ARMA 11-273

Revisiting borehole stresses in anisotropic elastic media: comparison ofanalytical versus numerical solutions

Karpfinger, F., Prioul, R.Schlumberger-Doll Research, Cambridge, MA, USAGaede, O.University of Western Australia, Perth, WA, AustraliaJocker, J.Schlumberger DCS, The Hague, The Netherlands

Copyright 2011 ARMA, American Rock Mechanics AssociationThis paper was prepared for presentation at the 45th US Rock Mechanics / Geomechanics Symposium held in San Francisco, CA, June 26-29, 2011.This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of thepaper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members.Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited.Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must containconspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: We present a rigorous validation of the analytical Amadei solution for the stress concentration around arbitrarily orientatedborehole in general anisotropic elastic media. First, we revisit the theoretical framework of the Amadei solution and present analyticalinsights that show that the solution does indeed contain allspecial cases of symmetry, contrary to previous understanding, provided thatthe reduced strain coefficientsβ11 andβ55 are not equal. It is shown from theoretical considerations and published experimental datathat theβ11 andβ55 are not equal for realistic rocks. Second, we develop a 3D finite-element elastic model within a hybrid analytical-numerical workflow that circumvents the need to rebuild and remesh the model for every borehole and material orientation. Third, weshow that the borehole stresses computed from the numericalmodel and the analytical solution match almost perfectly for differentborehole orientations (vertical, deviated and horizontal) and for several cases involving isotropic and transverse isotropic symmetries. Itis concluded that the analytical Amadei solution is valid with no restrictions on the borehole orientation or elastic anisotropy symmetry.

1 Introduction

The calculation of stresses and displacements around cav-ities is required in some of the most important subsurfacegeotechnical engineering problems such as for boreholes,tunnels and mine excavations. For example, the presence ofa borehole in a stressed subsurface rock formation alters thelocal principal stress directions and magnitudes around theborehole and away from it over a distance of several bore-hole diameters. For isotropic elastic homogeneous rocks,borehole stresses are given by the classical elastic solutionby [1] or its generalized version for nonaligned boreholeand stress directions by [2, 3] and [4]. Borehole stressesdepend on the far-field stress, the orientation of the bore-hole with respect to the stress field directions, the wellborepressure and the material Poisson’s ratio. These solutionsare very convenient for practical purposes as all the bore-hole stress components except the axial component are in-dependent of the material elastic properties, and the axialcomponent is only dependent on the Poisson’s ratio by thevirtue of the plane strain assumption. Consequently, these

classical elastic solutions are widely used for engineeringand research applications.Most wells drilled for the purpose of natural oil and gas ex-traction encounter anisotropic shale formations during thedrilling process, either in the overburden for conventionalreservoirs or in the reservoir itself for unconventional shalereservoirs. For conventional clastic reservoirs, it has beenreported that shales constitute about 75% of the clastic fillof sedimentary basins [5] and for unconventional reservoirs,the recent exploration and production of US gas shale reser-voirs has put a renewed focus on drilling and hydraulic frac-turing in shale formations [6]. Shales are known to exhibitanisotropic properties not only for their elastic behavior[7, 8, 9, 10, 11] but also for their strength due to their lami-nated structure [12, 13]. [14] gives a thorough review of ex-isting experimental data in shales. Today, most wells drilledin highly deviated or horizontal directions are penetratingstrongly transverse isotropic formations or lower symme-tries such as orthorhombic or monoclinic if fractures arepresent. Consequently, in principle, most rock mechanicsanalysis in such anisotropic environments, for example for

wellbore stability and hydraulic fracturing design, shouldinvolve two key steps: first, the calculation of boreholestresses for anisotropic rocks, and second, a stress-relatedfailure criteria for anisotropic rocks.The fundamentals for the stress analysis in anisotropic me-dia were established by [15]. [16] used this approachbased on generalized plane strain assumption to calculatethe stress concentration around arbitrarily oriented bore-holes in arbitrary anisotropic rock formations. Althoughwell established from a theoretical point of view since morethan two decades, this fundamental analytical solution israrely used for practical applications in anisotropic mediaexcept by a few authors [17, 18, 19, 14] or is replaced bynumerical computation [20]. Alternative theoretical deriva-tion has also been obtained [21]. We speculate that there areseveral reasons for the unfortunate low use or acceptance ofthe Amadei solution: (1) the solution involves three coor-dinate systems (stress field, borehole and material orienta-tions) and is more complicated in form than the isotropicsolution; (2) it has been attributed some ”unjustified” se-vere shortcoming; (3) no numerical verification of the solu-tion has been presented in the literature and (4) in practice,the elastic and strength anisotropic material properties maybe difficult to obtain in-situ. From a measurement point ofview, modern sonic logging tools [22] can measure three orfour out of five elastic constants of transversely isotropicmedia [23, 24] and can potentially be used for boreholestress computations in such media. From a theoretical pointof view, despite a very thorough and comprehensive anal-ysis of the Amadei solution, [18] and [19] have statedthat the Amadei solution does not reduce to the Kirsch solu-tion for isotropic media or for transverse isotropic (heraftercalled TI) media when the borehole axis coincides with theTI symmetry axis. This apparent shortcoming is severe butunjustified, as shown in this paper.The purpose of our paper is to present a rigorous valida-tion of the Amadei solution using a numerical finite-elementanalysis for several cases involving different anisotropicsymmetries and well orientations. First, we revisit the the-ory of borehole stresses in anisotropic elastic media andpresent analytical insights that show that the Amadei solu-tion does indeed contain all special cases of symmetry, con-trary to previous understanding, provided that the reducedstrain coefficientsβ11 andβ55 are not equal. Second, wedevelop a 3D finite element elastic model within a hybridanalytical-numerical workflow that circumvents the need torebuild and remesh the model for every borehole and mate-rial orientation. Third, we show that the borehole stressescomputed from the numerical model and the analytical so-lution match almost perfectly for different borehole orienta-tions (vertical, deviated and horizontal) and for several casesinvolving isotropic and transverse isotropic symmetries.

2 Borehole stresses in anisotropic elasticmedia

Here, we give the framework within which the stressesaround a fluid-filled borehole in anisotropic homogeneouselastic media are derived. We only provide the governingequations and the final result. As this work was pioneeredby [15] and [16] we recommend their work for a moredetailed understanding of the problem.

2.1 General Assumptions

We consider an infinite formation of arbitrary anisotropywhich is homogeneous and continuous in all directions. In-ternally this body is bounded by a cylindrical borehole ofradiusa.

2.1.1 Geometry and coordinate systems

Figure 1: Schematic of the geographic and borehole ref-erence frames and the principal stress directions. The ge-ographic reference frame is the north-east-vertical (NEV)frame whose x-axis points to the north, y-axis points to theeast, and z-axis points downward in vertical direction. Theborehole frame is the top-of-hole (TOH) frame whose z-axis points along the borehole in the direction of increasingdepth. The x-axis is in the cross-sectional plane and pointsto the most upward direction, and the y-axis is found byrotating the x-axis90o in the cross-sectional plane in a di-rection dictated by the right-hand rule. The principal stressdirections are chosen such that one component is parallel tothe vertical NEV axis and the maximum horizontal compo-nent is rotated by the angleγ with respect to the north axis.

In the far-field an in-situ stress field is applied where the

principal stress tensor takes the form

σ =

σH 0 00 σh 00 0 σv

, (1)

whereσH and σh are the maximum and minimum hori-zontal stresses, respectively, andσv is the vertical stress(see Figure 1). For the sake of simplicity, but withoutloss of generality, we assume that the vertical stressσv isalways aligned with the vertical component (V) of the NEV(north-east-vertical) coordinates system. The horizontalstress field can be rotated by an angleγ measured betweenN (north) andσH towards E (east). The regional stress fieldis first rotated into the NEV frame. For the computationof the borehole stress concentration it is convenient torotate the stress field in the NEV frame into the top-of-holeborehole coordinate system, hereafter called TOH (seeFigure 1 for definition). The orientation of the boreholeis defined by the deviation angleαD and the azimuthangleαA. All these coordinate transforms are defined by[25]. Here and in the rest of the paper we assume, forconvenience, that the in-situ stress field is aligned withthe NEV frame (i.e.,γ = 0). The solution of the stressconcentration is obtained in the TOH frame in Cartesiancoordinates [16]. Due to geometry the borehole problem itis natural to transform the borehole stress components intocylindrical coordinates [see 18, eq. A.15].

2.1.2 Generalized plane strain

For the solution of the stress concentration and displace-ment problems some assumptions can be made which sim-plify the general solution but are still reasonable approxi-mations. We can assume that the borehole is infinite andhomogeneous in the axial direction; this is referred to asa plane strain formulation which can be applied if solu-tions are sought far enough from the ends of the boreholeas well as interfaces. For an isotropic medium this formu-lation requires that the strainsεzz = εxz = εyz = 0, fromwhich follows that the axial displacement isw = 0. For ananisotropic body this assumption is not valid as the equationof equilibrium and Hooke’s law would not be satisfied [15].Thus, we have to apply the generalized plain strain formu-lation [16] which requires the displacement components tobe functions ofx andy only:

∂u

∂z=

∂v

∂z=

∂w

∂z= 0. (2)

From equation 2 it follows that the only strain which is zerois εzz. Based on this assumption, we will introduce the ba-sic equations that are necessary to determine the stressesinduced by a far-field stress field around a borehole in anarbitrarily anisotropic formation.

2.1.3 Governing equations

For all elastostatic problems, the stress, strain and displace-ment components must satisfy the constitutive relations, theequations of equilibrium, the equations of compatibility forstrains and the strain-displacement relations, as well as theboundary conditions.

The strain componentsǫij are related to the stress com-ponentsσij via the constitutive relation:

ǫij = Sijklσkl, (3)

whereSijkl is the compliance tensor. Although the theo-retical developments presented here are valid for arbitraryanisotropy symmetry, for the sake of clarity, we considerhere only the following symmetries for the compliance ten-sor: (1) isotropic with two elastic constants (hereafter calledISO) and (2) transverse isotropy with five elastic constants(i.e.,TI). In addition, depending on the orientation of theTIsymmetry axis, we distinguish two TI symmetries: TI witha vertical axis of symmetry (hereafter called VTI) and TIwith tilted axis of symmetry (hereafter called TTI).

As all measurements are obtained in the borehole, it isconvenient to rotate the compliance tensor into the TOHframe. This is done by applying two Bond transformationsto the6×6 Voigt notation compliance matrixsij giving aij.The definition of this transformation can be found in [18].The orientation of a TTI material is defined by the dip angleβD and the azimuthβA (see Figure 2).At any position around the borehole, the strain is now re-lated to the stress in Cartesian coordinates via the constitu-tive relation

ǫxx

ǫyy

0γyz

γxz

γxy

=

a11 a12 a13 a14 a15 a16

a12 a22 a23 a24 a25 a26

a13 a23 a33 a34 a35 a36

a14 a24 a34 a44 a45 a46

a15 a25 a35 a45 a55 a56

a16 a26 a36 a46 a56 a66

︸ ︷︷ ︸

a

σxx

σyy

σzz

τyz

τxz

τxy

,

(4)

whereγyz = 2ǫyz etc. Theaij are the components of thecompliance tensora for a general anisotropic medium in theborehole frame. After rotating the compliance tensors intothe borehole frame, all components ofa can be nonzero.The only assumption made at this point is thatǫzz = 0.For generalized plain strain problems very often the reducedstrain coefficientβij is used which is defined as

βij = aij −ai3aj3

a33

i, j = 1...6. (5)

In addition to Hooke’s law, we also require the equationsof equilibrium which can be found in [see 26, Chap. 5.5].As there are five strain components but only three displace-ment components, we need, in addition, two strain compati-bility equations which can be found in [see 26, Chap. 2.13].

Figure 2: The material coordinate system for transverseisotropic medium with tilted symmetry axis (called TTI)whereβD is the dip of the transverse isotropy plane andβA is the dip azimuth.

2.2 General Solution

A general solution for the stresses around a borehole in ananisotropic medium can be found by using the concept ofAiry stress functions [26, Chap. 5.7]. The general expres-sions for the borehole-induced stressesσbi which can be su-perimposed onto the corresponding components of the far-field in-situ stress tensor in the TOH frameσTOH to get theborehole stress tensorσBH are

σxx,BH = σxx,TOH + σxx,bi

= σxx,TOH + 2Re[µ2

1φ′

1(z1) + µ2

2φ′

2(z2) + λ3µ2

3φ′

3(z3)]

σyy,BH = σyy,TOH + σyy,bi

= σyy,TOH + 2Re[φ′

1(z1) + φ′

2(z2) + λ3φ′

3(z3)]

τxy,BH = τxy,TOH + σxy,bi

= τxy,TOH − 2Re[µ1φ′

1(z1) + µ2φ′

2(z2) + λ3µ3φ′

3(z3)]

τxz,BH = τxz,TOH + σxz,bi

= τxz,TOH + 2Re[λ1µ1φ′

1(z1) + λ2µ2φ′

2(z2) + µ3φ′

3(z3)]

τyz,BH = τyz,TOH + σyz,bi

= τyz,TOH − 2Re[λ1φ′

1(z1) + λ2φ′

2(z2) + φ′

3(z3)], (6)

whereφ′

zi(i = 1, 2, 3) are the spatial derivatives of three

analytic functions which are defined by [18]. The problemcan now be fully solved by finding solutions to the analyticfunctionsφ′

ziby applying the correct boundary conditions

in the far-field as well as on the borehole wall. This is de-scribed in detail by [16] and [18].From equation (4), it is obvious due to the generalized plain

strain assumption that the axial stressσzz,BH can be writtenas

σzz,BH = σzz,TOH −1

a33

×

(a31σxx,bi + a32σyy,bi + a34τyz,bi + a35τxz,bi + a36τxy,bi) .(7)

σzz,BH can be computed after the other induced stress com-ponents are obtained from equation (6).

2.3 Special cases of anisotropy

There are several cases to consider in which a degenera-tion of the general solution happens [16]: (i) orthorhom-bic medium with one plane of elastic symmetry perpendic-ular to the hole axis (the two other planes being parallelto it), (ii) transverse isotropic medium with the plane ofisotropy striking parallel to the hole axis, (iii) transverseisotropic medium with the plane of isotropy perpendicularto the hole axis and (iv) isotropic medium. This is the casewhen the system of coupled differential equations in [see18, Eq.3.3.3] is decoupled (L3 = 0). Thus the elastic ten-sor takes the following form

A =

a11 a12 a13 0 0 0a12 a22 a23 0 0 0a13 a23 a33 0 0 00 0 0 a44 0 00 0 0 0 a55 00 0 0 0 0 a66

. (8)

This is the elastic tensor for an orthorhombic material withnine different elastic components (three Young’s moduli,three shear moduli and three Poisson’s ratios).Now the problem is decoupled into two problems: a plainstrain problem involvingσxx, σyy andτxy and a longitudi-nal shear problem involvingτxz andτyz.After the decoupling, it also follows that the resulting char-acteristic equation for elastic media in which there is no az-imuthal symmetry ((iii) and (iv)), is [18, eq.3.3.5]

l4 = β11(µ4 + 2µ2 + 1) = 0 (9)

l2 = β55(µ2 + 1) = 0. (10)

It has been stated by [18] and [19] that a solution for thisproblem can only be found for the above cases (i) and (ii),and that for cases (iii) and (iv), no solution can be found dueto coincident roots which result in singularities. We wouldlike to show here that this is not the case, and the solutionactually works for any symmetry.The roots of this characteristic equation are identical toµ = i only if β11 = β55 6= 0 [18, eq.3.6.4]. For thatcase the solution would have singularities due to coincidingroots. We investigated this assumption about the identity

of the reduced strain coefficientsβ11 andβ55. The identityβ11 = β55 would imply, for transversely isotropic media,the relationshipsν2

v = Ev(1/Eh − 1/Gv), i.e. Gv ≥ Eh

and for isotropic mediaν2 = 1−E/G, i.e. G ≥ E; both in-equalities are unlikely for realistic rocks. Furthermore,wehave calculated the reduced strain coefficients from an ex-tensive ultrasonic velocity and anisotropy data set publishedby [11]. The data set includes a wide variety of sedimen-tary rocks (reservoir rocks and seals) from oil and gas fieldsaround the world. We have taken the entire data set into con-sideration, consisting of 17 brine-saturated shale samples,one gas and brine-saturated coal sample, 8 brine-saturatedsands, 12 gas-saturated sands, 32 gas-saturated carbonatesamples and 25 brine-saturated carbonate samples. In Fig-ure 3, we have plottedβ11 againstβ55 for the entire data set.The dashed line represents the assumption thatβ11 = β55.One can see thatβ11 is never equal toβ55, and in the caseof the shalesβ55 can be up to three times larger thanβ11. Itis important to note that the numerical instability mentionedby [18] only occurs whenβ11 is exactly equal toβ55.

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

β11

β55

Carbonates GasCarbonates BrineSandsShales

Figure 3: Reduced strain coefficientsβ11 andβ55 for var-ious shales, sandstones and carbonates (gas and brine sat-urated); obtained from experimental ultrasonic velocities[11].

In the following section we show that the Amadei solu-tion is in agreement with our numerical model for the spe-cial cases (iii) and (iv) as the conditionβ11 = β55 is neverfulfilled for any realistic combination of elastic parameters.A close inspection of the values of the roots of the character-istic equations for the isotropic cases shows that these rootsare very close to the imaginary numberi but always have asmall real part and an imaginary part slightly different fromi. We can only speculate that the reason why Ong found lim-itations to the Amadei solution is that single-precision com-putation would yieldµ = i when double-precision compu-

tation would giveµ 6= i. This small difference always givesa stable result for all the special cases considered providedour computations are done with double precision variables.

3 Numerical Method

In order to validate the analytical solution described above,we utilized the commercial finite element solver COMSOL[27]. We have developed a 3D finite element model whichcomputes the stresses around a deviated borehole in an ar-bitrarily anisotropic medium. This problem cannot be mod-eled as a 2D problem as for 2D the out-of-plane stress com-ponents would be neglected, and the far-field boundary con-dition acting parallel to the borehole could not be applied.The model also cannot be reduced to one quadrant usingsymmetries in the geometry as is usually done for isotropicsimulations because for an arbitrary anisotropic materialthesymmetries are not known a priori. Thus, in order to com-pute borehole stresses, we use a cubical model with an edgelength of5 m. The borehole is placed in the center of themodel with a radius of0.1 m. The structured mesh is il-lustrated in Figure 4 and consists of 11400 hexahedral ele-ments with 294840 degrees of freedom. The computationaltime for such a model is around three minutes on a work-station with a 3 GHz processor and 32 GB RAM. The meshis refined in a cylindrical region around the borehole, whichhas a radius of six borehole diameters (0.6 m). In this re-gion, the mesh density is increased linearly. The size of theinnermost elements is5 times smaller than the size of theoutermost elements of the cylindrical region. The Dirichlet

Figure 4: View of the mesh of the 3D numerical model anda 2D cross section perpendicular to the borehole axis.

boundary conditions are applied as stresses. As stated be-fore, we assumed that the coordinate system of our modelis aligned with the in-situ stressσ and therefore representsthe NEV-frame. This means that for a vertical borehole onlynormal stress components are applied to the boundaries ofthe model. The linear solver MUMPS [28] is used.

First, we validated our numerical model against the

Figure 5: Comparison of borehole stresses computed usingfinite element and Kirsch solutions for a vertical well in anisotropic medium (ν = 0.32), and a stress fieldσV = 30MPa,σH = 20 MPa,σh = 10 MPa withPw = 5 MPa. Theradial, tangential and axial stresses are plotted as functionsof radial position away from the borehole in the direction ofσh (borehole azimuthθ = 0). The stress concentration dueto the borehole abates within the densely meshed region ofthe numerical model.

Kirsch solution [1] for a vertical well in an isotropic for-mation. In Figure 5 the borehole stresses,σrr, σzz andσθθ,from the numerical model medium are plotted together withthe corresponding Kirsch solution radially away from theborehole. It shows a very good agreement between the twomodels. It also shows that the chosen geometry and meshgive steady results and that the size of the model is bigenough to avoid that the stress concentration is influencedby the size of the model.

In order to keep the same geometry and the same mesh forall models of interest we decided to mimic the borehole de-viation by applying appropriate stress boundary conditionson the surface of the block Figure 6-a and 6-b. This can beachieved by computing the components of the rotated stresstensor for a given borehole deviation and azimuth. The re-sulting stress tensor can have up to six different components(three normal and three shear stresses). These componentsare applied as stress boundary conditions on the surface ofthe block.The components of the elastic tensor are influenced by thematerial symmetry and orientation as well as the boreholeorientation. The Bond transformation gives the full elastictensor for a given material dip angle and azimuth (for ex-ample for a TTI medium) as well as borehole deviation andazimuth. We use the rotated tensor in the finite element sim-ulation in order to mimic the material orientation relativetothe borehole (Figure 6-a and 6-b).

(a)

(b)Figure 6: Schematic of the stress and elastic tensors trans-formations required to set up the boundary conditions of a”borehole centric” finite element model mimicking the sit-uation of an arbitrarily oriented borehole in an arbitrarilyoriented anisotropic medium. Subfigure (a) depicts the in-situ conditions and (b) the ”borehole centric” finite elementmodel with appropriate boundary conditions.

4 Validation of the Amadei solution

4.1 Boundary Conditions and Material Parame-ters

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

Figure 7: Borehole stressesσrr, σθθ, σzz, σθz around theborehole wall from the analytical (lines) and numericalfinite element (dots) solutions for a vertical well in anisotropic medium with Poisson’s ratioν = 0.079.

We validated the Amadei solution for isotropic, VTI andTTI media in an normal stress regime (σv = 30 MPa,σH = 20 MPa andσh = 10 MPa) and a wellbore pres-sure ofPw = 5 MPa.For the VTI medium we chose a representative shale (Sam-ple E5) from the experimental data collection published by[11]. The values for the five elastic parameters are listedin Table 1. The elastic tensor for the TTI medium is de-fined by tilting the VTI medium via a bond transformationwith a material dip ofβD = 30o and a material azimuth ofβA = 30o. The various borehole and material orientationsare summarized in Table 2.

Geomechanics Geophysics Stiffness

Eh = 31.17 GPa Vp0 = 3340 m/s C11= 45.2 GPaEv = 15.42 GPa Vs0 = 1675 m/s C33 =28 GPaνh = 0.079 ǫ = 0.3065 C44 = 7.05 GPaνv = 0.32 γ = 0.234 C66 = 14.4 GPaGv = 7.05 GPa δ = 0.5244 C13= 19.67 GPa

Table 1: Anisotropic elastic properties for the VTI andTTI models reported in several notations: rock mechanics(Young’s and shear moduli and Poisson’s ratio), geophysics(velocities and [29]’s parameters) and stiffness. Values arefrom shale E5 from [11] where the density is2.535 g/cm3

and porosity is5%.

4.2 Validation Results

Figures 7 to 9 summarize the results of the borehole stressesat the borehole wall and around it for the seven chosenmodels (Table 3) using both the numerical model and theAmadei solution. We observe that the agreement betweenthe two solutions is excellent. The degeneration of theAmadei solution when a plane of elastic symmetry is per-pendicular to the borehole axis (Figure 8a) as described by[19] was not observed. In particular, Figure 7 shows thatthe Amadei solution is also valid for isotropic media as ex-pected from the theoretical section. Furthermore, we ob-tained identical results for any borehole orientation in anisotropic medium using the Kirsch and the Amadei solu-tions; we chose not to plot them here as the curves are indis-tinguishable. Furthermore For all these special cases and forarbitrary borehole and material orientations we have foundno limitations of the Amadei solution and therefore havevalidated this closed-form solution up to TI anisotropy.

5 Discussion

Every borehole stability analysis has to address two essen-tial questions; first, what is the stress distribution aroundthe borehole and away from it and, second, at what stress

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(a)

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(b)

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(c)

Figure 8: Borehole stressesσrr, σθθ, σzz, σθz around theborehole wall from the analytical (lines) and numerical fi-nite element (dots) solutions for a VTI medium and wellorientations: (a) vertical, (b) deviated and (c) horizontal.Corresponding material properties are given in Table 1 andorientations angles in Table 2.

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(a)

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(b)

0 20 40 60 80 100 120 140 160 180−10

0

10

20

30

40

50

60

70

80

Angle θ

Str

ess

in M

Pa

Analytical σrr

Analytical σθθ

Analytical σzz

Analytical σθ z

Numerical

(c)

Figure 9: Borehole stressesσrr, σθθ, σzz, σθz around theborehole wall from the analytical (lines) and numerical fi-nite element (dots) solutions for a TTI medium (TI planehas30o dip angle and azimuth) and well orientations: (a)vertical, (b) deviated and (c) horizontal. Corresponding ma-terial properties are given in Table 1 and orientations anglesin Table 2.

Model αD αA βD βA FigureISO vert 0 0 0 0 7VTI vert 0 0 0 0 8(a)VTI dev 45 45 0 0 8(b)VTI hor 90 0 0 0 8(c)TTI vert 0 0 30 30 9(a)TTI dev 45 45 30 30 9(b)TTI hor 90 0 30 30 9(c)

Table 2: Summary of the models used for validation withthe orientations of the well (αD andαA) and the TI medium(βD andβA) as well as the figure numbers where the resultsare displayed.

does the formation fail. As shown above, the analyticalAmadei solution answers the first question correctly for anarbitrarily oriented borehole in a general anisotropic elasticmedium subjected to an anisotropic in-situ stress field. Thisnumerical verification is of great importance, because therehas been some debate over the general validity of the solu-tion. This debate stemmed from the claimed loss of gen-erality of the solution for special cases of anisotropy andisotropy and the lack of an independent verification of thesolution as such. This and the fact that in the past mea-surements of the formation anisotropy were difficult to ob-tain in-situ led often to the unfortunate situation in whichthe Amadei solution is not used in common practice. Froma measurement point of view, modern sonic logging tools[22] can measure three or four out of five elastic constants oftransversely isotropic media [23, 24] and can potentially beused for borehole stress computations in such media. Mostwells drilled today for the purpose of natural oil and gas ex-traction encounter anisotropic shale formations during thedrilling process either in the overburden for conventionalreservoirs or in the reservoir itself for unconventional shalereservoirs. Variations in elastic anisotropy lead to changesin both the field stress conditions [30] and the boreholestresses responsible for initiating hydraulic fractures andborehole breakouts, changing both the upper and the lowerlimit of the mud weight required for drilling. This highlightsthe importance of characterizing formation anisotropy andthe need to incorporate this knowledge into borehole stabil-ity analysis. The Amadei solution is a powerful closed formsolution as the computation time is practically negligible,and it is therefore an obvious choice for a stability optimiza-tion or inversion analysis. The second question for boreholestability analysis, the failure criterion of anisotropic elasticrocks [26, 31, 14], is a broad topic that is considered outsideof the scope of the paper. The borehole stress solution pre-sented here addresses only elastic media and therefore cou-pled physical processes such as described in poroelasticityor porothermoelasticity theory are not included. In general,it is possible to extend the analytical solution for such cou-

pled problems and first steps have been taken in this direc-tion [32], although numerical solutions have also been de-veloped [20]. As soon as the borehole starts to fail and timedependent processes occur, the stresses will be redistributedand static elastic solutions such as the one presented hereare no longer valid. Dynamic stress redistributions have tobe analyzed using modern numerical methods.

6 Conclusion

We have presented a rigorous validation of the analyticalAmadei solution for the stress concentration around arbi-trarily orientated boreholes in general anisotropic elasticmedia. First, the review of the theory provided analytical in-sights that the solution does indeed contain all special casesof symmetry contrary to previous understanding providedthe reduced strain coefficientsβ11 and β55 are not equal.Next, we have shown from theoretical considerations andpublished experimental data that theβ11 and β55 are notequal for realistic rocks. Second, we developed an efficienthybrid analytical-numerical workflow using a 3D finite el-ement elastic model that circumvents the need to rebuildand remesh the model for every borehole and material ori-entation. Third, we have shown that the borehole stressescomputed from the numerical model and the analytical so-lution match almost perfectly for different borehole orien-tations (vertical, deviated and horizontal) and for severalcases involving isotropic and transverse isotropic symme-tries. It is concluded that the analytical Amadei solutionis valid with no restrictions on the borehole orientation orelastic anisotropy symmetry provided the generalized planestrain assumption is met.

Acknowledgements

The authors would like to thank Schlumberger for allowingus to publish this paper.

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