26.6. Infiltration Equations and Models

Developed by Robert E. Horton:

f(t)=f_0 + (f_c - f_0)e^{-kt}

Where:

• f(t): infiltration rate at time t
• f_0: initial infiltration rate
• f_c: final/constant infiltration rate
• k: decay constant
• t: time

Horton’s model fits well with field data and represents the declining infiltration rate over time.

Horton’s Equation is a mathematical formula used to describe how the rate of infiltration changes over time. Initially, when water begins infiltrating the soil, the rate is high, often represented by f_0. As time passes, environmental conditions such as soil saturation cause this rate to decrease, approaching a final steady value, f_c. The decay constant, k, determines how quickly this decrease occurs. By using this equation, engineers can predict how quickly and efficiently water will infiltrate into various soil types, which is crucial for water management.

Think of Horton’s equation like a sponge soaking up water. When you first pour water on a dry sponge (initial infiltration), it absorbs rapidly. But as the sponge becomes wetter, it absorbs more slowly until it can hold no more water (steady infiltration). This equation helps engineers understand how quickly and how much water will enter the ground compared to that initial rapid absorption.

Based on capillarity and gravity forces:

f(t) = S t^{-1/2} + A

Where:

• S: sorptivity (depends on soil suction and porosity)
• A: steady infiltration rate due to gravity

It is useful for short-duration infiltration analysis.

Philip’s Equation accounts for both the capillary action of the soil and the gravitational forces acting on water. The term 'S' reflects how quickly water can enter the soil based on its properties, while 'A' represents a constant rate of infiltration due to gravity once the soil is saturated. This equation is particularly useful for analyzing how much water will infiltrate during short rain events, allowing engineers to estimate immediate soil absorption rates effectively.

Imagine trying to pour syrup on a stack of pancakes. At the start, the syrup can soak in quickly due to the light, fluffy pancake texture (sorptivity), but as you continue to pour (increased saturation), it starts to flow off (steady rate). Philip’s equation helps predict how this process works for soil and water.

A physically based model assuming a sharp wetting front:

f = K rac{ ext{ψ}( heta_s - heta_i)}{F}

Where:

• K: hydraulic conductivity
• ψ: wetting front suction head
• θ_s: saturated moisture content
• θ_i: initial moisture content
• F: cumulative infiltration

The Green-Ampt Equation models infiltration by considering how a wetting front moves through the soil. The hydraulic conductivity (K) measures how easily water can flow through the soil, while the wetting front suction head (ψ) indicates how deeply the water is penetrating. This equation provides a more comprehensive understanding of water movement in soil, taking both initial moisture content and total infiltrated water into account. It is particularly accurate for scenarios where there is a clear transition from dry to wet soil, known as a wetting front.

Consider when you water a dry lawn. Initially, the water quickly saturates the top layer (the wetting front), pushing down through the soil. The Green-Ampt Equation helps predict how fast and deep that water will travel, similar to the way a wave moves through sand at the beach, creating layers of wet and dry sand.

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