11th Congress of the International Society for Rock Mechanics – Ribeiro e Sousa, Olalla & Grossmann (eds)© 2007 Taylor & Francis Group, London, ISBN 978-0-415-45084-3

Effect of rocks anisotropy on deviation tendencies of drilling systems

Effet de l’anisotropie des roches sur les tendances déviationnelles des systèmes de forage pétrolier

R. Boualleg, H. Sellami, A. Rouabhi & S. MenandEcole des Mines de Paris, Centre de Geosciences, Fontainebleau, France

C. SimonDrillScan, Montreuil, France

ABSTRACT: Cases history have demonstrated that inter-bedded formations are a major cause of borehole deviation and tortu-osity. This phenomenon induces higher torque and drag, running tubular problems, stabilizers wear, pipe damage and trajectorycontrolling problems. In some fields, shale formations have a tendency to cause wellbore deviations to undesired directions.The formation anisotropy modifies the rock-bit and the bit-string interactions. Those interactions have to be fully understoodin order to eliminate the undesired well deviations and tortuosities. To do so, an experimental drilling program has been carriedout on a full-scale bench using various commercial drilling bits in different formations including anisotropic shale at several dipangles. Due to the limitation of the existing criterion to describe the failure of anisotropic rocks, a new criterion was developed.It introduces the notion of fictitious isotropic material obtained by using a linear anisotropic operator. Coupling a 3D bit-rockmodel with a 3D bottomhole assembly (BHA) model, enables to predict the occurrence of tortuosity (due to the anisotropy)at small and large scales (micro and macro-tortuosity) and to evaluate feet by feet the response of the commercial directionaldrilling systems (rotary, steerable motors or rotary steerable systems). A post analysis of some real drilling cases showed that,when the anisotropy of the formations is known, it is possible to select the best drilling system minimizing the well tortuosity.

1 INTRODUCTION

The effect of the anisotropy of drilled rock formations on devi-ation tendencies of drilling systems has been observed since along time, both in laboratories and fields (Reich et al., 2003).In fact, field observations showed that laminated formationshave caused well deviations from planed trajectories.This phe-nomenon is considered as problematic by the drillers sinceit requires correctional operations and the use of expensivedirectional drilling systems.

Drillers have observed that drilling through inter-beddedformations causes inevitable and unwanted undulationsaround the planned well trajectory. These deviations, calledtortuosity, have been recognized as one of the critical factorsin extended-reach wells because they induce a high torqueand drag, drillstring buckling, running tubular problems andBHA (Bottom-Hole Assembly) wear. In specific applica-tions, excessive tortuosity in horizontal wells can even impairproductivity. When drilling these anisotropic formations (lam-inated or inter-bedded rock), the first cause for deviation isattributed to the cutter-rock interaction hence to the drillingbit-rock interaction.

The goal of this paper is to describe a theoretical modeldeveloped and calibrated on full-scale drilling benches inorder to describe those interactions and show how they canaffect the directional behaviour of drilling systems. Thismodel uses a new failure criterion that describes the fail-ure of anisotropic rocks, during the rock cutting process, byintroducing the notion of fictitious isotropic material.

2 CUTTER-ROCK INTERACTION MODEL

The model presented here relates to the PDC cutter (Poly-crystalline Diamond Compact) (Figure 1). The objective of

such model is to estimate the forces applied on a single cutterknowing its geometry (back rake angle, side rake angle, diam-eter, chamfer. . .) and the mechanical failure characteristics ofthe rock.

2.1 Isotropic rocks

In the literature, the most models utilise shear rock failure cri-terion such as Coulomb criterion in which the rock is describedby its cohesion C and internal angle of friction φ, derived fromtriaxial compressive tests.

Ecole des Mines de Paris has developed a complete model,detailed in Gerbaud et al. (2006).The model assumes that eachcutter of the drilling bit cuts the rock at a significant depth ofcut and produces chips with a given geometry. It takes intoaccount build-up edge of crushed materials, chamfer and backcutter force, uses limit analysis and Mohr-Coulomb criterionto calculate the specific energy Req defined as the ratio ofhorizontal cutting force over the cutting area (Figure 1). Reqcan be written formally as:

Figure 1. Cutting forces model.

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Figure 2. Cutting configurations in laminated rock.

where ωc is the PDC back rake angle and θf the friction anglebetween the PDC and the rock.

For laminated (orthotropic) rocks, the problem is differentbecause the specific energy depends also on the orientationof the formation dip defined by the unit normal −→e . As wecan see in Figure 2, it is logically easier to cut the rock inconfiguration (1) than configuration (2).

Since the various cutting tests conducted in laboratory haveshown that the utilisation of Mohr-Coulomb criterion cannot capture the cutting force variation in laminated rocks,a new single cutter model has been developed based on anappropriate transverse failure criterion.

2.2 Proposed transverse failure criterion

The general form of a failure surface can be expressed by ascalar-valued function F(σ) = 0, which describes the failureenvelope in six-dimensional stress space. In order to respectthe material frame-indifference, F must necessarily be invari-ant relative to the material symmetry group. The materialstructure can be defined by the symmetric tensor:

It can be shown that by adding the structural tensor e to theoriginal arguments of F , one can construct a function which isinvariant relative to the orthogonal second-order tensors space(Zheng, 1994).Thus, for transversely isotropic material, F canbe expressed as:

where σ = �(σ, e), F and � are invariant relative to the orthog-onal second-order tensors space. Considering this generaliza-tion, one can take all the advantages of the well-establishedtheory of the representation of isotropic functions. σ can beinterpreted as the stress tensor within a fictitious isotropicsolid. The real and the fictitious stresses are related by thetransformation function �. In this work we will use the rela-tionship established by Rouabhi et al. (2007) and which canbe expressed in terms of the first and the second invariant ofσ as: p is the mean or the hydrostatic stress, q is the Von Misesequivalent stress and N and T are respectively the intensityof the normal and the shear stress acting across the cut plannormal to the unit vector −→e .

In the transformation above, the material is characterizedonly by two material constants α andβ. The advantage of usingF(σ) instead of F(σ) is to use failure surfaces developed forisotropic materials. In order to respect the material symmetry,

F must only depend on the principal invariants of σ. In geome-chanics, failure of isotropic material is described, in the mostcase, by the well-known Drucker-Prager and Mohr-coulombsurfaces. In this work, we define the Drucker-Prager failurecriterion for the transversely isotropic material.

In the principal fictitious stress space, the Drucker-Pragersurface is a cone with a circular deviatoric section centered onthe hydrostatic axis:

where γ and λ are two material constants.In order to write the failure surface for the real material,

we have to replace in the equation above p and q by theirexpressions in terms of p, q, T and N, and thus

As a result, we obtain a failure surface with four materialparameters (α, β, γ , λ).

2.3 Application to the cutter-rock interaction model

In order to determine the forces applied on the single cutterwhen cutting a laminated rock (Figure 2), we assume:

i) A fine layer of crushed rock exists between the cutter andthe produced chip.

ii) The stress state in the chip is homogeneous and equilibratesthe cutter action and the drilling mud pressure Pb. It isexpressed as:

where σ0 is the unknown stress applied by the crushed zoneto the chip and tg(θf ) a friction coefficient.

iii) The chip formation is governed by the transverse failurecriterion presented above:

The resolution of the previous equation allows us tocalculate σ0 as a function of (−→e , θf , α, β, γ , λ).

The rock failure parameters are usually evaluated from clas-sical triaxial compressive tests. However, for a better descrip-tion of the rock cutting process, we have designed a specialcircular cutting test, taking into account the 3D variation ofdip orientation during the cutter action. This test consists incutting a circular groove (Figure 3) in an orthotropic rock.

During a test on a rock sample at a given dip angle, the vari-ation of the cutter angular position θ, allows the cutter to loadthe rock in different configurations. Thus, one test is enoughto adjust the parameters (α, β, γ , λ). Once these parametersdetermined they are kept constant for any other dip angles.Figure 3 presents a comparison between the model and theexperimental result for 45◦ dip angle. We can note, in thiscase of orthotropic rock, that the specific energy is very sen-sitive to the angular position and the model can describe wellthis sensitivity.

3 DRILLING BIT-ROCK INTERACTION MODEL

The aim of developing a drilling bit-rock interaction model isto describe a relation between the forces applied on a PDC bit

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Figure 3. Model-experiment comparison at 45◦ dip angle inTourne-mire shale.

Figure 4. Side force during crossing rocks interface at 45◦.

(Figure 4) and its resulting displacements. Based on a kine-matical approach, the model uses the cutter-rock model todetermine the force on each PDC and adds all those forces toobtain the global forces on the bit.

For inter-bedded rocks, during the revolution of the PDCbit, the cutters may be either on the hard rock or on the soft one.The model checks the radial position of each cutter relative tothe interface and uses the adequate parameters of the rock.For laminated rock, the model computes the orientation of theformations dip relative to the cutting direction of each cutterbefore force calculation.

4 DIRECTIONAL LABORATORY DRILLING TESTS

The PDC bit-rock model has been validated by various testson the full scale directional drilling bench of Ecole des Minesde Paris. The drilling bench allows testing drill bit undersimulated downhole conditions. Two sets of strain gaugesare mounted on the drilling shaft to measure the bendingmoments. The total resulting lateral force value and directionare computed.

4.1 Inter-bedded rocks drilling tests

These tests are carried out in order to study the effects of theinterface angle, and the bit profile when crossing a hard/softor soft/hard rocks interfaces. The test procedure consists in

Figure 5. WOB and side force in 30◦ dip angle shale sample.

drilling a sequence of 2 different rocks with an inclinedinterface, recording the forces on the bit and measuring thedeviations of the borehole. In this campaign different rocksand bits have been used.

As long as the bit drills only through the soft rock vertically,it does not generate a side force on the shaft, but when it startsto cut the hard rock, the anisotropy side force is created makingit deviates. Figure 4 shows plots of the shaft side force andanisotropic force versus the bit position.

The shaft side force increases up to a limit Fmax and beginsto fall as the cutting structure drills more in hard rock. At thismoment the shaft side force makes the bit back to the verticalposition. For most cases, the bit deviates initially toward thesoft rock and then moves to the hard rock.

The theoretical and experimental results showed that moredeviations are occurred when the interface between the rocksis close to the vertical. In addition, the results indicate thatthe deviation decreases when the stabilisation gauge of the bitincreases.

4.2 Laminated rocks drilling tests

The tests in laminated rocks consist in drilling a vertical holethrough a cylindrical shale sample with a diameter 20 cm andlength 45 cm.The dip angle is changed from 5◦ to 85◦ . Figure 5presents the variation of the WOB (Weight On Bit) and sideforce measured on the bit when drilling a shale sample witha dip angle of 30◦ . These results prove clearly the existenceof a side force which is mainly due to the rock anisotropicbehaviour. The sample length is probably too short to obtainside force stabilization. From these tests, we noticed that at0◦ and 90◦ dip angle the side force is low. As the dip angleis around 50◦ , the side force is maximal and reaches 16% ofWOB. For most tests, the side force direction varies from −20◦to 45◦ around the down to up direction.

5 TRAJECTORY PREDICTION MODEL

The parameters that affect directional system behaviour canbe classified in two groups related to:

– Bit-rock interaction: bit steerability, walk angle, rockanisotropy (Boualleg, 2006).

– Drill string mechanics: drill collars, stabilizers, etc . . .

A new 3D code, based on direct integration of the stiffstrings behaviour and equilibrium equations without using thefinite element method, has been developed at Ecole des Minesde Paris (Menand et al., 2005). This code, enabling to simulateany BHA or drillstring is coupled to the 3D rock-bit model.

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Figure 6. Effect of WOB on the inclination of drilled hole whencrossing inter-bedded rocks at 75◦ interface angle.

The calculations are made stepwise with an algorithm in orderto predict drilling trajectory feet by feet. This algorithm hasbeen validated on data from actual wells.

5.1 Deviation in inter-bedded rocks

In order to analyse the effect of drilling an inter-bedded rockson the well trajectory, we simulate the case of a BHA#1(Figure 6) coupled with a bit having a given steerability andwalk angle. Initially this system drills a soft rock and enterssuddenly another one 5 times harder, at a 60◦ interface angle.Figure 6 presents the well inclination variation as a functionof the measured depth with various WOB when crossing aninter-bedded rocks at 75◦ . We note that in the soft rock theinclination is constant and a sudden deviation (dogleg) appearswhen crossing the interface. The well inclination increasesof 3.5◦ in less than 0.4 m. Three meters following this firstdeviation, a second one occurs even if the bit drills on homo-geneous rock. Indeed, the sliding of the first stabilizer throughthe first dogleg causes a second local dogleg. These undu-lations are repeated each time the first stabilizer crosses adogleg. The bit behaviour, looking like a harmonic move-ment, can be amplified or attenuated, depending on BHA, bitand operation parameters. Figure 6 shows also that increasingWOB may reduce considerably the amplification of trajec-tory undulations. Other simulations have demonstrated thatthe first stabilizer position has a strong correlation with theoscillations period as observed in the field.

5.2 Deviation in laminated rocks

The difference between inter-bedded and laminated rockseffect on trajectory behaviour is explained by the side forceorigin. For an inter-bedded rock, the side force is createdlocally and for a laminated rock it is continuously createdas the bit remains in the rock. For this reason, drilling lami-nated rock should have a more significant large scale effect.Figure 7 presents the evolution of the hole inclination with thedrilled depth for two different BHA, a dropping BHA#2 anda building BHA#3.

The two BHAs drill in an isotropic formation and thenencounter into a shale formation with dip angle 45◦ , at 62 m.We notice that the slope of the curve changes from 2.7◦/30 m to4.3◦/30 m for BHA #3 and from −2.75◦/30 m to 1.64◦/30 mfor BHA#2. These findings show that rock anisotropy haschanged completely the behaviour of the directional systemsleading to hazardous trajectory control.

Figure 7. Effect of anisotropy on trajectory build/drop rate.

6 CONCLUSION AND FIELD RECOMMENDATIONS

An interaction bit-rock interaction model based on a sin-gle cutter model has been developed to calculate the sideforce on PDC bit drilling anisotropic formations. The mod-els has been validated and calibrated on full-scale drillingbenches. Coupling the 3D bit-rock model with BHA mechan-ical behaviour model allows predicting trajectories throughisotropic, inter-bedded and laminated rock formations.

Simulations have shown that when the nature of the drilledformations is known, it is possible to take some actions tominimise the formation anisotropy consequence:

• WOB: to minimise the trajectory oscillations risk in inter-bedded rocks, we recommend increasing WOB. Howeverfor laminated rocks it is recommended to decrease WOB inorder to reduce well deviations.

• Bit design: as deviations are caused by the side force,decreasing side cutting ability of drilling bit may minimisedeviation in anisotropic formations.

• The risk of amplification of trajectory oscillations is con-trolled mainly by the nearest stabilizer to the bit. Increasingthe distance between the bit and the first stabilizer maydecrease this risk.

• In laminated rocks, stiff BHA with a near bit stabilizer mayresist more to the lateral deviations.

REFERENCES

Boualleg R., 2006. Etude du comportement des systemes de forageen formations geologique anisotropes, PhD thesis of Ecole desMines de Paris.

Gerbaud L., Menand S., Sellami H., 2006. All comes from the cut-ter rock interaction. SPE paper N◦ 98988. IADC/SPE DrillingConference, Miami, Florida.

Menand S., Sellami H., Tijani M., Stab O., Dupuis D., Simon C.,2005. Advancement in 3D drillstring mechanics: from the bit tothe Topdrive. paper SPE 98965. IADC/SPE Drilling Conference,Miami, Florida.

Reich M., Oesterberg M., Montes H., Treviranus J., 2003. Straightdown to success. Paper SPE 84451. SPE annual Technical Confer-ence, Denver, Colorado.

Rouabhi A., Tijani M., Rejeb A., 2007. Triaxial behaviour of trans-versely isotropic materials – Application to sedimentary rocks.Accepted for publication in the International Journal for Numericaland Analytical Methods in Geomechanics.

Simon C., 1996. Modelisation of PDC bit directional behaviour inanisotropic formation. PhD thesis of Ecole des Mines de Paris.

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