EP0677641A1 - Method for optimizing the characteristics of an axial fluid flow in a variable annular space around the string - Google Patents

Abstract

A method of optimising fluid circulation characteristics in a pref. narrow annular space around a tubular element, e.g. a rod length or string with variable eccentricity caused by deformation, involves creating a flow model taking into account the varying shape of the annular space along the rod length and the actual rheological properties of the fluid, e.g. variation of viscosity with shear, in order to evaluate the velocity field and pressure at any point along the annular space. <IMAGE>

Description

The present invention relates to a method for optimizing the characteristics of a circulation of fluid which is established in an annular space around a tubular element and in particular of a rotating tubular element, such as a long rod or a train of rods, dynamically taking into account the deformations undergone in operation by this tubular element and therefore the variation of the annular space around it.

The method according to the invention is particularly suitable in the case of narrow annulars where the ratio between the diameter of the internal tubular element and the diameter of the external duct is greater than 0.5.

The method according to the invention finds applications in particular in the context of oil drilling or in geotechnics, where it makes it possible to determine the speed field of a drilling fluid circulating in space around a drill string as well that the pressure losses resulting from friction, for complex geometries of this space, consecutive to the movements and deformations of the rods.

The method is particularly suitable for optimizing the circulation conditions of fluids in boreholes carried out in narrow wells according to the so-called "slim hole" technique where, due to the reduced annular dimensions, pressures can be generated, which endanger stability of the crossing training.

There are a number of publications reporting theoretical or practical studies on the circulation of fluids in wells around a stationary or rotating drill string and variations in flow parameters due to the eccentricity of a drill string and its deformations, particularly in narrow wells.

• - Vaughn, R.D., 1965, "Axial Laminar Flow of non Newtonian Fluids in Narrow Eccentric Annuli," S.P.E., Vol.5, Dec.;
• - Bourgoyne, A.T et al, 1986, "Applied Drilling Engineering," in SPE Text Book Series, Vol.2;
• - Markatos N.C.G.et al, "Flow in an annulus of non-uniform gap" in Trans. IChemE, vol. 56;
• - Reed, T.D et al, 1993, "A new model for laminar, Transitional and Turbulent Flow in Drilling Fluids," in SPE 25456, Proceedings of the Prod. Operations Symposium, Oklahoma City, OK
• - Marken, C. D. et al, 1992, "The influence of drilling conditions on annular pressure losses, article SPE 24598
• - Dodge D.W. and Metzner A.B., 1959, "Turbulent flow of Non-Newtonian Systems, AIChE Journal, vol 5, p33.

We can quote for example:

• - Vaughn, RD, 1965, "Axial Laminar Flow of non Newtonian Fluids in Narrow Eccentric Annuli," SPE, Vol.5, Dec .;
• - Bourgoyne, AT et al, 1986, "Applied Drilling Engineering," in SPE Text Book Series, Vol.2;
• - Markatos NCGet al, "Flow in an annulus of non-uniform gap" in Trans. IChemE, vol. 56;
• - Reed, TD et al, 1993, "A new model for laminar, Transitional and Turbulent Flow in Drilling Fluids," in SPE 25456, Proceedings of the Prod. Operations Symposium, Oklahoma City, OK
• - Marken, CD et al, 1992, "The influence of drilling conditions on annular pressure losses, article SPE 24598
• - Dodge DW and Metzner AB, 1959, "Turbulent flow of Non-Newtonian Systems, AIChE Journal, vol 5, p33.

The resolution method generally used to model the behavior of a fluid circulating in an eccentric annular, consists in assimilating the space around the rod to a juxtaposition of slots. It has hitherto been considered either that the rod was centered in the duct, or that its eventual eccentricity was uniform throughout this rod. Under this assumption, the slots are considered to be parallel and of constant thickness over their entire length.

The circulation of drilling fluids in a narrow hole (slim hole) where a drill string turns, is a complex phenomenon difficult to model. Among the important factors influencing the pressure losses, for a given type and flow of mud, one can cite the speed of rotation of the drill string, as well as the geometry of the annular space around and all along the gear train. rods due in particular to the eccentricity of the latter in the hole, its movements, its flexions, etc.

Most often, in fact, the respective axes of the drilled hole and the drill string are offset from one another due to the deviations of one and / or bending of the other. On this offset which varies along the drill string, depends the eccentricity of the annular space between them.

In many cases therefore, and in particular for drilling wells, existing models based on the assumption that the relative positioning of the rod relative to the conduit is uniform over its entire length, do not properly account for the complexity of the phenomena. In addition, the existing models do not take account of the significant modifications to the mud circulation brought about by the coupling between the effects of the rotation of the rod and its variable eccentricity with respect to the conduit or the hole.

The existing models therefore do not allow the driller to safely predict the pressure losses and the actual speed field resulting from all of these parameters: rotation of the rod, effective rheological properties of the fluids used in practice, variable offset non-uniform etc, and therefore to optimize the circulation of fluid to be established: flow rate, rheology, taking into account the speed of rotation.

The object of the method according to the invention is to construct a representative model of the velocity field of a fluid circulating in a conduit around a tubular rod with variable eccentricity, both in laminar and turbulent conditions, as well as in the distribution annular pressure losses as a function of flow rates.

It thus makes it possible to optimize the characteristics of a circulation of fluid which is established in an annular space around a tubular element whose eccentricity is variable, such as a long rod or a train of rods, subjected to deformations, especially in the case where this annular space is relatively narrow.

The method according to the invention is characterized in that it includes the modeling of the flow in the annular space considering that its shape is variable all along the tubular element and taking into account real rheological properties of the fluid (variation of the viscosity with the shear rate for example), so as to determine the value of the velocity field and the value of the pressure at any point along this annular space.

The method can also include the application to these values obtained for a tubular element with variable eccentricity, of an adimensional corrective factor depending on the Reynolds number (Re) and the Taylor number (Ta) of the fluid used, to take account of the variations in pressure losses in the annular generated by the speed of rotation of the tubular element.

When the respective ratios of the inertial and viscous effects, in the axial direction and the azimuthal direction respectively, are greater than a determined value, for example, one can determine the dimensionless corrective factor to be applied by the relation:

A, c and d are parameters whose values can be chosen within defined ranges.

According to an embodiment of the method, usable when the sinuosity of the tubular element is relatively low, account is taken of possible dynamic modifications of the shape of the tubular element, by application of another substantially constant corrective factor. and independent of the shape of the tubular element, included for example in a range

The method according to the invention takes good account of the two essential factors which govern the evolution of the pressures in the narrow annulars: the variable eccentricity and the rotation of the tubular element. It therefore makes it possible to reliably relate the annular pressure to the operating parameters: geometry, flow rate, rotation speed, as well as to the rheology of the fluid in circulation.

The distribution of the pressure drops which are determined by application of the method according to the invention, as defined above, in complex cases where any fluid circulates in a narrow annular around a rotating rod subject to deformations, especially when the annular space around it is narrow, is in good agreement with the practical results that we have been able to measure.

In the context of drilling operations in particular, it is therefore possible to define, by application of the method, the optimal rheology of the fluid to maintain a high flow rate making it possible to obtain good removal of the cuttings without the annular pressures leaving a range of safety and damage the hole. The method thus makes it possible to define rules on rheology and therefore on the composition of fluids and in particular fluids without solid particles used in "slim hole".

• - les Fig.1 et 2 montrent schématiquement un élément tubulaire allongé soumis à des déformations respectivement à variation sinusoïdale et linéaire;
• - la Fig.3 montre schématiquement en coupe transversale, un tube excentré dans un conduit tel qu'un puits;
• - les Fig.4 et 5 représentent schématiquement un annulaire sinueux respectivement en position refermée et déployée;
• - la Fig.6 montre schématiquement la variation en fonction de la sinuosité d'un élément tubulaire, d'un facteur correctif à appliquer aux pertes de charge obtenues en supposant un excentrement nul ou invariable, prédite par la méthode et corroborées expérimentalement;
• - la Fig.7 montre schématiquement la variation en fonction de l'indice de rhéo-fluidification du fluide, du même facteur correctif;
• - la Fig.8 montre comment la contrainte de cisaillement τ varie avec le taux de cisaillement y , dans le cas de fluides newtoniens et de fluides non-newtoniens; et
• - la Fig.9 montre la distribution des vitesses V et les iso-valeurs de pression dans un annulaire dont la configuration varie d'une façon sinusoïdale le long de son axe.

Other characteristics and advantages of the method according to the invention will appear on reading the description below, with reference to the accompanying drawings in which:

• - Fig.1 and 2 schematically show an elongated tubular element subjected to deformations with sinusoidal and linear variation respectively;
• - Fig.3 shows schematically in cross section, an eccentric tube in a conduit such as a well;
• - Fig.4 and 5 schematically show a sinuous annular respectively in the closed and deployed position;
• - Fig.6 schematically shows the variation as a function of the sinuosity of a tubular element, of a corrective factor to be applied to the pressure losses obtained by assuming zero or unchanging eccentricity, predicted by the method and corroborated experimentally;
• - Fig.7 schematically shows the variation as a function of the rheo-fluidification index of the fluid, of the same corrective factor;
• - Fig.8 shows how the shear stress τ varies with the shear rate y, in the case of Newtonian fluids and non-Newtonian fluids; and
• - Fig.9 shows the distribution of velocities V and the pressure iso-values in a ring finger whose configuration varies in a sinusoidal fashion along its axis.

• - l'excentricité qui est un nombre sans dimension valant 0 quand la tige et le conduit sont concentriques et 1 quand la tige touche la paroi intérieure du conduit, est définie par la relation:
  
  • R_o et R sont respectivement les rayons du conduit et de la tige (Fig.1), et 0 est l'écartement de leurs axes respectifs.
• - l'excentricité maximale: qui est inférieure à 1 si la tige est pourvue de centreurs (de rayon RJ qui l'empêchent de toucher la paroi du conduit;
  
• - la sinuosité (skewness) S, qui est le rapport entre le diamètre moyen de l'espace annulaire d_avg=2r_avg et l'intervalle entre deux inflexions successives de la tige AL,
  

In a well 1, the shape of the drill string 2 driving the tool generally varies from one location to another (Fig. 1 to 3). It depends on the deviation of the drilled hole, the tension or compression exerted on the rod etc. We define the actual configuration of a drill string by three geometric parameters:

• - the eccentricity which is a dimensionless number equal to 0 when the rod and the conduit are concentric and 1 when the rod touches the interior wall of the conduit, is defined by the relation:
  
  • R _o and R are respectively the radii of the conduit and the rod (Fig. 1), and 0 is the spacing of their respective axes.
• - the maximum eccentricity: which is less than 1 if the rod is provided with centralizers (of radius RJ which em avoid touching the wall of the duct;
  
• - the sinuosity (skewness) S, which is the ratio between the mean diameter of the annular space d _avg = 2r _avg and the interval between two successive inflections of the rod AL,
  

For the calculations, according to a known formalism, the annular space 3 is assimilated to a series of juxtaposed slots of variable thickness as a function of the real eccentricity. The annular space around the stem is de- ployed ⁹ (Fig 4, 5) and it is considered that the fluid flows between a number of variable spacer plates in the axial direction (Fig. 9).

avec e_max défini par ₍₂₎ et _ravg = 0.5dav^g. We can show that, for 0 ≦ x 0 ≦ AL, and 02 y Z 2πr _avg , the equations giving in Cartesian coordinates, the thickness of the slots in the case of a sinuous annular with sinusoidal variation (Fig. 1) or rectilinear (Fig. 2), have as an expression:

with e _max defined by ₍₂₎ and _rav g = 0.5dav ^g.

qui relie entre eux la pression P, la vitesse axiale u, la vitesse azimutale v, la viscosité η, les coordonnées x (axiale) et y (azimutale), le rapport S défini par (3), le coefficient de frottement f et le nombre de Reynolds Re.To model the flow of the fluid in a relatively thin annular space, whose eccentricity is variable, the following equation of motion is used:

which connects the pressure P, the axial speed u, the azimuthal speed v, the viscosity η, the coordinates x (axial) and y (azimuthal), the ratio S defined by (3), the coefficient of friction f and the Reynolds Re number.

We also use two relations connecting the pressure and the components of the speed averaged in a radial direction:

All parameters are made dimensionless. So:

The dimensionless viscosity ηD is equal to 1 for Newtonian fluids. For non-Newtonian fluids, the flow velocity and the transverse length scale are taken into account to calculate this dimensionless viscosity. P _D = P / P ₀ where P _o which represents the pressure losses of the fluid circulating in a concentric annular similarly reduced to a slit, is calculated by the relationships established by Reed et al in the publication already cited.

Numerical model:

We apply the previous relationships, to a numerical model of finite difference type such as that defined by Markatos et al, in the publication already cited, where the annular space is divided into grids, each representing a slit as we have them. defined above, the thickness of which is determined by solving equations (4) or (5).

The Markatos model, which was applied to Newtonian fluids, is improved as will be seen below, to take into account all of the rheological laws to which drilling fluids obey.

For a given AP, at the abscissa x = 0, the model initially considers a linear variation profile of the pressure and an average velocity field based on a certain flow rate for each slot. The Reynolds Re number, the coefficient of friction (f) and the viscosity are calculated.

For a Newtonian fluid in laminar regime for example, the product f.Re is equal to 24 and the viscosity is independent of the shear. In this case, equations (6) to (8) are simplified.

The modeling method according to the invention makes it possible to extend the field of application of the previous model to non-laminar flow regimes of any fluids whose rheo-fluidification index is generally less than 1, which better correspond to the circulations that we have to model in practice.

ou Test la contrainte de cisaillement et K' et n' sont les paramètres définis dans la publication de Dodge, D.W et al déjà citée.For a non-Newtonian fluid, this relation must be modified. It is necessary first of all to integrate into equations (6) to (8) of the model, the product f.Re of the corrective factor f by the Reynolds number Re, this for any fluids. The viscosity of fluids generally varies with the shear rate y (Fig. 8). They obey the rheological law:

or Test the shear stress and K 'and n' are the parameters defined in the publication of Dodge, DW et al already cited.

To take account of this type of fluid, the definition of the Reynolds number Re is modified and the method described by Reed, TD et al, in the publication already cited, is used between this modified Re number and the friction coefficient f, for model resolution

The so-called GNM method described by Reed et al in the publication already cited, is used to evaluate the diffusion term z ³ _D , fRe _ηD in equation (6). This diffusion term being known for each grid, the discretized form of equation (6) is solved to obtain a new pressure field. A new velocity field is calculated from the same equations (7) to (8). The calculations are repeated until a convergence of the calculated speed fields is obtained. An example of a velocity field obtained with a sinusoidal eccentricity is shown in Fig. 9.

In all cases where the flow becomes turbulent in the wider parts of the annular space while remaining laminar in the narrow parts, due to a rapid variation of the diffusion term, from one grid to another, a method relaxation known per se, is used. It increases stability but it decreases the speed of convergence.

When convergence is reached, the digital integration of the axial velocity field gives the flow.

By simple modifications of equations (4) and (5), we adapt the model to cases of zero or uniform eccentricity of the rod.

Eccentricity corrections

Estimating the velocity field and the distribution of pressure drops in a narrow ring is very complex because of the possible diversity of the winding and the eccentricity of the rod under real operating conditions.

The method according to the invention however makes it possible to model the speed field and the pressure drop distributions around a rod of great length in rotation.

In the general case of a rod whose sinuosity coefficient (1 / S) is low, the distribution of the pressure drops is determined by integrating into the model defined by relations 6 to 8, the analytical expressions of the variation of l thickness of the slots corresponding to the real shape of the rod, and for example equations (4, 5) if the deformation of the rod is of sinusoidal or linear type. This leads to complex calculations.

The method according to the invention provides a much simpler solution in the case where the sinuosity of the rod is low (factor 1 / S higher).

A corrective factor R is defined as the ratio, for the same flow rate, between the pressure drops per unit length (AP / AL) e, generated by an eccentric rod and the corresponding pressure drops (AP / AL) c generated by the same centered rod.

The curves in Fig. 6, determined by modeling in accordance with the method, show that the corrective factor R varies asymptotically, and remains practically invariable regardless of the actual form (in the form of a broken line or in the form of a sinusoid) taken by the rod, when the sinuosity coefficient 1 / S reaches fairly high values (1 / S> 10), which corresponds to a slight deformation of the rod.

Fluid circulation tests were carried out in the laboratory with an installation comprising a conduit whose internal diameter was 24 mm and an internal rod curved sinusoidally, whose external diameter was 18 mm. In Fig. 6 point a with coordinates 1 / S = 12, R = 0.66 and point b with coordinates 1 / S = 18, R = 0.64, respectively correspond to values obtained during the tests. The comparison shows that the agreement between the model predictions and the experimental results is excellent.

Under normal conditions of use, the deformation of a rod in a narrow pipe or hole, in particular of a drill string in an oil well, is generally little accentuated. This is the most common mode of operation in practice. The pressure losses resulting from an eccentricity of the rod (AP / AL) e, can therefore be calculated simply, in a manner substantially independent of the sinuosity coefficient 1 / S in the case of ring fingers of large length.

The calculation can be generalized for the case of non-Newtonian fluids, whatever their degree of rheo-fluidification. The variation of the corrective factor R which must be introduced when the rod is eccentric but slightly sinuous (1 / S important) as a function of the rheohluidification index n is shown in FIG. 7 for the case of a sinuosity of linear type or sinusoidal.

The introduction of this corrective factor R therefore makes it easy to obtain, for weak sinuosities, the results in an eccentric configuration from the results obtained in the case of a centered rod, for any rheological law of the type given by the relation ( 9).

Rotation corrections

The values obtained by modeling, must then be modified to take into account the effects generated by the rotation of the rod.

Tests relating to the flow of a rheo-fluidizing fluid around a rod centered in a duct, have shown that the evolution of the pressure drop as a function of the rotation speed W, depends on the flow regime generated by the coupling of the axial movement and the tangential movement.

où W est la vitesse de rotation et ₁₁, la viscosité.Specialists know that the tangential flow between two coaxial cylinders is characterized by the magnitude of the Taylor number defined by

where W is the speed of rotation and ₁₁ , the viscosity.

In the absence of axial movement, the tangential flow is governed by the Taylor Ta number. When Ta <41.3, the flow is laminar. For Taylor numbers between 41.3 and about 400, we find that stable vortices, called Taylor vortices, are superimposed on the circular current lines. The flow becomes turbulent for values of Ta> 400.

We also know that, superimposed on a tangential flow, an axial flow modifies the values of Ta corresponding to the limits of the transition zone. Likewise, a tangential flow, superimposed on an axial flow, can affect the Re values corresponding to a laminar-turbulent transition in the axial direction.

To identify the overall trends of the effects of rod rotation and the value of the corrective factors to bring to the previous results, to take account of the type of flow, it is useful to put the different parameters in an adimensional form.

Thus, we use the Reynolds number defined by Dodge et al for a non-Newtonian fluid in the publication already cited.

The Taylor number characterizing the ratio of inertial effects to viscous effects along the azimuthal direction is given by relation (10) provided that its definition is extended to include the effects of rheofluidification. It is necessary for that to replace in this relation, the Newtonian viscosity by the viscosity of a fluid in law of power, calculated at the value of the rate of shearing in the internal wall (against the rod) corresponding to the speed of rotation W.

We also define the ratio Rp of the annular pressure for a given speed of rotation W, related to the annular pressure with zero rotation at the same flow rate, by the relation:

où U_m la vitesse débitante. R_w est une mesure du rapport des taux de cisaillement respectivement suivant les directions azimutale et axiale.From the Taylor and Reynolds numbers, we construct another dimensionless parameter:

where U _m the flow rate. R _w is a measure of the ratio of the shear rates respectively along the azimuthal and axial directions.

• 1) Faibles nombres de Reynolds (Re < 200):
  • Le rapport Rp diminue quand le paramètre Rw augmente et il devient significatif quand Rw devient supérieur à 2. La chute de la pression que l'on constate, est liée à la diminution de la viscosité du fluide liée à la superposition du cisaillement azimutal au cisaillement axial. La diminution des pertes de charge que l'on constate pour les valeurs faibles du nombre de Reynolds sont la conséquence de la chute de viscosité apportée par le mouvement tangentiel.
• 2) Faibles nombres de Taylor (Ta < 200)
  • Le rapport de pression Rp tend asymptotiquement vers une limite Rp₁ supérieure à 1 pour les valeurs du nombre de Taylor Ta < 200.
• 3) Nombres de Taylor et de Reynolds supérieurs à 200

The results obtained vary according to the values taken by the Reynolds and Taylor numbers:

• 1) Low Reynolds numbers (Re <200):
  • The ratio Rp decreases when the parameter Rw increases and it becomes significant when Rw becomes greater than 2. The drop in pressure which is observed, is linked to the decrease in the viscosity of the fluid related to the superimposition of the azimuth shear on the axial shear. The reduction in pressure losses which is observed for the low values of the Reynolds number are the consequence of the drop in viscosity brought about by the tangential movement.
• 2) Weak Taylor numbers (Ta <200)
  • The pressure ratio Rp tends asymptotically towards a limit Rp ₁ greater than 1 for the values of the Taylor number Ta <200.
• 3) Taylor and Reynolds numbers greater than 200

où les coefficients A, c, d sont compris dans les fourchettes suivantes.

avec des valeurs typiques suivantes: A = 0,29, c = 0,17 et d = 0,067.For this range of variation, we could establish an empirical relation of the form

where the coefficients A, c, d are included in the following ranges.

with the following typical values: A = 0.29, c = 0.17 and d = 0.067.

The application to the results of this second corrective factor depending on the speed of rotation and the type of flow, therefore makes it possible to obtain the pressure drops all along the rod.

Claims (5)

1) Method for optimizing the characteristics of a fluid circulation which is established in an annular space (3), and in particular in a relatively narrow annular, around a tubular element (2), such as a long rod or a drill string whose eccentricity varies due to deformations, characterized in that it includes the modeling of the flow in the annular space by considering that the shape of the latter is variable all along the tubular element and taking into account real rheological properties of the fluid such as the variation of the viscosity with the shear rate, so as to determine the value of the velocity field and the value of the pressure at any point along this annular space . 2) Method according to claim 1, characterized in that it comprises the application to said values obtained for a tubular element with variable eccentricity, of a dimensionless corrective factor (Rp) depending on the Reynolds number (Re) and the number of Taylor (Ta) of the fluid used, to take account of the variations in pressure losses in the ring finger caused by the speed of rotation (W) of the tubular element. 3) Method according to claim 2, characterized in that the dimensionless corrective factor (Rp) is determined by the application of the relation:

and

when the respective ratios of the inertial and viscous effects, in the axial direction and the azimuthal direction respectively, are greater than a determined value. 4) Method according to one of the preceding claim, characterized in that account is taken of possible dynamic modifications of the shape of the tubular element, by application of another corrective factor (R) substantially constant and independent of the shape of the tubular element, as long as the sinuosity (S) thereof remains relatively low. 5) Method according to claim 4, characterized in that one chooses the other corrective factor (R) in the interval

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