Petroleum & Geothermal Wellbore Surveying, MWD Directional Drilling & Torque/Drag
Directional Drilling Dogleg Severity & Minimum Curvature Formula
Calculate 3D Spherical Dogleg Angle beta (deg), Normalized Dogleg Severity DLS (deg/30 m), Minimum-Curvature Ratio Factor RF, and True Vertical Depth Step Delta_TVD (m).
Turning azimuth at high inclination spikes 3D Dogleg Severity—even when inclination barely changes!
Measurement-While-Drilling (MWD) directional surveys record Measured Depth interval Delta_MD (m), Inclination from vertical I_1, I_2 (deg), and Azimuth A_1, A_2 (deg) at successive pipe stands! Industry-standard Minimum Curvature Surveying fits a smooth spherical circular arc between the two 3D tangent unit vectors via Exact Dogleg Angle beta = ACOS(COS(I_1) * COS(I_2) + SIN(I_1) * SIN(I_2) * COS(A_2 - A_1))! Normalizing beta to a 30-meter stand gives Dogleg Severity DLS = (beta_deg / Delta_MD) * 30 (deg/30 m—where excessive doglegs fatiguing drill-pipe tool joints or hanging up casing strings trigger reaming alarms!), while smoothing straight-chord average cosines by the Minimum-Curvature Ratio Factor RF = (2 / beta_rad) * TAN(beta_rad / 2) (with RF = 1 when beta = 0) yields Exact Vertical Depth Increment Delta_TVD = (Delta_MD / 2) * (COS(I_1) + COS(I_2)) * RF!
Verify on a 30.0 m Build-and-Turn MWD Stand—Course_Length Delta_MD = 30.0 m (B2), Upper_Inclination I_1 = 15.0° (B3), Lower_Inclination I_2 = 18.0° (B4), Upper_Azimuth A_1 = 45.0° (B5), and Lower_Azimuth A_2 = 55.0° (B6): 3D Dogleg Angle beta hits 4.121° (not 3.0°!, E2), Normalized DLS equals 4.12 deg/30 m (E3), Minimum-Curvature Ratio Factor RF equals 1.00043 (E4), and True Vertical Depth Step Delta_TVD adds 28.77 m (E5)!
| MWD Minimum-Curvature KPI (dMD=30 m, I: 15°->18°, Az: 45°->55°) | Formula | Verified Output |
|---|---|---|
| 3D Dogleg Angle beta (deg) | DEGREES(ACOS(COS(I1)*COS(I2)+SIN(I1)*SIN(I2)*COS(dA))) | 4.121° (10.858° @ I2=21°, Az2=75°) |
| Dogleg Severity DLS (deg/30 m) | (beta_deg / Course_dMD_m) * 30 | 4.12 °/30m (10.86 °/30m sharp!) |
| Minimum-Curvature Factor RF | IF(beta=0, 1, (2/beta_rad)*TAN(beta_rad/2)) | 1.00043 (1.00300 @ sharp turn) |
| True Vertical Depth Step dTVD (m) | (dMD / 2) * (COS(I1) + COS(I2)) * RF | 28.77 m (28.58 m @ sharp turn) |
API / SPE Minimum-Curvature Dogleg Severity & Delta_TVD Formula
Validate Course_Length Delta_MD > 0, Inclinations in [0°, 180°], Azimuths in [0°, 360°], and guard beta_rad < 1e-6 cleanly so straight tangent sections evaluate RF = 1 without division by zero!
=LET(md_m, B2, ia_d, B3, ib_d, B4, aa_d, B5, ab_d, B6, IF(OR(md_m <= 0, ia_d < 0, ia_d > 180, ib_d < 0, ib_d > 180, aa_d < 0, aa_d > 360, ab_d < 0, ab_d > 360), "Invalid Survey Input", LET(dr_f, PI() / 180, ia_r, ia_d * dr_f, ib_r, ib_d * dr_f, da_r, (ab_d - aa_d) * dr_f, cs_b, MIN(1, MAX(-1, COS(ia_r) * COS(ib_r) + SIN(ia_r) * SIN(ib_r) * COS(da_r))), bt_r, ACOS(cs_b), rf_m, IF(bt_r < 0.000001, 1, (2 / bt_r) * TAN(bt_r / 2)), ROUND((md_m / 2) * (COS(ia_r) + COS(ib_r)) * rf_m, 2))))Aggressive Rotary-Steerable Build & 30° Azimuth Turn Check (Set I_2 = 21° & Az_2 = 75°)
Steer harder across the 30 m stand to I_2 B4 = 21.0° and Azimuth A_2 B6 = 75.0°: 3D Dogleg Angle E2 jumps to 10.858°, Dogleg Severity E3 spikes to 10.86 deg/30 m, Ratio Factor E4 rises to 1.00300, and Delta_TVD E5 foreshortens to 28.58 m!
=IF(ISTEXT(E5), E5, LET(dr_f, PI() / 180, ia_r, B3 * dr_f, ib_r, B4 * dr_f, da_r, (B6 - B5) * dr_f, bt_d, ACOS(MIN(1, MAX(-1, COS(ia_r) * COS(ib_r) + SIN(ia_r) * SIN(ib_r) * COS(da_r)))) / dr_f, ROUND((bt_d / B2) * 30, 2)))Try the example
Use the original values or change them to test your understanding.