Non-Newtonian Fluid Mechanics, API 13D Herschel-Bulkley & Bingham Plug Flow
Herschel-Bulkley Yield Stress & Pipe Plug-Flow Formula
Calculate Herschel-Bulkley Total Shear Stress tau (Pa), Apparent Viscosity (mPa·s / cP), Unsheared Central Plug Radius Ratio rp/R, and Sheared Flowing Area (%).
Why xanthan drilling muds and concrete slurries travel down pipe centers as a solid, un-sheared cylindrical plug!
Drilling fluids, mine paste backfill, cement slurries, and ketchup combine a finite **Yield Stress `tau_y`** (`Pa`, needed to suspend barite weighting solids and rock cuttings the instant mud pumps shut off!) with shear-thinning power-law flow (`Flow Behavior Index n < 1`, `Consistency Index K` Pa·s^n). **W. H. Herschel & R. Bulkley (`API RP 13D`)** unified Bingham plastics (`n = 1`) and Ostwald power-law fluids (`tau_y = 0`) into **`tau = tau_y + K * gamma_dot^n`** (`Pa`), giving **Apparent Viscosity `mu_{app} = (tau / gamma_dot) * 1,000`** (`mPa·s` = `cP`)! Inside cylindrical pipe flow of Wall Shear Stress `tau_w = (R * Delta_P) / (2 * L)` (`Pa`), radial shear stress drops linearly toward zero at the pipe axis (`tau(r) = tau_w * (r / R)`): wherever `tau(r) <= tau_y`, the fluid **cannot shear at all**—moving as a rigid **Unsheared Central Plug of Radius Ratio `r_p / R = MIN(1, tau_y / tau_w)`** surrounded by a **Sheared Annular Envelope `(1 - (r_p / R)^2) * 100%`**!
Verify on an API RP 13D Xanthan-Polymer Drilling Mud at Fann `300` RPM (`511 s⁻¹`)—Yield_Stress `tau_y` = 8.40 Pa (B2), Consistency `K` = 0.42 Pa·s^n (B3), Shear-Thinning_Index `n` = 0.68 (B4), Viscometer_Shear_Rate `gamma_dot` = 511 s⁻¹ (B5), and Pipe_Wall_Shear_Stress `tau_w` = 24.0 Pa (B6): Shear Stress equals `37.57` Pa (E2), Apparent Viscosity equals `73.53` cP (E3), Central Plug Radius Ratio `r_p / R` equals `0.350` (E4), leaving `87.75%` of pipe cross-section actively sheared (E5)!
| API RP 13D Herschel-Bulkley KPI (K=0.42, n=0.68, gamma=511 s⁻¹, tau_w=24 Pa) | Formula | Verified Output |
|---|---|---|
| Total Shear Stress tau (Pa) | Yield_tau_y_Pa + K * (Shear_Rate ^ n) | 37.57 Pa (48.97 Pa @ gelled 19.8 Pa) |
| Apparent Viscosity (mPa·s / cP) | (Shear_Stress_Pa / Shear_Rate_inv_s) * 1000 | 73.53 cP (95.84 cP @ gelled 19.8 Pa) |
| Central Plug Radius Ratio (rp / R) | MIN(1, Yield_tau_y_Pa / Wall_tau_w_Pa) | 0.350 (0.825 massive core plug @ 19.8 Pa!) |
| Sheared Annular Area (%) | (1 - (Plug_Radius_Ratio ^ 2)) * 100 | 87.75% (31.94% thin wall slip @ 19.8 Pa!) |
API RP 13D Herschel-Bulkley Constitutive Law & Cylindrical Plug-Core Formula
Validate `Yield_Stress tau_y >= 0` Pa, `Consistency K > 0.001` Pa·s^n, `Behavior_Index n` in `(0.10, 2.00]`, `Shear_Rate gamma_dot > 0.1` s⁻¹, and `Wall_Shear tau_w > 0.1` Pa!
=LET(ty_p, B2, kc_p, B3, nb_i, B4, gd_s, B5, tw_p, B6, IF(OR(ty_p < 0, kc_p <= 0.001, nb_i <= 0.1, nb_i > 2, gd_s <= 0.1, tw_p <= 0.1), "Invalid Rheology Input", ROUND((1 - (MIN(1, ty_p / tw_p) ^ 2)) * 100, 2)))Flocculated High-Yield Bentonite Gel (`tau_y = 19.8` Pa) 82.5% Plug-Core Check (Set Yield Stress B2 = 19.8 Pa)
Contaminate bentonite mud with cement calcium so Yield Stress jumps to `tau_y` B2 = 19.8 Pa: Central Plug Radius Ratio E4 swells to **`0.825`** (`82.5%` of pipe diameter moves as an un-mixed solid piston!) and Sheared Envelope Area E5 shrinks to `31.94%`—promoting flat-profile cuttings transport at low velocity!
=IF(ISTEXT(E5), E5, ROUND(MIN(1, B2 / B6), 3))Try the example
Use the original values or change them to test your understanding.