EP0743423A1 - Metod for determining drilling conditions using a model - Google Patents

Abstract

A method of improving drilling performance involves (a) using a drilling model which takes into account the rock destruction effects of one or more cutters fixed on a rotary tool body and the debris flushing effect of a drilling mud by calculating a material balance from (i) debris prodn. by the cutter after penetration into the rock to a depth (delta), (ii) the debris bed covering the rock to a thickness (I), (iii) a mud layer of thickness (h) and debris concn. (c) between the debris bed and the tool body, (v) control parameters and (v) environmental parameters; and (b) determining the drilling conditions as a function of the response of the model to determined values of these parameters.

Description

The present invention relates to a method for determining the drilling conditions of a tool comprising several cutters interacting with a rock. The method involves the use of a drilling model based on the coupling of the effects of destruction of the rock by the cutters and the effects of the removal of the cuttings by a fluid. The invention preferably applies to the study of the phenomenon of jamming of a tool of the PDC type. Jamming is a dysfunction frequently observed by the driller and very harmful because it can very significantly reduce the speed of advancement of the drilling and sometimes even, in certain terrains, irreversibly annihilate the effects of drilling.

Several works have already been presented, but none takes into account the release of matter as does the representation modeled in this method. The main works are cited in the list of references included.

• de la production de débris de roche par le taillant ayant pénétré dans la roche d'une profondeur δ,
• d'un lit de débris recouvrant la roche sous une épaisseur l,
• d'une lame de fluide d'épaisseur h comprise entre le lit de débris et le corps, la lame fluide ayant une concentration c en débris,
• de paramètres de commandes,
• de paramètres d'environnement.

Thus, the present invention relates to a method for improving drilling performance in which a drilling model is implemented. The model takes into account the effects of destruction of a rock by at least one cutter fixed on a tool body driven in rotation and the effects of evacuation of rock debris by a fluid, by calculating a material balance from:

• the production of rock debris by the cutter having penetrated into the rock with a depth δ,
• a bed of debris covering the rock under a thickness l ,
• a fluid blade of thickness h between the debris bed and the body, the fluid blade having a debris concentration c,
• command parameters,
• environment settings.

By the method, the drilling conditions are determined as a function of the response of the model for determined values of said parameters.

At least one of the parameters: weight on the tool, speed of rotation of the tool and fluid flow rate, can be a control parameter.

In the model, the lift W of the tool can be broken down into a solid component Ws and a hydraulic component Wh which depends in particular on the fluid blade.

We can consider a particle size spread of the debris distributed according to a normal law as a function of the depth δ of the notch, of average µ related to the ductility of the rock and of a dispersion characterized by the standard deviation σ.

The solid matter balance B (t) can be such that B (t) = B ⁺ (t) -B ^- (t), in which B ⁺ (t) is a debris production term depending on δ and corresponding to the rhythm of destruction of the rock, and B ^- (t) is an expulsion term depending on l and h.

We can represent drilling as a dynamic system comprising, in the classic internal representation by state variables x, inputs u which will be those of a "weight on tool" control system, rod rotation speed, hydraulic power, system also subject to uncontrollable disturbances v associated with the variability of rock properties. By means of this model, the system is observed thanks to the output variables y which can be, among others, the torque at the tool, the speed of penetration in the axis of the hole, indicators linked to the vibratory level such as the enlargement of the diameter of the hole, wear indicators of the bits of the drill head, unfortunately difficult to design, all of these output variables being able to be disturbed by a noise w.

• d'éviter les risques relatifs à des écueils localisés, par exemple, liés à des intercalations de roche très dures ou, à l'autre extrême, susceptibles d'entraîner le bourrage de l'outil,
• d'avoir une stratégie cohérente à l'échelle du forage: par exemple, la détermination du nombre et de la durée d'utilisation optimaux des outils de forage, ou la nécessité d'une adaptation de la conduite du forage au fur et à mesure de l'usure des taillants.

Optimizing drilling can thus be the search for a control strategy that allows the driller:

• avoid the risks relating to localized pitfalls, for example, linked to very hard rock insertions or, at the other extreme, likely to cause the tool to jam;
• to have a coherent strategy on a drilling scale: for example, determining the optimal number and duration of use of drilling tools, or the need to adapt the conduct of drilling as and when wear of the cutting edges.

It is also clear that the present method can make it possible to assist in determining the structure of the drilling tools: for example, shape and location of the cutters, determination of the hydraulic flows in the vicinity of the destruction of the rock.

The following references can be used to illustrate the technological background of the field concerned as well as to complement the description of the present invention.

Andersen E.E. and Azar J.J., 1990, "PDC performance under simulated borehole conditions" SPE 20412, New Orleans Sept. 1990.

Cheatham C.A. and Nahm J.J., 1990, "Bit balling in water-reactive shale during full-scale drilling rate tests" IADC / SPE No. 19926, Houston.

Déliac E.P., 1986, "Optimization of slaughtering machines with peaks" Doctoral Dissertation, U. Paris 6 ed by ENSMP / CGES France.

Detournay E. and Atkinson C., 1991, "Influence of pore pressure on the drilling response of PDC bits", Rock Mechanics as a Multidisciplinary Science, Roegiers (ed.), Rotterdam.

Detournay E. and Defourny P., 1992, "A Phenomenological Model for the Drilling Action of Drag Bits", Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Flight. 29, No. 1, p13-23.

Falconer I.G., Burgess T.M. and Sheppard M.C., 1988, "Separating Bit and Lithology Effects from Drilling, Mechanics Data", IADC / SPE Drilling Conference, Dallas, feb 28-march 2, 1988.

Garnier AJ and van Lingen NH, 1958, "Phenomena affecting drilling rates at depth" SPE fall meeting, Houston.

Glowka D.A., 1985, "Implications of Thermal Wear Phenomena for PDC Bit Design and Operation", 60th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers in Las Vegas, Sept 22-25, 1985, SPE 14222.

Karasawa H. and Misawa S., 1992, "Development of New PDC Bits for Drilling of Geothermal Wells - Part 1: Laboratory Testing", Journal of Energy Resources Technology, Dec 1992, vol.114 p 323.

Pessier R.C. and Fear M.J., 1992, "Quantifying common drilling problems with mechanical specific energy and a bit specific coefficient of sliding friction" SPE 24584.

Pessier R.C., Fear M.J., and Wells M.R., 1994, "Different shales dictate fundamentally different strategies in hydraulics, bit selection, and operating practices".

Pierry J. and Charlier R., 1994, "Finite element modeling of shear band localization and application to rock cutting by a PDC tool" SPE / ISRM Eurock Conference, Delft.

Putot C., 1995, "A drilling model taking into account the effects of rock destruction and removal of spoil", 2nd National Colloquium in Structural Design, Giens.

Sellami H., 1987, "Study of worn peaks, application to slaughter machines" Doctoral Dissertation ENSMP / CGES France.

Sellami H., Fairhurst C., Déliac E. and Delbast B., 1989, "The Role of in-situ Stresses and Mud Pressure on the Penetration of PDC bits" Rock at Great Depth, Maury & Fourmaintraux eds, Rotterdam 1989.

Sinor A. and Warren T.M., 1989, "Drag Bit Wear Model", SPE Drilling Engineering, June 1989, p 128.

Sinor A., Warren T.M., Behr S.M., Wells M.R. and Powers J.R., 1992, "Development of an anti-whirl core bit", SPE 24587.

Wardlaw HWR, 1971, "Optimization of Rotary Drilling Parameters" PHD Dissertation, U. of Texas.

Warren T.M. and Winters W.J., 1986, "Laboratory Study of Diamond-Bit Hydraulic Lift", SPE Drilling Engineering, aug 1986.

Warren T.M., 1987 "Penetration-Rate Performance of Roller-Cone Bits", SPE Drilling Engineering, march 1987.

Warren T.M. and Armagost W.K. "Laboratory drilling performance of PDC bits" SPE Drilling Engineering, june 1989.

Warren T.M. and Sinor A., "Drag-bit performance modeling", SPE Drilling Engineering, june 1989.

Wells R., "Dynamics of rock-chip removal by turbulent jetting" SPE Drilling Engineering, june 1989.

Zijsling D.H. "Single cutter testing: a key for PDC bit development" SPE 16529 Offshore Europe Aberdeen, 1987.

• Les figures 1A et 1B représentent le modèle physique dans les conditions initiales et en cours d'évolution à l'instant t.
• La figure 2 représente la courbe d'équilibre obtenue dans le cas d'une application particulière du modèle selon l'invention.

The present invention will be better understood from the following description, illustrated by the appended figures, among which:

• FIGS. 1A and 1B represent the physical model under the initial conditions and in the course of evolution at time t.
• FIG. 2 represents the equilibrium curve obtained in the case of a particular application of the model according to the invention.

The model presented below is a non-linear evolution model with, in a first variant, three independent variables supposed to completely characterize the state of the drilling system. It is in fact a so-called "local" cutter model whose operation suffices to describe, in this variant, an average of the overall behavior of the drilling tool.

In FIG. 1B, the cutter is in interaction with the virgin rock 2 and the current penetration δ constitutes a first state variable. Figure 1A shows the initial conditions where the cutter of height H, fixed on a body 3, has penetrated from the depth δ ₀ into the rock. Specific studies are also conducted on the cutting process which show the difficulty of taking into account and the diversity of modes of representation: more or less guaranteed independence of the cutting and abutment effects, not necessarily one-to-one link of penetration and of the normal force, justified by the theory of plasticity, influence of successive recoveries (work hardening).

The hypothesis retained in this work consists in being satisfied with a one-to-one link between normal effort exerted on the cutting edge and sinking. Let W _{S be} the so-called "solid" vertical force associated with this penetration. The link of W _S with δ will be explained below.

Each of the N _C equivalent cutting edges constituting the tool produces rock chips and this instantaneous production, assumed to be proportional to δ, is partially evacuated into the annular space, partially stored in the immediate vicinity of the cutting edge in the form of a bed of debris, the l 'current thickness is the second state variable of our formulation, called l; this debris bed is assumed to line the rock front evenly.

   où γ est la hauteur de lame usée, grandeur lentement évolutive que nous considérerons en fait comme paramètre. L'évacuation est également entravée lorsque la viscosité équivalente de la suspension est augmentée, en raison de l'accroissement de concentration en particules solides. Ces deux effets sont traduits par la relation suivante : $${\mathit{\text{W}}}_{\mathit{\text{H}}}\text{=}\frac{\mathit{\text{N}}\text{δη}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{4}}}{{\mathit{\text{h}}}^{\text{3}}}$$

The residual space between the tool body and the debris bed allows the evacuation of rock chips. This evacuation is made difficult when the residual space is limited; we will note h the thickness of the fluid blade, obviously linked to the total height H of the cutting edge in new condition by the relation: $$\text{H =}\mathit{\text{h}}\text{+ δ +}\mathit{\text{l}}\text{+ γ}$$

where γ is the height of the worn blade, a slowly evolving quantity that we will in fact consider as a parameter. Evacuation is also hampered when the equivalent viscosity of the suspension is increased, due to the increased concentration of solid particles. These two effects are translated by the following relation: $${\mathit{\text{W}}}_{\mathit{\text{H}}}\text{=}\frac{\mathit{\text{NOT}}\text{δη}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{4}}}{{\mathit{\text{<figure-callout id="3" label="h" filenames="imgf0001.png" state="{{state}}">h</figure-callout>}}}^{\text{3}}}$$

We can refer to the article by Jordaan I.J., Maes M.A. and J.P. Nadreau, 1988, "The crushing and clearing of ice in fast spherical indentation tests", Offshore Mechanics and Arctic Engineering, Houston.

The third state variable is also very naturally introduced: it may be the concentration c of the suspension but the choice will be made of the "equivalent" dynamic viscosity associated η or the equivalent kinematic viscosity ν (at distinguish from the viscosity ν ₀ of the fluid proper).

• le poids sur outil W
• la vitesse de rotation N
• le débit fluide ou la puissance hydraulique ; en fait, dans le présent modèle,

la vitesse de l'écoulement en sortie de duse v _n.As has been announced, quantities are defined as commands on which intervention is possible or desirable; these are essentially:

• the weight on tool W
• the rotation speed N
• fluid flow or hydraulic power; in fact, in this model,

the flow velocity at the outlet of nozzle v _n .

In the present example, these quantities are supposed to be constant and therefore comparable to the many parameters of the problem. We can nevertheless consider the response of the system to a perturbation of this command, and consider various types of regulation associated with the variability of rock properties.

In the present model, the analysis of the decomposition of the weight on the tool is based on the principle of separation between a classical component called solid W _S subject to the usual representation formulas, and a hydraulic lift W _H which increases considerably when the thickness h of the fluid wafer decreases and the equivalent viscosity η increases; we write : $${\mathit{\text{W = W}}}_{\mathit{\text{S}}}{\mathit{\text{+ W}}}_{\text{H}}$$

• A _γ l'aire de butée de chaque taillant au stade d'usure γ
• A _c (δ) l'aire de coupe lorsque l'usure est γ, la pénétration solide δ
• S _p et S _c les résistances de la roche, respectivement en compression et au cisaillement
• N _c nombre de taillants
• D _B le diamètre de l'outil
• α et µ⁺ caractéristiques liées à l'interface outil/roche

The solid component W _S is explained according to the article by Kuru E. and Wojtanowsicz AK, 1988, "A Method for Detecting In-Situ PDC Dull and Lithology Change", IADC / SPE Drilling Conference, Dallas, feb 28, march 2, 1988.

• At _γ the abutment area of each cutter at the wear stage γ
• A _c (δ) the cutting area when the wear is γ, the solid penetration δ
• S _p and S _c the resistances of the rock, respectively in compression and shear
• N _c number of cutters
• D _B the diameter of the tool
• α and µ ⁺ characteristics linked to the tool / rock interface

We write: $$\frac{{\mathit{\text{W}}}_{\mathit{\text{S}}}}{{\mathit{\text{NOT}}}_{\mathit{\text{vs}}}{\mathit{\text{AT}}}_{\text{γ}}}\text{=}{\mathit{\text{S}}}_{\mathit{\text{p}}}\text{+}\frac{{\mathit{\text{AT}}}_{\mathit{\text{vs}}}\left(\text{δ}\right)}{{\mathit{\text{AT}}}_{\text{γ}}}{\mathit{\text{S}}}_{\mathit{\text{vs}}}{\text{(sinα + µ}}^{\text{+}}\text{cosα)}$$

   η viscosité (dynamique) équivalente de la suspension boue plus particules solides.The hydraulic component is explained according to the article by Jordaan IJ, Maes MAand JP Nadreau, 1988, "The crushing and clearing of ice in fast spherical indentation tests", Offshore Mechanics and Arctic Engineering, Houston. $${\mathit{\text{W}}}_{\mathit{\text{H}}}\text{=}\frac{\mathit{\text{NOT}}\text{δη}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{4}}}{{\mathit{\text{<figure-callout id="3" label="h" filenames="imgf0001.png" state="{{state}}">h</figure-callout>}}}^{\text{3}}}$$

η equivalent (dynamic) viscosity of the slurry suspension plus solid particles.

The thwarted circulation of the drilling fluid (enriched in particles) and in particular the pressure drop at the front of the tool are indicators of this lift effect.

The present invention also describes a rock rupture model integrated into the drilling model.

It is a representation model with an idealized diagram of parallelepipedal chip with thickness and square area, on the side mD _c , where D _c is the diameter hydraulic considered for evacuation. Despite the simplicity of this geometry, it is considered important to take into account a particle size spread.

• de la profondeur actuelle de l'entaille δ
• de la ductilité de la roche exprimée au travers du paramètre µ = E(D _c )/δ
• d'une dispersion caractérisée par l'écart-type σ.

We thus consider a Gaussian distribution of sizes D _c which takes into account:

• of the current depth of the notch δ
• of the ductility of the rock expressed through the parameter µ = E (D _c ) / δ
• of a dispersion characterized by the standard deviation σ.

E (D _c ) expresses the mean of the size distribution and µ reflects the degree of ductility of the broken rock under drilling conditions, a characteristic assumed to be independent of δ; m≥1 is a parameter linking hydraulic diameter and geometry; we often assume m = 1.

Rather than the variable D _c , we may prefer to introduce the number n of chips released by each of the N _c bits of a tool of diameter D _B during a revolution, so that:

In the article "A Dynamic Model for Rotary Rock Drilling", Journal of Energy Resources Technology, june 1982, vol 104 p 108, from the authors Eronini IE, Somerton WH and Auslander DM, 1982, we consider, for a tricone bit, a chip evacuation model which is reproduced here with, however, the introduction of a "spread" particle size.

The expression of the hydrodynamic forces exerted on the rock chip delimited by the rupture, used in the present model, is also described in the article above.

The foundations of the model are as follows:

To release the chip, it is first necessary to overcome the retention effect due to the difference in pressure existing between mud pressure and pore pressure whose effect is considerable compared to that of gravity; the associated effort is assumed to be overcome by the only lift effect F _L (L = lift), the expression of which is presented in appendix 1. The time constant τ _L of the process is extremely short and therefore neglected vis-à-vis that associated with the actual training effect ( F _D and τ _D ; D = drag). The chip is then accelerated from the position where it is conceptually taken out of its housing under the effect of the drag force F _D to the annular space;

   avec un modèle de représentation de ω _c dû à Eronini (1982), dont le détail n'est pas reproduit ici, condensé grâce au paramètre λ, en fonction notamment de la présence d'un cake dont la perméabilité est supposée connue. $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{≥1+}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{c}}}\mathit{\text{g}}}$$

   ρ _c masse volumique des particules solides.Let ω _{0 be} the self-weight of the rock chip of current size D _c and ω _c the suction force exerted on this fragment to retain it; the evacuation condition is written: $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{≥}\frac{{\text{ω}}_{\mathit{\text{vs}}}}{{\text{ω}}_{\mathit{\text{o}}}}$$

with a representation model of ω _c due to Eronini (1982), the details of which are not reproduced here, condensed thanks to the parameter λ, in particular as a function of the presence of a cake whose permeability is assumed to be known. $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{≥1 +}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{vs}}}\mathit{\text{g}}}$$

ρ _c density of solid particles.

In practice, the term 1 is completely negligible compared to the second.

sont expulsées, où D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{c}}}$

est la dimension de copeau réalisant exactement l'équilibre entre effort de succion et effet de portance : $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{=}\frac{{\text{ω}}_{\mathit{\text{c}}}}{{\text{ω}}_{\mathit{\text{o}}}}$$

Only the particles characterized by D _c ≤ D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

are expelled, where D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

is the chip dimension achieving the exact balance between suction force and lift effect: $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{=}\frac{{\text{ω}}_{\mathit{\text{vs}}}}{{\text{ω}}_{\mathit{\text{o}}}}$$

par rapport à la courbe granulométrique conditionne la proportion de particules "évacuées" par rapport à celles "produites". D's position $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

compared to the grain size curve conditions the proportion of particles "evacuated" compared to those "produced".

We will assume that the distributions are normal; the distribution of sizes D _c as a function of δ depends, of course, on the ductility of the rock but it has been assumed that there is no scale effect and that only the distribution D _c / δ is to be characterized .

• Le seuil de décollement est d'autant plus élevé que l'épaisseur à est plus petite.
• Le fractionnement en un nombre de copeaux élevé (roche ductile avec µ faible) favorise le décollement et donc les possibilités d'évacuation.
• L'augmentation du débit (à travers la vitesse en sortie de duse v _n ) et de la viscosité stimulent évidemment l'évacuation.

We show that:

• The separation threshold is higher the smaller the thickness.
• The fractionation into a high number of chips (ductile rock with low µ) promotes detachment and therefore the possibilities of evacuation.
• The increase in the flow rate (through the velocity at the outlet of nozzle v _n ) and in the viscosity obviously stimulate evacuation.

The mass balance is written as follows:

   où τ = τ _D puisque l'accélération du copeau s'effectue principalement sous l'effet des efforts de traînée. V _f est le volume élémentaire du copeau et N _c le nombre de sites de production, autrement dit, le nombre de taillants. V̇ _R , homogène à un volume par unité de temps, est le débit d'évacuation solide.Supposing for a moment that the particle size is not spread out. We then write:

where τ = τ _D since the acceleration of the chip takes place mainly under the effect of the drag forces. V _f is the basic volume of the chip and N _c the number of production sites, in other words, the number of cutters. V̇ _R , homogeneous at a volume per unit of time, is the solid evacuation rate.

   ce qui donne un bilan de progression, exprimé cette fois en unité de longueur par unité de temps : $$\dot{\mathit{\text{s}}}\text{=}\mathit{\text{N}}\text{δ}\mathit{\text{-}}\frac{{\dot{\mathit{\text{V}}}}_{\mathit{\text{R}}}}{\frac{\text{π}}{\text{4}}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{2}}}$$

   si ce bilan est positif, il y a accumulation de débris et enrichissement de la suspension ; si le bilan est négatif, les conclusions sont inversées en présence d'un fond enrichi de matières solides ; sinon, l'évacuation est parfaite et il n'y a pas lieu de se poser le présent problème.The rate of solid production (volume per unit of time) should be assumed to be equal to: $$\frac{\text{π}}{\text{4}}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{2}}\mathit{\text{NOT}}\text{δ}$$

which gives a progress report, this time expressed in units of length per unit of time: $$\dot{\mathit{\text{s}}}\text{=}\mathit{\text{NOT}}\text{δ}\mathit{\text{-}}\frac{{\dot{\mathit{\text{V}}}}_{\mathit{\text{R}}}}{\frac{\text{π}}{\text{4}}{\mathit{\text{D}}}_{\mathit{\text{B}}}^{\text{2}}}$$

if this balance is positive, there is accumulation of debris and enrichment of the suspension; if the balance is negative, the conclusions are reversed in the presence of a background enriched with solid matter; otherwise, the evacuation is perfect and there is no reason to ask the present problem.

, taille de copeau réalisant exactement l'équilibrage, et la distribution. On écrit donc : $${\dot{\mathit{\text{V}}}}_{\mathit{\text{R}}}\text{=χ}{\mathit{\text{N}}}_{\mathit{\text{c}}}\frac{\frac{{\text{ω}}_{\mathit{\text{o}}}^{\mathit{\text{o}}}}{{\text{ρ}}_{\mathit{\text{c}}}\mathit{\text{g}}}}{{\text{τ}}_{\mathit{\text{D}}}}$$

   ω $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{o}}}$

est le poids de copeau dont la taille est D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{c}}}$

(pour l'épaisseur δ)In this model we use a particle size distribution according to the normal law; more precisely D _c / δ is assumed to be distributed according to a normal law of mean µ and standard deviation σ. This results in a reduction factor χ (calculated in appendix 2) multiplier of N _c V _f / τ _D function, as said above, of the shift between D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

, chip size performing exactly balancing, and distribution. So we write: $${\dot{\mathit{\text{V}}}}_{\mathit{\text{R}}}\text{= χ}{\mathit{\text{NOT}}}_{\mathit{\text{vs}}}\frac{\frac{{\text{ω}}_{\mathit{\text{o}}}^{\mathit{\text{o}}}}{{\text{ρ}}_{\mathit{\text{vs}}}\mathit{\text{g}}}}{{\text{τ}}_{\mathit{\text{D}}}}$$

ω $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{o}}}$

is the chip weight whose size is D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

(for thickness δ)

The calculation path is presented in appendix 1.

• l'effort de portance F _L
• l'effort de traînée F _D et le temps caractéristique associé τ _D

It makes it possible to evaluate successively, for the chip of current size D _c :

• lift effort F _L
• the drag force F _D and the associated characteristic time τ _D

$${\dot{\mathit{\text{s}}}}^{\text{-}}\text{=}{\mathit{\text{m}}}_{\text{0}}\text{χ(δ)}$$

où m ₀ est défini plus loin.The balance sheet then takes the form: $${\dot{\mathit{\text{s}}}}^{\text{+}}\text{=}\mathit{\text{NOT}}\text{δ}$$

$${\dot{\mathit{\text{s}}}}^{\text{-}}\text{=}{\mathit{\text{<figure-callout id="0" label="m" filenames="imgf0001.png" state="{{state}}">m</figure-callout>}}}_{\text{0}}\text{χ (δ)}$$

where m ₀ is defined below.

• la vitesse ν _n
• la viscosité de la boue
• la pression de rétention λP
• la densité de boue essentiellement, la densité de copeaux accessoirement.

The evacuation term depends on δ only through χ and is conditioned, with fixed technology, by:

• the speed ν _n
• the viscosity of the mud
• the retention pressure λP
• essentially the density of mud, the density of chips incidentally.

= ṡ, nous notons B(δ) le bilan, homogène à une accumulation (longueur) par unité de temps.To avoid notation derived from type $\frac{\mathit{\text{ds}}}{\mathit{\text{dt}}}$

= ṡ, we denote B (δ) the balance, homogeneous to an accumulation (length) per unit of time.

• (i) la première est une modification de pure forme consistant, par souci d'homogénéité, à rendre adimensionnelle à en la remplaçant par y ₁ = δ/δ₀.
  On note encore B le bilan adimensionnel, homogène à l'inverse d'un temps, de sorte que : $${\mathit{\text{B(y}}}_{\mathit{\text{1}}}\text{) =}{\mathit{\text{Ny}}}_{\mathit{\text{1}}}\text{-}{\mathit{\text{m}}}_{\mathit{\text{1}}}{\mathit{\text{.χ(y}}}_{\text{1}}\mathit{\text{)}}$$
  
• (ii) la seconde est réalisée pour rendre compte assez correctement du phénomène de bourrage notamment. Elle consiste à reconnaître la dépendance du terme d'expulsion vis-à-vis des variables d'état l et h. Il nous est apparu assez commode, dans un premier temps, de rendre compte du phénomène en faisant dépendre le terme d'expulsion de la seule variable adimensionnelle y ₃ = l/δ₀ de sorte que : $${\mathit{\text{B(y}}}_{\mathit{\text{1}}}{\mathit{\text{,y}}}_{\mathit{\text{3}}}{\mathit{\text{) - Ny}}}_{\mathit{\text{1}}}{\mathit{\text{- m}}}_{\mathit{\text{1}}}{\mathit{\text{(y}}}_{\mathit{\text{3}}}{\mathit{\text{)χ(y}}}_{\mathit{\text{1}}}\text{)}$$
  
  où la dépendance m ₁ (y ₃ ) est explicitée en annexe 3.

In fact, we operate in the following two modifications:

• (i) the first is a purely formal modification consisting, for the sake of homogeneity, of making it dimensionless to by replacing it with y ₁ = δ / δ ₀ .
  We also denote B the dimensionless balance, homogeneous to the inverse of a time, so that: $${\mathit{\text{B (y}}}_{\mathit{\text{1}}}\text{) =}{\mathit{\text{Ny}}}_{\mathit{\text{1}}}\text{-}{\mathit{\text{m}}}_{\mathit{\text{1}}}{\mathit{\text{.χ (y}}}_{\text{1}}\mathit{\text{)}}$$
  
• (ii) the second is carried out to give a fairly good account of the jamming phenomenon in particular. It consists in recognizing the dependence of the expulsion term on state variables l and h. It seemed to us quite convenient, at first, to account for the phenomenon by making the expulsion term depend on the only dimensionless variable y ₃ = l / δ ₀ so that: $${\mathit{\text{B (y}}}_{\mathit{\text{1}}}{\mathit{\text{, y}}}_{\mathit{\text{3}}}{\mathit{\text{) - Ny}}}_{\mathit{\text{1}}}{\mathit{\text{- m}}}_{\mathit{\text{1}}}{\mathit{\text{(y}}}_{\mathit{\text{3}}}{\mathit{\text{) χ (y}}}_{\mathit{\text{1}}}\text{)}$$
  
  where the dependency m ₁ ( y ₃ ) is explained in appendix 3.

Strictly speaking, the expulsion term also visibly depends on the current residual thickness of the fluid blade, i.e. h, which is rather considered as a parameter in appendix 3.

$${\text{B}}^{\text{-}}{\text{(t) = B}}^{\text{-}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) roche expulsée}$$

$${\text{B(t) = B}}^{\text{+}}{\text{(t) - B}}^{\text{-}}\text{(t)}$$

$${\mathit{\text{y}}}_{\text{1}}\mathit{\text{(t) =}}{\text{δ(t)/δ}}_{\text{0}}\text{entaille réduite dans la roche vierge}$$

$${\mathit{\text{y}}}_{\mathit{\text{2}}}\mathit{\text{(t)}}{\text{= Log h(t)/h}}_{\text{o}}\text{viscosité équivalente de la suspension}$$

$${\mathit{\text{y}}}_{\text{3}}\text{(t) =}\mathit{\text{I(t)}}{\text{/δ}}_{\text{0}}\text{épaisseur réduite du lit de débris}$$

$${\text{B}}^{\text{-}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}{\mathit{\text{m}}}_{\text{0}}{\mathit{\text{f}}}_{\text{v}}\text{(}{\mathit{\text{y}}}_{\text{3}}{\text{)χ}}_{\text{µ/σ}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{) =}{\mathit{\text{m}}}_{\text{0}}{\mathit{\text{f}}}_{\text{ν}}^{\text{*}}\text{(}\mathit{\text{Z}}{\text{)χ}}_{\text{µ/σ}}^{\text{*}}\text{(X)}$$

• m_o "jauge" de l'évacuation, norme du terme d'expulsion
• χ $\genfrac{}{}{0ex}{}{\text{*}\hfill }{\text{µ/σ}\hfill }$
  
  (χ) dépendance, dite principale, vis-à-vis de la pénétration (y ₁) ; issue de la fonction de répartition de la loi normale
• f $\genfrac{}{}{0ex}{}{\text{*}\hfill }{\mathit{\text{ν}}\hfill }$
  
  (Z) modulation de l'expulsion selon l'épaisseur du lit de débris (y ₃ )
  
• a_d, a_c, a_l coefficients utilisés dans la formulation hydrodynamique et dont les valeurs peuvent être trouvées dans l'article d'Eronini
• d diamètre de duse ; v_n vitesse de fluide en sortie de duse
• D _B diamètre de l'outil
• ρ_m, ρ_c masses volumiques de la boue et de la roche respectivement
• λP effet de rétention par pression différentielle au travers du copeau
• x et z sont des variables associées respectivement à y ₁ et y ₃ permettant une écriture explicite (Annexes 2 et 3).

Ultimately, the solid material balance includes a production term B ⁺ corresponding to the rate of destruction of the rock and an expulsion term B ^- . With regard to the dependence on state variables y ₁ , y ₂ , y ₃ , we made the following choice: $${\text{B}}^{\text{+}}{\text{(t) = B}}^{\text{+}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{) destroyed rock}$$

$${\text{B}}^{\text{-}}{\text{(t) = B}}^{\text{-}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) expelled rock}$$

$${\text{B (t) = B}}^{\text{+}}{\text{(t) - B}}^{\text{-}}\text{(t)}$$

$${\mathit{\text{y}}}_{\text{1}}\mathit{\text{(t) =}}{\text{δ (t) / <figure-callout id="0" label="δ" filenames="imgf0001.png" state="{{state}}">δ</figure-callout>}}_{\text{0}}\text{reduced cut in virgin rock}$$

$${\mathit{\text{y}}}_{\mathit{\text{2}}}\mathit{\text{(t)}}{\text{= Log h (t) / h}}_{\text{o}}\text{equivalent viscosity of the suspension}$$

$${\mathit{\text{y}}}_{\text{3}}\text{(t) =}\mathit{\text{I (t)}}{\text{/ <figure-callout id="0" label="δ" filenames="imgf0001.png" state="{{state}}">δ</figure-callout>}}_{\text{0}}\text{reduced thickness of debris bed}$$

$${\text{B}}^{\text{-}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}{\mathit{\text{m}}}_{\text{0}}{\mathit{\text{f}}}_{\text{v}}\text{((}{\mathit{\text{y}}}_{\text{3}}{\text{) χ}}_{\text{µ / σ}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{) =}{\mathit{\text{m}}}_{\text{0}}{\mathit{\text{f}}}_{\text{ν}}^{\text{*}}\text{((}\mathit{\text{Z}}{\text{) χ}}_{\text{µ / σ}}^{\text{*}}\text{(X)}$$

• m _o "gauge" of evacuation, norm of the term of eviction
• χ $\genfrac{}{}{0ex}{}{\text{*}\hfill }{\text{µ / σ}\hfill }$
  
  ( χ ) dependence, said to be principal, on penetration ( y ₁ ); from the normal distribution function
• f $\genfrac{}{}{0ex}{}{\text{*}\hfill }{\mathit{\text{ν}}\hfill }$
  
  ( Z ) modulation of the expulsion according to the thickness of the debris bed (y ₃ )
  
• a _d , a _c , a _l coefficients used in the hydrodynamic formulation and whose values can be found in the article by Eronini
• d nozzle diameter; v _n fluid speed at the outlet of the nozzle
• D _B tool diameter
• ρ _m , ρ _c densities of mud and rock respectively
• λP retention effect by differential pressure through the chip
• x and z are variables associated respectively with y ₁ and y ₃ allowing explicit writing (Appendices 2 and 3).

Before reduction to three state variables, the problem a priori comprises five variables including three of geometric type : δ, l, h respectively depth of notch in the virgin rock, thickness of bed of debris and thickness of the fluid section. (γ worn blade height is a variable of slow evolution in comparison with those which will be studied in this problem; it therefore intervenes here as a parameter); then two state variables of the concentration of the suspension type; c the concentration, ν the associated "equivalent" kinematic viscosity (to be distinguished from the viscosity ν ₀ of the drilling fluid itself).

• d'une équation de conservation de la somme des épaisseurs des différentes tranches qui, exprimée sous forme différentielle sur les variables dimensionnelles δ,l,h s'énonce : $$\mathit{\text{dδ + dl + dh}}\text{= 0}$$
  
• d'une écriture de répartition du bilan de matière B(y ₁ , y ₃ ) ou taux d'accumulation ds/dt entre contributions partielles dues à :
  • (i) épaississement du lit de débris (dl)
  • (ii) augmentation de concentration de la suspension (h dc)
  • (iii) diminution d'épaisseur de la lame fluide (c dh)
  de sorte que : $$\mathit{\text{dl + cdh + hdc}}\text{=}\mathit{\text{ds}}$$
  
• de la loi de commande W = cte = W _S + W _H

The evolution equations result from the writing:

• of a conservation equation of the sum of the thicknesses of the different slices which, expressed in differential form on the dimensional variables δ, l , h is stated: $$\mathit{\text{dδ + dl + dh}}\text{= 0}$$
  
• an entry for the distribution of material balance B ( y ₁ , y ₃ ) or accumulation rate ds / dt between partial contributions due to:
  • (i) thickening of the debris bed (dl)
  • (ii) increase in suspension concentration (h dc)
  • (iii) decrease in thickness of the fluid blade (c dh)
  so that : $$\mathit{\text{dl + cdh + hdc}}\text{=}\mathit{\text{ds}}$$
  
• of the control law W = cte = W _S + W _H

   et on suppose que cette grandeur est invariante avec δ. La relation différentielle s'écrit donc : $$\text{dW = 0 soit:}\frac{\text{dδ}}{\text{δ}}\text{+}\frac{\text{dη}}{\text{η}}\text{-3}\frac{\text{dh}}{\text{h}}\text{+}\frac{{\text{W}}_{\text{S,δ}}}{{\text{W}}_{\text{H}}}\text{dδ=0}$$

• de relations de comportement simplifiées:

The expression of W _S and W _H mentioned above makes it possible to explain, still in differential form, this very particular command. We use the condensed notation: $${\text{W}}_{\text{S, δ}}\text{=}\frac{{\text{dW}}_{\text{S}}}{\text{from}}{\text{= S}}_{\text{vs}}{\text{NOT}}_{\text{vs}}{\text{(sin α + µ}}^{\text{+}}\text{cos α)}\frac{{\text{dA}}_{\text{vs}}}{\text{from}}$$

and we suppose that this quantity is invariant with δ. The differential relation is therefore written: $$\text{dW = 0 either:}\frac{\text{from}}{\text{δ}}\text{+}\frac{\text{dη}}{\text{η}}\text{-3}\frac{\text{dh}}{\text{h}}\text{+}\frac{{\text{W}}_{\text{S, δ}}}{{\text{W}}_{\text{H}}}\text{dδ = 0}$$

• simplified behavior relations :

$$\text{η dc - b(1-c) dη = 0}$$

We state in what follows two differential relationships, depending on the only parameters a and b, linking the concentration of the suspension c, equivalent viscosity η and fluid slice thickness h. Those are : $$\text{η dh + ah dη = 0}$$

$$\text{η dc - b (1-c) dη = 0}$$

Writing the evolution equations:

We ask: $$\text{K =}\frac{\frac{\text{1}}{\text{δ}}\text{+}\frac{{\text{W}}_{\text{S, δ}}}{{\text{W}}_{\text{H}}}}{\text{1 + 3a}}$$

This factor will be noted K (y ₁ , y ₂ ) in the final presentation. Manipulating the five relationships leads to the following reduction: $$\mathit{\text{-ds = dδ}}\text{{1+ (}\mathit{\text{at}}\text{+}\mathit{\text{b) (}}\text{1-}\mathit{\text{c) hK}}}$$

$$\dot{\mathit{\text{I}}}\mathit{\text{=}}\text{-(1+}\mathit{\text{ahK)f(X,u)}}$$

$$\frac{\dot{\text{η}}}{\text{η}}\text{=}\mathit{\text{-Kf(X,u)}}$$

$$\frac{\dot{\text{h}}}{\text{h}}\text{=aK f(X,u)}$$

$$\frac{\dot{\mathit{\text{c}}}}{\text{1-}\mathit{\text{c}}}\text{=}\mathit{\text{-bKf(X,u)}}$$

avec : $$\mathit{\text{f(X,u) =}}\frac{\text{-}\dot{\mathit{\text{s}}}\left(\mathit{\text{X,u}}\right)}{\text{1+}\mathit{\text{h}}\left(\text{1}\mathit{\text{-c}}\right)\left(\mathit{\text{a+b}}\right)\mathit{\text{K}}}$$

The five state variables thus evolve according to the diagram, where X denotes to simplify the state vector and u the command, very elementary here: $$\dot{\mathit{\text{δ}}}\mathit{\text{= f (X, u)}}$$

$$\dot{\mathit{\text{I}}}\mathit{\text{=}}\text{- (1+}\mathit{\text{ahK) f (X, u)}}$$

$$\frac{\dot{\text{η}}}{\text{η}}\text{=}\mathit{\text{-Kf (X, u)}}$$

$$\frac{\dot{\text{h}}}{\text{h}}\text{= aK f (X, u)}$$

$$\frac{\dot{\mathit{\text{vs}}}}{\text{1-}\mathit{\text{vs}}}\text{=}\mathit{\text{-bKf (X, u)}}$$

with: $$\mathit{\text{f (X, u) =}}\frac{\text{-}\dot{\mathit{\text{s}}}\left(\mathit{\text{X, u}}\right)}{\text{1+}\mathit{\text{h}}\left(\text{1}\mathit{\text{-vs}}\right)\left(\mathit{\text{a + b}}\right)\mathit{\text{K}}}$$

$$\mathit{\text{E = Log}}\frac{\text{η}}{{\text{η}}_{\text{0}}}\mathit{\text{E}}\text{=}{\mathit{\text{y}}}_{\mathit{\text{2}}}$$

$$\mathit{\text{L}}\text{=}\frac{\mathit{\text{l}}}{{\text{δ}}_{\text{0}}}\mathit{\text{L}}\text{=}{\mathit{\text{y}}}_{\mathit{\text{3}}}$$

$$\mathit{\text{H=-Log}}\frac{\mathit{\text{h}}}{{\mathit{\text{h}}}_{\mathit{\text{0}}}}$$

$$\mathit{\text{F = Log(1-c)}}$$

We explain on the one hand the similarity of the last three relations and use the adimensional forms: $$\text{Δ =}\frac{\text{δ}}{{\text{δ}}_{\text{0}}}\text{Δ =}{\mathit{\text{y}}}_{\mathit{\text{1}}}$$

$$\mathit{\text{E = Log}}\frac{\text{η}}{{\text{η}}_{\text{0}}}\mathit{\text{E}}\text{=}{\mathit{\text{y}}}_{\mathit{\text{2}}}$$

$$\mathit{\text{L}}\text{=}\frac{\mathit{\text{l}}}{{\text{δ}}_{\text{0}}}\mathit{\text{L}}\text{=}{\mathit{\text{y}}}_{\mathit{\text{3}}}$$

$$\mathit{\text{H = -Log}}\frac{\mathit{\text{h}}}{{\mathit{\text{h}}}_{\mathit{\text{0}}}}$$

$$\mathit{\text{F = Log (1-c)}}$$

$$\text{F(t) = bE(t)}$$

The differential equations then take the form reduced to three independent variables only since we obviously have: $$\text{H (t) = aE (t)}$$

$$\text{F (t) = bE (t)}$$

$${\dot{\mathit{\text{y}}}}_{\text{2}}\text{=}\mathit{\text{-K}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{)}{\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

$${\dot{\mathit{\text{y}}}}_{\text{3}}\text{= -(1+}\mathit{\text{afe}}{\text{}}^{{\mathit{\text{-ay}}}_{\mathit{\text{2}}}}\mathit{\text{K}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{))}{\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

où : $${\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\frac{\text{-}\mathit{\text{B}}\left({\mathit{\text{y}}}_{\text{1}}{\mathit{\text{,y}}}_{\text{3}}\right)}{\text{1+}\left(\mathit{\text{a+b}}\right)\frac{{\mathit{\text{h}}}_{\text{0}}}{{\text{δ}}_{\text{0}}}\mathit{\text{e}}{\text{}}^{\mathit{\text{-}}\left(\mathit{\text{a+b}}\right){\mathit{\text{y}}}_{\mathit{\text{2}}}}\mathit{\text{K}}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\right)}$$

avec: $$\mathit{\text{K}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{) =}\frac{\text{1+}\frac{{\mathit{\text{W}}}_{\mathit{\text{S,}}\text{δ}}\mathit{\text{e}}{\text{}}^{\mathit{\text{-}}\left(\text{1+3}\mathit{\text{a}}\right){\text{y}}_{\text{2}}}}{{\mathit{\text{W}}}_{\mathit{\text{H}}}^{\mathit{\text{0}}}{\text{/δ}}_{\mathit{\text{0}}}}}{\left(\text{1+3}\mathit{\text{a}}\right){\mathit{\text{y}}}_{\text{2}}}$$

   K(y ₁, y ₂) caractérise l'aptitude, compte tenu du bilan B, à canaliser les dépôts sur le lit de débris ; K est une forme explicite des paramètres.The evolution equations are then presented in the very particular form: $${\dot{\mathit{\text{y}}}}_{\text{1}}\text{=}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

$${\dot{\mathit{\text{y}}}}_{\text{2}}\text{=}\mathit{\text{-K}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{)}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

$${\dot{\mathit{\text{y}}}}_{\text{3}}\text{= - (1+}\mathit{\text{afe}}{}^{{\mathit{\text{-ay}}}_{\mathit{\text{2}}}}\mathit{\text{K}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{)))}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

or : $${\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\frac{\text{-}\mathit{\text{B}}\left({\mathit{\text{y}}}_{\text{1}}{\mathit{\text{, <figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}\right)}{\text{1+}\left(\mathit{\text{a + <figure-callout id="0" label="b h" filenames="imgf0001.png" state="{{state}}">b </figure-callout>}},\right)\frac{{\mathit{\text{h}}}_{}}{}\frac{{}_{\text{0}}}{{\text{δ}}_{\text{0}}}\mathit{\text{e}}{}^{\mathit{\text{-}}\left(\mathit{\text{a + b}}\right){\mathit{\text{y}}}_{\mathit{\text{2}}}}\mathit{\text{K}}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\right)}$$

with: $$\mathit{\text{K}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{) =}\frac{\text{1+}\frac{{\mathit{\text{W}}}_{\mathit{\text{S,}}\text{δ}}\mathit{\text{e}}{}^{\mathit{\text{-}}\left(\text{1 + 3}\mathit{\text{at}}\right){\text{y}}_{\text{2}}}}{{\mathit{\text{<figure-callout id="0" label="W H" filenames="imgf0001.png" state="{{state}}">W </figure-callout>}}}_{\mathit{\text{H}}}^{}{\text{/ <figure-callout id="0" label="δ" filenames="imgf0001.png" state="{{state}}">δ</figure-callout>}}_{\mathit{\text{0}}}}}{\left(\text{1 + 3}\mathit{\text{at}}\right){\mathit{\text{y}}}_{\text{2}}}$$

K ( y ₁ , y ₂ ) characterizes the ability, taking into account balance B, to channel deposits on the debris bed; K is an explicit form of the parameters.

By way of illustration, here is a coherent example of values that have enabled the resolution of the case shown in Figure 2.

The simulations consisted in varying the entry δ ₀ , initial notch depth in the absence of a debris bed (representative of the weight on the tool under ideal clearance conditions). The result of the calculation is δ *, notch at equilibrium - once the transient has passed - and which conditions the speed of penetration stabilized. The effectiveness of penetration can become zero, past a certain weight threshold, depending on the parameters of the problem (and this corresponds to the stuffing threshold). The degree of drilling efficiency is judged by comparing the "solid" and "hydraulic" lift effects.

The form, here very particular, of the evolution equations leads to a monotonic convergence of δ towards its equilibrium value δ * whereas, intuitively, one expects fluctuations (see comments in appendix 4).

• Paramètres de commande :
  • δ₀ pénétration initiale dans la roche vierge (lien avec le poids sur outil WOB) (varié dans la plage 0 à 1,26 mm)
  • N vitesse de rotation, supposée invariable (N = 0,7 tour/s)
  • ν _n vitesse du jet fluide, en sortie de duse (lien avec le débit de boue Q (ν _n = 50ms ^-1 ))
• Paramètres liés à l'outil
  • D _B diamètre de l'outil. (D _B = 0,2 m)
  • d diamètre de duse. (d = 0,01 m)
  • N_c nombre de taillants ; autant de "sites" producteurs de copeaux, autant de supports pour la reprise de l'effort vertical. (N_c = 81)
• Paramètre lié au taillant :
  • H hauteur efficace de taillant. (H = 2,65 mm)
  • Le paramètre conditionne la répartition initiale H = δ₀ + h_0.
• Paramètres liés à l'interface taillant/roche
  • Aγ aire caractéristique pour la représentation de l'effort vertical, (fonction de l'usure γ). (Aγ= 1 mm²)
  • A _c,δ terme proportionnel à la pénétration δ représentatif de l'effort de coupe. (A _c,δ = 5 mm, soit 5 mm² de variation d'aire par mm de pénétration) α et µ⁺ angle caractéristique de coupe ; coefficient de frottement,
  • le choix : sin α + µ⁺cosα = 1 a été fait
  • S_c résistance de "coupe" (cisaillement). (S_c = 500 MPa)
  • S_p résistance en "butée" (compression). (S_p = 500MPa)
• Paramètres liés au copeau de roche
     ρ_c masse volumique du copeau. (ρ_c = 2500 kg.m^-3)
• Paramètres liés à la découpe :
  • µ élancement moyen de copeau illustrant le degré de fragilité de la coupe
  • µ élevé, rupture fragile ; µ bas, rupture ductile (µ = 2)
  • σ resserrement granulométrique de la distribution (écart-type) (σ = 0,5)
• Paramètres liés à l'expulsion :
  • µ coefficient intervenant dans la définition du diamètre hydraulique
  • ν 0<ν<1 index de sensibilité au bourrage
  • (ν = 1 aucune sensibilité)
• Paramètre lié à l'interface boue/roche saine
     λP effet de maintien du copeau (λP = 1MPa)
• Paramètres liés à la boue
  • ρ_m masse volumique de la boue (ρ_m = 1250 kg.m^-3)
  • ν _0, viscosité cinématique de la boue ; à distinguer de la "viscosité équivalente" caractérisant la suspension notamment pour l'effet de portance hydraulique
  • ν _0, = 10^-3m²s^-1 (viscosité dynamique η=1,25 Pa.s)
• Paramètres constitutifs liant entre eux certains paramètres d'évolution au niveau des lois d'interface
  • a pour le lien entre viscosité équivalente et épaisseur de lame fluide (a = 1)
  • b pour le lien entre viscosité équivalente et concentration de la suspension (b = 1)

The following list therefore concerns the model entries required to identify the case. To facilitate reading, a classification of these entries has been made.

• Command parameters:
  • δ ₀ initial penetration in virgin rock (link with weight on WOB tool) (varied in the range 0 to 1.26 mm)
  • N speed of rotation, assumed to be invariable (N = 0.7 rpm)
  • ν _n speed of the fluid jet, at the outlet of the nozzle (link with the mud flow rate Q (ν _n = 50ms ^-1 ))
• Tool related settings
  • D _B tool diameter. ( D _B = 0.2 m)
  • d nozzle diameter. (d = 0.01 m)
  • N _c number of cutters; as many "sites" producing chips, as many supports for the resumption of the vertical effort. (N _c = 81)
• Parameter linked to the cutting edge:
  • H effective cutting height. (H = 2.65 mm)
  • The parameter conditions the initial distribution H = δ ₀ + h _0.
• Parameters related to the cutting / rock interface
  • Aγ characteristic area for the representation of the vertical force, (function of wear γ). (Aγ = 1 mm ² )
  • A _{c, δ} term proportional to penetration δ representative of the cutting force. ( A _{c, δ} = 5 mm, i.e. 5 mm ² of area variation per mm of penetration) α and µ ⁺ characteristic cutting angle; coefficient of friction,
  • the choice: sin α + µ ⁺ cosα = 1 has been made
  • S _c "cut" resistance (shear). (S _c = 500 MPa)
  • S _p resistance at "stop" (compression). (S _p = 500MPa)
• Rock Chip Parameters
  ρ _c density of the chip. (ρ _c = 2500 kg.m ^-3 )
• Parameters related to cutting:
  • µ average slenderness of the chip illustrating the degree of fragility of the cut
  • µ high, brittle rupture; µ low, ductile rupture (µ = 2)
  • σ particle size tightening of the distribution (standard deviation) (σ = 0.5)
• Parameters related to eviction:
  • µ coefficient involved in the definition of the hydraulic diameter
  • ν 0 <ν <1 jam sensitivity index
  • (ν = 1 no sensitivity)
• Parameter linked to the mud / healthy rock interface
  λP chip holding effect (λP = 1MPa)
• Parameters related to mud
  • ρ _m density of the mud (ρ _m = 1250 kg.m ^-3)
  • ν _0, kinematic viscosity of the mud; to be distinguished from the "equivalent viscosity" characterizing the suspension, in particular for the effect of hydraulic lift
  • ν _0, = 10 ^-3 m ² s ^-1 (dynamic viscosity η = 1.25 Pa.s)
• Constitutive parameters linking together certain evolution parameters at the level of interface laws
  • a for the link between equivalent viscosity and fluid blade thickness (a = 1)
  • b for the link between equivalent viscosity and concentration of the suspension (b = 1)

• en abscisse : la pénétration initiale,
• en ordonnée : la pénétration à l'équilibre.

The curve represented in FIG. 2 is therefore the translation of the behavior of the drilling tool in terms of efficiency for this particular choice of the 23 parameters. The curve shown in Figure 2 is the response of the drilling tool, to equilibrium, to the command: weight on tool. More precisely, in terms of evolution model, are brought:

• on the abscissa: the initial penetration,
• on the ordinate: penetration at equilibrium.

Note a separation into four characteristic regimes.

Regime 1 (R1): below a certain weight threshold, corresponding to an initial penetration threshold, the state slowly evolves towards complete clogging by the production of fine debris; the expulsion capacity is saturated by a surplus broken rock production regime.

Regime 2 (R2): here, on the contrary, it is the possibilities of evacuation of cuttings by hydraulics which are dominant, so that, under these conditions, only intervene to limit the performances in terms of penetration speeds the characteristics usual techniques linking weight on tool (WOB) and speed of penetration (ROP). The representative cases of regime 2 are obviously characterized by δ ₀ = δ *, since the debris bed cannot be reconstituted in a sustainable manner.

Regime 3 (R3): this is again (as in operating regimes 1 and 4) an operating case for which the evacuation capacity is, at all times evolution, lower than the production of broken rock. However, by moving from the initial state, the system reaches a configuration where the mass balance is balanced.

The release conditions gradually become more and more unfavorable vis-à-vis the rock production conditions, with the increase in weight on the tool (equivalent to the increase in δ0). The resumption of this weight is done more and more in the form of hydraulic lift W _H due to gradually more difficult conditions of expulsion of the drilling fluid enriched with particles (increasing pressure losses) to the detriment of the solid vertical force W _S assigned to the effective work of disintegration of virgin rock.

Regime 4 (R4): below a certain weight threshold, the functioning of the system involves a rapid evolution towards clogging by producing initially coarse debris, then gradually becoming finer.

As an example, and to complete the illustration of the case shown in Figure 2, the vertical force corresponding to a penetration of δ ₀ = 0.63 mm from each of the cutters (point B), taking into account the characteristics of the rock is 165 kN; for a penetration δ ₀ = 0.69 mm (point D), the weight on associated tool is 190 kN, the hydraulic contribution WH at equilibrium begins to become significant, of the order of 5 kN.

The tamping threshold δ ₀ ^THRESHOLD = 1.02 mm (in the case treated) (point C) corresponds to the condition of application of the weight on tool WOB = 245 kN, which leads irreparably in a few seconds to a complete clogging of the space between the tool body and the formation: the mass production / expulsion rock balance has become so unfavorable that no possibility of "dynamic equilibrium" (with δ *, non-zero penetration) exists.

It is clear that the determination of the value of δ ₀ at point (D) of Figure 2 gives the optimal operating point for the given parametric conditions. Indeed, the top of the bell curve represents the greatest forward speed, therefore the best performance of the drilling tool.

ANNEX 1

EXPRESSION OF HYDRODYNAMIC EFFORTS EXERCISING ON THE CHIP AND TIME CONSTANTS ASSOCIATED WITH THE CORRESPONDING MECHANISMS

Lifting effort: evaluation of the characteristic threshold size D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

   ρ_m est la masse volumique de boue, ν₀ sa viscosité et a _L une constante de proportionnalité.The basic formula due to Eronini expresses in a semi-empirical way the effect of lift exerted on a particle of hydraulic diameter D _c in the vicinity of a tool of diameter D _B when the fluid speed is v _n at the outlet of duse of diameter d.

ρ _m is the density of mud, ν ₀ has its viscosity and _L a constant of proportionality.

   de sorte que :

The chip of size D _c has a dead weight: $${\text{<figure-callout id="0" label="ω" filenames="imgf0001.png" state="{{state}}">ω</figure-callout>}}_{\text{0}}\text{=}{\mathit{\text{m}}}^{\text{2}}{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\text{2}}{\text{ρ}}_{\text{vs}}\mathit{\text{g}}$$

so that :

Rather than the variable D _c, we may prefer to introduce the number n of chips released by each of the N _c blades during a revolution:

The previous expression is then changed to:

de copeaux en nombre n _o , au seuil de décollement, c'est à dire réalisant : $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{=1+}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{c}}}\mathit{\text{g}}}\text{≈}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{c}}}\text{g}}$$

We are now interested in threshold quantities, characteristic size D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

chip number _No, the release threshold, i.e. realizing: $$\frac{{\mathit{\text{F}}}_{\mathit{\text{L}}}}{{\text{ω}}_{\mathit{\text{o}}}}\text{= 1 +}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{vs}}}\mathit{\text{g}}}\text{≈}\frac{\text{λ}\mathit{\text{P}}}{{\text{δρ}}_{\mathit{\text{vs}}}\text{g}}$$

We have :

We show, after some calculations, that:

au dessous de laquelle il y a libération du copeau est importante et plus le débit solide potentiel est élevé.Plus size threshold D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

below which there is release of the chip is important and the higher the potential solid flow rate.

Drag effort: estimation of the characteristic expulsion time and evacuation term

The drag effect F _D is evaluated by Eronini according to a formula analogous to that describing the lift effect.

est une estimation de la vitesse moyenne de circulation du fluide au front donnée par Eronini.The associated characteristic time is such that: $$\frac{\text{1}}{{\text{τ}}_{\mathit{\text{D}}}}\text{=}\frac{\frac{{\mathit{\text{F}}}_{\mathit{\text{D}}}}{{\text{ω}}_{\mathit{\text{o}}}}}{\frac{{\mathit{\text{v}}}_{\mathit{\text{vs}}}}{\mathit{\text{g}}}}{\mathit{\text{where v}}}_{\mathit{\text{vs}}}\mathit{\text{=}}\left(\frac{{\mathit{\text{at}}}_{\mathit{\text{vs}}}\mathit{\text{d}}}{{\mathit{\text{D}}}_{\mathit{\text{B}}}}\right){\mathit{\text{v}}}_{\mathit{\text{not}}}$$

is an estimate of the average speed of circulation of the fluid at the front given by Eronini.

So :

This characteristic time is independent of the particle size D _c .

The solid balance sheet is finally written, explaining the definition of the term expulsion: $${\dot{\mathit{\text{V}}}}_{\mathit{\text{R}}}\text{= χ}{\mathit{\text{NOT}}}_{\mathit{\text{vs}}}\frac{\frac{{\text{ω}}_{\mathit{\text{o}}}^{\mathit{\text{o}}}}{{\text{ρ}}_{\mathit{\text{vs}}}\mathit{\text{g}}}}{{\text{τ}}_{\mathit{\text{D}}}}$$

APPENDIX 2 CALCULATION OF THE MINORATION FACTOR χ Preliminary remark

Let F _{µ, σ be} the function of the normal law of mean µ and standard deviation σ; it is incorrect to write: $$\text{χ =}{\mathit{\text{F}}}_{\mathit{\text{µ, σ}}}\left(\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\mathit{\text{o}}}}{\text{δ}}\right)$$

δ :  épaisseur de coupe actuelle
D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{c}}}$

:  taille seuil pour l'hydraulique (indépendante de δ). :The writing would suppose a distribution in relative number of chips while we are looking for a weighting in volume :
µ: average particle size imposed by the type of cut µ = $\frac{{\mathit{\text{E (D}}}_{\mathit{\text{vs}}}\mathit{\text{)}}}{\text{δ}}$

δ: current cutting thickness
D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

: threshold size for hydraulics (independent of δ). :

Calculation of χ

, copeaux dont le volume élémentaire est alors $$\frac{{\text{ω}}_{\mathit{\text{o}}}^{\mathit{\text{o}}}}{{\text{ρ}}_{\mathit{\text{c}}}\mathit{\text{g}}}$$

It is necessary to relate the volume corresponding to chips of size distributed D _c to the volume of released material if the particle size was comparable to a distribution of Dirac on D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$

, shavings whose elementary volume is then $$\frac{{\text{ω}}_{\mathit{\text{o}}}^{\mathit{\text{o}}}}{{\text{ρ}}_{\mathit{\text{vs}}}\mathit{\text{g}}}$$

   de sorte que, pour une granulométrie étalée de moyenne µ et d'écart type σ :

   avec: $$\mathit{\text{t=}}\frac{{\mathit{\text{D}}}_{\mathit{\text{c}}}}{{\mathit{\text{D}}}_{\mathit{\text{c}}}^{\mathit{\text{o}}}}\frac{{\mathit{\text{D}}}_{\mathit{\text{c}}}^{\mathit{\text{o}}}}{\text{δ}}\mathit{\text{=}}\frac{{\mathit{\text{D}}}_{\mathit{\text{c}}}}{\text{δ}}$$

The elementary volume of the hydrodynamic size chip D _c is:

so that, for a spread size of mean µ and standard deviation σ:

with: $$\mathit{\text{t =}}\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}}{{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\mathit{\text{o}}}}\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\mathit{\text{o}}}}{\text{δ}}\mathit{\text{=}}\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}}{\text{δ}}$$

(It would have been more attractive to adopt a normal log law so as not to have to consider the negative t).

From where :

   avec:

Φ(x) fonction de répartition de la loi normale centrée réduite.Integration by parts gives, after some calculations:

with:

Φ (x) distribution function of the reduced centered normal law.

dépend uniquement du rapport de la moyenne sur l'écart-type de la distribution des tailles

• µ est lié à la ductilité de la roche
• σ à est lié au resserrement granulométrique.

depends only on the ratio of the mean to the standard deviation of the size distribution

• µ is related to the ductility of the rock
• σ to is linked to the particle size tightening.

(µ is a dimensionless variable reflecting in a way the average brittleness of the rock in the form of a "slenderness" of the fragment; one can also establish a link between this characteristic and the angle of friction of a material). $$\text{µ =}\frac{\mathit{\text{E}}\left({\mathit{\text{D}}}_{\mathit{\text{vs}}}\right)}{\text{δ}}$$

est le seuil, tenant compte essentiellement des conditions de pression, au-dessus duquel les particules n'ont pas à être comptabilisées dans le bilan d'évacuation. $$\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\mathit{\text{o}}}}{\text{δ}}$$

is the threshold, essentially taking into account the pressure conditions, above which the particles need not be counted in the evacuation balance.

Remarks

• If the conditions (viscosity ν ₀ and density ρ _m of sludge, differential pressure P) do not change, the threshold D $\genfrac{}{}{0ex}{}{\mathit{\text{o}}}{\mathit{\text{vs}}}$
  
  is invariant.
• If, moreover, the rock does not change its characteristics µ and σ the curve χ (δ) is invariant.

APPENDIX 3 MODULATING FUNCTION ACCORDING TO THICKNESS OF DEBRIS BED

• le mécanisme A est caractérisé par une diminution de l'effet de rétention du copeau en présence d'un lit de débris qui consacre notamment l'augmentation des performances à l'équilibre en termes de vitesse de pénétration ROP lorsqu'on augmente le poids sur l'outil WOB (δ < h).
• le mécanisme B correspond à une expulsion du copeau rendue difficile en raison de l'étroitesse de la lame fluide (h) vis-à-vis de la taille du copeau (δ) (δ > h).
• la transition (δ ≈ h) correspond à une compétition des mécanismes A et B.

It appeared necessary to modulate the term of expulsion according to the variable thickness of the debris bed to take into account the following mechanisms called A and B; mechanism A being dominant when the fluid blade h is large, mechanism B prevailing when this blade becomes narrow.

• Mechanism A is characterized by a reduction in the chip retention effect in the presence of a bed of debris which devotes in particular the increase in performance at equilibrium in terms of penetration speed ROP when the weight is increased over the WOB tool (δ <h).
• mechanism B corresponds to an expulsion of the chip made difficult due to the narrowness of the fluid blade (h) with respect to the size of the chip (δ) (δ> h).
• the transition (δ ≈ h) corresponds to a competition of mechanisms A and B.

   où v est un index dit de sensibilité au bourrage intervenant comme paramètre dans la fonction f _ν retenue :

$${\mathit{\text{f}}}_{\mathit{\text{v}}}^{\text{*}}\left(\mathit{\text{z}}\right)\text{=}{\mathit{\text{f}}}_{\mathit{\text{v}}}\left({\mathit{\text{y}}}_{\text{3}}\right)\text{avec}\frac{\text{1-}\mathit{\text{v}}}{\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\text{-1}}{\mathit{\text{y}}}_{\text{3}}\text{=}\mathit{\text{z}}$$

   où h₀ et δ₀ sont les paramètres associés aux valeurs initiales des variables h et δ.Formally, the evacuation report is written: $${\text{B}}^{\text{-}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{<figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}\text{=}{\mathit{\text{m}}}_{\mathit{\text{0}}}{\mathit{\text{f}}}_{\text{ν}}\text{((}{\mathit{\text{y}}}_{\text{3}}{\text{) χ}}_{\text{µ / σ}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{)}$$

where v is an index called sensitivity to stuffing intervening as a parameter in the function f _ν retained:

$${\mathit{\text{f}}}_{\mathit{\text{v}}}^{\text{*}}\left(\mathit{\text{z}}\right)\text{=}{\mathit{\text{<figure-callout id="3" label="f v y" filenames="imgf0001.png" state="{{state}}">f </figure-callout>}}}_{\mathit{\text{v}}}\left({\mathit{\text{y}}}_{}\right)\left({}_{\text{3}}\right)\text{with}\frac{\text{1-}\mathit{\text{v}}}{\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\text{-1}}{\mathit{\text{<figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}\text{=}\mathit{\text{z}}$$

where h ₀ and δ ₀ are the parameters associated with the initial values of the variables h and δ.

APPENDIX 4 LINEARIZATION IN THE VICINITY OF BALANCE REMINDER AND RATINGS

A system whose state variables are defined by the vector x and the main parameters grouped in the vector λ is described by evolution equations of the form: $$\dot{\mathit{\text{X}}}\text{=}\mathit{\text{F (X, λ)}}$$

It is often examined in the vicinity of an equilibrium position X ₀ , therefore satisfying: $${\mathit{\text{F (<figure-callout id="0" label="X" filenames="imgf0001.png" state="{{state}}">X</figure-callout>}}}_{\text{0,}}\text{λ) = 0}$$

   et le système différentiel initial se présente alors sous la forme dite linéarisée [si l'on fait abstraction du terme résiduel f (ξ,λ)]. $$\dot{\text{ζ}}\text{= A(λ)ζ+}\mathit{\text{f}}\text{(ζ,λ)}$$

   où

   et f(ξ,λ)=0(|ξ| ² ) contient des termes de degré supérieur ou égal à 2.We introduce the local coordinates ξ around X ₀ so that: $${\mathit{\text{X = <figure-callout id="0" label="X" filenames="imgf0001.png" state="{{state}}">X</figure-callout>}}}_{\text{0}}\text{+}\mathit{\text{ξ}}$$

and the initial differential system is then presented in the so-called linearized form [if we ignore the residual term f (ξ, λ)]. $$\dot{\text{ζ}}\text{= A (λ) ζ +}\mathit{\text{f}}\text{(ζ, λ)}$$

or

and f (ξ, λ) = 0 (| ξ | ² ) contains terms of degree greater than or equal to 2.

• (i) X₀ est stable si les valeurs propres de A(λ) ont des parties réelles négatives.
• (ii) X₀ est instable si au moins une valeur propre a une partie réelle positive.
• (iii) le cas critique est celui où la partie réelle de l'une ou plusieurs valeurs propres s'annule alors que les autres valeurs propres gardent leur partie réelle négative.

It follows from Liapunov's stability theorems that:

• (i) X ₀ is stable if the eigenvalues of A (λ) have negative real parts.
• (ii) X ₀ is unstable if at least one eigenvalue has a positive real part.
• (iii) the critical case is that where the real part of one or more eigenvalues is canceled while the other eigenvalues keep their real part negative.

Application: Jacobian writing

$${\dot{\mathit{\text{y}}}}_{\text{2}}\text{=}\mathit{\text{K}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{)}{\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

avec: $${\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\frac{\text{-B}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\right)}{\text{1+}\left(\mathit{\text{a+b}}\right)\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\mathit{\text{K}}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\right)\mathit{\text{e}}{\text{}}^{\text{-}\left(\text{a+b}\right){\text{y}}_{\text{2}}}}$$

A l'équilibre : $$\text{B(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{=⇔}{\mathit{\text{F}}}_{\text{1}}\text{(}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\mathit{\text{0}}$$

Le jacobien s'écrit :

The differential system comes in quite a special form: $${\dot{\mathit{\text{y}}}}_{\text{1}}\text{=}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

$${\dot{\mathit{\text{y}}}}_{\text{2}}\text{=}\mathit{\text{K}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{)}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{)}$$

with: $${\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\frac{\text{-B}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{<figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}\right)}{\text{1+}\left(\mathit{\text{a + b}}\right)\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\mathit{\text{K}}\left({\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\right)\mathit{\text{e}}{}^{\text{-}\left(\text{a + b}\right){\text{y}}_{\text{2}}}}$$

Equilibrium : $$\text{B (}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{<figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}\text{= ⇔}{\mathit{\text{F}}}_{\text{1}}\text{((}{\mathit{\text{y}}}_{\text{1}}\text{,}{\mathit{\text{y}}}_{\text{2}}\text{,}{\mathit{\text{y}}}_{\text{3}}\text{) =}\mathit{\text{0}}$$

The Jacobian is written:

$${\mathit{\text{a}}}_{\text{21}}\text{= -}{\mathit{\text{Ka}}}_{\text{11}}$$

$${\mathit{\text{a}}}_{\text{22}}\text{=}\mathit{\text{o}}$$

$${\mathit{\text{a}}}_{\text{23}}\text{= -}{\mathit{\text{Ka}}}_{\text{13}}$$

Taking into account the equilibrium relationships, the Jacobian matrix is presented in the following simplified form:

$${\mathit{\text{at}}}_{\text{21}}\text{= -}{\mathit{\text{Ka}}}_{\text{11}}$$

$${\mathit{\text{at}}}_{\text{22}}\text{=}\mathit{\text{o}}$$

$${\mathit{\text{at}}}_{\text{23}}\text{= -}{\mathit{\text{Ka}}}_{\text{13}}$$

$$\text{soit L* = 1+}\mathit{\text{a}}\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\mathit{\text{K}}\text{*}{\mathit{\text{e}}}^{\mathit{\text{-ay}}}{\text{}}^{{\text{}}_{\text{2}}}$$

   où les symboles * rappellent que le calcul doit être effectué pour un triplet (y ₁ , y ₂ , y ₃ ) satisfaisant la condition de l'équilibre.This results in the vicinity of equilibrium by the entries:

$$\text{either L * = 1+}\mathit{\text{at}}\frac{{\mathit{\text{h}}}_{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}\mathit{\text{K}}\text{*}{\mathit{\text{e}}}^{\mathit{\text{-ay}}}{}^{{}_{\text{2}}}$$

where the symbols * remind that the calculation must be carried out for a triplet ( y ₁ , y ₂ , y ₃ ) satisfying the condition of equilibrium.

   décrit le comportement linéarisé au voisinage de l'équilibre, l'allure est de type exponentiel (convergent) si :

So :

describes the linearized behavior near equilibrium, the pace is of the exponential (convergent) type if:

   définit, conjointement à B(y ₁,y ₃) = 0, la condition de bifurcation caractérisant le bourrage.The condition : $$\frac{\text{∂}\mathit{\text{B}}}{\text{∂}{\mathit{\text{y}}}_{\text{1}}}\text{-}\mathit{\text{L}}\text{*}\frac{\text{∂}\mathit{\text{B}}}{\text{∂}{\mathit{\text{<figure-callout id="3" label="y" filenames="imgf0001.png" state="{{state}}">y</figure-callout>}}}_{\text{3}}}\text{=}\mathit{\text{O}}$$

defines, jointly with B ( y ₁ , y ₃ ) = 0 , the bifurcation condition characterizing the stuffing.

The complete and explicit resolution of the bifurcation relationships can be achieved if one takes advantage of the fact that, when the transition to stuffing occurs, K (y ₁ , y ₂ ) is close to 1.

We then show that, in variables x and z (see appendices 2 and 3), the bifurcation condition is presented in the explicit form z function of x:

Equilibrium condition : (B = 0)

avec $$\mathit{\text{k}}\text{=}\frac{\text{1}}{\text{µ}}\frac{{\mathit{\text{D}}}_{\mathit{\text{c}}}^{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}$$

Bifurcation condition (dB = 0)

with $$\mathit{\text{k}}\text{=}\frac{\text{1}}{\text{µ}}\frac{{\mathit{\text{D}}}_{\mathit{\text{vs}}}^{\mathit{\text{o}}}}{{\text{δ}}_{\mathit{\text{o}}}}$$

Note that the bifurcation condition is independent of N.

NOTE

= y ₂, H=-Log $\frac{\mathit{\text{h}}}{{\mathit{\text{h}}}_{\mathit{\text{o}}}}$

, F = -Log(1 - x) implique un comportement linéarisé à une dimension, on aurait : $$\dot{\text{ζ}}{\text{= Aζ\hspace{1em}\hspace{1em}\hspace{1em}ζ = (ζ}}_{\text{1}}{\text{,ζ}}_{\text{2}}{\text{,ζ}}_{\text{3}}\text{)}$$

$$\mathit{\text{χ(s) = déterminant(sI - A)}}$$

   avec 3 valeurs propres complexes ou réelles et une convergence vers l'équilibre d'un type différent de celui supposé ici.In a less particular case than this one, where, in particular, the supposed proportionality of the magnitudes E = Log $\genfrac{}{}{0ex}{}{\text{η}}{{\text{η}}_{\mathit{\text{o}}}}$

= y ₂ , H = -Log $\frac{\mathit{\text{h}}}{{\mathit{\text{h}}}_{\mathit{\text{o}}}}$

, F = -Log (1 - x) implies a one-dimensional linearized behavior, we would have: $$\dot{\text{ζ}}{\text{= Aζ ζ = (ζ}}_{\text{1}}{\text{, ζ}}_{\text{2}}{\text{, ζ}}_{\text{3}}\text{)}$$

$$\mathit{\text{χ (s) = determinant (sI - A)}}$$

with 3 complex or real eigenvalues and a convergence towards equilibrium of a different type from that assumed here.

Claims (5)

Method for improving drilling performance in which a drilling model is implemented, characterized in that said model takes into account the effects of destruction (2) of a rock by at least one cutter (1) fixed on a tool body (3) driven in rotation and the effects of evacuation of rock debris by a fluid, by calculating a material balance from: - the production of rock debris by the cutter having penetrated into the rock with a depth δ, - a bed of debris covering said rock at a thickness l, a fluid blade of thickness h between said debris bed and said body, said fluid blade having a debris concentration c, - command parameters, - environmental parameters, and in that the drilling conditions are determined as a function of the response of said model for determined values of said parameters. Method according to claim 1, characterized in that at least one of said parameters: weight on the tool, speed of rotation of the tool and fluid flow rate, is a control parameter. Method according to one of the preceding claims, characterized in that, in said model, the lift W of the tool is broken down into a solid component W _S and a hydraulic component W _h depending in particular on the fluid blade. Method according to one of the preceding claims, characterized in that one considers a particle size spread of the debris distributed according to a normal law as a function of the depth δ of the notch, of average µ related to the ductility of the rock and of a dispersion characterized by the standard deviation σ. Method according to one of the preceding claims, characterized in that said solid matter balance B (t) is such that B (t) = B ⁺ (t) -B ^- (t), in which B ⁺ (t) is a debris production term dependent on δ and corresponding to the rate of destruction of the rock, and B ⁺ (t) is an expulsion term dependent on l and h.

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