Transient Stress Distribution and Failure Response of a Wellbore Drilled by a Periodic Load

energies

Article Transient Stress Distribution and Failure Response of a Wellbore Drilled by a Periodic Load

Xiaoyang Wang 1 , Mian Chen 1, Yang Xia 1, Yan Jin 1,* and Shunde Yin 2

1 State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China 2 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada * Correspondence: [email protected]

Received: 21 June 2019; Accepted: 6 September 2019; Published: 10 September 2019

Abstract: The poroelastodynamic failure of a wellbore due to periodic loading during drilling is an unsolved problem. The conventional poroelastic method to calculate the stress distribution around wellbore is for static loading cases and cannot be used for short-time dynamic-loading cases which result in wave propagation in the formation. This paper formulates a poroelastodynamic model to characterize dynamic stress and pressure wave due to periodic loadings and to analyze the transient failure of the suddenly drilled wellbore in a non-hydrostatic stress field. The fully coupled poroelastodynamic model was developed based on the equations of motion, fluid flow and constitutive equations to reflect stress and pressure waves that resulted from a periodic stress perturbation at the wellbore surface. The model was analytically solved by means of field expansions of the solutions, by performing a Laplace transform as well as some special techniques. Simulation results show that the pressure and stress responses inside the formation resemble a damped oscillator where the amplitude decays as the distance to wellbore increases. Especially the potential shear failure zone around the wellbore was computed and plotted. Influences of poroelastic parameters, in-situ stress and periodic load parameters on the shear failure responses were analyzed in a detailed parametric study, and the results provide fundamental insights into wellbore stability maintenance in different reservoirs.

Keywords: periodic stress; wellbore instability; poroelastodynamics; shear failure

1. Introduction Wellbore instability can cause many substantial problems during drilling, such as stuck pipe and lost circulation. Overall, problems related to wellbore collapse account for 5–10% of drilling costs in exploration and production, which are hundreds of million dollars per year, when considering time loss and the rental of drilling equipment [1]. Drilling a wellbore in a formation results in the redistribution of the stress around the borehole. The maximum compressive stress usually occurs as the hoop stress at the wellbore surface in the direction of the minimum principal stress. When the shear stress exceeds the shear strength of the rock near the wellbore, shear failure happens. If the failed rock exceeds the cleaning capacity of the mud circulation, the excessive failed rock can cause stuck drill pipe or bottom-hole assembly. Sometimes, wellbore collapse can result in sidetracked holes and abandoned wells, which means a great loss in oil and gas industry. Therefore, wellbore stability analysis is vital to safe and eﬀective drilling and production. Wellbore stability analysis requires the information on stress distribution around the wellbore. Study on stress distribution around a wellbore can be traced back to the Kirsch solution [2], which is used to calculate the stress distribution near a circular hole in an inﬁnite plate. Based on Kirsch’s

Energies 2019, 12, 3486; doi:10.3390/en12183486 www.mdpi.com/journal/energies Energies 2019, 12, 3486 2 of 21 work, Bradley [3] presented an analytical solution of wellbore stress distribution. This solution is used as the standard approach to obtain stress distribution by researchers [1,4,5]. However, it can only be used for static loading analyses. Several commonly used failure criteria for wellbore collapse have also been developed, such as Mohr-Coulomb failure criterion, Modified Lade failure criterion, von Mises criterion and Drucker-Prager failure criterion [1]. In fact, loadings inside the wellbore are more often dynamic than static. The periodic loads applied at the borehole surface during drilling can generate stress and pressure waves. In this regard, the inertial effect of the solid-fluid system has to be taken into account [6]. Detournay and Cheng [7] proposed an analytical method for calculation of wellbore stress distribution considering poroelastic effect and studied the failure initiation on the basis of stress distribution. Zamanipour et al. [8] investigated the wellbore stress distribution and wellbore stability when transient surge stress is generated in a directional wellbore during tripping process. Zhang et al. [9] conducted wellbore stability analysis by coupling transient swab/surge model and poromechanical solution during tripping and reaming processes and proposed a work flow to obtain safe mud-weight window. Meng et al. [10] developed a coupled poroelastodynamic model to obtain dynamic stress distribution around the wellbore. One of the salient features of the poroelastic response is the generation and dissipation of excess pore water pressure under applied loadings, and wave propagation is induced from various external disturbances such as periodic surge/swab pressure during drilling [11,12]. Biot developed the theory of wave propagation in poroelastic materials by adding the inertia terms to his three-dimensional consolidation theory [13–15]. Over the past decades, Biot’s theory of poroelastodynamics has been used by many researchers and the scientific groundwork for the model has been more firmly established through some experimental validations of its most fundamental predictions [16–18]. A detailed review on formulation and development of Biot’s theory of poroelastodynamics with some analytical solutions can be found in [19]. An important class of problems encountered in dynamic response of using poroelastodynamics is related to the study of wave propagation from an underground borehole. However, some studies on the poroelastodynamics are limited to axisymmetric cases, thus, the problems can be simplified to one dimension (1D) [20–22]. In reality, the original formation is subjected to a non-hydrostatic stress field and the stress release at the wellbore surface after drilling includes radial and shear stress. The early 1D simplification is incapable of analyzing the shear stress wave propagation in the formation so that it limits the application of the poroelastodynamic model in the engineering field. Two-dimensional (2D) problems are also investigated by a few researchers in recent years. Liu et al. [23] theoretically investigated the scattering of an elastic wave by a cylindrical shell embedded in a poroelastic medium. They employed the normal mode expansion technique for analyzing the scattering field. Hasheminejad and Kazemirad [24] studied the dynamic response of a permeable circular tunnel lining of circumferentially varying wall thickness buried in an unbounded porous elastic fluid-saturated formation. Particularly, they studied the two-dimensional dynamic interaction of monochromatic progressive plane compressional and shear seismic waves. They used the Helmholtz decomposition theorem to resolve the displacement fields as a superposition of longitudinal and transverse vector components. Xia et al. [25] presented an exact closed-form solution for poroelastodynamic response of a borehole in a non-hydrostatic stress field and the solution was decomposed into an axisymmetric mode and an asymmetric mode. However, they considered a constant pressure at the wellbore surface and the shear failure response was not analyzed. More recently, Senjuntichai et al. [26] used poroelastodynamic model to investigate three-dimensional (3D) dynamic response of a multilayered poroelastic medium subjected to time-harmonic loading in Cartesian coordinate system. Keawsawasvong and Senjuntichai [27] presented dynamic fundamental solutions of a transversely isotropic poroelastic half-plane subjected to time harmonic buried loads and fluid sources. In drilling engineering, a periodic load during drilling is a common occurrence. Although the equations of poroelastodynamics were put forward by Biot very early, it is still challenging to study the transient responses of a wellbore subjected to Energies 2019, 12, x FOR PEER REVIEW 3 of 21

Energies 2019, 12, 3486 3 of 21 subjected to a non-hydrostatic in-situ stress field and a periodic load during drilling. The problem becomes the superposition of an axisymmetric mode and an asymmetric mode, and the periodic aboundary non-hydrostatic condition in-situ further stress amplifies field and the a complexity periodic load of the during problem. drilling. To the The best problem of our becomes knowledge, the superpositionthis challenge of has an not axisymmetric been addressed mode directly. and an asymmetric mode, and the periodic boundary condition furtherIn amplifiesthis paper, the we complexity consider the of theporoelastodynamic problem. To the bestfailure of ourof a knowledge, wellbore due this to challenge periodic hasloading not beenduring addressed drilling; directly.the wellbore is embedded into a formation that is characterized by non-hydrostatic horizontalIn this far paper,-field we stress consideres. The the poroelastodynamic poroelastodynamic model failure is develop of a wellboreed based due on to Biot’s periodic theory loading, and duringwe present drilling; a novel the wellboresolution to is embeddedthe model. intoThe astress formation and pressure that is characterized waves in the byformation non-hydrostatic resulted horizontalfrom a periodic far-field stress stresses. perturbation The poroelastodynamic at the wellbore model surface is developed are analyzed based in on detail, Biot’s and theory, the andresults we presentshow that a novel the solution pressure to and the stress model. responses The stress inside and pressure the formation waves in resemble the formation a damped resulted oscillator from a periodicwhere the stress amplitude perturbation decays at the as wellborethe distance surface to wellbore are analyzed increases. in detail, It andis noteworthy the results show that the that pore the pressurepressure andreaches stress its responses peak at insidea certain the time formation and location, resemble while a damped the conventional oscillator where poroelastic the amplitude theory decaysalways as fails the distance to discover to wellbore this phenomenon increases. It is due noteworthy to the that lack the of pore consideration pressure reaches of solid its- peakfluid atacceleration. a certain time The and shear location, failure while response the conventionalhas also been poroelastic studied. Finally, theory the always influences fails to of discover poroelastic this phenomenonparameters, in due-situ to stress the lack and of considerationperiodic load ofparameters solid-fluid on acceleration. the shear failure The shear responses failure responseare analyzed has alsoin a beendetailed studied. parametric Finally, study. the influences The model of and poroelastic results presented parameters, in this in-situ paper stress provide and periodic fundamental load parametersinsights into on wellbore the shear instability failure responses in different are reservoirs analyzed, inand a detailedsuggestions parametric are made study. on maintaining The model andwellbore results stabi presentedlity. in this paper provide fundamental insights into wellbore instability in different reservoirs, and suggestions are made on maintaining wellbore stability. 2. Mathematical Formulation 2. Mathematical Formulation To find the governing equations for transient response of a wellbore subjected to dynamic loadings,To find a thefundamental governing knowledge equations for of transientporoelastodynamics response of a is wellbore required. subjected In this tosection, dynamic we loadings,consider aa fundamental vertical borehole knowledge drilled ofin poroelastodynamics a poroelastic formation is required. characterized In this section,by non-hydrostatic we consider horizontal a vertical boreholefar-field drilledstresses in, and a poroelastic a periodic formation load is characterizedapplied at the by borehole non-hydrostatic surface, horizontalas shown far-fieldin Figure stresses, 1. The anddefinitions a periodic of all load the is symbols applied that at the have borehole physical surface, meanings as shown are given in Figure in Table1. The A1 definitions in Appendix of all A. the yy symbols and  xx that are have the maximum physical meaningsand minimum are given horizontal in Table in-situ A1 stresses,in Appendix respectively.A. σyy andIt shouldσxx are be thenoted maximum that the and directions minimum of maximum horizontal and in-situ minimum stresses, horizontal respectively. in-situ It should stresses be in noted the figures that the of directionsstress or p ofore maximum pressure and distribution minimum and horizontal figures in-situof shear stresses failure in zone the figures distribution of stress are or in pore accordance pressure distributionwith Figure and1. figures of shear failure zone distribution are in accordance with Figure1.

FigureFigure 1. 1.Borehole Borehole in in a non-hydrostatic a non-hydrostatic stress stress ﬁeld field (from (from Xia et Xia al. [25 et ]), al. the [25] load), the applied load at applied the wellbore at the surfacewellbore is surface periodic. is periodic.

2.1.2.1. ConstitutiveConstitutive andand DynamicDynamic EquationsEquations AssumingAssuming the the rock rock to be to a belinear-elastic, a linear-elastic, isotropic isotropic and saturated and saturated porous medium porous and medium considering and theconsidering eﬀect of deformationthe effect of ondeformation the balance on of the mass, balance the constitutive of mass, the equations constitutive for linearequations poroelasticity, for linear couldporoelasticity, be written could as follows be written (note as that follows tension (note is here that takentension as is positive) here taken [7]: as positive) [7]:

σij==22µ ee ij+ +λ e eδ  ij −α p pδ  ij ((1)1) ij ij ij − ij pp==− M((ζαe) e) ((2)2) −

Energies 2019, 12, 3486 4 of 21

where σij and eij are the total stress and strain components, respectively; e is the volumetric strain; p is the excess pore water pressure; δij is the Kronecker delta, which equals 1 when i = j or equals 0 when i , j; λ and µ are the Lame constants and µ is also called shear modulus; ζ is the increment of pore fluid per unit volume; α and M are the Biot effective stress coefficient and Biot modulus, respectively, which have the relations α = 1 K/Ks and 1/M = φ/K + (1 φ)/Ks, where Ks and K are the bulk − f − f modulus of solid grains and pore fluid, respectively; K = λ + 2µ/3 is the drained bulk modulus of the porous medium. Compared to underground in-situ stress and pore pressure, body forces are negligible in the case studied. In the absence of body forces, the dynamic equations of motion can be written as [25]: 2 2 ∂σrr 1 ∂σrθ σrr σθθ ∂ ur ∂ wr + + − = ρ + ρ f (3) ∂r r ∂θ r ∂t2 ∂t2 2 2 1 ∂σθθ ∂σrθ 2σrθ ∂ uθ ∂ wθ + + = ρ + ρ f (4) r ∂θ ∂r r ∂t2 ∂t2 where ρ and ρ f are density of porous medium and fluid, respectively; t is time; r is radius defined in Figure1; ur, uθ, wr, wθ are the displacement components of solid and fluid, respectively; σrr, σθθ, σrθ are the radial stress, hoop stress and shear stress, respectively. The subscripts r and θ refer to the radial and azimuthal components, respectively. Substituting Equation (1) and Equation (2) into Equation (3) and Equation (4), we have the governing equations in displacement form in polar coordinates: ! 2 µ 2∂uθ ∂e ∂p .. .. µ ur + ur + (λ + µ) α = ρur + ρ wr (5) ∇ − r2 ∂θ ∂r − ∂r f ! µ 2∂u ∂e ∂p .. .. µ 2u u r + (λ + µ) α = ρu + ρ w (6) ∇ θ − r2 θ − ∂θ r∂θ − r∂θ θ f θ It is noted that two “ ” on top of a variable means the second derivative of the variable with respect · to time and similarly, one “ ” on top means the first derivative of the variable with respect to time. · 2.2. Fluid Flow Equation The rates of total changes of the porous medium with respect to time satisfy the following relationship: 1 ∂V 1 ∂V 1 ∂Vp ∂e = s + = (7) V ∂t V ∂t V ∂t ∂t where V, Vs, Vp are the volumes of the porous medium, solid grains and pores, respectively. The changes of volume of soil grains and pores are given by:

1 ∂V 1 φ ∂p 1 ∂σ s = − + m 0 (8) V ∂t − Ks ∂t 3Ks ∂t

1 ∂Vp φ ∂p = qw (9) V ∂t −∇ − K f ∂t where m = [1 1 1 0 0 0] and σ = σ αpδ denotes the effective stress vector of the porous medium; φ is 0 ij − ij the porosity of the porous medium; qw is the volumetric flux of pore fluid, which is the rate of volumetric flow across a unit area. Substituting the constitutive relation Equation (1) into Equation (8) gives:

1 ∂V 1 φ ∂p K ∂e s = − + (10) V ∂t − Ks ∂t Ks ∂t

Substituting Equations (8)–(10) into Equation (7) yields:

∂e 1 ∂p qw = α (11) ∇ − ∂t − M ∂t Energies 2019, 12, 3486 5 of 21

. Since we have qw = w, we can obtain the following formulation: ∇ ∇ · . . . p = M ζ αe (12) − In ﬂuid dynamics, the water ﬂux is given by [14]: ! κ .. aρ f .. qw = p + ρ u + w (13) − τ ∇ f φ

Substituting Equation (13) into Equation (11) yields:

κ ∂e 1 ∂p κρ f .. aρ f κ .. 2p = α + u w (14) τ ∇ ∂t M ∂t − τ ∇ − τφ ∇ where κ and τ are the permeability of the porous medium and viscosity of the ﬂuid, respectively. a is a non-dimensional tortuosity factor with (ρa + φρ f )/φ = aρ f , while ρa is the mass density of the ﬂuid added into the porous medium and ρa = cφρ f [20]. Combining Equation (2) and Equation (14) we have: τ . .. aρ f .. p = w ρ u w (15) ∇ −κ − f − φ

2.3. Dimensionless Governing Equations With the help of Equation (5), Equation (6) and Equation (15), the governing equations in the absence of the body force read:

.. ρ f .. (λ + 2µ) 2e α 2p = ρ ρ α e p (16) ∇ − ∇ − f − M ! κ ∂e 1 ∂p κρ f aα .. κρ f a .. 2p = α + + 1 e + p (17) τ ∇ ∂t M ∂t τ φ − τφM In order to simplify the expressions, the physical quantities are non-dimensionalized as follows: s r u 1 λ + 2µ α 1 r = , u = , t = t, p = p, σij = σij (18) rw rw rw ρ λ + 2µ λ + 2µ where rw is the radius of the borehole. It is noted that symbols with “–” on top are dimensionless parameters or quantities. The dimensionless form of Equation (16) and Equation (17) reads:

2 2 .. .. e p = Φ e + Φ p (19) ∇ − ∇ 1 2 2 ...... p = Ω e + Ω p + Φ e + Φ p (20) ∇ 1 2 3 4 where the dimensionless quantities are listed as follows:

2 ∂2 1 ∂ 1 ∂2 = + + (21) ∇ ∂r2 r ∂r r2 ∂θ2

2 p α τrw τrw λ + 2µ Ω1 = p , Ω2 = (22) κ ρ(λ + 2µ) Mκ √ρ

ρ αρ f ρ f (λ + 2µ) ρ f α(aα φ) ρ f a(λ + 2µ) Φ = − , Φ = , Φ = − , Φ = (23) 1 ρ 2 − ραM 3 ρφ 4 φMρ Energies 2019, 12, 3486 6 of 21

3. Solution Strategy

3.1. Field Expansions As mentioned before, we consider a vertical wellbore drilled in a formation characterized by a non-hydrostatic horizontal in-situ stress field, as shown in Figure1. The problem can be considered as an asymmetric plane strain problem in polar coordinates. The original stress and pressure in polar coordinates are: σ0 = P S cos 2θ (24) rr − 0 − 0 σ0 = P + S cos 2θ (25) θθ − 0 0 σ0 = S sin 2θ (26) rθ − 0 0 p = p0 (27) where P0 and S0 are the far-field mean and deviatoric parts of the stresses; p0 is the unperturbed pore pressure; θ is the angle defined in Figure1. The superscript 0 represents the original value. With the consideration of symmetry, as used in the poroelastic theory [28–30], the field expansions of the solutions should be sought to have the following form according to the far-field stresses: = (0) (0) (0) (0) (0) + (2) (2) (2) (2) (2) e, ur, σrr, σθθ, p e , ur , σrr , σθθ , p e , ur , σrr , σθθ , p cos 2θ (28) = (2) (2) uθ, σrθ uθ , σrθ sin 2θ (29) where the superscript (0) represents the axisymmetric mode solution, while (2) represents the asymmetric mode solution resulting from the far-field deviatoric part of the stress.

3.2. Solutions in the Laplace Domain The Laplace transform technique is used to solve the governing equations. By applying the Laplace transformation to Equation (19) and Equation (20), we obtain the transformed governing equations:   " # 2  " 2 2 #  1 1  een  s Φ1 s Φ2  een  −  ∇n  =   (30)  2  2 2  e  0 1  ep  sΩ1 + s Φ3 sΩ2 + s Φ4 pn ∇n n In each mode n(n = 0, 2), the transformed variables satisfy the dimensionless governing equations. Equation (30) can be decoupled and solved:

een = AnKn(β1r) + BnKn(β2r) (31)

e pn = χ1AnKn(β1r) + χ2BnKn(β2r) (32) where: q 2 ξ1 ξ1 4ξ2 β = ± − (33) i 2 2 2 βi s Φ1 χi = − (34) 2 + 2 βi s Φ2 2 ξ1 = s Ω1 + Ω2 + s Φ1 + Φ3 + Φ4 (35) ξ = s3 Φ Ω + Φ Φ s Φ Ω Φ Φ s (36) 2 1 2 1 4 − 2 1 − 2 3 Energies 2019, 12, 3486 7 of 21

An and Bn are parameters depending only on s and can be determined by boundary conditions. Combining Equations (5), (6) and (15) and performing non-dimensionalization and Laplace transformation result in the following coupled equations:

e 2 1 ∂een ∂pn 2 eurn 2neuθn + eurn + Ψ1 + Ψ2 = Ψ3eurn (37) ∇ − r2 ∂r ∂r

e 2 1 neen npn 2 euθn + euθn + 2neurn + Ψ1 + Ψ2 = Ψ3euθn (38) − ∇ r2 r r − where:

2 2 4 λ + µ 1 Φ1 (λ + 2µ)s λ + 2µ 2 λ + 2µ 1 Φ1 (λ + 2µ)s Ψ = , Ψ = − , Ψ = s2 − (39) 1 µ 2 µαη − µ 3 µ − µαη

Ω s Ω Φ s2 η = 1 + 1 4 (40) α αΩ2 A technique is used to eliminate the displacement curl that the displacement components can be mapped into the following functions [29]:

Πn = eurn eu (41) − θn

Λn = eurn + euθn (42) After some algebraic manipulations involving Equations (31), (32), (37) and (38), we can obtain the governing equations of the modiﬁed displacement functions Πn and Λn:

2 2 n 1 Ψ3 Πn = ∆1β1AnKn 1(β1r) + ∆2β2BnKn 1(β2r) (43) ∇ − − − −

2 2 Ψ Λn = ∆ β AnK + (β r) + ∆ β BnK + (β r) (44) ∇n+1 − 3 1 1 n 1 1 2 2 n 1 2 where: 2 2 β Ψ3 ∆ = Ψ + Ψ χ = i − (45) i 1 2 i − 2 βi Through the solution of ordinary diﬀerential equations, Equations (43) and (44), which involves some manipulations, we can obtain the general expressions for the displacement components in the Laplace transform domain, and the dynamic stress distribution ﬁnally yields: ( ) n λ λn2 eσrr = An [Kn 1(β1r) + Kn+1(β1r)] + 2 Kn(β1r) + (1 χ1)Kn(β1r) 2β1r − β2r − ( 1 ) λ λn2 (46) +Bn [Kn 1(β2r) + Kn+1(β2r)] + 2 Kn(β2r) + (1 χ2)Kn(β2r) 2β2r 2 − β2r − h i nλ nΨ3λ +Cn 2 Kn Ψ3r + Kn 1 Ψ3r + Kn+1 Ψ3r r 2r −

n e λ σθθ = An 4 [Kn 2(β1r) + 2Kn(β1r) + Kn+2(β1r)] + (1 χ1)Kn(β1r) − − − λ +Bn [Kn 2(β2r) + 2Kn(β2r) + Kn+2(β2r)] + (1 χ2)Kn(β2r) (47) − 4 − − h i nΨ3 nλ +Cn Kn 1 Ψ3r + Kn+1 Ψ3r 2 Kn Ψ3r − 2r − − r Energies 2019, 12, 3486 8 of 21

( ) n nλ λn eσrθ = An [Kn 1(β1r) + Kn+1(β1r)] + 2 Kn(β1r) 2β1r β2r ( − 1 ) nλ λn +Bn [Kn 1(β2r) + Kn+1(β2r)] + 2 Kn(β2r) 2β2r − β2r (48)  h 2 i   n2λ Ψ3λ   2 Kn Ψ3r + Kn 1 Ψ3r + Kn+1 Ψ3r   2r 4r −  +Cn 2 h i   Ψ3λ   + Kn 2 Ψ3r + 2Kn Ψ3r + Kn+2 Ψ3r  8 − 3.3. Boundary Conditions The analytical solutions presented above, together with the appropriate boundary conditions at the wellbore, can be used to analyze the transient failure responses. We consider the drilling of a vertical borehole by a bit generates periodic loads at the bottom hole (the Sinusoidal and Cosine loads are considered in this paper). In the axisymmetric mode (n = 0), C0 = 0 and the remaining two constants A0 and B0 can be computed by the radial stress and pore pressure boundary conditions. If we assume a Sinusoidal oscillation of the bottom-hole pressure around a constant value pw with an amplitude 2L and an angular frequency ω (as shown in Figure2), then the stress and pore pressure boundary conditions in the axisymmetric mode (n = 0) can be written in the Laplace domain:

0 1 ωL eσ (1, s) = P0 p (49) rr s − w − s2 + ω2

1 ωL ep (1, s) = p p + (50) 0 s w − 0 s2 + ω2 Energies 2019, 12, x FOR PEER REVIEW 9 of 21

FigureFigure 2. 2. ExampleExample of of a a sinusoidal sinusoidal bottom bottom-hole-hole pressure. pressure.

InIn the the asymmetric modemode( n(n==22),) the, the stress stress release release at theat the wellbore wellbore results results from from the deviatoric the deviatoric parts partsof the of far-field the far stress.-field stress. After drilling, After drilling the rotational, the rotational waves are waves coupled are coupledwith compressional with compressional waves to propagate into the formation. The three constants A2, B2, C2 should be determined by the trigonometric waves to propagate into the formation. The three constants A2 , B2 , C2 should be determined by part of radial stress, hoop stress and pore pressure applied at the wellbore surface after removing the the trigonometric part of radial stress, hoop stress and pore pressure applied at the wellbore surface far-field values, which can be expressed in the Laplace domain as: after removing the far-field values, which can be expressed in the Laplace domain as:

2 2 SS00 eσrr((1,1, ss))==− (51(51)) rr − ss

2 S e 2 S00 σrθ((1,1, ss)) = (52(52)) r ss 2 ep 2( s) = ps(1,1,) = 0 (53 (53)) These boundary conditions must be used for closed solutions. These boundary conditions must be used for closed solutions.

3.4. Numerical Method The solutions presented in the Laplace domain should be carefully inverted to the real time domain. Since the solutions reflect the wave propagation phenomenon, the method of Crump is employed which uses a truncated Fourier series to approximate the complex inversion integral. This method serves a more accurate algorithm particularly suitable for transient and highly oscillatory problems [31]. The method of Crump approximates the inversion integral using the following equation [25]:

at n ek1  k f( t) = F( a) +Re F a + i ( − 1) (54) tt2 k =1  where i =−1 , and the parameters a and n must be optimized, and generally we can set at = 4.5 and n =100 to have a good accuracy [25]. Figure 3 shows the modeling methodology with user-defined input parameters for a better understanding of the method proposed in this work.

Figure 3. Modeling methodology.

4. Stress Distribution and Failure Responses

Energies 2019, 12, x FOR PEER REVIEW 9 of 21

Figure 2. Example of a sinusoidal bottom-hole pressure.

In the asymmetric mode (n = 2) , the stress release at the wellbore results from the deviatoric parts of the far-ﬁeld stress. After drilling, the rotational waves are coupled with compressional

waves to propagate into the formation. The three constants A2 , B2 , C2 should be determined by the trigonometric part of radial stress, hoop stress and pore pressure applied at the wellbore surface after removing the far-ﬁeld values, which can be expressed in the Laplace domain as: S  2 (1, s) =− 0 (51) rr s S  2 (1, s) = 0 (52) r s

2 Energies 2019, 12, 3486 ps(1,) = 0 (539) of 21

These boundary conditions must be used for closed solutions. 3.4. Numerical Method 3.4. Numerical Method The solutions presented in the Laplace domain should be carefully inverted to the real time domain. The solutions presented in the Laplace domain should be carefully inverted to the real time Since the solutions reﬂect the wave propagation phenomenon, the method of Crump is employed domain. Since the solutions reflect the wave propagation phenomenon, the method of Crump is which uses a truncated Fourier series to approximate the complex inversion integral. This method employed which uses a truncated Fourier series to approximate the complex inversion integral. serves a more accurate algorithm particularly suitable for transient and highly oscillatory problems [31]. This method serves a more accurate algorithm particularly suitable for transient and highly Theoscillatory method of problems Crump approximates [31]. The method the inversion of Crump integral approximates using the the following inversion equation integral [using25]: the following equation [25]:   at  Xn !  e 1 kπ k f (t) = at  F(a) + Re n F a + i ( 1)  (54) ekt 12 t − k f( t) = F( a) +Rek=1 F a + i ( − 1) (54) tt2 k =1  where i = √ 1, and the parameters a and n must be optimized, and generally we can set at = 4.5 and where i =−− 1 , and the parameters a and n must be optimized, and generally we can set n = 100 to have a good accuracy [25]. Figure3 shows the modeling methodology with user-deﬁned at = 4.5 and n =100 to have a good accuracy [25]. Figure 3 shows the modeling methodology input parameters for a better understanding of the method proposed in this work. with user-defined input parameters for a better understanding of the method proposed in this work.

FigureFigure 3.3. Modeling methodology. methodology.

4. Stress4. Stress Distribution Distribution and and Failure Failure ResponsesResponses

In all the computations and plots, the stress components and pressure are scaled by the original pore pressure value. The values of the used dimensional parameters are listed in Table1. These values are selected from a gas well in Keshen block in Tarim oil ﬁeld.

Table 1. Parameter sets used in the calculation.

Parameter Value Parameter Value Shear modulus µ [GPa] 10 Lame constant λ [GPa] 7 Biot coefficient α 0.6 Biot modulus M [GPa] 10 3 3 Pore fluid density ρ [kg m− ] 300 Rock density ρ [kg m ] 2800 f · · − Rock permeability κ [mD] 0.8 Pore fluid viscosity τ [Pa s] 3 10 5 · × − Tortuosity factor a 1.6 Rock porosity φ 0.06

Maximum principal stress σH [MPa] 130 Minimum principal stress σh [MPa] 100

Initial pore pressure p0 [MPa] 50 Wellbore radius rw [m] 0.1

We assume a Sinusoidal oscillation of the bottom-hole pressure that holds: ph = pw + L sin ωt (55) where the dimensional bottom-hole pressure is 50 MPa and L = 0.2pw, ω = 10.

4.1. Model Verification Since the existing models and poroelastodynamic solutions in the literature cannot be directly used for our problem, we first considered a simplified case where the bottom-hole pressure is constant Energies 2019, 12, x FOR PEER REVIEW 10 of 21

In all the computations and plots, the stress components and pressure are scaled by the original pore pressure value. The values of the used dimensional parameters are listed in Table 1. These values are selected from a gas well in Keshen block in Tarim oil field.

Table 1. Parameter sets used in the calculation.

Parameter Value Parameter Value Shear modulus  [GPa] 10 Lame constant  [GPa] 7 Biot coefficient  0.6 Biot modulus M [GPa] 10 −3 −3 Pore fluid density  f [kg·m ] 300 Rock density  [kg·m ] 2800 Rock permeability  [mD] 0.8 Pore fluid viscosity  [Pa·s] 3 × 10−5 Tortuosity factor a 1.6 Rock porosity  0.06

Maximum principal stress  H [MPa] 130 Minimum principal stress  h [MPa] 100

Initial pore pressure p0 [MPa] 50 Wellbore radius rw [m] 0.1

We assume a Sinusoidal oscillation of the bottom-hole pressure that holds:

phw=+ p Lsin( t ) (55) where the dimensional bottom-hole pressure is 50 MPa and Lp= 0.2 w , =10 .

4.1. Model Verification EnergiesSince2019 ,the12, 3486existing models and poroelastodynamic solutions in the literature cannot be directly10 of 21 used for our problem, we first considered a simplified case where the bottom-hole pressure is andconstant the two and principal the two far-field principal stresses far-field are stresses the same, are and the thus, same, the and problem thus, becomes the problem axisymmetric. becomes Senjuntichaiaxisymmetric. and Senjun Rajapaksetichai [ 32 and] gave Rajapakse the axisymmetric [32] gave solutions the axisymmetric under constant solutions fluid pressure under constant that can befluid used pressure in this simplifiedthat can be case. used In in our this model, simplified we considered case. In theour complex model, drillingwe consider conditionsed the involving complex didrillingfferential conditions stress and involving periodic differential loads and stress used aand diff periodicerent approach loads and to solveused a the different problem. approach The final to solutionsolve the isproblem. the superposition The final solution of an axisymmetric is the superposition mode ( nof= an0 )axisymmetricand an asymmetric mode ( moden = 0)( nand= 2an), andasymmetric the two mode modes ( haven = 2) the, and same the formtwo modes of the solution.have the Tosame verify form our of solution,the solution. we comparedTo verify our the axisymmetricsolution, we modecompare(n d= the0) with axisymmetric the solution mode given by Senjuntichai with the and solution Rajapakse given [32 ] by using Senjuntichai the input parametersand Rajapakse defined [32] using in Table the1 .input The resultsparameters are shown defined in in Figures Table 41. and The5 resultsfor the are pressure shown and in Figure radial 4 stress,and Figure respectively, 5 for which the pressure demonstrates and radial that the stress axisymmetric, respectively, mode whichsolution demonstrates obtained in this that paper the agreesaxisymmetric well with mode the published solution obtained solution in[32 this] when paper the agrees pressure well is prescribed with the published at the wellbore solution surface, [32] althoughwhen the thepressure solution is prescribed strategy is diatff theerent. wellbore This comparison surface, although shows thethe validitysolution and strategy the robustness is different. of theThis model comparison and solution. shows the validity and the robustness of the model and solution.

Energies 2019, 12, x FOR PEER REVIEW 11 of 21

Figure 4. Comparison of pore pressure distribution using the simplified simpliﬁed axisymmetric solution with constantconstant bottom bottom-hole-hole pressure.

Figure 5. ComparisonComparison of of radial stress distribution using the simplified simpliﬁed axisymmetric solution with constantconstant bottom bottom-hole-hole pressure.

4.2. Dynamic Distributions of Stress Components and Pressure We first examined the profiles of stress components with the two-dimensional isobaric plots for the solutions. Figures 6–8 show the dynamic distribution of stress components in the vicinity of the wellbore. The values in all the figures in this paper are dimensionless. It can be seen from the results that the stress components are all characterized by wave-propagation properties during the drilling process, and the stress perturbation applied at the wellbore surface propagates into the formation. At an early time after drilling ( t = 0.2 and t = 0.5 in Figures 6–8), a disturbance zone is observed near the wellbore and expands as time increases. The stress components appear to fluctuate in the disturbance zone. The stress outside the disturbance zone keeps the original state and the disturbance boundary propagates into the formation. The propagation velocity of the disturbance boundary equals the propagation velocity of the stress wave. It can be clearly seen from the plots that the wave velocities of radial and hoop stress are larger than that of the shear stress. Figure 9 shows the dynamic pore pressure distribution in the vicinity of the wellbore, which also exhibits fluctuations in the disturbance zone. Due to the sudden release of stress at the wellbore surface, the pore pressure near the wellbore decreases significantly at the beginning of drilling, even to a negative value. Then, due to the existence of the differential stress, the pore pressure increases in the direction of the minimum principal stress and gradually reaches a much higher value compared to the initial pore pressure.

Figure 6. Radial stress distribution in the vicinity of the wellbore at diﬀerent times, from left to right: t = 0.2 , t = 0.5 , t =1, t = 2 .

Energies 2019, 12, x FOR PEER REVIEW 11 of 21

Figure 4. Comparison of pore pressure distribution using the simplified axisymmetric solution with constant bottom-hole pressure.

Energies 2019, 12, 3486 11 of 21 Figure 5. Comparison of radial stress distribution using the simplified axisymmetric solution with constant bottom-hole pressure. 4.2. Dynamic Distributions of Stress Components and Pressure 4.2. Dynamic Distributions of Stress Components and Pressure We first examined the profiles of stress components with the two-dimensional isobaric plots for We first examined the profiles of stress components with the two-dimensional isobaric plots thefor solutions. the solutions. Figures Figures6–8 show 6–8 show the dynamic the dynamic distribution distribution of stress of stress components components in in the the vicinity vicinity of of the wellbore.the wellbore. The values The values in all thein all figures the figures in this in paper this paper are dimensionless. are dimensionless. It can Itbe can seen be seen from from the results the thatresults the stress that the components stress components are all characterized are all characterized by wave-propagation by wave-propagation properties properties during during the drilling the process,drilling and process, the stress and perturbationthe stress perturbation applied at applied the wellbore at the surfacewellbore propagates surface propagates into the formation.into the Atformation. an early time At an after early drilling time after (t = drilling0.2 and ( tt ==0.20.5 andin Figures t = 0.56 –in8), Figures a disturbance 6–8), a disturbance zone is observed zone nearis observedthe wellbore near and the expands wellbore as and time expands increases. as time The increases. stress components The stressappear components to fluctuate appear in to the disturbancefluctuate in zone. the disturbance The stress outside zone. The the disturbancestress outside zone the keepsdisturbance the original zone keeps state andthe theoriginal disturbance state boundaryand the propagates disturbance into boundary the formation. propagates The propagation into the formation. velocity ofThe the propagation disturbance velocity boundary of equals the thedisturbance propagation boundary velocity equals of the stressthe propagation wave. It can velocity be clearly of the seen stress from wave. the plotsIt can that be clearly the wave seen velocities from ofthe radial plots and that hoop the stresswave velocities are larger of than radial that and of hoop the shear stress stress. are larger Figure than9 shows that of thethe dynamicshear stress. pore pressureFigure distribution9 shows the indynamic the vicinity pore ofpressure the wellbore, distribution which in also the exhibits vicinity fluctuations of the wellbore, in the which disturbance also zone.exhibits Due fluctuationsto the sudden in releasethe disturbance of stress at zone. the wellboreDue to the surface, sudden the release pore pressure of stress nearat the the wellbore wellbore decreasessurface, significantly the pore pressure at the beginning near the wellbore of drilling, decreases even to signia negativeficantly value. at the Then, beginning due to of the drilling existence, ofeven the di tofferential a negative stress, value. the poreThen, pressure due to the increases existence in the of the direction differential of the stress, minimum the pore principal pressure stress increases in the direction of the minimum principal stress and gradually reaches a much higher and gradually reaches a much higher value compared to the initial pore pressure. value compared to the initial pore pressure.

FigureFigure 6. 6.Radial Radial stress stress distribution distribution inin thethe vicinity of the the wellbore wellbore at at diﬀerent diﬀerent times, times, from from left left to toright: right: t = 0.2 , t = 0.5 , t =1, t = 2 . Energiest = 20190.2,,, 12t ,,= xx FOR0.5, tPEER= 1, REVIEWt = 2. 12 of 21

Figure 7. HoopHoop stress stress distribution distribution in in the the vicinity vicinity of of the the wellbore wellbore at at diﬀerent diﬀerent times, times, from from left left to to right: right: tt==0.2,0.2t ,,=t 0.5,= 0.5t =,,1,t t==1,,2.t = 2 ..

Figure 8. ShearShear stress stress distribution distribution in in the the vicinity vicinity of of the the wellbore wellbore at at diﬀerent diﬀerent times, times, from from left left to to right: right: t = 0.2, t ,,= 0.5, t =,,1, t = ,,2. ..

Figure 9. Pore pressure distribution in the vicinity of the wellbore at diﬀerent times,times, from from left left to to right: ,, ,, ,, ..

The maximum shear stress occurs in the direction of the minimum principal stress, and hence,, shear failure at the wellbore surface initiates in the direction of the minimum principal stress [5]. In order to further analyze the dynamic evolution of stress and pore pressure in the formation, we selected three locations along the direction of minimum principal stress to analyze the time variation of the stress and pressure. The coordinates of the selected locationslocations are (r ,, ) = ( 1,0 1,0) ,, (2,0) ,,(3,0) ,, respectively.respectively. Figure 10 shows the pressure fluctuation at the selected locationslocations.. TheThe pressure at r =1 isis thethe givengiven boundaryboundary conditioncondition thatthat isis aa sinusoidalsinusoidal wave.wave. InsideInside thethe wellborewellbore (( r = 2 and r = 3 ), the pore pressure decreases at the beginning and then increases to a peak, and finally approaches stable fluctuation. It is interestinginteresting toto findfind thethe fluctuationfluctuation amplitudeamplitude decreasesdecreases asas distance increases. The pore pressure reaches its peak at a certain time and location, while the conventional poroelastic theory always fails to discover this phenomenon due to the lack of consideration of solid-fluid acceleration. Figure 11 shows the variation of radial stress with time at selected locationslocations.. TheThe modelmodel inin thisthis paperpaper assumesassumes compression as negative, thus, the computed stresses are all negative. The radial stress at the wellbore surface equals the given pressure boundary condition. When the stress wave propagates to a certain location, the radial stress increases from the initiinitial value at the beginning and then decreases, and finally approaches to stable fluctuation. Figure 12 shows the hoop stress fluctuation at the selected locationslocations.. BecauseBecause thethe hoophoop stressstress isis notnot givengiven at the wellbore surface and the radial stress is suddenly released, the hoop compressive stress at the wellbore surface increases rapidly after drilling.. The The magnitudes magnitudes of of radial radial stress stress as as well well as as hoop hoop stress decrease as distance increases.

Energies 2019, 12, x FOR PEER REVIEW 12 of 21

Figure 7. Hoop stress distribution in the vicinity of the wellbore at diﬀerent times, from left to right: t = 0.2 , t = 0.5 , t =1, t = 2 .

EnergiesFigure2019, 128. ,Shear 3486 stress distribution in the vicinity of the wellbore at diﬀerent times, from left to right:12 of 21 , , , .

Figure 9. PorePore pressure pressure distribution distribution in in the the vicinity vicinity of ofthe the wellbore wellbore at di atﬀ erent diﬀerent times, times, from from left to left right: to right:t = 0.2, t = 0.5, ,t = 1, t = ,2. , .

The maximum shear shear stress stress occurs occurs in in the the direction of of the minimum principal principal stress, stress, and and hence hence,, shear failure failure at at the the wellbore wellbore surface surface initiates initiates in in the the direction direction of ofthe the minimum minimum principal principal stress stress [5]. [In5]. orderIn order to tofurther further analyze analyze the the dynamic dynamic evolution evolution of of stress stress and and pore pore pressure pressure in in the the formation, we selectselecteded threethree locations locations along along the the direction direction of minimum of minimu principalm principal stress to stress analyze to analyzethe time variationthe time variationof the stress of theand stress pressure. and Thepressure. coordinates The coordinates of the selected of the locations selected are locations(r, θ) = are(1, 0)(,r(,2,) 0=),(( 1,03, 0)),, respectively. Figure 10 shows the pressure fluctuation at the selected locations. The pressure at r = 1 is (2,0) , 3,0 , respectively. Figure 10 shows the pressure fluctuation at the selected locations. The the given( boundary) condition that is a sinusoidal wave. Inside the wellbore (r = 2 and r = 3), the pressurepore pressure at r decreases=1 is the atgiven the beginningboundary andcondition then increases that is a tosinusoidal a peak, and wave. finally Inside approaches the wellbore stable ( fluctuation.r = 2 and r It= is3 interesting), the pore topressu findre the decreases fluctuation at the amplitude beginning decreases and then as increases distance increases.to a peak, and The finallypore pressure approaches reaches stable its peak fluctuation. at a certain It is time interesting and location, to find while the fluctuation the conventional amplitude poroelastic decreases theory as distancealways fails increases. to discover The this pore phenomenon pressure reaches due to its the peak lack atof consideration a certain time of and solid-fluid location, acceleration. while the conventionalFigure 11 shows poroelastic the variation theory of radial always stress fails with to time discover at selected this phenomenon locations. The due model to in the this lack paper of considerationassumes compression of solid- asfluid negative, acceleration. thus, the Figure computed 11 shows stresses the arevariation all negative. of radial The stress radial with stress time at the at selectedwellbore locations surface equals. The model the given in pressurethis paper boundary assumes condition. compression When as thenegative, stress wavethus, propagatesthe computed to a stressescertain location,are all negative. the radial The stress radial increases stress at from the wellbore the initial surface value atequals the beginning the given andpressure then boundary decreases, condition.and finally When approaches the stress to stable wave fluctuation. propagates Figure to a certain 12 shows location, the hoop the radial stress fluctuationstress increases at the from selected the initilocations.al value Because at the beginning the hoop stress and then is not decreases, given at the and wellbore finally approaches surface and to the stable radial fluctuation. stress is suddenly Figure 12released, shows the hoop stress compressive fluctuation stress at atthe the selected wellbore locations surface. Because increases therapidly hoop stress after is drilling. not given The at the wellbore surface and the radial stress is suddenly released, the hoop compressive stress at the Energiesmagnitudes 2019, 12 of, x radialFOR PEER stress REVIEW as well as hoop stress decrease as distance increases. 13 of 21 wellbore surface increases rapidly after drilling. The magnitudes of radial stress as well as hoop stress decrease as distance increases.

FigureFigure 10. 10. TransientTransient response response of of pore pore pressure pressure at different diﬀerent locations.

Figure 11. Transient response of radial stress at different locations.

Figure 12. Transient response of hoop stress at different locations.

4.3. Failure Responses

EnergiesEnergies 20192019,, 1212,, xx FORFOR PEERPEER REVIEWREVIEW 1313 ofof 2121

Energies 2019, 12, 3486 13 of 21 FigureFigure 10.10. TransientTransient responseresponse ofof porepore pressurepressure atat differentdifferent locations.locations.

FigureFigureFigure 11.11. 11. TransientTransientTransient responseresponse ofof radialradial stressstress atat differentdifferent diﬀerent locations. locations.

FigureFigureFigure 12.12. 12. TransientTransientTransient responseresponse ofof hoophoop stressstress atat differentdifferent diﬀerent locations. locations.

4.3.4.3. FailureFailure Failure RR Responsesesponsesesponses We consider shear failure mode with the Mohr Coulomb criteria in this section to analyze the

transient failure response. The failure criteria can be written as [28]:

tan ϕ(σ1 + σ3) + 2Ct F = − σ + σ 0 (56) q − 3 1 ≤ tan2 ϕ + 1

where ϕ is the internal friction angle and Ct s the cohesion; σ1 and σ3 are the maximum and minimum principal stresses, respectively. We assume ϕ = 20◦ and Ct = 10 MPa. We first examine the dynamic profiles of the differential stress, i.e., σ σ at different times, and the results are shown in Figure 13. 1 − 3 The differential stress at the wellbore surface is small at the beginning of drilling, and becomes larger in the direction of minimum principal stress as time increases, which results in a high-risk zone of shear failure. Figure 14 shows the variation of the differential stress with time at three selected locations in the direction of minimum principal stress and the coordinates are (r, θ) = (1, 0), (1.2, 0), (1.5, 0). The differential stress also exhibits fluctuation characteristic at selected locations. However, different from the stress components, the amplitude of the differential stress almost keeps the same as distance Energies 2019, 12, x FOR PEER REVIEW 14 of 21

EnergiesWe consider2019, 12, x FOR shear PEER failure REVIEW mode with the Mohr Coulomb criteria in this section to analyze14 of 21 the transient failure response. The failure criteria can be written as [28]: We consider shear failure mode with the Mohr Coulomb criteria in this section to analyze the

transient failure response. The failure−tan criteria( 13 +can  be) + written 2Ct as [28]: F = − +  0 (56) 2 31 −tantan( 13 + + 1) + 2Ct F = −31 +  0 (56) tan2  + 1 where  is the internal friction angle and Ct s the cohesion; 1 and  3 are the maximum and minimumwhere  principal is the internal stresses, friction respectively. angle and We C assumet s the cohesion;  = 20 and1 andCt =10MPa3 are the. We maximum first examine and theminimum dynamic principal profiles stresses, of the differential respectively. stress, We assume i.e.,  13−= 20 at and different Ct =10MPa times, . andWe firstthe resultsexamine are shownthe dynamic in Figure profiles 13. The of the differential differential stress stress, at the i.e., wellbore13− at surface different is times, small and at thethe beginning results are of drillingshown, and in Figurebecomes 13 . larger The differential in the direction stress ofat minimumthe wellbore principa surfacel stress is small as time at the increases, beginning which of resultsdrilling in ,a and high becomes-risk zone larger of shearin the failure.direction Figure of minimum 14 shows principa the variationl stress as of time the increases,differential which stress withresults time inat athree high selected-risk zone locations of shear in failure. the direction Figure of14 minimumshows the principalvariation stressof the anddifferential the coordinates stress arewith (r , time ) = at( 1,0 three) , (1.2,0 selected) , ( 1.5,0locations) . The in the differential direction of stress minimum also exhibits principal fluctuation stress and the characteristic coordinates at Energies 2019, 12, 3486 14 of 21 selectedare (r locations, ) = ( 1,0. )However,, (1.2,0) , ( 1.5,0different) . The from differential the stress stress components, also exhibits the fluctuationamplitude of characteristic the differential at stressselected almost locations keeps. However, the same different as distance from increases. the stress Itcomponents, is in accordance the amplitude with expectation of the differential that the differentialincreases.stress almost It stress is in keeps accordance decreases the same with with as expectation distance ; increases.thus, that thethe diIt shearff iserential in accordancefailure stress is decreases most with likely expectation with todistance; occur that at the thus, the wellborethedifferential shear surface. failure stress isFigure most decreases likely15 shows with to occur the distance dynamic at the; thus, wellbore shear the failure shear surface. failurezone Figure distribution is most 15 shows likely in the the to dynamic occurvicinity at of shearthe the wellbore.failurewellbore zone The surface. distribution values Figure in the in 1 the5figures shows vicinity ofthe shear ofdynamic the failure wellbore. shear zone failure The distribution values zone distribution in thecorrespond figures in ofthe to shear vicinityF in failure Equation of the zone (56).distributionwellbore. When The correspondF is values greater in to thethanF in figures 0, Equation dynamic of shear (56). shear failure When failure zoneF is occurs greaterdistribution, and than a correspond 0, greater dynamic F to shearvalue F failurein means Equation occurs, more (56). When F is greater than 0, dynamic shear failure occurs, and a greater F value means more severeand a greatershear failure.F value At means the beginning more severe ( t = shear0.2 ), failure.the wellbore At the is beginningstable. As time (t = 0.2increases,), the wellbore the stress is severe shear failure. At the beginning ( t = 0.2 ), the wellbore is stable. As time increases, the stress wavestable. propagates As time increases, into the the formation, stress wave and propagates a symmetrical into the failure formation, zone appears and a symmetrical in the direction failure of zone the wave propagates into the formation, and a symmetrical failure zone appears in the direction of the minimumappears in principal the direction stress, of the and minimum the area principalof the failure stress, zone and gradually the area of increases. the failure In zone the graduallyfollowing minimum principal stress, and the area of the failure zone gradually increases. In the following subsections,increases. In the the influences following of subsections, poroelastic theparameters, influences in of-situ poroelastic stress and parameters, periodic load in-situ parameters stress and on subsections, the influences of poroelastic parameters, in-situ stress and periodic load parameters on the shear failure responses are analyzed in a detailed parametric study. periodicthe shear load failure parameters responses on are the analyz sheared failure in a detailed responses parametric are analyzed study. in a detailed parametric study.

FigureFigure 13. 13. DiDifferential Differentialfferential stressstress stress distributiondistribution distribution in inin the thethe vicinity vicinity of of the thethe wellbore wellborewellbore atat at didifferent differentfferent times,times, times, fromfrom from left left to to right: t = 0.2 , t = 0.5 , t =1, t = 2 . toright: right:t = t0.2,= t0.2= 0.5,, t =t =0.51, t, t==2.1, t = 2 .

Figure 14. Transient response of differential stress at different locations. Energies 2019, 12, x FORFigureFigure PEER 14. 14. REVIEW TransientTransient response response of of differential diﬀerential stress at didifferentﬀerent locations. 15 of 21

Figure 15. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent times, from left Figure 15. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent times, from left to to right: t = 0.2 , t = 0.5 , t = 2 , t =10 . right: t = 0.2, t = 0.5, t = 2, t = 10.

4.3.1.4.3.1. Eﬀ Effectect of of Poroelastic Poroelastic Parameters Parameters WeWe investigate investigate the the shear shear failure failure responseresponse underunder various sets sets of of poroelastic poroelastic parameters parameters,, while while keepingkeeping the the rest rest at at their their baseline baseline values: values: (i) (i) the theLam Laméé constantconstant λ; (ii); (ii) the the Biot Biot coe coefficientﬃcient α ; (iii); (iii) the the Biot modulusBiot modulusM; (iv) theM ; rock (iv) permeability the rock permeabilityκ. For a completely . For a completely dry porous dry medium porousM medium= 0, whereas M = 0 for, a materialwhereas with for incompressiblea material with constituentsincompressible we constituents have M we andhaveα M →1. The and shear  → failure1. The modes shear in failure modes in response to different values of poroelast→ic ∞ parameters→ are plotted in Figures 16–19, and each figure corresponds to variation in one of the four sets of parameters (i)–(iv). Among the four poroelastic parameters, the Lamé constant  has the slightest effect on the area of failure zone. A larger value of  or M slightly enlarges the area of failure zone in the direction of the minimum principal stress. It is interesting to observe that a small value of M (for example a gas reservoir) can cause a slight failure along the diagonal direction, which is isolated near the wellbore. A higher permeability  enables a higher speed of pressure diffusion so that it will shrink the failure area in the direction of the minimum principal stress and the wellbore becomes more stable.

Figure 16. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent values of  , from left to right:  = 2GPa ,  = 5GPa ,  = 20GPa .

Figure 17. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent values of  , from left to right:  = 0.5  = 0.7  = 0.9 .

Energies 2019, 12, x FOR PEER REVIEW 15 of 21 Energies 2019,, 12,, xx FOR PEER REVIEW 15 of 21

Figure 15. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent times, from left Figure 15. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent times, from left to right: t = 0.2 , t = 0.5 , t = 2 , t =10 . toto right:right: t = 0.2 ,, t = 0.5 ,, t = 2 ,, t =10 ..

4.3.1. Effect of Poroelastic Parameters 4.3.1. Effect of Poroelastic Parameters We investigate the shear failure response under various sets of poroelastic parameters, while We investigate the shear failure response under various sets of poroelastic parameters,, while keepingEnergies 2019 the12 rest at their baseline values: (i) the Lamé constant  ; (ii) the Biot coefficient  ; (iii) the keeping the, rest, 3486 at their baseline values: (i) the Lamé constant  ;; (ii)(ii) thethe BiotBiot coefficient  ;; (iii)(iii)15 of thethe 21 Biot modulus M ; (iv) the rock permeability  . For a completely dry porous medium M = 0 , Biot modulus M ;; (iv) (iv) the the rock rock permeability permeability .. For For a a completely completely dry dry porous porous medium medium M = 0 ,, whereas for a material with incompressible constituents we have M → and  →1. The shear whereasresponse for to dia ffmaterialerent values with ofincompressible poroelastic parameters constituents are we plotted have inM Figures→ and16– 19, and→1. each The figureshear failure modes in response to different values of poroelastic parameters are plotted in Figures 16–19, failurecorresponds modes to in variation response into onedifferent of the values four sets of poroelast of parametersic parameters (i)–(iv). are Among plotted the in four Figure poroelastics 16–19, and each figure corresponds to variation in one of the four sets of parameters (i)–(iv). Among the andparameters, each figure the Lamcorrespondsé constant toλ variationhas the slightest in one eofff ectthe on four the sets area of of parameters failure zone. (i) A–(iv). larger Among value oftheα four poroelastic parameters, the Lamé constant  has the slightest effect on the area of failure zone. fouror M poroelasticslightly enlarges parameters, the area the of Lam failureé constant zonein  the has direction the slightest of the effect minimum on the principalarea of failure stress. zone. It is A larger value of  or M slightly enlarges the area of failure zone in the direction of the A larger value of  or M slightly enlargesM the area of failure zone in the direction of the minimuminteresting principal to observe stress. that aIt small is interesting value of to (forobserve example that a gassmall reservoir) value of can M cause (for example a slight failure a gas minimum principal stress. It is interestinginteresting toto observeobserve thatthat aa smallsmall valuevalue ofof M (for example a gas reservoir)along the diagonalcan cause direction, a slight failure which along is isolated the diagonal near the direction, wellbore. which A higher is isolated permeability near theκ wellbore.enables a reservoir) can cause a slight failure along the diagonal direction, which is isolated near the wellbore. Ahigher higher speed permeability of pressure  di ffenablesusion so a that higher it will speed shrink ofthe pressure failure areadiffusion in the so direction that it of will the shrink minimum the A higher permeability  enables a higher speed of pressure diffusion so that it will shrink the failureprincipal area stress in the and direction the wellbore of the becomes minimum more principal stable. stress and the wellbore becomes more stable. failure area in the direction of the minimum principal stress and the wellbore becomes more stable.

Figure 16. Shear failure zone distribution in the vicinity of the wellbore at different values of  , Figure 16. Shear failurefailure zonezone distributiondistribution inin the the vicinity vicinity of of the the wellbore wellbore at at di ffdifferenterent values values of λof, from  ,, from left to right:  = 2GPa ,  = 5GPa ,  = 20GPa . fromfromleft to leftleft right: toto right:right:λ = 2  GPa,= 2GPaλ = 5,, GPa, = 5GPaλ = 20,, GPa. = 20GPa ..

Figure 17. Shear failure zone distribution in the vicinity of the wellbore at different values of  , Figure 17. Shear failurefailure zonezone distributiondistribution inin the the vicinity vicinity of of the the wellbore wellbore at at di ffdifferenterent values values of αof, from  ,, from left to right:  = 0.5  = 0.7  = 0.9 . fromfromleft to leftleft right: toto right:right:α = 0.5,  α==0.50.7,α==0.70.9. = 0.9 ..

Energies 2019, 12, x FOR PEER REVIEW 16 of 21

Figure 18. Shear failurefailure zonezone distributiondistribution in in the the vicinity vicinity of of the the wellbore wellbore at at di ﬀdiﬀerenterent values values of Mof, fromM ,

fromleft to left right: to right:M = 1M GPa,=1GPaM = 10, M GPa,=10GPaM = 100, M GPa.=100GPa .

Figure 19. Shear failurefailure zonezone distributiondistribution in in the the vicinity vicinity of of the the wellbore wellbore at diat ﬀdiﬀerenterent values values of κof, from  , fromleft to left right: to right:κ = 1  mD,=1mDκ = 0.1, mD,= 0.1mDκ = 0.01, mD.= 0.01mD .

4.3.2. Effect of In-Situ Stress In this subsection, we compute the shear failure responses under different in-situ stress states. Figure 20 shows the failure zone distribution in the vicinity of the wellbore with different deviatoric part of the stress while the mean part of the stress keeps the same. It can be seen from the plot that a larger differential stress results in a much more severe failure around the wellbore, especially in the diagonal direction. Figure 21 shows the failure zone distribution with different mean part of the stress while the deviatoric part of the stress keeps the same. It is interesting to find that a higher stress state with the same differential value results in a more severe failure response in the direction of the minimum principal stress; however, a smaller stress state can cause a slight failure along the diagonal direction, which is connected with the wellbore and greatly increases the area of the shear failure zone. Generally, a larger mean value of the original principal stress always indicates a deeper reservoir. We can conclude that a more severe wellbore instability occurs to a shallow reservoir compared to a deep reservoir, when the differential stress is the same.

Figure 20. Shear failure zone distribution in the vicinity of the wellbore at different in-situ stress states with different deviatoric parts of the stress.

Figure 21. Shear failure zone distribution in the vicinity of the wellbore at different in-situ stress states with different mean parts of the stress.

Energies 2019, 12, x FOR PEER REVIEW 16 of 21

EnergiesFigure 2019 18., 12 ,Shear x FOR failurePEER REVIEW zone distribution in the vicinity of the wellbore at diﬀerent values of 16M of , 21 from left to right: M =1GPa , M =10GPa , M =100GPa . Figure 18. Shear failure zone distribution in the vicinity of the wellbore at diﬀerent values of M , from left to right: M =1GPa , M =10GPa , M =100GPa .

EnergiesFigure2019, 1219., 3486Shear failure zone distribution in the vicinity of the wellbore at different values of 16, of 21 from left to right:  =1mD , = 0.1mD , = 0.01mD . Figure 19. Shear failure zone distribution in the vicinity of the wellbore at different values of  , from left to right:  =1mD , = 0.1mD , = 0.01mD . 4.3.2. Effect Effect of In In-Situ-Situ Stress 4.3.2.In Effectthis subsection, of In-Situ Stress we we compute compute the the shear shear failure failure responses responses under under different different in in-situ-situ stress stress states. states. Figure 20 shows the failure zone distribution in the vicinity of the wellbore with different deviatoric Figure In20 this shows subsection, the failure we zonecompute distribution the shear in failure the vicinity responses of the under wellbore different with in -differentsitu stress deviatoric states. part of the stress while the mean part of the stress keeps the same. It can be seen from the plot that partFigure of the 20 stress shows while the failure the mean zone partdistribution of the stress in the keeps vicinity the of same. the wellbore It can be with seen different from the deviatoric plot that a a larger differential stress results in a much more severe failure around the wellbore, especially in largerpart differentialof the stress stresswhile theresults mean in part a much of the more stress severe keeps failure the same. around It can the be wellbore,seen from especiallythe plot that in a the diagonalthelarger diagonal differential direction. direction. stressFigure Figure results 21 shows21 in shows a much the the failuremore failure severe zone zone failure distribution distribution around with the with wellbore, different different especially mean mean part part in ofthe of the stressdiagonal whilewhile direction. thethe deviatoricdeviatoric Figure part part21 ofshows of the the stress the stress failure keeps keeps zone the the same. distribution same. It is It interesting is with interesting different to find to mean thatfind a partthat higher ofa higher the stress stressstatestress with state while the with samethe the deviatoric disamefferential differential part value of the value results stress results inkeeps a more in the a severemoresame. severeIt failure is interesting failure response respon to in find these thatin direction the a higherdirection ofthe ofminimum stressthe minimum state principal with principal the stress; same however,stressdifferential; however, a smallervalue aresults smaller stress in state astress more can statesevere cause can failure a slightcause respon failurea slightse along in failure the the direction along diagonal the diagonaldirection,of the minimum direction, which is principal connectedwhich is stress connected with; however, the wellbore with a the smaller and wellbore greatly stress and increasesstate greatly can cause the increases area a slight of thethe failure sheararea alongof failure the theshear zone. failureGenerally,diagonal zone adirection,. larger Generally, mean which valuea is larger connected ofthe mean original with value the principal ofwellbore the stressoriginal and alwaysgreatly principal indicatesincreases stress the a deeper alwaysarea of reservoir. the indicates shear We a deepercanfailure conclude reservoir. zone that. Generally, a We more can severea conclude larger wellbore mean that valueinstability a more of thesevere occurs original wellbore to a principalshallow instability reservoir stress always occurs compared indicates to a to shallow a deep a reservoirreservoir,deeper compared reservoir. when the We ditoff a erential candeep conclude reservoir stress is that, thewhen a same. more the differentialsevere wellbore stress instabilityis the same occurs. to a shallow reservoir compared to a deep reservoir, when the differential stress is the same.

FigureFigure 20. 20. ShearShe Shearar failure failure failure zone zone zone distribution distribution distribution in in the the vicinity vicinity vicinity of of the of the thewellbore wellbore wellbore at di at atﬀ differenterent different in-situ in - insitu stress-situ stress stress states stateswithstates di withﬀ witherent different different deviatoric deviatoric deviatoric partsof parts parts the stress.ofof the stress.

FigureFigure 21. 21.Shear Shear failure failure zone zone distribution distribution in in the the vicinity vicinity of ofthe the wellbore wellbore at di atﬀ differenterent in-situ in-situ stress stress states Figure 21. Shear failure zone distribution in the vicinity of the wellbore at different in-situ stress withstates diﬀ werentith different mean parts mean of parts the stress.of the stress. states with different mean parts of the stress.

4.3.3. Effect of Periodic Loads In this subsection, we analyze the effect of the parameter value of the given periodic load on the failure responses. We assume the bottom-hole pressure is periodically fluctuating, which can be caused by trips or bit vibration during drilling. The load is applied at the wellbore surface and can be mathematically described by a simple Sinusoidal function as shown in Equation (55). We considered three parameters: the mean value of the load pw, the angular frequency of the Sinusoidal load ω and the amplitude of the Sinusoidal load L. The load is non-dimensionalized by the initial pore pressure p0. We first considered three cases: (i) pw = 0.9 < 1, corresponding to an underbalanced drilling; (ii) pw = 1, corresponding to a balanced drilling; (iii) pw = 1.1 > 1, corresponding to an overbalanced drilling. Figure 22 shows the shear failure responses under the three conditions. Unsurprisingly, increasing the bottom-hole pressure helps maintain the wellbore stability. We then investigate the Energies 2019, 12, x FOR PEER REVIEW 17 of 21 Energies 2019, 12, x FOR PEER REVIEW 17 of 21 4.3.3. Effect of Periodic Loads 4.3.3. Effect of Periodic Loads In this subsection, we analyze the effect of the parameter value of the given periodic load on In this subsection, we analyze the effect of the parameter value of the given periodic load on thethe failure failure responses.responses. WeWe assumeassume the bottom-hole pressurepressure isis periodicallyperiodically fluctuating fluctuating, ,which which can can be be causedcaused byby tripstrips oror bitbit vibrationvibration during drilling. TheThe loadload isis appliedapplied atat thethe wellbore wellbore surface surface and and can can bebe mathematically mathematically described described by a simple Sinusoidal function function as as shown shown in in EquationEquation (5(555).). We We considered three parameters: the mean value of the load pw , the angular frequency of the considered three parameters: the mean value of the load pw , the angular frequency of the SinusoidalSinusoidal loadload  andand thethe amplitude of the Sinusoidal loadload LL.. TheThe loadload is is non non-dimensionalized-dimensionalized by the initial pore pressure p0 . We first considered three cases: (i) pw =0.9 1, corresponding to by the initial pore pressure p0 . We first considered three cases: (i) pw =0.9 1, corresponding to an underbalanced drilling; (ii) p =1, corresponding to a balanced drilling; (iii) p =1.1 1 , an underbalanced drilling; (ii) pw =1, corresponding to a balanced drilling; (iii) pww=1.1 1 , correspondingcorresponding ttoo anan overbalancedoverbalanced drilling. Figure 2222 showsshows thethe shearshear failurefailure responses responses under under the the threethree conditions.conditions. Unsurprisingly,Unsurprisingly, increasing thethe bottombottom--holehole pressurepressure helpshelps maintain maintain the the wellbore wellbore stability.stability. WeWe thenthen investigateinvestigate the effect of the frequency ofof thethe periodicperiodic l loadoad on on the the failure failure response, response, andand thethe resultsresults areare shownshown in Figure 23. It is noteworthy thatthat thethe failurefailure isis more more severe severe at at a a certain certain valuevalue of of the the loadload frequency,frequency, and the area of the failure zonezone reachesreaches a a peak peak when when ==1010. .The The effect effect Energiesofof amplitudeamplitude2019, 12, of 3486of thethe SinusoidalSinusoidal load is also examined, whichwhich isis foundfound toto bebe negligiblenegligible on on the the shear17 shear of 21 failurefailure response response,, asas sshownhown in Figure 24. For furtherfurther investigation,investigation, wewe calculate calculate the the differential differential stress stress atat two two selectedselected locationslocations inin thethe direction of minimum principalprincipal stress,stress, and and the the results results are are shown shown in in eFigureFigureffect of 225 the5. . TheThe frequency amplitudeamplitude of the of periodic the differential load on stress the failure is is found found response, to to be be much andmuch the smallersmaller results than than are theshown the load load in Figureamplitude,amplitude, 23. It indicating indicatingis noteworthy thatthat that subtraction subtraction the failure of is the more principal severe at stress stress a certain reduces reduces value the the of the wave wave load amplitude amplitude frequency, and and thedecreasesdecreases area of thethe the riskrisk failure ofof wellborewellbore zone reaches instability a peak due when to waveω = propagation.propagation.10. The effect of amplitude of the Sinusoidal loadBased isBased also onon examined, thethe results results which of the is found parametric to be study, negligible we we foundfound on the thatthat shear thethe failure inin--situsitu response, stress,stress, especially especially as shown the the in Figuredifferentialdifferential 24. For stressstress further,, isis thethe investigation, mostmost criticalcritical we parameter calculate forfor the wellborewellbore differential stabilitystability stress. .Moreover atMoreover two selected, ,the the Biot locationsBiot coefficient coefficient in the direction of minimum principal stress, and the results are shown in Figure 25. The amplitude of the , , Biot Biot modulus modulus MM ,, permeability permeability  , bottom bottom--holehole pressurepressure ppww andand the the frequencyfrequency ofof the the diSinusoidalSinusoidalfferential stressload load  is found alsoalso impactimpact to be much the area smaller of shear than failure thefailure load zone.zone. amplitude, MMeanwhile,eanwhile, indicating the the Lamé Laméthat subtraction constant constant  of theandand principal the the amplitude amplitude stress reduces of of the the the Sinusoidal wave amplitude load 2L and havehave decreases the the slightest slightest the risk effectof effect wellbore on on the the instability shear shear failurefailure due to waveresponse.response. propagation. Thus, Thus, theirtheir effecteffect cancan be ignored in practical analysisanalysis ofof wellborewellbore stability. stability.

Figure 22. Shear failure zone distribution in the vicinity of the wellbore at different values of pw , Figure 22. Shear failurefailure zonezone distributiondistribution in in the the vicinity vicinity of of the the wellbore wellbore at at di ffdifferenterent values values of pofw, frompw , leftfrom to left right: to right:p = 0.9,pwp= 0.9= 1,, p w ==11.1., pw = 1.1. from left to right:w pw =w 0.9 , pww =1, pw = 1.1.

Figure 23. Shear failure zone distribution in the vicinity of the wellbore at different values of  , Figure 23.23. Shear failurefailure zonezone distributiondistribution in in the the vicinity vicinity of of the the wellbore wellbore at diat ffdifferenterent values values of ω of, from  , Energiesfrom 2019 ,left 12, tox FOR right: PEER  REVIEW=1, = 10 , =100 . 18 of 21 leftfrom to left right: to right:ω = 1,ω==110,,ω==10100., =100 .

Figure 24. Shear failurefailure zonezone distributiondistribution inin the the vicinity vicinity of of the the wellbore wellbore at diat ﬀdiﬀerenterent values values of Lof, from L , fromleft to left right: to right:L = 0.05, L =L0.05= 0.1,, LL ==0.10.2., L = 0.2 .

Figure 25. Response of differential stress at different times.

5. Conclusions This paper presents a poroelastodynamic model that is based on Biot’s theory to study transient stress distribution and failure response of a wellbore subjected to periodic loading during drilling. In response to the periodic load perturbation at the wellbore surface, pressure and stress waves are created and propagate into the formation. The wave phenomenon is characterized by introducing solid-fluid acceleration term into the coupling mass conservation equation. The solutions are obtained in the Laplace domain through a new method and inverted numerically to the real time domain using a reliable numerical scheme. The results show that a disturbance zone is created after drilling. Inside the disturbance zone, pressure and stress waves propagate; outside the disturbance zone, the pressure and stress keep the original state. The disturbance zone expands over time. When the stress waves propagate to a certain location, the pore pressure, radial and hoop stresses change rapidly at the beginning, and then reach a stable fluctuation. It was found that the wave velocities of radial stress, hoop stress and pressure are larger than that of the shear stress. It is noteworthy that pore pressure reaches its peak at a certain time and location, while the conventional poroelastic theory always fails to discover this phenomenon when a periodic load is applied on the wellbore due to the lack of consideration of solid-fluid acceleration. Through the detailed parametric study on shear failure responses, we found that a smaller value of Biot coefficient  or Biot modulus M can shrink the area of failure zone in the direction of the minimum principal stress and helps maintain wellbore stability. A higher permeability of a reservoir enables a higher speed of pressure diffusion so that it will also shrink the failure area and make the wellbore more stable. A larger differential stress of the original non-hydrostatic stress field results in a much more severe failure around the wellbore, especially in the diagonal direction. A larger mean value of the original non-hydrostatic stress field results in a more severe failure response in the direction of the minimum principal stress. However, a larger mean value of the

Energies 2019, 12, x FOR PEER REVIEW 18 of 21

EnergiesFigure2019, 1224., 3486Shear failure zone distribution in the vicinity of the wellbore at diﬀerent values of L18, of 21 from left to right: L = 0.05 , L = 0.1, L = 0.2 .

FigureFigure 25. 25. ResponseResponse of of differential diﬀerential stress stress at at different diﬀerent times times..

5. ConclusionsBased on the results of the parametric study, we found that the in-situ stress, especially the differential stress, is the most critical parameter for wellbore stability. Moreover, the Biot coefficient α, This paper presents a poroelastodynamic model that is based on Biot’s theory to study Biot modulus M, permeability κ, bottom-hole pressure pw and the frequency of the Sinusoidal load transient stress distribution and failure response of a wellbore subjected to periodic loading during ω also impact the area of shear failure zone. Meanwhile, the Lamé constant λ and the amplitude of drilling. In response to the periodic load perturbation at the wellbore surface, pressure and stress the Sinusoidal load 2L have the slightest effect on the shear failure response. Thus, their effect can be waves are created and propagate into the formation. The wave phenomenon is characterized by ignored in practical analysis of wellbore stability. introducing solid-fluid acceleration term into the coupling mass conservation equation. The solutions5. Conclusions are obtained in the Laplace domain through a new method and inverted numerically to the real time domain using a reliable numerical scheme. The results show that a disturbance zone is This paper presents a poroelastodynamic model that is based on Biot’s theory to study transient created after drilling. Inside the disturbance zone, pressure and stress waves propagate; outside the stress distribution and failure response of a wellbore subjected to periodic loading during drilling. disturbance zone, the pressure and stress keep the original state. The disturbance zone expands In response to the periodic load perturbation at the wellbore surface, pressure and stress waves are over time. When the stress waves propagate to a certain location, the pore pressure, radial and hoop created and propagate into the formation. The wave phenomenon is characterized by introducing stresses change rapidly at the beginning, and then reach a stable fluctuation. It was found that the solid-fluid acceleration term into the coupling mass conservation equation. The solutions are obtained wave velocities of radial stress, hoop stress and pressure are larger than that of the shear stress. It is in the Laplace domain through a new method and inverted numerically to the real time domain noteworthy that pore pressure reaches its peak at a certain time and location, while the using a reliable numerical scheme. The results show that a disturbance zone is created after drilling. conventional poroelastic theory always fails to discover this phenomenon when a periodic load is Inside the disturbance zone, pressure and stress waves propagate; outside the disturbance zone, the applied on the wellbore due to the lack of consideration of solid-fluid acceleration. pressure and stress keep the original state. The disturbance zone expands over time. When the stress Through the detailed parametric study on shear failure responses, we found that a smaller waves propagate to a certain location, the pore pressure, radial and hoop stresses change rapidly at value of Biot coefficient  or Biot modulus M can shrink the area of failure zone in the direction the beginning, and then reach a stable fluctuation. It was found that the wave velocities of radial of the minimum principal stress and helps maintain wellbore stability. A higher permeability of a stress, hoop stress and pressure are larger than that of the shear stress. It is noteworthy that pore reservoir enables a higher speed of pressure diffusion so that it will also shrink the failure area and pressure reaches its peak at a certain time and location, while the conventional poroelastic theory make the wellbore more stable. A larger differential stress of the original non-hydrostatic stress always fails to discover this phenomenon when a periodic load is applied on the wellbore due to the field results in a much more severe failure around the wellbore, especially in the diagonal direction. lack of consideration of solid-fluid acceleration. A larger mean value of the original non-hydrostatic stress field results in a more severe failure Through the detailed parametric study on shear failure responses, we found that a smaller value response in the direction of the minimum principal stress. However, a larger mean value of the of Biot coefficient α or Biot modulus M can shrink the area of failure zone in the direction of the minimum principal stress and helps maintain wellbore stability. A higher permeability of a reservoir enables a higher speed of pressure diffusion so that it will also shrink the failure area and make the wellbore more stable. A larger differential stress of the original non-hydrostatic stress field results in a much more severe failure around the wellbore, especially in the diagonal direction. A larger mean value of the original non-hydrostatic stress field results in a more severe failure response in the direction of the minimum principal stress. However, a larger mean value of the original non-hydrostatic stress field helps to maintain stability along the diagonal direction. It is also noted that the wellbore instability is most severe at a certain value of the load frequency, which is dependent on the reservoir Energies 2019, 12, 3486 19 of 21 properties. It is suggested that the simulation should be performed prior to drilling to optimize the engineering parameters. The model developed in this paper is 2D; however, 3D simulation is necessary for a directional well that is drilled in a deep reservoir. To solve 3D poroelastodynamics, Fourier transformation should be employed. One of the features in the deep reservoir is the extremely high temperature, which leads to the thermal dilation of the solid and fluid. Thus, the coupling of heat flux is an important future subject. In general, our model can be readily extended to 3D or thermal-poroelastodynamics.

Author Contributions: Conceptualization, investigation, writing-original draft, X.W.; methodology, formal analysis, Y.X. and X.W.; writing-review and editing, X.W., Y.X. and S.Y.; supervision, M.C., Y.J. and S.Y.; funding acquisition, M.C. and Y.X. Funding: The authors are grateful to the supports provided by the National Natural Science Foundation of China (NSFC) (NO. U1762215) and the China Postdoctoral Science Foundation (No. 2018M641598). Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Appendix A Table A1 shows the symbols with commonly used units and their deﬁnitions.

Table A1. List of symbols and deﬁnitions.

Symbol Unit Definition r m Radius defined in Figure1 θ Angle defined in Figure1

rw m Radius of the borehole

P0, S0 MPa Far-ﬁeld mean and deviatoric parts of the stress

ur, uθ m Displacement components of the solid

wr, wθ m Displacement components of the ﬂuid

σij MPa Stress component

σrr, σθθ, σrθ MPa Radial stress, hoop stress and shear stress

eij Strain component e Volumetric strain p MPa Excess pore water pressure

pw MPa Constant bottom-hole pressure

ph MPa Transient bottom-hole pressure

δij Kronecker delta. δij = 1 when i = j and δij = 0 when i , j λ GPa Lame constant of the bulk material µ GPa Shear modulus of the bulk material ζ Increment of pore fluid per unit volume α Biot effective stress coefficient M GPa Biot modulus

Ks GPa Bulk modulus of solid grains

K f GPa Bulk modulus of pore ﬂuid K GPa Drained bulk modulus of the porous medium ρ Kg m 3 Density of porous medium · − 3 ρ f Kg m Density of porous ﬂuid · − Energies 2019, 12, 3486 20 of 21

Table A1. Cont.

Symbol Unit Deﬁnition t s Time 3 V, Vs, Vp m Volumes of the porous medium, solid grains and pores

σ0 MPa Effective stress vector of the porous medium 3 1 2 qw m s m Volumetric flux of pore fluid · − · − 3 ρa Kg m Mass density of the fluid added into the porous medium · − φ Porosity of the porous medium 15 2 κ mD(10− m ) Permeability of the porous medium τ mPa s Viscosity of the porous fluid · p0 MPa Unperturbed pore pressure 2L MPa Amplitude of stress wave 1 ω s− Angular frequency of stress wave

ϕ ◦ Internal friction angle

Ct MPa Cohesion

σH MPa Maximum principal stress

σh MPa Minimum principal stress a Non-dimensional tortuosity factor

η, Ω1, Ω2 Φ1, Φ2, Φ3, Φ4 Dimensionless quantities Ψ1, Ψ2, Ψ3

An, Bn, Cn Constants in diﬀerent mode solutions

References

1. Fjar, E.; Holt, R.M.; Raaen, A.M.; Risnes, R.; Horsrud, P. Petroleum Related Rock Mechanics; Elsevier: Amsterdam, The Netherlands, 2008. 2. Kirsch, C. Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Zeitschrift des Vereines Deutscher Ingenieure 1898, 42, 797–807. 3. Bradley, W.B. Failure of inclined boreholes. J. Energy Resour. Technol. 1979, 101, 232–239. [CrossRef] 4. Aadnoy, B.S.; Chenevert, M.E. Stability of highly inclined boreholes (includes associated papers 18596 and 18736). SPE Drill. Eng. 1987, 2, 364–374. [CrossRef] 5. Zoback, M.D. Reservoir Geomechanics; Cambridge University Press: Cambridge, UK, 2010. 6. Meng, M.; Zamanipour, Z.; Miska, S.; Yu, M.; Ozbayoglu, E.M. Dynamic stress distribution around the wellbore influenced by surge/swab pressure. J. Pet. Sci. Eng. 2019, 172, 1077–1091. [CrossRef] 7. Detournay, E.; Cheng, A.H.D. Poroelastic response of a borehole in a non-hydrostatic stress field. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 1988, 25, 171–182. [CrossRef] 8. Zamanipour, Z.; Miska, S.Z.; Hariharan, P.R. Effect of transient surge pressure on stress distribution around directional wellbores. In Proceedings of the IADC/SPE Drilling Conference and Exhibition, Fort Worth, TX, USA, 1–3 March 2016. 9. Zhang, F.; Kang, Y.; Wang, Z.; Miska, S.; Yu, M.; Zamanipour, Z. Transient coupling of swab/surge pressure and in-situ stress for wellbore-stability evaluation during tripping. SPE J. 2018, 23, 1019–1038. [CrossRef] 10. Meng, M.; Zamanipour, Z.; Miska, S.; Yu, M.; Ozbayoglu, E.M. Dynamic Wellbore Stability Analysis Under Tripping Operations. Rock Mech. Rock Eng. 2019, 1–21. [CrossRef] 11. Feng, Y.; Li, X.; Gray, K.E. An easy-to-implement numerical method for quantifying time-dependent mudcake effects on near-wellbore stresses. J. Pet. Sci. Eng. 2018, 164, 501–514. [CrossRef] 12. Chen, P.; Miska, S.Z.; Ren, R.; Yu, M.; Ozbayoglu, E.; Takach, N. Poroelastic modeling of cutting rock in pressurized condition. J. Pet. Sci. Eng. 2018, 169, 623–635. [CrossRef] Energies 2019, 12, 3486 21 of 21

13. Biot, M.A. General theory of three-dimensional consolidation. J. Appl. Phys. 1941, 12, 155–164. [CrossRef] 14. Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid, I: Low-frequency range. J. Acoust. Soc. Am. 1956, 28, 168–178. [CrossRef] 15. Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid, II: Higher frequency range. J. Acoust. Soc. Am. 1956, 28, 179–191. [CrossRef] 16. Rasolofosaon, P.N.J. Importance of interface hydraulic condition on the generation of second bulk compressional wave in porous media. Appl. Phys. Lett. 1988, 52, 780–782. [CrossRef] 17. Kelder, O.; Smeulders, D.M.J. Propagation and damping of compressional waves in porous rocks: Theory and experiments. SEG Technical Program Expanded Abstracts 1995. Soc. Explor. Geophys. 1995, 675–678. [CrossRef] 18. Gurevich, B.; Kelder, O.; Smeulders, D.M.J. Validation of the slow compressional wave in porous media: Comparison of experiments and numerical simulations. Transp. Porous Media 1999, 36, 149–160. [CrossRef] 19. Schanz, M. Poroelastodynamics: Linear models, analytical solutions, and numerical methods. Appl. Mech. Rev. 2009, 62, 030803. [CrossRef] 20. Schanz, M.; Cheng, A.H.D. Transient wave propagation in a one-dimensional poroelastic column. Acta Mech. 2000, 145, 1–18. [CrossRef] 21. Xie, K.; Liu, G.; Shi, Z. Dynamic response of partially sealed circular tunnel in viscoelastic saturated soil. Soil Dyn. Earthq. Eng. 2004, 24, 1003–1011. [CrossRef] 22. Gao, M.; Wang, Y.; Gao, G.Y.; Yang, J. An analytical solution for the transient response of a cylindrical lined cavity in a poroelastic medium. Soil Dyn. Earthq. Eng. 2013, 46, 30–40. [CrossRef] 23. Liu, G.B.; Xie, K.H.; Yao, H.L. Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium. Int. J. Numer. Anal. Methods Geomech. 2005, 29, 1109–1126. 24. Hasheminejad, S.M.; Kazemirad, S. Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation. Soil Dyn. Earthq. Eng. 2008, 28, 277–292. [CrossRef] 25. Xia, Y.; Jin, Y.; Chen, M.; Chen, K. Poroelastodynamic response of a borehole in a non-hydrostatic stress field. Int. J. Numer. Anal. Methods Geomech. 2017, 93, 82–93. [CrossRef] 26. Senjuntichai, T.; Keawsawasvong, S.; Plangmal, R. Three-dimensional dynamic response of multilayered poroelastic media. Mar. Georesour. Geotechnol. 2018, 37, 424–437. [CrossRef] 27. Keawsawasvong, S.; Senjuntichai, T. Poroelastodynamic fundamental solutions of transversely isotropic half-plane. Comput. Geotech. 2019, 106, 52–67. [CrossRef] 28. Xia, Y.; Jin, Y.; Wei, S.M.; Chen, M.; Zhang, Y.Y.; Lu, Y.H. Transient failure of a borehole excavated in a poroelastic continuum. Mech. Res. Commun. 2019.[CrossRef] 29. Mehrabian, A.; Abousleiman, Y.N. Generalized poroelastic wellbore problem. Int. J. Numer. Anal. Methods Geomech. 2013, 37, 2727–2754. [CrossRef] 30. Hasheminejad, S.M.; Hosseini, H. Dynamic stress concentration near a fluid-filled permeable borehole induced by general modal vibrations of an internal cylindrical radiator. Soil Dyn. Earthq. Eng. 2002, 22, 441–458. [CrossRef] 31. Hassanzadeh, H.; Pooladi-Darvish, M. Comparison of different numerical Laplace inversion methods for engineering applications. Appl. Math. Comput. 2007, 189, 1966–1981. [CrossRef] 32. Senjuntichai, T.; Rajapakse, R.K.N.D. Transient response of a circular cavity in a poroelastic medium. Int. J. Numer. Anal. Methods Geomech. 1993, 17, 357–383. [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).