; Smith, 1990; Moore et al., 1993; MacKenzie et al., 1994; Kramer et al., 1995). This organism poses serious problems because it takes very few individuals to establish infection (DuPont et al., 1995) and expensive filtration is generally considered the only effective barrier to C. Parvum. The large quantity of excreted C. Parvum by infected hosts (Current, 1986) and the robustness of C. Parvum oocysts (Robertson et al. 1992), including resistance to a wide variety of treatment practices (Madore et al., 1987; Peeters et al., 1989; Korich et al., 1990; Smith, 1990; Parker et al., 1993; Mayer and Parker, 1996; Chauret et al., 1996), facilitate their transmission in runoff (West, 1991) and make C. Parvum contamination particularly worrisome. The transport mechanisms of oocysts in runoff are currently not well understood, research in this area often showing contradictory results (Brush, 1997). In lieu of better understanding of entrainment, transport, and survivorship at the microorganism scale, it is arguable, for the time being, that C. Parvum be treated as a conservative pollutant in transport modeling.
While the approach used in this project assumes non-reactive contaminants, its application is not limited to these. The theory presented here is general, applicable to other substances, and easily modified to account for important pollutant transformations. Phosphorus (P), which has received particularly acute attention in recent years, is considered in this study because it has been well linked with eutrophication and associated ecosystem deterioration which seriously impairs the value of water bodies as recreation areas and drinking water sources (Bouldin et al., undated, Sharpley and Smith, 1992) A major source of P loading to surface waters in the United States Northeast is dairy farm manure spreading; this problem is exasperated by a trend towards higher P levels in feed rations over the past 50 years (Klausner and Bouldin, 1983) and increasing animal intensity (Sharpley et al., 1994). Better understanding of contaminant transport processes from surface sources, especially field-spread manure, will ultimately lead to better solutions for controlling the associated problems.
Figure 1 diagrams
the conceptual model. The model simulates three processes: overland flow,
horizontal pollutant convection, and vertical diffusion or convection.
It is assumed that the source height, H, is much greater than the overland
flow depth, h. The depth of flow through the source, hb, is defined as
h/n where n is the source porosity. The source matrix is stable and static.
The possibility of a crust over the source, as is occasionally observed
over dairy manure, is considered in this model; like the source matrix,
the crust is stable and static. At time, t=0, the source is essentially
saturated so that there is no wetting time or change in volumetric storage
within the source.
Overland flow is modeled with the St. Venant equation:
Where h is the depth of flow (m), q is the discharge per width (m2/s), i is the rainfall intensity (m/s), t is time (s), and x is downhill distance (m). The kinematic assumption applied to the momentum equation yields (Henderson and Wooding, 1964):
Where a and m are flow resistance and flow regime parameters respectively. For turbulent flow these parameters can be approximated with:
Where n is the Manning roughness factor (-) and s is the surface slope (m/m).
Using the method of characteristics, the time from the start of rainfall to reach steady flow, ts, at any point x is:
Assuming the time at which the rainfall ends, tr, is greater than ts, and using the method of characteristics and equations 1 and 2, the flow at any point, x, can be described by equation 5 through 7.
Rising Limb of Hydrograph (0<t<ts):
Steady Flow portion of Hydrograph (ts<t<tr):
Recession Limb of Hydrograph (tr<t):
The partial equilibrium situation,
tr<ts, can be analytically described, yielding different expressions
for equations 6 and 7, but is not presented here because the experimental
design did not allow for this situation to be adequately evaluated.
The vertical, upper region of the source, and horizontal, bottom region of the source, pollutant transport mechanisms are addressed as independent processes. Parameters associated with the bottom region are designated with a subscript "b". The source, a porous medium, is divided into two regions dependent on the discharge, q, the saturated conductivity of the source, Ks, and the bed slope, s. Applying the Boussinesq equation with the Dupuit assumption, flow in the bottom region is expressed as:
Where hb is the depth of flow through the source (m) (see figure 1) and defines the plane separating the upper and bottom regions. Invoking the kinematic approximation, the gradient of the depth with respect to x in equation 8 is approximated as the ground surface slope, s (negative).
This pollutant release model assumes two distinct temporal regimes of pollutant release; an early, quick pollutant release period is dominated by horizontal convection from the bottom region due to runoff passing through the source and a later, more gradually releasing period, during which pollutant moves vertically from the upper region. In the following derivation, much emphasis is given to determining time tb, the time at which essentially all pollutant is flushed from the bottom region.
Horizontal Model (bottom region):
Xin (1996) showed pollutant transport in the bottom region is dominated by convection; i.e. dispersive terms are negligible. The differential equation describing convective transport can be expressed as:
Where c is the concentration of pollutant (m mole/m3), t is time (s), u is the convective velocity (q/h: m/s), h is the depth of water flow (n(hb): m), i is the rainfall intensity, and Jb is the rate of solute uptake from the source into the flow (m mole m-2 s-1). The source porosity is n. The bottom region is flushed of pollutant at t=tb. As long as pollutant exists in the bottom region, t<tb, and rainfall has not stopped, t<tr, equation 9 can be solved for concentration, c, of solution leaving the bottom region using the relationships for h from equations 2, 5 and 6.
This solution assumes source matrix has negligible affect on the mass flow of water, i.e. there is no net storage. The cumulative mass leaving from the bottom region, Mb, as a function of time, for t<tb is:
Where w is the source width (m) perpendicular to flow. If a crust exists over the source, the mathematical description of flow at the source is violated, but because the contribution of runoff near the source is very small relative to flow from a watershed, simply due to the vastly larger spatial extent of a basin or field relative to a pollutant source, discontinuities near the source resulting from possible crusting are negligible. As long as no significant rainfall is stored on the crust, q can be visualized as the cumulative flow over and through the source and the concentration is the flow-weighted average concentration of the contaminated through-flow and clean "over-crust" flow.
The mechanism for rate of pollutant uptake from the source into the flow, Jb, may be largely dependent on the characteristics of a particular contaminant, e.g. its dissolving, dissolution, or desorption rates, and is beyond the scope of this study. As shown later in this paper, for some pollutants this mechanism is very rapid and its value can be estimated indirectly without full understanding of uptake mechanics. Also, the fundamental basis for the model presented here can be analytically extended to incorporate a larger range of flushing situations; i.e. it is not mathematically limited to the situation where the period of rainfall, tr, is greater than tb. Unfortunately, experimental limitations inhibited investigation of other scenarios.
Vertical Model (upper region):
Pollutant is transferred downward from the upper region to the bottom region via diffusion when the source is crusted and via convective-dispersion when there is significant vertical water flux. Parameters specific to the upper region are denoted with the subscript "u" and the subscripts "d" for diffusion and "c" for convection where appropriate.
Because diffusion requires a concentration gradient, the process is modeled by assuming no diffusive movement until the bottom region’s concentration is effectively zero, i.e. t³ tb. Assuming the upper region can be approximated as a semi-infinite region, bounded on the lower end by the top of the bottom region, hb (see Figure 1), where the concentration is zero for t³ tb, Carslaw and Jaeger (1959) showed that the concentration gradient in the upper region can be written as:
Where co is the initial concentration, assumed to be initially homogenous in the upper region (m mole/m3) and D is the diffusivity (m2/s). Fick’s law can be used to describe diffusive flux from the upper region, Ju
Replacing the differential in equation 13 with equation 12 and integrating, the cumulative mass removed from the upper region by diffusion, Mud, is
Where A is the source’s horizontal cross-sectional area (m2), assumed constant for this study.
When water is passing vertically through the source, convective-dispersion will be the dominant transport mechanism. The simplest modeling approach is to assume the concentration of solution released from the source follows a step function.
While pollutant is available in upper region of the source (15)
When all available pollutant is removed from the upper region of the source
The cumulative mass leaving the upper region by convective-dispersion, Muc, is
Where c is determined by equations 14 and A is the source’s horizontal cross-sectional area (m2). Unlike diffusion, convective transport is assumed to occur throughout the experiment, not just after the bottom region is flushed.
Three conditions were examined in the laboratory: the case with no crust on the source, the case with complete or full crusting, and an intermediate, 50% crusting, scenario. Each experiment was 30 minutes long and subject to a rainfall intensity of 2.4 cm/hr. A rectangular sponge, pre-soaked in a solution of KCl, was used to simulate a pollutant source. The initial concentration in the sponge was approximately 1 m mole/l; actual values ranged from 1.0 to 1.3 mmole/l. Figure 2 shows the flume and source set-up used to test the model. The slope of the flume is adjustable; for this study s=0.065. Mannings n was taken as 0.05 (Baltzer and Lai, 1968). Pollutant crusts were simulated using metal covers. The complete crust was a metal cover over the entire source; the 50% crust was a metal cover over half the source’s top surface. Diffusivity was taken as 2x10-5 cm2/s (typical for ionic solutes in water).
Effluent from the flume was collected every 10 seconds for the first minute, once per minute for the next 29 minutes, and again every 10 seconds for the recession period after rainfall ceased. Each sampling duration was 10 seconds. KCl concentration was measured using a pre-calibrated conductivity meter; care was taken to maintain constant room temperature for all conductivity measurements to avoid errors arising from conductivity’s thermal sensitivity. Due to the sponge’s large pores, substantial and prolonged drainage was observed. Experimental data were adjusted using an empirical drainage curve developed from laboratory sponge-drainage measurements, for mass removed via drainage. Before and after the set of experiments simulated rainfall uniformity was checked using Christian’s coefficient, Cu (James, 1988). Values for Cu were 0.90 and 0.91 indicating good uniformity. The kinematic assumptions and parameters were tested during the first two experiments even though daming the flume with the sponge was expected to attenuate the rising and falling legs of the hydrograph; results are shown in the Laboratory Results section.
Xin (1996) demonstrated in similar experiments that the time required to flush the bottom region of its original pollutants is short, depending primarily on the time to establish a steady bottom region. By the time steady state in the bottom region was fully established the region was essentially flushed. Jb can be approximated as the total mass of salt in the bottom region per unit area divided by the time to establish steady flow in the bottom region. It follows then, that tb, the time to flush pollutants from the bottom region, can be estimated as the time to establish steady flow in the bottom region:
Where vb (m3) is the volume of solution in the bottom region of the source, vo (m3) is volume of water applied to the flume-watershed until the time steady state overland flow is obtained, ts (equation 4), and v’o (m3) is the backwater volume. The rate of flow in the system is approximated as iLw; steady flow conditions implicitly assume ts<<tb. Combining equations 2 and 4 for h, multiplying by the flume’s width, w (m), and integrating over the flume’s length, L (m), vo is:
Where the function ts(L) is the time to steady overland flow at x=L. With the assumption that the volume of water discharged while t<tb is small relative to the associated static volumes of vb and v’o, these volumes can be easily estimated. Due to the flume’s simple geometry, the volume of the bottom region, vb, is:
Where H is the total source height (m) (see figure 1) and vt is the total solution volume in the source (m3); measured values ranged from 300 ml to 450 ml depending on drainage. By continuity, once steady state is reached, the flow at x=L, q, is iL. Using equation 8 and assuming steady state conditions are valid for this approximation:
Based on work done by Xin (1996), Ks was taken as 50 cm/min. The backwater volume is estimated assuming simple daming-up behind the source of runoff water, i.e. the backwater surface is level. With a constant slope, s, v’o is estimated by:
By the definition discussed earlier, the mass transfer flux for the bottom region is:
Assuming q is approximated by iL for 0<t<tb (the duration of the hydrograph rising limb is small compared to flushing period), from equations 10, 11, and 21:
for t = tb (23)
Figures 3 and 4 show the agreement between the kinematic model for runoff in the flume and experimental data. A statistical comparison between predicted and observed values is shown in Table 1. The experiment from which figure 4 was gleaned involved a "sponge dam" for which this kinematic model does not explicitly apply.
Figure 5 shows the direct cumulative mass results for the three experiments.
There was experimental variation in initial KCl concentration and initial solution volume in the sponge, +/ 0.05 mmole/ml (approx. 5%) and +/-75 ml (20%) respectively. Therefore, cumulative export results are shown in terms of "% of total."
Field observations were made in Delaware County, part of the Catskills region of New York State. Rainfall, runoff and phosphorus concentration data were collected from a manure spread field for the June 19 and 20, 1996 rainfall event. Field parameterization was difficult, therefore, order of magnitude estimates and system simplifications were used to apply the model. This investigation was to find evidence that the modeled processes were observable in the field rather than to quantify predictions with great precision. The runoff contributing area was 2.5 ha which for simplicity was considered roughly rectangular. The longest flow path length and average slope were 302 m and 11% respectively. Rainfall intensity was assumed constant over the event, 0.25 mm/hr. Manure was spread (17.8 T/ha) simultaneously with the rainfall so source crusting was assumed negligible. Both phosphorus samples and runoff measurements were automated; phosphorus samples were taken hourly and runoff discharge measurements were made every 15 minutes. The samples were tested in the laboratory for soluble phosphorus (P). The storm commenced before manure application began, therefore the model was run starting one hour before phosphorus first appeared at the sampling station (i.e. the last zero phosphorus sample). This starting time happened to correspond to a slight increase in the hydrograph (Figure 5).
The mathematical development used for the laboratory experiments was directly applied to the field with the exception of tb determination. Because of the complicated source geometry in the field a simple empirical relationship was used to estimate tb based on the laboratory results.
This relationship assumed similar conductivities between field and lab pollutant sources. Based on qualitative visual observations, the source was assumed to be well distributed across the lower portion of the contributing area in saturated clumps approximately 5 cm high and occupied roughly 1% of the total runoff-contributing area. Manure conductivity of 50 cm/min was used. Phosphorus diffusivity was assumed 2x10-5 cm2/s (typical of ionic solutes in water). At the time of preparing this document, field measurements, unfortunately, are only available for an uncrusted source. Also, the authors recognize the that the choice of phosphorus was, in many ways, non-ideal because of its strong sorption, desorption mechanisms but used it anyway because of data availability. The field data were used primarily to corroborate the general modeled mechanisms.
The field results are summarized in figures 4 and 5 and statistical comparisons are shown in Table 3. Figure 6 shows the agreement between the kinematic model and observed flow. Figure 7 shows observed and predicted cumulative pollutant release.
The kinematic flow equations corroborated well with the measured data. The oscillating nature of the lab data, seen in figures 3 and 4, can be largely attributed to the rainfall simulator’s oscillating mechanism, traveling up and down the flume. The no crust experiment, which shows the most obvious flow oscillations (squares in figure 4), was run immediately before the 50% crust experiment, which shows much less dramatic oscillatory characteristics presumably due to damping associated with the presence of a crust. It is possible, then, that some of the flow variability is inherent to the experimental apparatus, especially the rain simulator, which is oscillatory in nature.
The attenuation of the rising hydrograph limb in the 50% data (circles) in figure 4 show the "daming" effect of the sponge. The end of the early steep, linear section of the cumulative pollutant release curves in figure 5 are at t=tb, estimated with equations 16 - 20, and visually correlate well with the time at which the observed flow (circles in figure 4) reaches steady conditions. This suggests that the approximations, discussed earlier, in estimating vb and v’o are adequate for this experiment. The somewhat weaker statistical correlation between the experimental flow and kinematic equation relative to the high and low intensity experiments (table 1) may be partially due to the daming period for which, as discussed earlier, the runoff equations are not designed.
The field data agreed well with the kinematic equation, perhaps largely due to the fact that preliminary rainfall had saturated the contributing area making it essentially impermeable; i.e. flume-like.
The pollutant transport model corroborates well with the general observed trends in both field and laboratory observations. The field results show particularly good agreement in that they were much better than anticipated. The early (t<tb), linear portions of the lab curves, show good agreement and suggest that the highly simplified nature of equation 21 is adequate within the boundaries and precision of these experiments.
The full crust lab experiment showed the worst statistical agreement between measurements and predictions. While the full crust experiment corroborated well with predictions during the initial flushing stage, the total cumulative mass release prediction was about 20% lower than observed. The discrepancy in total cumulative pollutant release is due to an under-prediction of the delayed, vertical flux, stage of release. During this period, the model underpredicted by approximately 30%. This suggests that diffusion is not the only vertical transport mechanism or that the diffusion coefficient was too low for the very concentrated solution used in these experiments. One possible explanation is the presence of vertical convection via rainwater entering the upstream face of the sponge, which was unprotected (uncrusted). Assuming the exposed upstream face corresponds to a 10% uncrusted area of the source, and that convected pollutant from this area can be added to the diffused-out pollutant, corroboration of predicted pollutant release by measurements improves; R2 is 0.98 and relative difference is 4.8%. Though the effects of increased solute concentration on the diffusion coefficient were not investigated, increasing the diffusion coefficient four-fold makes the prediction-observation agreement similar to the other two lab experiments; R2 is 0.98 and relative difference is 5.3%. Xin (1996) suggested a dispersion mechanism along the boundary between the upper and bottom regions to explain the greater-than-diffusion-alone pollutant release. This explanation may justify a higher "effective" diffusion coefficient.
In the 0% and 50% crust lab cases the model’s rate of mass removal for t>tb is much more linear than the observations (figure 5). This suggests a missing or inadequately described mechanism in the model, which is present in "sponge" leaching. Overly linear model results relative to observations are also present in the field results. One possible explanation is that this model is the missing dispersion mechanism, which would presumably add curvilinear characteristics to the predictions. However, the strong statistical agreement between observed data and predicted results (tables 1 and 2) with the current simplified model did not warrant additional complication for this study.
This study shows drastic pollutant release restriction potential for crusted pollutant sources, such as field spread manure. As shown in figure 5, the crusted source released only 25-30% as much pollutant at the fully uncrusted source. For the lab conditions used, increase in pollutant release was roughly linear with degree of uncrusted area. Using the model to predict fully crusted pollutant release from the field scenario, predicted differences in fully crusted and uncrusted pollutant releases were 67%, similar to laboratory results.
Good corroboration of the model results with both field and laboratory observations suggests that the model theory correctly accounts for the primary mechanisms of pollutant transport from field-spread-manure-type pollutant sources, including potentially crusted sources. Crusts had an environmentally beneficial affect on pollutant results in the laboratory. Application of the model to a field scenario suggested a similar benefits from source crusts in the agricultural environment; while the lack of data to substantiate this makes the proposal of implementing manure management strategies that promote crusting premature, these results suggest the potential for worthwhile research in this area.
Work is needed in accurately parameterizing field information for the model to expand on model corroboration. Also, further thought should be given to developing methods to test extended derivations of the model shown here to account for other situations; for example, where the pollutant in the bottom region doesn’t flush-out quickly or the rainfall ends before steady state flow is established. Results could be somewhat improved with the addition of dispersion to the model, though, as stated earlier, predictions are fairly good without this complication. Though further field testing is needed, the results shown above suggest potential for adaptation of this kind of mechanistic approach to farm management research and other field situations.
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List of Symbols
† : N = number of observations
‡ : RMSE = Root Mean Square Error:
Triangles and diamonds = Observed
Figure 5: Pollutant transport results, data and model (laboratory results).
Triangles = Observed Data for No Crust Experiment
Squares = Observed Data for 50% Crust Experiment
Circles = Observed Data for Complete Crust Experiment
Lines = Respective Model Predictions