Effect of hydrodynamics on axial and torsional oscillations of a drillstring with using a positive displacement motor

Journal of Petroleum Science and Engineering 183 (2019) 106423

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Effect of hydrodynamics on axial and torsional oscillations of a drillstring with using a positive displacement motor

T

Vadim S. Tikhonova,∗, Fedor D. Baldenkob, Olga S. Bukashkinac, Valery Yu. Liapidevskiid a

Weatherford, 125047 Moscow, Russia Gubkin University, 119991 Moscow, Russia c Weatherford, 195251 St-Petersburg, Russia d Lavrenty'ev Institute of Hydrodynamics, SB RAS, 630090 Novosibirsk, Russia b

A R T I C LE I N FO

A B S T R A C T

Keywords: Drillstring oscillations PDM Mathematical model Hydrodynamics Numerical solution Dynamic analysis

A model and a method of dynamic analysis of a drilling system with a Positive Displacement Motor (PDM) have been proposed. The relevance of the work is related to the need to consider the interaction of the models of the drillstring oscillations, well hydrodynamics, PDM, bit-rock interaction, and a drawworks. The drilling with PDM distinguishes from rotary drilling, where the flow in the drillstring and in the annulus affects only hydraulic losses. Strange as it may seem, there is no complete model for the operation of such a system, although the need for it is rather high. The mathematical model takes into account wave processes in fluid and in the drillstring, effect of fluid pressure waves on axial drillstring motion, nonlinearity of mechanical and hydro-mechanical PDM characteristics, deformation of the cross-sectional area of the string and wellbore wall, viscous and Coulomb friction, PDC bit-rock interaction, drawworks dynamics and other effects. The Godunov's method is used for the numerical integration of the model equations. The computational scheme is universal and provides convergence of the numerical process in a wide range of system parameter variations. Calculations of dynamic responses to various disturbances and control actions during drilling wells with arbitrary profile have been provided. The paper provides analysis of previously unstudied processes related to the interaction of drillstring oscillations, flows in the well and the downhole motor, when the bit from the soft rock formation enters into the formation of more hard rocks, when the bit is fed into the well using a drawworks, etc.

1. Introduction

subsystem processes:

Today PDM has become an important part of state-of-the-art drilling technologies because of its high performance. Its effectiveness and the tendency towards drilling process automation require development of a mathematical model of the system as a whole and its analysis method. This model will enable a more precise evaluation of the main process parameters including the Rate Of Penetration (ROP), Weight On Bit (WOB) and Torque On Bit (TOB), pressure losses in the hydraulic system, to analyze motor operation stability and durability, assign parameters of the bit feed, thr mud flow rate, the well backpressure control system et al. One of the PDM operational aspects is that the basic drilling parameters depend not only on the motor mechanical characteristic but on the variation of the fluid flow rate through the motor and on the angular velocity of the drillstring. It necessitates consideration of the drilling process as a whole with regard for the interaction of its main

− − − − −

∗

Drillstring dynamics; Hydrodynamic processes in the pipe and in the annulus; Downhole motor mechanical characteristic; Bit-rock interaction; Drawworks dynamics.

The subsystem interaction scheme is shown in Fig. 1. Another distinctive feature of the system is that the distance between the power pump and the motor can be several thousand meters, and disturbances and control actions are transferred via the hydrodynamic and mechanical wave conductors with a delay. The issue of refined model development is growing more urgent with drilling of deeper wells, the need to increase ROP, use more powerful motors, etc. When a well is drilled with a PDM, fluid flows via a conduit inside the string to the PDM at a set flow rate that determines the hydraulic

Corresponding author. E-mail address: [email protected] (V.S. Tikhonov).

https://doi.org/10.1016/j.petrol.2019.106423 Received 2 January 2019; Received in revised form 30 July 2019; Accepted 25 August 2019 Available online 31 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 183 (2019) 106423

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Fig. 1. Scheme of the interaction of subsystem.

compressible viscoplastic model is used to determine pressure losses. A distinctive feature of the axial oscillation model is that it accounts for pressure waves in compressible fluid and Coulomb friction in deviated well sections. Wellbore and drillstring sectional areas of hydraulic line conduits (in the drillstring and in the annulus) are calculated with provisions for three-dimensional: hoop, radial and longitudinal deformations of the walls induced by pressure in the drillstring and the annulus, and axial force. The nonlinear equations of PDM mechanical and hydro-mechanical characteristics developed in the monograph of Baldenko et al. (2007), are considered as the downhole motor model. The cutting equations offered by Detournay and Defourny (1992), and Richard et al. (2002), are used to describe the PDC bit-rock interaction. The drawworks dynamics during drillstring running in the well is considered. The relation between the subsystems is set by the boundary conditions and discontinuity conditions. The paper suggests the numerical scheme based on one of the Godunov's method with approximation of contact discontinuities by the method of Rozhdestvesnky and Yanenko (1983), that enabled considerable reduces in running time without any essential loss of accuracy. The paper also provides an improved drillstring axial oscillation equation at variations of the pressure gradient of mud as well as refined PDM mechanical characteristic considering fluid flow rate deviations from the rated value and drillstring rotation. The paper presents examples of calculations of transient and steadystate processes during drilling wells with arbitrary profiles with typical disturbances and control actions, which analysis without the complete model review would be inadequate.

motor torque (Baldenko et al., 2007). The motor torque is transferred via the mechanical subsystem to the bit. On the other hand, changes in the WOB result in changes in the ROP, the conduit pressure, the flow rate, and, therefore, changes in motor torque and angular velocity. Besides, interaction of axial and torsional oscillations of the drillstring manifests itself in the bit cutting characteristic (Detournay and Defourny, 1992). Conventional models consider dynamic processes in the drillstring and hydrodynamic processes in the well separately. Such approach is more or less adequate for rotary drilling, but may not be applied to drilling with a PDM owing to essential dependence of PDM characteristics on the hydrodynamic processes in the system. Besides, oscillations of compressible fluid result in pressure waves and changes in fluid density and, therefore, affect the axial drillstring dynamics. There is only one known paper where authors attempted to develop a dynamic model of a similar system that considers interactions between the mechanical and hydrodynamic subsystems and PDM characteristics (Baldenko et al., 2007). This model enabled to assess the features of wave processes in the system, to study the effects of axial drillstring oscillations on transient PDM characteristics and classify the basic similarity criteria of the dynamic system. At the same time, this model lacks a drillstring torsional oscillations subsystem, which effect, as will be demonstrated below, plays an important role in the determination of dynamic characteristics of the system as a whole. The bit-rock interaction analysis was simplified based on the average specific torque and such approach prevents from reproducing the real process of rock cutting with a PDC bit. The model didn't account for the effect of liquid flow rate variations on the PDM mechanical characteristic. There was an error in the derivation of axial oscillations equations. The primary objective of this paper is to develop a complete model of the drilling system that includes interactions of all subsystems and an efficient numerical method to solve the system equations. The model of the hydrodynamic subsystem is represented by mass and momentum conservation equations for fluid flows in the drillstring and in the annulus. Such models are used in drilling hydrodynamics to analyze downhole surge and swab effects (Mitchell, 1988). A

2. System model 2.1. Hydrodynamic subsystem model Consider the scheme of flows in the pipe (into drillstring) and in the annulus (Fig. 2). Basic assumptions: 2

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The equation of state relating pressure and density of linearly compressible fluid is written as:

Pk = c 2 (ρk − ρ0 ) + P0,

(3)

where c is the sound velocity in stationary fluid; index “0” refers to variables in standard (atmospheric conditions) or unloaded state. The equations relating areas of conduits Ai and Ar with densities ρi , ρo (and axial force in the string) are defined by the ratios of the string and wellbore wall linear elasticity. Many expressions can be used to calculate pressure losses of viscoplastic fluid flow in the pipe and in the annulus (Chandrasekhar et al, 2013; Gjerstad et al, 2013). To be specific, this paper uses the standard model (API RP 13D, 2000, subsection 7.4). For the solution convenience, transform the set of Eqs. (1) and (2) by introducing new variables: − Mass of the fluid column per unit length:

mk = ρk Ak

(4)

− Fluid mass flux: (5)

Wk = mk vk − Fluid momentum flux:

Rk = 0.5vk2 + c 2 ln

ρk − gy, ρ0

(6)

s

where y =

∫ cos α (s ) ds is the TVD. 0

Based on the new variables, the set of Eqs. (1) and (2) can be represented as:

Fig. 2. Hydrodynamic subsystem flow scheme.

The problem setting is limited to 1D fluid flow, The fluid is linearly compressible, The flow is single-phase and fluid does not entrain solids or gas, Temperature in all well zones is the same, Lateral drillstring oscillations are disregarded, String and wellbore wall deformation follows the equations of linear elasticity, 7) The drillstring and the Bottom Hole Assembly (BHA) are comprised of pipe joints of various sizes; the cased hole and open hole diameters are different, 8) The fluid flows from the PDM to the annulus through the bit nozzles and slots (Fig. 2), 9) The fluid level in the pipe and in the annulus at the surface is maintained constant and equal to the True Vertical Depth (TVD) (Fig. 2).

1) 2) 3) 4) 5) 6)

(1)

∂ (ρAν )k ∂ (ρA)k = 0, + ∂s ∂t

(2)

(7)

∂mk ∂Wk + = 0, ∂t ∂s

(8)

where Φk = −pk / ρk . 2.2. Drillstring axial oscillations model Present the equations of the drillstring oscillations in the form a set of the equation of forces acting along a tangent to the drillstring center line (Fig. 3) and the equation of kinematic relation for the drillstring axial strain:

m0

∂u ∂T − = m 0 g cos α − (fc + pi Ai0 + po Ao0 ) ∂t ∂s

∂ε ∂u − =0 ∂t ∂s

The set of equations for the flows in the pipe and in the annulus includes well-known Euler's equations and mass flow conservation equations. For each of the two flows this set can be represented as:

p ∂vk ∂v 1 ∂Pk + vk k + = − k + g cos α ∂t ∂s ρk ∂s ρk

∂vk ∂Rk + = Φk , ∂t ∂s

(9) (10)

where u is the axial velocity, ε is the relative axial strain, m 0 = ρA0∗ is per unit length in undeformed state ; ρ is the metal density; A0∗ = Ao0 − Ai0 is the drillstring cross-sectional area; T is the axial force in the string; fc is Coulomb friction of the drillstring against the wellbore wall per unit length. Equations (9) and (10) should be supplemented with the equation of state relating the relative strain ε with the axial, hoop and radial stresses and representing the generalized Hooke's law:

where v is the flow velocity; ρ is the fluid density; P is the pressure; p is pressure losses per unit length; A is the conduit area: Ak = i = Ai is the cross sectional area of the pipe conduit; Ak = o = Ar is the annular crosssectional area; g is gravity acceleration; α is the inclination; s is the measured depth; t is the time; index k = i, o , “i” refers to pipe flow variables and “o” refers to annular flow variables.

ε=

1⎡T − ν (σr + σθ ) ⎤ ∗ ⎥ E⎢ ⎦ ⎣ A0

(11)

where E is Young's modulus; ν is Poisson's ratio; σr and σθ are radial and hoop stresses in the drillstring wall. 3

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managed pressure drilling generated by external action (backpressure) can be high enough and may not be disregarded in the general case. The Coulomb friction fc is determined by the conventional formula used in drilling (Johancsik et al., 1984; Tikhonov et al, 2014a,b):

fc = kt fn

u u2 + (0.5ωD0)2

(15)

where fn = (w cos ψ sin α − Te α′)2 + (Te sin αψ′)2 is the contact force (Fig. 3); kt is the friction factor in axial direction; w = g (m 0 − Ao0 ρo + Ai0 ρi ) is the weight of the drillstring in liquid per unit length; ψ is the well azimuth angle; D0 is the drillstring OD; ω is the angular velocity of drillstring. 2.3. Drillstring torsional oscillations model Write the set of drillstring torsional oscillation equations as:

ρIp

∂ω ∂B − = −b ∂t ∂s

∂θ ∂ω − =0 ∂t ∂s

(16) (17)

where Ip is the polar moment of inertia of the drillstring; B is the torque:

B = GIp θ

(18)

θ is the relative torsion angle; G is the shear modulus; b is the external torque per unit length: b = kθ fn

0.25ωD02 u2

+ (0.5ωD0)2

(19)

where kθ is the friction factor in hoop direction. The torque b (19) accounts for Coulomb friction when the drillstring comes into contact with the wellbore wall in inclined and curved hole sections. Hydraulic rotation friction is small and can be approximately considered as equal to zero. Fig. 3. Scheme of forces acting on the drillstring exposed to axial oscillations.

2.4. PDM model Using the Lame's formulas, represent the relative elongation ε as:

ε=

Te A0∗ E

whereTe = T + A ‾o0 Po − A ‾i0 Pi

The positive displacement motor is a BHA component and transfers torque through the spindle shaft to the bit being located in close proximity to the latter. The PDM consists of two basic parts (Samuel et al., 2015) (Fig. 4):

(12) (13)

1) Screw rotor, and 2) Stator with high-pressure and low-pressure areas connected to the BHA (Fig. 5).

is the effective axial force (Goodman and Breslin, 1976); A ‾i = 2νAi ; A ‾o = 2νAo . Effective force Te is the force at which the axial deformation of the drillstring under uniaxial loading is equal to the axial deformation under combined 3D loading at the axial force T and pressures Po and Pi . In a special case of deformation without volume changes, ν = 0.5, the values A ‾i = Ai , A ‾ o = Ao . The effective force Te is continuous along the string length including at discontinuities of its cross sections, A0∗. Expressing the force T from Eq. (13) and substituting it into Eq. (9), get:

m0

Owing to that the PDM length is short as compared to the drillstring length, assume that it is a material point coincident with the bottom end of the drillstring and the shaft torsional rigidity can be disregarded. The rotor and stator motion equations are provided in Tikhonov et al. (2014a):

∂u ∂T ∂Pi ∂Po − e =A −A + m 0 g cos α − (fc + pi Ai0 + po Ao0 ). ‾i0 ‾ o0 ∂t ∂s ∂s ∂s (14)

(Jr + Mr Δ2 z r2) Ω˙ − Mr δ 2z r z s ω˙ s = Bm − Bc

(20)

Bc = B (L)

(21)

where Jr is the effective moment of inertia of the shaft considering the moment of inertia of the rotor and the bit; Mr is the rotor mass; zr and zs is the number of lobes of the motor rotor and stator, respectively; Bm is the PDM torque; Bc is the TOB; Ω is the rotor angular velocity; ωs ≡ ω (L) is the stator angular velocity; L is the well depth; δ is the engagement eccentricity (Fig. 5). Mass inertia characteristics of the rotor should be taken into account only in high frequency torsional oscillation analysis (Tikhonov et al., 2014a). In other cases, the left side of Eq. (20) can be considered negligible and equated to zero.

The effect of pressure waves on axial drillstring oscillations manifests itself in the existence of the terms proportionate to the pressure gradients ∂Pi/ ∂s and ∂P0/ ∂s in the right side of Eq. (14), which, as it follows from Eq. (1), are related to fluid flow acceleration. Research people of axial drillstring oscillations often disregard this relation, assuming that fluid is incompressible. In such case the effect of the pressure gradient in the pipe and in the annulus practically comes to the buoyancy force. However, the pressure gradient in the well during 4

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the relative rotor angular velocity, Ω − ωs , instead of the angular velocity Ω. When the torque reaches the braking torque value, the motor will lose the degree of freedom. In this case, the rotor and the associated bit will stop if the well is drilled without drillstring rotation. If the drillstring starts to rotate, the motor will switch over to the pumping regime and will develop torque proportional to the angular stator velocity, ωs . When the value of ωs is “sufficiently” high, the bit will break free and will start to rotate with the drillstring at the angular velocity Ω = ωs . Based on the above, extend formula (22) to the general case:

⎧ ω + Ω Q‾ ⎡1 − I ⎪ s ⎣

Bm β Bmax

( ) ⎤⎦,   B

m

≤ Bmax ,   (a)

Ω = 0,   B ≥ B ∪ B > B ,   (b) m P m max ⎨ ⎪ < ∪ > ω ,   B B B B s m P m max ,  (c)  ⎩

(23)

where Q‾ = Qi (L)/ Qnom is the relative volumetric fluid flow rate in the PDM; Qnom is the rated fluid flow rate in the motor; BP is the motor torque in the pumping regime determined by the formula:

BP =

BI ωs ΩI

(24)

BI = ΔPI V /2π is the idling torque; ΔPI = ξd ρi (L) Qi (L) is pressure losses of the PDM in idle mode, i.e. without load, Bm = 0 ; ξd is drag coefficient of the PDM; V is the PDM capacity (Baldenko et al., 2007). Fig. 6 shows the dependence Bm (Ω) at various values of the relative volumetric fluid flow rate at the PDM inlet, Q‾, constructed based on Eq. (23a). The PDM hydro-mechanical characteristic sets the dependence between the pressure drop in the motor, ΔPd , and torque:

Fig. 4. Design of positive displacement motor.

ΔPd = ΔPI + kp Bm,

(25)

where kp is the slope of the motor hydro-mechanical characteristic curve. 2.5. Bit-rock interaction model The rock cutting model with a PDC bit proposed by Detournay and Defourny (1992), and Richard et al. (2002), was used as the bit-rock interaction model. The consideration is limited to a cutting when the axial velocity of the bit and angular bit velocity Ω stay positive during the entire transient process: u (L) > 0 and Ω > 0 (23a). In this case, the ROP, uR can be taken as equal to the velocity u (L) , and the cutting depth

Fig. 5. Motor stator/rotor pair engagement diagram.

Equations (20) and (21) of the PDM are to be supplemented with equations of motor mechanical and hydro-mechanical characteristics. The motor mechanical characteristic was proposed by Baldenko et al. (2007), in the form:

B Ω = 1 − ΩI ⎛ m ⎞ ⎝ Bmax ⎠ ⎜

β

⎟

(22)

where Bmax is the braking torque; ΩI is the angular velocity in idle mode; β is nonlinearity of the motor mechanical characteristic. All three parameters refer to the rated regime on the fluid flow rate in the motor. Note, that with β = 1, Eq. (22) will describe the mechanical characteristic of a downhole mud motor. Equation (22) is valid for the PDM operational regime and the rated values of the torque in braking mode, Bmax , and the angular velocity in idle mode, ΩI , corresponding to the rated value of the fluid flow rate. It disregards the effect of fluid flow rate variations at the motor inlet, stator rotation and doesn't reflect the PDM operation features during stop of rotor at bit stall typical of the stick-slip mode (Tikhonov et al., 2014a). The data of Schlumberger (2004), shows that actual motor rated parameter values are proportional to the relative flow rate. It is obvious also that when the stator rotates, Eq. (22) should be written in terms of

Fig. 6. PDM torque vs bit rpm at various relative volumetric fluid rates at PDM inlet. 5

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− the wellhead pressure,

Pg = c 2 [ρo (0) − ρ0 ] + P0;

(32)

− the shaft torque of the electric machine that controls the drawworks, B wg ; − the angular velocity of the drillstring rotary mechanism, ωg = ω (0) . In the general case, the above parameters can vary in the form of a prescribed function of time: Qg = Qg (t ) , Pg = Pg (t ) , B wg (t ) , ωg = ωg (t ) . The following equations are to be used at the bottomhole, s = L :

Fig. 7. Scheme of rock cutting with a PDC bit.

− mass flow conservationequation for fluid in the drillstring and in the annulus:

can be determined by the formula:

h=

2πuR . Ω

Wi (L) + Wo (L) = 0

(26)

The WOB and the TOB are determined based on the condition of equilibrium of forces acting on the bit cutters (Fig. 7) (Richard et al., 2002):

Gb =

Bc =

⎧ rb eh (ζ + κ ),  h < ⎨ rb e (ζh + l),  h ≥ ⎩

l , κ l , κ

⎧ 0.5rb2 eh (1 + μγκ ),  h <

− fluid momentum conservation equation:

Ri (L) = Ro (L) + Rd + Rh

where Rh = ΔPh/ ρi (L); ΔPh is pressure losses in bit nozzles (API RP 13D, 2010, subsection 7.5); Rd = ΔPd/ ρi (L) ; losses in the motor ΔPd are determined by Eq. (25);

(28)

− WOB equation:

where rb is the bit radius; e is intrinsic specific energy (the energy required to cut a unit volume of rock), characterizing rock strength; ζ is the ratio of the vertical cutting force projection to its horizontal projection for an absolutely sharp cutter; l is the overall wear flat width (Fig. 4); κ is the slope of the curve that relates characteristic contact length of the cutter with cutting depth in elastic area (Detournay et al., 2008); μ is the friction factor at the wear flat; γ is a bit constant, which encapsulates the influence of the orientation and distribution of the contact forces acting on the bit (for bits with flat profile, it can be assumed that 1 ≤ γ ≤ 1.33, Richard et al., 2002).

Te (L) = −Gb

− rotor and stator motion equations, (20) and (21), where PDM torque, Bm , is determined by Eq. (23). The following conditions shall be met at internal contact discontinuities in points s = sj :

Wk − = Wk +, Rk − = Rk +, u− = u+, Te − = Te +, ω− = ω+, θ− = θ+

(29)

4. Numerical method

where Jw is the effective moment of inertia of rotating drawworks parts including the reduced moment of inertia of the bit feed mechanism and transmission, as well as inertia of the swivel, hook and travelling block; Ba is the constant torque relating to the swivel, hook and travelling block weight; rw is the mean radius of drill line spooling on the drum; B w is the braking torque. The braking torque depends on the electric machine shaft torque and the type of bit feed control transmission:

Tw

dB w + B w = B wg dt

4.1. Steady-state solution Steady-state solution is required not only to set the initial conditions to calculate dynamic processes, but is significant as such because it enables to optimize drilling practices. In steady-state conditions, ω˙ (s ) = Ω˙ = 0 , ω0 = ωg , and the WOB, Gb0 , is specified. Here, as it follows from (20) and (21), Bc 0 = Bm0 . Thus, using Eqs. (22), (26)–(28), the following parameters may be determined:

(30)

− Angular bit velocity

where Tw is the brake electromagnetic time constant; B wg is control torque on the drawworks shaft.

β

1 ⎧ ⎛ Gb0 − a2 b1 + b2 ⎞ ⎤ ⎫ , Ω0 = ωg + Q‾0 ΩI 1 − ⎡ ⎢ Q‾0 Bmax ⎝ ⎨ a1 ⎠⎥ ⎣ ⎦ ⎬ ⎩ ⎭ ⎜

3. Boundary conditions The following conditions will be set at point s = 0 :

⎟

(37)

− ROP, u 0 (s ) = uR , where

− the fluid flow rate,

Qg = Ai (0) vi (0);

(36)

where the subscripts “-” and “+” refer to the points sj − 0 and sj + 0 , respectively.

Equation of moments on the drawworks (Baldenko et al., 2007):

d [u (0)/ rw] = rw Te (0) − (B w − Ba) dt

(35)

where Gb = Gb (h (L)) (see Eqs. (27) and (26));

2.6. Drawworks model

Jw

(34)

(27) l , κ

⎨ 0.5r 2 e (h + μγl),  h ≥ l , b. κ ⎩

(33)

uR = (31)

6

(Gb0 − a2) Ω0 2πa1

(38)

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oscillation equations, except for the boundary conditions at the point s = L is small (displaying only in the source terms pk of the function Φk depending on the drillstring velocity). Similarly, the effect of axial oscillations on torsional oscillations is sufficiently small (see Eq. (19)), except for the boundary conditions at the point s = L . Considering this, will solve the set of Eq. (41) by its splitting into subsets that include fluid motion equations, axial and torsional oscillation equations. Each of the introduced subsets is a hyperbolic second-order equations. Approximate values of characteristics of the split set:

− λk ± = ± c + vk , k = i, o,for the fluid flows in the drillstring and in the (42)

annulus

− Relative torsion angle

1 ⎛ Gb0 − a2 b1 + b2 ⎞ GIpN (L) ⎝ a1 ⎠ ⎜

(43)

− λt = ± ct ,for axial torsional oscillations

(44)

where ca = E / ρ is the sound speed in metal at axial oscillations; ct = G / ρ is the sound speed in metal at torsional oscillations. Various versions of the numerical mesh schemes proposed by Godunov are widely used to solve hyperbolic-type equation (41) (Kulikovski et al., 2001). Such schemes are multipurpose and rather efficient enabling a fairly simple solution of the first and higher order accuracy along the length with a minimum quantity of computing operations. The explicit scheme that was used to solve Eq. (40) is written as:

Fig. 8. ROP vs bit rpm.

θ0 (L) =

− λa = ± ca,for axial oscillations

⎟

(39)

Unj + 1 − Unj Δt

a1 = rb e

⎧ ζ + κ,  h < κ ,

a2 =

⎧ 0,  h < κ ,

b1 =

⎨ rb el,  h ≥ l ; ⎨ ζ ,  h ≥ l ; κ κ ⎩ ⎩ l l 1 + μγκ,  h < κ , 0,  h < κ , ⎧ ⎧ 0.5rb2 e b2 = ⎨1,  h ≥ l ; ⎨ 0.5r 2 eμγl,  h ≥ l . b κ κ ⎩ ⎩ Eliminating Gb0 from Eqs. (37) and (38), get the dependence of ROP vs angular bit velocity. In particular case of ideal sharp cutters, l = 0: 1/ β

uR =

Ω0 Q‾0 Bmax ⎛ Ω0 ⎞ 1− Q‾0 ΩI ⎠ πrb2 e ⎝ ⎜

⎟

.

(40)

Fig. 8 shows characteristic ROP vs bit rpm curves for ideal sharp and blunt cutters. Solving the boundary value problem, Eqs. (1) and (2) (∂vk / ∂t = ∂ρk / ∂t ≡ 0) , with the set boundary conditions (31)–(34), distribution of pressures Pk0 (s ) and fluid velocities vk0 (s ) in the drillstring and in the annulus can be determined. Integrating Eq. (14) over the length (∂u/ ∂t ≡ 0) with the boundary condition (33), get the distribution of the axial force Te0 (s ) along the drillstring length. Knowing Te0 (0) , can determine the drawworks braking torque B w0 from the Eq. (28). Integrating Eq. (16) over the length (∂ω/ ∂t ≡ 0) with the boundary condition B0 (L) = GIp (L) θ0 (L) , get the distribution of the torque B0 (s ) and the torsion angle θ0 (s ) .

Δt ≤

Δs

= Ψnj

(45)

KΔs λ max

(46)

where λ max = max[λk (s ), λa , λt ] = λa ; K ≤ 1 is the Courant constant. k

Eq. (41) solution details are provided in the paper of Tikhonov et al. (2016). 5. Analysis of system response to various disturbances A single-joint string with a BHA, which characteristics are provided in Table 1, was used as a test example.Other parameters of the subsystems are presented in Table 2. The following reference output parameters were used:

4.2. Numerical dynamic analysis

− ROP, uR (t ) ;

Using the vector form, write the set of equations of hydrodynamics in the pipe and in the annulus (7), (8), and axial and torsional drillstring oscillations, (10), (14) and (16), (17), respectively, as:

∂U ∂F + =Ψ ∂t ∂s

Fj + 1/2 − Fj − 1/2

where the integral index j = 1, 2, …, designates the values of the function related to the center of the j-th cell, and the half-integer subscripts j ± 1/2 designate the values of the mesh function at the boundaries of j and j ± 1 cells; the upper integer superscript n = 0, 1, 2, …, is point of time index; Δt is the time step; Δs is the mesh step. The flux values Fj + 1/2 at internal contact discontinuities of the domain were determined from linearized relations at the characteristics for Eqs. (7), (8), (10) and (17) written in the integrated form (Rozhdestvesnky and Yanenko, 1983). The differential equations (20), (21), (29) and (30) approximated using the explicit Euler's scheme were used to calculate fluxes at the boundaries. The scheme will satisfy the numerical stability requirement if the selected time step satisfies the Courant condition:

l

l

where

+

Table 1 String and well geometry. STRING

(41)

where U = (vi, mi , vo, mo , u, ε , ω, B / GIp) is the vector of variables; F = (Ri , Wi , Ro , Wo, −Te / m 0 , −u, −B / ρIp, −ω) is the vector of flux; Ψ is the vector of the right side of the equations containing the terms not included in the left side (41). The dependence of the fluid motion equations on axial and torsional

Length, m Inside diameter, mm Outside diameter, mm

7

WELL

DRILLSTRING

BHA

CASED HOLE

OPEN HOLE

2000 107 127

110 76 165

1600 225 –

2110 215.9 –

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characteristics of the split system without considering the interaction of various oscillation modes. Fig. 9 shows transient processes during drilling of a vertical well with a bit having ideal sharp cutters as it penetrates harder rocks with the intrinsic specific energy half as much as the initial energy, 180 MPa, characterizes hard sandstone. As follows from the figures, ROP sets substantially immediately at level of 1.5 times smaller of initial value according to the ratio of rock hardness values (see Eq. (40)) and remains practically unchanged during the whole process. The effect of high frequency axial drillstring oscillations appears during the first seconds of the process. When the bit penetrates into the formation with the rock that is 1.5 times harder, WOB and TOB also increase by 1.5 times substantially immediately, as they should from Eqs. (27) and (28). Increased TOB results in a decrease in motor rpm and the onset of torsional oscillations. Pressure losses in the motor, ΔPd (25), also increase, which initiates fluid oscillations in the pipe and in the annulus. The relative fluid flow rate at the motor inlet changes slightly. The drillstring torsional oscillations with a period of 1.5 s close to the estimated value, dominate. It is characteristic that SRP oscillations are at the torsional oscillations frequency and attenuate to the 30th-35th second only. The fluid oscillations mainly affect the bit rpm. According to Fig. 9, their period is 5.4 s, which is also close to the estimated value. It should be noted that the bit rpm with the amplitude of 4–4.5 rpm practically doesn't decay. It can be assumed that these self-oscillations are related to the system nonlinearity caused by the effect of bit-rock interaction and the dependence of the motor hydro-mechanical characteristic on the hydrodynamic processes in the pipe not unlike the stick-slip oscillations (Detournay et al., 2008). In order to remove doubts about the existence of self-oscillations, a similar test was conducted with the intrinsic specific energy jump of the smaller value = 150 MPa (Fig. 10). Based on the comparison with the processes shown in Fig. 9, it follows that both transient processes are similar; however, oscillation amplitude of the bit rpm remains practically unchanged, which confirms the self-oscillating nature of these oscillations. When the well is drilled by a bit with blunt cutters with a total width of 2.1 mm, Fig. 11, ROP in initial state drops to 20 m/h in the initial mode, and further more to 8 m/h after bit penetration into hard sandstone. The amplitude of SRP oscillations increases, and it value periodically drops to zero, and initially even to small negative values. Oscillations of the bit rpm set up substantially immediately after the start of drilling through hard sandstone, and the amplitude of the bit rpm, the bit load and pressure drop in the PDM decreases. Self-oscillations of the bit rpm with the period of 5.5 s again demonstrate the nonlinear relationship among the cutting, motor operation and well hydrodynamics processes. Fig. 12 shows transient processes during drilling of a J-shaped well when the bit with sharp cutters penetrates hard sandstone. Well profile:

Table 2 Subsystem parameters. Mud Density under standard conditions Yield stress Consistency measure Drillstring Angular velocity of the drillstring rotary mechanism Coulomb friction factor in cased hole Coulomb friction factor in open hole Mesh step PDM Rated motor inlet flow rate Motor inlet flow rate in initial state (static) Angular velocity in idle mode Braking torque Motor mechanical characteristic nonlinearity indicator Effective rotor moment inertia Drag coefficient Motor hydro-mechanical characteristic curve slope coefficient Bit WOB in initial state Diameter Total bit nozzles area Ratio of vertical and horizontal cutting force projections Bit cutting characteristic curve slope in the initial section Formation Intrinsic specific energy in initial state Drawworks Effective moment of inertia of the drawworks drum Mean diameter of drill line spooling on the drum Brake electromagnetic time constant

− − − − − − − −

1377 kg/m3 0.8 cP 0.3 12 rad/sec 0.27 0.4 10 m 25 L/sec 25 L/sec 20.94 rad/sec 5700 N⋅m 2 0.615 kg⋅ m2 100,000 m⋅ sec−1 12 rad/sec

37 kN 215.9 mm 500 mm2 0.7 5

120 MPa (medium sandstone) 600 kg⋅ m2 0.65 m 2 sec

Surface Rate of Penetration (SRP), u (0, t ) ; Hook load, Te (0, t ); WOB, Gb (t ) ; Bit rpm, 30Ω (t )/ π ; Stator rpm, 30ωs (t )/ π ; TOB, B (L, t ) ; Relative fluid rate at PDM inlet, Q‾ (t ) ; PDM pressure drop, ΔPd (t ) .

Due to absence of field measurements and experimental studies tests, as well as data on the transient process calculations in technical and scientific papers on analysis of dynamic bit feed systems, the program debugging was focused on verification of process parameter error at zero disturbances and control of their steady-state values upon the end of the transient process. For this purpose, disturbing and control actions were applied not at the zero time but after holding in the initial state until the first second. The steady state was verified by comparison with the static equation solution at such disturbances. It should be noted that pressure deviations are the most sensitive to dynamic changes. The error of the implemented numerical scheme after pressure setting in dynamics without disturbance did not exceed 0.8 kPa. To analyze the presented results, determine the periods of natural oscillations of each of the system modes. Using the sound velocities in fluid and metal: c (3), ca (43), ct (44), it is easy to determine the periods for

• Vertical section: 0 m–780 m; • Buildup section: 0.15°/30 m, 780 m to 1270 m; • Inclined section: 32°, 1270 m to 2110 m. The drillstring friction force, which main component in a deviated well is Coulomb friction, while such friction is absent in a vertical well, considerably increases the degree of axial oscillations damping, which damp out quickly enough. At the same time, the level of the bit rpm self-oscillations stays virtually the same. This again demonstrates that the source of self-oscillations is the nonlinear character of bit-rock interaction and nonlinearity of the mechanical characteristic of the motor. Fig. 13 exhibits processes that occur when the electric machine shaft torque that controls the drawworks drops from 83.5 kN m to 79 kN m during drilling of medium sandstone by a bit with sharp cutters. This change corresponds to the increase in the WOB by 15 kN. ROP

− Fluid oscillations – 6 s; − Drillstring axial oscillations – 0.8 s; − Drillstring torsional oscillations – 1.4 s. Naturally, these are not exact but estimated values of natural oscillation periods, as they are determined approximately based on the 8

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Fig. 9. Transient processes during vertical drilling through hard sandstone with intrinsic specific energy = 180 MPa. Bit with ideal sharp cutters.

oscillations are set up with the period of fluid natural oscillations and the amplitude of 11 m/h, which is associated with changes in the motor operation regime caused by increased TOB (almost by two times) and PDM pressure losses. The average ROP after self-oscillations are set increases from 64 m/h to 74 m/h. The amplitude of SRP self-oscillations increases to 6.5 mm, if express it in terms of the drillstring suspension point displacement. The amplitude of self-oscillations of a bit rpm increases to 18 rpm. PDM pressure losses and the fluid flow rate at the PDM inlet in the transient process increase considerably. The aperiodic nature of the process to the steady-state results from the drawworks inertia. It is impossible to get the similar transient process without considering the interaction of all processes in the system. So far, it has been customary for the drillstring axial oscillations analysis to set the drawworks drum torque (or hook load) but not the SRP. However, as show processes in Fig. 13, this approach is inappropriate, and to develop the bit feed control system with using a PDM, the system should be considered as a whole.

6. Conclusion 1) A dynamic model of the system has been developed that includes models of several subsystems: drillstring axial and torsion oscillations, hydrodynamic processes in the pipe and in the annulus, PDM, bit-rock interaction, and drawworks. 2) For developing of complete system model was used improved equations of drillstring axial oscillations in mud with regard for dynamic changes in the pressure gradient in the well and a new PDM model considering flow rate changes at the motor inlet and drillstring rotation. 3) A numerical scheme has been developed to calculate transient processes in the system. For lack of field measurements, experimental studies and data of other technical and scientific papers verification of the model has been conducted by comparison with results of steady state analysis. The maximum error of steady state parameter deviations calculated via dynamic model from their static values characteristic on pressure deviations did not exceed 0.8 kPa at absolute pressures of the order hundreds to thousands of kPa. 4) The numerical modeling results demonstrate system self-oscillations 9

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Fig. 10. Transient processes during vertical drilling through hard sandstone with intrinsic specific energy = 150 MPa. Bit with ideal sharp cutters.

characteristics of the drillstring when drilling using PDM, estimation of the influence of various disturbing factors in drilling process wells, selection the optimal drilling regimes, development of the control mechanism for the bit feed et al.

resulting from complex interaction of the drilling subsystems. 5) The ROP during drilling at penetrating in the harder rocks drops quickly, while the SRP during the process oscillates at considerably high amplitude and damps out slowly. 6) Damping of axial oscillations during deviated well drilling as a result of Coulomb friction effects increases considerably. However, the amplitude of self-oscillations stays virtually at the same level as in a vertical well. 7) Self-oscillations at penetrating in the harder rocks are mainly limited to the bottomhole area; however, they are also transferred to the SRP, which is sensitive to any actions. The dominating oscillation frequency of the majority of variables is the natural frequency of torsional oscillations of the system. At oscillations of the bit rpm, which largely depends on the fluid flow rate at the PDM inlet, the dominating oscillation frequency is the frequency of fluid oscillations in the well. 8) System self-oscillations during bit feeding by the drawworks are spread along the whole length of the drillstring, which requires more flexible braking torque control. ROP oscillations with sufficiently high amplitudes are set up at the fluid oscillation frequency; WOB and TOB amplitude increases considerably. 9) The proposed model can be used for expert study of the dynamic

Nomenclature A Ar A* B Ba Bc BP Bw Bwg Bmax D F E 10

area (m2) annular cross-sectional area (m2) drillstring cross-sectional area (m2) torque (N·m) torque relating to the swivel, hook and travelling block weight (N·m) TOB (N·m) torque of the PDM in pumping regime (N·m) drawworks braking torque (N·m) control torque on the drawworks shaft (N·m) braking torque (N·m) drillstring OD (m) vector of flux (m2/sec2, kg/sec, m2/sec2, kg/sec, m2/sec2, m/ sec, rad·m/sec2, rad/sec) Young's modulus (N/m2)

Journal of Petroleum Science and Engineering 183 (2019) 106423

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Fig. 11. Transient processes during vertical drilling through hard sandstone. Bit with blunt cutters.

G Gb Ip J K L M P Q Qnom Q‾ R T Te U V W b

shear modulus (N/m2) WOB (N) polar moment of inertia of drillstring (m4) effective moment of inertia (kg·m2) Courant constant well depth (m) mass (kg) pressure (Pa) volumetric flow rate (m3/sec) rated fluid flow rate in the PDM (m3/sec) relative volumetric fluid flow rate in PDM (m3/sec) fluid momentum flux (m2/sec2) axial force in the drillstring (N); time constant (sec) effective axial force (N) vector of variables (m/sec, kg/m, m/sec, kg/m, m/sec, rad, rad/sec, rad/m) PDM capacity (m3) fluid mass flux (kg/sec) external torque per unit length (N)

c ca ct e fc fn g h kp kt kθ l m p rw s t 11

sound velocity in stationary fluid (m/sec) sound speed in metal at axial oscillations (m/sec) sound speed in metal at torsional oscillations (m/sec) intrinsic specific energy (Pa) Coulomb friction of the drillstring against the wellbore wall per unit length contact force (N/m) gravity acceleration (m/sec2) cutting depth (m) slope of the PDM hydro-mechanical characteristic curve (1/ m3) the friction factor in axial direction the friction factor in hoop direction overall wear flat width (m) mass per unit length (kg/m) pressure losses per unit length (Pa/m) mean radius of drill line spooling on the drum (m) measured depth (m) time (sec)

Journal of Petroleum Science and Engineering 183 (2019) 106423

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Fig. 12. Transient processes during drilling of J-shaped well through hard sandstone.

u uR v w y z δ Δ ΔPd ΔPh ΔPI Φ Ψ Ω α β γ

δ

ε ζ

axial drillstring velocity (m/sec) ROP (m/sec) fluid velocity (m/sec) weight of the drillstring in liquid per unit length (N/m) true vertical depth (m) number of lobes of the PDM engagement eccentricity of the PDM (m) deviation sign pressure losses in the PDM (Pa) pressure losses in bit nozzles (Pa) pressure losses of the PDM in idle mode (Pa) source term in the Euler's equation (m/sec2) vector of the right side of the equations (m/sec2, Pa·sec, m/ sec2, Pa·sec, m/sec2, rad/sec, rad/sec2, rad/m·sec) angular rotor/bit velocity (rad/sec) drillstring inclination (rad) nonlinearity of the PDM mechanical characteristic bit constant, which encapsulates the influence of the orientation and distribution of the contact forces acting on the bit engagement eccentricity of the PDM (m)

θ κ λ ξd μ ν ρ σr σθ ψ ω ωs Subscripts 0 I – 12

relative elongation ratio of the vertical cutting force projection to its horizontal projection for an absolutely sharp cutter relative torsion angle slope of the curve that relates characteristic contact length of the cutter with cutting depth in elastic area characteristic (m/sec) drag coefficient of the PDM (1/m·sec) friction factor at the bit cutter wear flat Poisson's ratio density (kg/m3) radial stresses in the drillstring wall (Pa) hoop stresses in the drillstring wall (Pa) well azimuth angle (rad) angular velocity of drillstring (rad/sec) angular velocity of the PDM stator (rad/sec)

standard conditions; unloaded state idle left-hand limit

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Fig. 13. Transient processes during vertical drilling when the drawworks braking torque drops by 4.5 kN m.

+ a i j

k m o r

right-hand limit axial pipe flow mesh cell index 13

{i, o} motor annular flow rotor

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g specified s stator t torsional w drawworks Superscript n

experiment. Int. J. Rock Mech. Min. Sci. 45, 1347–1360. Gjerstad, K., Sui, D., Bjørkevoll, K.S., et al., 2013. Automatic prediction of downhole pressure surges in tripping operations. In: IPTC, Beijing, China, 26-28 March, IPTC 16974. Goodman, T.R., Breslin, J.P., 1976. Statics and dynamics of anchoring cables in waves. J. Hydronautics 10, 113–120. Johancsik, C.A., Friesen, D.B., Dawson, R., 1984. Torque and drag in directional wells – prediction and measurement. SPE J. Pet. Technol. 36, 987–992. Kulikovski, A.G., Pogorelov, N.V., Semenov, A.Yu, 2001. Mathematical aspects of numerical solutions of hyperbolic systems. Monogr. Surv. Pure Appl. Math. 118 Chapman and Hall/SRC. Mitchell, R.F., 1988. Dynamic surge/swab pressure predictions. SPE Drill. Eng. 325–333 September. Richard, T., Detournay, E., Fear, M., et al., 2002. Influence of bit-rock interaction on stickslip vibrations of PDC bits. In: SPE ATCE Conference and Exhibition, San Antonio, Texas, 29 Sept – 2 Oct, SPE 77616. Rozhdestvesnky, B.L., Yanenko, N., 1983. Systems of quasilinear equations and their applications to gas dynamics. A.M.S. Transl. Math. Monogr. 55. Samuel, R., Baldenko, D., Baldenko, F., 2015. Positive Displacement Motors – Theory and Application. SigmaQuadrant. Schlumberger, 2004. PowerPak Steerable Motor Handbook. Sugar Land, Texas. Tikhonov, V.S., Bukashkina, O.S., Gandikota, R., 2014a. A new model of high-frequency torsional oscillations (HFTO) of drillstring in 3-D wells. In: IPTC, Kuala Lumpur, Malaysia, 10-12 Dec, IPTC-17958-MS. Tikhonov, V., Valiullin, K., Nurgaleev, A., Ring, L., Gandikota, R., Chaguine, P., Chatham, C., 2014b. Dynamic model for stiff-string torque and drag. SPE Drill. Complet. 29, 279–294. Tikhonov, V., Bukashkina, O., Liapidevskii, V., Ring, L., 2016. Development of model and software for simulation of surge-swab process at drilling. In: SPE RPTC, Moscow, Russia, 24-26 Oct, SPE-181933-MS.

time index

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.petrol.2019.106423. References API Recommended Practice 13D, 2000. Rheology and Hydraulics of Oil-Well Fluids, sixth ed. pp. 27–32. Baldenko, D.F., Baldenko, F.D., Gnoevykh, A.N., 2007. Single-screw Hydraulic Machines. Positive Displacement Motors – V. 2. OOO “IRTs Gazprom”, Moscow, Russia (in Russian). Chandraeskhar, S., Bacon, W., Toldom, B., et al., 2013. Surge and swab effects due to vessel heave in deepwater wells: model development and benchmarking. In: IADC/ SPE Drilling Conference and Exhibition, Amsterdam, Netherlands, 5-7 March, IADC/ SPE 163545. Detournay, E., Defourny, P., 1992. A phenomenological model of the drilling action of drag bits. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 40, 13–23. Detournay, E., Richard, T., Shepherd, M., 2008. Drilling response of drag bits: theory and

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