Production Tubing Design and Force Analysis

Completions engineering skill for production tubing string design, force analysis, and buckling evaluation per Lubinski et al. (1962).

Important: Tubing force analysis is sensitive to completion geometry. Always verify packer type (free, anchored, set-on-tension/compression) before applying load case equations. For HP/HT wells, use dedicated software (WellCat, StressCheck, or equivalent).

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Module 1 — Tubing Properties

API Tubing Grades and Common Sizes

OD (in)	Weight (lb/ft)	ID (in)	Wall (in)	Grade	Yield (psi)
1.900	2.90	1.610	0.145	J-55	55,000
2⅜	4.70	1.995	0.190	J-55	55,000
2⅜	4.70	1.995	0.190	N-80	80,000
2⅞	6.50	2.441	0.217	J-55	55,000
2⅞	6.50	2.441	0.217	N-80	80,000
3½	9.30	2.992	0.254	J-55	55,000
3½	9.30	2.992	0.254	N-80	80,000
4	11.00	3.476	0.262	N-80	80,000

def tubing_cross_section_areas(od_in, id_in):
    """
    Returns pipe body cross-sectional area (steel) and internal area (in²).
    """
    import math
    A_i = math.pi / 4.0 * id_in**2   # internal (fluid) area
    A_o = math.pi / 4.0 * od_in**2   # outer area
    A_s = A_o - A_i                   # steel area
    return {"A_steel_in2": A_s, "A_inner_in2": A_i, "A_outer_in2": A_o}

Material Constants (Steel)

Property	Value	Units
Young's modulus (E)	30 × 10⁶	psi
Thermal expansion (α)	6.9 × 10⁻⁶	1/°F
Poisson's ratio (ν)	0.30	—
Density (steel)	489.5	lb/ft³

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Module 2 — Four Force Effects (Lubinski et al., 1962)

The total force change on a free-moving tubing string equals the sum of four independent effects. Each produces an axial force change ΔF (positive = tension, negative = compression).

Effect 1: Temperature

def delta_F_temperature(delta_T_avg_f, od_in, id_in, E=30e6, alpha=6.9e-6):
    """
    ΔF_T = -E * A_steel * alpha * ΔT
    Positive ΔT (heating during production): compressive force (negative)
    delta_T_avg_f: average temperature change along string (°F)
    Returns: ΔF_T in lbf (negative = compression added)
    """
    import math
    A_s = math.pi / 4.0 * (od_in**2 - id_in**2)
    return -E * A_s * alpha * delta_T_avg_f

Effect 2: Ballooning

def delta_F_ballooning(delta_Pi_psi, delta_Po_psi, od_in, id_in, nu=0.30):
    """
    ΔF_B = -2ν * (ΔPi * Ai - ΔPo * Ao)
    Positive ΔPi (pressure increase inside): compressive (shortens tubing, piston)
    delta_Pi_psi: change in average internal pressure (psi)
    delta_Po_psi: change in average annulus (external) pressure (psi)
    Returns: ΔF_B in lbf
    """
    import math
    Ai = math.pi / 4.0 * id_in**2
    Ao = math.pi / 4.0 * od_in**2
    return -2.0 * nu * (delta_Pi_psi * Ai - delta_Po_psi * Ao)

Effect 3: Piston Effect

def delta_F_piston(delta_Pi_psi, delta_Po_psi, od_in, id_in):
    """
    ΔF_P = ΔPo * Ao - ΔPi * Ai   (force at packer / area change)
    Occurs at any cross-section change: packer bore, tubing end, plug
    delta_Pi_psi: change in tubing pressure at packer
    delta_Po_psi: change in annulus pressure at packer
    Returns: ΔF_P in lbf (positive = tension)
    """
    import math
    Ai = math.pi / 4.0 * id_in**2
    Ao = math.pi / 4.0 * od_in**2
    return delta_Po_psi * Ao - delta_Pi_psi * Ai

Effect 4: Total Force Change and Buckling Check

def total_force_change(dF_T, dF_B, dF_P):
    """Returns total ΔF = ΔF_T + ΔF_B + ΔF_P (lbf)."""
    return dF_T + dF_B + dF_P

def buckling_check(F_initial_lbf, dF_total_lbf, W_buoyed_lb_ft, depth_ft):
    """
    Neutral point depth = (F_effective) / W_buoyed
    Below neutral point: compression -> buckling risk
    F_initial: initial tubing weight in fluid at packer (lbf, typically tension)
    """
    F_final = F_initial_lbf + dF_total_lbf
    neutral_point_ft = F_final / W_buoyed_lb_ft if W_buoyed_lb_ft > 0 else 0
    buckled_length = depth_ft - neutral_point_ft
    return {
        "F_final_lbf": F_final,
        "neutral_point_ft": neutral_point_ft,
        "buckled_length_ft": max(0.0, buckled_length),
        "buckling": buckled_length > 0
    }

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Module 3 — Buckling Analysis (Lubinski)

Critical Buckling Loads

def sinusoidal_buckling_force(w_e_lb_ft, r_c_in, EI_lbf_in2):
    """
    Sinusoidal buckling onset:
    F_cr_sin = 2 * sqrt(E*I * w_e / r_c)
    w_e_lb_ft: buoyed weight per unit length (lb/ft)
    r_c_in:    radial clearance between tubing OD and casing ID (inches)
    EI_lbf_in2: flexural rigidity = E * I
    Returns: F_cr_sin in lbf (compressive force to initiate sinusoidal buckling)
    """
    w_e_in = w_e_lb_ft / 12.0   # convert to lb/in
    r_c_ft = r_c_in / 12.0
    import math
    return 2.0 * math.sqrt(EI_lbf_in2 * w_e_in / r_c_in)

def helical_buckling_force(w_e_lb_ft, r_c_in, EI_lbf_in2):
    """Helical onset ≈ 2.83 * F_cr_sin (Dawson-Paslay, 1984)."""
    return 2.83 * sinusoidal_buckling_force(w_e_lb_ft, r_c_in, EI_lbf_in2)

def moment_of_inertia(od_in, id_in):
    """I = pi/64 * (OD^4 - ID^4) (in^4)"""
    import math
    return math.pi / 64.0 * (od_in**4 - id_in**4)

Helical Pitch and Lateral Force

def helical_pitch(F_comp_lbf, EI_lbf_in2, r_c_in):
    """
    Pitch of helix (in): p = 2*pi * sqrt(2*E*I / (F * r_c))
    Shorter pitch = tighter helix = more casing wear
    """
    import math
    return 2.0 * math.pi * math.sqrt(2.0 * EI_lbf_in2 / (F_comp_lbf * r_c_in))

def lateral_contact_force(w_e_lb_ft, r_c_in, p_in):
    """
    Lateral contact force per unit length (lb/ft):
    w_n = (4*pi^2 * w_e * r_c) / p^2
    """
    import math
    return (4 * math.pi**2 * w_e_lb_ft * r_c_in) / (p_in**2)

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Module 4 — Seal Assembly Stroke

def seal_assembly_stroke(total_movement_in, safety_factor=1.5):
    """
    Required seal assembly stroke = total tubing movement * SF
    total_movement_in: calculated free-end movement (inches)
    Return: minimum seal assembly length (in)
    """
    return abs(total_movement_in) * safety_factor

def tubing_movement_temperature(L_ft, delta_T_avg_f, alpha=6.9e-6):
    """
    Free thermal movement (in): delta_L = alpha * L * delta_T * 12
    Positive ΔT (heating): tubing elongates (moves up at surface, pushes down at packer)
    """
    return alpha * L_ft * delta_T_avg_f * 12.0

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Module 5 — Velocity String Sizing

def velocity_string_recommendation(q_gas_mscfd, p_wellhead_psi, T_wellhead_f,
                                   sigma=60.0, rho_l=62.4):
    """
    For a liquid-loading gas well, find the largest tubing ID that keeps
    gas velocity above Turner critical velocity.
    Returns: max ID (inches) for unloading.
    """
    import math
    T_R = T_wellhead_f + 459.67
    rho_g = (28.97 * 0.65 * p_wellhead_psi) / (10.73 * T_R * 0.9)
    v_crit = 1.593 * sigma**0.25 * (rho_l - rho_g)**0.25 / rho_g**0.5
    # q_gas (scf/day) = v_crit (ft/s) * A (ft2) * 86400 * (P/14.7) * (520/T_R)
    # Solve for A_max
    scfd = q_gas_mscfd * 1000.0
    A_max_ft2 = scfd / (v_crit * 86400.0 * (p_wellhead_psi / 14.7) * (520.0 / T_R))
    id_max_in = math.sqrt(4.0 * A_max_ft2 / math.pi) * 12.0
    return {"v_crit_ft_s": v_crit, "A_max_ft2": A_max_ft2, "ID_max_in": id_max_in}

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Module 6 — Packer Selection and DHSV

Packer Type Selection

Situation	Packer Type	Notes
Permanent (stimulation)	Permanent set, milled to retrieve	Best isolation; costly retrieval
Workover anticipated	Retrievable (slip, hydraulic)	Release with string manipulation
High-pressure differential	Permanent or production packer	Higher pressure rating
Multiple zones	Retrievable straddle or permanent	Zone-specific stimulation
ESP completion	Bagpipe / tandem seal	Accommodates ESP motor below

DHSV Setting Depth

def dhsv_minimum_depth_ft(wellhead_shut_in_psi, fluid_grad_psi_ft=0.433):
    """
    DHSV must be set deep enough that casing burst is protected if tubing fails.
    Minimum depth: DHSV setting depth > P_wellhead / fluid_gradient
    Regulation: typically 100–200 ft below surface casing shoe or as specified.
    Returns: minimum TVD for DHSV (ft)
    """
    return wellhead_shut_in_psi / fluid_grad_psi_ft

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Output Format

## Tubing String Design — [Well Name / API]
**String:** X-in OD, Grade N-80 / P-110 | **Set depth:** X,XXX ft
**Packer:** [Free / Anchored / Semi-anchored]

### Force Analysis Summary
| Load Case | ΔF_T (klbf) | ΔF_B (klbf) | ΔF_P (klbf) | ΔF_Total (klbf) |
|-----------|------------|------------|------------|----------------|
| Installation → Stimulation | | | | |
| Installation → Production | | | | |
| Production → Workover | | | | |

### Buckling Assessment
| Load Case | F_final (klbf) | Neutral Point (ft) | Buckled Length (ft) | Risk |
|-----------|---------------|-------------------|---------------------|------|
| | | | | |

### Movement and Seal Assembly
| Load Case | Free Movement (in) | Required Stroke (in) |
|-----------|-------------------|---------------------|
| | | |

### Velocity String Check (if applicable)
Required ID max: X.XX in | Selected: X-in OD, X.XXX ID

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Error Handling

Condition	Cause	Action
Neutral point above packer	Large compressive load	Verify packer type; increase set weight
Buckled length > tubing length	Extreme load case	Use anchored packer; check stimulation pressure
Movement exceeds seal stroke	Underdimensioned seal assembly	Specify longer seal; consider anchored completion
Negative torque on helix	Tension everywhere	No buckling; helix equations don't apply

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Caveats

• Lubinski four-effect analysis assumes straight wellbore and uniform temperature gradient. For deviated wells, contact forces and friction modify the analysis significantly — use dedicated software.
• Free-tubing analysis gives maximum movement; anchored packer absorbs movement and transfers force to the wellhead. Semi-anchored falls between.
• The Lubinski (1962) sinusoidal and helical buckling onset loads assume uniform tubing properties and no friction. Deviated wells have lower onset buckling loads due to gravity components.
• Thermal effects dominate in steam injection and high-GOR wells. Ensure temperature profile is accurately modeled for these cases.
• API RP 5C3 burst and collapse ratings apply; always check that tubing survives stimulation (frac) and production pressure differentials.