Civil, Structural & Geotechnical Engineering
Kostiakov-Lewis (Modified Kostiakov) Irrigation Infiltration Formula: Z(t) & Intake Rate
Design furrow, border, and basin surface irrigation opportunity times using A. N. Kostiakov (1932) and M. R. Lewis / Mezencev’s Modified Kostiakov-Lewis Cumulative Intake Z(t) = k * (t^a) + f_0 * t and Instantaneous Rate I(t).
Fix Kostiakov’s long-time zero-rate flaw by adding steady basic intake rate f_0!
Russian soil hydrologist Alexei N. Kostiakov (1932) proposed the classical empirical power-law cumulative infiltration depth Z_K(t) = k * (t^a) (with 0 < a < 1), whose derivative dZ/dt = a * k * t^(a - 1) erroneously decays toward zero as irrigation opportunity time t grows to several hours! Adding steady basic infiltration rate f_0 (mm/min or m^3/m/min) yields the industry-standard Modified Kostiakov-Lewis (Mezencev, 1948; FAO Paper 45 / USDA NRCS WinSRFR) surface irrigation equation:
Cumulative Infiltrated Depth Z(t) = k * (t^a) + f_0 * t (mm), and Instantaneous Intake Rate I(t) = dZ/dt = a * k * (t^(a - 1)) + f_0 (mm/min)! Because Z(t) integrates both early transient macropore/matric filling k * t^a and steady deep percolation f_0 * t, irrigation engineers verify whether intake opportunity time t_opp delivers target root-zone depth Z_req!
| Opportunity Time t (k = 8.5 mm/min^a, a = 0.50, f_0 = 0.25 mm/min) | Transient Term k * t^a | Steady Term f_0 * t | Cumulative Z(t) (mm) | Instantaneous Rate I(t) |
|---|---|---|---|---|
| t = 16 min | 34.00 mm | 4.00 mm | 38.00 mm | 1.3125 mm/min |
| t = 64 min | 68.00 mm | 16.00 mm | 84.00 mm | 0.7813 mm/min |
| t = 100 min | 85.00 mm | 25.00 mm | 110.00 mm | 0.6750 mm/min |
Modified Kostiakov-Lewis Cumulative Infiltration Depth Z(t) (mm)
Evaluates k * (t^a) + f_0 * t across empirical coefficient k > 0 (B2), exponent 0 < a < 1 (B3), basic intake rate f_0 >= 0 (B4, mm/min), and opportunity time t > 0 (B5, min).
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B4 < 0, B5 <= 0), "Requires k > 0, 0 < a < 1, f_0 >= 0, and Time t > 0", ROUND(B2 * (B5 ^ B3) + B4 * B5, 2))Modified Kostiakov-Lewis Instantaneous Intake Rate I(t) = dZ/dt (mm/min)
Computes a * k * (t^(a - 1)) + f_0 at opportunity time t > 0.
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B4 < 0, B5 <= 0), "Requires k > 0, 0 < a < 1, f_0 >= 0, and Time t > 0", ROUND(B3 * B2 * (B5 ^ (B3 - 1)) + B4, 4))Try the example
Use the original values or change them to test your understanding.