**1. Introduction**

This study seeks to clarify the accuracy and possible limitations of the classical analytical solutions of Kirsch [1]—for the stress tensor field in linear-elastic plates pierced by one or more circular holes under certain far-field stress and internal pressure loads—when used in practical applications. These solutions are widely applied in wellbore-stability models and tunneling applications, and regularly involve the superposition of elastic displacements due to various boundary forces (far-field and internal pressure-loading) acting on the holes. Such situations have been systemically evaluated for linear-elastic isotropic and anisotropic rocks under a plane stress assumption [2–4].

However, recurrent concerns prevail related to the accuracy of results when using the solutions of [1] without modifications in wellbore-stability models. For example, the density of drilling mud pumped into the space between the wellbore and the drill string during drilling operations needs to be selected such that fracturing due to tensile and shear failure will not occur [5–7]. Obviously, the magnitude of the elastic stress concentrations and their orientation near the hole in the rock formation will rapidly vary when the pressure load of the mud is added to the borehole. When there is no internal pressure on the wellbore, the stress concentration factor for uni-axial far-field stress is always 3 (and for a biaxial compression reaches 4 [8]). However, when a net pressure is exerted on the wellbore's interior, the induced elastic deformation of the host rock increases or decreases the stress concentrations induced by the far-field stress, and therefore both contributions must be carefully evaluated, preferably in real-time, during drilling operations [9]. The need for real-time analysis is also the reason why superposed analytical solutions are still in vogue and cannot easily be replaced by solution methods that require gridding and have consequent longer computation times.

An additional concern is whether the standard plane stress solution of Kirsch [1] is accurate enough, whether a plane strain approach should be used, or any other approach. The plane stress solution is an end member solution for so-called thin plates; the other end member would be a thick-plate approach (plane strain assumption); each is often portrayed as 2D solution but in fact considers the state of 3D strain and stress, respectively, at all times. Although prior studies have evaluated the difference between plane strain and plane stress solutions, typically only the maximum stress concentrations are compared without analyzing the stress states further away from the boreholes. Also, the effect of the internal pressure loading superposed in the far-field stress anisotropy is normally only evaluated for arbitrarily chosen cases, which is why additional systematic evaluations in our study are merited.

Other concerns arise when multiple wells are drilled in close proximity from the same surface location and the stress interference due to the mutual interaction between the wells needs to be accounted for in the wellbore stability models. Our analysis considers both single-hole solutions and solutions for the superposition of multi-holes, all with or without individually varying pressure loads in addition to the far-field stress loading. The method of solution used is the linear superposition method (LSM) first named in ref. [3], which adds the elastic displacement vectors due to various contributions (usually boundary forces) to the elastic distortion and then solves the stress tensor field using an appropriate constitutive equation for linear elasticity.

The present analysis revisits the basic solutions for plane stress and plane strain, points out some earlier errors in displacement equations appearing in standard

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

textbooks, and then proceeds to compute and compare the two end-member solutions (plane stress and plane strain). We also quantify the delta between thin and thickplate solutions for multi-hole problems using the LSM method (analytical superposition of displacements) for which plane stress solutions were first given in ref. [3]. Plane strain solutions are developed in the present study to quantify the delta between the solutions due to the assumed boundary condition. Additionally, the LSM multihole results are validated against—(1) photo-elastic contour patterns for a 5-hole problem of Koslowska [10], and (2) a numerical example of Yi et al. [11]. The present study is limited to hole analysis based on a linear elasticity assumption for isotropic elastic plates; borehole analysis of anisotropic media was given in prior work [4, 9], applying equations developed for plane stress cases in [12].

## **2. Prior work**

The petroleum, mining, and geotechnical tunneling industry have embraced the Kirsch equations for stability analysis of cylindrical boreholes in isotropic media. The governing equations for the Kirsch solution are based on Airy's stress function. A fact easily overlooked and little emphasized is that the equations introduced by Kirsch [1] assume a boundary condition of plane deviatoric stress, which would strictly limit the use of the Kirsch equations to cases that comply with the original boundary condition of plane stress. In spite of this limitation, the Kirsch equations are routinely applied in wellbore stability computations that may potentially yield inaccurate results if initially assumed boundary conditions in the analytical solutions are not met in the field application. For example, one may suggest that it may be more appropriate for deep boreholes in thick formations to use a thick-plate analysis (plane strain boundary condition), rather than the thin-plate analysis that fully justifies the use of the original Kirsch equations.

In nature and in real manufactured materials, the plane stress solution would only be valid for very large holes in very thin plates, such as for rivets in thin airfoils used in aircraft. However, for very thick elastic media perforated by tiny holes, such as in the case of boreholes penetrating rock formations of several kilometers thickness, the plane strain boundary condition seems more appropriate. We, therefore, evaluated what may be the actual inaccuracy creeping into the analysis of the stress concentration magnitude due to variations in the boundary conditions. Unwarranted wellbore stability problems may occur if the stress state in the well appears to deviate from plane stress proxy solutions. The delta between the stress solutions for the thin and thick-plate approaches is fully quantified in the present study for a variety of cases.

#### **2.1 Evaluation by Clark**

The notion that considerable differences may arise between stress magnitudes in elastic plates due to different transverse boundary conditions (such as plane stress versus plane strain) if subjected to otherwise the same far-field stress has been long recognized. That the stress differential may become significant was quantified in a study by Clark [13] for a uniform plate of finite, uniform thickness (and no holes) loaded with time-dependent far-field sinusoidal stresses at a lateral edge of the plate. Three cases were highlighted in ref. [13], scaling the problem with a typical wavelength, *l*, of the sinusoidal load and the plate thickness, 2*h*. For <sup>2</sup>*<sup>h</sup> <sup>l</sup>* ! 0, we have plane stress (when the wavelength of stress applied is very large as compared to thickness). For <sup>2</sup>*<sup>h</sup> <sup>l</sup>* ! ∞, we have plane strain (when the wavelength of stress applied is very small as compared to thickness). When the wavelength is comparable to the thickness 2*h <sup>l</sup>* ¼ 1, the maximum stress at the edges of the plate may be up to 20% larger than for "elementary plane stress" (also termed "generalized plane stress" or "very thin plate theory"), which occurs when (2*<sup>h</sup> <sup>l</sup>* ! 0). The stress concentration values obtained for 2*h <sup>l</sup>* <sup>¼</sup> 1 exceed the plane strain solution (2*<sup>h</sup> <sup>l</sup>* ! ∞) by 31%. Clark [13] also emphasized that the generalized 2D plane stress solution for isotropic elastic plates is independent of the Poisson's ratio and neglects totally the transverse and normal stresses. Given the results of Clark [13], it is by no means obvious whether we may neglect the variations in both the stress concentrations and the stress transverse to an elastic plate with finite thickness and a boundary condition that is somewhere close to halfway between plane strain and plane stress. Below we discuss this quandary of the impact of boundary conditions on stress concentrations and transverse stresses for isotropic elastic plates with circular holes. Although the stress magnitude differentials according to ref. [13] may not be applicable to static loading cases (see later), borehole stability may be adversely impacted by seismic events, given the considerable differences in stress magnitude solutions when sinusoidal stress loads of different wavelengths are applied.

#### **2.2 Evaluation by Green**

Green [14] considered a linear elastic plate scaled by thickness 2*h* perforated by a hole of typical radius *a*. He introduced a practical dimensionless scaling parameter *<sup>a</sup> h* with hole radius in the nominator and plate half-thickness in the denominator. For a thick plate *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> , a plane strain boundary condition can be assumed (in this case the strain in the *<sup>z</sup>*-direction *<sup>ε</sup>zz* <sup>¼</sup> 0). For a thin-plate solution *<sup>a</sup> <sup>h</sup>* ! <sup>∞</sup> , a plane stress boundary condition can be assumed (in this case the stress in the *z*-direction *σzz* ¼ 0). The impact of boundary conditions—on the stress concentrations and transverse stresses in elastic plates with circular holes—that are midway between those leading to either perfect plane stress or perfect plane strain cases were reviewed by Green [14]. He posited that the case *<sup>a</sup> <sup>h</sup>* ¼ 1 would lie midway between the two extremes of plane strain *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> and plane stress *<sup>a</sup> <sup>h</sup>* ! <sup>∞</sup> boundary conditions. Based on 3D calculations for the case *<sup>a</sup> <sup>h</sup>* ¼ 1 ("midway" boundary condition), the stress concentration halfway the total depth of the plate at *z* ¼ 0 in the rim of a single hole in a location transverse to the applied far-field stress, *σxx*�<sup>∞</sup>, increased to 3*:*10 *σxx*�<sup>∞</sup>. If *σxx*�<sup>∞</sup> >0, we have a tension under the engineering sign convention, and hence 3*:*10 *σxx*�<sup>∞</sup> is an increased tension. At the plate's surface *<sup>z</sup>* <sup>¼</sup> *<sup>h</sup>* (for the case *<sup>a</sup> <sup>h</sup>* ¼ 1), the maximum stress concentration was less: 2*:*81 *σxx*�<sup>∞</sup>.

For both plane strain and generalized plane stress, the maximum stress concentration averaged over the thickness of the plate should be equal to 3*σxx*�<sup>∞</sup> [14]. This solution is exact for plane strain (thick plates) where—from a theoretical point of view—there may exist no variation in the maximum stress concentration near the hole for any depth *z*. However, for the finite-thickness plate (case with *<sup>a</sup> <sup>h</sup>* ¼ 1), only the averaged value will be 3*σxx*�<sup>∞</sup>, as is evident from [14] treatise. Again, at *z* ¼ 0 at the rim of a single hole, we have 3*:*10 *σxx*�<sup>∞</sup> (+3% different from 3*σxx*�<sup>∞</sup>), while at the surface of the plate at *z* ¼ *h*, the maximum stress concentration was lowered to 2*:*81*σxx*�<sup>∞</sup> (�6% different from 3 *σxx*�<sup>∞</sup>). The stress attenuation at the hole rim in the

*Asymptotic Solutions for Multi-Hole Problems: Plane Strain versus Plane Stress Boundary… DOI: http://dx.doi.org/10.5772/intechopen.105048*

longitudinal direction parallel to *σxx*�<sup>∞</sup> appeared to vary from �1*:*10 *σxx*�<sup>∞</sup> at *z* ¼ 0 (a compression +10% above �*σxx*�<sup>∞</sup>), while at the surface of the plate at *z* ¼ *h* the stress concentration was �0*:*81 *σxx*�<sup>∞</sup> (19% below �*σxx*�<sup>∞</sup>). Likewise, in ref. [15], Yang et al. found stress concentrations between two interacting holes in a finite-thickness elastic plate are maximum only at *z* ¼ 0, but decrease toward the surface of the plate ð Þ *z* ! *h* . Also, as the plates thicken, the maximum stress concentration shifts gradually to the surface of the plates.

According to [16] "generalized plane stress"-theorem, variations in stress concentration values throughout the thickness of a plate coinciding with the ð Þ *x*, *y* -plane can be neglected and only the average values of the remaining stress are estimated. However, as Green [14] showed for solutions at the rim of a circular hole, there will be variations in stress concentrations over depth 0 ≤*z*≤*h* when the plate has a finite thickness, characterized by *<sup>a</sup> <sup>h</sup>*. However, for *<sup>a</sup> <sup>h</sup>* ¼ 1, the average maximum stress concentration would only deviate about 3% from the average values. Green [14] concluded that the generalized plane stress theory gives "fairly good" estimations for the average values of stress concentrations at the hole in a stressed plate with boundary conditions "midway" between plane stress and plane strain (adopting *<sup>a</sup> <sup>h</sup>* ¼ 1 for this case).

#### **2.3 Other evaluations**

In our opinion, there can be little doubt that plane strain is the obvious boundary condition when boreholes are drilled in thick formations. So, the question is, what (if any) corrections are necessary when applying the Kirsch equations for plane stress to compute the stress concentrations near real-world boreholes. This question is addressed below considering two cases (A and B), as previously evaluated in ref. [4].

The plane stress boundary condition *<sup>σ</sup>zz* <sup>¼</sup> *<sup>σ</sup>xz* <sup>¼</sup> *<sup>σ</sup>yz* <sup>¼</sup> <sup>0</sup> assumed in the Kirsch solution in ref. [1] implies that the mean stress *σ* in the thin elastic plate *<sup>a</sup> <sup>h</sup>* ! <sup>∞</sup> is everywhere given by: *<sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup>xx*þ*σyy* <sup>3</sup> (Case A). For plane strain, the mean stress *σ* <sup>∗</sup> within the thick plate'<sup>s</sup> ð Þ� *<sup>x</sup>*, *<sup>y</sup>* plane of the thick plate *<sup>a</sup> <sup>h</sup>* ! <sup>0</sup> is given by *<sup>σ</sup>* <sup>∗</sup> <sup>¼</sup> ð Þ <sup>1</sup>þ*<sup>ν</sup> <sup>σ</sup>* <sup>∗</sup> *xx*þ*<sup>σ</sup>* <sup>∗</sup> ð Þ *yy* 3 (Case B), introducing the Poisson's ratio *ν*. The longitudinal stress is given by *σ* <sup>∗</sup> *zz* ¼ *ν σ* <sup>∗</sup> *xx* <sup>þ</sup> *<sup>σ</sup>* <sup>∗</sup> *yy* ; the longitudinal strain along the borehole is absent *<sup>ε</sup>* <sup>∗</sup> *zz* ¼ 0. For the special case of *ν* ¼ 0, the longitudinal stress will vanish, but rocks have Poisson's ratio closer to 0.25. We may assume that the mean stresses for adjacent plane strain and plane stress

sections of a borehole will be nearly identical stress concentration requirements such that we may equate the mean stress expressions for Case A and for Case B, from which it follows that *<sup>σ</sup>xx*þ*σyy σ* ∗ *xx*þ*σ* <sup>∗</sup> *yy* ¼ 1 þ *ν*. This relationship says that the magnitude of the principal stresses *σxx* þ *σyy* acting in a plane stress section of the borehole will be larger than the plane strain case *σ* <sup>∗</sup> *xx* <sup>þ</sup> *<sup>σ</sup>* <sup>∗</sup> *yy* by a factor 1 þ *ν* (about 125% in practice, using *ν* ¼ 0*:*25). For plane stress, the longitudinal strain component is given by *<sup>ε</sup>zz* ¼ � *<sup>ν</sup> <sup>E</sup> <sup>σ</sup>xx* <sup>þ</sup> *<sup>σ</sup>yy* , where *E* is Young's modulus of the material, which for practical situations with thinbedded rock strata are negligibly small strains that can be easily accommodated by the discontinuities in strain occurring due to variations in the elastic constants when the drill bit moves from one rock layer into the next.
