Jason Icerman

Journal of Undergraduate Research
Volume 7, Issue 1 - September/October 2005

High Frequency Irrigation Simulation (HFIS) Model: Development and Analysis

INTRODUCTION

Population growth in the U.S. continues to burden our water resources. Though improvement in water consumption rates has been seen since the early 1980s, rates in the U.S. remain at about 408 billion gallons of water per day, with irrigation consuming the most freshwater at 137 billion gallons per day (USGS, 2004). Hence, an important issue in agriculture is irrigation management coupled with water conservation. With irrigation being the primary use of developed water supplies, in many areas the irrigation water demand exceeds the available supply at current costs (Pitts et al., 1996). In agriculture, the goal is higher crop yield (higher profits). Water conservation, while preferred, will often be sacrificed to obtain higher yields in the belief that the profit gains will outweigh the extra water costs. This quandry perpetuates the never ending search for maximum profit from minimal water usage.

Optimizing irrigation scheduling can increase water use efficiency (Howell et al., 1975). Water requirements become even more important when water consumption is regulated, such as under the Water Management Districts in the State of Florida. Irrigation models are a useful tool for managing and scheduling irrigation when water is limited or regulated (Santos et al., 2000). Though users still request a water allotment based on crop requirements, often models are used by governing bodies to determine an appropriate figure. The goal of the consumptive use program in the State of Florida is to provide water for reasonable-beneficial uses while protecting the water resources of the Districts (SJRWMD, 2004). Currently, Consumptive Use Permits (CUPs) do not incorporate effective rainfall into their water limitations. Effective rainfall describes precipitation neither lost through runoff nor deep percolation. The inclusion of an accurate representative of effective rainfall in CUPs will aid in water conservation by creating the most precise crop water requirement possible. If permit allowances reflect the true requirement of a user, a more precise distribution of available water can occur — in effect allowing more access to more users. The creation of a model with the ability to determine crop water need throughout the growing season, using the most efficient and effective irrigation scheduling possible while focusing on accurate accounting of effective rainfall, should enhance the regulatory success of CUPs.

The objectives of this study were to first create a one-dimensional model capable of simulating crop water use, irrigation, drainage, and effective rainfall at hourly and daily time steps. Second, a sensitivity analysis was conducted to determine the effect of input changes on key outputs.

MATERIALS AND METHODS

Model Logic

The model, High Frequency Irrigation Simulation (HFIS), utilizes the Microsoft Excel spreadsheet interface to simulate crop water need throughout the growing season. The model receives a spreadsheet input along with 19 quantitative inputs from the user (Figure 1).

Figure 1. User input parameters as displayed in model, Kc Break Down, Root Zone Function, and Other Inputs tab displays can be seen in Figures A1, A2, and A3 in the Appendix.

The required spreadsheet input from the user had to meet the following format: contain date and time information in the first column, evapotranspiration (ET) data in millimeters in the second column, and rain data in millimeters in the third column, with the three columns of data left justified and beginning at row one. Once all input data were entered correctly, the data were stored in arrays and as single variables.

In general, the model is governed by a balance of the water in the crop root zone as follows:

(Equation 1)

where:

CropH₂Oreq is the water required by the crop (mm)
ET_c is the crop evapotranspiration (mm)
EffectiveRain is the depth of rain entering the root zone (mm)

(Equation 2)

where:

ΔS is the change in storage at a given time (mm)
Rain is the rain occurring at a given time (mm)
Irrigation is the scheduled irrigation at a given time (mm)
ET is the evapotranspiration at a given time (mm)
RunOff + Drainage is water unable to be stored in the root zone (mm)

However, these equations are broken into several functions in the model logic. The first function run by the model was the creation of a crop coefficient (K_c) function for the entire specified season. A simple step-function was assumed to represent the change in the crop coefficient over the season similar to the function defined in FAO - 56 (Allen et al., 1998). At time zero, the K_c function returns a value equal to the first entered threshold and continues to return that value until the first seasonal fraction is reached. Seasonal fractions are input by the user as decimals. They represent a break point in the season similar to ranges defining a combination function (Figure 2). Returning to the K_c function, from the first seasonal fraction to the second seasonal fraction, the second threshold is returned; from the second seasonal fraction to the third seasonal fraction, the third threshold is returned; and from the third season fraction to the end of the season, the fourth and final threshold is returned. A representative crop water need could then be calculated by Equation 3.

(Equation 3)

where:

ET_c is the crop evapotranspiration (mm)
REFET is the ET data from the spreadsheet input (mm)
K_c is the crop coefficient determined by the step function (mm)

The root zone function was developed by the summation of two linear functions and two thresholds. The representative equation returns a root zone depth output equal to the initial depth at time zero. The return value then increases linearly until it reaches a value equal to the first threshold at a time equal to the first threshold seasonal fraction. The return value remains at this threshold until the time equals the second threshold seasonal fraction, at which point the return value again begins increasing linearly until maximizing the output root zone depth at the second threshold, with a time equal to the third threshold seasonal fraction. The function returns the second threshold value until the end of the season (Figure 2).

Figure 2. Sample root zone function representation.

The available water storage was quantified at each time interval by multiplying the root zone function at that interval by the difference between the field capacity and permanent wilting point as entered by the user (Equation 4).

(Equation 4)

where:

AWS is the available water storage (mm)
FC is the field capacity (mm/mm)
PWP is the permanent wilting point (mm/mm)
RZ is the root zone depth (mm)

Now that a water storage volume and a crop water need at each time interval had been calculated according to user specifications, the soil moisture saturation depth was modeled through the season.

The model had two logical sequences at this stage, one for daily irrigation routines and one for hourly irrigation routines. The daily soil moisture logic, including irrigation and drainage events, will be described first due to the relative simplicity. For the daily simulations, the model compared the current day’s (for daily scheduling time steps were in days) parameters, as shown in Equation 5.

(Equation 5)

where:

Θ is the soil moisture content (mm)
ET is evapotranspiration (mm)
I is irrigation (mm)
R is rain (mm)

AWS is available water storage (mm)

The logic statement above accounts for possible drainage, with i representing the current time step. If the addition of water depth on the previous day exceeds the water storage capacity on the current day, drainage occurs and the soil moisture depth (Θ) equals the available water storage. If the soil moisture level falls below the maximum depletion threshold, as input by the user, an irrigation event would take place at that time interval (Equation 6).

(Equation 6)

where:

Θ is the soil moisture content (mm)
MAD is maximum allowable depletion (mm/mm)
AWS is available water storage (mm)
I is irrigation (mm)

Added into the above logic is the ability to specify a minimum irrigation depth. If the difference between the available water storage and soil moisture on day i is less than the specified minimal irrigation, no irrigation would be scheduled for day i and the drop below the minimal soil moisture level would be ignored.

(Equation 7)

where:

R is rain (mm)
ETc is crop evapotranspiration (mm)
I is irrigation (mm)
AWS is available water storage (mm)

If all the water present could not be stored in a given time interval, the excess water was represented as drainage (Equation 7). Calculation of drainage allowed for a calculated effective rainfall is shown in Equation 8.

(Equation 8)

where:

REF is effective rain (mm)

R is rain (mm)
D is drainage (mm)

The hourly scheduling simulation of soil moisture, and the associated irrigation and drainage, is complicated by the introduction of a user input drainage delay. In general, the above logic remained, but a drainage event no longer lasted for a single interval, but lasts as long as specified by the user. The time extension retains more water in the soil profile overall, since the soil moisture depth was at a maximum as long as drainage was occurring. The amount of drainage per time interval was determined by dividing the drainage depth equally among the time intervals (for example a drainage event of 3 mm with a 2 hr delay had a model output of three 1 mm drainage events, time i, time i + 1, time i + 2). If a second drainage event occurred during the delayed response of a previous event, the remaining drainage volume depth from the previous event was added to the depth of the new event and the “delay clock” was reset to zero. Accruing the drainage surplus in a storage variable separate from the drainage array allowed the model to time-release given amounts as stated above.

The model then prompts the user to select a high frequency or a daily irrigation simulation. Upon selection, the model output a new workbook containing three worksheets. The first worksheet displays a summary of the events scheduled according to the simulation selected, irrigation depths and event times. The second worksheet contains a table summary of parameters calculated during the simulation. The third worksheet is a graphical summary of parameters calculated during the simulation.

Sensitivity Analysis

After completion, the model was verified against an accepted irrigation scheduling spreadsheet. The model showed similar output values for both the daily and hourly simulations. Verification complete, sensitivity analysis was performed for all the user input variables over a compilation of almost two years (544 days) of rain and evapotranspiration data.

To find the impact of each variable on model calculations, a baseline simulation was created. The baseline values were changed in increments of 25% from -100% to 100%, with the baseline values representing a 0% change (Figure 3). The six variables compared for each input were the average soil water content over the season, total irrigation, drainage, and effective rainfall over the season, and the number of irrigation and drainage events over the season.

Figure 3. Sensitivity analysis parameters and values.

The comparison results were then graphed for each of the input changes. For the root zone function and crop coefficient function the seasonal fractions were not changed and the thresholds were treated as a single variable when changed.

DISCUSSION

Model Results

A notable observation seen during model runs was the consistently higher irrigation sums from the high frequency irrigation simulations (HFIS) relative to the daily irrigation simulations (DIS), 621mm to 582mm for the baseline simulations used in the sensitivity analysis. One would expect hourly simulations to conserve water; however, the reason this is not seen in the model is the presentation of irrigation as a depth (see the baseline simulations of Table A1 through Table A7). Upon further inspection, the relatively smaller percentage difference of 6.7% between the HFIS and DIS cumulative irrigation depths is outweighed by a larger difference in average soil moisture content over the season, 7.31 mm and 6.54 mm, respectively, for HFIS and DIS baseline simulations (see the baseline simulations of Table A1 through Table A7). The 11.8% change in average soil moisture content is almost double the change in irrigation depth applied in the two simulations, with a higher soil moisture level creating a more crop-friendly environment. It can be inferred that the DIS simulation with the lower average soil moisture content induces more crop stress, though crop stress is not accounted for or quantified in the model.

Sensitivity Analysis

As discussed in the Materials and Methods section, sensitivity analysis was performed on the quantitative inputs. The model output values of the soil water content average, irrigation sum, drainage sum, effective rain sum, irrigation count, and drainage count were measured at 25% intervals from a -100% to 100% increase. Visual representations of the sensitivity analyses can be seen in Figure 4 through Figure 10 and are discussed below.

Sensitivity of maximum allowable depletion (MAD) in the model calculations is best seen through the change in irrigation events and effective rain magnitude from both the daily and hourly simulations (Figure 4 & Table A5). Increasing irrigation events with decreasing MAD percentages is seen in both simulation figures. Rising event counts is explained by the shrinking root zone associated with a decreasing MAD, a shallow root zone will dry out faster than a deeper root zone. A root zone that dries more rapidly needs more frequent irrigation events of smaller magnitudes. This increase in events coupled with a marginal increase in irrigation depth is exhibited in Figure 4. Associated with the higher irrigation counts is the decrease in effective rainfall. With a quicker demand on irrigation events, and subsequent increase in events, there is less opportunity for infiltrating rain to be effective. Also, deeper root zones accompanied by larger time intervals between events allow for more rainfall to infiltrate the soil, resulting in more effective rainfall.

Figure 4. Sensitivity analysis of the Maximum Allowable Depletion parameter where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

Note: Maximum allowable depletion simulations were extrapolated from a baseline simulation with no minimum irrigation depth (minimum irrigation depth = 0). The reason for the change lies in the model logic. A minimum irrigation depth too large eliminated describable effects of the maximum allowable depletion percentage. All other input parameter analyses were developed from a baseline simulation having a minimum irrigation depth of 10 mm (see Materials and Methods section).

The effect of the crop coefficient function is quite opposite of the MAD on the model. For both the high frequency simulations and daily simulations, a clear increase in irrigation events and summation can be seen as the K_c values increase (see Figure 5 & Table A7). Also seen, though on a smaller magnitude, is an increase in effective rainfall as K_c values increased. Both aforementioned observations can be attributed to a higher crop water demand created by the increasing K_c thresholds. Since the baseline simulation had a relatively small root zone compared to the increasing K_c thresholds, it holds that a larger increase would be seen in irrigation events and depth. A larger root zone coupled with increasing crop water demand would take longer to deplete and allow for more effective rainfall to occur relative to required irrigation. Also seen in Figure 5 is a small decrease in drainage and average soil water content as K_c values increase. On the other side of the graphs, an increase is seen drainage and soil water content larger in magnitude than the decrease seen as K_c values increased. The difference in magnitude is most likely due to the K_c values approaching zero, causing the root zone to remain near saturation for longer periods of time.

Figure 5. Sensitivity analysis of the Crop Coefficient Function where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

Changing the root zone function thresholds had little effect on any model variables besides average soil water content (Figure 6 & Table A6). The average soil water content for both simulation methods increases as root zone function thresholds increased. The lack of differentiation seen in the other output variables may be explained by the relatively high minimum irrigation requirement. The high irrigation depth requirement would maintain a stable irrigation pattern through the simulations. The deeper root zone allows for more water accumulation in the soil, hence the rise in soil water content, with little change in irrigation events as discussed there would be a negligible effect on output parameters outside of average soil water content.

Figure 6. Sensitivity analysis of the Root Zone Function where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulatio

Unlike other user inputs, the drainage delay only affected high frequency irrigation simulations and, accordingly, sensitivity analysis for the drainage delay variable were only performed for hourly simulations. The largest change, as to be expected, was the rise in drainage events as the delay is increased. As written in the model code, the drainage delay creates more events through the delay at each time interval, so it follows an increase in drainage events occurs with the increasing delay (Figure 7 & Table A4). Using similar logic, an increase in effective rainfall is expected and again is observed. There is little change seen in the other variables. A proportional decrease in irrigation depth would be anticipated assuming an increase in effective rainfall; however, since analyses were performed on a percentage basis, the decrease in irrigation depth, while equal in depth to the increase in effective rainfall, was proportionally less.

Figure 7. Sensitivity analysis of the Drainage Delay parameter where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

The most dramatic changes in model output were observed during the sensitivity analysis performed on the minimum irrigation input (Figure 8 & Table A3). First, as predicted, decreasing the minimum irrigation requirement greatly increased the number of irrigation events scheduled for both daily and hourly simulations. Conversely, since increasing the minimum irrigation requirement effectively bypasses irrigation events, an increase in effective rainfall is expected and observed (Figure 8). The extended time between irrigation events, due to the increasing minimum irrigation depth, is reason for both observations reported above.

Figure 8. Sensitivity analysis of Minimum Irrigation Depth parameter where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

Similar to results obtained from the root zone function sensitivity tests, permanent wilting point (PWP) and field capacity (FC) analyses affected only one variable significantly, average soil water content (Figure 9, 10 & Table A1, A2); however, the individual results vary inversely. As field capacity increased through the analyses, so did the average soil water content as compared to increasing the permanent wilting point, which decreased the average soil water content. The increase in average soil water content observed with increasing FCs showed a differential around twice the magnitude of the decrease observed with increasing PWPs. The inverse relationship between the two input variables is expected, due to their opposite effect on the root zone depth. Increasing the FC yields a deeper usable root zone while increasing the PWP shrinks the root zone. Little change is seen in irrigation and drainage events since there was no change in crop water demand over the simulations.

Figure 9. Sensitivity analysis of the Permanent Wilting Point parameter where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

Figure 10. Sensitivity analysis of the Field Capacity parameter where DIS represents the Daily Irrigation Simulation and HFIS represents the High Frequency Irrigation Simulation.

CONCLUSION

The study illustrated that even a simple one-dimensional model could incorporate effective rainfall into irrigation scheduling. Extrapolating from this, future regulation can be drafted to include effective rainfall into allotment considerations, which would in turn encourage growers to reduce water usage. Further research is needed to determine all the implications of effective rainfall inclusion.

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