1 Introduction to Surface Irrigation

Surface irrigation is the oldest and most common method of applying water to crops. It involves the distribution of water across the soil surface by gravity.

1.1 Key Types

  1. Basin Irrigation: Water is applied to a level area surrounded by dikes.
  2. Border Irrigation: The field is divided into long, parallel strips separated by low ridges.
  3. Furrow Irrigation: Water flows through small channels (furrows) between crop rows.

2 Hydraulic Principles and Formulas

The design of surface irrigation systems relies on the Volume Balance Equation and Infiltration Models.

2.1 1. Infiltration (Kostiakov Equation)

The rate at which water enters the soil is critical for determining how long water must remain on the surface. \[I = k \cdot t^a\] Where: - \(I\) = Cumulative infiltration (mm or m) - \(t\) = Intake opportunity time (min) - \(k, a\) = Soil-specific empirical constants

2.2 2. Manning’s Equation (Flow Velocity)

Used to calculate the velocity of water flow in furrows or borders. \[V = \frac{1}{n} R^{2/3} S^{1/2}\] Where: - \(V\) = Velocity (m/s) - \(n\) = Manning’s roughness coefficient - \(R\) = Hydraulic radius (m) - \(S\) = Longitudinal slope (m/m)

2.3 3. Volume Balance Equation

For a given time \(t\), the volume of water applied equals the volume on the surface plus the volume infiltrated. \[Q \cdot t = A_s \cdot L + z_{avg} \cdot L\] Where: - \(Q\) = Inflow rate per unit width (\(m^3/s/m\)) - \(A_s\) = Average cross-sectional area of surface storage (\(m^2\)) - \(L\) = Advance distance (m) - \(z_{avg}\) = Average depth of infiltrated water (\(m\))

3 Design Criteria for Furrow Irrigation

When designing a furrow system, we must determine the Maximum Unit Stream (\(Q_{max}\)) to avoid erosion and the Minimum Unit Stream (\(Q_{min}\)) to ensure water reaches the end of the field.

3.0.1 Erosion Velocity Limit

\[Q_{max} = \frac{C}{S}\] *Typically, \(C = 0.6\) to \(0.63\) for many soils (where \(Q\) is in L/s and \(S\) is in %).

4 Real-World Design Example

4.1 Problem Statement

A farmer wants to design a furrow irrigation system for a maize field. - Soil Type: Loam (\(k = 0.0051\) m/min^a, \(a = 0.5\)) - Field Length (\(L\)): 200 m - Furrow Spacing (\(W\)): 0.75 m - Required Net Depth (\(d_{net}\)): 75 mm (0.075 m) - Slope (\(S\)): 0.1% - Inflow Rate (\(Q\)): 1.5 L/s per furrow

4.2 Calculation using R

We will calculate the Intake Opportunity Time (\(t_{req}\)) required to deliver 75mm of water and the Total Application Time.

# Constants
k <- 0.0051       # Kostiakov constant
a <- 0.5          # Kostiakov exponent
d_req <- 0.075    # Required depth in meters
L <- 200          # Length in meters
W <- 0.75         # Spacing in meters
Q_ls <- 1.5       # Inflow in Liters per second

# 1. Calculate required intake opportunity time (t_req)
# From d = k * t^a -> t = (d/k)^(1/a)
t_req <- (d_req / k)^(1/a)

# 2. Calculate Total Volume required per furrow (m3)
vol_req <- d_req * L * W

# 3. Calculate Application Time (T_app) in minutes
# T_app = Vol / Q
Q_m3_min <- (Q_ls / 1000) * 60  # Convert L/s to m3/min
t_app <- vol_req / Q_m3_min

# Results
cat("Required Opportunity Time (min):", round(t_req, 2), "\n")
## Required Opportunity Time (min): 216.26
cat("Total Application Time (min):", round(t_app, 2), "\n")
## Total Application Time (min): 125

5 Design Considerations & Efficiency

5.0.1 1. Application Efficiency (\(E_a\))

\[E_a = \frac{\text{Water Stored in Root Zone}}{\text{Water Delivered to Field}} \times 100\]

5.0.2 2. Deep Percolation Ratio (DPR)

Occurs when the water remains at the head of the field longer than necessary, causing water to move below the root zone.

5.0.3 3. Tailwater Runoff

Water that reaches the end of the furrow and exits the field. This can be recaptured using a Tailwater Recovery System.

6 Summary Checklist for Design

  1. Analyze Soil: Determine infiltration characteristics (\(k, a\)).
  2. Select Flow Rate: Ensure \(Q\) is below the erosion limit but high enough to reach the end.
  3. Determine Length: Adjust field length to match soil type (shorter for sands, longer for clays).
  4. Calculate Timing: Ensure \(t_{app} \geq t_{req}\) at the furthest point of the field.

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6.0.1 How to use this:

  1. Install RStudio.
  2. Open RStudio and create a new file: File > New File > R Markdown.
  3. Delete the default template and paste the code above.
  4. Click the Knit button to generate a beautiful, formatted document.

6.0.2 Why R Markdown for Irrigation Design?

  • Reproducibility: You can change the flow rate or soil constants in the R code chunk, and the “Real-World Example” results will update automatically.
  • Math Support: It uses LaTeX to render professional engineering formulas.
  • Documentation: You can keep your design logic and calculations in one single file.