Poroelastic modeling of cutting bottom-hole rock – Part I: Stress state of bottom-hole rock

Journal Pre-proof Poroelastic modeling of cutting bottom-hole rock – Part I: Stress state of bottom-hole rock Pengju Chen, Stefan Miska, Mengjiao Yu, Evren Ozbayoglu PII:

S0920-4105(20)30109-1

DOI:

https://doi.org/10.1016/j.petrol.2020.107014

Reference:

PETROL 107014

To appear in:

Journal of Petroleum Science and Engineering

Received Date: 20 September 2019 Revised Date:

13 January 2020

Accepted Date: 29 January 2020

Please cite this article as: Chen, P., Miska, S., Yu, M., Ozbayoglu, E., Poroelastic modeling of cutting bottom-hole rock – Part I: Stress state of bottom-hole rock, Journal of Petroleum Science and Engineering (2020), doi: https://doi.org/10.1016/j.petrol.2020.107014. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

CRediT author statement Pengju Chen – Modeling and Programming, writing original and revised manuscript, etc. Stefan Miska – Writing - Review & Editing, Funding acquisition, Validation Mengjiao Yu – Writing - Review & Editing, Funding acquisition, Validation Evren Ozbayoglu – Project administration, Funding acquisition, Validation

Poroelastic Modeling of Cutting Bottom-hole Rock – Part I: Stress State of Bottom-hole Rock Pengju Chen, Stefan Miska, Mengjiao Yu, and Evren Ozbayoglu, TUDRP, The University of Tulsa, Tulsa, OK, 74104, United States Summary This is the first part of a series of two papers modeling cutting bottom-hole rock during drilling. This paper studies the stress state of a bottom-hole rock that a downhole bit is drilling. A bottom-hole rock underground is subjected to far field in-situ stresses. During drilling process, the rock materials above the bottom-hole rock are gradually removed and replaced with drilling fluid, thus altering the stress state of the bottom-hole rock. In this paper, a model is developed based on the theory of Linear Poroelasticity to calculate the evolution of stresses and pore pressure in a bottom-hole rock during drilling process. The bottom-hole rock under consideration can be selected at any depth underground. To solve the model, superposition principle and finite difference method are employed. The model is also extended for inclined wellbores in arbitrary directions. A series of case studies with various drilling parameters in different formations are presented. The results show that during drilling bottom-hole rock will expand, leading to the pore pressure decrease if formation has a low permeability. The pore pressure decrease in a bottom-hole rock will increase differential pressure, eventually increasing difficulties in drilling. The results show that low permeability formation drilled with a greater rate of penetration will lead to a lower pore pressure in a bottom-hole rock. Moreover, during air drilling in hard formation, the pore pressure decreases significantly in bottom-hole rock.

Key words: Far field in-situ stresses; cutting rock; Poroelasticity; directional drilling; bottom-hole rock; MSE.

1 Introduction The rate of penetration (ROP) during drilling generally shows a steady decline as the depth increases. The reduction of ROP with depth can be ascribed to many reasons, such as high torques and drags during drilling, drillstring buckling, bad wellbore cleaning, wellbore instability (Huang et al, 2015; Chen et al, 2017). Another important issue that also leads to a low ROP is inefficient rock cutting. The rock cutting efficiency is quantified by the concept of mechanical specific energy (MSE), which is defined as the amount of work required to remove a unit volume of rock. A smaller MSE indicates a more efficient rock cutting, which would reduce the operation time and drilling cost. Unfortunately, large MSE is often observed during deep well drilling. The large MSE can be caused by improper bit design, bit wear and even bit failure. To study the mechanism behind the large MSE during deep well drilling, this paper mainly focuses on the cutter – rock interaction under bottom-hole condition, more precisely, the stress state of a bottom-hole rock during drilling.

To understand the reasons behind the large MSE, many studies on cutting rock were carried out. Miedema (1987) built a mechanistic model for cutting clay, sand and rock under hydrostatic pressure. The model can predict cutting force and MSE based on the assumption of force equilibrium and minimum external work principle during cutting. Their study is one of the most comprehensive and systematic work on rock cutting theory. The application of their study is cutting seafloor. Detournay and Defourny (1992) developed a mechanistic model for PDC bit. They first built a PDC single cutter model and established a linear relation between specific energy and drilling strength. Then the model is extended to PDC bit by mapping the weight on bit (WOB) and torque on bit (TOB) to the cutting forces on a single cutter. They concluded that even for a sharp cutter, the frictional contact between the cutter and the rock is a pervasive feature of drag bits. Detournay and Atkinson (2000) studied the influence of pore pressure response when drilling impermeable formation. Mass balance was applied to evaluate the pore pressure in cuttings in front of the cutter. Three pore pressure regimes were identified: low-speed, transient and high-speed regime. In the high-speed regime, the dilatancy in the shear zone can lead to the cavitation in the cuttings. Chen et al (2018) and Chen et al (2019b) developed a model to study the poroelastic effect in the rock during cutting process. They proposed that during cutting process, the cutter is not only fragmenting the rock matrix but also driving the pore fluid in front of the cutter. The pore pressure field in the rock is coupled with the stress field caused by cutting forces. The pore pressure build-up due to cutting force can decrease the local differential pressure, leading to a more efficient rock cutting. Previous studies do provide us with good insights on PDC cutter-rock interaction, however, they still have some drawbacks in modeling rock cutting during an oil and gas drilling. One of the reasons is that the rock being cut described in previous works is not really a bottom-hole rock during drilling.

Fig. 1 – Cutting model only considers hydrostatic pressure (modified from Chen et al, 2018)

Most previous rock cutting models developed for oil and gas drilling are based on the system as depicted Fig.1. The rock is under hydrostatic pressure
 and the cutting tool is exerting cutting forces on the tool-rock interface. The system in Fig.1 originates from Miedema (1987) and was originally designed for cutting seafloor. When using the system to study the cutting process during drilling, the intact rock is regarded as a bottom-hole rock, the hydrostatic pressure is considered as drilling fluid pressure and the cutting tool is replaced by a PDC single cutter.

Fig. 2 – State of bottom-hole rock

However, during drilling process, the bottom-hole rock is not only subjected to drilling fluid pressure
 but also to

the far field in-situ stresses in the formation, i.e. overburden stress  , maximum and minimum horizontal stresses  and

 (see Fig.2). Due to the far field in-situ stresses, the bottom-hole rock will be subjected to much larger confining pressure, thus increasing the rock strength drastically. Hence, ignoring in-situ stresses and considering only drilling fluid pressure on the bottom-hole rock would lead to a significant underestimation of the strength of bottom-hole rock. Moreover, based on the field experience, MSE required to drill a vertical wellbore is different from the MSE when drilling an inclined wellbore at the same depth. This is because, compared to a vertical wellbore, the bottom-hole rock in an inclined wellbore is under different stress state due to the inclination of the wellbore relative to the in-situ stress field. Therefore, if in-situ stresses are ignored, the cutting model would be unable to interpret the fact that MSE varies with wellbore directions even in a relatively homogenous formation. Hence, to study the rock cutting process during drilling, the influence of in-situ stresses must be considered. Our study focuses on the process of cutting bottom-hole rock. The study consists of two parts. In the first part, we make effort to model the stress state of a bottom-hole rock, and temporarily ignore the cutting forces by a PDC single cutter. In this part, a model is developed to calculate coupled stresses and pore pressure in the bottom-hole rock during drilling. The model (referred to as “bottom-hole rock model”) is based on the theory of Linear Poroelasticity. In the second part, we will consider the cutting forces and study the cutting process of a PDC single cutter. In this part, a cutting model for PDC single cutter is developed. The model (referred to as “cutter model”) considers a PDC cutter advancing at a steady speed and removing the rock materials over a constant depth of cut. The model is also based on the theory of Linear Poroelasticity and calculates the stresses and pore pressure in the rock caused by the cutting forces. Due to the linearity of the problem, the results of the two models can be superposed. The superposed results represent the stresses and pore pressure in a bottom-hole rock when there is a PDC cutter cutting it. Therefore, the combination of the two models can be used to study cutting bottom-hole rocks. The connection of the two models is illustrated in Fig.3. Eventually, the complete study can give good insights on cutting a bottom-hole rock during drilling. Furthermore, due to the short computation time, the two models can be easily extended to implement real-time cutting forces and MSE

prediction during drilling operations. This paper is the first part of our study. Moreover, for an easy reading, we moved all model derivations to the Appendix A, the readers are encouraged to read the main content first to have a general understanding of the paper, and then go to the appendix for detailed derivations.

Fig.3 – Connection of bottom-hole model and cutter model

2 Problem Statement Determining the stress state of a bottom-hole rock during drilling is not easy, since stresses and pore pressure in a bottom-hole rock at a certain depth will be altered even before the wellbore reaches the bottom-hole rock (see Fig.4).

Fig.4 – Schematic of drilling application

Assume that the bottom-hole rock under consideration is selected at the measured depth  . Since stresses and pore

pressure in the bottom-hole rock at  will be altered before the wellbore reaches the bottom-hole rock, we consider that

the wellbore is currently at the measured depth  and is approaching the bottom-hole rock at  .

In addition, instead of only focusing on the bottom-hole rock, we also consider a large formation at the same depth  . The bottom-hole rock under consideration is within this formation and the surface of the bottom-hole rock is at the upper boundary of the formation (see Fig.4). In the far field area, the formation at  is subjected to maximum and minimum horizontal stresses  and  ,

and overburden stress  (since the formation has a thin layer, the increase of overburden stress within the formation is ignored). During drilling process, the upper part of the rock column above the bottom-hole rock at  is gradually replaced

by lighter drilling fluid column. When the current wellbore (at measured depth ) is close to the bottom-hole rock, the bottom-hole rock becomes less compressed and expands. This will alter the stress state of the bottom-hole rock during

drilling. Furthermore, due to the solid-fluid interaction, pore pressure in the bottom-hole rock will also be affected. Once bottom-hole rock expands, the pore pressure in the bottom-hole rock tends to decrease, which leads to the flow of pore fluid from the surrounding rock into the bottom-hole rock. In addition, drilling fluid in the wellbore also penetrates to the bottom-hole due to the pressure difference between the drilling fluid and pore fluid (also see Fig.4). All these processes will eventually make the pore pressure in the bottom-hole rock deviate from the virgin pore pressure in studied formation, hence changing the differential pressure between drilling fluid and bottom-hole rock. In this paper, a “bottom-hole rock model” is presented. The model is based on the theory of Linear Poroelasticity and can simulate the evolution of coupled stresses and pore pressure in the formation containing the considered bottom-hole rock at  . The final stresses and pore pressure in the bottom-hole rock are obtained by extracting from the calculated stresses and pore pressure in the formation. The model can also be applied in wellbores in arbitrary directions. Case studies on drilling Mancos shale formation and air drilling are presented. The influences of ROP and formation permeability are also discussed. Note again that the model in this paper does not consider cutting forces on the bottom-hole rock exerted by a PDC cutter. In fact, the model results represent the stress state of a bottom-hole rock when the drill bit is picked up from the bottom-hole rock during drilling. The cutting forces of a PDC cutter (or drill bit) will be studied in “cutter model” in the second part of our study. The major assumptions in bottom-hole rock model are summarized below: (a) Both the formation at  and overburden formation are homogenous and isotropic; (b) The formation rocks are considered as poroelastic materials; (c) The formation fluid is brine. Currently, we don’t consider oil and gas as formation fluids. Considering oil and gas as the formation fluids will dramatically complicate the modeling work, e.g. the phase change of oil to gas. (d) The change of the permeability of the bottom-hole rock due to expansion is not considered. (e) The dynamics caused by the circulation of drilling fluid is ignored; (f) Sign convention: compression is taken as positive. 3 Stresses and Pore pressure in Bottom-Hole Rock In Fig.4, during drilling process, the rock column above the bottom-hole rock at  is gradually replaced by the lighter drilling fluid column. The bottom-hole rock will be under less compression and will expand, leading to the stresses and pore pressure variation in the bottom-hole rock. The bottom-hole rock model is designed to simulate this process. The model considers the formation at  as system domain (see Fig.4), which is large enough to eliminate boundary effects. Governing equations are Beltrami-Michell stress formulation without body force. The derivations of governing equations can be found in the literature (Wang, 2017).

∇  +

1    1 −  
− 2  +  ∇
 = 0 1 +    1 +   

  
∇
+  − !=0 3  

(1a) (1b)

where  is Skempton’s coefficient;  is Poisson ratio;  is diffusivity coefficient of the formation; / is the time

derivative in the fixed coordinate system # − $% on the formation (see Fig.A-1).

We first consider a vertical wellbore. Boundary conditions for bottom-hole rock model are shown in Fig.A-1. The entire boundary is denoted by Γ. The boundary Γ'( is the surface of the bottom-hole rock at  , i.e. wellbore region on the surface of the domain. The vertical stress and pore pressure on Γ'( , i.e. ) |+,- and
|+,- , will change when the

current wellbore is close to the formation at  . The in-situ stresses act on the boundary Γ − Γ'( . Zero initial conditions are applied. By solving the governing equations Eq.1 along with the boundary conditions Eq.A-1, the stresses and pore pressure in the formation at  can be obtained. The stresses and pore pressure in the bottom-hole rock under consideration can be

extracted from the calculated stresses and pore pressure in the formation at  .

Superposition principle and finite difference method are used to solve bottom-hole rock model. The detailed derivations and solution procedures can be found in Appendix A. The bottom-hole rock model is also extended to wellbores in arbitrary directions. The details are provided in Appendix A.4. In addition, it should be noted that the
in governing equations, initial conditions and boundary conditions represents excess pore pressure, which is defined as the pore pressure change caused by the change in applied external forces on the rock and/or the change in fluid content in pore space. Excess pore pressure is calculated by
=
. −
/ , where
. is

actual (or absolute) pore pressure after the changes happen to the rock; and
/ is original/virgin pore pressure in the rock

before rock undergoes any changes. In the following, a series of case studies are presented. At the beginning, a case study on drilling Mancos shale formation is presented to give an overview of the stresses and pore pressure in the bottom-hole rock. The expansion of bottom-hole rock is also investigated. In addition, the influence of formation permeability and ROP on the pore pressure in the bottom-hole rock is also studied. Finally, several case studies on air drilling are also presented.

4 Drilling Impermeable Formation – Mancos Shale

Fig.5 – Drilling Mancos shale formation

In this section, a case study on drilling Mancos shale is presented (see Fig.5). The depth of the Mancos shale formation is 9843ft and the virgin pore pressure is 4322psi. The gradients of in-situ stresses  ,  and  are 0.83, 0.70 and 1.01psi/ft respectively, and the corresponding far field in-situ stresses at the formation is 8170, 6890 and 9941psi. The wellbore diameter is 8.5in. Drilling fluid density is 10lbm/gal and ROP is taken as 49ft/hr. A vertical wellbore is studied first, and then horizontal wellbores will be considered. The properties of Mancos shale are summarized in Table B-1. 4.1 Model settings – grid and time step etc. Since the model is designed for real-time predictions, the computation time should be as short as possible. Thus, a relatively coarse grid (i.e. large grid step) is desired. However, it should be noted that the final objective of obtaining the stresses and pore pressure in a bottom-hole rock is to study the cutting process on a bottom-hole rock. For a cutting problem the MSE is very sensitive to the depth of cut, and thus the bottom-hole rock model must be able to output the stresses and pore pressure at the depth of cut 01 (see Eq.A-19). Therefore, the step of the grid in % direction must be small enough. In

this case study, the number of grids is 312312100. The steps in  and $ directions are 1.575in and the step in % direction is 0.08in, which is a very small step. (Later on, we will choose the depth of cut 01 as 0.08in.)

The starting calculation depth /34 (see Appendix A.5) is selected as 9839.90ft and the time step is determined by

Von-Neuman stability condition (Balsara, 1995). 4.2 Stresses and pore pressure distribution in formation at 567

Fig.6 ~ Fig.8 show the evolution of stresses and pore pressure in the formation at  while the current wellbore is

approaching that formation. The stresses and pore pressure within and nearby the wellbore region are different from those in the far field area, which proves the fact that the stresses and pore pressure in the formation at  has already been altered even before the current wellbore reaches the formation. In addition, the variation of stresses and pore pressure are localized and only around the wellbore region. Fig.6 shows the stress distribution of 8 on the plane at %=0.08in while the current wellbore is approaching the

formation at  . The disturbed stress field 8 within and nearby the wellbore region shows an octagon shape. This is

due to the coarse mesh in the  and $ directions. The step of the grid in the  and $ directions (1.575in) is relatively

large compared to the wellbore diameter (8.5in). If a finer mesh in the  and $ directions is employed, the disturbed stress distribution within and nearby the wellbore would have a circular shape. Also note, the stress distribution is not axisymmetric due to heterogeneous horizontal in-situ stresses  and  . In addition, the distribution of 8 in Fig.6(d) has a different pattern compared to Fig.6(a) ~ Fig.6(c). This is because in Fig.6(d) the current wellbore just reached the formation at  and drilling fluid pressure directly acts on the surface of the bottom-hole rock, changing the boundary condition in wellbore region drastically (see Fig.A-8 and Fig.A-9). Fig.7 shows the stress ) in #% plane in the formation. The stress ) in the wellbore region is smaller compared

to the ) in the far filed area, indicating the bottom-hole rock is under less compression and will expand (since compression is taken as positive). Once the bottom-hole rock expands, the pore pressure in the bottom-hole rock tends to decrease. To clearly show the change of pore pressure, Fig.8 only plots the excess pore pressure (the change of the pore pressure relative to virgin pore pressure
/ in the formation at  ). The negative excess pore pressure in the wellbore region indicates the decrease of pore pressure in the bottom-hole rock. The pore pressure decrease is caused both by the expansion of bottom-hole rock and by the extremely low permeability of Mancos shale – due to the extremely low permeability, the pore fluid in the surrounding rock and drilling fluid in the wellbore cannot flow into the bottom-hole rock immediately, thus leading to the pore pressure decrease. At the moment that current wellbore reaches the formation at  , the maximum pore pressure decrease is about

1300psi (i.e. minimum excess pore pressure
is -1300psi) at the depth of 0.13in below the surface of the bottom-hole rock

(see Fig.8(d)). The absolute pore pressure
. in the formation can be simply evaluated by
. =
/ +
.

(a) 56=9839.9ft

(b) 56=9841.5ft

(c) 56=9842.5ft

(d) 56=9843ft (=567 )

Fig.6 – Evolution of 9: at ;=0.08in in formation at 567

(a) 56=9839.9ft

(b) 56=9841.5ft

(c) 56=9842.5ft

(d) 56=9843ft (=567 )

Fig.7 – Evolution of 9< on =>; plane in formation at 567

(a) 56=9839.9ft

(b) 56=9841.5ft

(c) 56=9842.5ft

(d) 56=9843ft (=567 )

Fig.8 – Evolution of ? on =>; plane in formation at 567

4.3 Average stresses and pore pressure at depth of cut in bottom-hole rock For simplicity, the average stresses and pore pressure at the depth of cut in the bottom-hole rock is used when studying the cutting process on the bottom-hole rock. The average stresses and pore pressure can be evaluated by Eq.A-19. The basic idea is to extract the stresses and pore pressure within the wellbore region at the depth of cut in the formation at  and take the average value based on wellbore area. The results are shown in Table 1. Stresses & pore pressure

8

@

)





.

A8@

A8)

A@)

Average value (psi)

7770

6468

5325

-989

3333

0

0

0

Table 1 – Average stresses and pore pressure at ; = BC in bottom-hole rock (vertical well)

From Table 1, the vertical stress ) in the bottom-hole rock changes drastically. The original value of ) equals to 9941psi (the overburden stress) and now its average equals to 5325psi, which is close to the drilling fluid pressure on the bottom-hole rock at  . The large decrease of ) indicates that after the well is drilled, the bottom-hole rock is under less compression compared to its original state before the well is drilled. The average excess pore pressure is –989psi, indicating that there is a significant decrease in pore pressure due to bottom-hole rock expansion. This will increase the difficulty in drilling. One of the reasons is that the pore pressure decrease in the bottom-hole rock will increase the actual differential pressure between drilling fluid pressure and the pore pressure in the rock. For example, in this case study, the differential pressure we conventionally calculated is the difference between drilling fluid pressure and virgin pore pressure in the formation, which equals to
 −
/ = 864
GH. However, when considering the pore pressure decrease due to bottom-hole rock expansion, the actual differential pressure is
 −
. =
 − I
/ +
J = I
 −
/ J −
= 864 − I−989J = 1853
GH

(2)

Therefore, the decrease of pore pressure caused by bottom-hole rock expansion will increase differential pressure significantly, which will further increase challenges in wellbore cleaning (Warren & Smith, 1985). Moreover, the decrease of the pore pressure in bottom-hole rock will also strengthen the rock matrix and thus increase cutting forces during drilling. Note that effective stresses in a rock are calculated by  M =  − N


(3)

where  ′ is effective stress;
is excess pore pressure; N is Biot coefficient;  is Kronecker delta.

From Eq.3, for the same value of stresses  , the pore pressure decrease will increase the effective stresses in the

bottom-hole rock, which indicates that the rock matrix will be subjected to a larger compression (since compression is taken as positive). Once this happens, the rock matrix will be strengthened, increasing MSE in cutting rock (Chen et al, 2018). 4.4 Wellbores in arbitrary directions Two horizontal wellbores and an inclined wellbore in arbitrary direction are studied in this section. The two horizontal wellbores are drilled along the direction of  and  respectively. The rotation angles of the three wellbores are summarized in Table 2. Q

Rotation angle (°) Horizontal well 1, along 

R

Horizontal well 2, along 

0

90

90

90

Well 3, in arbitrary direction

30

70

Table 2 – Summary of rotation angles (see Appendix A.4 for the definitions of S and T)

Average stresses and pore pressure at the depth of cut in the bottom-hole rock are summarized in Table 3. By comparing the excess pore pressure in Table 1 and Table 3, it is easy to find that when drilling a horizontal well the pore pressure decrease in the bottom-hole rock is smaller than that when drilling a vertical well. This is because that, compared to a vertical well, when drilling a horizontal well, a smaller confining stress will be removed from the bottom-hole rock (e.g. see Fig.2, when drilling horizontal well 2, the confining stress removed from the bottom-hole rock is  , while when

drilling a vertical well, the confining stress removed is  ). In this case, a smaller expansion will happen to the bottom-hole rock, thus leading to a smaller pore pressure decrease in the bottom-hole rock.

Also note that shear stresses appear in wellbore 3, since after coordinate transformation, the stress tensor is no longer along its principle direction. 8

@

)





.

A8@

A8)

A@)

Horizontal well 2, along 

9480

6456

5267

-614

3708

0

0

0

9472

7743

5224

-333

3989

0

0

0

Well 3, in arbitrary direction

9240

6780

5264

-596

3726

-192

-665

-527

Average stresses & pore pressure (psi) Horizontal well 1, along 

Table 3 – Average stresses and pore pressure at ; = BC in bottom-hole rock (arbitrary wellbore directions)

5 Influences of Formation Permeability and ROP on Pore Pressure During drilling process, the stresses on the bottom-hole rock gradually reduces, resulting in the expansion of bottom-hole rock. Once the rock expands, the pore pressure in the bottom-hole rock will decrease. This will cause the pressure gradient that drives the pore fluid to flow from the surrounding rock into the bottom-hole rock. The result is that the pore pressure in the bottom-hole rock will recover. In addition, at the same time, the drilling fluid will also penetrate from the current wellbore at  into the bottom-hole rock at  and increase the pore pressure. Therefore, the final pore pressure in the bottom-hole rock depends on the extent of rock expansion and how fast the fluid can flow into the bottom-hole rock. If the bottom-hole rock expands and there is no sufficient fluid flowing into the bottom-hole rock within a certain time, the pore pressure in the bottom-hole rock will decrease. However, if there is

sufficient fluid flowing into the bottom-hole rock, the pore pressure in the bottom-hole rock will eventually build up. The two processes are controlled by ROP and formation permeability. If ROP is large, more rock materials will be drilled within a certain time, and the compression on the bottom-hole rock will decrease fast, leading to the bottom-hole rock expansion and pore pressure decrease; while if formation permeability is large, the pore fluid can flow into the bottom-hole rock immediately and recover the pore pressure in the bottom-hole rock. Moreover, from the perspective of mathematical modeling, ROP will influence the model results due to the change of the boundary conditions )'( and
'( at each iteration (see Appendix A.3); while the formation permeability

influences the model results by directly influencing the diffusivity coefficient  in the governing equation Eq.1b. Hence, the different ROPs and formation permeabilities will lead to different pore pressure in the bottom-hole rock. In this case study, formation rock is assumed to be Indiana limestone. The properties of Indiana limestone can be found in Table B-1. The actual permeability of Indiana limestone is 10md. However, to conduct this case studies, the permeability of Indiana limestone is hypothetically assigned from 10-4md to 10md. The ROP in the case study ranges from 15ft/hr to 160ft/hr, which is commonly seen during drilling operation. Also, a vertical well is studied. Other drilling parameters remain the same as in the case study in Section 4.

Fig.9 – Influence of ROP and formation permeability

Fig.9 shows the average excess pore pressure in the bottom-hole rock under different ROPs and permeabilities. When the formation has a relatively high permeability (>0.1md), ROP will not influence the pore pressure in the bottom-hole rock. In this situation, there is always sufficient pore fluid and drilling fluid flowing into the bottom-hole rock, and the pore pressure in the bottom-hole rock builds up. However, when the formation has a relatively low permeability (<0.01md), ROPs will affect the pore pressure in the bottom-hole rock. From Fig.9, the higher ROP will lead to lower excess pore pressure (i.e. lower absolute pore pressure) in the bottom-hole rock. This is because that when ROP is high, the current wellbore at  will reach the formation at  within a shorter time period and leave less time for drilling fluid and pore fluid to flow into the bottom-hole rock, thus leading to the lower pore pressure. Therefore, ROP only influences the pore pressure in a bottom-hole rock when the formation has a low permeability. Note that when ROP is 50ft/hr and permeability is 10-4md, the average excess pore pressure in the bottom-hole rock of Indiana limestone is -458psi, which is smaller in magnitude than that in the bottom-hole rock of Mancos shale (-989psi, see

Table 1). The reason is that compared to Mancos shale, the Indiana limestone is harder to expand, thus leading to a smaller pore pressure decrease. Also note that, in general, the influence of ROP on the pore pressure in a bottom-hole rock is very limited. 6 Air Drilling This section studies the stresses and pore pressure in a bottom-hole rock during air drilling. Air drilling is an underbalanced drilling (UBD) technique, where compressed air or nitrogen are used to cool the drill bit and remove the cuttings from a wellbore. It is often used when drilling horizontal section in an extended-reach well (Chen et al, 2015). In the first case study, a vertical well is drilled with air in Mancos shale formation. All the model parameters remain the same, except that the drilling fluid density is set as zero. The results of the case study are comparable to the results in Section 3, where the well is drilled with conventional drilling fluid. By following the same calculation procedures, the average stresses and excess pore pressure in the bottom-hole rock can be obtained. The result is summarized in Table 4. Stresses & pore pressure

8

@

)





.

A8@

A8)

A@)

Average value (psi)

10980

9683

436

-1846

2476

0

0

0

Table 4 – Average stresses and pore pressure at ; = BC in bottom-hole rock (air drilling – Mancos shale)

According to Table 4, compared to a conventional drilling, excess pore pressure decreases drastically during air drilling (also see Table 1), indicating a significant pore pressure drop in the bottom-hole rock. The result is as expected, since once air is used as drilling fluid, the compression on the bottom-hole rock reduces significantly and the bottom-hole rock expands drastically, eventually leading to a large pore pressure decrease in the bottom-hole rock. The actual differential pressure is
 −
. =
 − I
/ +
J = I
 −
/ J −
= −4322 − I−1846J = −2476
GH

(4)

which indicates an underbalanced drilling. In addition, by calculating effective stresses and using Modified Lade failure criterion, it is found that the bottom-hole rock will not fail under the stress state in Table 4. Since one of the most common air drilling applications is hard rock drilling where rate of penetration is less than 15 ft/hr (Hartley et al, 2011), it is also desirable to see the stresses and pore pressure in a bottom-hole rock at a hard formation. In the following, we consider air drilling applications in Westerly granite and Tennessee marble respectively. Only rock properties are changed (see Table B-1), and other model parameters remain the same. The average stresses and pore pressure in the bottom-hole rock are shown in Table 5. Average stresses & pore pressure (psi)

8

@

)





.

A8@

A8)

A@)

Westerly granite

8558

7261

368

-5910

-1588

0

0

0

Tennessee marble

8433

7136

340

-4482

-160

0

0

0

Table 5 – Average stresses and pore pressure at ; = BC in bottom-hole rock (air drilling – hard rock)

Note that in Table 5, the excess pore pressure in both formations shows a very large negative value, which indicates a drastic pore pressure drop in the bottom-hole rock. In Table 5, the absolute pore pressure in the bottom-hole rock is still calculated by using virgin pore pressure 4322psi. It is found that the absolute pore pressure has already become a negative

value, which is unrealistic. In this situation, the model cannot be used anymore and the absolute pore pressure in the bottom-hole rock is actually zero. 7 Conclusions In this paper, “bottom-hole rock model” is built to calculate the evolution of the stresses and pore pressure in the bottom-hole rock. The model is based on the theory of Linear Poroelasticity and is solved by superposition principle and Finite Difference Method. The model can be used in wellbores in arbitrary directions. A simplified model is also developed to predict the boundary conditions )'( and
'( in “bottom-hole rock model”. The model considers the rock column above the bottom-hole rock as a 1-D rod. The mechanical connection between the rock column and its surrounding rock is modeled as the wall shear stress. Eventually, the simplified model can predict )'( and
'( when current wellbore is at any measured depth above the bottom-hole rock.

When current wellbore is approaching the formation at  , the bottom-hole rock at the formation will expand. If the

formation has a low permeability, e.g. Mancos shale, the pore pressure in the bottom-hole rock will decrease, thus increasing the differential pressure between drilling fluid and bottom-hole rock. In addition, stresses and pore pressure in a bottom-hole rock also depends on wellbore directions. This is due to the rotation of in-situ stress tensor based on different wellbore directions. In impermeable formation, a fast ROP will lead to lower pore pressure in a bottom-hole rock, while in permeable formation, ROP will not affect the pore pressure in a bottom-hole rock. However, in general, the influence of ROP on the pore pressure in a bottom-hole rock is very limited. In air drilling, due to the negligible compression exerted by drilling fluid (air), the bottom-hole rock expands drastically, thus leading to a significant pore pressure decrease in bottom-hole rock. In hard formation, during air drilling the absolute pore pressure in the bottom-hole rock is very likely to drop to zero. The corresponding actual differential pressure is also zero, indicating a balanced drilling condition. Nomenclature

/


.





̅/ 01

 

X

=

drilling fluid pressure, psi

=

virgin pore pressure in the formation at  , psi

=

absolute pore pressure in the formation at  , psi

=

excess pore pressure, psi

=

Average excess pore pressure in bottom-hole rock, psi

=

depth of cut, in

=

Skempton’s coefficient

=

diffusivity coefficient, in2/s

=

permeability, md

1

=

consolidation time, s

=

time step, s

=

wellbore diameter, in

=

wellbore cross-section area, ft2

=

unconfined compressive strength of rock, psi

=

mechanical specific energy, psi

=

measured depth of current wellbore, ft

=

starting calculation depth, ft

=

true vertical depth of current wellbore, ft

=

measured depth of the bottom-hole rock under consideration, ft

`a

=

true vertical depth of the formation at  , ft



=

poroelastic stress coefficient

=

drained Poisson ratio

=

Biot coefficient

=

maximum horizontal stress, psi

=

minimum horizontal stress, psi

=

overburden stress, psi

=

average bottom-hole pressure, psi

=

wall shear stress between rock column and surrounding rock, psi

=

drilling fluid density, lbm/gal

=

average density of overburden rock, lbm/gal

=

rotation angle around %° axis

Δ

Z [Z

\]^

^_ 

/34

`a 

Greek Letters



N







bcd/ ddd AZ

e ef

Q

R

=

rotation angle around $ axis

Reference Balsara, D. S. (1995). Von Neumann stability analysis of smoothed particle hydrodynamics—Suggestions for optimal algorithms. Journal of Computational Physics, 121(2), 357-372. Chen, P., Gao, D., Wang, Z., & Huang, W. (2015). Study on multi-segment friction factors inversion in extended-reach well based on an enhanced PSO model. Journal of Natural Gas Science and Engineering, 27, 1780-1787. Chen, P., Gao, D., Wang, Z., & Huang, W. (2017). Study on aggressively working casing string in extended-reach well. Journal of Petroleum Science and Engineering, 157, 604-616. Chen, P., Miska, S. Z., Ren, R., Yu, M., Ozbayoglu, E., & Takach, N. (2018). Poroelastic modeling of cutting rock in pressurized condition. Journal of Petroleum Science and Engineering.

Chen, P., Meng, M., Miska, S., Yu, M., Ozbayoglu, E., & Takach, N. (2019). Study on integrated effect of PDC double cutters. Journal of Petroleum Science and Engineering. Chen, P., Meng, M., Ren, R., Miska, S., Yu, M., Ozbayoglu, E., & Takach, N. (2019). Modeling of PDC single cutter– Poroelastic effects in rock cutting process. Journal of Petroleum Science and Engineering, 183, 106389. Cheng, A. D., Abousleiman, Y., & Roegiers, J. C. (1993, December). Review of some poroelastic effects in rock mechanics. In International journal of rock mechanics and mining sciences & geomechanics abstracts (Vol. 30, No. 7, pp. 1119-1126). Pergamon. Detournay, E., & Defourny, P. (1992, January). A phenomenological model for the drilling action of drag bits. In International journal of rock mechanics and mining sciences & geomechanics abstracts (Vol. 29, No. 1, pp. 13-23). Pergamon. Detournay, E., & Atkinson, C. (2000). Influence of pore pressure on the drilling response in low-permeability shear-dilatant rocks. International Journal of Rock Mechanics and Mining Sciences, 37(7), 1091-1101. Fjar, E., Holt, R. M., Raaen, A. M., Risnes, R., & Horsrud, P. (2008). Petroleum related rock mechanics (Vol. 53). Elsevier. Haimson, B., & Chang, C. (2000). A new true triaxial cell for testing mechanical properties of rock, and its use to determine rock strength and deformability of Westerly granite. International Journal of Rock Mechanics and Mining Sciences, 37(1-2), 285-296. Hartley, R. C., Weisbeck, D. H., Robert, S., & Smith, M. A. (2011, January). The Successful Evolution Of An LWD Rotary Steerable System For Air Drilling. In SPE/IADC Drilling Conference and Exhibition. Society of Petroleum Engineers. Huang, H., Lecampion, B., & Detournay, E. (2013). Discrete element modeling of tool rock interaction I: rock cutting. International Journal for Numerical and Analytical Methods in Geomechanics, 37(13), 1913-1929. Huang, W., Gao, D., Wei, S., & Chen, P. (2015). Boundary conditions: a key factor in tubular-string buckling. SPE Journal, 20(06), 1-409. Miedema, S. A. (1987). Calculation of the cutting forces when cutting water saturated sand. Doctor thesis, Delft, Netherlands. Wang, H. F. (2017). Theory of linear poroelasticity with applications to geomechanics and hydrogeology. Princeton University Press. Warren, T. M., & Smith, M. B. (1985). Bottomhole stress factors affecting drilling rate at depth. Journal of petroleum technology, 37(08), 1-523. Appendix A Modeling of Bottom-hole Rock We first consider a vertical wellbore. The boundary conditions on the formation at  are shown in Fig.A-1. The

coordinate system # − $% is established at the center of the bottom-hole rock surface at the formation such that  and

$ axis points to  and  respectively and % axis is along the wellbore direction. In the wellbore region Γ'( on the

formation, the excess pore pressure
equals to
'( , and the vertical stress ) equals to )'( , where the subscript H

represents the iterations. Both of )'( and
'( need to be updated at each iteration H and are influenced by the drilling

fluid column and overburden rock in the wellbore region above the formation at  . The calculation of )'( and
'( is illustrated in Appendix A.3.

Fig.A-1 Boundary conditions for bottom-hole rock model

The boundary conditions in Fig.A-1 are summarized below 8 |+ =  @ g+ = 

) |+,- = )'(

) |h+,- = I + )'( J/2 ) |+i+,- = 
|+,- =
'(


|j+,- =
'( /2

(A-1a) (A-1b) (A-1c) (A-1d) (A-1e) (A-1f) (A-1g)


|+i+,- = 0

(A-1h)

A8@ , A8) , A@) g+ = 0

(A-1i)

 g+,mno = 0

(A-2a)

The initial conditions are zero stresses and pore pressure in system domain Ω


|+,mno = 0

(A-2b)

Since governing equations are linear, the superposition principle can be applied. The entire problem can be divided into two cases. In the first case, only the in-situ stresses in the formation at  are considered; while in the second case, only the stresses and pore press in the wellbore region are considered. The details of the two subcases are presented in Appendix A.1 and Appendix A.2 respectively. The stresses and pore pressure solved from the two cases can be added together to get the final stresses and pore pressure in the formation at  . Since the bottom-hole rock under consideration is located in the formation, eventually, the stresses and pore pressure in the bottom-hole rock can be obtained.

Appendix A.1 Case I – In-situ stresses Fig.A-2 shows the boundary conditions for Case I. The initial conditions are the same as Eq.A-2. The boundary conditions are 8 |+ = 

(A-3a)

@ g = 

(A-3b)

) |+ = 

(A-4c)

+


|+ = 0

(A-3d)

A8@ , A8) , A@) g = 0 +

(A-3e)

It should be noted that since initial conditions are zero, while the boundary conditions are non-zero, the calculated pore pressure at the early time in the formation will increase drastically due to the sudden applied boundary conditions, i.e. in-situ stresses. Then, pore pressure begins to diffuse. Assume p

p

is the time in Case I, and after a long geological time, i.e.

→ ∞, the pore pressure build-up at the beginning will eventually be diffused, leading to a homogenous pore pressure field.

Therefore, eventually the excess pore pressure
in the formation should be zero everywhere, and the corresponding

absolute pore pressure in the formation is the virgin formation pore pressure
/ . The whole process, to some extent, can be

compared to diagenesis.

Fig.A-2 – Boundary conditions for Case I

Note that for Case I, only the solution at

p

→ ∞ is needed. Since when drilling operation starts, the formation is

already at a stable state, which suggests that a long geological time has passed. Because excess pore pressure
is zero in the domain when ∇  +

p

→ ∞, the governing equations in Eq.1 reduces to

1    =0 1 +   

(A-4)

The equation is simply Beltrami-Michell stress formulation without body force in Elasticity. By combining governing equation Eq.A-4 and boundary conditions Eq.A-3, it is easy to directly write out the solutions for Case I: 8 =  @ =  ) = 

(A-5a) (A-5b) (A-5c)


, A8@ , A8) , A@) = 0

(A-5d)

Appendix A.2 Case II – Wellbore region The initial conditions for Case II are still the same as Eq.A-2. Fig.A-3 shows the boundary conditions for Case II. The boundary conditions are ) |+,- = )'(s

(A-6a)

) |j+,- = )'(s /2

(A-6b)

) |+i+,- = 0

(A-6c)


|+tu =
'(

(A-6d)


|j+tu =
'( /2

(A-6e)


|+i+tu = 0

(A-6f)

8 , @ , A8@ , A8) , A@) g+ = 0

(A-6g)

where )'(s equals to )'( −  and  at each iteration is updated by  =  iv + Δ 2 w#x, Δ is the time step during iterations.

Fig.A-3 – Boundary conditions for Case II

Stresses and pore pressure in Case II can be obtained by solving the governing equations Eq.1 along with the initial and boundary conditions Eq.A-2 and Eq.A-6. The final stresses and pore pressure in the formation at  can be obtained by superposing the stresses and pore pressure obtained in Case I and Case II. In addition, assume

pp

is the time in Case II. One should note that the time

have different time scales. Time

p

is in geological time scale while

pp

p

in Case I and the time

pp

in Case II

is in drilling operation time scale. Therefore, in

fact, the final solutions of stresses and pore pressure in the formation are obtained by superposing the solutions of Case I at p

→ ∞ and the solutions of Case II at a certain drilling operation time

Appendix A.3 Determination of 9
pp .

The boundary conditions in the wellbore region on the formation at  are the stress )'( and excess pore

pressure
'( . The )'( and
'( need to be updated during each iteration. Let’s first consider )'( . When the current measured depth  is zero, the )'( should equal to the overburden stress  ; while the current wellbore reaches the

bottom-hole rock, i.e.  =  , the )'( is supposed to be drilling fluid pressure
 at  . A simple idea is to use the stress caused by the weight of drilling fluid column and the rock column above the bottom-hole rock:

)'( = e { + ef {I −  J

(A-7)

However, using Eq.A-7 to evaluate )'( will lead to unrealistic results. The reason is that for Eq.A-7, even a small

change of  at a shallow depth can lead to the change of )'( on the bottom-hole rock at  , i.e. a small disturbance at a shallow depth would be propagated to the bottom-hole rock deep underground which is apparently against reality. In fact, the disturbance at the shallow depth will be damped down within a limited range, rather than being propagated to the deep formation. The biggest drawback of Eq.A-7 is that it ignores the “mechanical connections” of the rock column with the surrounding rock.

Fig.A-4 – Schematic of model for calculating 9
To estimate )'( , a simplified model is built. In the model, the rock column above the bottom-hole rock is considered as a 1-D rod (see Fig.A-4). The top of the rock column is subjected to drilling fluid pressure and the bottom of the rock column is connected to the surface of the bottom-hole rock at  . The rock column has mechanical connections with

surrounding rock and )'( and
'( are the stresses and pore pressure at the bottom of the rock column.

At the same depth, the rock in the rock column is under less compression compared to the rock in the surrounding rock. Thus, compared to the surrounding rock, the rock column tends to expand towards the drilling fluid along its axis (also see Fig.A-4). Due to the mechanical connections, the surrounding rock will exert downward forces to prevent the expansion of rock column.

Fig.A-5 Forces on small element in rock column

To model this problem, the system in Fig.A-5 is built. The local coordinate system #′%′ is built where the origin is at

 and #′%′ always points to the wellbore direction. Consider a small element with length 0%′ at location %′ in the rock column. According to the force equilibrium | − I| + 0|J + AZ ^} 0% M + ~ = 0

(A-8)

where | is internal force; AZ is wall shear stress; ^} is the circumference of the wellbore, 2€Z ; ~ is the weight of the small element and is calculated by ~ = ef {0% M [Z , where [Z is the cross section area, €Z. Dividing by 0%′ on both sides of Eq.A-8 and rearranging the terms yields

0| = AZ ^} + ef {[Z 0%′

(A-9)

Discretizing it by using explicit form gives | v = | + IAZ ^} + ef {[Z J0%′

(A-10)

The wall shear stress AZ at %′ is determined by the deformation of the rock column relative to the surrounding rock

at %′

AZ = ‚ƒ/ I„) − „)Z J

(A-11a)

where ‚ƒ/ is the shear modulus of overburden formation; „) and „)Z are the axial strain in the direction of #′%′ in the surrounding rock and rock column at %′. They can be evaluated by „) = ef {I + % M J/_ƒ/ = I |…M J/_ƒ/ „)Z = | /[Z _ƒ/

(A-11b) (A-11c)

where _ƒ/ is the elastic modulus of overburden formation.

At % M = 0, i.e. the surface of the rock column (or the current bottom-hole), the initial value of internal force is

|o =
 [Z = e { ⋅ [Z

(A-12)

Eq.A-9 to Eq.A-12 forms an initial value problem of first order ODE, which can be solved easily by iterative method. The calculation starts from % M = 0. At each iteration, %′ is increased by step 0%′ until % M =  − . During each

iteration, Eq.A-10 is evaluated to obtain internal force | v at the corresponding location on the rock column. After

obtaining the internal force | v at  , the boundary condition )'( can be calculated as )'( = I| v |…‡ nˆ‰Š iˆ‰ J/[Z

(A-13)

It is easy to prove that as long as the 0%′ is small enough, e.g. 0.0003ft, the calculated )'( will eventually equal to

the drilling fluid pressure when current wellbore reaches the formation at  , i.e. )'( = e { when  =  .

Once )'( is obtained, the boundary condition )'(s in Case II can be determined by )'(s = )'( −  |ˆ‰Š ,

where the overburden stress  at the measured depth  is used.

The excess pore pressure in the rock column is influenced by two processes. The first is the penetration of drilling fluid from wellbore to the bottom-hole rock. This can be considered as a typical consolidation problem and the analytic solution is already available. The pore pressure due to consolidation at any location %′ in the rock column can be estimated by

(Wang, 2017, Chapter 6)

%′
1 |…M = I
 −
/ JerfcI J 4 1

where

1

(A-14a)

is consolidation time and can be estimated by time step Δ ;
 is the drilling fluid pressure on the current

bottom-hole, i.e.
 = e {;
/ is still taken as the virgin pore pressure in the formation.

The second effect on the pore pressure is the expansion of rock column. Once rock column expands, the pore pressure will decrease. This is determined by Skempton coefficient and the change of stress state
= Δ /3 (compression is taken as positive in this section). Thus, the pore pressure caused by the rock expansion at any location %′ in the rock

column can be evaluated by
 |…M =

 I | ‡ −  |…‡ J 3 )…

(A-14b)

where ) |…‡ is the stress at location %′ in the rock column and is calculated by | /[Z .

The pore pressure at location %′ in the rock column can be obtained by adding Eq.A-14a and Eq.A-14b.


|…M =
1 |…M +
 |…M

(A-15a)


'( =
|…‡ nˆ‰Š iˆ‰

(A-15b)

The boundary condition
'( is determined by setting % M =  − 

So far, all the boundary conditions in bottom-hole rock model can be determined and Case II can be solved by Finite Difference Method. By adding the solutions to Case I and Case II, the final stresses and pore pressure in the formation at  at current iteration H can be determined. The detailed solution procedures are summarized in Appendix A.5.

Before going to the next section, two examples of calculating )'( and
'( in different formations are presented. In

both examples, a vertical wellbore of 8.5in is considered, and the drilling fluid density is 10lbm/gal. The overburden stress gradient is still taken as 1.01psi/ft. The formation is at the measured depth of 9843ft and the consolidation time

1

is taken

as 3.76s. In the first example, the formation of Indiana limestone is considered (see Table B-1 for the properties of Indiana limestone).

Fig.A-6 Distribution of 9< and ? along rock column when current wellbore is at 56 9830ft

Fig.A-6 shows the calculated vertical stress ) and excess pore pressure
along the rock column. The current

wellbore is at  9830ft. On the surface of the rock column (% M = 0), the vertical stress equals to the drilling fluid

pressure 5181psi (see Eq.A-12). When going deeper, the ) increases drastically, indicating a drastic increase in compression at a shallow depth in the

rock column. The drastic increase in compression is caused by the wall shear stress AZ which arises from the expansion of

rock column relative to the surrounding rock (see Eq.A-11a). Furthermore, if the rock column is long enough ( is small enough compared to  ), the ) in the rock column will eventually increase to the overburden stress in the surrounding rock at a certain depth, as shown in Fig.A-6. And the rock column below that depth is subjected to the same compression as the surrounding rock. In addition, the increase and decrease of the pore pressure in the rock column is caused by the combined effect of drilling fluid penetrating into rock column and rock column expansion.

Fig.A-7 Calculated 9
The )'( and
'( for wellbores at different  are shown in Fig.A-7. According to Fig.A-7, when the current

wellbore () is relatively far from the formation at  , the calculated )'( and
'( equal to their original values,

i.e. the overburden stress  and zero excess pore pressure. In this case, the formation will be undisturbed.

When the current wellbore is approaching the formation at  , the
'( increases due to the penetration of drilling

fluid from current wellbore into the formation. When the current wellbore is very close to the formation at  , the stress

on the bottom-hole rock )'( reduces, indicating the expansion of the bottom-hole rock. The decrease of
'( is caused by the rock column expansion above the bottom-hole rock. In addition, one should note that from Fig.A-7 only when the wellbore is very close to the formation at  (about 10ft above the formation in this example), the stresses and pore pressure in the formation will be disturbed.

Fig.A-8 Calculated 9
In the second example, an impermeable formation – Mancos shale is considered (see Table B-1 for the properties). The calculated )'( and
'( for the wellbore at different  are shown in Fig.A-8. The value of )'( shows a similar

trend as in the first example. However, the
'( continuously decrease until the current wellbore is very close to the

formation at  (0.03ft above the formation in this example). Then
'( increases drastically. The decrease of
'( is due to the rock expansion and the extremely low permeability. When current wellbore is approaching the formation, the surface of the bottom-hole rock gradually expands, leading to the decrease of
'( . Furthermore, due to the extremely low permeability, the penetration depth of drilling fluid in the rock column is very small, and the drilling fluid cannot reach the bottom-hole rock, hence
'( cannot be increased. However, when the current wellbore is very close to the bottom-hole

rock at  , even the penetration depth is limited, the drilling fluid is able to penetrate to the surface of the bottom-hole

rock, leading to the drastic increase of
'( (see Fig.A-9).

Fig.A-9 Enlargement of ?yz when 56 is very close to 567

Finally, we will give a simple discussion on the accuracy of predicted )'( and
'( by using the model in this

section. According to Fig.A-8, in Mancos shale formation, the )'( and
'( start to deviate from the initial values when

the current wellbore is at 2.3ft above the formation at  . This is to say that only after the current wellbore reaches 2.3ft

above the formation at  , the stresses and pore pressure in the formation will be disturbed. Hence, by using the simplified model in this section, we predict that the drilled wellbore can alter the stress field in the surrounding formation that is within 2.3ft around the wellbore, which is 3~4 times of the wellbore diameter. While in Fig.7(a), the stress filed in the formation starts to change when the current wellbore is at  9839.9ft,

which is 9843–9839.9=3.1ft below the current wellbore. Hence, based on the solutions of stress formulation Eq.1, we predict that the drilled wellbore can alter the stress field in the surrounding formation within 3.1ft from the wellbore. By comparing number of 2.3ft and 3.1ft, we can find that the two predictions are quite consistent, which indirectly proves that the simplified model to calculate )'( and
'( in this section can give reasonable results. Appendix A.4 Extension to wellbores in arbitrary directions Previous section only considers a vertical wellbore, while in this section, we will extend the model to wellbores in any directions. The key idea is to modify the boundary conditions based on the wellbore directions. The coordinate system # − $% is also redefined based on wellbore directions (see Fig.A-10).

Fig.A-10 Coordinate system > − =‘; for wellbore in arbitrary direction

In Fig.A-10, the coordinate system # − $% for inclined wellbore is still built at the center of the bottom-hole rock

surface where % axis points toward the wellbore direction,  axis points toward the uppermost radial direction of the

wellbore and $ axis is determined by right-hand rule. The original coordinate system # − $%° is built such that °, $°

and %° point toward to the direction of  ,  and  . The coordinate system # − $%° is also right-handed. The

transformation from # − $%° to # − $% can be obtained by

(a) A rotation Q around %° axis (anticlockwise is positive when observing against %°). (b) A rotation R around $ axis (anticlockwise is positive when observing against $).

Expressing in the coordinate system # − $%, the formation principle stresses at any depth are (Fjar, 2008)

   8 = ’88M  + ’8@M  + ’8)M 

   @ = ’@8M  + ’@@M  + ’@)M     ) = ’)8M  + ’)@M  + ’))M 

A8@ = ’88M ’@8M  + ’8@M ’@@M  + ’8)M ’@)M  A@) = ’@8M ’)8M  + ’@@M ’)@M  + ’@)M ’))M  A)8 = ’)8M ’88M  + ’)@M ’8@M  + ’))M ’8)M 

(A-16a) (A-16b) (A-16c) (A-16d) (A-16e) (A-16f)

where ’88M = cos Q cos R ; ’8@M = sin Q cos R ; ’8)M = − sin R ; ’@8M = −sin Q ; ’@@M = cos Q ; ’@)M = − sin R ; ’)8M = cos Q sin R; ’)@M = sin — sin R; ’))M = cos R.

It should be noted that in this model, the formation containing the bottom-hole rock is always perpendicular to the wellbore axis. Therefore, once the coordinate system is rotated, the considered formation will also be rotated as shown in Fig.A-10.

For Case I in Appendix A.1, when considering wellbores in any directions, the principle far field in-situ stresses in boundary conditions Eq.A-3 should be replaced by the same far field in-situ stresses but under rotated coordinate system # − $% (see Eq.A-16). Eventually, the solutions of Case I is  =  g

ˆ‰Š


=0

(A-17a) (A-17b)

where  g

ˆ‰Š

represents the rotated far field stresses at measured depth  .

For Case II in Appendix A.2, when calculating )'( for the wellbore in an arbitrary direction, the gravity is no longer along the wellbore axis and Eq.A-9 should be replaced by 0| = AZ ^} + ef {[Z cos ˜ 0%′

(A-18a)

where ˜ is wellbore inclination angle and equals to |H|.

Moreover, when calculating AZ , the in-situ stress ) should be used, i.e. Eq.A-11b should be replaced by

„) = I) g J/_ƒ/

(A-18b)

…M

where ) g

…M

is the ™ component of in-situ stress tensor under coordinate system # − $% at measured depth of

 + %′. Also, the measured depth  must be replaced by the corresponding true vertical depth `a when calculating

|o ,
 and in-situ stresses at a certain vertical depth in the formation. Similarly, when calculating
'( , Eq.14b should be replaced by


 |…M =

 I | ‡ − ) g ‡ J … 3 )…

(A-18c)

where ) |…‡ is still calculated by | /[Z .

Again, once )'( for the wellbore in an arbitrary direction is obtained, the boundary condition )'(s in Case II can

be determined by )'(s = )'( − ) g

ˆ‰Š

, where the ™ component of far field in-situ stress ) at the measured depth

 is used. Once Case II is solved, the final solution of stresses and pore pressure in the formation at  can be obtained by superposing the solutions in Case I and Case II. Appendix A.5 Solution procedures Although the model studies the bottom-hole rock at  , the computation of the model must start from the measured

depth above  . This is because stresses and pore pressure in the bottom-hole rock at  will be altered before the current wellbore reaches it. The starting measured depth /34 must ensure that the formation containing the bottom-hole

rock at  is under undisturbed state, i.e. )'(s and
'( in Case II must be zero. The detailed procedures are (also see Fig.A-11):

Step 1: specify the measured depth of bottom-hole rock  , select the starting calculation depth /34 and time

step Δ ;

Step 2: determine the boundary conditions )'( and
'( on the bottom-hole rock at  by following

Appendix A.3; further determine the boundary condition )'(s for Case II;

Step 3: solve Case II by finite difference method to obtain the solutions to Case II; then add the solutions to Case I and the solutions to Case II to obtain the stresses and excess pore pressure in the formation at  during current iteration H; Step 4: update measured depth  v and time

v

=



+Δ .

v

for the next iteration by  v =  + w#x 2 Δ and

Step 5: repeat Step 2 to Step 4 until  =  . Finally, evaluate the average stresses and excess pore pressure at the

depth 01 in the bottom-hole rock by bcd ddd

/

= š g 0[Z /[Z

(A-19a)

›œ


̅/ = š
|›œ 0[Z /[Z

(A-19b)

This can be done by numerical integration. The procedures are also summarized in Fig.A-11. The absolute pore pressure
.

in the formation at  can be calculated by
. =
/ +
.

Fig.A-11 Schematic of computation procedures

Appendix A.6 Discretization of governing equations Governing equation Eq.1a is written in compact form, the complete form of Eq.1a consists of seven equations ∇ 8 + ∇ @ + ∇ ) +

1    1 −  
− 2  + ∇
 = 0 1 +    1 +    1    1 −  
− 2  + ∇
 = 0 1 +  $  1 +  $ 

1    1 −  
− 2  + ∇
 = 0 1 +  %  1 +  % 

∇ A8@ + ∇ A8) +

1    1 −  
− 2 =0 1 +  $ 1 +  $

1    1 −  
− 2 =0 1 +  % 1 +  %

(A-20a) (A-20b) (A-20c) (A-20d) (A-20e)

∇ A@) +

1    1 −  
− 2 =0 1 +  $% 1 +  $%

(A-20f)

Solving seven equations simultaneously is time consuming. Fortunately, in this problem, the seven equations can be decoupled into two sets. The first set of equations includes Eq.A-20a~c and Eq.1b, corresponding to the variables 8 , @ , )

and
; while the second set of equations include Eq.A-20d~f, corresponding to the variables A8@ , A8) and A@) . When

solving these equations, we can first solve the first set of equations and obtain 8 , @ , ) and
, and then substitute these solutions into Eq.A-20d~f respectively and solve Eq.A-20d~f in serial to obtain A8@ , A8) and A@) . In this way, the

computation time can be largely reduced. A rectangular mesh is used for the implementation of Finite Difference Method. The system domain, i.e. the formation at  , in Case II is divided into rectangular grids. Standard central difference scheme similar in Chen et al, 2019 is then employed to discretize the governing equation Eq.A-20 and Eq.1b, and the implicit form of time integration is also used in Eq.1b. Appendix B Summary of Rock Properties The poroelastic properties of rocks are summarized in this section. Properties

Mancos shale

Indiana limestone

Westerly granite

Tennessee marble

Shear modulus G (GPa)

4.18

12.1

15

24

Drained Poisson ratio ν

0.22

0.26

0.25

0.25

Bulk modulus K (GPa)

6.07

21.2

25

40

Biot coefficient α

0.86

0.8

0.47

0.19

Undrained Poisson ratio νu

0.33

0.33

0.34

0.27

Undrained bulk modulus Ku (GPa)

14.61

31.2

42.0

44.0

Skempton coefficient B

0.68

0.41

0.85

0.51

UCS (psi)

9996

7000

29547

19110

Inner friction angle (deg)

28

29

30

27

Permeability (md)

1×10-4

10

4×10-4

1×10-4

Table B-1 – Rock properties (Wang, 2017; Haimson & Chang, 2000)

The paper studies the cutting process under bottom-hole condition, and this is the first part of the study. A model is built to calculate stresses and pore pressure in bottom-hole rock during drilling. Drilling impermeable formation leads to bottom-hole rock expansion, increasing the difficulty in drilling.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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