Understanding Horton's Equation for Infiltration Rate

Horton's equation is a widely used empirical model in hydrology to describe the process of infiltration, which is the movement of water from the ground surface into the soil. It models how the rate of infiltration changes over time during a rainfall event.

According to Horton's model, the infiltration rate starts at a high initial value and decreases exponentially over time, eventually reaching a constant minimum rate. This decline happens because the soil pores near the surface become saturated, reducing the capacity for further infiltration.

The mathematical form of Horton's equation for the infiltration rate \(F_t\) at any given time \(t\) is:

\(F_t = F_c + (F_o - F_c) e^{-kt}\)

Let's break down the terms in this equation:

• \(F_t\): Infiltration rate at time \(t\) (e.g., in cm/hr or in/hr). This is what the equation calculates.
• \(F_o\): Initial infiltration rate at the beginning of the rainfall (\(t = 0\)). This is typically the maximum rate.
• \(F_c\): Final or equilibrium infiltration rate. This is the constant minimum rate that the infiltration approaches as time becomes large.
• \(k\): A decay constant specific to the soil and cover conditions. It determines how quickly the infiltration rate decreases from \(F_o\) to \(F_c\). A larger \(k\) means faster decay.
• \(e\): The base of the natural logarithm (approximately 2.718).
• \(t\): Time elapsed since the start of the rainfall or infiltration process.

The term \((F_o - F_c)\) represents the difference between the initial and final infiltration rates, which is the amount by which the rate decreases. The exponential term \(e^{-kt}\) describes how this difference decays over time. As \(t\) increases, \(e^{-kt}\) decreases, causing \(F_t\) to approach \(F_c\).

Comparing this with the given options, the correct form is the one where the initial difference \((F_o - F_c)\) is added to the final rate \(F_c\), and this difference is multiplied by an exponential term with a negative exponent \(-kt\), indicating decay over time.