Philip Two-Term Soil Infiltration Formula: Sorptivity S, Gravity Factor A & Rate i(t)

Separate early capillary suction (S * t^0.5) from late gravitational drainage (A * t)!

Whereas Horton and Kostiakov infiltration equations are empirical curve fits, Australian soil physicist John R. Philip (1957, Soil Science) solved Richards’ nonlinear unsaturated PDE analytically as a power series in SQRT(t), retaining the physically transparent two-term law: Cumulative Infiltration Depth I(t) = S * SQRT(t) + A * t (cm or mm) and Instantaneous Infiltration Rate i(t) = dI/dt = 0.5 * S / SQRT(t) + A!

At early storm times (t -> 0), dry-soil matric suction dominates via Sorptivity S (cm / hr^0.5), pulling water into pores equally horizontally or vertically; at long storm times (t -> infinity), 0.5 * S / SQRT(t) decays to zero and vertical infiltration approaches the steady gravitational transmission parameter A (typically 0.33 * K_sat to 1.0 * K_sat)!

Philip Cumulative Infiltration Depth I(t) = S * SQRT(t) + A * t

Computes cumulative infiltrated water depth (cm) for sorptivity B2 >= 0, gravity parameter B3 >= 0, and storm time t > 0 in B4.

=IF(OR(B2 < 0, B3 < 0, B4 <= 0), "Requires S >= 0, A >= 0, and Time t > 0", ROUND(B2 * SQRT(B4) + B3 * B4, 4))

Philip Instantaneous Infiltration Capacity i(t) = 0.5 * S / SQRT(t) + A

Evaluates time derivative dI/dt (cm/hr) at elapsed storm time t > 0.

=IF(OR(B2 < 0, B3 < 0, B4 <= 0), "Requires S >= 0, A >= 0, and Time t > 0", ROUND(0.5 * B2 / SQRT(B4) + B3, 4))

Further reading

• USDA NRCS — National Engineering Handbook Part 630 Hydrology