Horton Infiltration Rate

Robert Horton watched soil accept water and wrote down what it did. The rate starts high on dry ground, falls off as the surface layer fills and the pores clog with the fines rain knocks loose, and settles at a steady value once the profile is wetted through: $f = f_c + (f_0 - f_c)e^{-kt}$. A soil going from 200 mm/h to a final 20 mm/h with a decay constant of 2 per hour is passing $20 + 180e^{-1} = 86$ mm/h half an hour in. The three parameters have plain readings — $f_0$ is the dry-soil capacity, $f_c$ is essentially the saturated hydraulic conductivity of the limiting layer, and $k$ sets how fast the first collapses into the second.

Be clear about what the equation returns, because this is the classic misuse. It gives infiltration CAPACITY, not actual infiltration. Actual infiltration is the lesser of the capacity and the rainfall rate, so during light rain the soil takes everything that falls and the curve is irrelevant; only once rainfall exceeds capacity does ponding begin and the curve start to govern. Worse, the exponent is a function of elapsed time rather than of water absorbed, so the equation cheerfully decays the capacity during a dry interval in the middle of a storm when nothing is infiltrating at all. Multi-burst storms need the time-compression fix — track cumulative infiltration and re-enter the curve at the time that matches it — or a model built on cumulative depth in the first place.

The parameters come from a double-ring infiltrometer or from back-fitting a measured hydrograph, and they are site values, not table values. Published ranges are wide enough to be nearly useless as defaults: $f_0$ from 25 mm/h on clay to 250 or more on sand, $f_c$ from under 1 to about 25, and $k$ most often between 0.5 and 5 per hour. Two field realities that matter more than the arithmetic. A crust forms on bare soil under raindrop impact, and a crusted silt can drop below a tenth of the conductivity you would measure on the same soil undisturbed — which is the mechanism connecting infiltration to erosion. And compaction from construction traffic does the same thing permanently. If you are modelling a site that has been driven on, $f_c$ belongs to the compacted layer and not to the soil survey. Season moves the numbers too, and by more than most people expect: the same plot infiltrates several times faster in late summer, dry and cracked and worked over by roots and worms, than it does in early spring on a profile still holding winter water, and frozen ground behaves like pavement. A single set of Horton constants describes one soil in one condition, which is worth writing on the calculation rather than leaving implied. Horton is a fit, which is its honest weakness: three constants that describe what one soil did once, with no route from soil properties to parameter values. Green-Ampt is the model that offers that route.