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Transient failure of a borehole excavated in a poroelastic continuum
Accepted Manuscript
Transient failure of a borehole excavated in a poroelastic continuum Yang Xia , Yan Jin , Shi M. Wei , Mian Chen , Ya Y. Zhang , Yun H. Lu PII: DOI: Reference:
S0093-6413(18)30455-5 https://doi.org/10.1016/j.mechrescom.2019.03.006 MRC 3359
To appear in:
Mechanics Research Communications
Received date: Revised date: Accepted date:
13 September 2018 20 March 2019 20 March 2019
Please cite this article as: Yang Xia , Yan Jin , Shi M. Wei , Mian Chen , Ya Y. Zhang , Yun H. Lu , Transient failure of a borehole excavated in a poroelastic continuum, Mechanics Research Communications (2019), doi: https://doi.org/10.1016/j.mechrescom.2019.03.006
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ACCEPTED MANUSCRIPT HIGHLIGHTS Poroelastodynamic response is analyzed in a non-hydrostatic stress field.
Analytical solutions for stress and pressure wave propagation are obtained.
Transient tensile and shear failure responses are investigated.
Permeable and impermeable borehole conditions are considered.
A comparison between poroelastic and poroelastodynamic theory is present.
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Transient failure of a borehole excavated in a poroelastic continuum Yang Xia1, Yan Jin1*, Shi M. Wei1, Mian Chen1, Ya Y. Zhang1, Yun H. Lu1 1 State Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum (Beijing), Beijing, China, 102249 *[email protected] Tel.: +81-13810986965.
Abstract
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In this paper, the poroelastodynamic theory is employed to study the transient failure of a suddenly excavated borehole in a poroelastic continuum subjected to a non-hydrostatic far-field stress state. By introducing four scalar potential functions, the solutions for the stresses and pore pressure are given as the product of an exact treatment of the full fluid-solid coupling through field expansions with emphasis on the effect of solid-fluid acceleration. Transient tensile and shear failure responses are investigated based on an overbalanced drilling for two types of boundary conditions: a permeable surface and an impermeable surface. It's found that the permeable borehole has a larger tens ile failure area than the impermeable borehole in the direction of maximum in-situ stress, and meanwhile, four symmetric shear failure areas are observed near the borehole for both internal boundary conditions. Influences of poroelastic parameters on the transient failure responses of a permeable borehole are analyzed in a detailed parametric study. The failure responses for both classical quasi-static poroelastic and poroelastodynamic theories are computed and compared to study the importance of the inertial effect in very early times. Noteworthy is that the borehole is more unstable in the poroelastodynamic theory compared to the classical poroelastic theory because the inertial effect enhances the transient response in the early times, increasing the minimum principal stress to a tensile state in the direction of maximum in-situ stress so as to result in transient generation of both tensile and shear failures in that direction. The results in this paper provide fundamental insights on the borehole instability under some dynamic bottom-hole conditions.
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© 2015 The Authors. Published by Elsevier Ltd.
Keywords: Poroelastodynamics, Transient failure, Analytical solution, Wellbore stability
1. Introduction
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There has been a progressing interest in transient response of a borehole excavated in fluid-saturated porous media due to its important applications in various technical and engineering processes, for example, analyzing dynamic wellbore instability, generation of microfractures surrounding the borehole as well as instantaneous fracture initiation. In these dynamic problems, the inertial effect of the solid-fluid system becomes important and it has to be taken into account [1]. The quasi-static poroelastic response of a borehole was introduced by Detournay and Cheng [2]. On the other hand, the propagation theory of elastic waves in fluid-saturated porous media proposed by Biot [3] suggests that the inertia effect of the solid-fluid system have a decisive effect on the transient response of the rock [4, 5]. In addition, the scientific groundwork for Biot's model of poroelastodynamics has been more firmly established through some experimental validations of its most fundamental predictions [6-8], leading to increasing interests of the dynamic theory. The wave phenomena in the fluid saturated formation are paid special attention in recent years [9-10], and the dynamic response of the borehole may lead to the generation of the microfractures surrounding the borehole [11]. Thus, the need for applying the dynamic theory to study the transient failure of the borehole has been arised. The dynamic extension of Biot's theory to three phases was first published by Vardoulakis and Beskos [12]. Further development of this kind of model to describe the dynamic behavior of rocks has been found in [13-15]. Local heterogeneities in porous materials have been considered by Wei and Muraleetharan [16]. Schanz and Cheng [17] examined the wave propagating behavior in a onedimensional colume. Recently, Xia et al. presented an exact
closed-form solution for poroelastodynamic response of a borehole in a non-hydrostatic stress field [18]. These works are great attempts for analyzing poroelastodynamic problems. In some highly non-linear dynamic problems, the complex boundary conditions in terms of heterogeneity of the poroelastic parameters can take advantage of the use of numerical methods [19]. However, analytical solutions are still very popular for most cases since they are much more convenient to implement for analyzing the general transient behavior. Today, the wellbore instability model has been well developed to incorporate the effects of poroelasticity, elastoplastic, chemical and thermal diffusions between the drilling mud and the formation, dual-porosity system [20-22]. Time-dependent wellbore instability becomes of great interest. Rahman et al. considered the increase of pore pressure around the borehole due to mud penetration to study the time-dependent borehole instability [23]. Schoenball et al. used an implementation of time-dependent brittle creep to study the breakout growth in time and time-delayed wellbore instability [24]. Qu et al. studied the time-delay borehole collapse in underbalanced drilling in coal seam due to the pore pressure change in the fractures [25]. These works suggest that the borehole remains stable at the beginning but would probably fail due to some time-delay effects. However, few works have been performed to analyze the transient failure in the short-term change of the loading boundary conditions, such as the transient response of a sudden pressing, unloading or excavation in the porous rock. In this paper, we study the transient failure of a suddenly excavated borehole in a poroelastic continuum in a nonhydrostatic stress field based on Biot's theory of poroelastodynamics. Both tensile and shear failure modes are studied for permeable and impermeable boundary conditions. In particular, the failure responses for both classical quasistatic poroelastic and poroelastodynamic theories are
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computed and compared to show the importance of the inertial effect in very early times. The results in this paper provide fundamental insights on the borehole instability under some dynamic bottom-hole conditions. 2. Problem formulation
Based on Biot's dynamic theory of poroelasticity, the medium around the borehole is taken to be a macroscopically homogeneous and isotropic two-component solid/fluid system. The stress tensor 𝜎𝑖𝑗 and the mean pore fluid pressure p are given by (note that tension is here taken positive): (1) ij (u e M ) ij 2eij
p M ( e)
(2)
s2 0 2 0 0 2 2 f s
2 u Ku 3
2 Ks K f Ku K K s ( ) K f
(9)
K Ks
(3)
In which is the shear modulus of the bare skeletal frame, the porosity, K , K s , K f the bulk modulus of the drained
w+ w k
the fast compressional, slow compressional, and the elastic shear waves, respectively, are given as:
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Fig. 1. A borehole subject to anisotropic stresses.
where and f are the density of porous solid and pore fluid, and k and are the permeability of the porous rock and viscosity of the fluid, respectively. a is a nondimensional tortuosity factor with ( a f ) / a f ,
2 f ,s
1 12 4 2 2
2f s 4 s a f s 2 2bs 2
2
(12) (13)
where 1
(u 2 )(a f s 2 b s) M s 2 2 M f s 2 (u 2 ) M
2
(5)
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porous rock, solid grains and pore fluid, respectively. e and are the dilatations of the solid and fluid, respectively. K u is the bulk modulus of the undrained system. u and w are the displacement vectors of the solid and the fluid, respectively. The governing equations governed by Biot's theory [26] in displacement form with dissipation of interstitial fluid taken into account take the following forms: (4) (u 2 ) u M w u = u + f w
a f
(11) 2 2 0 where f , s and designate the complex wave numbers of
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K f (Ks K ) Ks (Ks K f )
M u + M w f u +
a f s s k 2
By wave analysis, the above coupled equations can be manipulated to yield Helmholtz equations [27]: (10) 2 f , s 2f , s f , s 0
K f K s2
1
f s2
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Where
M
in the plot. In the following content, we state the direction of maximum in-situ stress and minimum in-situ stress by σ𝐻 direction and σℎ -direction. Helmholtz decomposition theorem allows us to resolve the displacement fields as superposition of longitudinal and transverse vector components [18]: (6) u (7) w Substituting the above resolutions into the field equations Eq. (4) and Eq. (5), we obtain two sets of coupled equations after performing Laplace transformation: s2 f s2 u 2 M 2 (8) a f s2 M 2 2 M s f s k
b
a f s 4 b 2 s 3 2f s 4 (u 2 ) M 2
rw k
(14) (15) (16)
Employing Eqs. 6-11, with some manipulations, the potentials , , , and have the following relations:
f s , f f s s , 0
(17)
where
f
M 2f f s 2 a f s 2 b s M 2f
(18)
s
M s2 f s 2 a f s 2 b s M s2
(19)
0
f s 2 a f s 2 bs
(20)
while a is the added fluid mass density and a c f [17].
3. Field expansions and boundary conditions
Consider a vertical borehole drilled in a formation that is characterized by a non-hydrostatic horizontal in-situ stress field, shown in Fig. 1, and the initial stress state is also shown
The transmitted fast dilatational wave, slow dilatational wave and the shear wave at the borehole surface into the
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f An K n ( f r )cos n
(21)
s Bn K n ( s r )cos n
(22)
n 0
n 0
Cn K n ( r )sin n
(u M f ) 2f f
(32) ur u ) r r The solutions of stresses, pore pressure can be expressed in a superposed manner of the axisymmetric and asymmetric modes:
(u M s ) s2 s 2 (
rr rr(0) rr(2) cos2 (0) (2) cos2 r r(2) sin2
(23)
n 1
The far-field stresses are composed of only two parts: a constant term and a 2𝜃-trigonometric term, thus, the infinite series of the potential functions can be simplified to the superposed form of an axisymmetric and an asymmetric function: (24) f A0 K0 ( f r ) A2 K2 ( f r )cos2
s B0 K0 ( s r ) B2 K2 ( s r )cos2
(25)
C2 K2 (r )sin2
(26) So, expressions for the frame and liquid dilatations can be obtained in the Laplace domain to yield: (27) e u = 2 2 f 2 s 2f f s2 s
w = f f s s 2 f
(u M s ) s2 B0 K 0 ( s r )
u rr (u M f ) f (u M s ) s 2 r r 2 s
u u r ( r r u ) r r
(u M s ) s2 B0 K 0 ( s r )
( s ) s2 MB0 K 0 ( s r )
(30)
M
2f A0
K 0 ( f r )
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K ( r ) K3 ( r ) 4 C2 K 2 ( r ) 2C2 1 r2 r K1 ( f r ) K3 ( f r ) (2) 2 2 (u M f ) f A2 K 2 ( f r ) (u M s ) s B2 K 2 ( s r ) f A2 r A2 K 2 ( f r ) B2 K 2 ( s r ) K1 ( s r ) K3 ( s r ) 4 K1 ( r ) K3 ( r ) s B2 2 C2 K 2 ( r ) 8 2C2 r r r2 r K1 ( f r ) K 3 ( f r ) A2 K 2 ( f r ) B2 K 2 ( s r ) K1 ( s r ) K3 ( s r ) (2) r 2 f A2 2 s B2 4 r r r2 2 K ( r ) K3 ( r ) 4 C2 [ K 0 ( r ) 2 K 2 ( r ) K 4 ( r )] 2 C2 K 2 ( r ) C2 1 4 r 2 r p(2) ( f ) 2f MA2 K2 ( f r ) ( s ) s2 MB2 K2 ( s r ) 2
K 4 ( f r )
2
K 0 ( s r )
B2 2 s
2
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(39)
(40)
K 4 ( s r )
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B2
2 s
p (0) ( f ) 2f MA0 K 0 ( f r )
rr(2) (u M f ) 2f A2 K 2 ( f r ) (u M s ) s2 B2 K 2 ( s r ) 2 s
(38)
A0 K 2 ( f r ) B0 K 2 ( s r ) 2 f
(31)
The solutions in the asymmetric mode read:
2f A2
(37)
2 s
(0) (u M f ) 2f A0 K 0 ( f r )
Utilizing Eqs. 1, 2, 17, 27, 28, pore fluid pressure and the stress components can be obtained: (29) p (M f M ) 2f f (M s M ) s2 s 2 f
(35)
A0 K 2 ( f r ) B0 K 2 ( s r ) 2 f
(28)
2 s
(34)
(2)
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2
(33)
(36) p p p cos2 where the superscript (0) and (2) represent the axisymmetric and asymmetric mode, respectively. Substituting Eqs. 24-26 into Eqs. 29-32, the solutions in the axisymmetric mode read: rr(0) (u M f ) 2f A0 K 0 ( f r ) (0)
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poroelastic continuum are, respectively, represented in the Laplace transform domain:
The radial position r and time t have been nondimensionalized by the radius of the borehole rw and a time scale rw / , respectively. In this work, we consider two types of boundary conditions: an ImPermeable Borehole (IPB) and a Permeable Borehole (PB). IPB condition means the rock is extremely tight and the fluid flow through the borehole surface into the rock can be neglected, thus the flux equals zero at the surface. PB condition means the fluid flow through the borehole surface into the rock is considered to maintain the bottom-hole pressure. The inner boundary conditions can be expressed as follows after performing Laplace transformation:
P0 pw (0) rr (1, s ) S0 sin2 s , r(2) (1, s ) S cos2 s (2) (1, s ) 0 rr s pw p0 (0) qr(0) (1, s) 0 p (1, s) PB : s ; IPB: (2) qr (1, s) 0 p (0) (1, s) 0
(41)
(42)
(43)
(44)
(45)
where the bottom-hole pressure is denoted by pw. The complete solution can be obtained by superposing the axisymmetric and asymmetric modes and adding the unperturbed field. The solution in real time domain can be obtained by using the Stehfest method. We consider two failure modes: tensile failure and shear
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tan ( 1 3 ) 2Ct
(47) 3 1 0 tan 2 1 Where St , Ct and are the tensile strength, cohesion and internal friction angle, respectively.
shear failure: F
4. Transient failure of the borehole In all the calculations and plots, the displacements and time are scaled by the radius of the borehole rw and rw / , respectively. The values of used parameters in the calculations are listed in Table. 1.
Table 1: Parameter sets used in the calculation.
4.1. Tensile failure
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We first examine the dynamic profiles of the minimum principal stress surrounding the borehole with the twodimensional isobaric plots. Fig. 2 shows four relatively higher compressive stress zones around the borehole are symmetrically generated along with the development of two tensile stress zones in the σ𝐻 -direction near the borehole. A higher tensile stress is observed near the permeable borehole surface compared to the impermeable borehole surface. Fig. 3 shows the tensile failure area surrounding the borehole after a long-time simulation. It’s found that the permeable borehole is more unstable under the same condition.
undrained Lame constant [GPa] Biot parameter M [GPa] rock density [kg/m3] pore fluid viscosity [Pa·s] rock porosity minimum principal stress σℎ [MPa] bottom-hole pressure pw [MPa] Cohesion Ct [MPa] Borehole radius rw [m]
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Fig. 2: Minimum principal stress surounding the borehole: (a) ̅ (b) ̅ ; (c) ̅ ; (d) ̅ 2 .
;
74.1 100 2000 1×10-4 0.5 37.5 30 5 0.1
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shear modulus [GPa] 5 Biot parameter 0.7356 pore fluid density [kg/m3] 1000 rock permeability k [mD] 34.1 tortuosity factor a 1.6 maximum principal stress 𝜎𝐻 [MPa] 62.5 initial pore pressure p0 [MPa] 25 Tensile strength St [MPa] 5 Internal friction angle 40
Fig. 3: Tensile failure area: (a) PB; (b) IPB
We then investigate the tensile failure modes for a permeable borehole under various sets of poroelastic parameters. Specifically, the baseline for all parameters are listed in Table. 1. We consider variations of the following sets of parameters while keeping the rest at their baseline values: (i). the Biot parameter ; (ii). the Biot parameter M; (iii). the rock permeability k; (iv). the undrained Lame constant . For a completely dry material M = 0, whereas for a material with incompressible constituents we have 𝑀 → ∞ and → . The tensile failure areas around the borehole for different values of poroelastic parameters are plotted in Fig. 4, each figure group corresponds to variation in one of the four sets of parameters (i)-(iv). If we focus on the failure area when the sets of parameter groups (i)-(iv) are varied, it is then observed that the undrained Lame constant has a more pronounced effect on the transient failure response. For a larger value of , the borehole is easier to fail and a larger failure area is observed in the 𝜎𝐻 -direction. A larger value of or permeability k will both result in a slightly larger area of tensile failure.
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We first examine the dynamic profiles of the differential stress |𝜎1 − 𝜎3 | for both types of boundary conditions, and the results are shown in Fig. 5.
Fig. 5: Differential stress surounding the borehole: (a) ̅ ̅ ; (d) ̅ 2 .
; (b) ̅
; (c)
for the two types of boundary conditions, we plot the Mohr’s circle at two selected points near the borehole at an early and large time, respectively, and the results are shown in Fig. 6. The first selected point is in the 𝜎ℎ -direction located at 𝑟̅ = 1.2 (Fig. 6(a)), and Mohr’s circles of the stress state at ̅ and ̅ for the two different boundary conditions are plotted. From the results it’s found that the borehole stays stable at early times and the impermeable borehole delay the failure response to a relatively large time ̅ . The second selected point is in the 𝜎𝐻 -direction located at 𝑟̅ = 1.2 (Fig. 6(b)), contrary to the results obtained in the σh-direction, the permeable borehole fails at an early time ( ̅ 2) but the impermeable borehole remains stable in the entire time range. Fig. 7 shows the shear failure area surrounding the borehole after a long-time simulation. It’s interesting to figure out four symmetric failure areas near the borehole surface. The permeable borehole has a larger failure area in the σHdirection but a smaller failure area in the 𝜎ℎ -direction.
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4.2. Shear failure
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To further investigate the difference of the failure response
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Fig. 6: Stress Mohr circles for PB and IPB condition at: (a) 𝑟̅
Fig. 7: Shear failure area: (a) PB; (b) IPB.
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The shear failure modes in response to different values of poroelastic parameters are also examined and plotted in Fig. 8, with each figure group corresponding to variation in one of the four sets of parameters (i)-(iv). The four parameters all have pronounced effects on the failure response. A larger value of α will slightly enlarge the four shear failure areas
2𝜃
; (b) 𝑟̅
2𝜃
around the borehole, especially when reaches to a much higher value, for example, , the shear failure symmetrically grows into the formation in the diagonal direction. The parameter M mainly affects the failure shape in the diagonal direction. It’s interesting to observe that a smaller value of M (M = 10GPa) can cause a serious shear failure along the diagonal direction. The rock permeability has a relatively slight influence on the failure response. A higher permeability κ will enlarge the failure area in the 𝜎𝐻 -direction and shrink the failure area in the 𝜎ℎ -direction. As for the influence of the undrained Lame constant λ , we find this parameter mainly affects the borehole stability in the 𝜎𝐻 direction, and a larger λ can obviously cause a larger failure area.
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Fig. 8: Effects of four poroelastic parameters on the dynamic shear failure around a permeable borehole: (a) α = 0.5, α = 0.7, α = 0.9; (b) M=10MPa, M=100MPa, M=1000MPa; (c) κ = 340mD, κ = 34mD, κ = 3.4mD; (d) λu = 60MPa, λu = 100MPa, λu = 150MPa.
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For the baseline parameters listed in Table. 1, the transient response for both quasi-static poroelastic and poroelastodynamic theories are computed in this section, and comparisons on the failure response for a permeable borehole between the two are made below. We first compare the pore pressure profiles in the 𝜎ℎ -direction and the results are shown in Fig. 9. Since the inertial effect and diffusion process are both considered in the poroelastodynamic formulation, the pressure response is governed by a transient wave propagation
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4.3. Comparison between poroelastodynamic and poroelastic response
in early times and transitions to diffusive behavior after the inertial effect weakens and diffusion dominates (Fig. 9(a)). The early-time pressure response in the poroelastodynamic theory resembles a damped oscillator, and a low-pressure zone (even lower than the initial pore pressure) can be observed. However, the pore pressure from the quasi-static poroelastic theory behaves purely diffusive (Fig. 9(b)). At a much longer time ( ̅ ), the poroelastodynamic solution approaches the quasi-static poroelastic solution (Fig. 9(c)), as the early time inertia-induced wave-like behavior transitions to diffusive behavior.
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Fig. 9: Radial pore pressure distribution at the direction of 𝜎ℎ : (a) poroelastodynamic solution at different times; (b) poroelastic solution at different times; (c) comparison bewteen poroelastodynamic and poroelastic solution at ̅ .
shear modes in that direction.
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The tensile and shear failure responses are also compared, shown in Fig. 10 and Fig. 11, respectively. In terms of the tensile failure mode, the quasi-static poroelastic theory shows the borehole stays stable while the current poroelastodynamic theory shows failure appears in the 𝜎𝐻 -direction. Fig. 11 shows the difference of shear failure mode between the two theories. It’s interesting to find the difference in the σHdirection. Poroelastodynamic solution shows four symmetrical failure areas near the borehole surface while poroelastic solution shows only two. To further analyze the failure in the σH-direction, we plot the Mohrs circles at a selected point (where a shear failure has been marked in the poroelastodynamic calculation) for both theories at different times, and the results are shown in Fig. 12. It’s important to observe the stress Mohrs circles in the poroelastodynamic solution significantly expand and become much larger than that in the poroelastic solution in very early times so as to result in a transient tensile and shear failure after the sudden excavation. The comparison indicates that the inertial effect enhances the transient response in the early times, increasing the minimum principal stress to a tensile state in the 𝜎𝐻 direction and resulting in transient failure of both tensile and
Fig. 10: tensile failure area for: (a) poroelastic solution; (b) poroelastodynamic solution.
Fig. 11: Shear failure area for: (a) poroelastic solution; (b) poroelastodynamic solution.
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Fig. 12: Comparison of the stress Mohr circles at 𝑟̅
,𝜃
.
5. Conclusions
References
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This paper obtains poroelastodynamic solutions for dynamic stress and pressure fields to study the transient failure of a borehole excavated in a fluid-saturated poroelastic continuum subjected to a non-hydrostatic stress field. In response to a sudden excavation of a borehole under an overbalanced drilling, steep stress and pressure fronts are created at the borehole surface and they propagate into the porous medium and during the transient period, the rock fails. This wave-diffusion characteristic is significantly different from the classical purely diffusive theory in the absence of acceleration effect, where the pore pressure decreases montonically in time. The results show that the minimum principal effective stress surrounding the borehole reaches to a tensile state in the 𝜎𝐻 -direction so that the borehole is likely to experience a tensile crazing. By contrast, the impermeable borehole is more stable than the permeable borehole in terms of the tensile failure mode. The differential stress (𝜎1 − 𝜎3) surrounding the borehole is examined to find that a big differential stress is developed in the 𝜎ℎ -direction so as to result in a shear failure along that direction. By contrast, the permeable borehole is more stable in the σh-direction but more unstable in the σH-direction in terms of the shear failure mode. A detailed parametric study about the influences of four poroelastic parameters , M, k and on the failure responses of a permeable borehole has been investigated. A direct comparison between the classical quasi-static poroelastic theory and current poroelastodynamic theory is presented. Typical for diffusion problems, the quasi-static pore pressure has a very high maximum at the earliest time, while the poroelastodynamic pore pressure delays the response, which indicates the importance for inertial effect in very early times. In regard to the failure analysis, it’s interesting to find that the borehole is more unstable in the poroelastodynamic solution because the inertial effect enhances the transient response in the very early times, increasing the minimum principal stress to a tensile state in the 𝜎𝐻 -direction and resulting in transient generation of both tensile and shear failures around the borehole.
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China Postdoctoral Science Foundation (No. 2018M641598) and Chinese National Natural Science Foundation: the State Key Program (No. 51234006).
Acknowledgements The authors are grateful to the supports provided by the © 2015 The Authors. Published by Elsevier Ltd.
ACCEPTED MANUSCRIPT [24] M. Schoenball, D.P. Sahara, T. Kohl. Time-dependent brittle creep as a mechanism for time-delayed wellbore failure, Int J Rock Mech Min Sci, 70 (2014) 400–406. [25] P. Qu, R. Shen, L. Fu, et al. Time delay effect due to pore pressure changes and existence of cleats on borehole stability in coal seam, Int J Coal Geol, 85 (2) (2011) 212–218. [26] M.A. Biot. Theory of propagation of elastic waves in a fluid-saturated porous solid. i. low-frequency range, J Acoust Soc Am, 28 (2) (1956) 168– 178. [27] S.M. Hasheminejad, H. Hosseini. Dynamic stress concentration near a fluid-filled permeable borehole induced by general modal vibrations of an internal cylindrical radiator, Soil Dyn Earthq Eng, 22 (6) (2002) 441–458
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[19] G. Gazetas, N. Gerolymos, I. Anastasopoulos. Response of three athens metro underground structures in the 1999 parnitha earthquake, Soil Dyn Earthq Eng, 25 (7) (2005) 617–633. [20] X. Zhou, A. Ghassemi. Finite element analysis of coupled chemo-porothermo-mechanical effects around a wellbore in swelling shale, Int J Rock Mech Min Sci, 46 (4) (2009) 769–778. [21] S.L. Chen, Y.N. Abousleiman. Stress analysis of borehole subjected to fluid injection in transversely isotropic poroelastic medium, Mech Res Commun, 73 (2) (2016) 63–75. [22] S.L. Chen, Y.N. Abousleiman. Wellbore stability analysis using strain hardening and/or softening plasticity models, Int J Rock Mech Min Sci, 93 (2017) 260–268. [23] M.K. Rahman, D. Naseby, S.S. Rahman. Borehole collapse analysis incorporating time-dependent pore pressure due to mud penetration in shales, J Petrol Sci Eng, 28 (1) (2000) 13–31.
Transient failure of a borehole excavated in a poroelastic continuum Yang Xia , Yan Jin , Shi M. Wei , Mian Chen , Ya Y. Zhang , Yun H. Lu PII: DOI: Reference:
S0093-6413(18)30455-5 https://doi.org/10.1016/j.mechrescom.2019.03.006 MRC 3359
To appear in:
Mechanics Research Communications
Received date: Revised date: Accepted date:
13 September 2018 20 March 2019 20 March 2019
Please cite this article as: Yang Xia , Yan Jin , Shi M. Wei , Mian Chen , Ya Y. Zhang , Yun H. Lu , Transient failure of a borehole excavated in a poroelastic continuum, Mechanics Research Communications (2019), doi: https://doi.org/10.1016/j.mechrescom.2019.03.006
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ACCEPTED MANUSCRIPT HIGHLIGHTS Poroelastodynamic response is analyzed in a non-hydrostatic stress field.
Analytical solutions for stress and pressure wave propagation are obtained.
Transient tensile and shear failure responses are investigated.
Permeable and impermeable borehole conditions are considered.
A comparison between poroelastic and poroelastodynamic theory is present.
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Transient failure of a borehole excavated in a poroelastic continuum Yang Xia1, Yan Jin1*, Shi M. Wei1, Mian Chen1, Ya Y. Zhang1, Yun H. Lu1 1 State Key Laboratory of Petroleum Resource and Prospecting, China University of Petroleum (Beijing), Beijing, China, 102249 *[email protected] Tel.: +81-13810986965.
Abstract
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In this paper, the poroelastodynamic theory is employed to study the transient failure of a suddenly excavated borehole in a poroelastic continuum subjected to a non-hydrostatic far-field stress state. By introducing four scalar potential functions, the solutions for the stresses and pore pressure are given as the product of an exact treatment of the full fluid-solid coupling through field expansions with emphasis on the effect of solid-fluid acceleration. Transient tensile and shear failure responses are investigated based on an overbalanced drilling for two types of boundary conditions: a permeable surface and an impermeable surface. It's found that the permeable borehole has a larger tens ile failure area than the impermeable borehole in the direction of maximum in-situ stress, and meanwhile, four symmetric shear failure areas are observed near the borehole for both internal boundary conditions. Influences of poroelastic parameters on the transient failure responses of a permeable borehole are analyzed in a detailed parametric study. The failure responses for both classical quasi-static poroelastic and poroelastodynamic theories are computed and compared to study the importance of the inertial effect in very early times. Noteworthy is that the borehole is more unstable in the poroelastodynamic theory compared to the classical poroelastic theory because the inertial effect enhances the transient response in the early times, increasing the minimum principal stress to a tensile state in the direction of maximum in-situ stress so as to result in transient generation of both tensile and shear failures in that direction. The results in this paper provide fundamental insights on the borehole instability under some dynamic bottom-hole conditions.
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© 2015 The Authors. Published by Elsevier Ltd.
Keywords: Poroelastodynamics, Transient failure, Analytical solution, Wellbore stability
1. Introduction
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There has been a progressing interest in transient response of a borehole excavated in fluid-saturated porous media due to its important applications in various technical and engineering processes, for example, analyzing dynamic wellbore instability, generation of microfractures surrounding the borehole as well as instantaneous fracture initiation. In these dynamic problems, the inertial effect of the solid-fluid system becomes important and it has to be taken into account [1]. The quasi-static poroelastic response of a borehole was introduced by Detournay and Cheng [2]. On the other hand, the propagation theory of elastic waves in fluid-saturated porous media proposed by Biot [3] suggests that the inertia effect of the solid-fluid system have a decisive effect on the transient response of the rock [4, 5]. In addition, the scientific groundwork for Biot's model of poroelastodynamics has been more firmly established through some experimental validations of its most fundamental predictions [6-8], leading to increasing interests of the dynamic theory. The wave phenomena in the fluid saturated formation are paid special attention in recent years [9-10], and the dynamic response of the borehole may lead to the generation of the microfractures surrounding the borehole [11]. Thus, the need for applying the dynamic theory to study the transient failure of the borehole has been arised. The dynamic extension of Biot's theory to three phases was first published by Vardoulakis and Beskos [12]. Further development of this kind of model to describe the dynamic behavior of rocks has been found in [13-15]. Local heterogeneities in porous materials have been considered by Wei and Muraleetharan [16]. Schanz and Cheng [17] examined the wave propagating behavior in a onedimensional colume. Recently, Xia et al. presented an exact
closed-form solution for poroelastodynamic response of a borehole in a non-hydrostatic stress field [18]. These works are great attempts for analyzing poroelastodynamic problems. In some highly non-linear dynamic problems, the complex boundary conditions in terms of heterogeneity of the poroelastic parameters can take advantage of the use of numerical methods [19]. However, analytical solutions are still very popular for most cases since they are much more convenient to implement for analyzing the general transient behavior. Today, the wellbore instability model has been well developed to incorporate the effects of poroelasticity, elastoplastic, chemical and thermal diffusions between the drilling mud and the formation, dual-porosity system [20-22]. Time-dependent wellbore instability becomes of great interest. Rahman et al. considered the increase of pore pressure around the borehole due to mud penetration to study the time-dependent borehole instability [23]. Schoenball et al. used an implementation of time-dependent brittle creep to study the breakout growth in time and time-delayed wellbore instability [24]. Qu et al. studied the time-delay borehole collapse in underbalanced drilling in coal seam due to the pore pressure change in the fractures [25]. These works suggest that the borehole remains stable at the beginning but would probably fail due to some time-delay effects. However, few works have been performed to analyze the transient failure in the short-term change of the loading boundary conditions, such as the transient response of a sudden pressing, unloading or excavation in the porous rock. In this paper, we study the transient failure of a suddenly excavated borehole in a poroelastic continuum in a nonhydrostatic stress field based on Biot's theory of poroelastodynamics. Both tensile and shear failure modes are studied for permeable and impermeable boundary conditions. In particular, the failure responses for both classical quasistatic poroelastic and poroelastodynamic theories are
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computed and compared to show the importance of the inertial effect in very early times. The results in this paper provide fundamental insights on the borehole instability under some dynamic bottom-hole conditions. 2. Problem formulation
Based on Biot's dynamic theory of poroelasticity, the medium around the borehole is taken to be a macroscopically homogeneous and isotropic two-component solid/fluid system. The stress tensor 𝜎𝑖𝑗 and the mean pore fluid pressure p are given by (note that tension is here taken positive): (1) ij (u e M ) ij 2eij
p M ( e)
(2)
s2 0 2 0 0 2 2 f s
2 u Ku 3
2 Ks K f Ku K K s ( ) K f
(9)
K Ks
(3)
In which is the shear modulus of the bare skeletal frame, the porosity, K , K s , K f the bulk modulus of the drained
w+ w k
the fast compressional, slow compressional, and the elastic shear waves, respectively, are given as:
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Fig. 1. A borehole subject to anisotropic stresses.
where and f are the density of porous solid and pore fluid, and k and are the permeability of the porous rock and viscosity of the fluid, respectively. a is a nondimensional tortuosity factor with ( a f ) / a f ,
2 f ,s
1 12 4 2 2
2f s 4 s a f s 2 2bs 2
2
(12) (13)
where 1
(u 2 )(a f s 2 b s) M s 2 2 M f s 2 (u 2 ) M
2
(5)
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porous rock, solid grains and pore fluid, respectively. e and are the dilatations of the solid and fluid, respectively. K u is the bulk modulus of the undrained system. u and w are the displacement vectors of the solid and the fluid, respectively. The governing equations governed by Biot's theory [26] in displacement form with dissipation of interstitial fluid taken into account take the following forms: (4) (u 2 ) u M w u = u + f w
a f
(11) 2 2 0 where f , s and designate the complex wave numbers of
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K f (Ks K ) Ks (Ks K f )
M u + M w f u +
a f s s k 2
By wave analysis, the above coupled equations can be manipulated to yield Helmholtz equations [27]: (10) 2 f , s 2f , s f , s 0
K f K s2
1
f s2
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Where
M
in the plot. In the following content, we state the direction of maximum in-situ stress and minimum in-situ stress by σ𝐻 direction and σℎ -direction. Helmholtz decomposition theorem allows us to resolve the displacement fields as superposition of longitudinal and transverse vector components [18]: (6) u (7) w Substituting the above resolutions into the field equations Eq. (4) and Eq. (5), we obtain two sets of coupled equations after performing Laplace transformation: s2 f s2 u 2 M 2 (8) a f s2 M 2 2 M s f s k
b
a f s 4 b 2 s 3 2f s 4 (u 2 ) M 2
rw k
(14) (15) (16)
Employing Eqs. 6-11, with some manipulations, the potentials , , , and have the following relations:
f s , f f s s , 0
(17)
where
f
M 2f f s 2 a f s 2 b s M 2f
(18)
s
M s2 f s 2 a f s 2 b s M s2
(19)
0
f s 2 a f s 2 bs
(20)
while a is the added fluid mass density and a c f [17].
3. Field expansions and boundary conditions
Consider a vertical borehole drilled in a formation that is characterized by a non-hydrostatic horizontal in-situ stress field, shown in Fig. 1, and the initial stress state is also shown
The transmitted fast dilatational wave, slow dilatational wave and the shear wave at the borehole surface into the
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f An K n ( f r )cos n
(21)
s Bn K n ( s r )cos n
(22)
n 0
n 0
Cn K n ( r )sin n
(u M f ) 2f f
(32) ur u ) r r The solutions of stresses, pore pressure can be expressed in a superposed manner of the axisymmetric and asymmetric modes:
(u M s ) s2 s 2 (
rr rr(0) rr(2) cos2 (0) (2) cos2 r r(2) sin2
(23)
n 1
The far-field stresses are composed of only two parts: a constant term and a 2𝜃-trigonometric term, thus, the infinite series of the potential functions can be simplified to the superposed form of an axisymmetric and an asymmetric function: (24) f A0 K0 ( f r ) A2 K2 ( f r )cos2
s B0 K0 ( s r ) B2 K2 ( s r )cos2
(25)
C2 K2 (r )sin2
(26) So, expressions for the frame and liquid dilatations can be obtained in the Laplace domain to yield: (27) e u = 2 2 f 2 s 2f f s2 s
w = f f s s 2 f
(u M s ) s2 B0 K 0 ( s r )
u rr (u M f ) f (u M s ) s 2 r r 2 s
u u r ( r r u ) r r
(u M s ) s2 B0 K 0 ( s r )
( s ) s2 MB0 K 0 ( s r )
(30)
M
2f A0
K 0 ( f r )
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K ( r ) K3 ( r ) 4 C2 K 2 ( r ) 2C2 1 r2 r K1 ( f r ) K3 ( f r ) (2) 2 2 (u M f ) f A2 K 2 ( f r ) (u M s ) s B2 K 2 ( s r ) f A2 r A2 K 2 ( f r ) B2 K 2 ( s r ) K1 ( s r ) K3 ( s r ) 4 K1 ( r ) K3 ( r ) s B2 2 C2 K 2 ( r ) 8 2C2 r r r2 r K1 ( f r ) K 3 ( f r ) A2 K 2 ( f r ) B2 K 2 ( s r ) K1 ( s r ) K3 ( s r ) (2) r 2 f A2 2 s B2 4 r r r2 2 K ( r ) K3 ( r ) 4 C2 [ K 0 ( r ) 2 K 2 ( r ) K 4 ( r )] 2 C2 K 2 ( r ) C2 1 4 r 2 r p(2) ( f ) 2f MA2 K2 ( f r ) ( s ) s2 MB2 K2 ( s r ) 2
K 4 ( f r )
2
K 0 ( s r )
B2 2 s
2
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(39)
(40)
K 4 ( s r )
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B2
2 s
p (0) ( f ) 2f MA0 K 0 ( f r )
rr(2) (u M f ) 2f A2 K 2 ( f r ) (u M s ) s2 B2 K 2 ( s r ) 2 s
(38)
A0 K 2 ( f r ) B0 K 2 ( s r ) 2 f
(31)
The solutions in the asymmetric mode read:
2f A2
(37)
2 s
(0) (u M f ) 2f A0 K 0 ( f r )
Utilizing Eqs. 1, 2, 17, 27, 28, pore fluid pressure and the stress components can be obtained: (29) p (M f M ) 2f f (M s M ) s2 s 2 f
(35)
A0 K 2 ( f r ) B0 K 2 ( s r ) 2 f
(28)
2 s
(34)
(2)
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2
(33)
(36) p p p cos2 where the superscript (0) and (2) represent the axisymmetric and asymmetric mode, respectively. Substituting Eqs. 24-26 into Eqs. 29-32, the solutions in the axisymmetric mode read: rr(0) (u M f ) 2f A0 K 0 ( f r ) (0)
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poroelastic continuum are, respectively, represented in the Laplace transform domain:
The radial position r and time t have been nondimensionalized by the radius of the borehole rw and a time scale rw / , respectively. In this work, we consider two types of boundary conditions: an ImPermeable Borehole (IPB) and a Permeable Borehole (PB). IPB condition means the rock is extremely tight and the fluid flow through the borehole surface into the rock can be neglected, thus the flux equals zero at the surface. PB condition means the fluid flow through the borehole surface into the rock is considered to maintain the bottom-hole pressure. The inner boundary conditions can be expressed as follows after performing Laplace transformation:
P0 pw (0) rr (1, s ) S0 sin2 s , r(2) (1, s ) S cos2 s (2) (1, s ) 0 rr s pw p0 (0) qr(0) (1, s) 0 p (1, s) PB : s ; IPB: (2) qr (1, s) 0 p (0) (1, s) 0
(41)
(42)
(43)
(44)
(45)
where the bottom-hole pressure is denoted by pw. The complete solution can be obtained by superposing the axisymmetric and asymmetric modes and adding the unperturbed field. The solution in real time domain can be obtained by using the Stehfest method. We consider two failure modes: tensile failure and shear
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tan ( 1 3 ) 2Ct
(47) 3 1 0 tan 2 1 Where St , Ct and are the tensile strength, cohesion and internal friction angle, respectively.
shear failure: F
4. Transient failure of the borehole In all the calculations and plots, the displacements and time are scaled by the radius of the borehole rw and rw / , respectively. The values of used parameters in the calculations are listed in Table. 1.
Table 1: Parameter sets used in the calculation.
4.1. Tensile failure
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We first examine the dynamic profiles of the minimum principal stress surrounding the borehole with the twodimensional isobaric plots. Fig. 2 shows four relatively higher compressive stress zones around the borehole are symmetrically generated along with the development of two tensile stress zones in the σ𝐻 -direction near the borehole. A higher tensile stress is observed near the permeable borehole surface compared to the impermeable borehole surface. Fig. 3 shows the tensile failure area surrounding the borehole after a long-time simulation. It’s found that the permeable borehole is more unstable under the same condition.
undrained Lame constant [GPa] Biot parameter M [GPa] rock density [kg/m3] pore fluid viscosity [Pa·s] rock porosity minimum principal stress σℎ [MPa] bottom-hole pressure pw [MPa] Cohesion Ct [MPa] Borehole radius rw [m]
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Fig. 2: Minimum principal stress surounding the borehole: (a) ̅ (b) ̅ ; (c) ̅ ; (d) ̅ 2 .
;
74.1 100 2000 1×10-4 0.5 37.5 30 5 0.1
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shear modulus [GPa] 5 Biot parameter 0.7356 pore fluid density [kg/m3] 1000 rock permeability k [mD] 34.1 tortuosity factor a 1.6 maximum principal stress 𝜎𝐻 [MPa] 62.5 initial pore pressure p0 [MPa] 25 Tensile strength St [MPa] 5 Internal friction angle 40
Fig. 3: Tensile failure area: (a) PB; (b) IPB
We then investigate the tensile failure modes for a permeable borehole under various sets of poroelastic parameters. Specifically, the baseline for all parameters are listed in Table. 1. We consider variations of the following sets of parameters while keeping the rest at their baseline values: (i). the Biot parameter ; (ii). the Biot parameter M; (iii). the rock permeability k; (iv). the undrained Lame constant . For a completely dry material M = 0, whereas for a material with incompressible constituents we have 𝑀 → ∞ and → . The tensile failure areas around the borehole for different values of poroelastic parameters are plotted in Fig. 4, each figure group corresponds to variation in one of the four sets of parameters (i)-(iv). If we focus on the failure area when the sets of parameter groups (i)-(iv) are varied, it is then observed that the undrained Lame constant has a more pronounced effect on the transient failure response. For a larger value of , the borehole is easier to fail and a larger failure area is observed in the 𝜎𝐻 -direction. A larger value of or permeability k will both result in a slightly larger area of tensile failure.
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We first examine the dynamic profiles of the differential stress |𝜎1 − 𝜎3 | for both types of boundary conditions, and the results are shown in Fig. 5.
Fig. 5: Differential stress surounding the borehole: (a) ̅ ̅ ; (d) ̅ 2 .
; (b) ̅
; (c)
for the two types of boundary conditions, we plot the Mohr’s circle at two selected points near the borehole at an early and large time, respectively, and the results are shown in Fig. 6. The first selected point is in the 𝜎ℎ -direction located at 𝑟̅ = 1.2 (Fig. 6(a)), and Mohr’s circles of the stress state at ̅ and ̅ for the two different boundary conditions are plotted. From the results it’s found that the borehole stays stable at early times and the impermeable borehole delay the failure response to a relatively large time ̅ . The second selected point is in the 𝜎𝐻 -direction located at 𝑟̅ = 1.2 (Fig. 6(b)), contrary to the results obtained in the σh-direction, the permeable borehole fails at an early time ( ̅ 2) but the impermeable borehole remains stable in the entire time range. Fig. 7 shows the shear failure area surrounding the borehole after a long-time simulation. It’s interesting to figure out four symmetric failure areas near the borehole surface. The permeable borehole has a larger failure area in the σHdirection but a smaller failure area in the 𝜎ℎ -direction.
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4.2. Shear failure
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To further investigate the difference of the failure response
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Fig. 6: Stress Mohr circles for PB and IPB condition at: (a) 𝑟̅
Fig. 7: Shear failure area: (a) PB; (b) IPB.
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The shear failure modes in response to different values of poroelastic parameters are also examined and plotted in Fig. 8, with each figure group corresponding to variation in one of the four sets of parameters (i)-(iv). The four parameters all have pronounced effects on the failure response. A larger value of α will slightly enlarge the four shear failure areas
2𝜃
; (b) 𝑟̅
2𝜃
around the borehole, especially when reaches to a much higher value, for example, , the shear failure symmetrically grows into the formation in the diagonal direction. The parameter M mainly affects the failure shape in the diagonal direction. It’s interesting to observe that a smaller value of M (M = 10GPa) can cause a serious shear failure along the diagonal direction. The rock permeability has a relatively slight influence on the failure response. A higher permeability κ will enlarge the failure area in the 𝜎𝐻 -direction and shrink the failure area in the 𝜎ℎ -direction. As for the influence of the undrained Lame constant λ , we find this parameter mainly affects the borehole stability in the 𝜎𝐻 direction, and a larger λ can obviously cause a larger failure area.
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Fig. 8: Effects of four poroelastic parameters on the dynamic shear failure around a permeable borehole: (a) α = 0.5, α = 0.7, α = 0.9; (b) M=10MPa, M=100MPa, M=1000MPa; (c) κ = 340mD, κ = 34mD, κ = 3.4mD; (d) λu = 60MPa, λu = 100MPa, λu = 150MPa.
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For the baseline parameters listed in Table. 1, the transient response for both quasi-static poroelastic and poroelastodynamic theories are computed in this section, and comparisons on the failure response for a permeable borehole between the two are made below. We first compare the pore pressure profiles in the 𝜎ℎ -direction and the results are shown in Fig. 9. Since the inertial effect and diffusion process are both considered in the poroelastodynamic formulation, the pressure response is governed by a transient wave propagation
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4.3. Comparison between poroelastodynamic and poroelastic response
in early times and transitions to diffusive behavior after the inertial effect weakens and diffusion dominates (Fig. 9(a)). The early-time pressure response in the poroelastodynamic theory resembles a damped oscillator, and a low-pressure zone (even lower than the initial pore pressure) can be observed. However, the pore pressure from the quasi-static poroelastic theory behaves purely diffusive (Fig. 9(b)). At a much longer time ( ̅ ), the poroelastodynamic solution approaches the quasi-static poroelastic solution (Fig. 9(c)), as the early time inertia-induced wave-like behavior transitions to diffusive behavior.
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Fig. 9: Radial pore pressure distribution at the direction of 𝜎ℎ : (a) poroelastodynamic solution at different times; (b) poroelastic solution at different times; (c) comparison bewteen poroelastodynamic and poroelastic solution at ̅ .
shear modes in that direction.
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The tensile and shear failure responses are also compared, shown in Fig. 10 and Fig. 11, respectively. In terms of the tensile failure mode, the quasi-static poroelastic theory shows the borehole stays stable while the current poroelastodynamic theory shows failure appears in the 𝜎𝐻 -direction. Fig. 11 shows the difference of shear failure mode between the two theories. It’s interesting to find the difference in the σHdirection. Poroelastodynamic solution shows four symmetrical failure areas near the borehole surface while poroelastic solution shows only two. To further analyze the failure in the σH-direction, we plot the Mohrs circles at a selected point (where a shear failure has been marked in the poroelastodynamic calculation) for both theories at different times, and the results are shown in Fig. 12. It’s important to observe the stress Mohrs circles in the poroelastodynamic solution significantly expand and become much larger than that in the poroelastic solution in very early times so as to result in a transient tensile and shear failure after the sudden excavation. The comparison indicates that the inertial effect enhances the transient response in the early times, increasing the minimum principal stress to a tensile state in the 𝜎𝐻 direction and resulting in transient failure of both tensile and
Fig. 10: tensile failure area for: (a) poroelastic solution; (b) poroelastodynamic solution.
Fig. 11: Shear failure area for: (a) poroelastic solution; (b) poroelastodynamic solution.
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Fig. 12: Comparison of the stress Mohr circles at 𝑟̅
,𝜃
.
5. Conclusions
References
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This paper obtains poroelastodynamic solutions for dynamic stress and pressure fields to study the transient failure of a borehole excavated in a fluid-saturated poroelastic continuum subjected to a non-hydrostatic stress field. In response to a sudden excavation of a borehole under an overbalanced drilling, steep stress and pressure fronts are created at the borehole surface and they propagate into the porous medium and during the transient period, the rock fails. This wave-diffusion characteristic is significantly different from the classical purely diffusive theory in the absence of acceleration effect, where the pore pressure decreases montonically in time. The results show that the minimum principal effective stress surrounding the borehole reaches to a tensile state in the 𝜎𝐻 -direction so that the borehole is likely to experience a tensile crazing. By contrast, the impermeable borehole is more stable than the permeable borehole in terms of the tensile failure mode. The differential stress (𝜎1 − 𝜎3) surrounding the borehole is examined to find that a big differential stress is developed in the 𝜎ℎ -direction so as to result in a shear failure along that direction. By contrast, the permeable borehole is more stable in the σh-direction but more unstable in the σH-direction in terms of the shear failure mode. A detailed parametric study about the influences of four poroelastic parameters , M, k and on the failure responses of a permeable borehole has been investigated. A direct comparison between the classical quasi-static poroelastic theory and current poroelastodynamic theory is presented. Typical for diffusion problems, the quasi-static pore pressure has a very high maximum at the earliest time, while the poroelastodynamic pore pressure delays the response, which indicates the importance for inertial effect in very early times. In regard to the failure analysis, it’s interesting to find that the borehole is more unstable in the poroelastodynamic solution because the inertial effect enhances the transient response in the very early times, increasing the minimum principal stress to a tensile state in the 𝜎𝐻 -direction and resulting in transient generation of both tensile and shear failures around the borehole.
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China Postdoctoral Science Foundation (No. 2018M641598) and Chinese National Natural Science Foundation: the State Key Program (No. 51234006).
Acknowledgements The authors are grateful to the supports provided by the © 2015 The Authors. Published by Elsevier Ltd.
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