Expansion of horizontal wellbore stability model for elastically...

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1. INTRODUCTION

Horizontal well sections drilled in shale (Figure 1a) that

successfully avoid incurring any local imperfections

arguably will be better producers. Ideally, the wellbore

should stay perfectly circular cylindrical along its full

length. However, when the weight of the drilling fluid

does not properly balance the natural stress load on the

wellbore, some well damage may develop, due to brittle

failure (Figure 1b, c). Such damage may include borehole

breakout and drilling induced tensile failure, which may

(1) in open hole completions preclude the placement of

stage isolation sleeves at regular spacing during the frac

job, and (2) cause local perforation failure where cement

casing is thickened due borehole washout.

The risk of imperfect wellbore gauges is particularly high

when drilling in anisotropic shale, which results in near-

wellbore stress concentrations that are higher than in

isotropic formations, and therefore the safe drilling

window in shale will be narrower (Li and Weijermars,

2019). A modern state-of-the-art WBS model must

account for the effects of three types of anisotropy: (1) the

regional stress anisotropy, (2) the elastic anisotropy of the

local formation, and (3) local strength anisotropy, due to

anisotropy of the shale fabric. In particular, shale

anisotropy requires modification of failure criteria used

for isotropic rock (Li and Weijermars, 2019) and can be

augmented with a criterion for weak bedding plane slip

(Jaeger et al., 2007; Zhang, 2013).

The general trend of a decrease in wellbore stability due

to the anisotropy of both rock elasticity and principal

stresses increases has been noted in prior studies

(Aadnoy, 1988; Abousleiman and Cui, 1998; Ekbote and

Abousleiman, 2006; Karpfinger et al., 2011; Serajian and

Ghassemi, 2011; Setiawan and Zimmerman, 2018). Not

only does the elastic anisotropy cause stress re-

orientations near the wellbore, but also the likelihood of

bedding plane splitting and slip is known to increase when

the wellbore is inclined more than 60° from the vertical

axis (Okland and Cook, 1998; Zhang, 2013).

All prior analyses of stress concentrations and wellbore

stability in anisotropic shale formations compute the local

stress state using the traditional Lekhnitskii-Amadei

solution (Amadei, 1983; Lekhnitskii, 1963), which can be

ARMA 19–2062

Expansion of horizontal wellbore stability model for

elastically anisotropic shale formations with

anisotropic failure criteria: Permian Basin case study

Wang, J.

New Mexico Tech, Socorro, New Mexico, USA

Texas A&M University, College Station, Texas, USA

Weijermars, R.

Texas A&M University, College Station, Texas, USA

Copyright 2019 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 53rd US Rock Mechanics/Geomechanics Symposium held in New York, NY, USA, 23–26 June 2019. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper 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 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: The analytical wellbore stability model proposed in this study allows for a more accurate calculation of the safe

drilling window for horizontal wells drilled in anisotropic shale formations. The stress distributions around the horizontal wellbore,

based on Green and Taylor equations for stress concentrations around holes in elastically orthotropic plates, are compared with the

result from the traditional Kirsch equations. The comparison reveals that the elastic anisotropy of shales induces larger magnitudes

of tangential stress concentrations at the potential shear and tensile failure locations, i.e. 𝜋/2 or 0 from the maximum in-situ stress

direction. The anisotropic wellbore stability (WBS) model is expanded with a failure criterion that accounts for the anisotropy of

shale. The critical stresses for the safe drilling window depend on (1) the anisotropy of the in-situ stress conditions, (2) the elastic

anisotropy of the local formation, and (3) local strength anisotropy, due to anisotropy of the shale fabric. The model can be practically

applied in real-time wellbore stability analyses, to determine the stress concentrations and trajectories near the wellbore, as well as

for calculation of the safe drilling window. Two case study wells for the Permian Basin (West Texas and New Mexico) are included

in our study.

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applied to situations of wellbores either inclined or

parallel to the principal stress directions (e.g.,Setiawan

and Zimmerman, 2018). In the present study, we apply a

more concise and faster analytical anisotropic WBS

model based on stress function solutions of Green and

Taylor, 1945 (see Appendix A). Figure 2 gives a succinct

overview of the physical system attributes of interest as

they appear in the subsurface, with the typical acquisition

methods for determining characteristic system

parameters. Also included in Figure 2 are the scaled

representations of the physical system in two model

systems, which can be computed by applying certain

solution methods using the characteristic inputs. The

workflow schedule captured in Figure 2 will be the basis

for the structure and approach in this study.

The application of the stress solutions is limited to

horizontal wellbore sections (Figure 1a) aligned with the

direction of a principal stress direction. An additional

assumption is that the horizontal well coincides with the

isotropic plane of transverse isotropic elastic shale

(Figure 3a, b). With these conditions in place, the fast

WBS model for anisotropic shale can be applied to predict

the safe drilling window for horizontal hydrocarbon wells

in any basin that fulfills the assumed boundary conditions.

The two case study wells included are from the Permian

Basin, which spans broadly across West Texas and New

Mexico, US. The regional stress map for the Permian

Basin indicates that the native principal stress changes

gradually from the Delaware Basin to the Midland Basin

(Snee and Zoback, 2016). The present analysis first

develops stress concentration plots and stress trajectory

plots, using well data for the two locations, which are each

considered representative for the disparity in the regional

stress state across the Permian Basin. One drilling

location is selected in the Delaware Basin, and another in

the Midland Basin. Appropriate failure criteria are

applied, and the result for the safe drilling window in each

location is then compared when using (1) a standard

isotropic failure criterion, and (2) an anisotropic failure

criterion. A typical operational depth of 3,048 m (10,000

ft) is assumed, but results can be readily transposed to

either deeper or shallower horizontal wells.

This article proceeds as follows. Section 2 analyzes the

regional state of stress in the Permian Basin and identifies

all required input parameters to compare the stress

Figure 1. (a) Principle sketch of horizontal laterals in stacked

play with multi-bench development of producing intervals.

Horizontal wells are all mutually parallel and aligned with

the regional minimum stress. (b)&(c) Potential failure

locations along unbalanced wellbore, due to either under-

balancing (b) or over-balancing (c) of the native stress load

on the wellbore. For details on terminology used, see Wang

and Weijermars, 2019.

Figure 2. Key methods for the acquisition of the data on

initial state of the physical system (upper box), and the

solution methods for subsequent computations of the model

system (lower box), using the input data for the physical

system quantities of interest (upper box). This study uses

input data for two case study wells, compiled based on

previous characterization of key parameters. The new

contribution of our study is the application of the novel

solution methods to compute anisotropic shale WBS models

(Appendix A) for two representative horizontal well sections

in the Permian Basin.

Figure 3. Principle sketch of transverse isotropic shale with

horizontal isotropic fabric, and wellbore within the isotropic

plane. This study assumes a 3D in-situ stress resulting in

either (a) bi-axial deviatoric stress (with two non-vanishing

deviatoric stresses, red arrows) acting in the plane normal to

the wellbore, or (b) uni-axial deviatoric stress (with just one

non-vanishing deviatoric principal stress, red arrow) acting

in the plane normal to the wellbore. Case (a) applies to the

Delaware Basin well case study, and (b) to the Midland Basin

well case study (see discussion on in-situ stress in Section

2.2).

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concentrations of horizontal wells in rocks with isotropic

elastic moduli and anisotropic elasticity (using transverse

isotropic symmetry). With the stress concentrations

quantified in Section 2, Section 3 adopts an anisotropic

failure criterion and compares the results with an isotropic

failure criterion when applied to a wellbore stability

analysis of two case study wells in an anisotropic shale

section. Section 4 gives a sensitivity analysis of the

anisotropic WBS model for the horizontal well section in

the shale target zone. Section 5 is a discussion and Section

6 gives conclusions.

2. APPLICATIONS OF STRESS SOLUTIONS

TO WELLBORES IN PERMIAN BASIN

Sections 2.1 and 2.2 outline the characteristics and

regional stress state of the Permian Basin, respectively.

Section 2.3 outlines the stress function solution based on

Green and Taylor, 1945. Sections 2.4 (Delaware Basin

well) and 2.5 (Midland Basin well) show the stress

concentrations in the two assumed wells in the selected

locations.

2.1. Permian Basin drilling activity The geology and petroleum system(s) of the Permian

Basin have been described at length in numerous prior

studies (Dutton et al., 2005; Fairhurst et al., 2015; Hills,

1983; Hobson et al., 1985; Montgomery, 1996; Ross,

1986; Scholle et al., 2007). Figure 4 highlights the

regional subdivision of the Permian Basin in the eastern

Midland Basin and the western Delaware Basin. The

importance of perfect horizontal well completions is

evident from the relatively recent renewal of drilling

activity throughout the Permian Basin.

To give some context, the first hydrocarbon well in the

Midland Basin, Santa Rita-1, was drilled in 1923 and

remained in production until 1990. Major drilling activity

in the Midland Basin only picked up after the 1940s

(Blomquist, 2016). From the 1940s till about 2010, the

Spraberry was the primary target. Infill drilling in the

1980s, together with a reduction of vertical well spacing

to 0.16 km2 (40 acres) in 1997 and to 0.08 km2 (20 acre)

units in 2008, has led to sustained production from the

Spraberry formation. Before 2010 development of the

Spraberry target zone was primarily developed with

vertical wells. Midland Basin drilling activity steeply

accelerated in 2011 with 57 new wells in the Wolfcamp

Formation. By 2016 over 3,000 new horizontals were

completed, mainly in the Wolfcamp Formation, Midland

Basin, all with fracture treatment for enhanced recovery

(Blomquist, 2016).

The Delaware Basin (Figure 4) is also experiencing a

revitalized drilling campaign, due to the advent of new

technologies such as horizontal drilling and multistage

hydraulic fracturing. In 2016, over 1,000 new horizontal

wells were spud in the Delaware Basin (Mire and

Moomaw, 2017). The main targets of most drilling

campaigns are the unconventional Wolfcamp and Bone

Spring formations, collectively known as the Wolfbone

play (Figure 4). The overlying Delaware Mountain Group

is an exploration target for conventional hydrocarbon

accumulations. Before multiple horizontal wells are

drilled from a single pad (Figure 1a), typically a vertical

pilot well is drilled first. Such pilot wells not only provide

information about the hydrocarbon potential, but also

provide valuable insight about the geomechanical

properties of the subsurface. The regional dips in the

Permian Basin are generally only 1º or 2 º, which means

most wellbores stay perfectly in the target zone when

drilled horizontally.

2.2. In-situ stress Permian Basin, pore pressures

and elastic anisotropy target formation The required input data, characterizing the initial state of

the physical system, can be categorized into three groups

(Figure 2): (1) magnitude and orientation of the in-situ

stress and its anisotropy, (2) elastic anisotropy of the shale

target formation, and (3) anisotropic failure criteria for the

target zone. This section addresses the first two sets of

parameters, which will suffice to construct the model of

stress concentrations and stress orientations around the

wellbores in both locations (Delaware and Midland

Basins). A later section discusses anisotropic failure

criteria and the required input parameters, which are

needed to modify the anisotropic stress model into an

anisotropic WBS model.

(i) In-situ stress

Mechanical Earth Models (MEM) use well log data,

regional information, and published empirical

correlations to calibrate through history matching the in-

situ stresses, pore pressure, and rock mechanical

properties such as elastic moduli and rock strength

throughout the geologic column (Plumb et al., 2000). The

MEM can constrain the input parameters for the safe

drilling margin in order to prevent both borehole breakout

Figure 4. Map of the greater Permian Basin showing

subdivisions. The stratigraphic column is for the Delaware

Basin, using smoothed petrophysical logs to distinguish the

main geologic groups.

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and drilling induced tensile failures. If the wellbore for

which the MEM is constructed shows clear breakouts,

these may help constrain the orientation of the principal

stress directions, but the quality of caliper logs may be

inconclusive in many cases.

Rather than relying on a single, local MEM, a recent

regional stress regime study for the Permian Basin (Snee

and Zoback, 2016; Snee and Zoback, 2018) is used to

constrain the direction of the local largest horizontal stress

direction. The regional stress map for the Permian Basin

(Figure 5a) is based on several hundred local observations

of drilling-induced tensile fractures and borehole

breakouts detected in wellbore image logs, maximum

horizontal shear wave velocity from crossed-dipole sonic

logs, and hydraulic fractures from micro-seismic data.

The two wells studied here are located in respectively

zones 5 (Delaware Basin well, Reeves County) and 9

(Midland Basin well, Upton County). The direction of the

maximum horizontal principal stress is the same in both

zones: N085E. The horizontal laterals studied here are

assumed to be oriented perpendicular to the maximum

horizontal stress.

Although the stress regime map (Figure 5a) is a very

practical tool, care must be taken not to use the map

blindly. For example, a regional stress model using E-W

compression and know fault boundaries in the Delaware

Basin (Umholtz et al., 2018), suggests significant

reorientations of the principal stresses occur in the

vicinity of the faults (Figure 5b). At the same time, such

stress deviations are poorly constrained as they will vary

with the assumed properties of the faults.

The regional stress regime map (Fig. 5a) further scales the

regional stress anisotropy by a scaling parameter, 𝐴𝜙,

defined as follows (Simpson, 1997):

𝐴𝜙 = 𝑛 + 0.5 + (−1)𝑛(𝜙 − 0.5) (1)

where, 𝜙 is determined by total principal stresses 𝜎1,2,3 as

follows,

𝜙 =𝜎2 − 𝜎3𝜎1 − 𝜎3

(2)

𝑛 is a scaling factor related to Andersonian normal

faulting (𝑛=0), strike-slip faulting (𝑛=1), and reverse

faulting (𝑛=2). The values of 𝐴𝜙 range between 0 and 3,

with dominant Andersonian component as follows:

normal faulting (0<𝐴𝜙<1), strike-slip faulting (1<𝐴𝜙<2),

and reverse faulting (2<𝐴𝜙<3). Cartoons for stress states

depicted by certain values of 𝐴𝜙 are given in Figure A1

in Snee and Zoback, 2016. No effort was made in the

original work (Simpson, 1997) to exclude the overlapping

categorization for 𝐴𝜙 of 1 and 2. Note that for 𝑛=0, we

can equate 𝐴𝜙=𝜙. For the two study wells, located in

respectively zones 5 (Delaware Basin well, Reeves

County) and 9 (Midland Basin well, Upton County), the

𝐴𝜙 is 0.6 and 0.8, respectively.

For the Delaware Basin well, a value of 𝐴𝜙=0.6 indicates

the existence of nearly perfect Andersonian normal

faulting conditions (𝐴𝜙=0.5). As a starting point we may

assume a vertical stress gradient of 𝑆𝑉/𝛥𝐿=22.6 kPa/m

(=1.1 psi/ft). For a horizontal lateral at 𝐿=𝑇𝑉𝐷=3,048 m

(=10,000 ft), the vertical pressure at the wellbore is

𝑆𝑉=𝜎1=75.8 MPa (=10,000 psi). For perfect Andersonian

normal faulting in the Delaware Basin section of interest

Figure 5. (a) Stress state in the area of interest. Black solid

lines show the direction of the maximum horizontal stress,

color-coded map – stress regime in terms of 𝐴𝜙 (after Snee

and Zoback, 2018). (b) Deflection of regional stress near the

fault networks in the Delaware Basin (Ruppel et al., 2005),

according to geomechanical material point method (MPM)

simulation (Umholtz et al., 2018). Red colors indicate NW-

SE deflection of the input E-W stress field; blue colors

indicate NE-SW deflection of the input stress field.

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to occur, it must be in a state of plane stress (by definition

with 𝜏2=0 and 𝑃=𝜎2). Henceforth, 𝑆𝐻𝑚𝑎𝑥=𝜎2=69.0 MPa

(=10,000 psi), 𝑆ℎ𝑚𝑖𝑛=𝜎3=62.1 MPa (=9,000 psi). Note

that the deviatoric stresses (𝜏1,2,3=𝜎1,2,3- 𝑃) must be

𝜏1=6.9 MPa (=1,000 psi) and 𝜏3=-6.9 MPa (=-1,000 psi)

confirming that the plane stress condition is fulfilled (with

𝜏2=0 because 𝑃=𝜎2=68.9 MPa(=10,000 psi) and

remember 𝑃=(𝜎1 + 𝜎2 + 𝜎3)/3. For the Delaware Basin

well it is thus justified to use 𝑆𝑉=𝜎1=75.8 MPa (=11,000

psi), 𝑆𝐻𝑚𝑎𝑥=𝜎2=69.0 MPa (=10,000 psi), 𝑆ℎ𝑚𝑖𝑛=𝜎3=62.1

MPa (=9,000 psi). There will be a biaxial deviatoric stress

in a vertical plane parallel to the horizontal wellbore

aligned with 𝑆ℎ𝑚𝑖𝑛=𝜎3 (Figure 3a). However, a uniaxial

deviatoric stress occurs in a plane perpendicular to the

horizontal wellbore section when the well itself is

assumed aligned with 𝑆ℎ𝑚𝑖𝑛=𝜎3 (Figure 3b).

For the Midland Basin well, 𝐴𝜙=0.8 indicates a so-called

transtensional stress regime, an oblique vector

combination of strike-slip a normal faulting (Sanderson

and Marchini, 1984). Prior studies proposed the region of

the Wolfcamp well in the Midland Basin is in a normal

faulting stress regime (Agharazi, 2016; Patterson, 2017;

Wilson et al., 2016), with a very modest horizontal stress

anisotropy (ANI) ranging from 3% (Wilson et al., 2016)

to 18% (Patterson, 2017, for Wolfcamp-B), with ANI

defined as follows (Svatek, 2017, p.43):

𝐴𝑁𝐼 =𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛

𝑆ℎ𝑚𝑖𝑛 (3)

Because we are still in a normal faulting regime (𝑛 = 0)

we can equate 𝐴𝜙 = 𝜙, such that (Parsegov et al., 2018):

𝜙 =𝜎2 − 𝜎3𝜎1 − 𝜎3

=𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛𝑆𝑉 − 𝑆ℎ𝑚𝑖𝑛

= 𝐴𝑁𝐼𝑆ℎ𝑚𝑖𝑛

𝑆𝑉 − 𝑆ℎ𝑚𝑖𝑛

(4a)

Eq. (4a) can be rewritten as:

𝐴𝑁𝐼 = 𝜙𝑆𝑉 − 𝑆ℎ𝑚𝑖𝑛𝑆ℎ𝑚𝑖𝑛

(4b)

As we have 𝐴𝜙=𝜙=0.8, then assume 𝑆𝑉/𝛥𝐿=24.9 kPa/m

(=1.1 psi/ft) and 𝑆ℎ𝑚𝑖𝑛/𝛥𝐿=18.1 kPa/m (=0.8 psi/ft)

(from DFIT test), it follows for a well at 𝐿=𝑇𝑉𝐷=3,048 m

(=10,000 ft) that the horizontal stress anisotropy

𝐴𝑁𝐼=0.375. Given 𝑆ℎ𝑚𝑖𝑛=55.2 MPa (=8,000 psi) and

using Eq. (3) solves for 𝑆𝐻𝑚𝑎𝑥=71.7 MPa (=10,400 psi),

which implies Δ𝑆=(𝑆𝐻𝑚𝑎𝑥-𝑆ℎ𝑚𝑖𝑛)=16.5 MPa (=2,400

psi). For the Midland Basin, well it is thus justified to use

𝑆𝑉=𝜎1=75.8 MPa (=11,000 psi), 𝑆𝐻𝑚𝑎𝑥=𝜎2=71.7 MPa

(=10,400 psi), 𝑆ℎ𝑚𝑖𝑛=𝜎3=55.2 MPa (=8,000 psi).

Remember that 𝑃=(𝜎1 + 𝜎2 + 𝜎3)/3, which for the

Midland Basin well gives 𝑃=67.6 MPa (=9,800 psi). The

deviatoric stress magnitudes are (using 𝜏1,2,3=𝜎1,2,3-𝑃):

𝜏1=8.3 MPa (=1,200 psi), 𝜏2=4.1 MPa (=600 psi), and

𝜏3=-12.4 MPa (=-1,800 psi). We have no vanishing

deviatoric stress in the plane perpendicular to the wellbore

(unlike the Delaware Basin well). The state of stress will

be biaxial in that plane (Figure 3b), with 𝜏1=8.3 MPa

(=1,200 psi) and 𝜏2=4.1 MPa (=600 psi), assuming the

wellbore is in the direction of the least principal stress.

(a) Vertical stress

We further validate the overburden stress due to the

weight of the overlying rock units, which can be

calculated using a depth integrated density log (Figure 6):

𝑆𝑉 = ∫ 0.433𝜌(𝑧) 𝑑𝑧𝑧

0

(5)

Where 𝑆𝑉Sv is the overburden stress, and 𝜌(𝑧)ρ the

formation density (in grams per cubic centimeter) at a

specified depth, z (ft), and conversion factor 0.433

converts the density into a pressure gradient of psi/ft. The

estimated vertical stress gradient for the wells in our study

is assumed to be 24.9 kPa/m (1.1 psi/ft).

(ii) Formation pore pressure

(a) Delaware Basin

Pore pressures in the Delaware Basin vary rapidly with

depth and prior studies have shown several overpressured

zones. Rapid burial from early Pennsylvanian to Late

Permian age explains overpressures in the Woodford and

Guadalupian Formations across the basin. Kerogen

maturity is assumed a factor in overpressure of the

Wolfcamp Formation (Sinclair, 2007). Rittenhouse et al.,

2016 compiled a large database or pore pressure

measurements in the Delaware Basin from various

sources (diagnostic fracture injection tests (DFITs), drill

stem test (DST), modular dynamics testers (MDTs), mud

weights etc.) and was able to create a regional pore

pressure earth model in the region. The results of Figure

7a, b show that typically the formation pressure increases

with depth. The steepest vertical formation pressure

gradients occur in region B marked in Figure 7a. A full

suite of pore pressure gradients is given in Figure 7b. For

the location of our study well in Reeves County of the

Delaware Basin, a pressure gradient of 15.8 kPa/m (0.7

psi/ft) seems appropriate.

(b) Midland Basin

Pore pressures in the Midland Basin vary rapidly with

formation in a manner comparable to the Delaware Basin.

Histograms of in-situ pressure data from a three-county

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study (Upton, Reagan and Irion counties, Midland Basin)

are given in Fig. 8. There are large differences in the

pressure gradients (bottom scale) between the various

reservoirs: both Guadalupian and Leonardian reservoirs

include wells that are slightly under-pressured (Figure 8),

which in part may be attributed to fluid withdrawals

during hydrocarbon production. However, the occurrence

of underpressure was already noted very early in the

development of these fields (Elkins, 1953). Rather than

being underpressured, the Wolfcamp shale shows

substantial overpressuring, which is consistent with its

role as a source rock. Slight under-pressuring observed in

the deeper Ordovician reservoirs, suggests that downward

fluid migration from the Wolfcamp shale and other over-

pressured source rocks has been very limited (Engle et al.,

2016). For the Wolfcamp horizontal lateral in Upton

County, Midland Basin, using a pressure gradient of 13.6

kPa/m (0.6 psi/ft) seems appropriate.

(iii) Elastic anisotropy

The present study analyzes borehole stability under the

assumptions of both fully isotropic and transversely

isotropic elasticity. Measurements of the elastic

stiffnesses on shale have confirmed these are commonly

transverse anisotropic (Laubie and Ulm, 2014). Rather

than measuring the stiffnesses under laboratory

conditions, sonic dipole logging tools can measure the

five non-vanishing stiffness tensor components

(𝐶11, 𝐶13, 𝐶33, 𝐶44 and 𝐶55) directly in the borehole (e.g.

see Aderibigbe et al., 2016; Brooks et al., 2015; Chen

Valdes et al., 2016). The methods to obtain such

parameters from the sonic dipole tool have been detailed

in Herwanger and Koutsabeloulis, 2011. The stiffnesses

can be translated using standard conversion expressions

to obtain the elastic engineering constants (in our model

𝐸𝑥, 𝐸𝑦, 𝐸𝑧, 𝐸ℎ and 𝐸𝑉; 𝜈𝑥𝑦, 𝜈𝑦𝑧, 𝜈𝑥𝑧, 𝜈ℎ and 𝜈𝑉). In this

study, the stiffness tensor elements (𝐶11, 𝐶12, 𝐶13, 𝐶33 and

𝐶44) were obtained from the most anisotropic section of

the well log corresponding to the Upper Wolfcamp

Formation (red rectangle in Figure 9; after Chen Valdes

et al., 2016). The anisotropic coefficients (various

combinations of 𝛼1 and 𝛼2) required for the stress

distribution analysis (Appendix A) were calculated based

on the stiffnesses using Eqs. (A4) and (A6), and the

results are listed in Table 1.

2.3. Stresses distribution around wellbore for

elastically anisotropic formation This section introduces a set of stress solutions around a

horizontal wellbore penetrating an elastically anisotropic

formation, based on the stress functions derived by Green

and Taylor, 1945, recently adapted for use in shale

basins(Weijermars et al., 2019). Note that the positive-

Figure 6. Calculated vertical stress based on density log and

extrapolation.

Figure 7. (a) Pore pressure distribution map for the Delaware

Basin modified from Rittenhouse et al., 2016 showing the

approximate location of the study well. (b) Area B is located

in a section of the Basin that is typically associated with high

pore pressures.

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compressive sign convention is used in the present study.

When an elastically anisotropic material is under a bi-

axial stress condition, the radial (𝜎𝑟), tangential (𝜎𝜃) and

shear stress (𝜎𝑟𝜃) distribution around a penetrating

wellbore are determined by Eqs. (6a-c).

𝜎𝑟(𝑟, 𝜃) = 𝜎𝑟,∞(𝑟, 𝜃) + 𝜎𝑟,∞′ (𝑟, 𝜃 +𝜋

2) + 𝜎𝑟,𝑤(𝑟, 𝜃) (6a)

𝜎𝜃(𝑟, 𝜃) = 𝜎𝜃,∞(𝑟, 𝜃) + 𝜎𝜃,∞′ (𝑟, 𝜃 +𝜋

2) + 𝜎𝜃,𝑤(𝑟, 𝜃) (6b)

𝜎𝑟𝜃(𝑟, 𝜃) = 𝜎𝑟𝜃,∞(𝑟, 𝜃) + 𝜎𝑟𝜃,∞′ (𝑟, 𝜃 +𝜋

2) + 𝜎𝑟𝜃,𝑤(𝑟, 𝜃) (6c)

where, 𝜎𝑟,∞(𝑟, 𝜃) and 𝜎𝑟,∞′(𝑟, 𝜃 + 𝜋/2) are the radial

stresses around the wellbore induced by the in-situ

stresses parallel and perpendicular to the bedding plane,

respectively. 𝜎𝑟,𝑤(𝑟, 𝜃) Sv = ∫ 0.433ρ(z). dzz

0 is the

radial stress induced by the wellbore pressure, generated

by drilling fluid weight, taking into account any net

pressure due to the interaction with the formation pore

pressure. Accordingly, the sums of 𝜎𝜃,∞(𝑟, 𝜃), 𝜎𝜃,∞′(𝑟, 𝜃 + 𝜋/2) and 𝜎𝜃,𝑤(𝑟, 𝜃), and 𝜎𝑟𝜃,∞(𝑟, 𝜃), 𝜎𝑟𝜃,∞′(𝑟, 𝜃 + 𝜋/2) and 𝜎𝑟𝜃,𝑤(𝑟, 𝜃) are, respectively, the

tangential and shear stresses, which are induced by the

interaction of the in-situ stresses and the wellbore

pressure. 𝑟 is the distance from the center of the wellbore

and 𝜃 is the angle from the bedding plane. The full

solutions of the stress functions in an elastically

anisotropic formation are described in Appendix A.

Using the stress function solutions described in Section

2.3 (and Appendix A), we computed and visualized the

stress distribution around two wellbores horizontally

drilled in elastically anisotropic formations located in the

Delaware and Midland Basins (Sections 2.4 and 2.5). The

results for the anisotropic elastic shale are compared with

the results for isotropic elastic rocks to reveal the effect of

the formation anisotropy. For the stress distribution

analysis, the deviatoric stress concept was adopted to

focus on the effective stress directly responsible for rock

deformation and failure.

2.4. Stress concentration for Delaware Basin well In Section 2.4 and 2.5, the deviatoric stress distributions

around the horizontal wellbore in the Delaware and

Midland Basins are determined, respectively, using the

in-situ stress conditions described in Section 2.2.

Assuming that the wellbore trajectory is aligned with the

minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛), the stress

concentration maps around a wellbore drilled at 3,048 m

(10,000 ft) depth are computed. Input parameters are

described in Figure 2. For the underbalanced and

overbalanced wellbore cases, we selected the wellbore

pressure that has a difference of 34.5 MPa (5,000 psi)

from the pore pressure, i.e. 𝑃𝑤-𝑃𝑃=-34.5 MPa (-5,000 psi)

for the underbalanced case and 𝑃𝑤-𝑃𝑃=34.5 MPa (5,000

psi) for the overbalanced case. The tangential stress is

always maximum and the radial stress is always minimum

at the potential shear failure location (𝜃=0) when the well

Figure 8. Pore pressure histograms for the Midland Basin

based on reservoir shut-in pressures from wells in Upton,

Reagan, and Irion counties, Texas. After Engle et al., 2016.

Figure 9. Well logs from the Upper Wolfcamp formation

with tracks from sonic dipole instrument logging the

transverse isotropic stiffness tensor elements 𝐶11, 𝐶12, 𝐶13,

𝐶33 and 𝐶44 (after Chen Valdes et al., 2016).

Table 1. Stiffnesses of the Upper Wolfcamp and derived

representative for stratigraphic interval indicated in Figure

9.

𝐶11

(GPa)

𝐶12

(GPa)

𝐶13

(GPa)

𝐶33

(GPa)

𝐶44

(GPa)

60.0 8.0 8.0 36.0 12.0

𝛼1𝛼2 𝛼1 + 𝛼2 𝛼1′𝛼2′ 𝛼1′ + 𝛼2′

0.590 2.704 1.687 4.399

8

is underbalanced. In addition, the radial and tangential

stresses at the potential tensile failure location (𝜃= 𝜋/2)

are always maximum and minimum for the overbalanced

wellbore, respectively.

(i) Underbalanced case

The maximum and minimum principal stress magnitudes

around an underbalanced wellbore (𝑃𝑤=13.8 MPa; 2,000

psi) drilled in the anisotropic Delaware Basin (Figure 10a-

d) is compared with the stress response of an isotropic

formation under the same regional stress conditions

(Figure 10e-h). Note that the x-axes of the stress

magnitude plots are aligned with the bedding plane and

the y-axes are vertical, and coincide with the maximum

principal stress directions. In Figure 10 a, c, e and g, the

deviatoric stress magnitudes (𝜏1 and 𝜏2) are plotted for a

region ranging from -5 to 5 times wellbore radius. The

stress concentration at the possible failure locations are

magnified for shear failure (𝜃=0; Figure 10b and f) and

for tensile failure (𝜃=𝜋/2; Figure 10d, h). The overall

contour pattern for the stress magnitudes near the

wellbore in the anisotropic formation do not significantly

differ from that of the isotropic formation (Figure 10a, e).

However, the magnitude of the stress peaks at the

potential shear failure location (Figure 10b, f) is higher if

elastic anisotropy is taken into account. The maximum

principal stress concentration for the anisotropic

formation is 65.4 MPa (9,491.9 psi), which is 18.6%

higher than for the isotropic case (55.2 MPa; 8,000 psi)

(see Table 3). The minimum principal stress

concentrations at the potential tensile failure location

remain identical for both cases (-34.5 MPa; -5,000 psi).

However, the contour shapes of the minimum principal

stress in Figure 10c, g are different, which implies that

that the formation anisotropy redirects the local stress

distributions.

(ii) Overbalanced case

Figure 11 shows the maximum and minimum principal

stress magnitudes around an overbalanced wellbore

(𝑃𝑤=82.7 MPa; 12,000 psi). The stress magnitude

contours are mapped in Figure 11a, c, e, and g, and the

stress concentrations at the potential failure locations in

Figure 11b, d, f and h. Since the wellbore is overbalanced,

the minimum stress concentrations appear larger in

magnitude at 𝜃=𝜋/2 (Figure 11c, g). In addition, the

tensile stress concentration for the anisotropic formation

is larger than for the isotropic formation. Table 3 indicates

that the minimum principal stress concentration at 𝜃=𝜋/2

for the anisotropic formation is 36.6% larger (-56.5 MPa;

-8,193.5 psi) in magnitude than the isotropic formation (

-41.4 MPa; -6,000 psi). However, the maximum principal

stresses at 𝜃=0 are same for both anisotropic and isotropic

formation (34.5 MPa; 5,000 psi).

(iii) Subconclusion

Larger stress concentrations are induced at the potential

failure locations when the formation possesses elastic

anisotropy. This occurs at both the shear failure location

(𝜃=0) and tensile failure location (𝜃=𝜋/2). Therefore, the

effect of the elastic anisotropy needs to be incorporated in

a WBS analysis to obtain a more realistic result.

Negligence of the elastic anisotropy would underestimate

the magnitude of both the tensile and compressional stress

concentrations at the wellbore wall.

(iv) Stress trajectories

We distinguish above two conditions of wells penetrating

a formation, where the pore pressure is either (1)

overbalanced or (2) underbalanced by the pressure due to

the weight of the drilling fluid when penetrated by the

well. The distinction is relevant because the

underbalanced wellbore develops stress trajectory

Figure 10. Principal stress magnitudes around an

underbalanced wellbore (𝑃𝑤=13.8 MPa; 2,000 psi and

𝑃𝑝=48.3 MPa; 7,000 psi) in Delaware Basin for anisotropic

(a-d) and isotropic formations (e-h). For both cases, the

maximum (a, e) and minimum (c, g) principal stress

magnitudes are shown for a region up to within -5 to 5 radii

away times from the wellbore radius. Stress concentrations

at the potential shear failure location (𝜃=0) are magnified in

(b) and (f). Potential tensile failure location (𝜃=𝜋/2) are

enlarged in (d) and (h).

9

patterns near the wellbore with an elliptical region

through the neutral points outlining a so-called fracture

cage (Figure 12a). The overbalanced wellbore typically

has stress trajectories outlining a so-called stress cage

(Figure 12c), with the elliptical region through the neutral

points rotated 90° with respect to the fracture cage case of

Figure 12a. The balanced wellbore has 4 neutral points

located at the wellbore rim (Figure 12b) with neither a

fracture cage nor a stress cage. Such balanced wells will

still develop local stress concentrations, but lower than for

the overbalanced and underbalanced cases outlined in

Figure 10 and 11, respectively. Balanced stress

concentrations are not further analyzed here, but an in-

depth discussion of fracture and stress cages and the

conditions that lead to balanced, overbalanced and

underbalanced wellbores is given in our prior studies

(Thomas and Weijermars, 2018; Wang and Weijermars,

2019; Weijermars, 2016; Weijermars et al., 2013).

2.5. Stresses distribution around wellbore for

Midland Basin well For a horizontal well drilled in the Midland Basin, the in-

situ and pore pressure gradients determined in Section 2.2

were adopted. The basin has a higher maximum

horizontal stress gradient and a lower pore pressure

gradient than the Delaware Basin (Table 2). Assuming the

Table 2. In-situ stress state of Delaware and Midland Basins

at 3,048 m (10,000 ft) depth.

Quantity Delaware Basin,

MPa, psi

Midland Basin

MPa, psi

𝑆𝑉(𝜏𝑉) 75.8 (6.9)

11,000 (1,000)

75.8 (8.3)

11,000 (1200)

𝑆𝐻𝑚𝑎𝑥(𝜏𝐻) 68.9 (0)

10,000 (0)

71.7 (4.1)

10,400 (600)

𝑆ℎ𝑚𝑖𝑛(𝜏ℎ) 62.1 (-6.9)

9,000 (-1,000)

55.2 (-12.4)

8,000 (-1,800)

𝑃𝑝 48.3, 7,000 41.4, 6,000

𝑃𝑤,𝑢𝑛𝑑𝑒𝑟𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 13.8, 2,000 6.9, 1,000

𝑃𝑤,𝑜𝑣𝑒𝑟𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 82.7, 12,000 75.8, 11,000

Table 3. Deviatoric stress concentration at potential shear

failure location (𝜃=0) and at tensile failure location (𝜃=𝜋/2)

of under- and overbalanced wells in Delaware and Midland

Basins. For the anisotropic (A) and isotropic (I) formations.

The ratios indicate A/I.

Underbalanced,

MPa (psi)

Overbalanced,

MPa (psi)

Angle 𝜃=𝜋/2 𝜃=0 𝜃=𝜋/2 𝜃=0

Delaware

A 65.5

(9,491.9)

-34.5

(-5,000.0)

34.5

(5,000.0)

-56.5

(-8,193.5)

I 55.2

(8,000.0)

-34.5

(-5,000.0)

34.5

(5,000.0)

-41.4

(-6,000.0)

Ratio 1.19 1 1 1.37

Midland

A 66.5

(9,638.3)

-34.5

(-5000.0)

34.5

(5,000.0)

43.1

(-6,244.8)

I 55.2

(8,000.0)

-34.5

(-5,000.0)

34.5

(5,000.0)

-30.3

(-4,400.0)

Ratio 1.20 1 1 1.42


overbalanced wellbore (𝑃𝑤=82.7 MPa; 12,000 psi and

𝑃𝑝=48.3 MPa; 7,000 psi) in Delaware Basin for anisotropic

(a-d) and isotropic formations (e-h). For both cases, the


magnitudes are shown within -5 to 5 times the wellbore

radius. Stress concentrations at the potential shear failure

location (𝜃=0) are magnified in (b) and (f). Potential tensile

failure location (𝜃=𝜋/2) are enlarged in (d) and (h).

Figure 12. Principal stress trajectories for (a) underbalanced,

(b) balanced, and (c) overbalanced wellbore. Red dots are

neutral points and red curves outline fracture cage (in a) and

stress cage (in c). Blue trajectories are for largest principal

stress (deviatoric compression). Green trajectories are

smallest principal stress (deviatoric tension).

10

Wolfcamp Formation of the Midland Basin possesses the

same elastic anisotropy as in the Delaware Basin (Section

2.4), the anisotropy coefficients were again determined by

the stiffness tensor elements from Chen Valdes et al.

(2016) (Figure 9 and Table 1). The stress distributions

around an under- and overbalanced wellbore horizontally

drilled at 3,048 m (10,000 ft) depth of the Midland basin

were computed and compared with the isotropic

formation.

(i) Underbalanced case

Figure 13 shows the principal stress distributions around

an underbalanced horizontal wellbore drilled in the

Midland Basin (𝑃𝑤=75.8 MPa; 11,000 psi). The

maximum stress contours of both anisotropic and

isotropic formations indicate the largest stress

concentrations occur at the potential shear failure

locations (𝜃=0 and 𝜃=𝜋) (Figure 13a, e). The maximum

stress concentration for the anisotropic case observed at

𝜃=0 or 𝜃=𝜋 (𝑃𝑤=66.5 MPa; 9,638.3 psi) is 20.5% higher

than for the isotropic case (𝑃𝑤=55.2 MPa; 8,000 psi;

Figure 13b, f; Table 3). The stress concentration contours

of the anisotropic formation are highly deviated as

compared to the contours for the isotropic formation, but

the largest negative (tensile) stress concentrations at the

wellbore wall appear the same for both cases (𝑃𝑤=-34.5

MPa; -5,000 psi; Figure 13c, g).

(ii) Overbalanced case

The overbalanced wellbore case in the anisotropic

formation induces a minimum stress (𝑃𝑤=-43.1 MPa;

-6,244.8 psi) that is 41.9% larger than for the isotropic

formation (𝑃𝑤=-30.3 MPa; -4,400 psi). Figure 14c, d

show that higher stress concentrations occur at 𝜃=𝜋/2 and

3𝜋/2 when the formation anisotropy is taken into account.

However, the maximum stress at the wellbore wall is not

affected by the formation anisotropy (Figure 14a, b, e, f).

(iii) Subconclusion

In accordance with the previous Delaware Basin example,

the larger stress concentrations are observed at the

potential failure locations of both under- and

overbalanced wellbores in the Midland Basin. Therefore,

the elastic anisotropy of the formation needs to be taken

into account for the wellbore stability analysis when

drilling in highly anisotropic shale formations.

3. EXPANSION OF WBS MODEL WITH

ANISOTROPIC FAILURE CRITERIA

Section 3.1 reviews the most practical failure criteria,

including those for bedding plane slip. Sections 3.2 and

3.3 present the WBS model results for, respectively, the

wells in the Delaware and Midland Basins. In Section 3.2

and 3.3, both the standard Coulomb and JPW criteria are

integrated with the stress magnitudes at the wellbore wall

to determine the critical wellbore pressure that induces

shear failure. The in-situ stress, pore pressure and

anisotropic coefficients of Delaware and Midland Basins

were incorporated to investigate the effect of the

anisotropy on the critical wellbore pressure. Section 3.4

compares the WBS outcomes when using standard

isotropic and when using anisotropic failure criteria.

3.1. Brief review of failure criteria for anisotropic

shales Wellbore stability analysis in shale prompts for the use of

anisotropic failure criteria (Li and Weijermars, 2019).

The latter study applied the traditional Lekhnitskii-

Amadei equations (Amadei, 1983; Lekhnitskii, 1963) to

compute the near-wellbore stress concentrations in shale,

and accounted for anisotropic failure by adjustments to

the parameters of both tensile and shear failure using

Hoek-Brown failure criteria (Hoek and Brown, 1980).

Other studies have advocated the use of different failure


overbalanced wellbore (𝑃𝑤=6.9 MPa; 1,000 psi and 𝑃𝑝=41.4

MPa; 6,000 psi) in Midland Basin for anisotropic (a-d) and

isotropic formations (e-h). For both cases, the maximum (a,

e) and minimum (c, g) principal stress magnitudes are shown

within -5 to 5 times the wellbore radius. Stress

concentrations at the potential shear failure location (𝜃=0)

are magnified in (b) and (f). Potential tensile failure location

(𝜃=𝜋/2) are enlarged in (d) and (h).

11

criteria, such as Jaeger’s Plane of Weakness (JPW) model

based on the original model of Jaeger, 1960) and

experimental work by Donath, 1961). The criterion is

renowned for its straightforwardness and applicability as

physical properties of the weak planes are directly

incorporated into the failure envelope establishment with

the conventional Mohr-Coulomb parameters (Setiawan

and Zimmerman, 2018; Zhang, 2013). Laboratory test

were carried out by Ambrose et al., 2014, complemented

with data from prior studies to evaluate the JPW model

predictions with lab failure data. An alternative slip model

by Pariseau, 1968, was also evaluated and Ambrose et al.,

2014, results suggest that for certain shale formations

Pariseau’s model is more accurate than the JPW model,

while for others the JPW model is better than the Pariseau

model. Ambrose and Zimmerman, 2015, applied the JPW

model to Bossier shale and Vaca Muerta shale with

reasonable convergence. Zhang, 2013, uses anisotropic

failure criteria (including the JPW bedding slip criterion)

and unequal far -field stresses ("anisotropic" stress) but

neglects the redistribution of those stresses due to elastic

anisotropy. In effect, Zhang, 2013, uses isotropic elastic

stress state (using Kirsch equations) with anisotropic

strength (for which he uses the term transverse

anisotropy, somewhat confusingly). In our study we add

a new dimension to the analysis of wellbore stability in

shales by applying the more straightforward stress

equations of Green and Taylor, 1945, in combination with

the anisotropic JPW failure criterion. We compare the

anisotropic stress concentrations with isotropic stress

concentration and also juxtapose the results using both a

JPW criterion and a classical, “isotropic” Mohr-Coulomb

shear failure criterion.

According to the JPW model, the strength of a rock with

a weak bedding planes can be described by (Jaeger et al.,

2007; Jaeger, 1960):

𝜎1 − 𝑃𝑝 = (𝜎3 − 𝑃𝑝) +2 (𝑆𝑤 + 𝜇𝑤(𝜎3 − 𝑃𝑝))

(1 − 𝜇𝑤𝑐𝑜𝑡 𝛽)𝑠𝑖𝑛 2𝛽 (7)

where, 𝜙𝑤 is the angle of internal friction (𝐼𝐹𝐴) of the

weak plane. Even if shear failure does not occur on the

weak bedding planes, it still can be achieved on another

plane when the maximum principal stress reaches the

value determined by the Coulomb criterion.

𝜎1 − 𝑃𝑝 = 2𝑆𝑜𝑡𝑎𝑛 𝛽𝑜 + (𝜎3 − 𝑃𝑝)(𝑡𝑎𝑛 𝛽𝑜)2 (8)

where 𝑆𝑜 is the cohesion of the rock and 𝛽𝑜 is given by

𝑡𝑎𝑛 2𝛽𝑜 = 𝑡𝑎𝑛 (𝜙𝑜 +𝜋

2) = −

1

𝜇𝑜 (9)

where 𝜇𝑜 is a coefficient of the internal friction of the

rock. As the weak planes have lower strength, it can be

assumed that 𝑆𝑤<𝑆𝑜 and 𝜇𝑤<𝜇𝑜.

When well conditions reach the critical values for shear

failure, the maximum compressional principal stress at

the wellbore wall generally is the tangential stress. If a

wellbore trajectory is aligned with the maximum

horizontal stress, the axial stress can be higher than the

tangential stress. However, this is generally not the case

as it is preferred to drill a horizontal wellbore along the

minimum horizontal stress for better induced fracture

geometries. As can be seen in Figure 15a, the angle 𝜃


overbalanced wellbore (𝑃𝑤=75.8 MPa; 11,000 psi and

𝑃𝑝=41.4 MPa; 6,000 psi) in Midland Basin for anisotropic (a-

d) and isotropic formations (e-h). For both cases, the


magnitudes are shown within -5 to 5 times the wellbore

radius. Stress concentrations at the potential shear failure

location (𝜃=0) are magnified in (b) and (f). Potential tensile

failure location (𝜃=𝜋/2) are enlarged in (d) and (h).

12

(∠𝐴𝐵𝐶) is always 90°. If 𝛽 is overlaid in the figure, 𝛽

(∠𝐵𝐶𝐷) and 𝜃 are same. For example, if 𝜃=0, the

tangential stress is perpendicular to the weak bedding

plane and 𝛽=0 (Figure 15b). On the other hand, if we are

interested in the wellbore wall at 𝜃= 𝜋/2, the tangential

stress is aligned with the bedding plane, and thus, the 𝛽

value is also 𝜋/2. Consequently, as long as the coordinate

for the stress distribution calculation is aligned with the

bedding plane, the angle 𝜃 can be substituted with 𝛽 in Eq

(6) to integrate the anisotropic failure criterion with the

proposed WBS model. The lower-critical wellbore

pressure that induces shear failure at the wellbore wall can

be determined by substituting the maximum stress (𝜎1) in

Eqs. (7) and (8) with the tangential stress calculated from

Eq. (A1b) for an anisotropic formation, and from Eq.

(A7b) for an isotropic formation.

The upper-critical wellbore pressure at tensile failure can

be calculated by Eq. (10)

𝜎𝜃|𝜃=𝜋2− 𝑃𝑝 = −𝑇0 (10)

where 𝑇0 is the tensile strength of the formation, which is

assumed as 0 in this study, and pore pressures are given

in Table 2.

3.2. WBS for Delaware Basin well Physical properties of rock and weak planes were

extensively analyzed by Ambrose and Zimmerman, 2015,

who experimentally measured the JPW criterion

properties of Bossier shale and Vaca Muerta shale (Table

4). In this study, it is assumed that the 𝐼𝐹𝐴 and the

cohesion for the shale in the Wolfcamp case study wells

are respectively 30° and 4000 psi, and that the 𝐼𝐹𝐴 and

the inherent shear strength of the weak plane are 80% of

that of the rock matrix (Table 4).

(i) Coulomb failure criterion

The tangential stress magnitude around a horizontal

wellbore in the Delaware Basin at the moment of shear

failure is illustrated in Figure 16. Figure 16a compares the

tangential stress at the wellbore wall in the anisotropic

shale (red solid) with an isotropic (blue solid) formation,

while the dashed lines indicate the failure criteria. When

the original Coulomb criterion is adopted for the critical

wellbore pressure calculation, shear failure is expected to

occur at 𝜃=0 or 𝜋 for both the elastically anisotropic and

isotropic formations. This is because i) the Coulomb

criterion does not incorporate the weak bedding plane

effect on rock strength, ii) the critical tangential stress is

constant regardless of the angle of the bedding plane

(dashed lines in Figure 16a) and iii) the highest tangential

stress concentration is always achieved at 𝜃=0 or 𝜋. In

addition, the lower-critical wellbore pressure is higher for

the anisotropic formation than for the isotropic formation

as larger tangential stress is induced by the elastic

anisotropy.

As Figure 16a indicates, the lower-critical wellbore

pressure is 41.6 MPa (6,027.5 psi) when the formation

anisotropy is taken into account, while shear failure is not

expected to occur at the wellbore pressure higher than

39.9 MPa (5,785.9 psi) for the isotropic formation.

Figure 16b and c display the tangential stress distributions

around a wellbore in isotropic and anisotropic formations

in Delaware Basin at the critical wellbore pressure

(𝑃𝑤,𝑐𝑟𝑖𝑡=39.9 MPa; 5,785.9 psi and 41.6 MPa; 6,027.5

psi, respectively). The shear failure is expected to occur

at 𝜃=0, where the largest tangential stress is concentrated

(Figure 16b). According to the Coulomb criterion, the

shear failure will also take place at 𝜃=0 or 𝜋 for the

anisotropic formation (Figure 16c). Therefore, shear

failure on the wellbore wall is always likely to occur at

the angle with the largest tangential stress concentration,

i.e. 𝜃=0 or 𝜋, when the original Coulomb failure criterion

is applied.

Figure 15. Tangential stress direction at the wellbore wall

penetrating an elastically anisotropic formation with weak

bedding planes. (a) 𝜃 is the angle between the bedding plane

and the location of interest. 𝛽 is the angle between the normal

to the bedding plane and the tangential stress direction. (b)

and (c) indicate the cases for 𝜃=0 and 𝜃=𝜋/2, respectively.

Table 4. Properties of rock and weak plane.

Parameter

Bossier

shale

Vaca Muerta

shale This study

(after Ambrose and

Zimmerman, 2015)

𝜙𝑜, ° 29 27 30

𝜙𝑤, ° 24 26 24

𝑆𝑜, MPa

(psi)

25.9

(3,750)

33.4

(4,850)

27.6

(4,000)

𝑆𝑤, MPa (psi)

14.1

(2,050)

18.3

(2,650)

22.1

(3,200)

13

Figure 16d compares the safe drilling window for the

isotropic (blue arrow) and anisotropic (red arrow)

formations. The upper-critical wellbore pressures at

tensile failure calculated by Eq. (10) are 82.7 MPa

(12,000 psi; isotropic formation) and 81.5 MPa (11,824.5

psi; anisotropic formation). The larger tangential stress

concentration is achieved for the overbalanced wellbore

when the elastic anisotropy is taken into account. Since

the safe window is calculated by the difference between

the lower and upper-critical wellbore pressures, the safe

window of the isotropic formation is wider than for the

anisotropic formation. This is because not only the lower-

critical wellbore pressure for the anisotropic formation is

higher than that of the isotropic formation, the former has

the smaller upper-critical wellbore pressure at tensile

failure. In other words, a narrower range of safety window

needs to be used when a conventional isotropic wellbore

stability analysis is adopted for an elastically anisotropic

formation.

(ii) Jaeger plane of weakness (JPW) failure criterion.

Figure 17 shows the tangential stress around a wellbore at

critical wellbore pressure determined by the JPW

criterion for the anisotropic (red solid) and isotropic (blue

solid) formations. Since the effect of the weak bedding

plane on failure is incorporated in the criterion, the failure

envelope contains a combination of a concave upward

curve and a horizontal line, which represent failure on a

weak plane and on another plane, respectively (dashed

lines in Figure 17a). Although it appears artificial to

combine a model of stress concentrations assuming

Figure 16. (a) Tangential stress at the moment of shear failure

determined by the Coulomb criterion for elastically

anisotropic (red solid) and isotropic (blue solid) formations

in Delaware Basin. The dashed lines indicate the critical

tangential stress. (b) Tangential stress distribution around the

wellbore in the isotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=39.9 MPa

(5,785.9 psi). (c) Tangential stress distribution around the

wellbore in the anisotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=41.6 MPa

(6,027.5 psi). (d) Safe window of the isotropic (blue arrow)

and the anisotropic formations (red arrow).


determined by the JPW criterion for elastically anisotropic (red

solid) and isotropic (blue solid) formations in Delaware Basin.

The dashed lines indicate the critical tangential stress. (b)

Tangential stress distribution around the wellbore in the

isotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=41.0 MPa (5,948.1 psi). (c)


anisotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=41.6 MPa (6,027.5 psi). (d)

Safe window of the isotropic (blue arrow) and the anisotropic

formations (red arrow).

14

elastic isotropy with a failure criterion of weak bedding

planes, we actually compare the results when either

elastically anisotropic or isotropic stress distributions

affect wellbore stability by triggering weak bedding

planes to slip. In accordance with the results from the

Coulomb criterion, shear failure occurs at the lower-

critical wellbore pressure of 41.0 MPa (5,948.1 psi) for

the isotropic formation, while the wellbore in the

anisotropic formation has a narrower safe window as it

meets the lower-critical wellbore pressure at 41.6 MPa

(6,027.5 psi). The shear slip angle for which the failure

occurs in the isotropic rock is 53.2°, although the largest

tangential stress still appears at 𝜃=0 or 𝜋 (Figure 17b).

This is because the shear failure in the isotropic formation

is expected to occur on the bedding plane, as indicated by

the concave upward portion of the failure envelope in

Figure 17a.

The safe window calculated by the JPW criterion in

Figure 17d shows that the safe window of the isotropic

formation is still wider (41.7 MPa; 6,051.9 psi) than for

the anisotropic formation (40.0 MPa; 5,797.1 psi)

although the lower-critical wellbore pressure of the

isotropic formation is expected to be increased by the

weak bedding plane.

(iii) Subconclusion for Delaware Basin well

For both the isotropic and anisotropic formations, the

highest stress concentrations occur always at 𝜃=0 or 𝜋

(underbalanced wellbore). Since a critical tangential

stress calculated by the Coulomb criterion is constant,

regardless of the angle, 𝜃, the lower-critical wellbore

pressure is highest for the anisotropic formation, which

yields stress concentrations at the potential shear failure

locations (𝜃=0 or 𝜋) that are larger than for the isotropic

formation. When the JPW is adopted, shear failure of the

isotropic formation occurs at 𝜃𝑓=53.2° due to the

existence of the weak bedding planes. In addition, the

lower-critical wellbore pressure, for this case, will also

increase from 39.9 MPa (5,785.9 psi) to 41.0 MPa

(5,948.1 psi). Therefore, incorporating the anisotropic

failure criterion narrows the safe drilling fluid weight

window of the isotropic formation. However, the lower-

critical wellbore pressure and the failure angle of a

wellbore in the anisotropic formation are not affected by

the bedding plane. Shear failure at the wellbore wall is

expected at 𝑃𝑤,𝑐𝑟𝑖𝑡=41.6 MPa (6,027.5 psi) at 𝜃𝑓=0° for

the anisotropic formation, according to both criteria. In

addition, the safe drilling fluid weight window of the

anisotropic formation is narrower than for the isotropic

case, regardless of the effect of the bedding plane,

although the JPW criterion indicates that the lower-

critical wellbore pressure of the isotropic formation is

increased by the weak bedding plane effect. The effect of

the elastic anisotropy is more important than the

anisotropic JPW shear failure criterion (in the Delaware

Basin case). Additionally, the weak bedding plane failure

criterion should even be used for rock intervals with

elastic isotropy, if layered with weak bedding planes, in

order to result in a more reliable outcome of the wellbore

stability analysis.

3.3. WBS for Midland Basin well (i) Coulomb failure criterion

The critical wellbore pressure and tangential stress

distribution around the wellbore in Midland Basin is first

investigated using the Coulomb criterion. As Figure 18a

shows, the largest tangential stress concentration is

obtained at 𝜃=0 or 𝜋 for both the isotropic and anisotropic

formations. The shear failure will occur at 𝜃=0 or 𝜋, for

both the isotropic and anisotropic formations, as the

tangential stress magnitude at shear failure determined by

the Coulomb criterion is constant. The lower-critical

wellbore pressure of the anisotropic formation (37.9 MPa;

5,489.8 psi) is higher than that of the isotropic formation

(35.8 MPa; 5,185.9 psi) due to the higher tangential stress

concentration induced by the formation anisotropy

(Figure 18b, c). In the same manner, the upper-critical

wellbore pressure at tensile failure is lower for the

anisotropic formation due to the excessive tangential

stress concentration induced by the elastic anisotropy.

Consequently, the safe window is narrower for the

anisotropic formation (57.8 MPa; 8,376.8 psi) than for the

isotropic formation (62.1 MPa; 9,014.1 psi) (Figure 18d).

Therefore, according to the Coulomb failure criterion, the

safe drilling fluid weight window becomes narrower,

when the elastic anisotropy are taken into account.

(ii) Jaeger plane of weakness (JPW) failure criterion.

As shown in Figure 19a, the concave upward portion of

the failure envelope constructed by the JPW criterion

deteriorates the wellbore stability of the isotropic

formation in Midland Basin. Since the shear failure of the

isotropic formation is expected to occur on the weak

bedding plane, adopting the JPW criterion increases the

lower-critical wellbore pressure of the isotropic formation

(Figure 19b). In addition, the critical wellbore pressure of

the isotropic formation is higher than that of the

anisotropic formation, which implies that the drilling fluid

weight needs to be designed denser for the isotropic

formation when using a failure criterion of weak bedding

planes. However, shear failure in the anisotropic

formation is still achieved at 𝜃=0 or 𝜋 with the critical

wellbore pressure of 37.9 MPa (5,489.8 psi). Failure takes

place at the location with the maximum tangential stress

concentration, and the slip on the bedding plane would

require a higher shear stress than at 𝜃=0 (Figure 19c).

Therefore, the lower-critical wellbore pressure of the

isotropic formation is higher than for the anisotropic

formation with the effect of the weak bedding plane. On

the other hand, the safe window of the isotropic formation

is still wider than for the isotropic formation (Figure 19d).

This is because the upper-critical wellbore pressure of the

anisotropic formation is lower (95.6 MPa; 13,866.6 psi)

15

than for the isotropic formation (97.9 MPa; 14,200 psi),

which makes the safe window of the former still narrower.

(iii) Subconclusion for Midland Basin well

In accordance with the Delaware Basin case, using the

JPW criterion will affect both the lower-critical wellbore

pressure and failure angle for a wellbore drilled in the

isotropic formation. The safe drilling window of the

anisotropic formation will not be affected by the weak

bedding plane. When the anisotropic failure criterion is

used for the safe drilling window calculation, the lower-

critical wellbore pressure of the elastically isotropic

formation exceeds that of the elastically anisotropic

formation. The wellbore pressure needs to be maintained

higher if the weak bedding plane exists, even when the

formation is elastically isotropic. Consequently, it is

concluded that the effect of the anisotropic failure

criterion is more dominant for the lower-critical wellbore

pressure calculation for the Midland Basin well (Section

3.3) than for the Delaware Basin well (Section 3.2).

3.4. Comparison of outcomes from isotropic and

anisotropic failure criteria In Sections 3.2 and 3.3, the lower-critical wellbore

pressures at the moment of shear failure were investigated

using isotropic and anisotropic failure criteria, i.e. the

Coulomb and JPW criteria, respectively. For both the

Delaware and Midland Basins, the lower-critical wellbore

pressure determined by the Coulomb criterion strongly

depends on the tangential stress induced by the wellbore

pressure. Shear failure is always expected to occur at the


determined by the Coulomb criterion for elastically

anisotropic (red solid) and isotropic (blue solid) formations

in Midland basin. The dashed lines indicate the critical

tangential stress. (b) Tangential stress distribution around the

wellbore in the isotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=35.8 MPa

(5,185.9 psi). (c) Tangential stress distribution around the

wellbore in the anisotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=37.9 MPa

(5,489.8 psi). (d) Safe window of the isotropic (blue arrow)

and the anisotropic formations (red arrow).


determined by the JPW criterion for elastically anisotropic (red

solid) and isotropic (blue solid) formations in Midland basin.

The dashed lines indicate the critical tangential stress. (b)


isotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=39.5 MPa (5,731.6 psi). (c)


anisotropic formation at 𝑃𝑤,𝑐𝑟𝑖𝑡=37.9 MPa (5,489.8 psi). (d)

Safe window of the isotropic (blue arrow) and the anisotropic

formations (red arrow).

16

locations where the largest tangential stress concentration

is achieved (𝜃=0 or 𝜋). Since the tangential stress

concentration is larger for the anisotropic formation, the

wellbore pressure needs to be maintained higher to

prevent the shear failure on the wellbore wall.

Consequently, the safe window becomes narrower when

the elastic anisotropy is incorporated with the standard

Coulomb criterion.

If the JPW failure criterion is integrated with the stress

distribution around a wellbore, the failure on the weak

bedding plane can be investigated. For both Delaware and

Midland Basin cases, the isotropic formation is expected

to fail on the bedding plane, and thus, the lower-critical

wellbore pressure is increased by incorporating the JPW

criterion. When the Coulomb criterion is adopted for the

critical wellbore pressure calculation, the isotropic

formation in Delaware Basin reaches shear failure at the

wellbore pressure of 39.9 MPa (5,785.9 psi; Table 5).

However, the JPW criterion yields a higher critical

wellbore pressure 41.0 MPa (5,948.1 psi), as failure is

expected to occur on the bedding plane. Similarly, the

critical wellbore pressure of the isotropic formation in the

Midland Basin is 35.8 MPa (5,185.9 psi) according to the

Coulomb criterion, while the JPW criterion indicates that

the wellbore is unstable due to shear failure being

triggered at 39.5 MPa (5,731.6 psi) wellbore pressure or

lower. In addition, the results show (Figure 17, 19) that

the failure in the isotropic formations occurs at 53.2° (Delaware Basin well) and 55.0° (Midland Basin well)

although the locations are not at the largest tangential

stress. On the other hand, the weak bedding plane does

not affect the critical wellbore pressure of the anisotropic

formations. Therefore, it is expected that the failure will

always occur at the locations where the largest tangential

stress concentration is observed, i.e. 𝜃=0 or 𝜋.

For both the Delaware and Midland Basin wells, the safe

drilling fluid weight window is narrower when the elastic

anisotropy is taken into account regardless of the selected

failure criterion. However, the results show that the

isotropic formation of the Midland Basin well has the

higher lower-critical wellbore pressure than the

anisotropic formation if the JPW criterion is integrated. In

other words, the elastic anisotropy is more dominant in

the wellbore stability analysis for the Delaware Basin

case, while the analysis in the Midland Basin is more

sensitive to the incorporated failure criterion in terms of

the potential shear failure. Although, the anisotropy of the

elastic properties and the failure criterion do not always

yield a different outcome for the wellbore stability

analysis, the comparisons evaluated here indicate that

both effects should be taken into account in the wellbore

stability analysis to achieve the most reliable results.

4. SENSITIVITY ANALYSIS

In Sections 2 and 3 we have developed stress

concentration maps and WBS models, respectively, for a

horizontal well at an assumed depth of 3,048 m (10,000

ft) in both elastically isotropic and anisotropic formations.

The anisotropic Wolfcamp shale in Delaware Basin was

selected as the base case for the sensitivity analyses in this

section, which investigates the effect of uncertainty about

the specific values of certain input parameters, such as

formation pore pressure, 𝐼𝐹𝐴 of the rock matrix, elastic

anisotropy, and friction of the bedding plane. Overall

results of the sensitivity analyses are summarized in Table

6.

4.1. Pore pressure changes Figure 20 shows the tangential stress distribution around

the wellbore at critical wellbore pressure for pore pressure

of 41.4 MPa (6,000 psi; red), 48.3 MPa (7,000 psi; green;

base case) and 55.2 MPa (8,000 psi; blue). The failure

envelope for each pore pressure is shown in dashed

curves, which contain a horizontal line and a concave

upward curve. As the formation pore pressure increases,

the wellbore pressure needs to be higher to ensure the

wellbore will not develop shear failure during the drilling

operation. The critical wellbore pressure is 38.3 MPa

(5,561.4 psi) for the 41.4 MPa (6,000 psi) pore pressure,

while it is increased with the pore pressure and the

wellbore pressure needs to be maintained higher than 41.6

MPa (6,027.5 psi) and 44.8 MPa (6,493.5 psi) for 48.3

MPa (7,000 psi) and 55.2 MPa (8,000 psi) psi pore

pressure, respectively. In addition, the tangential stress

concentration at the shear failure locations will decrease

because the increased pore pressure deteriorates the rock

strength. Although the pore pressure is changed for the

well at 3,048 m (10,000 ft) depth, the failure always

occurs at 𝜃=0. Therefore, changes in the formation pore

pressure affect the wellbore pressure, but do not alter the

shear failure plane location.

4.2. Angle of internal friction (𝐼𝐹𝐴) variations The effect of changes in the 𝐼𝐹𝐴 is evaluated in Figure 21.

When the IFA is low (40°; red curve in Fig. 21), the

critical wellbore pressure is the lowest (39.5 MPa; 5,731.2

psi). Since the IFA indicates the strength increment by

depth, the formation with the higher IFA decreases the

lower-critical wellbore pressure. For the IFA of 30° and

Table 5. Critical wellbore pressure of isotropic and anisotropic

formations in Delaware and Midland basins determined by the

Coulomb and JPW criteria.

Criterion Para-

meter

Delaware Midland

I A I A

Coulomb

𝑃𝑤,𝑐𝑟𝑖𝑡 , MPa

(psi)

39.9

(5,785.9)

41.6

(6,027.5)

35.8

(5,185.9)

37.9

(5,489.8)

𝜃𝑓(°) 0 0 53.2 0

JPW

𝑃𝑤,𝑐𝑟𝑖𝑡 , MPa

(psi)

41.0

(5,948.1)

41.6

(6,027.5)

39.5

(5,731.6)

37.9

(5,489.8)

𝜃𝑓(°) 0 0 55.0 0

17

20°, shear failure at the wellbore wall is faced at 41.6 MPa

(6,027.5 psi) and 44.7 MPa (6,477.0 psi) wellbore

pressure, respectively. However, the change in the IFA

does not alter the location of the failure plane, as the shear

failure always occurs at 𝜃=0 and is not affected by

presence of the bedding plane.

4.3. Elastic anisotropy variations The degree of elastic anisotropy has a considerable effect

on the stress concentration and associated triggering of

shear failure. Figure 22 illustrates the tangential stress

distribution for a range of elastic anisotropy, by varying

one stiffness tensor element, 𝐶11, between 45 and 75 GPa.

In addition, a completely isotropic formation was also

included to compare the results. Figure 22 indicates that

the higher formation anisotropy yields larger tangential

stress concentrations at both 𝜃=0 or 𝜋/2. Therefore, the

lower-critical wellbore pressure is higher for the more

anisotropic formations, and shear failure is expected to

occur at 𝜃=0 for all anisotropic cases considered. In

contrast, for the isotropic case, shear failure occurs on the

weak bedding plane at 𝜃=53.2°, because the tangential

stress distribution curve is more concave downward when

the formation is more isotropic (purple solid curve in

Figure 22).

4.4. Strength of weak bedding planes The weak plane properties are significantly influencing

the location and occurrence of failure planes. For the base

case, when the IFA and inherent shear strength of the

bedding planes are 80% of that of the bedding planes,

Figure 20. Tangential stress around a wellbore in elastically

anisotropic Wolfcamp Formation, Delaware Basin, for pore

pressure of 55.2 MPa (8000 psi; blue), 48.3 MPa (7000 psi;

green) and 41.4 MPa (6000 psi; red). Despite of the

formation pore pressure variations, the failure is expected at

𝜃=0 for all the pore pressure cases.


anisotropic Wolfcamp Formation, Delaware Basin, for 𝐼𝐹𝐴

of 20° (blue), 30° (green) and 40° (red). Despite of the 𝐼𝐹𝐴

variations, the failure is expected at 𝜃=0 for all the 𝐼𝐹𝐴

cases.


anisotropic Wolfcamp Formation, Delaware Basin, for the

stiffness tensor element 𝐶11 of 75 GPa (blue), 60 GPa (green)

and 45 GPa (red). The completely isotropic formation is

expected to reach the shear failure on the bedding plane at

53.2° (purple).


anisotropic Wolfcamp Formation, Delaware Basin, for the

lower strength of the weak bedding planes. When the IFA

and inherent shear strength of the weak bedding planes are

80% of that of the rock matrix (blue), the failure is expected

at 𝜃=0° at 41.6 MPa (6,027.5 psi) wellbore pressure. When

the bedding plane’s IFA is 50% (green) and the bedding

plane’s shear strength is 50% (red), the failure is expected to

occur on the bedding plane at 41.7 MPa (6,042.1 psi) and

47.4 MPa (6,871.3 psi) wellbore pressure, respectively.

18

shear failure is expected to occur at 41.6 MPa (6,027.5

psi) wellbore pressure and at 𝜃=0°, instead of the bedding

plane (green solid curve in Figure 23). However, when the

IFA of the bedding plane is 50% of that of the rock matrix,

the failure would occur on the weak bedding plane for a

wellbore pressure of 41.7 MPa (6,042.1 psi) (blue solid

curve in Figure 23). Since the lower-critical wellbore

pressure of the formation containing bedding planes with

the low IFA does not show much difference from that of

the base case, it can be concluded that the weak plane’s

IFA does not noticeably affect the safe drilling fluid

weight window, although failure is likely to occur on the

bedding plane. However, when the inherent shear strength

of the weak bedding plane is 50% of the rock matrix (red

solid curve in Figure 23), failure occurs on the weak

bedding plane and the critical wellbore pressure will be

much higher (47.4 MPa; 6,871.3 psi) than for the basic

case (green solid curve in Figure 23). Consequently, the

shear strength of the bedding plane has a significant effect

on the failure, in terms of both the failure plane and the

safe drilling fluid weight window.

5. DISCUSSION

5.1. Model insights Analytical solutions for the stress distribution due to

anisotropic elasticity around the wellbore drilled in an

elastically anisotropic formation were combined with a

weak bedding plane failure criterion to assess the stability

of the horizontal section of wellbores in the Wolfcamp

shale completed in the central Delaware and Midland

Basins. A major insight from our systematic study is that

failure in isotropic shale with weak bedding planes may

not be limited to the traditional locations of shear failure.

When the tangential compressive stress is high, shear

failure may occur on the weak bedding planes, resulting

in partial collapse of the wellbore. The stress distributions

around the horizontal wellbore, based on Green and

Taylor equations, are compared with the result from the

traditional Kirsch equations for formations that are elastic

isotropic. The comparison reveals that the formation with

elastic anisotropy induces larger magnitudes of tangential

stress concentrations at the potential shear and tensile

failure locations, i.e. 𝜋/2 or 0 from the maximum in-situ

stress direction.

The elastic anisotropy of the Wolfcamp shale increases

the stress concentrations around the wellbore. The

computations are based on the estimated regional in-situ

stress and pore pressure gradients for a wellbore at 3,048

m (10,000 ft) depth in Delaware and Midland Basins. For

wellbores where the pore pressure is either underbalanced

or overbalanced by the pressure due to the drilling fluid,

larger stress concentrations were observed. The increased

stress concentrations occur at the potential failure

locations for shear failure (i.e. 𝜃=0 and 𝜋 from the

bedding plane) and for tensile failure (𝜃=𝜋/2 and 3𝜋/2

from the bedding plane) if the formation anisotropy is

incorporated into the stress distribution analysis. The

weak bedding plane failure criterion (the JPW criterion)

was adopted for a determination of the critical wellbore

pressure.

The results show that the shear failure of an otherwise

isotropic formation (no anisotropy of elastic moduli) is

expected to occur on the weak bedding plane, while that

of the anisotropic formation may still slip at 𝜃=0. The

reason being that for the anisotropic formation the stress

concentration is higher at 𝜃=0 and therefore favors shear

failure at the maximum stress location over slip on the

bedding plane, which would occur at angles where the

shear stresses does not reach high enough to cause slip.

However, whether slip occurs first on weak bedding

planes or elsewhere is quite sensitive variations in key

parameters. The failure properties of the weak bedding

plane appear to have the largest effect on the shear failure

on the plane. When either the 𝐼𝐹𝐴 or the inherent shear

strength of the bedding plane is lower, the failure is more

likely to occur on the weak bedding plane instead of

elsewhere in the rock matrix (Figure 23, Section 4.4).

Moreover, the shear strength of the weak bedding plane

yields very sensitive outcomes, because it potentially

changes not only the failure plane location but also the

lower-critical wellbore pressure. Therefore, when the

effect of the weak bedding plane is incorporated into the

wellbore stability analysis, it is necessary to reduce the

uncertainty in the local input parameters via dedicated

laboratory tests to increase the accuracy of the anisotropic

wellbore stability analysis.

Table 6. Overall results of the sensitivity analysis on formation

pore pressure, 𝐼𝐹𝐴 of the rock matrix, elastic anisotropy, and

properties of the bedding plane.

𝜙𝑜,

° 𝜙𝑤,

°

𝑆𝑤,

MPa

(psi)

𝑃𝑝,

MPa

(psi)

𝐶11,

GPa

𝑃𝑤,𝑐𝑟𝑖𝑡, MPa

(psi)

𝜃𝑓,

°

Base case 30 24 22.1

(3,200)

48.3

(7,000) 60

41.6

(6,027.5) 0

𝜙𝑜=20° 20 16 22.1

(3,200)

48.3

(7,000) 60

44.7

(6,477.0) 0

𝜙𝑜=40° 40 32 22.1

(3,200)

48.3

(7,000) 60

39.5

(5,731.2) 0

𝑃𝑝=41.4 MPa

(=6,000 psi) 30 24

22.1

(3,200)

41.4

(6,000) 60

38.3

(5,561.4) 0

𝑃𝑝=55.2 MPa

(=8,000 psi) 30 24

22.1

(3,200)

55.2

(8,000) 60

44.8

(6,493.5) 0

𝐶11=75 GPa 30 24 22.1

(3,200)

48.3

(7,000) 75

41.9

(6,075.3) 0

𝐶11=45 GPa 30 24 22.1

(3,200)

48.3

(7,000) 45

41.1

(5,964.0) 0

Isotropic 30 24 22.1

(3,200)

48.3

(7,000) 36

41.0

(5,948.1) 53.2

𝜙𝑤=50% 30 15 22.1

(3,200)

48.3

(7,000) 60

41.7

(6,042.1) 47.3

𝑆𝑤=50% 30 24 13.8

(2,000)

48.3

(7,000) 60

47.3

(6,871.3) 53.0

19

5.2. Applicability of anisotropic WBS model Much of our motivation to improve the WBS model for

shale and avoid imperfections in the wellbore shape when

drilling horizontal laterals is to support successful

completions and accomplish the highest well

productivities. Great strides have already been made in

year-over-year improvement in the drilling performance

in unconventional reservoirs (Willis et al., 2018). Drilling

speed remains a key driver for cost control, and in

addition to improvement of wellbore quality and

prevention of lost circulation. At the same time, increased

drill speeds may lead to twist-offs and ultra-heavy bit

wear (broken cutters and blade fracture), which requires

multiple runs to drill the well. Part of the remedy is a

streamlining of rotation rates, which need to remain slow

until weight on bit is fully applied (Hood et al., 2015). In

addition to practical changes in well rates, it is important

to understand why friction may increase at certain depths.

We suggest that the anisotropic WBS model can provide

timely alerts for increased instability risk.

Improving wellbore stability in anisotropic shale

formations by using the approach advocated in this study

may help to both understand and also reduces the risk of

lost circulation due to wellbore failure (and prevents total

well loss due to blowout). In particular, fracture initiation

has been reported for the Delaware Basin (private

communication operator) to occur at certain depths for

relatively low mud loads. Elimination of lost circulation

will substantially improve the economics of the local

drilling operations. Although lost circulation may occur

when a well crosses a natural fracture system with a

certain aperture, the existence of such conductive

fractures is but one plausible cause of lost circulation

events. Pressure management challenges persist when

drilling the vertical well section, even before arriving at

the horizontal landing zone. The challenges occur mostly

due to the variable pore pressure across formations

(Figure 7, 8). For example, lost circulation remains a

major concern when navigating from over-pressured to

under-pressured formation depths. Additionally, high

pressure formations due to water disposal injection

programs provide cementing challenges, which require an

elaborate two-stage cementing collar with casing annulus

packer (Thibodeaux et al., 2018). A prior study of our

team has emphasized that cutting shape can alert for the

need to change casing depth (Wang and Weijermars,

2019). Our present study suggest drilling losses are likely

related to shale horizons with relatively high elastic

anisotropy, which leads to stress concentrations higher

than in isotropic rocks for otherwise the same in-situ

stress. In addition, local decreases in failure strength due

to bedding plane splitting along a so-called weak bedding

plane will further lead to a narrower safe drilling window.

The main focus of our WBS model is to account for all of

the anisotropy effects (stress, elasticity, failure) that may

affect the quality of the horizontal wellbore section.

Ultimately, each completion design program needs to be

continuously fine-tuned by responsive learning to result

in drilling solutions better tailored to the characteristics of

both the vertical well section and the landing zone

section(s) with their spatial variability in key parameters

(Jaripatke et al., 2018; Pink et al., 2017). An emerging

area of interest is drill bit geomechanics, which utilizes

drill bit vibrations to infer mechanical rock properties

(Haecker et al., 2017). Finally, when the perfect wellbore

has been landed, flow models of the fractured reservoir

need to be used to determine the optimum fracture and

well spacing, as well as the ideal proppant load.

6. CONCLUSIONS

The anisotropic wellbore stability model developed here

uses Green and Taylor, 1945 equations in combination

with Jaeger’s Weak-bedding Plane (JPW) failure

criterion. The principal insights, developed are based on a

systematic analysis using conditions for two synthetic

wells located in the Delaware and Midland Basins, as

follows:

(i) The stress concentrations at the potential failure

locations are increased by the elastic anisotropy in

two case study wellbores in the Permian Basin.

(ii) Whether shear failure occurs by slip along the weak

bedding planes or in slip planes at other angles to

the wellbore is highly dependent on the specific

properties.

(iii) We concluded for the Delaware and Midland Basin

wells that the elastic anisotropy and the anisotropic

JPW failure criterion need to be incorporated for

more reliable wellbore stability analysis.

(iv) Based on the sensitivity analysis, changes in the

properties of the weak bedding plane have the

biggest impact on the formation failure

characteristics when the JPW criterion is used. The

critical wellbore pressure and the failure angle

strongly depend on the shear strength of the weak

plane.

(v) Applying the proposed anisotropic WBS model

provides advantages in field applications, as it is

fast and does not require excessive computational

power. The model can be practically applied in real-

time wellbore stability analyses, to determine the

stress concentrations and stress trajectories near the

wellbore, as well as for calculation of the safe

drilling window.

20

APPENDIX A. SOLUTIONS FOR ANALYTICAL

STRESS FUNCTIONS OF ELASTICALLY

ANISOTROPIC FORMATION

Stress distributions around a horizontal wellbore

penetrating an elastically anisotropic formation can be

derived from the stress solutions proposed by Green and

Taylor, 1945. More detailed information including in-

depth analysis of the approach and validation against the

Lekhnitskii-Amadei solution are given in a companion

paper (Weijermars et al., 2019). Under a uni-axial stress

condition, the radial (𝜎𝑟), tangential (𝜎𝜃) and shear stress

(𝜎𝑟𝜃) distribution around a wellbore are determined by

equations (A1a-c).

𝜎𝑟,∞(𝑟, 𝜃)

𝜎∞=(1 + 𝛽1)(1 + 𝛽2)

4𝛽1𝛽2𝑐𝑜𝑠 2𝜃

+1

4(𝛽1 − 𝛽2)

{

(1 + 𝛽2)(1 − 𝛽1𝑒2𝑖𝜃)2

𝛽1𝑒2𝑖𝜃√1 −

4𝑎2𝛽1𝑒2𝑖𝜃

𝑟2(1 + 𝛽1𝑒2𝑖𝜃)2

−(1 + 𝛽1)(1 − 𝛽2𝑒

2𝑖𝜃)2



𝑟2(1 + 𝛽2𝑒2𝑖𝜃)2

}

(A1a)

𝜎𝜃,∞(𝑟, 𝜃)

𝜎∞=(1 + 𝛽1)(1 + 𝛽2)

4𝛽1𝛽2𝑐𝑜𝑠 2𝜃

+1

4(𝛽1 − 𝛽2)

{

(1 + 𝛽2)(1 − 𝛽1𝑒2𝑖𝜃)2



𝑟2(1 + 𝛽1𝑒2𝑖𝜃)2

−(1 + 𝛽1)(1 − 𝛽2𝑒

2𝑖𝜃)2



𝑟2(1 + 𝛽2𝑒2𝑖𝜃)2

}

(A1b)

𝜎𝑟𝜃,∞(𝑟, 𝜃)

𝜎∞= −

(1 + 𝛽1)(1 + 𝛽2)

4𝛽1𝛽2𝑠𝑖𝑛 2𝜃

−𝑖

4(𝛽1 − 𝛽2)

{

(1 + 𝛽2)(1 − 𝛽12𝑒4𝑖𝜃)2



𝑟2(1 + 𝛽1𝑒2𝑖𝜃)2

−(1 + 𝛽1)(1 − 𝛽2

2𝑒4𝑖𝜃)2



𝑟2(1 + 𝛽2𝑒2𝑖𝜃)2

}

(A1c)

where, 𝛽1 and 𝛽2 are anisotropic coefficients, 𝜃 is the

angle from the uni-axial stress direction, 𝑎 is the wellbore

radius, and 𝑟 is the distance from the center of the

wellbore.

On the other hand, the stress distribution around a

pressurized wellbore are calculated by equations (A2a-c).

𝜎𝑟,𝑤(𝑟, 𝜃)

𝑃𝑤 − 𝑃𝑝= −1 +

𝛽1 + 𝛽22𝛽1𝛽2

𝑐𝑜𝑠 2𝜃

+1

2(𝛽1 − 𝛽2){𝛽2(𝑒

𝑖𝜃 − 𝛽1𝑒−𝑖𝜃)2

𝛽1𝑋1

−𝛽1(𝑒

𝑖𝜃 − 𝛽2𝑒−𝑖𝜃)2

𝛽2𝑋2}

(A2a)

𝜎𝜃,𝑤(𝑟, 𝜃)

𝑃𝑤 − 𝑃𝑝= −1 −

𝛽1 + 𝛽22𝛽1𝛽2

𝑐𝑜𝑠 2𝜃

+1

2(𝛽1 − 𝛽2){𝛽2(𝑒

𝑖𝜃 + 𝛽1𝑒−𝑖𝜃)2

𝛽1𝑋1

−𝛽1(𝑒

𝑖𝜃 + 𝛽2𝑒−𝑖𝜃)2

𝛽2𝑋2}

(A2b)

𝜎𝑟𝜃,𝑤(𝑟, 𝜃)

𝑃𝑤 − 𝑃𝑝= −

𝛽1 + 𝛽22𝛽1𝛽2

𝑠𝑖𝑛 2𝜃

+𝑖

2(𝛽1 − 𝛽2){𝛽2(𝑒

2𝑖𝜃 − 𝛽12𝑒−𝑖2𝜃)

𝛽1𝑋1

−𝛽1(𝑒

2𝑖𝜃 − 𝛽22𝑒−2𝑖𝜃)

𝛽2𝑋2}

(A2c)

where, 𝑃𝑤 is the wellbore pressure and 𝑃𝑝 is the pore

pressure. 𝑋1 and 𝑋2 are the complex numbers determined

from the anisotropic coefficients 𝛽1 and 𝛽2.

Consequently, the stress distributions around a

pressurized wellbore under a bi-axial principal stresses

can be calculated by superimposing equations (6a-c).

𝜎𝑟(𝑟, 𝜃) = 𝜎𝑟,∞(𝑟, 𝜃) + 𝜎

𝑟,∞′ (𝑟, 𝜃 +

𝜋

2) + 𝜎𝑟,𝑤(𝑟, 𝜃) (6a)

𝜎𝜃(𝑟, 𝜃) = 𝜎𝜃,∞(𝑟, 𝜃) + 𝜎

𝜃,∞′ (𝑟, 𝜃 +

𝜋

2) + 𝜎𝜃,𝑤(𝑟, 𝜃) (6b)

𝜎𝑟𝜃(𝑟, 𝜃) = 𝜎𝑟𝜃,∞

(𝑟, 𝜃) + 𝜎𝑟𝜃,∞

′ (𝑟, 𝜃 +𝜋

2) + 𝜎𝑟𝜃,𝑤(𝑟, 𝜃) (6c)

The anisotropic coefficients 𝛽1 and 𝛽2 are calculated by

equations (A3a and b).

𝛽1 =√𝛼1 − 1

√𝛼1 + 1 (A3a)

𝛽2 =√𝛼2 − 1

√𝛼2 + 1 (A3b)

The coefficients 𝛼1 and 𝛼2 are obtained from elements of

the compliances, 𝑆𝑖𝑗.

𝛼1𝛼2 = 𝑆11/𝑆22 (A4a)

21

𝛼1 + 𝛼2 = (𝑆66 + 2𝑆12)/𝑆22 (A4b)

The complex numbers 𝑋1 and 𝑋2 can be calculated from

equations (A5a and b).

𝑋1 = √1 −4𝑎2𝛽1

(𝑧 + 𝛽1�̄�)2 (A5a)

𝑋2 = √1 −4𝑎2𝛽2

(𝑧 + 𝛽2�̄�)2 (A5b)

where, the complex values 𝑧 and �̄� are given by

𝑟(cos𝜃 + 𝑖 sin𝜃) and 𝑟(cos𝜃 − 𝑖 sin𝜃), respectively.

For the stress components induced by the far-field stress

90°-rotated (𝜎𝑟,∞′, 𝜎𝜃,∞′ and 𝜎𝑟𝜃,∞′ in equations (6a-c)),

the anisotropy coefficients, 𝛽1 and 𝛽2, need to be

calculated from 𝛼1′ and 𝛼2′ of the updated coordinate as

follows,

𝛼1′𝛼2′ = 𝑠22/𝑠11 (A6a)

𝛼1′ + 𝛼2′ = (𝑠66 + 2𝑠12)/𝑠11 (A6b)

For the completely isotropic rock (𝛼1𝛼2=1.0 and 𝛼1 +𝛼2=2.0), the stress distributions can be obtained by the

traditional Kirsch equations as follows (Kirsch, 1898),

𝜎𝑟 = (𝑃𝑤 − 𝑃𝑝)𝑎2

𝑟2+1

2(𝜎𝐻𝑚𝑎𝑥 + 𝜎𝑉) (1 −

𝑎2

𝑟2)

+1

2(𝜎𝐻𝑚𝑎𝑥 − 𝜎𝑉) (1 − 4

𝑎2

𝑟2+ 3

𝑎4

𝑟4) 𝑐𝑜𝑠 2𝜃

(A7a)

𝜎𝜃 = −(𝑃𝑤 − 𝑃𝑝)𝑎2

𝑟2+1

2(𝜎𝐻𝑚𝑎𝑥 + 𝜎𝑉) (1 +

𝑎2

𝑟2)

−1

2(𝜎𝐻𝑚𝑎𝑥 − 𝜎𝑉) (1 + 3

𝑎4

𝑟4) 𝑐𝑜𝑠 2𝜃

(A7b)

𝜎𝑟𝜃 = −1

2(𝜎𝐻𝑚𝑎𝑥 − 𝜎𝑉) (1 + 2

𝑎2

𝑟2− 3

𝑎4

𝑟4) 𝑠𝑖𝑛 2𝜃 (A7c)

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