## Summary#

In this section we describe the calculation machinery behind axial loads calculations in Oliasoft WellDesign™. There are essentially two inputs that control the calculations, i.e. internal & external pressure and a temperature profile. The calculations are relative to some condition, normally the initial conditions which usually consist of the hydrostatic pressure and the geothermal temperature. The calculations can be divided into five separate calculations, ballooning- & piston- effects due to change in internal/external pressure, thermal effects, bending forces from the trajectory, and buckling effects. Axial loads calculations can be applied to arbitrary string configurations, and both packers, cement, and combinations are included.

NOTE!
In this documentation we denote any tubular as tubing. All calculations however encompass any tubular, such as tubings, casings, liners, tie-backs etc.

### Printable Version#

Oliasoft Technical Documentation - Axial Loads

## Input#

Axial loads calculations are done on a per string basis, meaning that the string under consideration is not affected by other casing strings in the design. Necessary input, is a complete description of the casing string. That is, a complete description of the wellbore trajectory where the casing string is defined, including measured depth, true vertical depth, inclination, azimuth, and dogleg severity, and also where, if any, there are packers or cement. For packers, information about allowable movement and seal bore area is also necessary. Also, the dimensions of the casing are needed, i.e. inner- and outer- diameters, weight per meter, yield, anisotropy, and asymmetry. In addition, initial internal- & external- pressure profile, initial temperature profile, and initial axial loads are needed, which usually stem from initial conditions. When all these are given, the axial loads calculations are based on three inputs, internal- & external- load case pressure profiles, and load case temperature profile. All the calculations are relative to the initial pressure and temperature profile.

NOTE!
For a detailed description of the calculations in this document, we refer to .

## Stress and Strain#

Consider a tubular with inside- and outside- diameter $d_i$ and $d_o$, respectively, implying an inside- and outside- area of:

$A_i = \frac{\pi}{4}d_i^2, \qquad A_o = \frac{\pi}{4}d_o^2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$

Let $A_x = A_o - A_i$ denote the cross sectional area. If an axial force, $F_z$, is applied to the tubing, then the axial stress $\sigma_z$ is given by:

$\sigma_z = \frac{F_z}{A_x} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$

Such a force also elongates or deforms the tubing, and the corresponding axial strain, $\epsilon_z$, is defined as:

$\epsilon_z = \frac{\Delta L}{L}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$

where $L$ is the original length of the tubing, and $\Delta L$ is the length change due to $F_z$.

Please note, in this document axial tensile forces are defined positive, while axial compressive forces are defined negative.

### Hooke's Law#

Elasticity is the property of a material to retain its original shape once the load is relieved. Hooke’s law states that the stress in a material is proportional to the strain which produced it, up to a limit called yield stress. The relation is expressed as:

$\sigma = E \epsilon\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4)$

here the proportionality constant, $E$, is called Young’s modulus.

### Poisson's Ratio#

An axial force, $F_z$, applied to a tubing, not only generates an axial strain, $\epsilon_z$, but also generates a radial strain, $\epsilon_r$. In the elastic regime, where the tubing is not permanently deformed if the load is relieved, these quantities are proportional, with proportionality constant, $\mu$, called Poisson's ratio, i.e.:

$\mu = -\frac{\epsilon_r}{\epsilon_z} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(5)$

## Assumptions#

1. We consider a tubing of length $L$, and we assume all the dimensions of the the tubing to be known. Specifically, let $A_i$ denote the inside area of the tubing, $A_o$ denote the outside area of the tubing, and $A_x = A_o - A_i$.
2. Assume the material in the tubing has Young's modulus, $E$, Poisson's ratio, $\mu$, and coefficient of (linear) thermal expansion, $C_T$.
3. Initially, the tubing is exposed to internal- and external- pressure profiles, $p_{i, 0}$ and $p_{e, 0}$, respectively. All pressure related calculations below are relative to these pressures.
4. Initially, the tubing is exposed to a temperature profile, $T_0$. All temperature related calculations are relative to this temperature profile.
5. Initially, the tubing has a known axial profile (usually from initial conditions). All axial loads calculations are relative to this initial profile.
6. For the pressure related calculations, we assume the tubing is exposed to an internal- and external- pressure profile, $p_{i, 1}$ and $p_{e, 1}$, respectively. Let $\Delta pi = p_{i,1} - p_{i,0}$ and $\Delta p_e = p_{e,1} - p_{e,0}$.
7. For the temperature related calculations, we assume the tubing is exposed to a temperature profile, $T_1$. Let $\Delta T = T_1 - T_0$.

## Ballooning#

Radial length change of a tubing, caused by a change in pressure, also results in a length change of the tubing, given by:

$\Delta L_B = \frac{-2\mu L}{EA_x}\left(A_i\Delta p_i - A_o\Delta p_e\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)$

In this calculation, we assume in addition that the tubing has a constant outer diameter, and a uniform wall thickness. If the tubing consists of sections with varying outer diameter and wall thickness, the calculation above are done per section, and then added together.

If the tubing is free to move, there is no axial force associated with ballooning. If, on the other hand, the tubing is fixed, this length change is converted to force through Hooke's law.

$F_B = 2\mu\left(A_i\Delta p_i - A_o\Delta p_e\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(7)$

### Piston Effects#

Piston effects are pressure-area effects that both results in change of axial forces and movement (if allowed). This effect, occurs where there is a change in geometry of the tubing, i.e. at any crossovers, at the bottom, at plugs inside the tubing, and at expansion devices. In other words, if there is a pressure, $p$, across an area, $A$, the resulting force, normal to the surface defined by $A$, is given by:

$F_p = A p \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(8)$

If movement is allowed, the length change due to this force, is given by Hooke’s law, equation 4:

$\Delta L = \frac{LF}{EA_x} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(9)$

### Crossovers#

A crossover occurs whenever there is a change in the diameters of the string, e.g. from a larger tubing above to a smaller tubing underneath, or vice versa. Also, if the outside diameter remains the same, and the inner diameter changes, internal pressure generates a point load. Denote by $\Delta A_i$ and $\Delta A_o$, the differential inner- and outer- area, respectively, across such a crossover. Then, the crossover force is given by:

$F_{XO} = \Delta p_i \Delta A_i - \Delta p_e \Delta A_o \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(10)$

This force act at the crossover, hence creates a discontinuity in axial load at the crossover.

### Plugs and Base of Tubing#

A tubing plug occupies the internal area, $A_i$, of the tubing, and if there is a differential pressure across the plug, a piston force occurs. This force is simply given by:

$F_{plug} = \Delta p_{plug} A_i \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(11)$

where $\Delta p_{plug} = \Delta p_i^+ - \Delta p_i^- = p{i,1}^+ - p_{i,1}^-$, and the $+$ and $-$ denote above and below the plug, respectively. The force act at the plug and creates a discontinuity in axial load at the plug.

At the base of the tubing, assumed open ended, we get a compressive axial force due to pressure acting on the cross sectional area, $A_x$, given by:

$F_{base} = -\Delta p_{e, base} A_x \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(12)$

where $\Delta p_{e, base}$ denotes the differential external pressure at the base of the tubing.

### Expansion Devices#

Expansion devices come in different flavors, however, the piston force due to such a device is device independent, and is given by:

$F_{ED} = \Delta p_e (A_{ED} - A_o) - \Delta p_i (A_{ED} - A_i) \;\;\;\;\;\;\;\;\;\;\;(13)$

where $A_{ED}$ is the seal bore area.

### Thermal Effects#

If the temperature profile around a tubing is changed, then the length of the tubing changes according to:

$\Delta L_T = C_T L \Delta T \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(14)$

As for ballooning, if the tubing is free to move, there is no axial force associated with a change in temperature. Also, if the tubing is fixed, this length change is converted to force through Hooke's law, equation 4,

$F_T = -C_T E A_x \Delta T \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(15)$

### Remark#

Cement is assumed to prevent axial movement, hence, in cemented areas of a string, ballooning, piston effects, and thermal effects directly create stress.

With the framework described above, a new axial load can be calculated based on the change in pressures and temperature. This is done using the following algorithm:

a) Divide the casing string into subsections, e.g. in 100 m intervals, and use the end points of these intervals as calculation points. In addition, any change in the casing configuration needs to be included, including change in wellbore trajectory, i.e. calculation points at packers, cement, crossovers, plugs, change in dogleg severity, etc.

b) Calculate the length change of the tubing, if free to move, from ballooning and thermal effects.

c) Calculate axial forces and associated length changes of the tubing, if free to move, from piston effects.

d) Identify all points in the casing string which are fixed or have limited allowable movement, typically packers or cemented areas. Then convert the cumulative movement of the string, from ballooning, thermal-, and piston- effects, to an axial force at these points, using Hooke's law, equation 4. Note, if there is limited movement, and the cumulative movement is greater than the allowable movement, the calculation of a restoring force has to take the allowable movement into account.

e) Calculate a final axial load, not including bending from dogleg, by summing up all the contributions and add it to the initial axial load. At this point, it is important to acknowledge the direction of the forces, and also which parts of the tubing string to be affected by the different restoring forces.

In deviated wellbores, the effect of wellbore curvature must be considered when the axial load on a tubing, inside the wellbore, is calculated. When a tubing is forced to bend, a tensile/positive stress occurs on the outside of the bend, and a compressive/negative stress occurs on the inside of the bend. Assume the pipe takes the form of a circular arc, with radius of curvature $R$. The radius of curvature is the inverse of the dogleg severity, $\beta$, i.e $R = 1/\beta$. Then, the bending stress at the outside of the pipe is:

$\sigma_b^{\pm} = \pm \frac{E d_o}{2R} = \pm \frac{E d_o\beta}{2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$

where $+$ and $-$ sign indicates outside and inside of the bend, respectively.

NOTE!
Replace $d_o$ with $d_i$ to get the bending stress at the inside of the pipe.

The associated bending force is given by:

$F_b^{\pm} = \sigma_b^{\pm} A_x \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(17)$

Let $Y_{nb}$ denote axial load no bending, as calculated above. Axial load with bending, $Y_b$, from dogleg is then given by:

$Y_b = \begin{cases} Y_{nb} + F_b^+,& Y_{nb} > 0 \\ Y_{nb} + F_b^-,& Y_{nb} < 0\end{cases} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(18)$

We observe that bending stress is a purely geometrical effect, and is different from zero if the dogleg severity is different from zero, making it also a local effect.

## Buckling#

Buckling is a more complex phenomenon than the effects described above. In the following, a short description of the mathematics involved is given, together with an algorithm for calculating the effects of buckling. For a more comprehensive description, reference is made to  and .

When a tubing is in compression, it will shorten according to Hooke's law. However, this is not the only effect of compression, if the compression force is big enough. At some point a critical compressional force is reached where the tubing is in an unstable condition, and will tend to buckle. It should be remarked that buckling is also dependent on internal- and external- pressure, and in theory, a tubing under tension can buckle if the internal pressure is big enough.

The following quantities are involved in buckling calculations:

In the presence of internal- and external- pressures, $p_i$ and $p_o$, respectively, the effective axial load on the tubing is given by:

$F_{eff} = -Y_{nb} + p_iA_i - p_oA_o \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(19)$

where $Y_{nb}$ denotes the axial load without bending from dogleg.

### Effective Buoyed Weight and Casing Contact Load#

When a tubing is displaced in a fluid, the effective buoyed pipe distributed weight is given by:

$W_e = \rho_T g + \rho_{f, i} g A_i - \rho_{f, e} g A_o \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(20)$

where $\rho_T$ is the weight of the tubing in $kg/m, \rho_{fi}$ and $\rho{f, e}$ are the internal- and external fluid density, respectively, and $g$ is the gravitational constant.

From this we calculate the casing contact load, given by:

$W_c = \sqrt{\left(W_e\sin(\alpha) + F_{eff}\frac{d\alpha}{dt}\right)^2 + \left(F_{eff}\sin(\alpha)\frac{d\epsilon}{dt}\right)^2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(21)$

where $\alpha$ and $\epsilon$ denote the inclination and azimuth of the wellbore trajectory, respectively, and the derivations are with respect to curve length, i.e. measured depth.

### Pasley Buckling Force#

The critical compression force is given by:

$F_p = \sqrt{\frac{EIW_c}{r}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(22)$

where $I$ is the second moment of area, given by $I = \frac{\pi}{64}(d_o^4 - d_i^4)$, and $r$ is the radial clearance, i.e. the difference in radius between the inside of the wellbore wall and the outside of the tubing.

It is the ratio between $F_{eff}$ and $F_p$ that determines whether there is buckling, or not, and also what type of buckling. In axial load calculations, we distinguish between no buckling/neutral $(N)$, lateral buckling $(L)$, and helical buckling $(H)$, and the conditions are:

$\text{Buckling mode} = \begin{cases} N , \quad &\frac{F_{eff}}{F_p} < 1 \\ L , \quad 1 \leq &\frac{F_{eff}}{F_p} < 2\sqrt{2} \\ H , \quad &\frac{F_{eff}}{F_p} \geq 2\sqrt{2} \end{cases} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(23)$

Buckling related quantities we calculate are, the helix angle and the resulting dogleg severity, $\Theta_{buc}$ and $\beta_{buc}$, strain, $\epsilon_{buc}$, bending moment, $M_{buc}$, and bending stress, $\sigma_{buc}$. The formulas for these quantities are related to the buckling mode. In the neutral mode, there is no buckling, hence all of these are zero.

### Helix Angle and Dogleg Severity#

The bend of the helix, aka the helix angle, is not constant for lateral buckling, hence the maximum is approximated. For helical buckling, on the other hand, this angle is constant.

The approximate solution for the maximum helix angle for lateral buckling is 

$\Theta_{buc, max}^L = \frac{1.1227}{\sqrt{2EI}}F_{eff}^{0.04}(F_{eff} - F_p)^{0.46} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(24)$

The helix angle for helical buckling is given by:

$\Theta_{buc, max}^H = \sqrt{\frac{F_{eff}}{2EI}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(25)$

The resulting dogleg severity is given by:

$\beta_{buc} = r\,\Theta_{buc}^2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(26)$

where $\Theta_{buc}$ is either $\Theta_{buc, max}^L$ or $\Theta_{buc, max}^H$, depending on the buckling mode.

### Buckling Strain#

Buckling reduces the length of the tubing, and the related buckling strain, $\epsilon_{buc}$, is useful, which is a function of the helix angle and radial clearance.

For lateral buckling, the helix angle is not constant, and an average is used for the buckling strain:

$\epsilon_{buc}^L = -0.7285\frac{r^2}{4EI}F_{eff}^{0.08}(F_{eff} - F_p)^{0.92} \;\;\;\;\;\;\;\;\;\;\;\;\;(27)$

For helical buckling, the buckling strain is:

$\epsilon_{buc}^H = -\frac{r^2}{4EI}F_{eff} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(28)$

### Bending Moment and Bending Stress#

The bending moment is directly related to the dogleg severity, and is given by:

$M_{buc} = EI\Theta_{buc} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(29)$

where $\Theta_{buc}$ is either $\Theta_{buc, max}^L$ or $\Theta_{buc, max}^H$, depending on the buckling mode.

The related bending stress on the outside of the pipe is given by:

$\sigma_{buc} = \frac{M_{buc} d_o}{2I} = \frac{Ed_o\Theta_{buc}}{2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(30)$

NOTE!
Replace $d_o$ with $d_i$ to get the bending stress at the inside of the pipe.

### Algorithm#

Since bending stress from buckling is dependent on the axial load, and the quantity we want to compute is axial load including bending stress from buckling, the effect of buckling needs to be calculated in an iterative manner. One way to proceed, is to iterate on the length change due to buckling, and stop when the difference in length change between two iterations is less than some tolerance. Explicitly:

1. Calculate axial load no bending, $Y_{nb}^0$

2. Calculate the length change from buckling, $\Delta L_{buc}^0$, due to $Y_{nb}^0$

3. Calculate a revised axial load, $Y_{nb}^1$ due to $\Delta L_{buc}^0$ (if restricted movement)

4. Calculate the length change from buckling, $\Delta L_{buc}^1$, due to $Y_{nb}^1$

5. Calculate a revised axial load, $Y_{nb}^2$ due to $\Delta L_{buc}^1$ (if restricted movement).

6. Iterate until convergence, i.e. $\Delta L_{buc}^m \rightarrow 0$, from $Y_{nb}^m$

7. Finally, add the bending stress from buckling to $Y_{nb}^m$