- Email: [email protected]
Modeling of PDC single cutter – Poroelastic effects in rock cutting process
Journal of Petroleum Science and Engineering 183 (2019) 106389
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Modeling of PDC single cutter – Poroelastic effects in rock cutting process Pengju Chen , Meng Meng, Rui Ren, Stefan Miska, Mengjiao Yu, Evren Ozbayoglu, Nicholas Takach
T
∗
TUDRP, The University of Tulsa, Tulsa, OK, 74104, United States
ARTICLE INFO
ABSTRACT
Keywords: Poroelastic effect PDC single cutter Rock cutting Cavitation
When cutting a saturated rock under pressure, the PDC cutter is not only fragmenting the rock matrix, but also driving the pore fluid ahead of it. Because of the solid-fluid coupling in rock, different pore pressures induced by cutter will affect rock failure and lead to different MSE. The fact that cutting process is influenced by the pore pressure response in the rock is referred to as poroelastic effects in this paper. This paper continues the research in our previous work (Chen et al., 2018) and gives more insights into the poroelastic effects during rock cutting process. The influences of rock diffusivity coefficient and cutter speed are studied. The results show that the two parameters will affect pore pressure response in rock and further affect rock failure and MSE during cutting process. Based on the results, the cutting process can be identified as three conditions: undrained, drained and a transition zone between undrained and drained condition. In undrained and drained condition, MSE will be independent of cutter speed; while in transition condition, MSE decreases with increasing cutter speed. The transition boundaries for the three conditions are given. Cavitation in intact rock during cutting process is also studied. The results show that cavitation is easy to occur when cutting a hard rock with low original pore pressure. Cutting tests were conducted on Torrey Buff sandstone and Carthage marble to verify the poroelastic effects in cutting process. A good agreement between the model results and experiments is found. In general, the results in this paper can give a good understanding on the combined influence of formation permeability, depth of cut and RPM on cutting rock during drilling.
1. Introduction An inefficient rock cutting can slow down the rate of penetration (ROP) and increase drilling risk and cost, especially during shale gas well drilling (Chen et al., 2015, 2017). The concept of mechanical specific energy (MSE) is often used to measure how efficient a rock cutting process is. MSE is defined as the amount of energy consumed to drill out a unit volume of rock, and a small MSE indicates a high efficient cutting. MSE is related to the strength of a rock. However, it is often observed that MSE is excessively large compared to the unconfined compressive strength (UCS) of a rock, especially under a large hydrostatic pressure. Several studies on rock cutting theory have been published. In the early years, Evans (1965) and Nishimatsu (1972) studied cutting rock in dry and atmospheric condition, which is essentially different as compared to the cutting under hydrostatic pressure. Hence, these theories are not applicable in drilling condition. Miedema (1987) built a mechanistic model for cutting clay, sand
∗
and rock in both atmospheric and hyperbaric conditions. The model is based on plane strain assumption and considers that a cutting tool is advancing at a stable speed and removing rock materials under water. During cutting process, the rock debris accumulate in front of the cutting tools. The model can predict cutting force and specific energy based on the assumption of force equilibrium and minimum external work principle during cutting. The model can also estimate pore pressure in cuttings and give the criterion for the cavitation to happen during cutting process. The influence of the various types of cuttings is also discussed. Their study is one of the most comprehensive and systematic work on cutting sea bed formation and provide the foundation for the many studies on rock cutting theory. However, the pore pressure in the model is not coupled with stress field. Glowka (1989) conducted numerous single cutter tests and concluded that cutting force on a worn PDC cutter has a linear relationship with the wear-flat area in contact with rock. Cutting force measured during the tests shows at the same depth of cut larger cutters are more efficient than smaller ones. Their test data were used to obtain very
Corresponding author. E-mail address: [email protected] (P. Chen).
https://doi.org/10.1016/j.petrol.2019.106389 Received 27 August 2018; Received in revised form 13 August 2019; Accepted 14 August 2019 Available online 21 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
useful correlations, and eventually they developed a mechanistic model for PDC bit. The model can predict weight on bit (WOB), torque on bit (TOB) and bit life accurately. Detournay and Atkinson (2000) studied the influence of pore pressure on drilling response in low permeability rocks. This is one of the excellent works that considers the pore fluid flow during cutting process. The idea of mass balance was applied to evaluate the pore pressure in chips produced 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. However, they also didn't consider the coupling between stresses and pore pressure in the rock round the cutter. When modeling the pore fluid flowing from intact rock to chips, the original pore pressure is still used. This may lead to derivations, since the pore pressure field around the cutter is already changed due to the large cutting forces. Kaitkay and Lei (2005) experimentally studied the influence of hydrostatic pressure on rock cutting with a PDC single cutter on Carthage marble. They found that the cutting process is transformed from a brittle to ductile-brittle mode with increasing hydrostatic pressure. The chip formation during cutting under hydrostatic pressure may have some similarities to that in metal cutting – long chips are generated. These long chips will reduce the ROP and the depth of cut of a drill bit. Ledgerwood (2007) used discrete element method (DEM) to investigate cutting process under hydrostatic condition. They simulated the cutting process on two types of numerical rock sample, which consists of numerous small particles: On one rock sample they considered the bonds between each particle, on the other rock sample, they removed the bonds between particles. Eventually they found that the simulated MSE almost the same. The simulation provides a strong evidence that during cutting process under high pressure, the mechanical properties of crushed rock detritus are more important than that of the elastic intact rock. A significant amount of energy will be consumed when the cutter deforms and extrudes crushed rock detritus. He also suggested that measuring inelastic rock behavior under pressure could be a more reasonable measure of rock drillability under high pressure condition. Though the simulation was conducted under hydrostatic pressure, but they didn't consider the influence pore pressure in the rock. Huang et al. (2013) also used DEM to study the critical depth of cut controlling the transition from ductile to brittle failure mode of a rock. Frictional contact between cutter face and the rock was also investigated. The interesting thing is that the friction angle between the cutter fact and the rock is in fact not a constant. The friction angle between cutter and rock cannot be explained by the classical concept of friction. It is in fact controlled by a multi-directional materials flow on the cutter face. During the cutting process, the inclination of the cutter face, i.e. back rake angle, determines the how much rock debris flows upward and how much rock debris flow to the base of the cutter. This multi-directional material flow determines the friction angle between the cutter and the rock. Grima et al. (2015) used a large depth of cut to experimentally study the influence of hydrostatic pressure and cutter speed on cutting process. Their experiments reveal that the effect of speed on cutting rock under hydrostatic pressure is significant on MSE. The analytic model they presented is an extension of the models by Miedema (1987) under totally drained and undrained condition. However, the coupling of stress and pore pressure is not considered and the solutions between drained and undrained conditions are not given. Most of previous studies didn't consider the solid-fluid coupling in the rock during cutting process, and this would lead to some deviations. In fact, during cutting process, the cutter is not only fragmenting the rock matrix, but also driving the pore fluid ahead of it. The pore fluid flow driven by the cutter highly depends on the rock diffusivity coefficient and the cutter speed. Different rock diffusivity coefficients and
cutter speeds will lead to different flows in front of the cutter, which will lead to different pore pressure responses. The different pore pressure responses will further affect the effective stresses in the rock and eventually will affect rock failure and MSE. The fact that cutting process is influenced by the pore pressure responses in the rock is referred to as poroelastic effects in this paper. To study the poroelastic effects in cutting process, a cutting model was built based on the theory of Linear Poroelasticity (Wang, 2017). The model considers different forces on the cutter and calculates coupled stresses and pore pressure in rock during cutting process. The Mohr-Coulomb failure criterion is introduced in the model to determine the failure of the rock being cut. Eventually, the model can predict the required cutting force and MSE. The model is already published in our previous work (Chen et al., 2018). However, we haven't got a chance to give a comprehensive study on the poroelastic effects in that work. Therefore, a complete study on the poroelastic effects during cutting process is present in this paper. The pore pressure distributions in front of the cutter are examined under different rock diffusivity coefficients and cutter speeds. Based on the pore pressure response, a cutting process can be identified as three conditions: undrained, drained and a transition zone between drained and undrained condition. For undrained and drained conditions, they have been covered in several studies (Miedema, 1987; Grima et al., 2015); while for transition zone, it hasn't been discussed in many literatures. Therefore, in this paper, detailed discussions are presented on the pore pressure responses under transition condition. A dimensionless number is also introduced to characterize the three conditions, and the transition boundaries for the three conditions are proposed. The cavitation during cutting process is also investigated. The difference is that we studied the cavitation in the intact rock (the part of the rock that is not fragmented by the cutter) rather than the cavitation in cuttings, i.e. crushed rock materials (Detournay and Atkinson, 2000). The results show that the cavitation in intact rock is easy to happen when cutting a hard rock with low original pore pressure. In general, the results in this paper can give a good understanding on the combined influence of formation permeability, depth of cut and RPM on cutting rock during drilling. In the following, we will first briefly summarize the model in our previous work, and then move to the poroelastic effects in cutting process. The major assumptions in the model are: i) only a sharp round cutter is considered. The chamfers and wear flats are ignored; ii) the cutter is moving at a constant cutter speed, i.e. cutting process is in a steady state. The corresponding cutting force is also constant; iii) the influence caused by fluid dynamics is ignored (Li and Gao, 2019; Zhu et al., 2017). 2. Forces on PDC single cutter Fig. 1 shows a sharp PDC single cutter at the middle of rock cutting
Fig. 1. Schematic of forces on PDC single cutter in cutting process. 2
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
process under hydrostatic pressure. A coordinate system x o y is used. The cutter is inclined forward with a back-rake angle , and is advancing at a constant speed v and removing rock materials over a constant depth of cut dc . Drilling fluid provides hydrostatic pressure pm on the surface of the rock. The rock is originally saturated with pore pressure pb . Due to the hydrostatic pressure, long chips (or cuttings) are produced and extruded by the cutter during cutting process. The cuttings will be removed not only through the cutter face, but also under the cutter as shown in Fig. 1 (Ledgerwood, 2007, Zhang et al, 2018). Forces on the cutter in general can be classified into two categories: 1) forces between the cutter and the intact rock, i.e. contact force N and friction f ; 2) forces between the cutter and cuttings, fc1 and fc2 . The fluid resistance due to cutter motion is ignored since it is not the main topic in this paper. Note that only the forces between the cutter and the intact rock will really contribute to fragmenting the rock matrix; while the forces between the cutter and cuttings should be only considered as resistance, since these forces act on cuttings and will not be used to break the rock. The contact area between the cutter and the intact rock is Acut . The contact area between the cutter and cuttings on the cutter face is Ac1, while the contact area between the cutter and cuttings under the cutter is Ac2 (see Fig. 2). The horizontal and vertical component of real cutting force is RH and RV , they can be calculated by decomposing N and f in the horizontal and vertical direction
RH = N cos
RV = N sin
f sin
+ f cos
Fig. 3. Schematic for calculating stresses and pore pressure in intact rock.
FV distributed along the z direction perpendicular to the paper (also see Fig. 3). The line loads FH and FV can be estimated by distributing the cutting force RH and RV along the contact edge between the cutter and the rock surface z (see Fig. 2). FH =
RH z
(3a)
FV =
RV z
(3b)
(1a)
z = 2 rc2
(1b)
The relations between all the forces in Fig. 1 are derived in our previous work (Chen et al., 2018). However, the total cutting force TH and TV and the real cutting force RH and RV cannot be determined by only using these force relations due to an extra freedom. To formulate the solutions, another constraint is needed, i.e. during cutting process the stresses and pore pressure in the rock induced by cutting forces must satisfy the rock failure criterion.
To obtain coupled stresses and pore pressure in the intact rock during cutting process, the theory of Linear Poroelasticity is employed. Fig. 3 is a schematic for calculating coupled stresses and pore pressure, where cuttings are already removed. Both fixed coordinate system XOY and moving coordinate system xoy are introduced. The moving coordinate system is attached at the cutter. By using the moving coordinate system, the stresses and pore pressure under the cutter can be easily described. The relations between the two coordinate systems are (2a)
Y=y
(2b)
(3c)
where rc is the radius of the PDC cutter. For the purpose of modeling, Fig. 3 can be simplified to Fig. 4, where the depth of cut dc is temporarily neglected. Neglecting dc at the beginning can largely simplify the boundary shape without leading to large error, since dc is small compared to the whole body of rock being cut. After obtaining stresses and pore pressure, the depth of cut will be re-considered to predict cutting force and MSE. Tension is taken as positive throughout the paper. The governing equations of Linear Poroelasticity in plane strain space in fixed coordinate system XOY are
3. Stresses and pore pressure in rock
X = x + vt
2
dc cos
rc
2(
2F
2(
c
(4a)
+ 2 p) = 0
2F
+ p) =
D ( Dt
2F
+ p)
(4b)
where c is diffusivity coefficient, defined as k / µS ; k is permeability, µ is viscosity, S is uniaxial specific storage and is calculated by (1 3 u)(1 2 ) ; is defined as , B is Skempton's coefficient, is BKu (1
)(1
2 u)
B (1 + u )
drained Poisson ratio, u is undrained Poisson ratio, is poroelastic stress coefficient; t is consolidation time and F is Biot stress function 2F
2F
2F
= 2 , y = 2 and yx = X Y and D /Dt is the time derivative in Y X fixed coordinate system XOY . The corresponding boundary conditions are x
From the view of mathematic modeling, the cutting force RH and RV are the boundary conditions when calculating the stresses and pore pressure in the rock. However, it should be noted that RH and RV are point forces and, in fact, they do not exist in a plane strain space. The counterparts of RH and RV in plain strain space are the line loads FH and
y (X , yx (X ,
0, t ) =
(pm
0, t ) =
FH
p (X , 0, t ) = pm
Fig. 2. Geometry of PDC single cutter.
pb ) (X
FV
(X
vt )
vt )
pb
Fig. 4. Simplified cutting process for modeling purpose. 3
(5a) (5b) (5c)
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
where (x ) is Dirac delta function. Since cutter is moving, it will be more convenient if we study the stresses and pore pressure under moving coordinate system xoy . By using chain rule, Eq. (4) can be transferred into moving coordinate system. Moreover, if we consider a steady state cutting process, the governing equations under coordinate system xoy finally become 2(
c
2(
2F
(6a)
+ 2 p) = 0
2F
+ p) =
v
x
(
2F
+ p)
(6b)
The corresponding boundary conditions in moving coordinate system are y (x ,
0) =
FV (x )
(7a)
FH (x )
(7b)
pb
(7c)
yx (x , 0) =
p (x , 0) = pm
Fig. 5. Microscopic view of cutter driving pore fluid.
also driving the pore fluid to flow forward. From a microscopic view, pore space is a bundle of capillaries (Li et al., 2018; Zhong et al., 2018) and there are two mechanisms for the cutter driving pore fluid: (i) the cutter directly pushes the pore fluid away from the cutter; (ii) the cutter compresses pore spaces and squeezes the pore fluid away from the cutter (see Fig. 5, where the far field rock indicates the locations in the rock that are far away from the cutter). The first mechanism can be regarded as convection, which is influenced by the cutter speed, v ; while the second is diffusion, which is influenced by rock's diffusivity coefficient c . Therefore, different rock diffusivity coefficients and cutter speeds will lead to different pore fluid responses in front of the cutter and eventually affect the rock failure. This is also reflected in governing equations, since c and v are the two coefficients in Eq. (6). Furthermore, when v is large, the pore fluid tends to accumulate in front of the cutter and lead to pore pressure increase; when c is large, the pore fluid tends to be diffused rapidly and lead to a flat pore pressure distribution ahead of the cutter. Hence, the cutter speed v can be seen as a measure of the pore pressure build-up effect, while the value of c is a measure of the pore pressure diffusion effect. To quantify the relative importance of the effects of pore pressure build-up vd = 4cc is introduced and diffusion, a dimensionless quantity (Detournay and Atkinson, 2000). It should be noted that the flow in front of the cutter is difficult to observe due to the tremendous flow resistance in the rock. In fact, there is only a small amount of pore fluid that is driven by the cutter and flows forward in the pore space, and most of the pore fluid is removed from the rock along with cuttings. However, the pore pressure response during this process is significant. In the following, a series of case studies is presented to illustrate how poroelastic effects will affect the cutting process. Since now we are focusing on the poroelastic effects in the rock, we only study the stresses and pore pressure in the rock and use real cutting force to calculate intrinsic MSE. Frictions caused by cuttings are neglected. The results in this section will provide a good understanding on the combined effects of formation permeability, RPM and depth of cut on a real drilling process. The rock in the case studies is Indiana limestone, the properties are shown in Table 1. The permeability of Indiana limestone commonly ranges from 10 md to 50 md, however, to obtain a wide range of diffusivity coefficients, the permeability of Indiana limestone in the following case studies will be imaginarily adjusted from a very small value to a very large value. The reason that we choose to adjust permeability rather than directly adjusting diffusivity coefficient is that permeability is more familiar to us. The hydrostatic pressure in the case study pm is set to 1470 psi, and original pore pressure pb in rock is set to 735 psi. A sharp cutter with the diameter of 0.512 in and the back-rake angle of 20° is considered. The depth of cut dc is 0.036 in. The cutter-rock friction angle is 10°. Cutter speed is changed from 0.20 in/s to 40 in/s, and the largest cutter speed corresponds to the velocity of an outermost cutter on an 8.5in
It should be noted that the pore pressure p is in fact excess pore pressure, which quantifies the pore pressure variation due to the alterations in applied external load on the rock (in this paper, cutting force) and the alterations in fluid content in pore spaces. Excess pore pressure is calculated by p = pa pb , where pa is absolute pore pressure after the alterations happen to the rock; and pb is original pore pressure in the rock before rock undergoes any change. Stresses and pore pressure in the rock can be solved from Eq. (6) and Eq. (7) by using superposition principle and Fourier transform (Huang et al., 2015; Xi et al., 2015). The detailed solution procedures can be found in our previous work (Chen et al., 2018). 4. Cutting force and MSE prediction Now, what we have is a mechanistic model describing the forces on the PDC cutter, and a poroelastic model calculating the coupled stresses and pore pressure in the rock. The two sub-models are connected by Eq. (7), namely, the real cutting force in the mechanistic model provides the boundary conditions for the poroelastic model. However, as explained in previous section, we are unable to determine the real cutting force because of an extra freedom. To formulate the solutions, another constraint is needed, that is, during cutting process the stresses and pore pressure in the rock must satisfy the rock failure criterion. Mohr-Coulomb failure criterion is introduced to the model, and a trial-and-error method is employed to predict cutting force and MSE. The basic idea is that we assume a total cutting force to calculate real cutting force, then, use it as boundary conditions to calculate stresses and pore pressure in the rock (Chen et al., 2019). Finally, we examine whether the rock failure criterion can be satisfied under calculated stress state. If it is satisfied, the assumed total cutting force is the actual total cutting force. Once cutting forces are determined, the intrinsic MSE can be evaluated by real cutting force RH .
MSEin =
RH Acut cos
(8)
where MSEin represents intrinsic MSE. Total MSE can be evaluated by total cutting force TH .
MSE =
TH Acut cos
(9)
The model is also verified by the published experiment data (Rafatian et al., 2010) in our previous work. 5. Poroelastic effects during cutting process As explained in previous section, in a rock cutting process under hydrostatic pressure, the cutter is not only fragmenting rock matrix, but 4
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 1 Poroelastic properties of Indiana limestone (Wang, 2017). Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg)
12.1 0.26 21.2 0.8 0.33 31.2 0.41 7000 38
PDC drill bit under 100 RPM.
Fig. 7. Stresses and effective stresses distribution ahead of cutter, undrained condition.
5.1. Undrained condition In this case study, a very small permeability of Indiana limestone, 10−4 md, is used. The corresponding rock diffusivity coefficient is 3.00 × 10−7 ft2/s. This case study can be compared to drilling a shale formation (Yang et al., 2016; Li et al., 2019). Pore pressure distributions at different depths in the rock ahead of the cutter (i.e. at x > 0, y = dc ) are plotted in Fig. 6. At each depth in the rock, the pore pressure distributions are calculated under different cutter speeds. Also note that absolute pore pressure rather than excess pore pressure is plotted. From Fig. 6, the pore pressure distributions under different cutter speeds are the same, which proves that the undrained limit is reached. Also note that the far field pore pressure (x ) is not equal to the original pore pressure due to the hydrostatic pressure on the rock. Because of the extremely low rock diffusivity coefficient, pore fluid in front of the cutter will not be diffused away immediately but will accumulate in front of the cutter. In this case, even a very slow cutter speed (0.4 in/s) can lead to the pore pressure build up in front of the cutter. Moreover, the pore pressure build-up is more drastic at the shallow depth y = 0.02in in the rock. However, when going deeper in the rock, the pore pressure decreases drastically. The pore pressure distribution tends to be flat at y = 0.08in . Therefore, the cutting force can only alter the pore pressure field at a shallow depth. The stresses and effective stresses at the depth of cut in the rock ahead of the cutter (i.e. at x > 0, y = dc ) are shown in Fig. 7. Both Figs. 6 and 7 shows the stresses and pore pressure concentration in front of the cutter. The values of Mohr-Coulomb failure function Eq. (10) at the depth of cut are shown in Fig. 8. From Fig. 8, the failure criterion is satisfied at the location of x = 0.032 in , which indicates the failure location at the depth of cut.
Fig. 8. Values of Mohr-Coulomb failure function at different depths in the rock.
MC =
1
UCS
3
tan2
4
+
2
(10)
Note that In Fig. 8, the values of MC at other two different depths in the rock are also plotted. At the depth 0.02in, the maximum value of MC function already exceeds zero, indicating that the rock will be broken; while at the depth 0.04in, the values of MC are all under zero, indicating that the rock at the depth larger than dc will not be broken. In this way, we can control the actual depth of cut in the model equal to the specified depth of cut. Intrinsic MSE during cutting process is shown in Fig. 9. According to Fig. 9, the intrinsic MSE remains the same under different cutter speeds. Therefore, intrinsic MSE will not be influenced by cutter speeds in undrained condition.
Fig. 6. Pore pressure distributions ahead of cutter, undrained condition.
Fig. 9. Intrinsic MSE in cutting process, undrained condition. 5
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. 10. Pore pressure distributions ahead of cutter, drained condition.
5.2. Drained condition
Fig. 12. Pore pressure distributions ahead of cutter, transition condition.
to the drained limit. As the cutter speed increases from 0.2 in/s to 2.0 in/s, more pore fluid right next to the cutter can be driven away. The pore fluid content right next to the cutter will decrease significantly, thus leading to a lowpressure region next to the cutter (also see Fig. 12). It should be noted that the low-pressure region is only caused by the pore fluid reduction rather than the dilation in the intact rock. In fact, under the large hydrostatic pressure and cutting force, the rock is always under compression. Fig. 13 shows the volumetric strain of the rock at the depth of cut when cutter speed is 0.2 in/s. The negative volumetric strain indicates that the rock is under compression (Note tension is taken as positive). When cutter speed is larger than 2.0 in/s, the effect of pore pressure build-up becomes dominant. In this case, the pore pressure right next to the cutter begins to increase and a high-pressure region forms. Eventually, when cutter speed increases to 12 in/s, undrained limit is reached. In this case, further increasing cutter speed will not increase the pore pressure anymore. Intrinsic MSE under different cutter speeds is shown in Fig. 14. The decrease of intrinsic MSE with increasing cutter speed is due to the pore pressure build-up in front of the cutter. Based on the definition of effective stress, ij = ij + p ij , once excess pore pressure increases, effective stresses will also increase, which indicates that rock is under less compression. In this condition, the Mohr circle of rock will shift to the right and the rock is easy to be broken (see Fig. 15). Another way to interpret Fig. 15 is that once the pore pressure builds up in front of the cutter, the local differential pressure right next to the cutter will decrease, and thus the rock will become easier to be broken.
0
In this case, the permeability of the rock is set to 200 md to obtain a very large diffusivity coefficient, 0.6 ft2/s. Pore pressure distributions under different cutter speeds at different depths in the rock are shown in Fig. 10. From Fig. 10, all the pore pressure distributions are the same (the curves are overlapped) and have a flat profile. This is caused by the large diffusion effect in the rock. Due to the large diffusivity coefficient, drilling fluid can penetrate to the rock easily, resulting in the similar pore pressure profiles at different depths in the rock. Moreover, because of the large diffusivity coefficient, any pore pressure build-up in front of the cutter will be diffused rapidly, resulting in a flat pore pressure profile. In this situation, even a very large cutter speed (40 in/s) is still unable to build up the pore pressure in front of the cutter. This case corresponds to a drained condition. Fig. 11 shows the intrinsic MSE under different cutter speeds. According to Fig. 11, the intrinsic MSE in drained condition will not be affected by the cutter speed. 5.3. Transition condition The permeability of the rock now is adjusted to 0.1 md and the corresponding diffusivity coefficient is 3.00 × 10−4 ft2/s. In this case, the effect of pore pressure build-up is comparable to the effect of pore pressure diffusion, and the cutting process is in transition condition, which is in between undrained and drained condition. The pore pressure distributions ahead of the cutter, i.e. along x > 0, y = dc , under different cutter speeds are shown in Fig. 12. When cutter speed is very slow, i.e. less than 0.2 in/s, only a very small amount of pore fluid can be driven away by the cutter and pore fluid content right next to the cutter only deceases a little. In this case, the effect of pore pressure diffusion is dominant. Pore pressure right next to the cutter only decreases a little and the entire pore pressure distribution in front of the cutter tends to be flat. This situation is close
5.4. Influence of depth of cut In this section, the influence of depth of cut is studied. The same rock diffusivity coefficient in transition case is used, i.e.
Fig. 11. Intrinsic MSE in cutting process, drained condition.
Fig. 13. Volumetric strain in the rock when cutter speed is 0.2 in/s. 6
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. 14. Intrinsic MSE in cutting process, transition condition.
Fig. 18. Pore pressure distributions ahead of cutter, d c = 0.2 in.
and intrinsic MSE after increasing to 12 in/s. In this case, the undrained limit is reached when cutter speed is 12 in/s. However, we should also note that in this case, the drained condition probably does not exist. According to Fig. 17, the steep slope at the left end suggests that the intrinsic MSE would further decrease if cutter speed continue to decrease. However, in this case, the cutter speed is already very slow. Further decreasing cutter speed may lead to a discontinuous cutting process, which violates the assumption of the model – we assume a steady cutting process. When the depth of cut is changed to 0.08 in, the pore pressure distributions at the depth of cut in the rock is plotted in Fig. 18, and the corresponding intrinsic MSE is shown in Fig. 19. From Figs. 18 and 19, cutter speed will not affect pore pressure and intrinsic MSE after increasing to 8 in/s. In this case, the undrained limit is reached when cutter speed is 8 in/s. By comparing the two cases of depths of cut of 0.08 in and 0.2 in, it is easy to find that the depth of cut will influence the cutter speed at which the undrained limit is reached. Similarly, the steep slope at the left end in Fig. 19 suggests that, the drained condition doesn't exist either when the depth of cut is 0.2 in. Fig. 20 shows the intrinsic MSE under different depths of cut. The cutter speed is set to 0.8 in/s, The depths of cut ranges from 0.04 in to 0.20 in. From Fig. 20, it shows that at the beginning intrinsic MSE will decrease with increasing depth of cut. However, after a certain point, if the depth of cut further increases, intrinsic MSE will increase slightly. The similar trend has been found in many literatures (Rajabov et al., 2012).
Fig. 15. Shift of Mohr circle due to pore pressure increase.
Fig. 16. Pore pressure distributions ahead of cutter, d c = 0.08 in.
5.5. Boundaries between different conditions By calculating intrinsic MSE for different rock diffusivities under different depths of cut and cutter speeds, we can evaluate the divd mensionless quantity = 4cc and identify the cutting process as the drained, undrained and transition zone between drained and undrained
Fig. 17. Intrinsic MSE in cutting process, d c = 0.08 in.
3.00 × 10−4 ft2/s, and the depth of cut is changed to 0.08 in. The pore pressure distributions at the depth of cut in the rock ahead of the cutter is plotted in Fig. 16, and the corresponding intrinsic MSE is shown in Fig. 17. From Figs. 16 and 17, cutter speed will not affect pore pressure
Fig. 19. Intrinsic MSE in cutting process, d c = 0.2 in. 7
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 2 Poroelastic properties of Westerly granite. Property
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Diffusivity coefficient ft2/s
15 0.25 25 0.47 0.34 42 0.85 29547 32 2.37 × 10−4
Fig. 20. Influence of depth of cut on intrinsic MSE.
conditions. Eventually, we proposed the following transition boundaries for undrained, transition and drained cutting condition: ● If > 10 , cutting process is in undrained condition. Cutter speed has no influence on intrinsic MSE; ● If 0.1 < < 10 , cutting process is in transition zone, which is in between undrained and drained condition. Intrinsic MSE will decrease with increasing cutter speed. ● If < 0.1, cutting process is in drained condition. Cutter speed has no influence on intrinsic MSE. Fig. 21. Pore pressure distributions ahead of cutter, Westerly granite.
6. Cavitation in intact rock Cavitation occurs when absolute pore pressure in the rock ahead the cutter becomes nil. In this situation, the differential pressure reaches the maximum, i.e. the hydrostatic pressure. Once this happens, MSE will increase drastically. The cavitation in cuttings or chips has already been covered by several literatures (Miedema, 1987; Detournay and Atkinson, 2000; Grima et al., 2015). In this section, we will study the cavitation in the intact rock. If cutting process is in undrained condition, pore pressure will directly build up in front of the cutter; while if cutting process is in drained condition, pore pressure distribution will be homogenous. In both cases, the pore pressure in front of the cutter will not decrease, and thus cavitation will not occur. However, when cutting process is in transition condition, there could be a low-pressure region ahead of the cutter. Once the absolute pore pressure in the low-pressure region decreases to zero, the cavitation will occur in the intact rock. In addition, cavitation is also likely to occur in a rock with low original pore pressure. Note that absolute pore pressure is calculated by pa = p + pb , and excess pore pressure p can be negative, thus, once original pore pressure pb is low enough, absolute pore pressure can reach zero and cavitation will occur. Moreover, it is also found that the cavitation in the intact rock is very likely to happen when cutting a hard rock. When cutting a hard rock, the required cutting force is very large. The large cutting force will severely compress the pore space, squeeze the pore fluid, and completely drain out the pore fluid ahead of the cutter, leading to the cavitation. A case study of cutting Westerly granite is present. The properties of Westerly granite are shown in Table 2. It is a very hard rock with UCS of 29547 psi (Haimson and Chang, 2000). The diffusivity coefficient of Westerly granite is 2.37 × 10−4 ft2/s. The hydrostatic pressure pm is set to 5145 psi, and original pore pressure pb is 4410 psi. Depth of cut dc is 0.036 in. The pore pressure distributions ahead of the cutter at x > 0 and y = dc are shown in Fig. 21. From Fig. 21, The pore pressure right next to the cutter drops to zero when cutter speed is 0.8 in/s. In this situation, the cutter causes a drastic pore pressure drop in the rock. Even though the rock has a large original pore pressure, the cavitation still occurs. The corresponding in this
case is 0.2 (Note, use SI units to calculate ). 7. Experimental validation In this section, cutting tests were conducted to validate the poroelastic effects during cutting process. The test facility is High Pressure Single Cutter Test facility at the University of Tulsa, TUDRP. The description of the test facility and test procedures can be found in the literature Rafatian et al. (2010). Rock samples used in cutting tests were Torrey Buff sandstone and Carthage marble, the poroelastic properties are measured by Tri-axial Rock Mechanics Testing Facility, GCTS at the University of Tulsa, TUDRP. The measured poroelastic properties are summarized in Table 3 and Table 4. A sharp PDC cutter was used in the tests. The diameter of the cutter is 0.512 in and the back-rake angle is 20°. Mineral oil was used to provide hydrostatic pressure on the rock surface. The hydrostatic pressure pm was 400 psi and pore pressure in rock samples was 200 psi. (To create pore pressure of 200 psi in the rock sample, we first install the rock sample in the high-pressure cell and pump the mineral oil in the cell to create the cell pressure of 200 psi. At the beginning the cell pressure may drop a little due to the penetration of the fluid to the rock Table 3 Poroelastic properties of Torrey Buff sandstone (see Appendix A).
8
Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Permeability (mD)
4.28 0.13 4.36 0.86 0.35 12.8 0.76 5000 29 20
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 4 Poroelastic properties of Carthage marble (see Appendix A). Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Permeability (mD)
17.1 0.22 24.7 0.45 0.25 29.5 0.36 16000 32 1 × 10
5
Fig. 23. Results of cutting Carthage marble.
diffusivity coefficient, the cutting process can be regarded as in drained or undrained condition, respectively. In both cases, cutter speed has no influence on MSE. However, due to the lack of suitable rock sample, at current stage we are temporarily unable to verify the conclusion for cutting process in transition condition by experiment, i.e. in transition condition, MSE will decrease with increasing cutter speed. A special rock sample with suitable diffusivity coefficient (neither too large nor too small) is a prerequisite for a transition cutting condition. 8. Conclusions This paper studies the poroelastic effects on cutting rock under hydrostatic pressure. Rock diffusivity coefficient and cutter speed can influence the MSE during cutting process. The former measures the effect of pore pressure diffusion, while the latter measures the effect of is introduced to pore pressure build-up. A dimensionless number quantify the relative importance of the effect of pore pressure diffusion and build-up. Based on the results, the cutting process can be identified to three conditions: undrained condition ( > 10 ), transition condition (0.1 < < 10 ) and drained condition ( < 0.1). Both in undrained and drained conditions, cutter speed has no influence on the intrinsic MSE; while in transition condition, intrinsic MSE decreases with increasing cutter speed, and the reason can be attributed to the pore pressure build-up in front of the cutter. The cavitation in intact rock is also studied. Cavitation in intact rock can only happen in a transition cutting process. In addition, cavitation is easy to happen when cutting a hard rock with low original pore pressure. MSE will increase once cavitation happens. Cutting tests were conducted at different cutter speeds on Torrey Buff sandstone and Carthage marble to verify the poroelastic effects during cutting process. The results show that the measured MSE during cutting tests remains almost constant with increasing cutter speed. This is consistent with model results. Since Torrey Buff sandstone is very permeable, while Carthage marble is nearly impermeable, the cutting tests on the both rock samples are either in drained or in undrained conditions, in both cases, cutter speed has no influence on MSE.
Fig. 22. Results of cutting Torrey Buff sandstone.
sample. Once cell pressure drops, we increase the cell pressure and keep the cell pressure at 200 psi. These procedures will be repeated until the cell pressure won't drop and can be maintained at 200 psi.) The average depth of cut was 0.065 in. Cutter speed ranged from the lowest speed, 0.18 in/s, to the highest speed, 21 in/s. Fig. 22 presents the results of cutting Torrey Buff sandstone at different cutter speeds. It shows a nearly constant MSE at different cutter speeds. The results are expected, since Torrey Buff sandstone is very permeable (see Table 3) and cutting process is in drained condition. In this case, cutter speed has no influence on MSE. The corresponding is 0.075 when cutter speed is highest, 21 in/s. The corresponding friction = 10° c1 = 15° and angles back-calculated from test data are c2 = 25° . Fig. 23 presents the results of cutting Carthage marble, which also shows a nearly constant MSE under different cutter speeds. This is because that Carthage marble has an extremely low permeability and cutting process is in undrained condition. In this case, cutter speed still has no influence on MSE. The corresponding is 362 when cutter speed is lowest, 0.18 in/s. The friction angles back-calculated from test data are = 18° c1 = 25° and c2 = 33°. The results of cutting tests on Torrey Buff sandstone and Carthage marble proved that if the rock being cut has a very large or very small Nomenclature
pm pb p v F B c t f
hydrostatic pressure, psi original pore pressure in rock before cutting force is applied, psi excess pore pressure, psi cutter speed, in/s Biot stress function, psi.in2 Skempton's coefficient diffusivity coefficient, in2/s time, s friction between cutter and rock, lbf 9
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
fc1 friction between cutter and cuttings flow on cutter face, lbf fc2 friction between cutter and cuttings flow under cutter, lbf N contact force between cutter and rock, lbf FH horizontal component of cutting force, lbf/in FV vertical component of cutting force, lbf/in TH horizontal component of total cutting force, lbf TV vertical component of total cutting force, lbf RH horizontal component of real cutting force, lbf RV vertical component of real cutting force, lbf dc depth of cut, in rc cutter diameter, in UCS unconfined compressive strength of rock, psi MC Mohr-Coulomb failure function, psi MSE mechanical specific energy, psi MSEin Intrinsic specific energy, psi Greek Letters
u
poroelastic stress coefficient drained Poisson ratio undrained Poisson ratio Biot coefficient back rake angel, degree
Appendix A. Poroelastic properties of Torrey Buff sandstone and Carthage marble Appendix A introduces the method (Wang, 2017) to obtain the poroelastic properties of Torrey Buff sandstone and Carthage marble (see Tables 3 and 4). Typically, drained moduli are measured from experiments, and the solid-grain modulus can be found in handbooks of mineral properties (Li et al., 2019). The undrained modulus can be calculated based on the assumption that the solid-grain modulus Ks is equal to both the unjacketed bulk modulus Ks and the unjacketed pore incompressibility K . In this paper, the Ks for Torrey Buff sandstone is taken as 33 GPa; while the Ks for Carthage marble is taken as 45 GPa. The compressibility of the mineral oil Kf in cutting tests is 1.8 GPa. The drained bulk modulus K and drained Poisson ratio c an be measured in a drained tri-axial test. The test facility is GCTS tri-axial test system at TUDRP. The test facility and core samples are shown in Figs. A-1 and A-2. The confining pressure and pore pressure during the tests were controlled at 1000 psi and 500 psi respectively, and the strain rate was controlled at/s 2 × 10 6 . Average stress vs. volumetric strain during the tests are plotted in Figs. A-3 and A-4.
Fig. A-1. GCTS tri-axial test facility at the University of Tulsa, TUDRP.
10
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. A-2. Core samples, Torrey Buff sandstone (left) and Carthage marble (right).
Fig. A-3. Average stress vs. volumetric strain, Torrey Buff sand stone.
Fig. A-4. Average stress vs. volumetric strain, Carthage marble.
According to the definition, drained bulk modulus K equals to the slope of the linear part of the curve before core failure. For Torrey Buff sandstone, the value of K is about 4.36 GPa; while for Carthage marble, it is about 14.7 GPa. Drained Poisson ratio can be calculated directly from raw data based on its definition. Drained Poisson ratio vs. volumetric strain during the tests are plotted in Figs. A-5 and A-6. From Fig. A-6, it is obvious that drained Poisson ratio for Carthage marble is 0.22. However, for Torrey Buff sandstone, drained Poisson ratio increases slowly with increasing volumetric strain. This is due to inelastic effects during the test. In this situation, an average value, 0.13, is taken as the drained Poisson ratio for Torrey Buff sandstone. Once K and are obtained, other poroelastic properties can be calculated based on their relations, which can be easily found in many literatures (Cheng et al., 1993; Wang, 2017).
11
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. A-5. Poisson ratio vs. volumetric strain, Torrey Buff sandstone.
Fig. A-6. Poisson ratio vs. volumetric strain, Carthage marble.
Appendix B. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2019.106389.
conditions. In: 1st Canada-US Rock Mechanics Symposium. American Rock Mechanics Association. Li, X., Gao, D., 2019. Study on mud weight window prediction model and safety additional value of horizontal well in shales considering effects of Well's extended-reach limit. J. Pet. Sci. Eng. 173, 579–587. Li, C., Shen, Y., Ge, H., Zhang, Y., Liu, T., 2018. Spontaneous imbibition in fractal tortuous micro-nano pores considering dynamic contact angle and slip effect: phase portrait analysis and analytical solutions. Sci. Rep. 8 (1), 3919. Li, X., Feng, Y., El Mohtar, C.S., Gray, K.E., 2019. Transient modeling of borehole breakouts: a coupled thermo-hydro-mechanical approach. J. Pet. Sci. Eng. 172, 1014–1024. Miedema, S.A., 1987. Calculation of the Cutting Forces when Cutting Water Saturated Sand. Doctoral thesis. (Delft, Netherlands). Nishimatsu, Y., 1972, March. The mechanics of rock cutting. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 9 (No. 2), 261–270 Pergamon. Rafatian, N., Miska, S.Z., Ledgerwood, L.W., Yu, M., Ahmed, R., Takach, N.E., 2010. Experimental study of MSE of a single PDC cutter interacting with rock under simulated pressurized conditions. SPE Drill. Complet. 25 (01), 10–18. Rajabov, V., Miska, S.Z., Mortimer, L., Yu, M., Ozbayoglu, M.E., 2012, January. The effects of back rake and side rake angles on mechanical specific energy of single PDC cutters with selected rocks at varying depth of cuts and confining pressures. In: IADC/SPE Drilling Conference and Exhibition. Society of Petroleum Engineers. Wang, H.F., 2017. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press. Xi, B., Gao, D., Chen, P., 2015. Uncertainty analysis method for intersecting process of U-shaped horizontal wells. Arabian J. Sci. Eng. 40 (2), 615–625. Yang, R., Huang, Z., Yu, W., Li, G., Ren, W., Zuo, L., Sheng, M., 2016. A comprehensive model for real gas transport in shale formations with complex non-planar fracture networks. Sci. Rep. 6, 36673. Zhang, F., Miska, S., Yu, M., Ozbayoglu, E., 2018. A unified transient solid-liquid two-phase flow model for cuttings transport-modelling part. J. Pet. Sci. Eng. 166, 146–156. Zhong, R., Miska, S., Yu, M., Ozbayoglu, E., Takach, N., 2018. An integrated fluid flow and fracture mechanics model for wellbore strengthening. J. Pet. Sci. Eng. 167, 702–715. Zhu, H., Zhu, J., Zhang, J., Zhang, H.Q., 2017, November. Efficiency and critical velocity analysis of gravitational separator through CFD simulation. In: ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers (pp. V007T09A067-V007T09A067).
References 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. J. Nat. Gas Sci. Eng. 27, 1780–1787. Chen, P., Gao, D., Wang, Z., Huang, W., 2017. Study on aggressively working casing string in extended-reach well. J. Pet. Sci. Eng. 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. J. Pet. Sci. Eng. 169, 623–635. https://doi.org/10. 1016/J.PETROL.2018.06.009. Chen, P., Meng, M., Miska, S., Yu, M., Ozbayoglu, E., Takach, N., 2019. Study on integrated effect of PDC double cutters. J. Pet. Sci. Eng. 178, 1128–1142. https://doi.org/10.1016/J. PETROL.2019.04.024. 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. Pergamon, pp. 1119–1126 No. 7. Detournay, E., Atkinson, C., 2000. Influence of pore pressure on the drilling response in lowpermeability shear-dilatant rocks. Int. J. Rock Mech. Min. Sci. 37 (7), 1091–1101. Evans, I., 1965. March). The force required to cut coal with blunt wedges. In: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 2. Pergamon, pp. 1–12 No. 1. Glowka, D.A., 1989. Use of single-cutter data in the analysis of PDC bit designs: Part 1-development of a PDC cutting force model. J. Pet. Technol. 41 (08), 797–849. Grima, M.A., Miedema, S.A., van de Ketterij, R.G., Yenigül, N.B., van Rhee, C., 2015. Effect of high hyperbaric pressure on rock cutting process. Eng. Geol. 196, 24–36. 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. Int. J. Rock Mech. Min. Sci. 37 (1–2), 285–296. Huang, H., Lecampion, B., Detournay, E., 2013. Discrete element modeling of tool‐rock interaction I: rock cutting. Int. J. Numer. Anal. Methods Geomech. 37 (13), 1913–1929. Huang, W., Gao, D., Wei, S., Chen, P., 2015. Boundary conditions: a key factor in tubular-string buckling. SPE J. 20 (06), 1–409. Kaitkay, P., Lei, S., 2005. Experimental study of rock cutting under external hydrostatic pressure. J. Mater. Process. Technol. 159 (2), 206–213. Ledgerwood III, L.W., 2007, January. PFC modeling of rock cutting under high pressure
12
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Modeling of PDC single cutter – Poroelastic effects in rock cutting process Pengju Chen , Meng Meng, Rui Ren, Stefan Miska, Mengjiao Yu, Evren Ozbayoglu, Nicholas Takach
T
∗
TUDRP, The University of Tulsa, Tulsa, OK, 74104, United States
ARTICLE INFO
ABSTRACT
Keywords: Poroelastic effect PDC single cutter Rock cutting Cavitation
When cutting a saturated rock under pressure, the PDC cutter is not only fragmenting the rock matrix, but also driving the pore fluid ahead of it. Because of the solid-fluid coupling in rock, different pore pressures induced by cutter will affect rock failure and lead to different MSE. The fact that cutting process is influenced by the pore pressure response in the rock is referred to as poroelastic effects in this paper. This paper continues the research in our previous work (Chen et al., 2018) and gives more insights into the poroelastic effects during rock cutting process. The influences of rock diffusivity coefficient and cutter speed are studied. The results show that the two parameters will affect pore pressure response in rock and further affect rock failure and MSE during cutting process. Based on the results, the cutting process can be identified as three conditions: undrained, drained and a transition zone between undrained and drained condition. In undrained and drained condition, MSE will be independent of cutter speed; while in transition condition, MSE decreases with increasing cutter speed. The transition boundaries for the three conditions are given. Cavitation in intact rock during cutting process is also studied. The results show that cavitation is easy to occur when cutting a hard rock with low original pore pressure. Cutting tests were conducted on Torrey Buff sandstone and Carthage marble to verify the poroelastic effects in cutting process. A good agreement between the model results and experiments is found. In general, the results in this paper can give a good understanding on the combined influence of formation permeability, depth of cut and RPM on cutting rock during drilling.
1. Introduction An inefficient rock cutting can slow down the rate of penetration (ROP) and increase drilling risk and cost, especially during shale gas well drilling (Chen et al., 2015, 2017). The concept of mechanical specific energy (MSE) is often used to measure how efficient a rock cutting process is. MSE is defined as the amount of energy consumed to drill out a unit volume of rock, and a small MSE indicates a high efficient cutting. MSE is related to the strength of a rock. However, it is often observed that MSE is excessively large compared to the unconfined compressive strength (UCS) of a rock, especially under a large hydrostatic pressure. Several studies on rock cutting theory have been published. In the early years, Evans (1965) and Nishimatsu (1972) studied cutting rock in dry and atmospheric condition, which is essentially different as compared to the cutting under hydrostatic pressure. Hence, these theories are not applicable in drilling condition. Miedema (1987) built a mechanistic model for cutting clay, sand
∗
and rock in both atmospheric and hyperbaric conditions. The model is based on plane strain assumption and considers that a cutting tool is advancing at a stable speed and removing rock materials under water. During cutting process, the rock debris accumulate in front of the cutting tools. The model can predict cutting force and specific energy based on the assumption of force equilibrium and minimum external work principle during cutting. The model can also estimate pore pressure in cuttings and give the criterion for the cavitation to happen during cutting process. The influence of the various types of cuttings is also discussed. Their study is one of the most comprehensive and systematic work on cutting sea bed formation and provide the foundation for the many studies on rock cutting theory. However, the pore pressure in the model is not coupled with stress field. Glowka (1989) conducted numerous single cutter tests and concluded that cutting force on a worn PDC cutter has a linear relationship with the wear-flat area in contact with rock. Cutting force measured during the tests shows at the same depth of cut larger cutters are more efficient than smaller ones. Their test data were used to obtain very
Corresponding author. E-mail address: [email protected] (P. Chen).
https://doi.org/10.1016/j.petrol.2019.106389 Received 27 August 2018; Received in revised form 13 August 2019; Accepted 14 August 2019 Available online 21 August 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
useful correlations, and eventually they developed a mechanistic model for PDC bit. The model can predict weight on bit (WOB), torque on bit (TOB) and bit life accurately. Detournay and Atkinson (2000) studied the influence of pore pressure on drilling response in low permeability rocks. This is one of the excellent works that considers the pore fluid flow during cutting process. The idea of mass balance was applied to evaluate the pore pressure in chips produced 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. However, they also didn't consider the coupling between stresses and pore pressure in the rock round the cutter. When modeling the pore fluid flowing from intact rock to chips, the original pore pressure is still used. This may lead to derivations, since the pore pressure field around the cutter is already changed due to the large cutting forces. Kaitkay and Lei (2005) experimentally studied the influence of hydrostatic pressure on rock cutting with a PDC single cutter on Carthage marble. They found that the cutting process is transformed from a brittle to ductile-brittle mode with increasing hydrostatic pressure. The chip formation during cutting under hydrostatic pressure may have some similarities to that in metal cutting – long chips are generated. These long chips will reduce the ROP and the depth of cut of a drill bit. Ledgerwood (2007) used discrete element method (DEM) to investigate cutting process under hydrostatic condition. They simulated the cutting process on two types of numerical rock sample, which consists of numerous small particles: On one rock sample they considered the bonds between each particle, on the other rock sample, they removed the bonds between particles. Eventually they found that the simulated MSE almost the same. The simulation provides a strong evidence that during cutting process under high pressure, the mechanical properties of crushed rock detritus are more important than that of the elastic intact rock. A significant amount of energy will be consumed when the cutter deforms and extrudes crushed rock detritus. He also suggested that measuring inelastic rock behavior under pressure could be a more reasonable measure of rock drillability under high pressure condition. Though the simulation was conducted under hydrostatic pressure, but they didn't consider the influence pore pressure in the rock. Huang et al. (2013) also used DEM to study the critical depth of cut controlling the transition from ductile to brittle failure mode of a rock. Frictional contact between cutter face and the rock was also investigated. The interesting thing is that the friction angle between the cutter fact and the rock is in fact not a constant. The friction angle between cutter and rock cannot be explained by the classical concept of friction. It is in fact controlled by a multi-directional materials flow on the cutter face. During the cutting process, the inclination of the cutter face, i.e. back rake angle, determines the how much rock debris flows upward and how much rock debris flow to the base of the cutter. This multi-directional material flow determines the friction angle between the cutter and the rock. Grima et al. (2015) used a large depth of cut to experimentally study the influence of hydrostatic pressure and cutter speed on cutting process. Their experiments reveal that the effect of speed on cutting rock under hydrostatic pressure is significant on MSE. The analytic model they presented is an extension of the models by Miedema (1987) under totally drained and undrained condition. However, the coupling of stress and pore pressure is not considered and the solutions between drained and undrained conditions are not given. Most of previous studies didn't consider the solid-fluid coupling in the rock during cutting process, and this would lead to some deviations. In fact, during cutting process, the cutter is not only fragmenting the rock matrix, but also driving the pore fluid ahead of it. The pore fluid flow driven by the cutter highly depends on the rock diffusivity coefficient and the cutter speed. Different rock diffusivity coefficients and
cutter speeds will lead to different flows in front of the cutter, which will lead to different pore pressure responses. The different pore pressure responses will further affect the effective stresses in the rock and eventually will affect rock failure and MSE. The fact that cutting process is influenced by the pore pressure responses in the rock is referred to as poroelastic effects in this paper. To study the poroelastic effects in cutting process, a cutting model was built based on the theory of Linear Poroelasticity (Wang, 2017). The model considers different forces on the cutter and calculates coupled stresses and pore pressure in rock during cutting process. The Mohr-Coulomb failure criterion is introduced in the model to determine the failure of the rock being cut. Eventually, the model can predict the required cutting force and MSE. The model is already published in our previous work (Chen et al., 2018). However, we haven't got a chance to give a comprehensive study on the poroelastic effects in that work. Therefore, a complete study on the poroelastic effects during cutting process is present in this paper. The pore pressure distributions in front of the cutter are examined under different rock diffusivity coefficients and cutter speeds. Based on the pore pressure response, a cutting process can be identified as three conditions: undrained, drained and a transition zone between drained and undrained condition. For undrained and drained conditions, they have been covered in several studies (Miedema, 1987; Grima et al., 2015); while for transition zone, it hasn't been discussed in many literatures. Therefore, in this paper, detailed discussions are presented on the pore pressure responses under transition condition. A dimensionless number is also introduced to characterize the three conditions, and the transition boundaries for the three conditions are proposed. The cavitation during cutting process is also investigated. The difference is that we studied the cavitation in the intact rock (the part of the rock that is not fragmented by the cutter) rather than the cavitation in cuttings, i.e. crushed rock materials (Detournay and Atkinson, 2000). The results show that the cavitation in intact rock is easy to happen when cutting a hard rock with low original pore pressure. In general, the results in this paper can give a good understanding on the combined influence of formation permeability, depth of cut and RPM on cutting rock during drilling. In the following, we will first briefly summarize the model in our previous work, and then move to the poroelastic effects in cutting process. The major assumptions in the model are: i) only a sharp round cutter is considered. The chamfers and wear flats are ignored; ii) the cutter is moving at a constant cutter speed, i.e. cutting process is in a steady state. The corresponding cutting force is also constant; iii) the influence caused by fluid dynamics is ignored (Li and Gao, 2019; Zhu et al., 2017). 2. Forces on PDC single cutter Fig. 1 shows a sharp PDC single cutter at the middle of rock cutting
Fig. 1. Schematic of forces on PDC single cutter in cutting process. 2
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
process under hydrostatic pressure. A coordinate system x o y is used. The cutter is inclined forward with a back-rake angle , and is advancing at a constant speed v and removing rock materials over a constant depth of cut dc . Drilling fluid provides hydrostatic pressure pm on the surface of the rock. The rock is originally saturated with pore pressure pb . Due to the hydrostatic pressure, long chips (or cuttings) are produced and extruded by the cutter during cutting process. The cuttings will be removed not only through the cutter face, but also under the cutter as shown in Fig. 1 (Ledgerwood, 2007, Zhang et al, 2018). Forces on the cutter in general can be classified into two categories: 1) forces between the cutter and the intact rock, i.e. contact force N and friction f ; 2) forces between the cutter and cuttings, fc1 and fc2 . The fluid resistance due to cutter motion is ignored since it is not the main topic in this paper. Note that only the forces between the cutter and the intact rock will really contribute to fragmenting the rock matrix; while the forces between the cutter and cuttings should be only considered as resistance, since these forces act on cuttings and will not be used to break the rock. The contact area between the cutter and the intact rock is Acut . The contact area between the cutter and cuttings on the cutter face is Ac1, while the contact area between the cutter and cuttings under the cutter is Ac2 (see Fig. 2). The horizontal and vertical component of real cutting force is RH and RV , they can be calculated by decomposing N and f in the horizontal and vertical direction
RH = N cos
RV = N sin
f sin
+ f cos
Fig. 3. Schematic for calculating stresses and pore pressure in intact rock.
FV distributed along the z direction perpendicular to the paper (also see Fig. 3). The line loads FH and FV can be estimated by distributing the cutting force RH and RV along the contact edge between the cutter and the rock surface z (see Fig. 2). FH =
RH z
(3a)
FV =
RV z
(3b)
(1a)
z = 2 rc2
(1b)
The relations between all the forces in Fig. 1 are derived in our previous work (Chen et al., 2018). However, the total cutting force TH and TV and the real cutting force RH and RV cannot be determined by only using these force relations due to an extra freedom. To formulate the solutions, another constraint is needed, i.e. during cutting process the stresses and pore pressure in the rock induced by cutting forces must satisfy the rock failure criterion.
To obtain coupled stresses and pore pressure in the intact rock during cutting process, the theory of Linear Poroelasticity is employed. Fig. 3 is a schematic for calculating coupled stresses and pore pressure, where cuttings are already removed. Both fixed coordinate system XOY and moving coordinate system xoy are introduced. The moving coordinate system is attached at the cutter. By using the moving coordinate system, the stresses and pore pressure under the cutter can be easily described. The relations between the two coordinate systems are (2a)
Y=y
(2b)
(3c)
where rc is the radius of the PDC cutter. For the purpose of modeling, Fig. 3 can be simplified to Fig. 4, where the depth of cut dc is temporarily neglected. Neglecting dc at the beginning can largely simplify the boundary shape without leading to large error, since dc is small compared to the whole body of rock being cut. After obtaining stresses and pore pressure, the depth of cut will be re-considered to predict cutting force and MSE. Tension is taken as positive throughout the paper. The governing equations of Linear Poroelasticity in plane strain space in fixed coordinate system XOY are
3. Stresses and pore pressure in rock
X = x + vt
2
dc cos
rc
2(
2F
2(
c
(4a)
+ 2 p) = 0
2F
+ p) =
D ( Dt
2F
+ p)
(4b)
where c is diffusivity coefficient, defined as k / µS ; k is permeability, µ is viscosity, S is uniaxial specific storage and is calculated by (1 3 u)(1 2 ) ; is defined as , B is Skempton's coefficient, is BKu (1
)(1
2 u)
B (1 + u )
drained Poisson ratio, u is undrained Poisson ratio, is poroelastic stress coefficient; t is consolidation time and F is Biot stress function 2F
2F
2F
= 2 , y = 2 and yx = X Y and D /Dt is the time derivative in Y X fixed coordinate system XOY . The corresponding boundary conditions are x
From the view of mathematic modeling, the cutting force RH and RV are the boundary conditions when calculating the stresses and pore pressure in the rock. However, it should be noted that RH and RV are point forces and, in fact, they do not exist in a plane strain space. The counterparts of RH and RV in plain strain space are the line loads FH and
y (X , yx (X ,
0, t ) =
(pm
0, t ) =
FH
p (X , 0, t ) = pm
Fig. 2. Geometry of PDC single cutter.
pb ) (X
FV
(X
vt )
vt )
pb
Fig. 4. Simplified cutting process for modeling purpose. 3
(5a) (5b) (5c)
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
where (x ) is Dirac delta function. Since cutter is moving, it will be more convenient if we study the stresses and pore pressure under moving coordinate system xoy . By using chain rule, Eq. (4) can be transferred into moving coordinate system. Moreover, if we consider a steady state cutting process, the governing equations under coordinate system xoy finally become 2(
c
2(
2F
(6a)
+ 2 p) = 0
2F
+ p) =
v
x
(
2F
+ p)
(6b)
The corresponding boundary conditions in moving coordinate system are y (x ,
0) =
FV (x )
(7a)
FH (x )
(7b)
pb
(7c)
yx (x , 0) =
p (x , 0) = pm
Fig. 5. Microscopic view of cutter driving pore fluid.
also driving the pore fluid to flow forward. From a microscopic view, pore space is a bundle of capillaries (Li et al., 2018; Zhong et al., 2018) and there are two mechanisms for the cutter driving pore fluid: (i) the cutter directly pushes the pore fluid away from the cutter; (ii) the cutter compresses pore spaces and squeezes the pore fluid away from the cutter (see Fig. 5, where the far field rock indicates the locations in the rock that are far away from the cutter). The first mechanism can be regarded as convection, which is influenced by the cutter speed, v ; while the second is diffusion, which is influenced by rock's diffusivity coefficient c . Therefore, different rock diffusivity coefficients and cutter speeds will lead to different pore fluid responses in front of the cutter and eventually affect the rock failure. This is also reflected in governing equations, since c and v are the two coefficients in Eq. (6). Furthermore, when v is large, the pore fluid tends to accumulate in front of the cutter and lead to pore pressure increase; when c is large, the pore fluid tends to be diffused rapidly and lead to a flat pore pressure distribution ahead of the cutter. Hence, the cutter speed v can be seen as a measure of the pore pressure build-up effect, while the value of c is a measure of the pore pressure diffusion effect. To quantify the relative importance of the effects of pore pressure build-up vd = 4cc is introduced and diffusion, a dimensionless quantity (Detournay and Atkinson, 2000). It should be noted that the flow in front of the cutter is difficult to observe due to the tremendous flow resistance in the rock. In fact, there is only a small amount of pore fluid that is driven by the cutter and flows forward in the pore space, and most of the pore fluid is removed from the rock along with cuttings. However, the pore pressure response during this process is significant. In the following, a series of case studies is presented to illustrate how poroelastic effects will affect the cutting process. Since now we are focusing on the poroelastic effects in the rock, we only study the stresses and pore pressure in the rock and use real cutting force to calculate intrinsic MSE. Frictions caused by cuttings are neglected. The results in this section will provide a good understanding on the combined effects of formation permeability, RPM and depth of cut on a real drilling process. The rock in the case studies is Indiana limestone, the properties are shown in Table 1. The permeability of Indiana limestone commonly ranges from 10 md to 50 md, however, to obtain a wide range of diffusivity coefficients, the permeability of Indiana limestone in the following case studies will be imaginarily adjusted from a very small value to a very large value. The reason that we choose to adjust permeability rather than directly adjusting diffusivity coefficient is that permeability is more familiar to us. The hydrostatic pressure in the case study pm is set to 1470 psi, and original pore pressure pb in rock is set to 735 psi. A sharp cutter with the diameter of 0.512 in and the back-rake angle of 20° is considered. The depth of cut dc is 0.036 in. The cutter-rock friction angle is 10°. Cutter speed is changed from 0.20 in/s to 40 in/s, and the largest cutter speed corresponds to the velocity of an outermost cutter on an 8.5in
It should be noted that the pore pressure p is in fact excess pore pressure, which quantifies the pore pressure variation due to the alterations in applied external load on the rock (in this paper, cutting force) and the alterations in fluid content in pore spaces. Excess pore pressure is calculated by p = pa pb , where pa is absolute pore pressure after the alterations happen to the rock; and pb is original pore pressure in the rock before rock undergoes any change. Stresses and pore pressure in the rock can be solved from Eq. (6) and Eq. (7) by using superposition principle and Fourier transform (Huang et al., 2015; Xi et al., 2015). The detailed solution procedures can be found in our previous work (Chen et al., 2018). 4. Cutting force and MSE prediction Now, what we have is a mechanistic model describing the forces on the PDC cutter, and a poroelastic model calculating the coupled stresses and pore pressure in the rock. The two sub-models are connected by Eq. (7), namely, the real cutting force in the mechanistic model provides the boundary conditions for the poroelastic model. However, as explained in previous section, we are unable to determine the real cutting force because of an extra freedom. To formulate the solutions, another constraint is needed, that is, during cutting process the stresses and pore pressure in the rock must satisfy the rock failure criterion. Mohr-Coulomb failure criterion is introduced to the model, and a trial-and-error method is employed to predict cutting force and MSE. The basic idea is that we assume a total cutting force to calculate real cutting force, then, use it as boundary conditions to calculate stresses and pore pressure in the rock (Chen et al., 2019). Finally, we examine whether the rock failure criterion can be satisfied under calculated stress state. If it is satisfied, the assumed total cutting force is the actual total cutting force. Once cutting forces are determined, the intrinsic MSE can be evaluated by real cutting force RH .
MSEin =
RH Acut cos
(8)
where MSEin represents intrinsic MSE. Total MSE can be evaluated by total cutting force TH .
MSE =
TH Acut cos
(9)
The model is also verified by the published experiment data (Rafatian et al., 2010) in our previous work. 5. Poroelastic effects during cutting process As explained in previous section, in a rock cutting process under hydrostatic pressure, the cutter is not only fragmenting rock matrix, but 4
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 1 Poroelastic properties of Indiana limestone (Wang, 2017). Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg)
12.1 0.26 21.2 0.8 0.33 31.2 0.41 7000 38
PDC drill bit under 100 RPM.
Fig. 7. Stresses and effective stresses distribution ahead of cutter, undrained condition.
5.1. Undrained condition In this case study, a very small permeability of Indiana limestone, 10−4 md, is used. The corresponding rock diffusivity coefficient is 3.00 × 10−7 ft2/s. This case study can be compared to drilling a shale formation (Yang et al., 2016; Li et al., 2019). Pore pressure distributions at different depths in the rock ahead of the cutter (i.e. at x > 0, y = dc ) are plotted in Fig. 6. At each depth in the rock, the pore pressure distributions are calculated under different cutter speeds. Also note that absolute pore pressure rather than excess pore pressure is plotted. From Fig. 6, the pore pressure distributions under different cutter speeds are the same, which proves that the undrained limit is reached. Also note that the far field pore pressure (x ) is not equal to the original pore pressure due to the hydrostatic pressure on the rock. Because of the extremely low rock diffusivity coefficient, pore fluid in front of the cutter will not be diffused away immediately but will accumulate in front of the cutter. In this case, even a very slow cutter speed (0.4 in/s) can lead to the pore pressure build up in front of the cutter. Moreover, the pore pressure build-up is more drastic at the shallow depth y = 0.02in in the rock. However, when going deeper in the rock, the pore pressure decreases drastically. The pore pressure distribution tends to be flat at y = 0.08in . Therefore, the cutting force can only alter the pore pressure field at a shallow depth. The stresses and effective stresses at the depth of cut in the rock ahead of the cutter (i.e. at x > 0, y = dc ) are shown in Fig. 7. Both Figs. 6 and 7 shows the stresses and pore pressure concentration in front of the cutter. The values of Mohr-Coulomb failure function Eq. (10) at the depth of cut are shown in Fig. 8. From Fig. 8, the failure criterion is satisfied at the location of x = 0.032 in , which indicates the failure location at the depth of cut.
Fig. 8. Values of Mohr-Coulomb failure function at different depths in the rock.
MC =
1
UCS
3
tan2
4
+
2
(10)
Note that In Fig. 8, the values of MC at other two different depths in the rock are also plotted. At the depth 0.02in, the maximum value of MC function already exceeds zero, indicating that the rock will be broken; while at the depth 0.04in, the values of MC are all under zero, indicating that the rock at the depth larger than dc will not be broken. In this way, we can control the actual depth of cut in the model equal to the specified depth of cut. Intrinsic MSE during cutting process is shown in Fig. 9. According to Fig. 9, the intrinsic MSE remains the same under different cutter speeds. Therefore, intrinsic MSE will not be influenced by cutter speeds in undrained condition.
Fig. 6. Pore pressure distributions ahead of cutter, undrained condition.
Fig. 9. Intrinsic MSE in cutting process, undrained condition. 5
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. 10. Pore pressure distributions ahead of cutter, drained condition.
5.2. Drained condition
Fig. 12. Pore pressure distributions ahead of cutter, transition condition.
to the drained limit. As the cutter speed increases from 0.2 in/s to 2.0 in/s, more pore fluid right next to the cutter can be driven away. The pore fluid content right next to the cutter will decrease significantly, thus leading to a lowpressure region next to the cutter (also see Fig. 12). It should be noted that the low-pressure region is only caused by the pore fluid reduction rather than the dilation in the intact rock. In fact, under the large hydrostatic pressure and cutting force, the rock is always under compression. Fig. 13 shows the volumetric strain of the rock at the depth of cut when cutter speed is 0.2 in/s. The negative volumetric strain indicates that the rock is under compression (Note tension is taken as positive). When cutter speed is larger than 2.0 in/s, the effect of pore pressure build-up becomes dominant. In this case, the pore pressure right next to the cutter begins to increase and a high-pressure region forms. Eventually, when cutter speed increases to 12 in/s, undrained limit is reached. In this case, further increasing cutter speed will not increase the pore pressure anymore. Intrinsic MSE under different cutter speeds is shown in Fig. 14. The decrease of intrinsic MSE with increasing cutter speed is due to the pore pressure build-up in front of the cutter. Based on the definition of effective stress, ij = ij + p ij , once excess pore pressure increases, effective stresses will also increase, which indicates that rock is under less compression. In this condition, the Mohr circle of rock will shift to the right and the rock is easy to be broken (see Fig. 15). Another way to interpret Fig. 15 is that once the pore pressure builds up in front of the cutter, the local differential pressure right next to the cutter will decrease, and thus the rock will become easier to be broken.
0
In this case, the permeability of the rock is set to 200 md to obtain a very large diffusivity coefficient, 0.6 ft2/s. Pore pressure distributions under different cutter speeds at different depths in the rock are shown in Fig. 10. From Fig. 10, all the pore pressure distributions are the same (the curves are overlapped) and have a flat profile. This is caused by the large diffusion effect in the rock. Due to the large diffusivity coefficient, drilling fluid can penetrate to the rock easily, resulting in the similar pore pressure profiles at different depths in the rock. Moreover, because of the large diffusivity coefficient, any pore pressure build-up in front of the cutter will be diffused rapidly, resulting in a flat pore pressure profile. In this situation, even a very large cutter speed (40 in/s) is still unable to build up the pore pressure in front of the cutter. This case corresponds to a drained condition. Fig. 11 shows the intrinsic MSE under different cutter speeds. According to Fig. 11, the intrinsic MSE in drained condition will not be affected by the cutter speed. 5.3. Transition condition The permeability of the rock now is adjusted to 0.1 md and the corresponding diffusivity coefficient is 3.00 × 10−4 ft2/s. In this case, the effect of pore pressure build-up is comparable to the effect of pore pressure diffusion, and the cutting process is in transition condition, which is in between undrained and drained condition. The pore pressure distributions ahead of the cutter, i.e. along x > 0, y = dc , under different cutter speeds are shown in Fig. 12. When cutter speed is very slow, i.e. less than 0.2 in/s, only a very small amount of pore fluid can be driven away by the cutter and pore fluid content right next to the cutter only deceases a little. In this case, the effect of pore pressure diffusion is dominant. Pore pressure right next to the cutter only decreases a little and the entire pore pressure distribution in front of the cutter tends to be flat. This situation is close
5.4. Influence of depth of cut In this section, the influence of depth of cut is studied. The same rock diffusivity coefficient in transition case is used, i.e.
Fig. 11. Intrinsic MSE in cutting process, drained condition.
Fig. 13. Volumetric strain in the rock when cutter speed is 0.2 in/s. 6
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. 14. Intrinsic MSE in cutting process, transition condition.
Fig. 18. Pore pressure distributions ahead of cutter, d c = 0.2 in.
and intrinsic MSE after increasing to 12 in/s. In this case, the undrained limit is reached when cutter speed is 12 in/s. However, we should also note that in this case, the drained condition probably does not exist. According to Fig. 17, the steep slope at the left end suggests that the intrinsic MSE would further decrease if cutter speed continue to decrease. However, in this case, the cutter speed is already very slow. Further decreasing cutter speed may lead to a discontinuous cutting process, which violates the assumption of the model – we assume a steady cutting process. When the depth of cut is changed to 0.08 in, the pore pressure distributions at the depth of cut in the rock is plotted in Fig. 18, and the corresponding intrinsic MSE is shown in Fig. 19. From Figs. 18 and 19, cutter speed will not affect pore pressure and intrinsic MSE after increasing to 8 in/s. In this case, the undrained limit is reached when cutter speed is 8 in/s. By comparing the two cases of depths of cut of 0.08 in and 0.2 in, it is easy to find that the depth of cut will influence the cutter speed at which the undrained limit is reached. Similarly, the steep slope at the left end in Fig. 19 suggests that, the drained condition doesn't exist either when the depth of cut is 0.2 in. Fig. 20 shows the intrinsic MSE under different depths of cut. The cutter speed is set to 0.8 in/s, The depths of cut ranges from 0.04 in to 0.20 in. From Fig. 20, it shows that at the beginning intrinsic MSE will decrease with increasing depth of cut. However, after a certain point, if the depth of cut further increases, intrinsic MSE will increase slightly. The similar trend has been found in many literatures (Rajabov et al., 2012).
Fig. 15. Shift of Mohr circle due to pore pressure increase.
Fig. 16. Pore pressure distributions ahead of cutter, d c = 0.08 in.
5.5. Boundaries between different conditions By calculating intrinsic MSE for different rock diffusivities under different depths of cut and cutter speeds, we can evaluate the divd mensionless quantity = 4cc and identify the cutting process as the drained, undrained and transition zone between drained and undrained
Fig. 17. Intrinsic MSE in cutting process, d c = 0.08 in.
3.00 × 10−4 ft2/s, and the depth of cut is changed to 0.08 in. The pore pressure distributions at the depth of cut in the rock ahead of the cutter is plotted in Fig. 16, and the corresponding intrinsic MSE is shown in Fig. 17. From Figs. 16 and 17, cutter speed will not affect pore pressure
Fig. 19. Intrinsic MSE in cutting process, d c = 0.2 in. 7
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 2 Poroelastic properties of Westerly granite. Property
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Diffusivity coefficient ft2/s
15 0.25 25 0.47 0.34 42 0.85 29547 32 2.37 × 10−4
Fig. 20. Influence of depth of cut on intrinsic MSE.
conditions. Eventually, we proposed the following transition boundaries for undrained, transition and drained cutting condition: ● If > 10 , cutting process is in undrained condition. Cutter speed has no influence on intrinsic MSE; ● If 0.1 < < 10 , cutting process is in transition zone, which is in between undrained and drained condition. Intrinsic MSE will decrease with increasing cutter speed. ● If < 0.1, cutting process is in drained condition. Cutter speed has no influence on intrinsic MSE. Fig. 21. Pore pressure distributions ahead of cutter, Westerly granite.
6. Cavitation in intact rock Cavitation occurs when absolute pore pressure in the rock ahead the cutter becomes nil. In this situation, the differential pressure reaches the maximum, i.e. the hydrostatic pressure. Once this happens, MSE will increase drastically. The cavitation in cuttings or chips has already been covered by several literatures (Miedema, 1987; Detournay and Atkinson, 2000; Grima et al., 2015). In this section, we will study the cavitation in the intact rock. If cutting process is in undrained condition, pore pressure will directly build up in front of the cutter; while if cutting process is in drained condition, pore pressure distribution will be homogenous. In both cases, the pore pressure in front of the cutter will not decrease, and thus cavitation will not occur. However, when cutting process is in transition condition, there could be a low-pressure region ahead of the cutter. Once the absolute pore pressure in the low-pressure region decreases to zero, the cavitation will occur in the intact rock. In addition, cavitation is also likely to occur in a rock with low original pore pressure. Note that absolute pore pressure is calculated by pa = p + pb , and excess pore pressure p can be negative, thus, once original pore pressure pb is low enough, absolute pore pressure can reach zero and cavitation will occur. Moreover, it is also found that the cavitation in the intact rock is very likely to happen when cutting a hard rock. When cutting a hard rock, the required cutting force is very large. The large cutting force will severely compress the pore space, squeeze the pore fluid, and completely drain out the pore fluid ahead of the cutter, leading to the cavitation. A case study of cutting Westerly granite is present. The properties of Westerly granite are shown in Table 2. It is a very hard rock with UCS of 29547 psi (Haimson and Chang, 2000). The diffusivity coefficient of Westerly granite is 2.37 × 10−4 ft2/s. The hydrostatic pressure pm is set to 5145 psi, and original pore pressure pb is 4410 psi. Depth of cut dc is 0.036 in. The pore pressure distributions ahead of the cutter at x > 0 and y = dc are shown in Fig. 21. From Fig. 21, The pore pressure right next to the cutter drops to zero when cutter speed is 0.8 in/s. In this situation, the cutter causes a drastic pore pressure drop in the rock. Even though the rock has a large original pore pressure, the cavitation still occurs. The corresponding in this
case is 0.2 (Note, use SI units to calculate ). 7. Experimental validation In this section, cutting tests were conducted to validate the poroelastic effects during cutting process. The test facility is High Pressure Single Cutter Test facility at the University of Tulsa, TUDRP. The description of the test facility and test procedures can be found in the literature Rafatian et al. (2010). Rock samples used in cutting tests were Torrey Buff sandstone and Carthage marble, the poroelastic properties are measured by Tri-axial Rock Mechanics Testing Facility, GCTS at the University of Tulsa, TUDRP. The measured poroelastic properties are summarized in Table 3 and Table 4. A sharp PDC cutter was used in the tests. The diameter of the cutter is 0.512 in and the back-rake angle is 20°. Mineral oil was used to provide hydrostatic pressure on the rock surface. The hydrostatic pressure pm was 400 psi and pore pressure in rock samples was 200 psi. (To create pore pressure of 200 psi in the rock sample, we first install the rock sample in the high-pressure cell and pump the mineral oil in the cell to create the cell pressure of 200 psi. At the beginning the cell pressure may drop a little due to the penetration of the fluid to the rock Table 3 Poroelastic properties of Torrey Buff sandstone (see Appendix A).
8
Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Permeability (mD)
4.28 0.13 4.36 0.86 0.35 12.8 0.76 5000 29 20
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Table 4 Poroelastic properties of Carthage marble (see Appendix A). Properties
Value
Shear modulus G (GPa) Drained Poisson ratio ν Bulk modulus K (GPa) Biot coefficient α Undrained Poisson ratio νu Undrained bulk modulus Ku (GPa) Skempton coefficient B UCS (psi) Inner friction angle (deg) Permeability (mD)
17.1 0.22 24.7 0.45 0.25 29.5 0.36 16000 32 1 × 10
5
Fig. 23. Results of cutting Carthage marble.
diffusivity coefficient, the cutting process can be regarded as in drained or undrained condition, respectively. In both cases, cutter speed has no influence on MSE. However, due to the lack of suitable rock sample, at current stage we are temporarily unable to verify the conclusion for cutting process in transition condition by experiment, i.e. in transition condition, MSE will decrease with increasing cutter speed. A special rock sample with suitable diffusivity coefficient (neither too large nor too small) is a prerequisite for a transition cutting condition. 8. Conclusions This paper studies the poroelastic effects on cutting rock under hydrostatic pressure. Rock diffusivity coefficient and cutter speed can influence the MSE during cutting process. The former measures the effect of pore pressure diffusion, while the latter measures the effect of is introduced to pore pressure build-up. A dimensionless number quantify the relative importance of the effect of pore pressure diffusion and build-up. Based on the results, the cutting process can be identified to three conditions: undrained condition ( > 10 ), transition condition (0.1 < < 10 ) and drained condition ( < 0.1). Both in undrained and drained conditions, cutter speed has no influence on the intrinsic MSE; while in transition condition, intrinsic MSE decreases with increasing cutter speed, and the reason can be attributed to the pore pressure build-up in front of the cutter. The cavitation in intact rock is also studied. Cavitation in intact rock can only happen in a transition cutting process. In addition, cavitation is easy to happen when cutting a hard rock with low original pore pressure. MSE will increase once cavitation happens. Cutting tests were conducted at different cutter speeds on Torrey Buff sandstone and Carthage marble to verify the poroelastic effects during cutting process. The results show that the measured MSE during cutting tests remains almost constant with increasing cutter speed. This is consistent with model results. Since Torrey Buff sandstone is very permeable, while Carthage marble is nearly impermeable, the cutting tests on the both rock samples are either in drained or in undrained conditions, in both cases, cutter speed has no influence on MSE.
Fig. 22. Results of cutting Torrey Buff sandstone.
sample. Once cell pressure drops, we increase the cell pressure and keep the cell pressure at 200 psi. These procedures will be repeated until the cell pressure won't drop and can be maintained at 200 psi.) The average depth of cut was 0.065 in. Cutter speed ranged from the lowest speed, 0.18 in/s, to the highest speed, 21 in/s. Fig. 22 presents the results of cutting Torrey Buff sandstone at different cutter speeds. It shows a nearly constant MSE at different cutter speeds. The results are expected, since Torrey Buff sandstone is very permeable (see Table 3) and cutting process is in drained condition. In this case, cutter speed has no influence on MSE. The corresponding is 0.075 when cutter speed is highest, 21 in/s. The corresponding friction = 10° c1 = 15° and angles back-calculated from test data are c2 = 25° . Fig. 23 presents the results of cutting Carthage marble, which also shows a nearly constant MSE under different cutter speeds. This is because that Carthage marble has an extremely low permeability and cutting process is in undrained condition. In this case, cutter speed still has no influence on MSE. The corresponding is 362 when cutter speed is lowest, 0.18 in/s. The friction angles back-calculated from test data are = 18° c1 = 25° and c2 = 33°. The results of cutting tests on Torrey Buff sandstone and Carthage marble proved that if the rock being cut has a very large or very small Nomenclature
pm pb p v F B c t f
hydrostatic pressure, psi original pore pressure in rock before cutting force is applied, psi excess pore pressure, psi cutter speed, in/s Biot stress function, psi.in2 Skempton's coefficient diffusivity coefficient, in2/s time, s friction between cutter and rock, lbf 9
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
fc1 friction between cutter and cuttings flow on cutter face, lbf fc2 friction between cutter and cuttings flow under cutter, lbf N contact force between cutter and rock, lbf FH horizontal component of cutting force, lbf/in FV vertical component of cutting force, lbf/in TH horizontal component of total cutting force, lbf TV vertical component of total cutting force, lbf RH horizontal component of real cutting force, lbf RV vertical component of real cutting force, lbf dc depth of cut, in rc cutter diameter, in UCS unconfined compressive strength of rock, psi MC Mohr-Coulomb failure function, psi MSE mechanical specific energy, psi MSEin Intrinsic specific energy, psi Greek Letters
u
poroelastic stress coefficient drained Poisson ratio undrained Poisson ratio Biot coefficient back rake angel, degree
Appendix A. Poroelastic properties of Torrey Buff sandstone and Carthage marble Appendix A introduces the method (Wang, 2017) to obtain the poroelastic properties of Torrey Buff sandstone and Carthage marble (see Tables 3 and 4). Typically, drained moduli are measured from experiments, and the solid-grain modulus can be found in handbooks of mineral properties (Li et al., 2019). The undrained modulus can be calculated based on the assumption that the solid-grain modulus Ks is equal to both the unjacketed bulk modulus Ks and the unjacketed pore incompressibility K . In this paper, the Ks for Torrey Buff sandstone is taken as 33 GPa; while the Ks for Carthage marble is taken as 45 GPa. The compressibility of the mineral oil Kf in cutting tests is 1.8 GPa. The drained bulk modulus K and drained Poisson ratio c an be measured in a drained tri-axial test. The test facility is GCTS tri-axial test system at TUDRP. The test facility and core samples are shown in Figs. A-1 and A-2. The confining pressure and pore pressure during the tests were controlled at 1000 psi and 500 psi respectively, and the strain rate was controlled at/s 2 × 10 6 . Average stress vs. volumetric strain during the tests are plotted in Figs. A-3 and A-4.
Fig. A-1. GCTS tri-axial test facility at the University of Tulsa, TUDRP.
10
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. A-2. Core samples, Torrey Buff sandstone (left) and Carthage marble (right).
Fig. A-3. Average stress vs. volumetric strain, Torrey Buff sand stone.
Fig. A-4. Average stress vs. volumetric strain, Carthage marble.
According to the definition, drained bulk modulus K equals to the slope of the linear part of the curve before core failure. For Torrey Buff sandstone, the value of K is about 4.36 GPa; while for Carthage marble, it is about 14.7 GPa. Drained Poisson ratio can be calculated directly from raw data based on its definition. Drained Poisson ratio vs. volumetric strain during the tests are plotted in Figs. A-5 and A-6. From Fig. A-6, it is obvious that drained Poisson ratio for Carthage marble is 0.22. However, for Torrey Buff sandstone, drained Poisson ratio increases slowly with increasing volumetric strain. This is due to inelastic effects during the test. In this situation, an average value, 0.13, is taken as the drained Poisson ratio for Torrey Buff sandstone. Once K and are obtained, other poroelastic properties can be calculated based on their relations, which can be easily found in many literatures (Cheng et al., 1993; Wang, 2017).
11
Journal of Petroleum Science and Engineering 183 (2019) 106389
P. Chen, et al.
Fig. A-5. Poisson ratio vs. volumetric strain, Torrey Buff sandstone.
Fig. A-6. Poisson ratio vs. volumetric strain, Carthage marble.
Appendix B. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2019.106389.
conditions. In: 1st Canada-US Rock Mechanics Symposium. American Rock Mechanics Association. Li, X., Gao, D., 2019. Study on mud weight window prediction model and safety additional value of horizontal well in shales considering effects of Well's extended-reach limit. J. Pet. Sci. Eng. 173, 579–587. Li, C., Shen, Y., Ge, H., Zhang, Y., Liu, T., 2018. Spontaneous imbibition in fractal tortuous micro-nano pores considering dynamic contact angle and slip effect: phase portrait analysis and analytical solutions. Sci. Rep. 8 (1), 3919. Li, X., Feng, Y., El Mohtar, C.S., Gray, K.E., 2019. Transient modeling of borehole breakouts: a coupled thermo-hydro-mechanical approach. J. Pet. Sci. Eng. 172, 1014–1024. Miedema, S.A., 1987. Calculation of the Cutting Forces when Cutting Water Saturated Sand. Doctoral thesis. (Delft, Netherlands). Nishimatsu, Y., 1972, March. The mechanics of rock cutting. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 9 (No. 2), 261–270 Pergamon. Rafatian, N., Miska, S.Z., Ledgerwood, L.W., Yu, M., Ahmed, R., Takach, N.E., 2010. Experimental study of MSE of a single PDC cutter interacting with rock under simulated pressurized conditions. SPE Drill. Complet. 25 (01), 10–18. Rajabov, V., Miska, S.Z., Mortimer, L., Yu, M., Ozbayoglu, M.E., 2012, January. The effects of back rake and side rake angles on mechanical specific energy of single PDC cutters with selected rocks at varying depth of cuts and confining pressures. In: IADC/SPE Drilling Conference and Exhibition. Society of Petroleum Engineers. Wang, H.F., 2017. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press. Xi, B., Gao, D., Chen, P., 2015. Uncertainty analysis method for intersecting process of U-shaped horizontal wells. Arabian J. Sci. Eng. 40 (2), 615–625. Yang, R., Huang, Z., Yu, W., Li, G., Ren, W., Zuo, L., Sheng, M., 2016. A comprehensive model for real gas transport in shale formations with complex non-planar fracture networks. Sci. Rep. 6, 36673. Zhang, F., Miska, S., Yu, M., Ozbayoglu, E., 2018. A unified transient solid-liquid two-phase flow model for cuttings transport-modelling part. J. Pet. Sci. Eng. 166, 146–156. Zhong, R., Miska, S., Yu, M., Ozbayoglu, E., Takach, N., 2018. An integrated fluid flow and fracture mechanics model for wellbore strengthening. J. Pet. Sci. Eng. 167, 702–715. Zhu, H., Zhu, J., Zhang, J., Zhang, H.Q., 2017, November. Efficiency and critical velocity analysis of gravitational separator through CFD simulation. In: ASME 2017 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers (pp. V007T09A067-V007T09A067).
References 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. J. Nat. Gas Sci. Eng. 27, 1780–1787. Chen, P., Gao, D., Wang, Z., Huang, W., 2017. Study on aggressively working casing string in extended-reach well. J. Pet. Sci. Eng. 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. J. Pet. Sci. Eng. 169, 623–635. https://doi.org/10. 1016/J.PETROL.2018.06.009. Chen, P., Meng, M., Miska, S., Yu, M., Ozbayoglu, E., Takach, N., 2019. Study on integrated effect of PDC double cutters. J. Pet. Sci. Eng. 178, 1128–1142. https://doi.org/10.1016/J. PETROL.2019.04.024. 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. Pergamon, pp. 1119–1126 No. 7. Detournay, E., Atkinson, C., 2000. Influence of pore pressure on the drilling response in lowpermeability shear-dilatant rocks. Int. J. Rock Mech. Min. Sci. 37 (7), 1091–1101. Evans, I., 1965. March). The force required to cut coal with blunt wedges. In: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, vol. 2. Pergamon, pp. 1–12 No. 1. Glowka, D.A., 1989. Use of single-cutter data in the analysis of PDC bit designs: Part 1-development of a PDC cutting force model. J. Pet. Technol. 41 (08), 797–849. Grima, M.A., Miedema, S.A., van de Ketterij, R.G., Yenigül, N.B., van Rhee, C., 2015. Effect of high hyperbaric pressure on rock cutting process. Eng. Geol. 196, 24–36. 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. Int. J. Rock Mech. Min. Sci. 37 (1–2), 285–296. Huang, H., Lecampion, B., Detournay, E., 2013. Discrete element modeling of tool‐rock interaction I: rock cutting. Int. J. Numer. Anal. Methods Geomech. 37 (13), 1913–1929. Huang, W., Gao, D., Wei, S., Chen, P., 2015. Boundary conditions: a key factor in tubular-string buckling. SPE J. 20 (06), 1–409. Kaitkay, P., Lei, S., 2005. Experimental study of rock cutting under external hydrostatic pressure. J. Mater. Process. Technol. 159 (2), 206–213. Ledgerwood III, L.W., 2007, January. PFC modeling of rock cutting under high pressure
12