Copyright @ 2005 Patron EditoreIn the percussively deep drilling, the rock is modeled by an infinite elastic medium
with a semi-infinite cylindrical bore-hole having a curvilinear bottom. First, the stationary indentation is formulated as a non-classical non-linear free-boundary contact problem with unknown rupturing and non-rupturing parts of the bore-hole boundary. Then the stationary-periodic per-
cussive drilling problem is reduced to the stationary one on the rupture progression stage of the cycle and to the classical contact problem on the reverse and progression-before-rupture stages of the cycle. This provides a nonlinear progression - force diagram for the bit dynamics prediction.This work is supported by the research grant GR/R85556/01 "Nonlinear Dynamics and Rock
Contact Fracture Mechanics in Modelling of Vibration Enhanced Drilling" of the Engineering and Physical Sciences Research Council, UK

Mikhailov, SE

Namestnikova, IV

English

Brunel University Research Archive

Proceedings of the 11th International Conf. on Computer Methods and Advances in Geomechanics(Eds. G.Barla and M.Barla), Bologna: Patron Editore, Vol.1, 2005, 103-109Quasi-static stationary-periodic model of percussive deepdrillingS.E. Mikhailov and I.V.NamestnikovaDept. of Mathematics, Glasgow Caledonian Univ., Cowcaddens Road, Glasgow, G4 0BA, UKKeywords: elasticity, boundary rupture, free-boundary problem, contact problemABSTRACT: In the percussively deep drilling, the rock is modeled by an infinite elastic mediumwith a semi-infinite cylindrical bore-hole having a curvilinear bottom. First, the stationary inden-tation is formulated as a non-classical non-linear free-boundary contact problem with unknownrupturing and non-rupturing parts of the bore-hole boundary. Then the stationary-periodic per-cussive drilling problem is reduced to the stationary one on the rupture progression stage of thecycle and to the classical contact problem on the reverse and progression-before-rupture stagesof the cycle. This provides a nonlinear progression - force diagram for the bit dynamics prediction.1 IntroductionThe progression in the percussive drilling is caused by a material rupture under the action ofa drilling bit applied at the bore-hole boundary points y(t) moving in time t due to rupture.This boundary loading generates stress σij(y, t) and strain εij(y, t) tensors at all material pointsy. Let a material point y has Cartesian coordinates (y1, y2, y3) in the non-deformed state.The radius-vector of the same material point y in a deformed state at a time t is y˜(y, t) =y + u(y, t), where u(y, t) is the displacement vector. We will use all equations in terms thenon-deformed (reference) coordinates y and refer the boundary conditions to the non-deformedboundary surfaces (Lagrange approach).Let us consider stationary-periodic percussive drilling of a half-infinite bore-hole, ΩH(t), spreadingto y3 = ∞ in an infinite elastic space. Let y3-axis of the coordinate system coincide with thebore-hole axes, and the drill bit progressive-periodic motion occurs only in the y3 direction. LetΩ(t) = IR3\ΩH(t) be the domain occupied by the material (i.e. the infinite space with drilledbore-hole) and ∂Ω(t) be the bore-hole surface in the non-deformed state (or, the same, in thereference coordinates y), while Ω˜H(t), Ω˜(t) = IR3\Ω˜H(t) and ∂Ω˜(t) be their counterparts inthe deformed state. If the rupture front ∂FΩ(t) constitutes only a finite part of the boundary∂Ω(t), then the borehole is a semi–infinite (not necessarily circular) cylinder with a curvilinearbottom being the rupture front ∂FΩ(t). Otherwise, the bore-hole has a monotonously wideningshape. If the bit is axially-symmetric then the bore-hole is axially symmetric as well. Let B(t)103S.E.Mikhailov and I.V.Namestnikovabe the domain occupied by the bit at the instant t, and ∂B(t) be its surface.y3y2y1Figure 1: Stationary-periodic percussive drillingTo describe the material strength for a point y, we will use an instant strength condition at apoint y at an instant t written asΛ(σ(y, t)) < 1, y ∈ Ω(t), (1)where the function Λ(σ) is associated with the von Mises, Coulomb–Mohr, Drucker–Prager oranother appropriate strength condition.We suppose that the rupture appears in the form of a rupture front ∂FΩ(t). The rupture frontis a part of the bore-hole boundary ∂Ω(t). The rupture front equation can be taken asΛ(σ(y), t) = 1, y ∈ ∂FΩ(t). (2)To formulate the simplest model, let us make the followingModel assumptions:(i) The deformation gradient is small.(ii) The material is homogeneous, i.e. its elastic moduli Cijkl = const.(iii) The bit is rigid.(iv) The borehole surface is loaded by the bit at the contact surfaces and is free of tractionsat all other points.(v) The bit action can be reduced to a pressure p(x).(vi) The ruptured material, for which strength condition (1) is violated, disappears (is washedaway) thus leaving the fresh rupture front (the bottom of the bore-hole) either free oftractions or in contact with the bit bottom.Under the model assumptions, the bore-hole boundary ∂Ω generally consists of four non-overlappingparts: a free of traction non-rupturing part ∂00Ω, a contact non-rupturing part ∂c0Ω, a free oftraction part of the rupture front ∂0FΩ, and a contact part of the rupture front ∂cFΩ.104S.E.Mikhailov and I.V.Namestnikova2 Stationary indentation modelLet us consider in this section an auxiliary problem of stationary indentation of an infinite elasticspace by a rigid indentor with a constant progression rate h˙3 < 0 in the y3 direction. Thenh˙ = (0, 0, h˙3) is the progression rate vector.In the stationary problem, the displacements, strains, and stresses are the same at the corre-sponding points at the corresponding instants,ui(y, t) = ui(y − th˙, 0), εij(y, t) = εij(y − th˙, 0), σij(y, t) = σij(y − th˙, 0), y ∈ Ω(t). (3)This impliesu˙i(y, t) = −h˙3ui,3(y, t), ε˙ij(y, t) = −h˙3εij,3(y, t), σ˙ij(y, t) = −h˙3σij,3(y, t), (4)Λ(σ(y, t)) = Λ(σ(y − th˙, 0)), y ∈ Ω(t). (5)The dot over u(y, t), ε(y, t) and σ(y, t) means partial derivative with respect to t, which forthe chosen Lagrange approach coincides with the material derivative in time, while the subscriptafter comma means partial derivative with respect to the corresponding coordinate.The boundary ∂Ω moves with the velocity h˙3 in the y3 direction, the corresponding movingboundary points are related asy(t) = y(0) + th˙, y(t) ∈ ∂Ω(t), y(0) ∈ ∂Ω(0).This prescribes the normal velocities of the material points on the non-rupturing part of theboundary,u˙i(y, t)η˜i(y) = h˙3η˜3(y), y ∈ ∂00Ω(t) ∪ ∂c0Ω(t), (6)where η˜j(y) is a unit outward (i.e. directed inward the bore-hole) normal vector to the boundary∂Ω˜(t) in the deformed state.On the other hand, the normal movements of the material points on the rupture front should beslower than the normal movement of the front surface itself,u˙i(y, t)η˜i(y) > h˙3η˜3, y ∈ ∂0FΩ(t) ∪ ∂cFΩ(t), (7)since otherwise there will be creation of material instead of its rupture there, to fill the gap ifthe material movement is faster, or there will be no rupture if the movements coincide.Taking in mind the first relation in (4), the time derivative can be replaced with the spacederivative, reducing (6) and (7) to the form independent of h˙3,ui,3(y, t)η˜i(y) = −η˜3, y ∈ ∂00Ω(t) ∪ ∂c0Ω(t), (8)ui,3(y, t)η˜i(y) > −η˜3, y ∈ ∂0FΩ(t) ∪ ∂cFΩ(t). (9)Relation (8) can be rewritten asy˜i,3(y, t)η˜i(y) = 0, y ∈ ∂00Ω(t) ∪ ∂c0Ω(t). (10)Equation (10) is satisfied if y belongs to a cylindrical surface parallel to the y3 axis, since thevector y˜i,3(y, t) is then tangent to the deformed surface which y˜i belongs to. This implies thatcondition (9) can not be satisfied on any cylindrical part of ∂Ω, that is, there is no rupture on105S.E.Mikhailov and I.V.Namestnikovathe cylindrical part of ∂Ω. On the other hand, this means that we may replace condition (8) bythe condition of the cylindrical non-rupturing surface,η3(y) = 0, y ∈ ∂00Ω(t) ∪ ∂c0Ω(t), (11)where ηj(y) is a unit outward boundary normal vector to the non-deformed boundary ∂Ω(t).On the contact surfaces we have generally the boundary inclusion y˜ = y + u(y) ∈ ∂B, y ∈∂c0Ω ∪ ∂cFΩ. Assuming the displacements u(y) are small, the boundary condition can belinearized asui(y)ηi(y) = d(y), y ∈ ∂c0Ω ∪ ∂cFΩ, (12)where d(y) is a (positive or negative) distance between the point y and the bit boundary ∂Bin the η(y) direction in the non-deformed state, and it is known if the contact surface is knownand is to be determined otherwise.To solve the stationary indentation problem, it is sufficient to consider it only for t = 0. Thus,taking into account relations (3)-(12) and dropping the argument t = 0 for brevity, we arrive atthe following non-classical non-linear free boundary problem,σij,j(y) = 0, Λ(σ(y)) < 1, y ∈ Ω; (13)σij(y)ηj(y)ξi(y) = 0, σij(y)ηj(y)ζi(y) = 0, ui(y)ηi(y) = d(y),η3(y) = 0, σij(y)ηj(y)ηi(y) < 0, Λ(σ(y)) < 1, y ∈ ∂c0Ω; (14)σij(y)ηj(y)ξi(y) = 0, σij(y)ηj(y)ζi(y) = 0, ui(y)ηi(y) = d(y),Λ(σ(y)) = 1, σij(y)ηj(y)ηi(y) < 0, ui,3(y)η˜i(y) > −η˜3(y), y ∈ ∂cFΩ; (15)σij(y)ηj(y) = 0,η3(y) = 0, ui(y)ηi(y) < d(y), Λ(σ(y)) < 1, y ∈ ∂00Ω; (16)σij(y)ηj(y) = 0,Λ(σ(y)) = 1, ui(y)ηi(y) < d(y), ui,3(y)η˜i(y) > −η˜3(y), y ∈ ∂0FΩ; (17)ui(y)→ 0, y →∞. (18)Hereσij(y) = σ0ij + Cijklεkl(y), εkl(y) = (uk,l + ul,k)/2, (19)and the constant stiffness tensor Cijkl and the constant stress σ0ij in the rock without the drillhole are know; ξj(y), ζj(y) are unit vectors orthogonal to to the normal vector ηj(y) and toeach other. Condition (18) is understood on almost any straight ray originating from y = 0,thus permitting a non-zero limit of the displacements as y →∞ parallel to the bore-hole.All the four boundary parts ∂00Ω, ∂0FΩ, ∂c0Ω, ∂cFΩ, and consequently d(y), are generallyunknown in this setting and the corresponding ”excessive” boundary equalities and inequalitiesare provided in (14) and (17) to allow their determination.Different strategies can be chosen to solve this problem, see e.g. Eliot & Ockendon (1982),Crank (1984, Section 8), Johnson (1985). One of the possibilities is the iteration algorithmdescribed below. It consists of iterations, each solving a linear mixed boundary value problem(13)-(18), (19) with some fixed boundaries, ∂00Ω, ∂0FΩ, ∂c0Ω, ∂cFΩ, and consequently d(y).106S.E.Mikhailov and I.V.NamestnikovaThen conditions (13)-(17) are checked and the boundaries are changed to alleviate the violationof inequalities, and the next iteration starts.On the first iteration one can reasonably assume that the rupture front coincides with the contactpart of the bit, ∂cB, which in turn coincides with the bit bottom, ∂bB, (consisting of the bitsurface points with algebraically smallest y3 coordinate, over the points with the same (y1, y2)coordinates), i.e. ∂cFΩ = ∂cB = ∂bB, ∂0FΩ = ∅, and there is no contact without rupture, i.e.∂c0Ω = ∅. Those assumptions imply that the borehole free boundary ∂0Ω is the semi-infinitecylindrical surface ended by the bit bottom, on the first iteration.After the iterations converge, the integration of the component σ3j(y)ηj(y) of the contacttraction gives the total axial force P (t) applied to the bit during the progression,P(t) =∫∂cΩσ3j(y)ηj(y) dS(yc). (20)In the case when condition (iii) is not satisfied, the rupture front, the contact surface and thepressure are unknown and we need to take into account elasticity of the bit. If condition (v) isviolated, that is, the bit interacts with the material not only by pressure, one has to introducesome friction contact, describing it e.g. by the Coulomb–More law.Note that in all the cases, the obtained solution and particularly the total force P is independentof the progression rate h˙3 or the progression itself.3 Stationary-periodic indentation modelLet the instant bit progression h3(t) be the lowest y3 coordinate of the bit boundary. LetT be the cycle period, dh3(t)/dt = h˙3(t) = −|h˙(t)| ≤ 0 be instant progression rate andh˙T3 = [h3(t) − h3(t − T )]/T = −|h˙T | ≤ 0 be average progression rate over cycle, in the y3direction, where h˙ = (0, 0, h˙3) and h˙T = (0, 0, h˙T3) are the instant and average progression ratevectors, respectively. In the stationary-periodic problems, h˙T does not depend on t, the strains,stresses and boundary tractions are independent of the cycle number m in the correspondingpoints at the corresponding instants,εij(y +mTh˙T , t+mT ) = εij(y, t), y ∈ Ω(t); (21)σij(y +mTh˙T , t+mT ) = σij(y, t), y ∈ Ω(t) (22)for any positive or negative integer m.Rupture in this model does not in fact depend on the natural time t but depends only on theactual state in the considered material point.In addition to the Model assumptions (i)-(vi), let as make the followingReverse assumption(vii) The rupture do not proceed during the reverse stage of the bit motion, i.e. the boreholeboundary consists of the same material points until the load reaches the extremum valueduring the next cycle.Due to assumption (vii), the stress and strain return to the same states during the reverseand the following progressive stages of the bit motion up to the rupture restarts. This impliesthe reverse and the following progressive stages can be considered as some interruptions of thestationary progression process, analyzed in the previous section, and moreover, the interruptionsdo not influence the material rupture. This means the relation between the total force and the107S.E.Mikhailov and I.V.Namestnikovabit progression looks as on Fig. 2, that is the elastic loading stage is followed by the rupturestage followed by the elastic unloading stage. Generally the elastic stages are non-linear on theloading and unloading stages due to the changing contact surface between the bit and the drilledmaterial, and the force is constant on the rupture stage. The extremum force and strains inthe process coincide with those obtained in the stationary indentation analysis in the previoussection.hPFigure 2: h− P diagram in the stationary-periodic instant rupture model.Then solution of the stationary indentation problem from the previous section fixes the bore-hole boundary for the reverse stage and gives the extremum values of the contact distributionp(y, tex), whose integral (20) gives the maximum force Pex = P(tex) on the bit.To predict the curvilinear part of the h−P diagram, one has to solve the classical linearly elasticcontact problem with a material boundary ∂Ω known in the non-deformed state, for both theprogressive (before rupture restart) and reverse stages of the cycle (which do coincide). Theproblem consists of the following equations,σij,j(y) = 0, y ∈ Ω; (23)σij(y)ηj(y)ξi(y) = 0, σij(y)ηj(y)ζi(y) = 0,ui(y)ηi(y) = d(y, h3), σij(y)ηj(y)ηi(y) < 0, y ∈ ∂cΩ(h3); (24)σij(y)ηj(y) = 0, ui(y)ηi(y) < d(y, h3), y ∈ ∂0Ω(h3); (25)ui(y)→ 0, y →∞. (26)The overall boundary ∂Ω here is known from the end of the previous progression-rupture stage,although the boundary partition into the free and contact parts is to be determined for each h3.Classical conforming contact problem (23)-(26), (19) can be solved by any of the well knownmethods, see e.g. Johnson (1985). Particularly, one can use the iteration algorithm similar tothe on on the progression-rupture stage described above, that is, to choose some reasonablepartition of ∂Ω onto ∂0Ω and ∂cΩ, solve mixed elasticity problem (23)-(26), (19), modify thepartition of ∂Ω to alleviate the violation of inequalities in (24)-(26) and start the next iteration.4 ConclusionsA stationary-periodic quasi-static model of percussive drilling is obtained. The cycles of thebit progression – force diagram consist of three stages: elastic loading, constant-force ruptureprogression, and elastic unloading parallel to the loading. The problem is split into a stationary108S.E.Mikhailov and I.V.Namestnikovafree-boundary non-linear problem for the rupture stage of the cycle, and a classical contactproblem for the rest of the cycle. Some iteration algorithms are described reducing the solutionto a sequence of linear problems of elasticity. Those problems are to be solved by a generalnumerical method, e.g. the FEM or the Boundary Integral Equation Method. As a result, thisprovides a nonlinear progression-force diagram, which is to be used in the bit dynamic motionprediction.5 AcknowledgementThis work is supported by the research grant GR/R85556/01 ”Nonlinear Dynamics and RockContact Fracture Mechanics in Modelling of Vibration Enhanced Drilling” of the Engineeringand Physical Sciences Research Council, UK.ReferencesEliot C.M., Ockendon J.R. 1982.Weak and Variational Methods for Moving Boundary Problems, Pitman, London(UK).Crank J. 1984. Free and Moving Boundary Problems, Oxford University Press, Oxford (UK).Johnson K. 1985. Contact Mechanics, Cambridge University Press, Cambridge (UK).109

Quasi-static stationary-periodic model of percussive deep drilling

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