Engineering Hydrology · Soil Physics

Infiltration Equations: Horton, Philip, Green-Ampt, Ponding Time and the φ-Index (with Interactive Curves)

Q: What is the difference between infiltration rate and infiltration capacity?

Infiltration capacity is the maximum rate at which a soil can absorb water under given conditions. The actual infiltration rate equals the capacity only when water is supplied faster than the soil can take it (ponded conditions); when rainfall intensity is lower than the capacity, the infiltration rate equals the rainfall intensity.

Q: What is Horton's infiltration equation?

f(t) = fc + (f0 − fc) e^(−kt), where f0 is the initial infiltration capacity, fc the final (steady) capacity and k a decay constant. Cumulative infiltration is F(t) = fc·t + (f0 − fc)(1 − e^(−kt))/k. The parameters are empirical and are fitted to infiltrometer data.

Q: What are the Green-Ampt parameters?

The Green-Ampt model needs the saturated hydraulic conductivity K, the wetting-front suction head ψ and the moisture deficit Δθ = (1 − Se)·θe, where θe is the effective porosity and Se the initial effective saturation. Rawls, Brakensiek and Miller (1983) tabulated typical values by USDA soil texture class.

Q: What is the φ-index?

The φ-index is the constant loss rate such that rainfall above it equals the observed direct runoff. It is found by trial: assume φ, subtract φ·Δt from each rainfall interval, ignore negative values, and adjust φ until the remaining rainfall excess equals the runoff depth.

Q: What is the difference between the φ-index and the W-index?

The φ-index is a constant loss rate that lumps all losses, including initial losses, into one number. The W-index separates out the initial loss Ia: W = (P − R − Ia) / te, where te is the duration of rainfall excess. Textbooks define te slightly differently, so always state the definition you used.

Q: What is ponding time?

Ponding time is the time after the start of rainfall when the soil's infiltration capacity drops to the rainfall intensity and water begins to pond and run off. For constant intensity i in the Green-Ampt model, tp = Kψ Δθ / [i (i − K)]. No ponding occurs if i ≤ K.

Infiltration decides how much rain becomes runoff. This guide explains the three classic infiltration models, gives the Rawls Green-Ampt parameters for all 11 soil textures, shows how to compute ponding time, and works through the φ-index and W-index. Move the sliders to see the models respond, and compute a φ-index from your own storm.

3 models comparedRawls soil tablePonding timeφ-index calculator

By Bimal Ghimire · Published October 28, 2026 · 17 min read

Concept

Infiltration Basics

Infiltration is the entry of water through the soil surface. Its rate starts high on a dry soil, because capillary suction pulls water in, and decreases toward a steady value close to the saturated hydraulic conductivity as the wetted zone deepens.

Infiltration capacity f_p

The maximum rate the soil can absorb (cm/h or mm/h) at a given moment.

Actual rate f

f = min(i, f_p): equal to rainfall intensity i until the soil can no longer keep up.

Cumulative infiltration F(t)

Total depth absorbed since the start; f = dF/dt.

Rainfall excess

i − f once ponding occurs. This is the input to runoff models such as the unit hydrograph.

Field

Factors Affecting Infiltration and How It Is Measured

Soil texture and structure: sands infiltrate tens of times faster than clays; macropores from roots and worms add fast pathways.
Initial moisture: a wet soil has less suction and less storage left, so infiltration is lower (the reason for antecedent moisture classes in the SCS-CN method).
Surface condition: crusting under raindrop impact, compaction, frozen ground and vegetation cover.
Water quality and temperature: viscosity and suspended sediment that clogs pores.

Measurement: the double-ring infiltrometer (ASTM D3385) keeps a constant ponded head in an inner ring while the outer ring limits lateral spreading; readings of water added over time give f(t), to which Horton or Philip parameters are fitted. Rainfall simulators and tension infiltrometers are used for research.

Model 1

Horton's Infiltration Equation

$$f_p(t)=f_c+(f_0-f_c)\,e^{-kt},\qquad F(t)=f_c\,t+\frac{f_0-f_c}{k}\left(1-e^{-kt}\right)$$

$f_0$ initial capacity, $f_c$ final (steady) capacity, $k$ decay constant (1/h). Empirical (Horton 1933, 1940).

Fitting from field data: rearranging gives $\ln(f-f_c)=\ln(f_0-f_c)-kt$, a straight line. Plot $\ln(f-f_c)$ against t; the slope is −k.

Example: with f₀ = 6 cm/h, f_c = 0.6 cm/h and k = 4 h⁻¹, after 1 hour $f_p=0.6+5.4e^{-4}=0.70$ cm/h and $F=0.6+\frac{5.4}{4}(1-e^{-4})=1.93$ cm.

Model 2

Philip's Two-Term Equation

$$F(t)=S\,t^{1/2}+K\,t,\qquad f(t)=\tfrac12\,S\,t^{-1/2}+K$$

$S$ = sorptivity (cm h⁻¹ᐟ²), a measure of capillary uptake; $K$ ≈ hydraulic conductivity. A truncation of Philip's (1957) series solution of Richards' equation.

Early on, the $S\sqrt t$ term dominates (capillarity); later, gravity (K t) takes over. A useful link to Green-Ampt: $S\approx\sqrt{2K\psi\Delta\theta}$, which the interactive tool uses to derive Philip parameters from the selected soil.

Model 3

The Green-Ampt Model

Green and Ampt (1911) idealised the wetted zone as a saturated "piston" with a sharp wetting front at depth L, where a constant suction head ψ pulls water down. Darcy's law across the wetted zone, with $F=L\,\Delta\theta$, gives:

$$f=K\left(\frac{\psi\,\Delta\theta}{F}+1\right),\qquad F-\psi\Delta\theta\,\ln\!\left(1+\frac{F}{\psi\Delta\theta}\right)=K\,t,\qquad \Delta\theta=(1-S_e)\,\theta_e$$

The second equation is implicit in F; solve it by Newton's method or successive substitution (the tool does this exactly).

Rawls, Brakensiek & Miller (1983) parameters

Worked example (silt loam, S_e = 0.30, ponded from t = 0): Δθ = (1 − 0.30) × 0.486 = 0.340, ψΔθ = 16.68 × 0.340 = 5.67 cm, K = 0.65 cm/h. Solving $F-5.67\ln(1+F/5.67)=0.65\times1$ gives F = 3.17 cm after 1 hour, and $f=0.65(5.67/3.17+1)=$ 1.81 cm/h.

Runoff Onset

Ponding Time Under Constant Rainfall

Under rain of constant intensity $i$, all rain infiltrates until the Green-Ampt capacity falls to $i$. Setting $f=i$ with $F=i\,t_p$ (Mein & Larson 1973):

$$F_p=\frac{K\,\psi\Delta\theta}{i-K},\qquad t_p=\frac{K\,\psi\Delta\theta}{i\,(i-K)}\qquad(i>K)$$

After ponding, $F-F_p-\psi\Delta\theta\ln\!\frac{\psi\Delta\theta+F}{\psi\Delta\theta+F_p}=K(t-t_p)$. For the silt loam above under 5 cm/h of rain: $F_p=0.65\times5.67/4.35=0.85$ cm and $t_p=0.17$ h (10 minutes). If $i\le K$, the soil never ponds and there is no infiltration-excess runoff.

Interactive Infiltration Model Comparison

Green-Ampt

Soil texture (Rawls 1983)

Initial effective saturation S_e

Horton

f₀ (cm/h)

f_c (cm/h)

k (1/h)

Philip

Derive from Green-Ampt soil

S (cm/h^½)

K (cm/h)

Rainfall intensity i (cm/h, 0 = ponded)

Duration shown (h)

Evaluate at t (h)

Loss Indices

φ-Index and W-Index

For flood studies on gauged catchments, a simple constant loss rate often replaces infiltration equations. The φ-index is the uniform rate such that the rainfall above it equals the observed direct runoff R:

$$\sum_j \max\big(0,\;P_j-\varphi\,\Delta t\big)=R$$

$P_j$ = rainfall depth in interval j of length Δt. Solve by trial: assume all intervals contribute, compute φ = (P − R)/(n Δt), then drop intervals where $P_j<\varphi\Delta t$ and repeat.

$$W=\frac{P-R-I_a}{t_e}$$

The W-index removes initial losses $I_a$ (interception, depression storage) from the loss total; $t_e$ is the duration of rainfall excess. Definitions of $t_e$ vary between textbooks, so state the one you use.

Example: hourly rainfall 0.5, 1.2, 2.4, 1.8, 0.9 and 0.3 cm (total 7.1 cm), runoff 3.0 cm. First trial φ = (7.1 − 3.0)/6 = 0.68 cm/h, but the first and last hours are below it. Second trial with the four middle hours: φ = (6.3 − 3.0)/4 = 0.825 cm/h, and all four exceed it, so φ = 0.825 cm/h.

φ-Index Calculator

Rainfall depth per interval (cm or mm, comma separated)

Interval Δt (h)

Direct runoff R (same unit)

Initial loss I_a (for W)

Choosing

Which Infiltration Model Should You Use?

Model	Type	Parameters	Best for
Horton	Empirical	f₀, f_c, k (fitted)	Sites with infiltrometer data; urban models (SWMM)
Philip	Physically based (approx.)	S, K	Short to medium times, ponded infiltration analysis
Green-Ampt	Physically based	K, ψ, Δθ (from soil texture)	Ungauged sites, event models (HEC-HMS, SWMM, TUFLOW)
φ-index	Lumped loss rate	φ (from one observed storm)	Quick flood estimation on gauged catchments
SCS-CN	Empirical, depth-based	CN	Storm runoff volume; see the CN calculator

Keep Learning

Related Tools and Guides

FAQ

Frequently Asked Questions

What is the difference between infiltration rate and infiltration capacity?

Infiltration capacity is the maximum rate at which a soil can absorb water under given conditions. The actual infiltration rate equals the capacity only when water is supplied faster than the soil can take it (ponded conditions); when rainfall intensity is lower than the capacity, the infiltration rate equals the rainfall intensity.

What is Horton's infiltration equation?

f(t) = fc + (f0 − fc) e^(−kt), where f0 is the initial infiltration capacity, fc the final (steady) capacity and k a decay constant. Cumulative infiltration is F(t) = fc·t + (f0 − fc)(1 − e^(−kt))/k. The parameters are empirical and are fitted to infiltrometer data.

What are the Green-Ampt parameters?

The Green-Ampt model needs the saturated hydraulic conductivity K, the wetting-front suction head ψ and the moisture deficit Δθ = (1 − Se)·θe, where θe is the effective porosity and Se the initial effective saturation. Rawls, Brakensiek and Miller (1983) tabulated typical values by USDA soil texture class.

What is the φ-index?

The φ-index is the constant loss rate such that rainfall above it equals the observed direct runoff. It is found by trial: assume φ, subtract φ·Δt from each rainfall interval, ignore negative values, and adjust φ until the remaining rainfall excess equals the runoff depth.

What is the difference between the φ-index and the W-index?

The φ-index is a constant loss rate that lumps all losses, including initial losses, into one number. The W-index separates out the initial loss Ia: W = (P − R − Ia) / te, where te is the duration of rainfall excess. Textbooks define te slightly differently, so always state the definition you used.

What is ponding time?

Ponding time is the time after the start of rainfall when the soil's infiltration capacity drops to the rainfall intensity and water begins to pond and run off. For constant intensity i in the Green-Ampt model, tp = Kψ Δθ / [i (i − K)]. No ponding occurs if i ≤ K.

Sources

References

Horton, R. E. (1940). An approach toward a physical interpretation of infiltration-capacity. Soil Science Society of America Proceedings, 5, 399–417.

Philip, J. R. (1957). The theory of infiltration: 4. Sorptivity and algebraic infiltration equations. Soil Science, 84(3), 257–264.

Green, W. H., & Ampt, G. A. (1911). Studies on soil physics, 1. The flow of air and water through soils. Journal of Agricultural Science, 4(1), 1–24.

Rawls, W. J., Brakensiek, D. L., & Miller, N. (1983). Green-Ampt infiltration parameters from soils data. Journal of Hydraulic Engineering, 109(1), 62–70.

Mein, R. G., & Larson, C. L. (1973). Modeling infiltration during a steady rain. Water Resources Research, 9(2), 384–394.

Chow, V. T., Maidment, D. R., & Mays, L. W. (1988). Applied Hydrology, Chapter 4. McGraw-Hill, New York.

ASTM D3385. Standard Test Method for Infiltration Rate of Soils in Field Using Double-Ring Infiltrometer. ASTM International.