EP0677641B1 - Method for optimizing the characteristics of an axial fluid flow in a variable annular space around the string - Google Patents

Description

The present invention relates to a method for optimize the characteristics of a fluid circulation that one establishes in an annular space around a tubular member and in particular a rotating tubular element, such as a long rod or a string of rods, dynamically taking into account 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 inner tubular member and the diameter of the outer conduit is greater than 0.5.

The method according to the invention finds applications especially in the context of oil drilling or in geotechnical, where it allows to determine the velocity field of a drilling fluid circulating in space around a train of drill pipes as well as the pressure losses resulting from frictions, for complex geometries of this space, consecutive to the movements and deformations of the rods.

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

There are a number of publications reporting theoretical or practical studies on the circulation of fluids in wells around a stationary drill string or in rotation and variations in flow parameters due to the eccentricity of a drill string and its deformations, and especially 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 for model the behavior of a fluid circulating in a eccentric annular, consists of assimilating the space around the rod to a juxtaposition of slots. We have so far considered either that the rod was centered in the duct, that its eccentricity eventual was uniform throughout this rod. As part from 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 well (slim hole) where a drill string turns, is a complex phenomenon difficult to model. Among the important factors influencing pressure losses, for a given type and flow rate of sludge, may cite the speed of rotation of the drill string, as well as the geometry of the annular space around and all along the train of rods due in particular to the eccentricity of the latter in the hole, of his movements, his flexions etc.

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

In many cases therefore and especially for the drilling of well, 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 fully reflect the complexity of the phenomena. In addition, existing models do not ignore important changes to the circulation of mud brought by the coupling between the effects of the rotation of the rod and its variable offset relative to the leads or to the hole.

The existing models therefore do not allow the driller to safely predict pressure drop and field of real speeds resulting from all of these parameters: rotation of the stem, effective rheological properties of the fluids used in practice, non-uniform variable offset 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 circulating fluid in a conduit around an eccentric tubular rod variable, both laminar and turbulent, as well as of the distribution of annular pressure drops as a function of debits.

It thus optimizes 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 string 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 involves modeling the flow in the annular space considering that the shape of it is variable all along of the tubular element and taking into account properties actual rheological of the fluid (variation of viscosity with the shear rate for example), so as to determine the value of the velocity field and the value of the pressure in all point along this annular space.

The method can also include the application to these values obtained for a tubular element with variable eccentricity, a dimensionless correction factor depending on the number of Reynolds (Re) and the Taylor number (Ta) of the fluid used, for take into account variations in pressure drop in the ring finger 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: Rp = A. Re vs .Your d or
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 0.1 <R <10

The method according to the invention takes good account of the two essential factors governing the evolution of pressures in narrow annulars: variable eccentricity and rotation of the tubular element. It therefore makes it possible to connect in a reliable ring pressure at operating parameters: geometry, flow, speed of rotation, as well as the rheology of the circulating fluid.

The distribution of pressure losses which is determined by application of the method according to the invention, as it stands defined above, in complex cases where any fluid circulates in a narrow ring around a rotating rod subject to deformation, especially when space annular around it is narrow, is well in agreement with the practical results that we were able to measure.

In the context of drilling operations in particular, it is possible to therefore define by application of the method, the rheology of the fluid optimal to maintain a high flow rate to obtain a good removal of cuttings without annular pressures break out of a safety range 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 γ ˙, 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 show schematically 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 shows schematically 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 invariable 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 γ ˙, 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, the configuration of which 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: e = δ R0-Ri R₀ et R_i sont respectivement les rayons du conduit et de la tige (Fig.1), et δ est l'écartement de leurs axes respectifs.
• l'excentricité maximale: qui est inférieure à 1 si la tige est pourvue de centreurs (de rayon R_t) qui l'empêchent de toucher la paroi du conduit; emax = R0-Rt R0-Ri
• 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 ΔL, S = davg ΔL

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: e = δ R 0 -R i R ₀ and R _i are respectively the radii of the duct and the rod (Fig. 1), and δ is the spacing of their respective axes.
• the maximum eccentricity : which is less than 1 if the rod is provided with centralizers (of radius R _t ) which prevent it from touching the wall of the conduit; e max = R 0 -R t R 0 -R i
• 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 ΔL, S = d avg ΔL

For calculations, according to a known formalism, the space annular 3 is assimilated to a series of juxtaposed slots of variable thickness depending on the actual offset. Space ring around the rod is deployed (Fig 4, 5) and we consider that the fluid flows between a number of plates of variable spacing in the axial direction (Fig. 9).

avec e_max défini par (2) et r_avg = 0.5d_avg. We can show that, for 0≤ x 0≤ ΔL, and 0≤ y ≤ 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 the expression:

with e _max defined by (2) and r _avg = 0.5d _avg .

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: U D = 24 z 2 D ∂P D fReη D ∂x D V D = 24 Z 2 D ∂P D fReη D ∂y D

All parameters are made dimensionless. So: x D = x / ΔL, y D = y / πr avg , z D = z / (D 0 - D i ) (D i = 2R i and D 0 = 2R 0 ).

The dimensionless viscosity η _D equals 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 ₀ 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 relations, to a model finite difference type numeric such as that defined by Markatos et al, in the publication already cited, where space ring finger is divided into grids, each representing a slit as defined above, whose thickness is determined by solving equations (4) or (5).

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

For a given ΔP, at the abscissa x = 0, the model considers initially a linear variation profile of the pressure and a average velocity field based on a certain flow for each slot. Reynolds number Re, coefficient of friction (f) and 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 allows extend the scope of the previous model to non-laminar flow regimes of any fluids including the rheofluidification index is generally less than 1, which correspond better to the circulations that we have to model In practice.

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 γ ˙ (Fig. 8). They obey the rheological law: τ = K ' γ not' where τ is the shear stress and K 'and n' are the parameters defined in the publication of Dodge, DW et al already cited.

To take this type of fluid into account, the definition of Reynolds number Re and we use the method described by Reed, T.D. et al, in the publication already cited, between this modified number Re and the coefficient of friction f, for the model resolution

The so-called GNM method described by Reed et al in the aforementioned publication is used to evaluate the diffusion term z 3 / D / fReη _D 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 remaining laminar in narrow parts, due to rapid variation in the diffusion term, from one grid to another, a method of known relaxation, is used. It increases the stability but it decreases the speed of convergence.

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

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

Eccentricity corrections

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

The method according to the invention however makes it possible to model the velocity field and the loss distributions of load around a long rod in rotation.

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

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

A correction factor R is defined as the ratio, for a same flow, between pressure drops per unit length (ΔP / ΔL) e, generated by an eccentric rod and the losses of corresponding load (ΔP / ΔL) c generated by the same rod centered.

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

Fluid circulation tests were performed at laboratory with an installation comprising a duct, the inner diameter was 24mm and an inner rod bent sinusoidally, the outside diameter of which was 18 mm. Sure Fig. 6 point a with coordinates 1 / S = 12, R = 0.66 and point b with coordinates 1 / S = 18, R = 0.64, correspond respectively at 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 a rod in a narrow pipe or hole, especially a train of drill pipe in an oil well, is generally low accentuated. This is the most common mode of operation in the practice. The pressure losses resulting from an offset of the rod (ΔP / ΔL) e, can therefore be calculated simply, from substantially independent of the sinuosity coefficient 1 / S in the case of long annulars.

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) in function of the rheohluidification index n is shown on the Figure 7 for the case of a linear type winding or sinusoidal.

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

Rotation corrections

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

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

Specialists know that the tangential flow between two coaxial cylinders is characterized by the magnitude of the Taylor number defined by Ta = ρWD i (D 0 -D i ) 4η 1/2 D 0 -D i D i where W is the speed of rotation and η, 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 included between 41.3 and around 400, we see that stable vortices so-called Taylor vortices, superimposed on the current lines circular. The flow becomes turbulent for values of Ta> 400.

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

To identify overall trends from the effects of rotation of the stem and the value of the corrective factors to be brought to previous results, to take into account the type of flow, it is useful to put the different parameters in a form dimensionless.

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

The Taylor number characterizing the effects ratio inertial to viscous effects in the azimuthal direction, is given by relation (10) on condition of extending its definition of so as to include the effects of rheo-fluidification. This requires replace in this relation, the Newtonian viscosity by the viscosity of a fluid in power law, calculated at the value of shear rate at the inner 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: R p = ΔP W ΔP W = 0

From the Taylor and Reynolds numbers, we construct another dimensionless parameter: R W = WR i U m 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 R_p1 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 that we observe is linked to the decrease in the viscosity of the fluid linked 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 R _p1 greater than 1 for the values of the Taylor number Ta <200.

3) Taylor and Reynolds numbers greater than 200

For this range of variation, we could establish an empirical relation of the form R p = ARe vs Your d where the coefficients A, c, d are included in the following ranges. 0 <A <10 0 <c <2 0 <d <2 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 (3)

1. Method for optimising the characteristics of a fluid circulation established in an annular space (3), and notably in a relatively narrow annulus, about a tubular element (2), such as a long pipe or a string of pipes whose eccentricity varies by virtue of deformations, characterised in that it comprises modelling the flow in the annular space so as to determine the value of the velocity field and the value of the pressure at any point along this annular space taking account of actual rheological properties of the fluid and bearing in mind that the form of the latter is variable over the entire length of the tubular element, comprising the application to the variations in pressure of a dimensionless correction factor (Rp) which depends on the Reynolds number (Re) and the Taylor number (Ta) of the fluid used and conforms to the relation: Rp = A.Rec.Tad, in which 0<A<10, 0<c<2 and 0<d<2, when the respective ratios of the inertial and viscous effects, in the axial direction and the azimuthal direction respectively, are greater than a given value, to take account of the variations in the losses of head in the annulus produced by the speed of rotation (W) of the tubular element.
2. Method according to the preceding claim, characterised in that account is taken of possible dynamic modifications of the shape of the tubular element, by applying another correction factor (R) which is substantially constant and independent of the shape of the tubular element, as long as the skewness (S) of the latter remains relatively low.
3. Method according to claim 2, characterised in that the other correction factor (R) is selected within the range 0.1<R<10.

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