Buoyancy effect dualism on drill string axial forces calculation

This paper discusses the use of two buoyancy schools in determining the axial forces in the drill string, which leads to the duality of the solution. The difference in the results in determining the true and effective force and their relationship with each other using the strength of stability were revealed. It is shown that only the true force can be used to calculate the stresses and deformations of the drill string elements, since their actual values correspond to this force. At the same time, taking into account the effective forces, it is possible to more fully use the actual stability of the drill string, increasing the permissible loads on the bit and increasing the penetration rate. Thus, the presence of two buoyancy schools and, accordingly, two types of axial forces, makes it possible to more accurately and comprehensively assess the operability of the drill string.


Introduction
Perhaps, the most important operability condition for the drill string in the well is its buckling under external loads.With even small compressive forces, an extended drill string (up to 3 km or more) can easily lose stability, followed by its jamming in the well and even destruction.The buckling condition is satisfied when the values of the internal compressive axial forces in the drill string do not exceed some known critical value.A special feature of the calculation of internal axial forces in the drill string is the presence of continuous circulation of drilling mud in the well, which completely covers the drill string, so it will be in a state of buoyancy.Buoyancy can be accounted in two different ways, resulting in two values of axial forces in the drill string and confusion when working with them [1].In this paper, we will consider the difference between the two approaches to buoyancy, the results obtained and the peculiarities of their use in the calculation of drilling strings.

Methods
Buoyancy is described by Archimedes' law, which states that a body immersed in a fluid medium is always affected by a force directed against the gravity and equal to [2]: where  -fluid density; S -cross-sectional area; L -body height; V -body volume.
When calculating the axial force in the drill string, it is necessary to correctly take into account the method of applying the pushing force (1) in the computed model and here there are disagreements that led to the emergence of two different ways, called schools: 1) The school of the force method (the piston force method, the force-area method) believes that the pushing force acts on the lower part of the submerged body and is directed upward.This is how buoyancy in physics is usually explained [3].
Both methods exist, since neither can fully describe all the observed demonstrations of buoyancy and this leads to different results in calculating the axial force inside the submerged body.
Consider the drill string as a simple pipe suspended vertically in air and in fluid (figure 1).When the pipe is in the air (figure1, a), the load is only its own specific gravity w [kg/m] and the value of the axial force in the arbitrary section of the pipe will be equal to: The axial force Fair varies linearly in the height of the pipe from zero to the total weight of the pipe at the top Fair (L) = wL = W, as shown in the figure 1, a.

Results
Now we will immerse the pipe in fluid by the level of its surface in order to simplify the solution, since in this case the pipe will not be additionally affected by the fluid column above it (figure 1, b, c).Let's determine the axial force in the pipe according to two schools.

The piston force method and "true" force
According to the force method, when the pipe is immersed in fluid, its density and specific gravity will be like in air, and only the concentrated buoyancy force FA occurs, which acts on the lower end (figure 1, b).Then the value of the axial force in the arbitrary section of the pipe at a distance x from its lower point will be equal to: Graphically, the axial force will also be linear and with the same slope from the Archimedes force value FA to the value Ftrue (L) = -FA + wL, as in air (figure 1, a, b).The axial force by this method is called the "true" force.

The law of Archimedes and "effective" force
When immersing a pipe in a fluid according to the Archimedes principle (figure 1, c), it is considered that each layer of the pipe, each of its volumes, is equilibrated in water, which can be taken into account by its new, reduced, specific gravity ws in the fluid, defined as: where ws -specific gravity of the pipe in the fluid; S -pipe cross-sectional area.
Then the axial force will be determined as well as for the case of a pipe in air: ( The axial force by this method is called the "effective" force, it also changes linearly along the height of the pipe, but with a different slope, reaching a value on the surface Feff(L)=wsL.
It can be noted that, taking into account the expressions (1-5), the values of the effective and true forces will be equal on the surface, differing at other points of the pipe.There is a relationship between the two approaches, which in the literature is referred to as stability force FBS (buckling stability force, Lubinski's "fictitious" force), which is equal to the difference between effective and true force: The values of the true and effective forces differ by a value equal to the pressure of the fluid at the considered level multiplied by the cross-sectional area of the pipe at the same level; or, in other words, equal to the weight of the fluid displaced by the pipe above the considered level.This definition is valid for standing fluid in the absence of circulation [5], when only hydrostatic pressure acts (figure 2, a).

Mud circulation
In the case of the most operations of the drill string, it is subjected to continuous circulation of mud in the well, the values of the fluid pressures inside and outside the drill string at the same depth will be different and the expression (8) will take a more general form [6]: where рi, рo -pressure inside and outside the drill string; Si, So -drill string cross-section area by its inner and outer diameters.With normal circulation of mud fed through the drill string into the well, the pressure inside the drill string will be greater than the pressure outside at free discharge.Therefore, in this case, the value of the effective force in the drill string at the surface level will be slightly less than the true force.Also, when mud is circulated through the bit nozzles, a FIF jet impact force occurs that also compresses the bottom of the drill string (figure 2,b).The ratio of true and effective forces can be reversed, for example, with reverse circulation through the annulus, a small gap between the drill string and the walls of the well, etc.

Discussion
In mechanics, there is a theorem on the uniqueness of the solution, that is, one possible internal force state arises in the body at a given load and support conditions.Here, the presence of two buoyancy schools leads to an ambiguous solution for axial forces.The question arises, which of the two axial forces is correct?Both axial forces have proved to be in demand in the oil industry, as they determine the different effects of mud on the operability of the drill string and its components.
The true force Ftrue is used to calculate the stress-strain state of the drill string elements, since the actual stresses and strains correspond to this force [7].The resulting stresses provide a check of the static and fatigue strength of the material of the drill string elements, and deformations allow determining changes in its dimensions under load.
The effective force Feff is usually derived in calculations from the magnitude of the true force by taking into account the stability force Fbs according to expressions (6,8).Studies of the drill string stresses in the fluid have shown [8,9] that accounting for the stability force Fbs leads to the exclusion of the hydrostatic component from the stresses in the drill string, which does not affect the buckling of the drill string, as well as the strength of the plastic materials from which the drill string elements are usually made.Therefore, effective force is widely used to check buckling conditions, which is one of the most important requirements for the operability of the drill string.For this reason, in some software systems for mechanical calculating drill strings, only the effective force is given in the results [10].The true force in these programs does not appear explicitly, but only participates in the calculation of stresses and deformations of a drill string.
Another important condition for the operability of the drill string is the position of its neutral stress point, which should be located within the bottom of the drill string assembly.When calculating the axial forces in figure 2,b, we get two neutral points, A and B. The neutral point A is obtained by crossing the true forces graph with the vertical axis it corresponds to the unloaded area of the drill string with true zero stress and deformation.However, for the buckling conditions of the drill string, it is reasonable to take into account its insensitivity to hydrostatic stresses, therefore, the more relevant neutral place becomes point B. Point B is located well below point A, reducing the size of the compressed part and the compression magnitude of the drill string.Therefore, consideration of effective forces allows for more complete use of actual stability, increasing the allowable bit loads and increasing the penetration rate.Summing up the results, the presence of two buoyancy schools makes it possible to more accurately and comprehensively estimate operability of the drill string by its different parameters: strength, stiffness and buckling.

Conclusion
The paper considers different approaches to calculation of axial forces at buoyancy of drill string and duality of obtained results in the form of true and effective force.Also, we showed the need to take into account both forces to estimation the operability of the drill string and their relationship with each other.