Science
Dupuit-Forchheimer Unconfined Aquifer Seepage & Parabolic Water Table
Calculate steady unconfined groundwater seepage per unit width q' = K * (h1^2 - h2^2) / (2 * L) (m^2/day) and parabolic phreatic water-table elevation h(x) between two canals or earthen dam faces.
Quantify earthen levee seepage and phreatic water-table profiles!
In an unconfined water-table aquifer resting on an impermeable base between two parallel canals or reservoir boundaries separated by distance L (m), saturated thickness h(x) shrinks along the flow path—so hydraulic gradient steepens into a Dupuit Parabola rather than a straight line!
Under Jules Dupuit and Philipp Forchheimer's horizontal-flow assumption, steady discharge per meter of shoreline width integrates cleanly to the difference of SQUARED heads: q' = (K * (h_1^2 - h_2^2)) / (2 * L) (in m^2/day), and phreatic water-table elevation halfway across (x = L / 2, with zero recharge) equals the root-mean-square head h_mid = SQRT((h_1^2 + h_2^2) / 2)—always higher than the naive linear average (h_1 + h_2) / 2!
| Unconfined Seepage Case (Sand K = 12 m/d, Width L = 200 m) | Boundary Heads h_1, h_2 (m) | Discharge q' (m^2/day/m) | Midpoint Water Table h(L/2) |
|---|---|---|---|
| Steep Canal Drawdown | h_1 = 15.0 m, h_2 = 9.0 m | 4.320 m^2/d | 12.37 m (vs 12.00 m linear!) |
| Moderate Head Drop | h_1 = 15.0 m, h_2 = 12.0 m | 2.430 m^2/d | 13.58 m |
| Reversed Gradient | h_1 = 9.0 m, h_2 = 15.0 m | -4.320 m^2/d (Flow right-to-left) | 12.37 m |
Dupuit-Forchheimer Unit-Width Seepage Discharge q' (m^2/day)
Evaluates (K * (h1^2 - h2^2)) / (2 * L) from hydraulic conductivity B2 (m/d), upstream head B3 (m), downstream head B4 (m), and aquifer span B5 (m).
=IF(OR(B2 <= 0, B3 < 0, B4 < 0, B5 <= 0), "Invalid Aquifer Input", ROUND((B2 * ((B3 ^ 2) - (B4 ^ 2))) / (2 * B5), 3))Mid-Span Phreatic Water-Table Elevation h(L / 2) = SQRT((h1^2 + h2^2) / 2)
Computes parabolic water-table height halfway between the two constant-head boundaries.
=ROUND(SQRT(((B3 ^ 2) + (B4 ^ 2)) / 2), 2)Try the example
Use the original values or change them to test your understanding.