Geotechnical Engineering, Cofferdam Dewatering, Terzaghi Quicksand & Piping Heave
Terzaghi Critical Hydraulic Gradient & Quicksand Formula
Calculate Terzaghi Critical Hydraulic Gradient i_cr = (Gs - 1)/(1 + e), Cofferdam Upward Exit Gradient dh/L, Piping Factor of Safety FS, and Sand-Boiling Heave Audit.
Quicksand is NOT a special mineral—it is ordinary sand where upward seepage drag cancels submerged grain weight to ZERO effective stress!
Pump out a sheet-pile river cofferdam or levee toe trench under Total Hydraulic Head Difference `Delta_h` (m) across Subsurface Seepage Path Length `L` (m), and upward pore-water seepage exerts frictional drag `i * gamma_w` (`kN/m³`) against gravity on the submerged sand skeleton (`gamma’ = ((G_s - 1) / (1 + e)) * gamma_w`, where Quartz Specific Gravity `G_s ≈ 2.65–2.68` and Void Ratio `e ≈ 0.55–0.80`). Karl Terzaghi proved that when **Exit Hydraulic Gradient `i_exit = Delta_h / L`** reaches **Critical Hydraulic Gradient `i_{cr} = (G_s - 1) / (1 + e)`** (`≈ 0.95–1.05` in clean sands!), Effective Vertical Intergranular Stress `sigma’_v` drops to **ZERO**—turning dense sand into weightless, boiling **"quicksand"** that blows out cofferdam bottoms! Because local permeability lenses concentrate exit flow, USACE / Eurocode 7 dewatering rules demand **Piping Factor of Safety `FS = i_{cr} / i_{exit} >= 2.5` to `3.0`**!
Verify on a Dewatered River Sheet-Pile Cofferdam—Sand_Specific_Gravity `G_s` = 2.66 (B2), In-Situ_Void_Ratio `e` = 0.68 (B3 -> Critical Gradient `i_{cr} = 0.988` in E2), River-to-Excavation_Head `Delta_h` = 4.20 m (B4), and Sheet-Pile_Flow_Path `L` = 11.50 m (B5): Upward Exit Gradient equals `0.365` (E3), Piping Safety Factor reaches `2.71 >= 2.5` (E4), confirming `"SAFE AGAINST PIPING (FS>=2.5)"` (E5)!
| Terzaghi Seepage Piping KPI (Gs=2.66, Void Ratio e=0.68, Head dh=4.2 m, L=11.5 vs 6.0 m) | Formula | Verified Output |
|---|---|---|
| Critical Gradient i_cr (Boiling) | (Specific_Gravity_Gs - 1) / (1 + Void_Ratio_e) | 0.988 (near unity for quartz sand) |
| Upward Exit Gradient i_exit | Total_Head_Diff_dh_m / Flow_Path_L_m | 0.365 (0.700 @ shallow L=6.0 m cutoff!) |
| Piping Factor of Safety FS | Critical_Gradient_icr / Exit_Gradient_i | 2.71 (1.41 < 2.5 @ shallow L=6.0 m!) |
| Cofferdam Sand-Boil Audit | FS >= 2.5 USACE Dewatering Threshold | SAFE AGAINST PIPING (FS>=2.5) |
Terzaghi Zero-Effective-Stress Critical Gradient & Cofferdam Piping FS Formula
Validate `Specific_Gravity G_s` in `[2.30, 3.20]`, `Void_Ratio e` in `[0.25, 1.50]`, `Head_Diff Delta_h > 0` m, and `Seepage_Path L > 0` m right on `E2` (checking `ISTEXT(E2)` on `E3..E4` so `E5` status text stays independent)!
=LET(gs_s, B2, vr_e, B3, dh_m, B4, lp_m, B5, ic_r, (gs_s - 1) / (1 + vr_e), ie_x, dh_m / lp_m, fs_p, ic_r / ie_x, IF(OR(gs_s < 2.3, gs_s > 3.2, vr_e < 0.25, vr_e > 1.5, dh_m <= 0, lp_m <= 0), "Invalid Seepage Input", IF(fs_p >= 2.5, "SAFE AGAINST PIPING (FS>=2.5)", "QUICKSAND BOILING RISK")))Under-Driven Sheet-Pile Cutoff (L = 6.0 m) Sand-Boiling Blowout Check (Set Seepage Path B5 = 6.0 m)
Stop sheet-pile driving early at Cobble Refusal so Seepage Path shrinks to `L` B5 = 6.0 m: Exit Gradient E3 nearly doubles to `0.700`, Piping Safety Factor E4 plunges to `1.41`, and E5 triggers `"QUICKSAND BOILING RISK"`—requiring an inverted graded-gravel filter blanket or deep relief wells before pumping down!
=IF(ISTEXT(E2), E2, LET(ic_r, (B2 - 1) / (1 + B3), ie_x, B4 / B5, ROUND(ic_r / ie_x, 2)))Try the example
Use the original values or change them to test your understanding.