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Analytical solution of a circular opening in an axisymmetric elastic-brittle-plastic swelling rock Mohsen S. Masoudian*, Mir Amid Hashemi Nottingham Centre for Geomechanics, Faculty of Engineering, The University of Nottingham, Nottingham, NG7 2RD United Kingdom

a r t i c l e i n f o

a b s t r a c t

Article history: Received 16 June 2016 Received in revised form 8 August 2016 Accepted 30 August 2016 Available online 31 August 2016

Unconventional gas reservoirs such as coal and shale have been increasingly considered for methane production and CO2 sequestration, over the last decades. In these reservoirs, methane and/or CO2 are usually in an adsorbed state which is associated with swelling and/or shrinkage. There exist a number of experimental and theoretical studies on the effect of swelling and/or shrinkage in prediction of permeability, stress, and displacement distribution. However, most of these studies have only considered the elastic deformation of the reservoir. The plastic deformations within brittle reservoir rock can have signiﬁcant implications for production, injectivity and stability and the wellbore and the reservoir. Therefore, development of improved models to estimate the distribution of stress and deformation around the wellbore is of great importance. A large number of analytical solutions for axisymmetric opening problem have been presented in the literature where different models are used for material behaviour. This paper aims to include the effect of swelling/shrinkage in the elasto-plastic formulations around the wellbore. The reservoir is assumed to behave as a linear elastic material up to the yield point, which is identiﬁed by the Mohr-Coulomb failure criterion. The post-failure brittle behaviour of the rock is modelled by deﬁning the residual strength parameters and employing a non-associated ﬂow rule. Strains are decomposed into mechanical elastic, elastic swelling/shrinkage, and mechanical plastic parts. Although the swelling/shrinkage strains are considered to be elastic, their distributions are closely linked with distributions of plastic strains through sophisticated integral and differential relationships. The swelling/shrinkage is deﬁned using a Langmuirlike curve, which is directly related to the pore pressure distribution within the reservoir. The model is then used to study the distributions of stress and strain around the wellbore, in both elastic and plastic zones and veriﬁed against a numerical solution. A parametric study is also conducted by deﬁning different values for swelling parameters, and pre-failure and residual strength parameters. The provided model can be useful to estimate the failed zone around the wellbore, where the formation is irreversibly damaged. On the other hand, the estimated distributions and radial and tangential stress from this model can help develop new permeability models for unconventional reservoirs. © 2016 Elsevier B.V. All rights reserved.

Keywords: Geomechanics Wellbore stability Analytical solution Brittle failure Swelling and shrinkage Unconventional gas

1. Introduction Methane production from unconventional gas reservoirs such as coal and shale has signiﬁcantly increased over the last decade, and they are also considered for CO2 sequestration purposes. In these reservoirs, methane is an adsorbed state and its production is associated with a desorption-induced shrinkage. On the other hand, during CO2 sequestration, adsorption of CO2 in matrix of rock,

* Corresponding author. E-mail addresses: [email protected], nottingham.ac.uk (M.S. Masoudian). http://dx.doi.org/10.1016/j.jngse.2016.08.076 1875-5100/© 2016 Elsevier B.V. All rights reserved.

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leads to swelling of the matrix blocks. The multi-physical processes associated with CO2 sequestration and methane production from unconventional reservoirs include thermo-chemo-hydromechanical interactions at different nano, micro, meso and macro scales. These processes and their interactions have been discussed and reviewed in by many investigations of coal (e.g. Busch and Gensterblum, 2011; Liu et al., 2011; Masoudian, 2016; Masoudian et al., 2013c) and shale (e.g. Huang et al., 2015; Meakin et al., 2013) and hydrate-bearing (e.g. Moridis et al., 2011; Rutqvist et al., 2012) reservoirs and they continue to attract many research efforts. Among all, the sorption-induced swelling/shrinkage has received a great attention, mainly due to its effect on permeability

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of the reservoir, and numerous models are suggested to relate the permeability of the unconventional reservoirs to the change in the stress distribution or the change in the fracture porosity as reviewed and discussed by many studies (e.g. Amann-Hildenbrand et al., 2012; Pan and Connell, 2012; Spencer, 1989). There exist a large number of experimental and theoretical studies on the effect of swelling and/or shrinkage in prediction of permeability and stress distribution within the reservoir (e.g. Chen et al., 2010; Connell, 2009; Cui et al., 2007; Masoudian et al., 2013a, 2016a; Sherwood and Bailey, 1994). However, most of these studies have only considered the elastic deformation of the reservoir. The plastic deformations within the reservoir rock can have signiﬁcant implications for production, injectivity, and stability of the wellbore and the reservoir. The need for elastoplastic analysis and the signiﬁcance of the plastic deformations around wellbores have been presented in the literature (e.g. Han and Dusseault, 2003). Recently, Lu and Connell (2016) have conducted a theoretical study and suggested that coal is more likely to undergo failure during gas production than other reservoirs. In addition, the stability of the wellbores is an important issue in any oil and gas production project. Therefore, improved models are needed to properly estimate the deformation and stress distributions around the wellbore. A large number of analytical solutions for axisymmetric opening problem have been presented in the literature using different models of material behaviour, such as the elastic-perfectly plastic, elasticebrittleeplastic (Park and Kim, 2006) and elastic-strain softening/hardening (Chen et al., 2012; Zhang et al., 2012) models, along with the linear MohreCoulomb (MeC) and the nonlinear HoekeBrown (HeB) (Sharan, 2003) and modiﬁed Cam Clay (Chen and Abousleiman, 2013) failure envelopes. Most of these solutions are developed for circular tunnels or boreholes in rocks with no ﬂuid interactions and therefore, the stresses and deformations are simply mechanical. Three of the existing models are of more signiﬁcance to this paper. Han and Dusseault (2003) provided a poro-elastic-perfectly plastic solution for a wellbore subjected to steady-state radial ﬂow using M-C failure criterion. Park and Kim (2006) developed a closed-form solution for both M-C and H-B models to the problem of circular opening within an elastic-brittle-plastic continuum subjected to constant stresses at the boundaries. In addition, Zareifard and Fahimifar (2015) have also developed a poro-elastic-brittle-plastic solution for a deep tunnel in presence of steady-state groundwater ﬂow. However, the existing models may not be directly applicable to unconventional gas recovery and/or CO2 sequestration, mainly due to the complex nature of the physico-chemical phenomena in unconventional reservoirs. Firstly, swelling is a major player in hydro-mechanical performance of the unconventional reservoirs. Secondly, the experimental studies have revealed the brittle mechanism of failure in shale (e.g. Amann et al., 2011; Hull et al., 2015; Lisjak et al., 2014) and coal (Deisman et al., 2008; Gentzis et al., 2007; Masoudian-Saadabad et al., 2012; Masoudian et al., 2013b, 2014), and therefore, including the brittle failure mechanism in the models seems necessary. In order to address the shortcomings of the existing models, this paper aims to include the effect of swelling/shrinkage strain and the brittle failure mechanism in poro-elastic-plastic solution of a wellbore. To achieve this, the reservoir was assumed to behave as linear elastic up to the yield point, identiﬁed by the linear MohreCoulomb (MeC) failure criterion. The post-failure brittle behaviour of the rock is considered through the use of residual strength parameters and employing a non-associated ﬂow rule. In the plastic region, the total strains are decomposed into mechanical elastic, elastic swelling/shrinkage, and mechanical plastic parts. Although the swelling/shrinkage strains are considered to be elastic, their distributions can closely affect the distributions of

plastic strains through sophisticated integral and differential coupled relationships. The swelling/shrinkage is deﬁned using a Langmuir-like curve, which is directly related to the pore pressure distribution within the reservoir. The model was then used to study the distributions of stress and strain around the wellbore, in both elastic and plastic zones. The sensitivity of the elasto-plastic deformations was also studied by considering different values for swelling parameters, and pre-failure and residual strength parameters. The provided model can be useful to estimate the failed zone around the wellbore, where the formation is irreversibly damaged. On the other hand, the estimated distributions and radial and tangential stress from this model can contribute to developing new permeability models for unconventional reservoirs. 2. Problem deﬁnition When considering the elastic-plastic deformations, it is of great importance to recognise the role and cause of different components of strain. The strain within the plastic zone has two components of elastic (εe) and plastic (εp) strains. However, the elastic strain can be decomposed into elastic mechanical strain, εe,m, and elastic swelling (or shrinkage) strain, εe,s, as stated below

ε ¼ εe þ εp ¼ εe;m εe;s þ εp

(1)

Note that throughout this paper, the term ‘swelling strain’ is used to refer to εe,s, but it can represent the adsorption-induced swelling (e.g. CO2 sequestration) or desorption-induced shrinkage (e.g. coalbed methane recovery), or their combination (CO2-enhanced coalbed methane recovery). Also note that the adsorption induces a positive swelling strain while desorption induces a negative swelling (shrinkage) strain. This is the opposite of the common signs conventions in geomechanics and hence the sign for swelling strain term is negative in Equation (1). On the other hand, the two most commonly-used failure criteria are the linear Mohr-Coulomb (M-C) and nonlinear Hoek-Brown (HB), which can be written in light of Biot's effective stress deﬁnition (s0 ¼saP), as below 0 s01 ¼q gsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3þY ﬃ MC 0 0 0 2 s1 ¼ s3 þ ms3 sc þ ssc H B

(2)

where s01 and s03 are the major and minor principal effective stresses at failure, respectively. sc is the uniaxial compressive strength of the intact rock, and m and s are the Hoek-Brown constants which depend on the properties of the rock. g and Y can be deﬁned based on the cohesion, c, and the friction angle, f, of the rock as

g¼

1 þ sin f 1 sin f

Y¼

2c cos f 1 sin f

(3)

Fig. 1 shows the plane view of a vertical borehole within a continuous homogenous isotropic elastic-plastic reservoir, under in-situ stress and pore pressure of s0 and p0, respectively. The wellbore is subjected to a ﬁxed internal pressure of pw, and the radial stress is kept at its initial level at the outer boundary. The plastic zone is the region immediately in the vicinity of the wellbore in which the rock undergoes failure and plastic deformation, while the elastic zone is the region in which the stresses do not exceed the strength of the rock and therefore the deformations remain elastic. Due to the axial symmetry of the problem, the major principal

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485

Fig. 1. Plane-view of a vertical wellbore in an elastic-plastic reservoir with ﬁxed radial stress at both inner and outer boundaries.

stresses are the tangential effective stress, s0qq , and radial effective stress, s0rr , and therefore, the failure criteria can be re-written as below 0 s0qq ¼q gsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ MC rr þ Y ﬃ 0 0 0 2 sqq ¼ srr þ msrr sc þ ssc H B

(4)

The material behaviour chosen for this study is shown in Fig. 2. The rock exhibits a linear elastic behaviour before failure. After the failure at peak strength, the strength of the rock suddenly drops and follows the post-yield behaviour. For a brittle rock, the failure criteria for the post-peak strength is modiﬁed to substitute the residual strength parameters to give

s0qq ¼q grﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s0rr þ Yr MC 0 0 2 sqq ¼ srr þ mr srr sc þ sr sc H B 0

(5)

where mr and sr are the residual H-B constants. gr and Yr are also explained with Equation (3) in which the residual cohesion, cr, and residual friction angle, fr substitute cohesion and friction angle. In this paper, we do not employ the nonlinear H-B failure criterion since its use in coupled formulations will result in a nonlinear nonhomogeneous ﬁrst order differential equation and ﬁnding an exact solution is not easy, although it has been widely used for mechanical-only (no ﬂuid ﬂow) problems (e.g. Park and Kim, 2006). Therefore, to simplify the analytical approach, we employed the MohreCoulomb criterion when solving the stress and strain in plastic zone to avoid the nonlinearity of the differential equations. 3. Solution of elastic zone The analytical solution of wellbore in an axisymmetric poroelastic reservoir with and without swelling behaviour has been previously studied (Cui et al., 2007; Han and Dusseault, 2003; Masoudian et al., 2016b) and it has been implemented in

numerical reservoir simulation studies. Studies have provided a solution that included the isotropic swelling effect, where the stress and displacement within an isotropic rock under plane-strain condition are given as

ð1 2yÞ FP E Fε þ ð1 yÞ r 2 3ð1 yÞ r 2 C ð1 2yÞ F E Fε εe;s sqq ¼ C1 þ 22 þ aP 2P þ v 2 ð1 yÞ 3ð1 yÞ r r r

srr ¼ C1

C2 r2

þ

ð1 þ yÞð1 2yÞ ð1 þ yÞ C2 rðC1 s0 þ ap0 Þ þ E E r ð1 þ yÞð1 2yÞ FP ð1 þ yÞ Fε ð1 þ yÞ e;s rεv;0 þ Eð1 yÞ 3ð1 yÞ r 3 r

(6)

u¼

where E is the elastic modulus, y is the Poisson's ratio, a is the Biot's is the coefﬁcient, and P is the reservoir pore pressure, and εe;s v volumetric swelling strain, u is the radial displacement which is positive towards the centre of the wellbore. It should be also noted that the appearance of in-situ stress, s0, in-situ pore pressure, p0, and in-situ swelling strain, εe;s v;0 , is because the displacement we are interested in, are only those induced by gas production and therefore the displacements induced by loading the initial conditions are deduced from the equations. Note that C1 and C2 are the integration constants which can be obtained using speciﬁed boundary conditions. Also, note that the borders of the elastic zone within the reservoir are the outer boundary, R0, and the elasticplastic interface, Rep. It is important to note that the value of Rep is not known yet, but the procedure for ﬁnding the elastic-plastic interface will be discussed later in this paper. For constant radial stresses at the outer boundary of the reservoir (i.e. equal to the inep situ stress) s0, and at the elastic-plastic interface, srr , the integration constants can be found as

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C1 ¼ C2 ¼

1 R20 R2ep 1 R20 R2ep

E ep ð1 2yÞ FP Rep FP ðR0 Þ þ Fε Rep Fε ðR0 Þ þ ð1 yÞ R20 s0 R2ep srr 3 ð1 yÞ

E ep ð1 2yÞ R20 FP Rep R2ep FP ðR0 Þ þ R20 Fε Rep R2ep Fε ðR0 Þ þ R20 R2ep s0 srr ð1 yÞ 3 ð1 yÞ

Equation (8) as below

Fp and Fε are deﬁned as

Z Fp ¼

aPrdr

Z

(8)

εe;s v rdr

Fε ¼

In order to ﬁnd Fp and Fε, a steady state ﬂow can be assumed so that the radial proﬁle of pore pressure can be approximated using the following logarithmic equation (Cui et al., 2007)

PðrÞ ¼ p0 þ Q ln

r R0

r2 r2 Fp ¼ aQ þ aP 4 2 " # εsL 2 2R20 1 þ bL p0 1 þ bL P exp 2 $Ei 2 Fε ¼ r bL Q 2 bL Q bL Q

where Ei(x) is the exponential integral function deﬁned as Z x et =tdt. EiðxÞ ¼ For an axisymmetric isotropic reservoir under a plane-strain condition (i.e. εezz ¼ 0), the strains are related to radial displacement, u, as below

vu vr u εqq ¼ r εrr ¼

(10)

In addition, the amount of gas present in the matrix of the rock, Vs, can be estimated using a sorption isotherm, such as Langmuir's:

ð1 þ yÞð1 2yÞ FP ð1 þ yÞ Fε ð1 þ yÞð1 2yÞ ð1 þ yÞ C2 ð1 þ yÞ e;s ðC1 s0 þ ap0 Þ þ εv;0 εeqq ¼ þ þ Eð1 yÞ E E 3 r 2 3ð1 yÞ r 2 r2

Vs ¼

VLs bL P 1 þ bL P

4. Solution of plastic zone 4.1. Solution of stress

where is the maximum sorption capacity, and bL is the Langmuir's isotherm constant. The volumetric swelling strain changes e;s from the initial state of εe;s v;0 to εv , and it is considered to be only elastic and that a linear relationship with concentration of gas in the matrix can be valid as below

bL P 1 þ bL P

(12)

Where εsL is the maximum swelling strain that can only takes place when the gas within the matrix is equal to the sorption capacity. Note that the term εe;s v;0 introduced in Equation (6) is used to account for the in-situ equilibrium state of the reservoir at its in-situ pressure of p0 and can be estimated using Equation (12) to give

εe;s v;0

¼

εsL

bL p 0 1 þ bL p0

(16)

(11)

VLs

s εe;s v ¼ εL

(15)

Combining this equation with displacement in Equation (6), the radial and tangential strains in the elastic region can be written as

ð1 þ yÞð1 2yÞ FP ð1 þ yÞ Fε ð1 þ yÞð1 2yÞ ð1 þ yÞ C2 ð1 þ yÞ e;s e;s ðC1 s0 þ ap0 Þ εv;0 þ a P þ ε þ v ð1 yÞE 3ð1 yÞ r 2 E E 3 r2 r2

εerr ¼

(14)

∞

(9)

where Q is deﬁned as a function of the well pressure and uniform initial reservoir pressure, p0, as below

pw p0 Q¼ lnðRw =R0 Þ

(7)

(13)

Using Equation (9), Equation (11), and Equation (12), we can solve

To ﬁnd the solution of stress distribution within the plastic region, we can start with the equation of equilibrium

vsrr sqq srr ¼ vr r

(17)

The equation of equilibrium can be rewritten in terms of the effective stress and pore pressure as below

vs0rr s0qq s0rr vP ¼ a vr vr r

(18)

Substituting the post-failure yield criterion (Equation (5)) into this equation gives

vs0rr s0 Yr vP þ ð1 gr Þ rr ¼ a vr vr r r

(19)

Given the steady-state pressure distribution, the equation can be

M.S. Masoudian, M.A. Hashemi / Journal of Natural Gas Science and Engineering 35 (2016) 483e496

gr 1 Yr aQ Yr aQ r þ Rw gr 1 gr 1 gr 1 Y aQ Yr aQ r þ gr s0qq ¼ Yr gr r Rw gr 1 gr 1

487

s0rr ¼

(23)

The relationships for total stresses will then become

gr 1 Yr aQ Yr aQ r þ þ aP Rw gr 1 gr 1 gr 1 Y aQ Yr aQ r þ gr sqq ¼ Yr gr r þ aP Rw gr 1 gr 1

srr ¼

(24)

This equation and the stress solution of the elastic zone (Equation (6)) give the stress distribution within the reservoir. However, we still need to ﬁnd the extent of the plastic zone (Rep) in order to complete the stress solution. To achieve this, we can consider the continuity of the radial stress at the interface, sep rr , while satisfying the peak strength yield criterion (Equation (4)). In other words, the radial stresses calculated from both elastic and plastic solutions should be equal at the elastic-plastic boundary. From Equation (6), we can easily arrive at the following relationship ep sep qq ¼ srr þ 2C1 þ

ð1 2yÞ E εe;s aP þ ð1 yÞ 3ð1 yÞ v

(25)

Substituting this equation into Equation (4) leads to the radial stress at the elastic-plastic boundary interface as below

sep rr ¼

1 y E 2C1 þ g εe;s aP Rep þ R Y ep ð1 þ gÞ 3ð1 yÞ v 1y (26)

On the other hand, substituting r ¼ Rep into Equation (23), can also give the stress at the elastic-plastic boundary as below

Fig. 2. The elastic-brittle-plastic behaviour of the rock (Park and Kim, 2006).

vs0rr s0 Yr aQ þ ð1 gr Þ rr ¼ r vr r r

(20)

This is a ﬁrst order nonhomogeneous linear differential equation which can be solved to give

(21)

where C3 is the integration constant that can be found through applying a boundary condition. If we assume that the total stress at the wellbore wall (r¼Rw) is equal to the well pressure, pw, (i.e. zero effective stress) C3 can be expressed as below 1gr

Yr aQ þ 1 gr 1 gr

gr 1 Rep Yr aQ Yr aQ þ þ aP Rep gr 1 gr 1 Rw 1 y E 2C1 þ g εe;s aP Rep þ ¼ v Rep Y ð1 þ gÞ 3ð1 yÞ 1y (28)

Yr aQ C ¼ þ 3 1 gr 1 gr r 1gr

C3 ¼ R w

(27)

The sep rr calculated from both Equation (26) and Equation (27) must be equal to satisfy the continuity of the radial stress. As such, we will have

re-written as

s0rr

gr 1 Rep Yr aQ Yr aQ þ þ aP Rep gr 1 gr 1 Rw

sep rr ¼

(22)

The radial stress can then be rewritten, replacing C3 into Equation (21). Substituting s0rr back into the yield criterion, can also give the tangential stress. Thus, the radial and tangential stresses can be expressed as below

Solving this equality for Rep, one can estimate the radius of the plastic zone. Unfortunately, this equation cannot be solved to give an explicit closed-form solution. However, an iterative scheme can be used to solve the equation for Rep. In this paper, all the equations were implemented in a MATLAB code where a trial and error scheme is employed to ﬁnd the value of Rep and its corresponding stress, sep rr . 4.2. Solution of displacement As explained earlier in problem deﬁnition, the strains in plastic zone, can be decomposed into elastic and plastic components. To ﬁnd the displacement ﬁeld in the plastic zone, we employ a nonassociated ﬂow rule. Additionally, we can assume that the elastic

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strains are relatively small compared to the plastic strains. As such, the plastic components of radial and tangential strains can be related through a plane-strain condition as below

εprr þ bεpqq ¼ 0

(29) f ðrÞ ¼ εerr þ bεeqq ¼ AK

whereb is a function of the dilation angle, j, as below

b¼

1 þ sin j 1 sin j

r gr 1 ð1 þ yÞð1 þ bÞ e;s εv þ BYr AK Rw 3 (33)

where

vu u þ b ¼ f ðrÞ vr r

(31)

Equation (31) is a ﬁrst-order nonhomogeneous linear differential equation which can be solved to give the displacement within the plastic zone as below

(30)

On the other hand, using Equation (15) and Equation (29) we will have

u ¼ uep

(29), f ðrÞ will be a function of the elastic parts of the strains, only. Therefore, Equation (24) can be replaced in Hooke's elastic law to give the elastic strains within the plastic zone and then f(r) can be given as below

ZRep Rep b 1 b rb f ðrÞdr r r

A ¼ 1 y ygr þ bðgr y ygr Þ B ¼ b by y

Substituting Equation (33) into Equation (32) and solving the integral term leads to the equation below

(32)

r

The boundary condition for this solution is the displacement at the elastic-plastic interface, uep. Using the obtained value for Rep from Equation (28), we can ﬁnd uep by continuity of the displacement, using the elastic solution of the displacement (Equation (6)). On the other hand, in light of Equation (1), Equation (15) and Equation

(34)

Yr aQ K¼ gr 1

u ¼ uep "

Rep b 1 þ y r Er b

#r¼Rep

rbþgr rbþ1 ð1 þ bÞE Fε;b ðrÞ AK þ ðBY AKÞ r g 1 3 bþ1 ðb þ gr ÞRwr

r¼r

(35) where

Fig. 3. The cross comparison of the numerical and analytical solutions - (a) distribution of effective stress in the reservoir; (b) distribution of effective stress near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore.

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489

Table 1 The values used in the sensitivity analysis of the developed analytical solution. Parameters

Symbol

Young modulus Peak cohesion Peak friction angle Residual cohesion Residual friction angle Dilation angle Maximum sorption-induced swelling Poisson ratio Langmuir's isotherm constant In-situ horizontal stress Initial reservoir pressure Wellbore pressure Biot's coefﬁcient Wellbore radius Outer radius

"

E c

f cr

fr j

εsL

y

bL

s0

p0 pw

a

Rw R0

r 1þb 1 þ bL p 0 exp ð1 þ bÞ Fε;b ¼ εsL 1 þ b bL Q bL Q # 1 þ bL P Ei ð1 þ bÞ bL Q

Value Low

Mid

High

2.0 2.2 30.3 1.5 25.0 0.0 0.5

3.0 3.4 33.5 2.2 30.3 10.0 0.926 0.3 0.4184 25.0 10.4 1.0 1.0 0.1 100.0

4.0 6.0 40.0 3.4 33.5 30.0 1.5

Unit

Reference for mid value

GPa MPa deg MPa deg deg % e 1/MPa MPa MPa MPa e m m

(Gentzis et al., 2007) (Gentzis et al., 2007) (Gentzis et al., 2007) (Gentzis et al., 2007) (Gentzis et al., 2007) (Park and Kim, 2006) (Cui et al., 2007) (Gentzis et al., 2007) (Cui et al., 2007) Corresp. to 1000 m depth (Gentzis et al., 2007) Corresp. to 1000 m depth (Gentzis et al., 2007) e e e e

tangential strains within the plastic zone can be obtained by substituting Equation (35) into Equation (15).

b R1þ 0

(36)

Note that, the upper and lower bounds of the integral needs to be utilised in the solution, i.e. r and Rep. In addition, the radial and

5. Veriﬁcation and example application In this section, we ﬁrst verify the accuracy of our analytical solution. In order to do that, we constructed a numerical model using Abaqus where the stress and displacement distributions within a disk shaped reservoir is estimated for an elastic-plastic rock. Note

Fig. 4. Case I (blue lines) is elastic-plastic solution, Case II (red lines) is elastic-plastic solution with swelling, Case III (black lines) is elastic-brittle-plastic solution with swelling - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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M.S. Masoudian, M.A. Hashemi / Journal of Natural Gas Science and Engineering 35 (2016) 483e496

Fig. 5. The effect of elastic modulus, E [GPa] ¼ 2 (blue), 3 (black), 6 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

that, in this task the rock does not represent a brittle failure in order to simplify the numerical model construction, i.e. the residual strength parameters are equal to the initial strength parameters. Naturally, the values of all input parameters are identical for both analytical and numerical models, but they do not need to correspond to the ﬁeld values as we are only interested in understanding the relative accuracy of the results obtained from the analytical solution against those of the numerical model. Also, note that the swelling strains in numerical model is implemented using the thermal expansion option where it is related to the pore pressure using a Langmuir curve. The cross-comparison of the numerical and analytical solutions shows a perfect match as depicted in Fig. 3, which veriﬁes the accuracy of our newly developed analytical solution. The physical phenomena behind these results will be explained in this section where the parametric study and example applications are conducted. In order to present the practical application of the solution, and to study the effects of various parameters on stress and displacement, a range of values were used for parameters as listed in Table 1, whose mid values mainly correspond to medium volatile bituminous coals of the Foothills and Mountains reported by Gentzis et al. (2007). The depth of the coal seam and the wellbore pressure are assumed to be 1000 m and 1.0 MPa, respectively, and the adsorption and swelling parameters are borrowed from Cui et al. (2007). Note that the parametric analysis was performed through changing one of the parameters at a time, using the values given as High and Low in Table 1. Before discussing the parametric study, we compare the results of three different cases with mid values to understand the effect of

two main considerations of the solution: brittle failure and swelling/shrinkage. Case I represents the elastic-perfectly-plastic solution with neither swelling nor brittle failure. Case II presents the elastic-perfectly-plastic solution with swelling effect. Case III is the elastic-brittle-plastic solution with swelling effect. Fig. 4 illustrates the stresses and displacements within the reservoir and their magniﬁcation near the wellbore. In Case I (blue curves), the existence of the circular opening in the rock, results in movement of the medium towards the wellbore (positive displacements in Fig. 4c,d), and consequently the radial and tangential stresses are redistributed within the reservoir. The redistributed tangential and radial stresses form the major and minor principal stresses and if the combination of these satisfy the yield criterion, the rock undergoes a plastic regime. Under a perfectly-plastic mode, there exists a location, beyond which the rock remains in elastic regime. This is the location of the elastic-plastic boundary which corresponds to the maxima in tangential stress proﬁle (Fig. 4a,b). When swelling is introduced as an elastic strain component (Case II, red curves), some changes occur. In this case, the desorption-induced shrinkage produces a negative stress component so that the stress concentration is reduced within the porous medium, hence the extent of the failure (plastic) zone reduces. Note that shrinkage is an elastic component of the strain and thus, it does not change the stress state within the plastic zone in Case II, compared to Case I (see Fig. 4a and b). The ﬁgures clearly show that the stress maxima and the radius of the plastic zone are both smaller in Case II, compared to Case I. However, the reduced stress concentration in the elastic zone induced by the gas desorption results in a larger deformation of the rock towards the wellbore. In

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Fig. 6. The effect of peak cohesion, c [MPa] ¼ 2.2 (blue), 3.4 (black), 6.0 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

other words, the two effects of mechanical deformation and swelling-induced components of strain compete with each other but the shrinkage component on strains and displacement is much larger than the mechanical component and hence, the total displacement is larger in the second case, especially within the elastic zone (e.g. Masoudian et al., 2013a). However, the displacement within the plastic zone is smaller in Case II, since the effect of mechanical strain component is dominant within the failed zone in the close vicinity of the wellbore (see Fig. 4c,d). The brittle-plastic behaviour of the rock (Case III, black curves) is identiﬁed by a post-failure strength that is smaller than the peak strength. As a result, the rock within the plastic zone can bear a smaller radial and tangential stresses, compared with Case I and Case II. However, note that the rock exactly located on the elasticplastic interface is on the verge of failure and its strength is the peak strength and it can still bear a larger stress than what the failed rock does. This explains the discontinuity of the tangential stress at the elastic-plastic interface. On the other hand, as explained earlier, the continuity of the radial stress and deformation was imposed to the model, in order to ensure the validity of the continuum mechanics approach which means the rock can be still considered as a continuum throughout the domain of the reservoir. In addition, the radius of the plastic zone in Case III is the largest among the three cases, mainly due to the fact that the radial stress (minor principal stress) in this case is the smallest, and hence smaller tangential stress is needed to satisfy the M-C yield condition. From Fig. 4d, it can also be seen that the displacement of the rock in the close vicinity of the wellbore is the highest among the

three cases, and then it immediately drops to become very similar to Case II in elastic zone, as seen in Fig. 4c. This is because the brittle-plastic rock facilitates the plastic ﬂow of the failed rock. However, the higher level of stress state within the elastic zone (compared to Case II) results in slightly larger displacements, but further within the elastic zone the displacement of Case III becomes almost identical to that of Case II. It should be noted that the estimated displacements near the outer boundary of the reservoir are largest. This reﬂects the limitation of the assumption that the radial stress at the outer boundary is constant under the plane-strain condition. This assumption seems to be more relevant for reservoirs which are cut by low angle faults at the outer boundary. Despite this limitation, the elastic solution has provided valuable information about the coupled reservoir and geomechanical behaviour of the rock and it can be very helpful in understanding the interaction of different phenomena. As a result, the assumption of constant stress at the outer boundary has been used in many reservoir and geomechanical studies (e.g. Cui et al., 2007; Han and Dusseault, 2003; Masoudian et al., 2016b). In order to understand the effect of important parameters on distribution of stress and displacement in the reservoir, results of the solution with different values for elastic, plastic, and swelling characteristics are discussed. The elastic modulus is an important parameter in geomechanical performance of reservoirs as shown by Masoudian et al. (2016a) who used a 1D deformation elastic solution. The analysis of the elastic-brittle-plastic solution provided in this paper shows that, although the elastic modulus does not

Fig. 7. The effect of friction angle, f [deg] ¼ 30.3 (blue), 33.5 (black), 40 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 8. The effect of residual cohesion, cr [MPa] ¼ 1.5 (blue), 2.2 (black), 3.4 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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Fig. 9. The effect of residual friction angle, fr [deg] ¼ 25.0 (blue), 30.3 (black), 33.5 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

have a signiﬁcant effect on stress distribution in the plastic region (Fig. 5a,b), it has an important role in the estimated radius of the failed region and the estimated displacements throughout the reservoir (Fig. 5c,d). As expected, elastic modulus does not change the stress distribution within the plastic zone near the wellbore. Fig. 5a shows that a higher value of elastic modulus results in smaller plastic zone and lower stress at elastic-plastic interface. This is because a rock with a larger elastic modulus means the effect of shrinkage induced component becomes larger, which in turns

leads to a larger reduction in the level of stress concentration (Masoudian et al., 2013a). As a result of reduced stress concentration in case of a larger elastic modulus, the rock will undergo a larger displacement (see Fig. 5c,d). This is consistent with any other mechanical analysis where a material with lower elastic modulus exhibit larger deformations. Cohesion and friction angle determine the strength of a rock and its likelihood of being failed and therefore, play important roles in geomechanical behaviour of the reservoir. Fig. 6 shows the results

Fig. 10. The effect of dilation angle, j [deg] ¼ 0.0 (blue), 10.0 (black), 30.0 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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Fig. 11. The effect of swelling factor, εL[%] ¼ 0.5 (blue), 0.926 (black), 1.5 (red) - (a) effective stress distribution in the reservoir; (b) effective stress distribution near the wellbore; (c) radial displacement in the reservoir; (d) radial displacement near the wellbore. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

for different values of peak cohesion. It can be seen that peak cohesion mainly inﬂuences the position of the elastic-plastic interface and its corresponding stress. A larger cohesion means the rock is less prone to failure and therefore it can bear larger stresses before failure. Hence, the plastic zone is smaller and the level of stress is higher when a larger cohesion is used. Although the displacement within the reservoir is not highly affected by the value of cohesion, the displacement in the vicinity of the wellbore is slightly smaller with a larger cohesion, mainly due to a smaller radius for the plastic zone. The peak effect of friction angle on stress and displacement is very similar to that of peak cohesion, as demonstrated in Fig. 7. A larger friction angle, is an indication of a stronger rock which can experience a higher stress level without being failed, and hence yields similar results as larger cohesion and the physical implications and explanations are also similar. An important signiﬁcance of this paper is to understand the inﬂuence of brittle failure on geomechanical response of unconventional gas reservoirs. Thus, the role of residual cohesion and friction angle on estimation of the stress and displacement within the reservoir has been examined. Figs. 8 and 9 depict the results for testing different values of residual cohesion and residual friction angle, respectively. It can be seen that a higher value for residual cohesion or friction angle results in a smaller plastic zone, while the level of tangential stress is not affected. For larger values of residual parameters, the level of both radial and tangential stress within the plastic zone is higher while it is lower within the elastic zone. To explain these effects, it should be noted that a larger value of residual parameters correspond to a less brittle behaviour of the rock, as the residual parameters are getting closer to the peak values. In

other words, in case of higher residual parameters, the rock mass within the failed zone can withstand higher levels of stress and yield smaller deformations. Hence, the plastic zone becomes smaller. The stress at the elastic-plastic interface remains unchanged because it does not depend on the residual parameters but on the peak strength parameters. The estimated displacements within the reservoir for different values of residual parameters are not very different from each other. However, near the wellbore and more obviously within the plastic zone, the higher residual cohesion and friction angle results in smaller displacements. This is because the failed zone with higher residual strength parameters can better resist against plastic deformations. Dilatancy can be deﬁned as a change in volume resulting from the shear distortion of an element in a material (Alejano and Alonso, 2005). As a result, it does not have any effect on the stress distribution within the reservoir or the displacement within the elastic zone. A dilation angle of zero deﬁnes a rock whose volume is preserved, which is chosen as the low value in this study. Larger values of dilation angle represent the rocks that show larger volume change during shear. Thus, displacements of the elastic zone is estimated to be the same for all values of dilation angle, as illustrated in Fig. 10a. On the other hand, larger dilation angles resulted in larger deformation within the plastic zone (Fig. 10b). Note that the mid and high values chosen for dilation angle are 10 and 30 , respectively. Hoek and Brown (1997) have suggested that the dilation angle of rocks may be signiﬁcantly smaller than the friction angle (between 0 and f/4), but we have used a high value of 30 to get a better picture of its role on deformations in a planestrain case.

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In order to investigate the role of desorption-induced shrinkage on the elasto-plastic behaviour of the reservoir, we have used different values for the maximum adsorption-induced swelling factor and presented the results in Fig. 11. From Fig. 11a and b, it can be seen that for a larger swelling factor, the plastic zone is smaller and the tangential stress at the elastic-plastic interface is lower. The radial stress within the plastic zone, however, is not affected but a larger swelling factor leads to a smaller radial (and tangential) stress within the elastic zone. In addition, a larger swelling factor leads to generally larger displacements, as depicted in Fig. 11c and d. In order to explain these observations, one should note that a larger swelling factor means that the shrinkage-induced component of strains is larger and hence the displacement increases, especially within the elastic region. However, within the failed zone, the displacements are smaller in case of a larger shrinkage factor, mainly because of a smaller failure region (blue curve in Fig. 11c). As the swelling factor increases, the effect of shrinkageinduced reduction in stress level becomes larger. As a result, both radial and tangential stresses within the elastic zone decrease, while the stress level within the plastic zone remains unchanged. The latter is due to the fact the swelling/shrinkage is an elastic component of the strain and thus, it does not change the stress state within the plastic zone. Since a larger swelling factor leads to a lower stress level, the M-C failure criterion is satisﬁed in a shorter distance from the wellbore, i.e. the radius of the plastic zone decreases. In other words, the shrinkage of the coal matrix, provides more space for larger deformations and hence the stress concentration reduces. Therefore, when a larger shrinkage is adopted, the maximum stress is lower. On the other hand, the reduced stress concentration means that the radial stress also decreases. The radial stress is actually the minor principal stress component that resist against the major principal stress (tangential) to prevent the shear failure. Since the friction angle is constant, a reduced radial stress means that coal can fail with a lower tangential stress level. To summarise, it must be noted that although in this paper swelling or shrinkage is not considered to directly change the mechanical properties of coal, it can inﬂuence the stress distribution within the coalbed. As a result of altered stress state within the reservoir, the production-induced shrinkage indirectly leads to a smaller failed zone, while an extended zone of failure can be expected when CO2injection is being carried out. However, this effect should not be confused with the sorption-induced changes in mechanical properties of coal due to the physico-chemical interactions of coal and gas molecules, as studied by various studies (e.g. Masoudian et al., 2014; Masoudian et al., 2016b). The phenomenon discussed here is based on a purely mechanical approach where the mechanical properties of coal are constant, i.e. the mechanical properties are independent of gas concentration in matrix blocks. 6. Conclusions This paper provided a novel analytical solution for a circular opening within an elastic-brittle-plastic rock with swelling/ shrinkage effect where a non-associated ﬂow rule was employed to develop the solution of both stress and deformation. The model can be useful for tunnelling analyses in swelling ground and wellbore stability in unconventional gas reservoirs. The latter was the main focus of this paper, and hence analyses conducted here were representative of those in unconventional reservoirs. Therefore, the values of parameters chosen in this paper are those of coalbeds, while discussing the results for shale gas reservoir can be an extension to this study. The signiﬁcances of the developed model are the consideration of swelling/shrinkage and brittle failure. To highlight the importance of these two signiﬁcances, a series of analyses were conducted, where three solutions were compared

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against each other: elastic-perfectly-plastic, elastic-perfectly-plastic with swelling, and elastic-brittle-plastic with swelling. This analysis showed that brittle failure and swelling/shrinkage can have important implications on wellbore stability in reservoirs, and the geomechanical performance of the reservoir as a whole. A parametric study was also undertaken, which showed that the extent of the plastic zone around the wellbore is inﬂuenced by both elastic and plastic parameters. The distributions of stress and displacement within the reservoir were studied and the qualitative importance of all parameters was investigated. The developed model can be very useful in many applications, including, but not limited to, veriﬁcation of the numerical models, and development of new permeability models for unconventional gas reservoirs. References Alejano, L.R., Alonso, E., 2005. Considerations of the dilatancy angle in rocks and rock masses. Int. J. Rock Mech. Min. 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