Closed-form solution of stress state and stability analysis of wellbore in anisotropic permeable rocks

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Contents lists available atScienceDirect

International Journal of

Rock Mechanics and Mining Sciences

journal homepage:www.elsevier.com/locate/ijrmms

Closed-form solution of stress state and stability analysis of wellbore in anisotropic permeable rocks

Duc-Phi Do

^a,^⁎

, Nam-Hung Tran

^b

, Hong-Lam Dang

^c

, Dashnor Hoxha

^a

aUniv. Orléans, Univ. Tours INSA CVL, Lamé, EA 7494, France

bLe Quy Don Technical University, Institue of Technique for Special Engineering, Hanoi, Vietnam

cUniversity of Transport and Communication, Civil engineering faculty, Geotechnical section, Hanoi, Vietnam

A R T I C L E I N F O Keywords:

Closed-form solution Transient flow Poro-elastic Permeable rocks Anisotropic strength Mud pressure window Wellbore stability

A B S T R A C T

This work aims to develop a closed-form solution of stress state around wellbore drilled in the anisotropic permeable rocks taking into account the transient effect of fluid flow. For this purpose, a simplified method was adopted to solve the anisotropic hydraulic diffusion equation. Knowing the explicit transient evolution of pore pressure, the hydraulic and hydro-mechanical potentials were proposed through which stress state around wellbore are determined thanks to the use of the well-known complex potential approach. The validity of this closed-form solution was verified by comparing with the result obtained from the numerical simulation. By combining the analytical solution of stresses on well wall with a chosen anisotropic failure criterion of rock mass, the stability analysis of wellbore was carried out. Through the parametric study, we elucidated the influence of different factors on the safe mud pressure window notably the anisotropic poro-elastic properties, the anisotropic strength of the rock masses as well as the anisotropic initial stress state and the inclined bedding plane of formation.

1. Introduction

The investigation of stress state in the surrounding rock masses and stability analysis of wellbore is one of the primary topics and the challenging issues in different industrial applications such as the pet- roleum or geothermal exploitation, the mining industry, the nuclear waste storage… In the engineering practice, thanks to its simplicity, the well-known solution of the assumed isotropic poro-elastic medium has been largely applied to study this kind of problem. However, very often, wellbores are drilled in deep geological formations having an inherent anisotropy due to their stratified structure as observed in the sedi- mentary and metamorphic foliated rock masses. Consequently, the application of the conventional method based on the assumption of isotropic formations could underestimate the stress state, which leads to inaccurate analysis of the wellbore stability.

Since the last two decades, the investigation of the wellbore's response in the anisotropic rocks has always been received a particular attention of the rock engineering community. Among numerous contributions, many interesting analytical solutions were presented. Due to the complexity of the problem, these analytical solutions are usually deduced from some simplified hypotheses which, however, cannot re- duce the utility of these closed-form solutions. For example, in,1–3the

scholars highlighted that the distribution of stress state around a circular opening, which is drilled in the anisotropic elastic medium, can be significantly different with the one calculated in the isotropic medium.

Following that, in an anisotropic formation, the stresses and displacements in the surrounding rock masses strongly depend on the orientation of bedding or foliation.^3–5Attempts have also been made to deal with the wellbore drilled in the anisotropic formation by accounting for the hydro-mechanical coupling. We can mention the pioneering work of Abousleiman and Cui⁶ who studied the behavior of a borehole in a transversely poro-elastic medium. Nevertheless, the study in6is limited in a particular case that the cross section of the borehole is parallel to the isotropic plane of medium. For the case that the longitudinal axis of the borehole lies on the isotropic plane, just only few works^7–9in- vestigated the effect of fluid flow on the behavior of the borehole. Note however that in these studies, the fluid flow is considered in the steady regime.

Stability analysis of wellbore consists of determining the safe mud pressure window that ensure the wellbore stability for drilling opera- tions. The instability of wellbore could occur when the surrounding stress exceeds the tensile or the shear strengths of the rock formation.

Therefore, the chosen mud weight window must be greater than the shear failure gradient and less than the fracture gradient.¹⁰In the past,

https://doi.org/10.1016/j.ijrmms.2018.11.002

Received 21 January 2018; Received in revised form 29 October 2018; Accepted 11 November 2018

⁎Corresponding author.

E-mail address:[email protected](D.-P. Do).

T

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almost the works analyzed the stability of wellbore by adopting the hypothesis of isotropic elastic properties as well as isotropic strength of rocks.^11–14It has been shown however that the ignorance of the anisotropic characteristic could induce an underestimate the mud pressure and hence, the stability of the borehole. In their works, Gupta and Zaman¹⁵and Jin et al.¹⁶used the analytical solutions to investigate the effects of different factors on the stability of wellbore. They concluded that the anisotropy of the elastic properties, of the initial stress and the bedding plane inclination can have a great effect on the fracture and collapse pressures. Using the weak plane failure criterion, some other authors^10,17–21elucidated the important effect of the anisotropic failure criteria on the prediction of mud pressure windows even in the case of formations owning an isotropic elastic behavior. Different scholars^14,22 also highlighted the considerable effect of fluid seepage on the wellbore stability. We can mention the work of Kanfar and Rahman²² who analyzed the time-dependent stress state around wellbore due to the transient fluid flow by using the finite element method. Note that, this last approach allows accounting for the fully hydro-mechanical coupling but demand a very fine mesh to give a precise result. Based on these numerical results, the important effect of the anisotropic poro- elastic properties on the stress around wellbore versus elapsed time was exhibited. Kanfar and Rahman²²indicated that the isotropic rock model underestimates the borehole collapse pressure and a remarkable difference is observed between the isotropic and anisotropic models for the estimated fracture initiation pressure. This discussion was con- firmed in our recent contribution²³ in which a closed-form solution based on the complex potential approach is utilized. The results in our previous paper highlighted that the fracture initiation pressure in the anisotropic poro-elastic rock could be significantly affected by different parameters like the anisotropic poro-elastic properties (such as Young's modulus, permeability) as well as the anisotropic initial stress and particularly, the anisotropic tensile strength. Furthers, the fracture initiation pressure calculated in the case of permeable and impermeable rocks could present a quite pronounced difference.

In this work, the extension of our previous works^23,24is conducted to deduce a closed-form solution of stress state around the wellbore taking into account the pore pressure effect due to the transient fluid flow. This one-way hydro-mechanical coupling solution is developed by using the complex potential approach. To this end, in this first step, the transient fluid flow in the weak anisotropic permeable rock is solved thanks to a simplified method. The hydraulic and hydro-mechanical potentials are then proposed from which the closed-form solution of stress state can be deduced. Then the verification test is done to validate this closed-form solution by comparing with the results obtained from the numerical simulation. In the next step, stability analysis of wellbore is carried out. The safe mud pressure window is determined by utilizing the analytical solution of stress state on the wellbore wall and a chosen anisotropic failure criterion. Through a parametric study, we elucidated the effect of different factors on the prediction of mud pressure window in both cases of permeable and impermeable rocks.

2. Closed-form solution of stress around wellbore in anisotropic permeable rocks with transient flow

In a previous work²³of the present authors, we presented a closed- form solution of stress state around a horizontal wellbore drilled in a transversely isotropic poro-elastic medium by using the complex potential approach. Under the 2D plane strain hypothesis, this analytical solution was developed by assuming that the longitudinal axis of wellbore coincides with one of the two principal horizontal stress axes of the porous rock. The saturated surrounding rock mass with an initial pore pressurep_ffis subjected at far-field to the principal stresses vff,

hff. In addition, the normal vector of the bedding plane (i.e., the isotropic plane) is inclined an angle with respect to the principal vertical

stress axis as shown inFig. 1a. Two problems were considered in the previous work²³which represent respectively the permeable and impermeable problem depending on the adopted hydraulic boundary condition at the well wall. Follow that, the wellbore is subjected on its surface to the mud weight pressurePwwhich can act at the same time as a radial stress and as a pore pressure if the well wall is permeable (case of highly permeable rock mass). Otherwise, if the well wall is impermeable (case of rock mass owning the very low permeability) the mud weight pressurePwcan act only as a radial stress. In this last case, pore pressure in the tight impermeable rocks is uniform, thus the total stresses can be deduced from the purely mechanical solution of wellbore drilled in dry rock. Concerning the permeable boundary case, due to the difference of pore pressure at the well wall and at far-field, fluid flow can occur. This could be an outflow (or inflow) if the mud weight pressurePwis higher (or smaller) than the initial pore pressurep_ffof the rock mass. The influence of this flow on the mechanical behavior of wellbore was detailed in our last work²³in which the complex potential approach was used to deduce the closed-form solution. To complete this analytical solution which is however limited at the steady state of fluid flow, in the present work, we consider the problem in the transient flow regime.

For the sake of clarity, all the formula of the one-way hydro-mechanical coupling in a transversely isotropic porous medium are briefly summarized as follows:

The equilibrium equation:

+ =0, + =0,

x^x y x y

xy yx y

(1) The strain compatibility equation:

= +

x y y x

2 ²^xy ²₂^x ²₂^y,

(2) The Hooke's law representing the stress and strain relationship in 2D plan strain conditions:

=

s s

s 0 0

0 0

,

x y xy

x y xy 11 12

21 22

33 (3)

The fluid flow in the transient state by combining the Darcy law and the conservation equation:

+ =

k p

x k p

y M

p t,

x 2 y w

2 2

2 (4)

The notion of effective stress based on the Biot's theory character- izing the influence of the hydraulic state on the mechanical response:

= + p, = + p,

x x x y y y (5)

The equations from(1)–(5)show that the poro-elastic behavior of a transversely isotropic rock is characterized by two hydraulic con- ductivitiesk k_x, _y, the Biot coefficients _x, _yand five elastic parameters composing the horizontal and vertical Young's modulus (Ex,Ey), the Poisson's ratios in the isotropic plane and anisotropic plane (xz, xy), the shear modulusG_xyin the anisotropic plane. The compliance coefficients appeared in Eq.(3)are related to these five elastic parameters through the following relationships:

= = = +

= =

=

s E s s

E s

G E

E

1 , (1 )

, 1

, 1 ,

,

xz x

xy xz

x

xy yx

y xy

yx xy y

x 11

2

12 21 22 33

(6) The procedure to determine the stress state around wellbore drilled in a highly permeable rock with transient fluid flow is similar as in the

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previous work²³thanks to using the complex potential approach. More precisely, the problem will be also decomposed into two sub-problems (Fig. 2). The first problem (problemI) corresponds to the purely mechanical problem in which the circular wellbore drilled in the saturated porous rock is subjected by a uniform pore pressurepff, the initial stress

vff, hffat far-field as well as the mud weight pressureP_won the wall. In the second problem (problemII), we calculate the stress state around the wellbore due to the transient evolution of pore pressure which ranges fromp₀=P_w-p_ffat the well wall top_∞= 0at infinity. Note that the problemIis exactly the problem of the wellbore drilled in the impermeable porous rock whose solution is recently detailed in23. Thus, in the following, we interest only on the problemII.

2.1. Closed-form solution of transient flow in anisotropic permeable rock using the simplified method

In this part, a closed-form solution of the transient fluid flow in an anisotropic saturated rock is presented. For this aim, the simplified method which was firstly introduced by25in the context of the isotropic porous medium is utilized. This last well-known problem of horizontal flow towards a wellbore was studied analytically in the first time by 26. However, based on the Green's function and integral transforms, this analytical solution is expressed as function of the first and second kind zero-order Bessel functions which makes it compli- cated and difficult for the practical and mathematical manipulation. To overcome these drawbacks, Perrochet²⁵ presented an alternative approach which is simpler in nature but yields essentially the same results as ones of26. The principal idea of this approach is that, at a given

time, pore pressure at a distance is almost unperturbed, then the transient solution of the radial diffusion equation can be computed as successive steady-state snapshots using a time-dependent radiusRw(t).

More precisely, it means that, at each instant, the perturbation of pore pressure induced by the circular opening is only occurred in the interior of a circle region with radius R_w(t)and beyond this distance, pore pressure equals to the initial water pore pressure. As a function of time, this radiusRw(t)will expand from the radius of wellborer0(instant t = 0) to the valueR_w(t =∞)=R_∞. This latter corresponds to the in- fluenced radius at the steady state, a distance far enough from the well wall as indicated in our previous work.²³The time-dependent solution of radiusR_w(t)as well as of pore pressure are presented by Perrochet²⁵ after some developments as follows:

= +

= ⁺

+

R t r e

s r t s

( ) , 1 ,

( , ) 1 ,

w k t

S r

R t r r

R t R t r

0 2

2 1 *

*

0

2 ( ) ln

2 ( ) ln ( )

w r

r

w Rw t

r w

02 2

0 2 02

2 ( )

0

2 02

(7) where:

= =

k k S

* , * M1

w (8)

In Eq.(7), we note pore pressure assto distinguish with the total pore pressurepof the previous hydro-mechanical coupling problem.

The corresponding fluid flux and cumulative volume of fluid across the perimeter of the wellbore can be deduced as:

Fig. 1.The initial problem of horizontal wellbore drilled in a transversely anisotropic formation whose bedding plane make an angle with the principle stress axis at the far-field (a), the equivalent problem after a rotation of angle (b).

Fig. 2.Decomposition of the equivalent problem (a) into two sub-problems: problem I (b) and problem II (c).

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=

= +

Q t k s

V t r S s

( ) 2 * ln 1

2 ,

( ) * 4 ln 1 4 ln 2 ,

0 1

1

0 0 2 1 1

1 2

2

2 2

(9) where the parameter s0 denotes pore pressure on the perimeter of wellbore.

It was shown that this closed-form solution can be extended in a more general context of anisotropic porous rock.^9,27In the hydrological field, accounting for the anisotropic aspect in the analytical solution of the transient diffusion problem is an interesting topic and a lot of contributions has been dedicated. For example, by using the Laplace transform, Mathias and Butler²⁸presented the analytical solution of the transient flow to a borewell in a horizontally anisotropic aquifer. The analytical solution is written in the Laplace domain in terms of Mathieu functions and then a numerical inversion is used to evaluate the response in time domain. Moreover, these authors showed that for large times, the problem can be approximated as ones in an equivalent isotropic domain by coordinate transformations. The approximation agrees well with the exact solution for moderately anisotropic systems (with the anisotropic ratiok kx/ y 25). This observation was also con- firmed in29who studied the transient flow in horizontally anisotropic multilayered aquifer systems. In addition, the study in29highlighted that the equivalent isotropic solution can give satisfactory results at observation points away from the injection/pumping wells even for highly anisotropic aquifer systems (with the anisotropic degree can reach tok kx/ y=1000).

Returning to our problem, in order to solve the following anisotropic transient flow:

+ =

k s

x k s

y M

s

x 2 y w t

2 2

2 (10)

we use the same transformation coordinate as proposed by Mathias and Butler²⁸:

= =

X x Y y k

, k^x

y (11)

Thus, the initial anisotropic transient fluid flow will be rewritten in the new coordinate system (X-Y) as:

+ =

k s

X k s

Y S k

k s

* t

e e y

x 2

2 2

2 (12)

This last equation presents the transient fluid flow in an isotropic medium with an equivalent isotropic permeabilityke=( k kx y)/w. The solution of this diffusion equation, must satisfy the initial and boundary conditions as noted in Eq.(13):

= = = =

s r( , 0) 0, s r t( , )₀ s₀, s( , )t s_ff 0 (13) Therefore, the initially anisotropic diffusion problem is now de- generated to the equivalent isotropic problem in the transformed domain. It is important to note that, as pointed out in 28–30with coordinate transformation, the circular tunnel of radius r0becomes an ellipse and in the transformed domain, the contours of pore pressure in the immediate vicinity of the tunnel have also elliptical shapes.

The conformal mapping technique ^23,24 will be applied now to transform the outside region of the ellipse in thezwplane (the complex variablezwis defined aszw=X + iY) onto the outside region of unit circle in wplane:

= = +

+ = =

Z w a b a b

a r b r k k

( ) 2 2 ; ; / ;

w w w w1 x y

0 0 (14)

In the conformal mapping plane w, the diffusion Eq. (12) is rewritten as:

+ =

k s

d dz S k

k s t 1

/ *

e w

e

w w w

y x 2

2 2

2 2 (15)

Mathematically, Eq.(15)has the same form deduced in the isotropic rock mass which means that we can directly apply the solution of the isotropic diffusion problem. However, it is essential to point out that the solution of the isotropic diffusion in the transient state was developed from the assumption of uniform source term, i.e. the right-hand side of the diffusion equation depends only on time and is uniform in space.²⁵ However, this assumption is not verified in the present problem where the term d _w/dzw in Eq.(15)is now function of the spatial variables (_w, _w) meaning that the right-hand side of Eq.(15)is not uniform. To overcome this problem, we use the following approximation:

+ =

k s k s r S k

k s ( ) * t

e w

y x 2

2 2

2 02

(16) where the parameterr0is defined as:

= = +

r d dz

r k k

lim 1

/

(1 / )

2

z w w

x y

0 0

w (17)

It is interesting to note here that, the same expression ofr₀ _was introduced by Kucuk and Brigham³¹and by Mathias and Butler.²⁸In fact, these authors called this parameter the effective radius of the equivalent circular opening to approximate the elliptical tunnel in the (X-Y) coordinate. And in their contribution, Mathias and Butler²⁸ showed that this approximation is good for large times but it can work well for small times as long as the anisotropic degree is moderate, i.e., k kx/ y 25.

With this approximation, the diffusion problem in Eq. (16) represents the radial fluid flow in the conformal mapping plane wof an equivalent isotropic medium with the corresponding permeabilityke. Thus, the solution of transient flow obtained from the simplified approach as presented in the previously paragraph can be straightforwardly applied. For example, the solution of pore pressure in the original problem is now calculated with respect to the time-dependent influence radiusRw( )t which are defined in _w-plane as follows (note that in the wplane, the radius of wellbore is equal to one):

+

R e k t

1 S

w **

R

R 1 e

w w

2 2

(18) withS**=( ) *r0 2S k ky/ x.

Correspondingly, the distribution of the pore pressure obtained in the conformal mapping plane wis:

= +

+

=

+ +

s t s R t

R t R t R t

s

R t R t

R t R t R t

( , ) 1 2 ( ) ln 1

2 ( ) ln ( ) ( ) 1

2 ( ) ln ( )

2 ( ) ln ( ) ( ) 1 ;

w w w w

w w w

w R t w w

w w w

0

2 2

0

2 ( ) 2 2

2 2

w w

(19) with _w= _w.

Besides, the fluid flux across the perimeterQ(t)and the cumulative volumeV(t)can be calculated through the following equations:

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=

+

Q t k s R

V t r S s R R

R

( ) 2 ln ( ) 1

2 ,

( ) ** ( ) 4 ln ( )

1 4 ln ( ) 2 ,

e w

R R

w w

R R

w R R 0

( )

( ) 1

1

0 0 2

( )

( ) 1

( )

( ) 1

1 w

w

w w

2 2

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2.2. Resolution of the hydro-mechanical coupling problem (problem II) The closed-form solution of pore pressure in the transient flow as presented above will be used in the problemIIto determine the induced stress state around wellbore. It should be noted here that the hydro- mechanical coupling is only one way when we consider the influence of fluid flow on the mechanical response of the wellbore.

Substituting the solution of pore pressure in the transient state (Eq.

(19)) in the problemIIwith the pore pressure (s0=Pw-pff) at the well wall, we obtain:

=

>

p P p + if R t

if R t

( ) , ( )

0 ( )

II w ff

R t R t

R t R t R t w w

w w

2 ( ) log ( )

2 ( ) log( ( )) ( ) 1

w w

R w t w w

w w w

2 ( ) 2 2

2 2

(21) Knowing the explicit expression of pore pressure, one can deduce its derivative and hence the hydraulic potential (see detail in^9,23,27) which takes the following form:

= +

z P p

R t R t

R t R t R t

( ) 1

2( ) . Re

2 ( ) log ( )

2 ( ) log( ( )) ( ) 1 ,

wIIpp

w ff

w R t w w

w w w

w

2 ( ) 2 2

2 2

w w

(22) in which ‘Re’ stands for the real part and the real coefficients is determined as:

= +

+ + +

= + = +

µ

s µ s s µ s

s s s s

(2 ) ,

with: x y; x y

1 w2 2 11 w4

12 33 w2 22

1 11 12 2 21 22 (23)

Because the hydraulic and hydro-mechanical potentials present the same order of effect on the stresses, the hydro-mechanical potentials take the same expression as the hydraulic one:

= +

z P p N

R t R t

R t R t R t

z P p N

( ) 1

2( ) Re

2 ( ) log ( )

2 ( ) log( ( )) ( ) 1 ,

( ) ( ) Re ,

IIpp w ff

R t

IIpp w ff

R t R t

R t R t R t

1 1 1

1 2

( ) 12

1 2

1 1 2

2 2 1

2 2

2 ( ) log ( )

2 ( ) log( ( )) ( ) 1

R t

1 1

2 2 2

2( ) 22 2 2

2 2 2 2 2

(24) The two potentials in Eq.(24)are written in the _k(k=1, 2)planes obtained from the conformal mapping technique which transforms the infinite domain outside the wellbore (of radiusr0) in thezkplane (with

= +

z_k x µ y_k ) to the infinite domain outside the unit circle in the k

plane. Similar to the Eq.(14), these transformations can be written as:

= = + +

= + +

=

z w r iµ r iµ

z z r µ

r iµ k

( ) (1 )

2

(1 )

2 , or

(1 )

(1 ) ( 1, 2)

k k k

k k

k

k k k

k

0 0 1

2 02 2

0 (25)

where the two parametersµ_k_{(k = 1, 2)}are the complex roots (with positive imaginary part) of the following characteristic equation²³:

+ + + =

s µ11 4 (2s s µ) s 0,

12 33 2

22 (26)

In addition, in Eq.(24), the parametersR k_k( =1, 2)represent the distance at which no flow is occurred in the corresponding kplane.

Through these hydraulic and hydro-mechanical potentials, stresses and displacements fields around the wellbore can be determined ac- cording to the following relations:

= + +

µ µ µ

U p p p

U q q q

2 Re[ ],

xIIpp IIpp IIpp

w wIIpp

yIIpp IIpp IIpp

wIIpp

xyIIpp IIpp IIpp

w wIIpp

xIIpp IIpp IIpp

w wIIpp

xIIpp IIpp IIpp

w wIIpp 12

1 22

2 2

1 2

1 1 2 2

1 1 2 2 (27)

where:

= + = + =

p_i s µ11 _i2 s , q_i s µ_i ^s_µ, (i 1, 2, )w

12 12

i

22 (28)

From Eq.(24)and(27), it shows that the stresses and displacements are the logarithmic functions of the complex variables ₁, ,₂ _w. How- ever, due to the multi-valued characteristics of logarithmic functions with respect to the complex variables, the requirement of single-valu- edness of the solution provides a system of equations through which we can calculate the real (N11,N21) and imaginary (N12,N22) parts of two complex constantsN N₁, ₂. It was shown from our previous work (see Appendix A in²³) that the imaginary parts of these constants are equal to zero (N12=N22=0) while their real parts are:

=

+

N N

and _{q µ}^q ^µ _{q µ}^µ _µ^{q µ} ,

q µ µ q µ q µ q µ µ

11 ( )

( )

21 ( )

( )

w w w

w

w w w

w

2 2 2 22 22 22

22 12 12 22 2

2 2 2 12 12 22

22 12 12 22 2 (29)

In Eq.(29), all the complex numbers are written in their explicit form: _i= _i₁+i _i₂.

Note that, in the problemII, the boundary conditions of zero total stresses at far-field and on the perimeter of the wellbore need to be verified. In the present work, we adopted the hypothesis that, at each instant, the transient fluid flow occurs only in the interior of the elliptical region around the wellbore (or the circle region with radius Rw( )t in the wplane). As consequent, the induced stress is also taken place only in this region meaning that the total stresses must be van- ished outside this region. Regarding the formula in Eqs.(24) and (27), one can state that these conditions are automatically satisfied when the distance _k= _k , (k=1, 2) approach R t_k( ). However, the stresses given by Eq.(27)introduce non-zero total normal and shear stresses at the well wall. Thus, to vanish these latter, at the circumference of the wellbore, the same normal and shear stresses but with opposite sign are imposed. Expressions of these stresses are:

= + +

=

cos sin

sin cos

1 2

1

2 2 2 ,

1

2 2 2 ,

rp

x y y x xy

p y x xy

0

0 (30)

where x, y, xyare stress components on the well wall obtained from Eq.(27):

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= + +

P p N µ N µ µ

P p N N

T P p N µ N µ µ

( )Re[ . ],

( )Re[ ],

( )Re[ . ],

x w ff w

y w ff

xy w ff w

1 12

2 22 2

1 2

1 1 2 2 (31)

This last case belongs to one of the well-known problems solved by Lekhnitskii³²who proposed the two following potentials (see also²³):

= +

r

µ µ iµ T µ i

r

µ µ iµ T µ i

1

2 [(1 ) ]1,

1

2 [(1 ) ]1

,

IIph xy y x

1 0

1 2 2 2

1

2 0

1 2 1 1

2 (32)

Their derivatives can be determined straightforwardly as follows:

= + +

+ +

= + +

+ +

µ µ

i µ T µ

µ µ

i µ T µ

µ µ

1

( )

[ (1 ) ( . )]

(1 ) 1 ,

1

( )

[ (1 ) ( . )]

(1 ) 1 ,

IIph xy x y

1

12 22

22 22

12 12 12

2

22 12

12 12

22 22

22 (33)

through which the stresses can be deduced:

= +

µ µ

2 Re[ ],

xIIph IIph IIph

yIIph IIph IIph

xyIIph IIph IIph

12

1 22

2

1 2

1 1 2 2 (34)

2.3. Final results of stress and pore pressure around wellbore

The final results of the original problem are obtained through the superposition of the solutions of problemIand problemIIso that the total stresses in the massif is given:

= + +

, , ,

x xI

xIIpp xIIph

y yI

yIIpp yIIph xy xyI

xyIIpp xyIIph

(35) For the effective stress, one combines the solutions of the total stresses (Eq.(35)) and of pore pressure in the formation by using the Biot's theory (Eq. (5)). The transient distribution of pores pressure in the massif is obtained by adding the uniform initial pore pressurepffin the solution of pore pressure calculated in the problemII:

= +

=

>

+

p p p

p P p if R t

p if R t

( ) , ( )

( )

ff II

ff w ff

R t R t

R t R t R t w w

ff w w

2 ( ) log ( )

2 ( ) log( ( )) ( ) 1

w w

R w t w w

w w w

2 ( ) 2 2

2 2

(36) It is worth to note also that, the solution of stresses (Eq.(35)) could be presented in a more convenient way in terms of principal stresses which, at well wall, coincide with the radial and tangential stresses. In difference to the isotropic medium, the effective tangential stress on the wall of wellbore drilled in an anisotropic medium depends not only on initial stress state (far-field stress) and wellbore pressure but also on the properties of the anisotropic poro-elastic medium while the effective radial stress at the well wall depends only on the wellbore pressurePw:

=

= +

E E G k k K p P

cos sin P

( , , , , , , , , , , , , , , ),

( . . 1) ,

x y xy xz xy x y vff x y ff w

r x y w

0

2 2 (37)

The explicit expression of these effective principal stresses on the well wall are similar to ones in the steady flow and can be found in Appendix B of our previous work.²³

3. Verification test

The closed-form solution of pore pressure and stress state as presented above is validated in this part by comparing with the numerical results calculated from the finite element software Code_Aster. In these simulations, we model a horizontal borehole of radiusr0=0.1mdrilled in a transversely isotropic formation whose poro-elastic properties are taken as follows: Ex=5600MPa, Ey=4000MPa, xz=0.3, yz=0.14,

=

G_xy 1600MPa, k_x= ×3 10 ¹³( / )m s, k_y=10 ¹³( / )m s, _x=0.6,

=0.7

y . The initial pore pressurep_ff =4.5MPaand the anisotropic far- field stresses vff = 12.5MPa, hff= 15MPa are respectively chosen while an arbitrary wellbore pressureP_w=10MPais used. The normal vector of the bedding plane forms to the principal vertical stress axis (σ_vff) an angle β = 0°. Due to the symmetry of the considered problem, only one quarter of the model is simulated as shown inFig. 3. All the necessary mechanical and hydraulic boundary conditions are also highlighted in this 2D plane strain model of dimension 10 m × 10 m.

InFig. 4are presented the results of pore pressure, effective radial and tangential stresses following the vertical and horizontal cuts at the center of wellbore. The evolution of pore pressure calculated from the closed-form solution and the numerical simulation at different instants of elapsed time presents a good agreement (Fig. 4a, b). The maximum difference about 6% was stated at the early instant of elapsed time t = 1 h for a point near the wellbore and in the horizontal direction.

Correspondingly, a moderate discrepancy is observed regarding with the effective stresses on the surrounding rock mass estimated from the analytical and numerical solutions as illustrated inFig. 4(c, d, e, f). The comparison shows a maximum difference about 5.7% and 3.5% for the effective radial and tangential stresses, respectively, which are also be found at the position near the wellbore and at the early transient flow state (t = 1 h). In general, we can state that the difference between the closed-form solution and the numerical simulation is more pronounced in the horizontal direction owning a higher permeability and as a function of elapsed time, the difference between the closed-form solution and the numerical simulation decreases.

As mentioned in the previous section, the closed-form solution of the transient fluid flow in anisotropic porous medium bases on the equivalent isotropic permeability concept. The discussions of Mathias and Butler²⁸ and Cihan et al.²⁹ elucidated that this approximation matches well in case of moderate anisotropic (k kx/ y 25) while discrepancy can become significant (particular for the observed points near the wellbore) with the increase of the anisotropic degree. This observation will be verified here with a focus on the mechanical Fig. 3.2D plane strain model used in the finite element simulation by Code_Aster.

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response of wellbore due to the hydro-mechanical coupling. For this aim, a parametric study was conducted with respect to the anisotropic permeability of rock masses representing by the ratiok kx/ y. InFig. 5are illustrated the numerical and analytical solutions of pore pressure, effective tangential and effective radial stresses near the wellbore which are calculated with different values of permeability ratio k kx/ y. To simplify the presentation, we highlight only the results in the horizontal cut and at the early transient fluid flow state (t = 1 h) at which the difference between the two approaches is maximum as observed previously.

Our results confirm the observation in28,29. More precisely, the difference of pore pressure obtained from the closed-form solution and the numerical simulation increases with respect to the anisotropic degree of permeability. A relative difference about 10% was noted at the anisotropic ratiok kx/ y=10which however increased more than 15% if the permeability anisotropy is higher than 25 (k kx/ y 25). Regarding with the mechanical response of wellbore, it seems that the influence of anisotropic permeability on the effective tangential stress near the wellbore is more pronounced in comparison with the effective radial

stress. Indeed, while the relative difference is monotonically increased with respect to the anisotropic degree of permeability for both pore pressure and the effective tangential stress, it is not the case of the effective radial stress. A maximum relative error about 15% was also noted at the high permeability ratiok k_x/ _y 25for the tangential stress while this value was lower than 8% for the radial stress.

This numerical verification test elucidates the validity of the closed form solution developed in this work particularly for the case of the surrounding rock mass owning a weak or moderate anisotropic hydraulic property.

4. Stability analysis of wellbore in the anisotropic porous rock 4.1. Chosen anisotropic failure criterion

In the literature, the stability of wellbore is usually analyzed by assuming an isotropic behavior of formation. This assumption was adopted not only for the poro-elastic properties but also for the strength of rock mass. In this context, the Mohr-Coulomb, Hoek-Brown, Mogi- Fig. 4.Pore pressure, effective radial and tangential stresses following the horizontal and vertical cuts at the center of wellbore: comparison between the analytical and numerical solutions.

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Coulomb and Drucker Prager criteria^11–14may be the most commonly used. If in the two former criteria (i.e., the Mohr-Coulomb, Hoek-Brown criteria), only the maximum and minimum principal stresses are considered, the two latter criteria take into account the effect of the intermediate principal stress on the rock strength.

The extension of these criteria in the case of anisotropic rock has been conducted since a long time15,21,33–36but this subject remains one of the most challenging issues in rock engineering. Among different contributions, we can mention the first attempt of Jaeger³⁷who used the Mohr-Coulomb failure criterion associated with the weak plane for a rock mass having a set of parallel planes of weakness. In this last model (also known as the single weakness plane theory), the failure is supposed to take place in the weak plane if the inclination angle (i.e., angle between the direction of the maximum compressive stress and the normal of the planes of weakness) varies from the angle of internal friction in the planes of weakness to 90°.¹⁰ Outside this range, the planes of weakness have no impact on the rock strength¹⁰ and the failure is decided by the strength of intact matrix rock. More sophisti- cated failure criteria have been presented in the literature employing the phenomenological15,21,33–36 as well as micromechanical approach.³⁸In their work, Pietruszczak and his collaborator^33–35used the phenomenological approach to extend the Mohr-Coulomb and Hoek- Brown criteria in the anisotropic rock by considering the orientation- dependent strength parameters. More precisely, they considered that the cohesion and the friction angle of the Mohr-Coulomb model (or correspondingly, the two parametersmands of the Hoek-Brown criterion) are not constant but are function of the spatial orientation. In comparison with the isotropic criteria model or the single weakness plane theory of Jaeger³⁷which suppose that the failure occurs in the direction parallel to the weakness plane, the extended models of Pie- truszczak and his collaborator^33–35consider that the failure may take place on the critical plane at which the failure function reaches the maximum. Moreover, this critical plane can have different orientation

with respect to the weakness plane.

In this work, for the stability analysis of wellbore, the non-linear failure criterion proposed by Mroz and Maciejewski³⁶is chosen. This choice can be explained by the fact that this criterion allows not only to characterize the anisotropic strength of rock masses but also to simulate the continuous transition between the compressive and tensile failure.

This last characteristic could facilitate the determination of mud pressure window in comparison with the classical procedure which bases usually on two separated programs to calculate the fracture initiation pressure and collapse pressure by distinguish the tensile cut-off strength from the shear failure criterion. Follow that, the expression of the non- linear failure criterion adopted in this work is as follows (see also³⁶):

= >

F a

St St St

( )_d n 1 n , , 0

m

n (38)

where _n, _ndenote the shear and the normal stress acting on an arbitrary plane representing by an inclined angle (angle between the normal vector of this plane and the vertical principal stress axis). Keep in mind that the compressive stresses are negative as assumed previously. Thus, the shear and the normal stresses are related to the vertical and horizontal principal stresses (1, 3) as follows³⁶:

= ( )sin[2 ] = +

2 , . sin [ ] . cos [ ] ,

n 1 3 n

3 2

1 2

(39) These relationships are deduced from the plane strain condition by considering that the strike of the bedding plane is parallel to the direction of the intermediate principal stress whose effect is neglected (hypothesis adopted in this work).

Applying on our problem of wellbore, at the well wall, the two extreme principal stresses are respectively the effective radial and ortho-radial stresses:

= ( , ),Pw = r( , )Pw

1 3 (40)

Fig. 5.Comparison between the analytical and numerical solutions of pore pressure (a), effective tangential stress (b), effect radial stress (c) following the horizontal cut at the center of wellbore and their maximum relative errors (d).

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In Eq.(38), the parameterSt denotes the tensile strength while the two other parameters must verify the conditionsa >0, 0<m<1. The chosen non-linear failure model presents an anisotropic characteristic due to the fact that the two parameters (St,a) are orientation dependent and are calculated as function of the damage parameter d(see³⁶):

= +

St St St

a a

St

a

St St

(1 ) . ;

(1 ) ;

t d d d

t tm

m

d d

dm m

d m

m

11 1

1 1

(41) In Eq.(41)a Stt, tanda Std, dcorrespond to the parameter and tensile strength of the intact and damaged states while the same parametermis taken in both limit surfaces of the intact and damaged states of rock mass.

In their work, Mroz and Maciejewski³⁶proposed an elliptical form of damage distribution function as follows:

= +

.

. sin [ ] . cos [ ] ;

d 1 2

12 2

22 2

(42) where the major ellipse axis 1inclines an angle with respect to the horizontal principal stress. In the local coordinate at the well wall, the horizontal principal stress represents in fact the radial stress, thus this major ellipse axis 1is parallel to the bedding plane (correspondingly, the normal vector of bedding plane coincides with the minor ellipse axis

2) as highlighted inFig. 6.

Under this condition, Mroz and Maciejewski³⁶ showed that the failure may occur on the critical plane which should satisfy the statio- narity property:

= =

=

F Pc( , , )w c max ( , , );F Pw c F 0;

c (43)

Thus, the inclined angle ccharacterizes the critical plane (Fig. 6) at which the functionFcis maximum at the considered point on the wall of wellbore (point characterized by the inclined angle and with a taken value of mud pressureP_w).

However, as the most important task, we must to determine the mud pressurePw at this last point so that the maximum of the functionFcis equal to zero:

= = = =

F P F P F

( , , ) max ( , , ) 0; | 0;

c w c w c c (44)

The resolution of this last equation presents in fact two solutions which are respectively the lower and upper bounds of mud pressure P_winf Pw Pwsupthat ensure the stability of wellbore at the considered plane . These lower and upper bounds correspond to the case that the effective radial stress or the effective ortho-radial stress plays the role as the maximum compressive stress, respectively:

=

F P with P P

( , , ) 0, ( , ) ( , )

c w c w r w

c w c r w w

inf inf inf

sup inf inf

(45)

Finally, the necessary minimumP_w^inf and maximumP_w^sup _{values of} the wellbore pressure P_w (known as the mud pressure window P_w^inf P_w P_w^sup) to ensure the stability of the wellbore for the whole range of inclined angle [0, ]are calculated as:

= =

P_w^inf max P_w , P_w min P_w

[0, /2]

inf sup

[0, /2]

sup

(46) The corresponding position angles ( ^inf and ^sup) at which the failure of wellbore initiated verify the condition:

= =

0; 0;

P_w^inf P_w inf

sup

sup (47)

4.2. Numerical applications

In this part, different numerical applications were conducted to reveal the anisotropic effect on the prediction of the mud pressure window. To this end, the same parameters as used in the previous verification test, like the anisotropic poro-elastic properties, the bedding plane angle (β = 0), the initial total stress and pore pressure, are taken while the chosen parameters of the non-linear model are respectively: a_t=5.8, St_t=3MPa, ad=1.8,St_d=0.5MPa, m=0.65,

=1,

1 ₂=0.45. With these chosen parameters, it shows inFig. 7a that the inclination of failure plane cin a classical biaxial test lies princi- pally in the range of 60–76° for different values of minor stress σ₃ ranging from 0 MPa to 10 MPa (noted that the absolute value of the stress is used in theFig. 7). This inclination angle c increases with respect to the magnitude of minor stress σ3as well as the inclination of weakness plane θ.Fig. 7b presents the depicted stress curves in σ3-σ1

spaces at failure for the intact rock and different damaged states of rock masses. With the same applied minor stress σ3the magnitude of major stress σ1is highest in the intact state of rock as expected while the lowest value is noted for the isotropic damage rule (η2=η₁= 1). With the same elliptical damage rule (η₂= 0.45,η₁= 1), the results show that the magnitude of major stress σ₁decreases when the inclination of weakness plane (θ) increases and it attains its minimum value at the inclination angle θ around 65°.

InFig. 8we present the variation of the failure functionFcalculated at the well wall with respect to the mud weight pressurePw and inclination angle plane θ. For each inclined plane θ, the lower and upper bounds (P_wînf,P_w^sup) of the mud pressure, which induce failure (i.e., the failure functionFequal to zero), were determined. For both case of permeable and impermeable rock, the same tendency can be observed when both the lower and upper bound (P_wînf,P_w^sup) increases with respect to the inclined angle. The results show that the mud pressure window (see the definition in Eq.(46)) determined in the permeable rock is bounded by the lower valueP_wînf=3.95MPaobtained at the inclined plane angle θ = 83.08° and the upper value P_w^sup=16.22MPa evaluated at the inclined plane angle θ = 0°. The corresponding mud pressure window in the impermeable rock is determined in the range of lower value P_wînf=P_wînf( =83.08 )° =4.23MPa and upper value Fig. 6.Critical plane characterized by the inclined angle cat which the non-linear failure function F is maximum and the elliptical rule of the damage parameter d

(for a representative point on the surface of wellbore with inclined plane θ).

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Fig. 7.Inclination of critical plane cversus inclination of weakness planes θ with different value of imposed minor stress σ3(a), stress conditions at failure depicted in σ3-σ1spaces of intact, isotropic damaged (η2=η1= 1) and elliptically damaged (η2= 0.45,η1= 1) states of rock mass with different inclined angle θ (b).

Fig. 8.Failure functionFcalculated on the well wall with respect to the mud pressurePwand inclination plane angle θ: results obtained in the permeable boundary case (a,c,e) and in the impermeable boundary case (b,d,f).