# Torque & Drag

## 1 Introduction (Soft String)#

The so-called “soft string model” for torque and drag calculations is one of the models implemented in Oliasoft WellDesign™. The basic idea behind the “soft string model” was first described in [1] and approximates the drill string as always lying on the bottom of the well bore in an inclined well. When the well is vertical the drill string is hanging freely in the center of the bore. The friction is thus governed by its weight and geometry, the “weight on bit” (WOB), the angular velocity, the geometry of the well bore and the well bore wall material.

NOTE!
The “stiff string model”, which takes into account that the drill string is stiff, but with elastic properties, is presented in a separate technical specification.

## 2 Basic Mechanics#

### 2.1 Basic Torque and Drag#

The set point for these calculations are the tip of the drill string. It can either be a compressive force on the drill bit towards the bed rock (the WOB) and a wanted torque, or the drill string tip can be freely “hanging” inside the well bore. From this, the forces in the string can be calculated. The side force is a function of the tension or compression in the string and the geometry of the well bore:

$\frac{dF_{\perp}}{dl} = \sqrt{{\left(F_d \sin \alpha \frac{d\phi}{dl}\right)}^2 + {\left(F_d\frac{d\alpha}{dl} + g \frac{dm}{dl} \sin\alpha \right)}^2} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)$

$dl = length\;of\;the\;element\\F_d = tension\;from\;the\;below\;element\\\alpha = inclination\;angle\;of\;the\;element\\\phi = azimuthal \;angle\\dm = mass\;of\;the\;element\\g = gravitational\;constant$

From a simple analysis of equation (1), one can see that $F_d$ can be both positive and negative. A negative tension is compression of the drill string. The side force does not depend on the sign of the turn, $d\phi/dl$ , but it does depend upon the sign of build, $d\alpha/dl$ . Tension in the subsequent element can thus be calculated, as the sum of the weight and the frictional force:

$\frac{dF_{d}}{dl} = g \frac{dm}{dl} \cos \alpha + \left(\mu \frac{dF_{\perp}}{dl} + \frac{dF_{mud}}{dl}\right)\sin\left(\tan^{-1}\left(\frac{v}{r_p \omega_{rot}} \right)\right) \;\;\;\;\;\;\;\;\;\; (2)$

$μ=friction\ coefficient\\ F_{mud} = frictional\ force\ on\ the\ pipe\ caused\ by\ mud\ flow\\ v=rate\ of\ penetration\ (ROP)\\r_p = outer\ radius\ of\ drill\ string\\ω_{rot} = angular\ frequency\ of\ drill\ string$

To account for the fact that the friction force is a vector, equation (2) depends upon the ratio of the ROP to rotation velocity. This modification term for the frictional part of the tension equation tends towards 1 when $\omega_{rot} \rightarrow 0$, and tends towards 0 when $v \rightarrow 0$.

Torque equation is as follows:

$\tau = \left(\mu F_{\perp} r_p + \tau_{mud} \right) \cos\left(\tan^{-1}\left(\frac{v}{r_p\omega_{rot}} \right)\right) \;\;\;\;\;\;\;\;\;\;\;\;\;\; (3)$

$\tau_{mud} = frictional\;torque\;on\;pipe\;caused\;by\;relative\;movement\;of\;mud\;on\;pipe\;surface$

The torque is basically proportional to the friction, since $v << r_p\cdot\omega_{rot}$ in most cases.

### 2.2 Mud Effects#

As can be seen in the equations (2) and (3), there is a friction component coming from the mud flow. The frictional forces caused by the moving mud is tested in [2], and is implemented as such. The paper also states that this additional friction causes minor changes to the simulation, but they do recommend such an approach, since it does account for an increased drag. It is implemented in Oliasoft WellDesign™ as follows:

$\frac{dF_{mud}}{dl} = \frac{dp}{dl} \frac{\pi\: (r_b^2 - r_p^2)\;r_p{}}{r_b - r_p} \;\;\;\;\;\;\;\;(4)$

$\frac{d\tau_{mud}}{dl} = \tau_s 2\pi r_p^2 \;\;\;\;\;\;\;\;(5)$

$\frac{dp}{dl} = pressure\;drop \\r_b = radius \;of \;open \;hole \\\tau_s = shear \;stress\; in \;the \;mud \\$

If the mud is not moving, these factors will be 0, and a simple order of magnitude analysis shows that in normal cases, we have the following:

$\frac{dF_{mud} / dl}{dF_d / dl} << 1 ,$

$\frac{dF_{mud} / dl}{dm / dl} << 1, \\$

$\frac{d\tau_{mud} / dl}{d\tau / dl} << 1$

Regardless if the mud is moving or not the mud will also create buoyancy in the element $d_m$. The buoyant weight is the elements weight in air minus the weight of the mud that the element displaces:

$\frac{dm}{dl} = \rho_p \pi (r_p^2 - r_{in}^2) \left(1 - \frac{\rho_m^{out} r_p^2 - \rho_m^{in} r_{in}^2}{\rho_p (r_p^2 - r_{in}^2)} \right) \equiv \frac{w}{g} \;\;\;\;\;\;\;(6)$

$\rho_p = density\;of\;pipe \\ \rho_m^{out} = density \;of \;mud \;in \;annulus \\ \rho_{min} = density \;of\;mud\;in\;pipe \\ r_{in} = inner \;radius\;of\;pipe$

In a non-flowing case the density of the mud would be the same inside the pipe as in the annulus, while this will not be the case if the mud is flowing. This difference in the inside and outside pressure also plays an important role when determining the buckling limits, which will be discussed later.

To determine the pressure in the pipe and the annulus, the pressure losses, $\frac{dp}{dl}$ must be determined throughout the flow path. This is done in the “hydraulics model”, also found in Oliasoft WellDesign™.

### Soft String 3D Model#

Aadnoy et al, [8], published a torque and drag model where the drag caused by the friction takes both build (changes in inclination) and turn (changes in azimuth) into account. For any straight section the standard Johancsik et al model is used, equation 5. The 3D model can be applied to an entire section with constant dogleg, $\Delta \theta$, the change in drag over such a section is:

$\Delta F = F_{low} \left(e^{\pm |\Delta \theta\cdot dl|} - 1\right) + \frac{dm}{dl}\cdot dh_{TVD}\;\;\;\;\;\;\;\;(7)$

where $F_{low}$ is the force action on the low side of the section, and  is the vertical height of the section. This model is also available in \os. This formula is very useful when the tension is high. When the tension is low it has a tendency to under predict frictional forces at the lowest part of the a bend, see eg. [reference 9], as gravity is ignored. It is clearly evident from equation 7, as the gravity part, while in reality it is opposite. This can be compensated for, see Section Stiff string emulation.

### 2.4 Friction - Contact Surface Effect#

Maidla and Wojtanowicz 10 concluded that friction between a rod and a flat surface is not the same as when the rod is lying inside a bigger pipe. They proposed a change in friction by $K_\mu$:

$d = \frac{\pi F_{\perp} r_{conn}}{12 E w_p} \\Y = 0.5 \frac{|r_b^2 - r_{conn}^2 + {\left(r_b - r_{conn} + d\right)}^2|} {r_b - r_{conn} + d} \\X = \sqrt{|r_{conn}^2 - Y^2|} \\ \gamma = \tan^{-1}{\frac{X}{Y - r_b + r_{conn}}} \\K_{\mu} = \frac{2\gamma}{\pi (4 / \pi - 1) + 1}$

### 2.5 Von Mises Stresses#

The Von Mises stress on en element $dl$ is calculated in a standard way. A 3-dimensional stress matrix is first calculated. The matrix is symmetric as there is no acceleration. The Von Mises stress is thus calculated as the double dot product of this matrix.

### 2.6 Beding Stress#

The bending stress is calculated from the compression of the drill string and its curvature coming from the curvature of the well bore:

$\sigma_{beam} = \frac{E}{2} \biggl(1 - \cos\left(\Delta\theta\cdot dl\right)\biggr)$

where E is Youngs Modulus for the material.

### 2.7 Bending Force - Stiff String Emulation#

Taking into account that the drill string has stiffness is called Stiff string modelling. An emulation of these effects are available in \os. When the pipe is bent through a build or turn section, one can calculate the force needed to bend the pipe. If the pipe has connectors with an outer radius larger than the pipe it is also possible to bend the pipe further, so not only the connectors touches the wall:over-bending''. The lateral force component can be calculated from the dogleg of the drill string segment:

$F^{bend}_{\perp} = \frac{4}{3} E \Delta\theta \left(r_{p}^3 -r_{in}^3\right) \cdot \sin\left(\Delta\theta \cdot dl\right)\;\;\;\;\;\;\;\;(8)$

The side forces, as calculated in equation 8, is a function of the tension in the string, thus a part of the lateral component will be absorbed by the elasticity of the material (as long as one is below the material yield strength). \os~has the option to include these calculations and thus reduce the normal force by the physical force required to bend a section.

As long as $\frac{dF{\perp}}{dl} > \frac{dF^{bend}{\perp}}{dl}$ , equation above is subtracted from the equation 1, if not is set to 0.

### 2.8 Buckling Force Limit#

Buckling limit calculations have been implemented as suggested by R. F. Mitchell, see 3. The approach is to calculate the critical limit from:

$F_c = \sqrt{\frac{4EIw_c}{r_c}} \;\;\;\;\;\;\;\;\;(9)$

$I = \frac{\pi}{4} \left(r_p^4 - r_{in}^4\right) \;\;\;\;\;\;\;\;\;(10)$

$r_c = r_b - r_p \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(11)$

$\frac{dw_c}{dl} = {\left(\frac{dF_\perp}{dl}\right)}_{F_d = F_c} \;\;\;\;\;\;(12)$

$E = Young´s \; modulus \\I = Second \; moment \; of \; area \\w_c = Contact \; force \; between \; pipe \; and \; bore \; wall \\r_c = Radial \; clearance \; between \; pipe \; and \; bore \; wall$

When the drill pipe is rotating, the contact force becomes:

$w_f = \frac{w_c}{\sqrt{1-\mu^2}} \;\;\;\;\;\;(13)$

2.8.1 Buckling Types

If the buckling force limit is exceed by compressive forces in the drill string the implementation classifies the buckled state as:

$1.38 \le \frac{F_d}{F_c} < 2.60 \;\;\;- \;\;\; Lateral\;Buckling \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (14)$ br/>

$2.60 \le \frac{F_d}{F_c} < 3.88 \;\;\;- \;\;\; Semi \;helical\;Buckling \;\;\;\;\;\;\;\;\; (15)$

$3.88 < \frac{F_d}{F_c} \;\;\;- \;\;\; Full\;helical\; Buckling \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (16)$

If compression in the drill string exceeds the full helical buckling limit, the contact forces increases, and they are then recalculated according to:

$\frac{dw_c}{dl} = \frac{r_c}{4EI}{\left(\frac{dF_d}{dl}\right)}^2 \pm \frac{r_c\tau \beta_{LB}}{2EI}\frac{dF_d}{dl} \;\;\;\;\;\;\;\;\; (17)$

$\beta_{LB} = \pm \sqrt{\frac{dF_d}{2EI}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\; \;\;\;\;\;(18)$

where $\beta_{LB}$ is the Lubinsky-Woods parameter, see 5 which accounts for chirality. It is positive if it is a right handed helix and negative if it is left handed.

### 2.9 Buckling Pitch and Period#

When the calculation determines that the drill string buckles, the pitch is also calculated for the section in question. When the lateral buckling criterion is satisfied, the pitch is calculated according to the “beam-column” model (see 6) as suggested in 7. This is done by calculating the period length for lateral buckling:

$L = \frac{1}{K_{SF}} \cdot \sqrt[3]{\frac{n^2 \pi^2 EI}{w}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;(19)$

where $K_{SF}$ is a safety factor in the range 1.1 - 1.2 with default 1.15, and n is the number of sinusoidal half periods.

If the criterion for helical buckling is satisfied, the pitch, $p$ , of the helix is calculated as follows:

$p = \sqrt{\frac{8 \pi^2 EI}{F_d}} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;(20)$

The maximum bending stress in this section of the buckled pipe is:

$\sigma_{\max} = \frac{r_b r_cF_d}{2I}\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\; \;\;\;\;\;(21)$

## 3 Simulation Modes#

There are 8 modes corresponding to the typical operations on a drilling rig. These can be separated into 2 groups: A set of operations where the drilling is not moving axially (called “Static”), and a set where the drilling string is moving axially in the well bore (called “Moving”):

• Static Simulation – axial velocity is always 0. Other fixed values are stated for each operation
• Hanging – Weight On Bit (WOB) is always 0, torque from the bit is always 0, and angular velocity is always 0
• Hanging Rotating – Weight On Bit (WOB) is always 0, and torque from the bit is always 0
• On Bottom – Angular velocity is always 0, and torque from the bit is always 0
• On Bottom Rotating

• Moving Simulation – Fixed values are stated for each operation
• Tripping in – Angular velocity of drill string is always 0, torque from the bit is always 0, and drill string will be moving down into the well
• Tripping out– Angular velocity of drill string is always 0, torque from the bit is always 0 and drill string will be moving out of the well
• Reaming in – torque from the bit is always 0, and drill string will be moving down into the well
• Reaming out – torque from the bit is always 0, and drill string will be moving out of the well
• Drilling – drill string will be moving into the well and the full set of input parameters are available
• Slide Drilling – drill string will be moving into the well, angular velocity will always be 0, torque from the bit is always at fixed input value

### 3.1 Dynamic Weight on Bit#

The implementation has 2 different algorithms where the WOB, which is an input parameter, is reduced, but keep as high as possible to maintain either a non-helical-buckled drill string or if the block weight is not big enough. The first situation is a user analysis option, while the second one will be formed regardless if “Dynamic WOB” is enabled or not. This means that in some special cases both algorithms might be used on the same simulation.

3.1.1 Buckling Prevnetion Option

The user can enable this option. It is available when running the following modes: “On Bottom”, “On Bottom Rotating”, “Drilling” and “Slide Drilling”. When enabled, the simulation will decrease the WOB if the drill string starts to buckle at a specific MD. The WOB will be decreased until the drill string does not buckle at any MD.

3.1.2 Block Weight Limitation on WOB

If the calculation results is a hookload ≤ 0, the block weight is not enough to maintain the requested WOB. The implementation will in these cases search for the maximum WOB resulting in a hookload = 0. To prevent this from happening, in a theoretical analysis case, the user can simply increase block weight until the WOB is constant at all MDs.

### 3.2 Tortuosity#

Since a long “elastic” pipe is not truly capable of drilling in a straight line in the real world, the simulation has optional randomness added to the geometry. The principle is basically to perturb the well bore geometry to create more friction. This has been shown to partially compensate for too low simulated force values compared to measured data. There are 3 options for perturbing the geometry, and thus increase the friction. Friction factors are also user input values, but are constant as a function of MD, which Tortuosity does not have to be. Enabling tortuosity will also create friction in vertical parts of the well bore. The following modes are implemented:

• Tortuosity On - Smooth helical circling the well path
• Tortuosity Random - Smooth random helical perturbation of well path
• Exxon - Sinusoidal changes to the ideal well path as suggested by R. Dawson.
• Exxon Random - Sinusoidal random changes

3.2.1 Continuous Perturbation

Tortuosity perturbation is implemented as a change to the well path, making it a helix where the central line is the well path. A helix, in local coordinates, is described by:

$x = r_T \cos \left(\Delta\cdot 2\pi \right) \;\;\;\;\;\;\;\;\; \;\;\;\;\;(22)$

$y = r_T \sin \left(\Delta\cdot 2\pi \right) \;\;\;\;\;\;\;\;\;\;\;\;\;(23)$

$z = \Delta \cdot p \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(24)$

where $\Delta$ is a free parameter ∈ [0,1>, and is the radius of the helix. This local offset point is then added to the the well path point in question. If the well path consists of survey points i, then $\Delta_i$ is:

$\Delta_i = \frac{x_{MD}}{p} \;\;\;\;\;\;\;\;\;\;\;\;\;(25)$

where $x_{MD}$ is the path length along the well trajectory (non-perturbed) between the 2 adjacent points.

The “Exxon” mode perturbation is implemented as:

$\delta = T\sin{ 2\pi\frac{x_{MD}}{p}} \;\;\;\;\;\;\;\;(26)$

where $T$ is the tortuosity amplitude, which is recommended to around 0.7$\degree$. $\delta$ is then added to the inclination and the azimuthal angle. The process involves making sure the inclination does not become negative. Note that the pitch, should not be a small multiple of the distance between the survey points:

$\frac{\vert \vec{x}_i - \vec{x}_{i-1}\vert}{p} \gg 1 \;\;\;\;\;\;\;\;(27)$

3.2.2 Perturbation with Randomness

By enabling “random” to the tortuosity it might resemble real world drilling in a better way. Random tortuosity is implemented as:

$r_{T_i}' = r_{T_{i-1}} + \frac{R-0.5}{|R|} R \cdot \Delta \;\;\;\;\;\;\;\;(28)$

$p_i = p_{i-1} + \frac{R-0.5}{|R|} R \cdot \Delta \;\;\;\;\;\;\;\;\;\;\;\;\;(29)$

where $R$ is a random number ∈ [0,1].

When the “Exxon” mode is chosen with randomness, the inclination and azimuthal angles are perturbed the following way:

$\alpha_p = \alpha + \delta \cdot R \;\;\;\;\;\;\;\;\;\;\;\;\;(30)$

$\phi_p = \phi + \delta \cdot R \;\;\;\;\;\;\;\;\;\;\;\;\;(31)$

## 4 Available Simulation Data#

For all the simulation modes, the well path data are available, perturbed or non-perturbed. When performing the static modes, the following output is available:

• Tension
• Torque
• Side force
• $\Delta$Tension
• $\Delta$Torque
• Von Mises stress
• Buckling force limits
• (Buckling force limits without totuosity, if tortuosity was applied)
• Buckling type

For the moving modes, the following is available:

• Torque at RKB
• Weight on bit
• Trip velocity . assumed to be proportional with the WOB

In addition to these moving-mode data, the data listed for static operation is also available, but the user needs to specify at which depth to view them. They can be presented at a specific point ($x_{MD}$) as the drill string moves past this point, or at a specified drill bit location (called “Road map”).

## Acronyms#

MD - Measured Depth - distance inside the well bore

RKB - Rotary Kelly Bushing

ROP - Rate Of Penetration

WOB - Weight On Bit