This article demonstrates a piston effect and how it will change the length of tubing with full detailed calculations.
For the analysis, Lubinski’s sign conventions are used.
Compression force = ⇑(+)
Tensile force = ⇓ (-)
Shorten in Length = (-)
Elongate in length = (+)
Note: The tubing is free to move.
Please see this in the reference section to get a full paper.
Total force change
Total length change
Where;
Ai = Tubing ID area (in2)
Ao = Tubing OD area (in2)
Ap = Packer Seal Bore Area (in2)
As = Cross sectional area of tubing (in2)
L = Length of tubing (in)
ΔPo = Change in annulus pressure @ packer
ΔPi = Change in tubing pressure @ packer
ΔF =Change in force (lb)
ΔLpiston = Change in length due to piston effect (in)
E = Young modulus of steel (30×106 psi)
Pi = pressure in tubing (psi)
Po= pressure in annulus (psi)
These formulas are suitable for both configurations shown in Figure 1.

Figure 1 – Packer Configuration
Example
Packer is set at 10,000 ft.
Tubing and packer are free to move.
The well is vertical.
4.5” Tubing
ID of tubing = 3.862”
Packer seal bore outside diameter = 5.0”
Weight per length = 17.7 lb/ft
Initial Condition
Fluid in annulus = 10.0 ppg
Fluid in tubing = 10.0 ppg
Tubing pressure = 0 psi
Annulus pressure = 0 psi
Final Condition
Fluid in annulus = 8.0 ppg
Fluid in tubing = 10.0 ppg
Tubing pressure = 1,500 psi
Annulus pressure = 0 psi

Figure 2 – Initial and Final Condition
Calculate Areas
Ai = Tubing ID area (in2)
Ai= (π÷4) × 3.8622 = 11.497 in2
Ao = Tubing OD area (in2)
Ao= (π÷4) × 4.52 = 15.904 in2
Ap = Packer Seal Bore Area (in2)
Ap= (π÷4) × 5.02 = 19.635 in2
As = Cross sectional area of tubing (in2)
As = Ao – Ai = 15.904 – 11.497 = 4.407 in2
Determine Pressure at Packer at Initial Condition
Po @ surface = 0 psi
Pi @ surface = 0 psi
Po @ packer = Po @ surface + Hydrostatic P in annulus = 0 + (0.052×10×10,000) = 5,200 psi
Pi @ packer = Pi @ surface + Hydrostatic P in tubing = 0 + (0.052×10×10,000) = 5,200 psi
Determine Pressure at Packer at Final Condition
Po @ surface = 0 psi
Pi @ surface = 1500 psi
Po @ packer = Po @ surface + Hydrostatic P in annulus = 0 + (0.052×10×10,000) = 5,200 psi
Pi @ packer = Pi @ surface + Hydrostatic P in tubing =1,500 + (0.052×8×10,000) = 5,660 psi
Determine Pressure Change (ΔP)
ΔPo = Change in annulus pressure @ packer
ΔPo = 5,200 – 5,200 = 0 psi
ΔPi = Change in tubing pressure @ packer
ΔPi = 5,660 – 5,200 = 460 psi
Determine Force Change (ΔF)
ΔF = 3,744 lb (compression force (+))
Determine Length Change (ΔLpiston)
ΔLpiston = – 3.4 inch (shorten)
Conclusion
Based on the given information, change in force acting at the packer is 3,744 lb (compression force (+)) and length change due to piston effect is 3.4 inch (shorten).
References
Jonathan Bellarby, 2009. Well Completion Design, Volume 56 (Developments in Petroleum Science). 1 Edition. Elsevier Science.
Wan Renpu, 2011. Advanced Well Completion Engineering, Third Edition. 3 Edition. Gulf Professional Publishing.
Ted G. Byrom, 2014. Casing and Liners for Drilling and Completion, Second Edition: Design and Application (Gulf Drilling Guides). 2 Edition. Gulf Professional Publishing.
Lubinski, A., & Althouse, W. S. (1962, June 1). Helical Buckling of Tubing Sealed in Packers. Society of Petroleum Engineers. doi:10.2118/178-PA