Natural Frequency

Key Takeaways

• Drillstring vibration and natural frequency resonance are among the most significant sources of downhole tool failure and reduced rate of penetration in rotary drilling operations: the drillstring (drill collars, heavy-weight drillpipe, and regular drillpipe extending from the surface to the bit) has multiple natural frequencies in each vibration mode -- torsional (the frequency at which the string twists and untwists, driven by bit torque variability and string stiffness), axial (the frequency at which the string bounces up and down, driven by bit bounce on the formation), and lateral (the frequency at which the drill collars whirl or bow sideways in the wellbore, driven by mass imbalance and eccentricity effects); when the rotary speed (RPM), which sets the excitation frequency as fractions or multiples of the rotation frequency (e.g., 2 blade tricone bit generates excitation at 2 x RPM/60 Hz per revolution), coincides with a natural frequency of the drillstring, resonance occurs and vibration amplitudes grow until limited by damping (from drilling fluid viscosity, formation contact, stabilizer contact) or until fatigue failure of a component terminates the resonance by physically separating the string; field symptoms of torsional resonance (stick-slip) include surface torque oscillations visible on the rig display, bit speed variations of 0 to 300 percent of average RPM at the bit (measured by near-bit MWD accelerometers), and premature tool joint and bit failure; lateral resonance (whirl) causes severe shock loads in the drill collars (up to 50 to 100g) visible on downhole vibration sensors and manifests at surface as erratic WOB and torque fluctuations.
• Acoustic resonance in the tubing fluid column is a source of pressure pulsations and vibration in production and injection wells: the tubing string filled with production fluid (oil, gas, water, or multiphase mixture) behaves as a closed-open acoustic resonator (closed at the wellhead by the surface choke or tree valve, open at the perforations where the wellbore fluid communicates with the reservoir), with natural frequencies determined by the acoustic velocity in the tubing fluid, the tubing length, and the boundary conditions at each end; the fundamental acoustic natural frequency for a quarter-wave resonator (closed at one end, open at the other) is f_n = (2n-1) * c / (4L), where n is the mode number (1, 2, 3...), c is the acoustic velocity in the fluid (approximately 1,000 to 1,500 m/s for liquid, 200 to 400 m/s for gas at wellhead conditions), and L is the tubing length; for a 3,000 m tubing string filled with oil (c approximately 1,200 m/s), the fundamental resonant frequency is approximately 0.1 Hz (one full pressure wave cycle per 10 seconds) -- well below the operating frequency of most surface pumps and compressors; however, the higher modes (harmonics at multiples of the fundamental) can fall in the range of pump or compressor operating frequencies and may excite dangerous pressure pulsations in the wellhead and surface equipment if not controlled by suction and discharge pulsation dampers on the surface pump system.
• Christmas tree and wellhead equipment acoustic and mechanical resonance have been the root cause of several reported fatigue cracking incidents in high-pressure, high-production-rate gas wells: gas production at high flow rates through a wellhead choke can generate acoustic noise (broadband pressure pulsations with energy at all frequencies from the turbulent flow noise) that excites acoustic resonance in connecting piping, instrument tubing, and small-bore connections on the tree; the resonance amplifies the cyclic stress on small-bore welds and threaded connections (which have stress concentration factors of 2 to 5 relative to plain pipe) until fatigue crack initiation occurs, typically at the weld toe or thread root where peak stress is highest; structural resonance of the tree body (excited by wind, compressor vibration, or production flow turbulence) at frequencies near the mechanical natural frequency of a tree fitting (typically 10 to 100 Hz for small-bore connection assemblies) can cause fretting fatigue at bolted connections and vibration-induced loosening of instrument connections; vibration monitoring of Christmas trees using accelerometers and regular baseline surveys (API RP 17K for subsea trees) provides early warning of developing resonance conditions and guides corrective action (adding vibration dampers, changing choke settings, or replacing susceptible connections with more robust designs).
• Sucker rod pumping unit natural frequency analysis is performed to identify critical pump speeds at which the rod string-pump system reaches resonance, which generates large cyclic loads that exceed the Goodman fatigue allowable and cause premature rod or coupling failure: the sucker rod string transmits the alternating polished rod load from the surface pumping unit to the downhole pump plunger as a longitudinal wave that propagates at the acoustic velocity for a solid steel rod (approximately 16,700 ft/s); for a rod string of length L (in feet), the fundamental torsional (actually longitudinal) natural frequency is f_n = c/(2L) Hz; for a 5,000-ft rod string (c = 16,700 ft/s), f_n = 1.67 Hz, corresponding to a pump speed of 100 strokes per minute; operating at or near this frequency places the pumping unit at the fundamental natural frequency and can cause resonant amplification of the dynamic loads by factors of 2 to 5 above the static design loads, causing rapid fatigue failure; API RP 11L rod string design calculations include identification of critical speeds to avoid, typically specifying operating ranges that exclude pump speeds within 10 to 15 percent of any natural frequency resonance; wave equation modeling of the rod string-pump system (using the Gibbs equation or equivalent wave propagation model) identifies these critical speeds for any specific rod string geometry and well depth, allowing the operating speed to be set safely below or above the resonance band.
• Pipeline and riser natural frequency analysis is required for offshore production systems where ocean current-induced vortex shedding (VIV, vortex-induced vibration) can excite lateral natural frequencies of free spans (unsupported pipe sections between seabed supports) and risers: when flow passes around a cylindrical pipeline or riser, alternating vortices are shed at a frequency proportional to flow velocity (Strouhal relationship: f_s = St * U/D, where St = 0.2 for a cylinder, U is the current velocity, and D is the pipe diameter), and if the vortex shedding frequency approaches a natural frequency of the pipe span, resonance occurs with cyclic lateral displacements that induce cyclic bending stress at the supported ends; the first natural frequency of a free-spanning pipeline section (for a pinned-pinned beam with span length L, pipe mass m per unit length, and bending stiffness EI) is f_1 = (pi/2L^2) * sqrt(EI/m) Hz; for a 12-inch pipeline free span of 30 meters in 100 m water depth, f_1 is approximately 0.2 to 0.5 Hz depending on wall thickness and content density, while the vortex shedding frequency at 0.5 m/s current is approximately 0.8 Hz -- potentially near resonance for deeper spans or longer spans where f_1 is lower; free span assessment (per DNV RP-F105) and VIV suppression (using helical strakes on the pipeline or riser that disrupt vortex formation) are standard elements of offshore pipeline and riser design.