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Numerical simulation of 3D fracture propagation in wellbore strengthening conditions
Journal of Petroleum Science and Engineering 156 (2017) 258–268
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Numerical simulation of 3D fracture propagation in wellbore strengthening conditions Jia Li, Zhengsong Qiu *, Dingding Song, Hanyi Zhong, Zhichuan Tang College of Petroleum Engineering, China University of Petroleum, Qingdao 266580, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wellbore strengthening Numerical simulation Fluid-solid coupling Finite-element Lost circulation
The principle of damage mechanics was adopted in order to solve the lost circulation problem when drilled low bearing capacity formation. According to quadratic nominal stress criterion and B-K criterion, cohesive element with zero thickness was inserted into fracture to simulate fracture initiation and propagation. A 3D fluid-solid coupling finite element model was established to analyze circumferential stress distribution of wellbore, hoop stress distribution of crack and geometry of fracture. Results show that closed fracture surface is still in the stage of complete fracture under the action of crustal stress and crack tip reopens after plugging particle failed. The fracture is difficult to widen for filling more plugging materials in a short time while the lost circulation rate and permeability are low. The wellbore strengthening treatment will makes the fracture become short and narrow under a larger filtration coefficient, but mud cake quality has an insignificant influence on fracture geometry when filtration coefficient increases to a certain value. The simulation results were supported by providing some cases of real wellbore condition.
1. Introduction During the drilling process, drilling fluid can balance formation pressure to prevent well collapse or lost circulation. In the meantime, it may fractures the formation with excessive pressure in low pressure drain well, which can lead to lost circulation (Alberty and McLean, 2001). If taking the method of reducing density of drilling fluid to prevent loss of circulation, it may results in overflow or blowout risks and it can also generate underground complex accidents of wellbore instability in sloughing formation. Lost circulation signifies a serious problem which can cause significant economic losses and increase nonproductive time (Wang et al., 2005). It is shown that approximately 2–4 billion dollars were wasted annually on handling lost circulation from some recent engineering accidents (Cook et al., 2011). With the development of lost circulation protection and sealing technology, the method of improving ground bearing capacity and widening safe drilling fluid density window by using proper drilling fluid technology has become a hot spot in global petroleum industry. For this reason, a large number of studies have been performed over the past decades (Morita et al., 1990; Alberty and McLean, 2004; Dupriest, 2005; Van Oort et al., 2011). The technological core consists of physical, chemical or mechanical methods which can enhance rock strength or strengthen wellbore stress status to improve ground bearing capacity.
* Corresponding author. E-mail address: [email protected] (Z. Qiu). http://dx.doi.org/10.1016/j.petrol.2017.06.010 Received 12 March 2017; Received in revised form 26 May 2017; Accepted 2 June 2017 Available online 4 June 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.
According to the principle of the methods, three types are included as follows: The first one is using temporary shielding and plugging technology to seal leak channel (You et al., 2013). The second one is using chemical reinforcement to enhance rock strength (Aston et al., 2007). The third one is utilizing wellbore strengthening method to improve formation fracture pressure or fracture reopening pressure (Van Oort et al., 2011). More recently, wellbore strengthening becomes the most commonly used method to expand safe drilling fluid density window which reduces the occurrence of lost circulation effectively. In order to optimize the design of wellbore strengthening technology, a series of numerical researches were performed. Salehi and Nygaard (2010) built a 3D finiteelement model to simulate fracture opening and growth under different rock permeability. The results show that cracks are widened more in higher permeability. However, the influence of bridging is not considered for the fracture geometry. Feng and Gray (2016) used a poroelastic finite-element numerical model to investigate the effect of several parametrics, and obtained the best bridge location and other useful field implications, but the simulation couldn't provide dynamic change process of fracture behavior during wellbore strengthening operations. Furthermore, most of wellbore strengthening models are based on 2D finite-element and the influence of overburden pressure is neglected, and the length of pre-existing fracture is default value in
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previous studies (Alberty and McLean, 2004; Wang et al., 2009, 2007a, 2007b), it can't be determined by the actual stress state of rock. Based on theory of damage mechanics, a 3D finite-element method was established. Cohesive element with zero thickness was inserted into fracture to simulate fracture initiation and propagation, and fluid flow model of cohesive element was utilized to simulate filtration of drilling fluid in fracture. So, this model has a much higher accuracy than previous 2D model in estimate fracture and wellbore behavior and it has significance of theoretical guidance in wellbore strengthening operations. 2. Fluid-solid coupling equation Fig. 1 shows the principle of effective stress. The concept of effective stress was proposed by Terzaghi et al. (1995). It is assumed that the internal stress of fluid-saturated porous medium consists of pore pressure and effective pressure. The normal stress acts on periphery of pore is called pore pressure and the stress which is transferred by interface of rock particles is described as effective pressure. When external load of porous medium changes, the rock matrixes will be deformed with external force which can lead to the change of porosity and permeability. In the meantime, due to the deformation of rock matrixes, the flowing state and pressure of pore fluid also change along with it. According to effective stress principle, the mechanical equilibrium equation of poroelastic medium can be written as follow (Roe and Siegmund, 2003):
∫ V ðσ pw IÞδεdV ¼ ∫ S tδvdS þ ∫ V fδvdV þ ∫ V φρw gδvdV
Fig. 2. Diagram of cohesive element.
where J is volume change of porous medium; t is computation time; nw is void ratio; x is space vector. 3. Cohesive element model Fig. 2 shows the spatial representation of three-dimensional cohesive element. The cohesive element consists of 12 nodes and it can be divided into top face, mid surface and bottom face which is formed by 4 nodes respectively. In addition, the relative motion of the bottom and top faces along the thickness direction represent opening or closing of the fracture. The relative movement of the bottom and top faces along the direction of perpendicular to thickness represents shear behavior of the cohesive element. The membrane strains is generated from tensile and shear motion of mid surface in the cohesive element, but the cohesive elements do not generate any stresses in a purely membrane response.
(1)
where σ is effective stress matrix; pw is pore pressure; δε is virtual strain rate matrix; V is integral region; t is surface force matrix; δv is virtual rate matrix; S is surface of integral region; f is volume force matrix; ρw is density of pore fluid. The fluid flow in rock pore obeys Darcy's law, so that the continuity equation of fluid is expressed as follow (Yao et al., 2010):
1 ∂ ∂ ðJρw nw Þ ¼ ðρw nw vw Þ J ∂t ∂x
3.1. Fracture damage initiation Before the damage of cohesive element, the stress and strain of cohesive element are assumed to satisfy linear elastic response, it can be described as follow (SIMULIA, 2012):
(2)
8 9 8 < tn = < Knn t ¼ ts ¼ Kε ¼ Kns : ; : tt Knt
where vw is flow rate, which is calculated by using Eq. (3):
1 ∂pw ρw g vw ¼ k⋅ nw gρw ∂x
(3)
Fig. 1. The schematic of effective stress theory. 259
Kns Kss Kst
98 9 Knt =< εn = Kst ε ;: s ; Ktt εt
(4)
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effective traction; d is effective displacement; G0 is elastic energy at the stage of damage initiation. Fig. 3 shows that the cohesive element is gradually damaging from bonding state under the external loading. It finally achieved the breaking state and accomplished interface debond.
where K is element stiffness matrix; t is stress vector; tn , ts and tt represent the normal and two shear tractions; ε is strain vector; εn , εs and εt represent the normal and two shear strains. The ε can be expressed as follow:
εn ¼
dn ds dt ; εs ¼ ; εt ¼ To To To
(5)
3.4. Fracture fluid flow equation
where dn , ds and dt represent the separation of normal and two shear directions; To is the original thickness of the cohesive element. When utilized cohesive element to simulate fracture before the damage of element, the real thickness of it was 0. It will generates singularity when the strain is calculated by using geometry thickness. In order to eliminate singularity, the stress and strain will be described by using original thickness. The original thickness is assumed as 1. This paper utilizes a quadratic interaction function involving the nominal stress ratio to determine whether the fracture will be opened. When the sum of square of critical stress ratio is equal to 1, the damage is assumed to be complete. This criterion (Camanho and Davila, 2002) can be represented as
〈tn 〉 tn0
2
2 2 ts tt þ 0 þ 0 ¼1 ts tt
As shown in Fig. 4, the flow of cohesive element consists of two directions. One is tangential direction, which is parallel to propagation direction, and it is used to simulate flow of drilling fluid in fracture during lost circulation. The other is normal direction, which is perpendicular to the upper and lower fracture surface, and it is used to model filtration of drilling fluid. The tangential flow is assumed as Newtonian fluid. The constitutive relation is expressed as follow (Zimmerman et al., 1991):
qd ¼
(6)
qt ¼ ct ðpi pt Þ qb ¼ cb ðpi pb Þ
3.2. Fracture damage mode
Gs þ Gt Gn þ Gs þ Gt
4. Wellbore strengthening model 4.1. Analysis steps
η ¼ GC
The simulation is divided into three analysis steps. The first step is insitu stress equilibrium. The internal stress field and gravity are applied to model by using geostatic step of ABAQUS which can obtain unstrained model under initial stress field. This step will brings the displacement of model up to a much higher precision. The second step is to simulate lost circulation. The lost circulation will be simulated by injecting fluid into cohesive element. The initiation and propagation of fracture will be simulated by actual stress state of rock. The third step is to simulate wellbore strengthening. The research shows that the best place to bridge the fracture is the fracture mouth near wellbore wall (Feng and Gray, 2016). Through defining velocity boundary condition in fracture mouth near wellbore wall, the sealing progress will be simulated to accomplish the goal of wellbore strengthening.
(7)
where GCn , GCs and GCt represent the fracture energy of normal and two shear directions; Gn , Gs and Gt represent work of stresses from normal and two shear directions which are acted on displacement; η is a material parameter. 3.3. Fracture damage evolution The stiffness degradation criterion is used to simulate fracture element damage evolution. This criterion can be represented as (Camanho and Davila, 2002):
ð1 DÞt n ; tn ; ts ¼ ð1 DÞts tt ¼ ð1 DÞt t tn ¼
tn 0 tn < 0
4.2. Initial and boundary conditions
(8) Due to the boundary of model default to impermeable, the special boundary condition needs to be defined when seepage free surface encounters the free draining profile, so the initial pore pressure and porosity and saturation will be defined by using predefined field. As shown in the Fig. 5, in order to satisfy mechanical equilibrium equation of fluid-solid coupling model, it also needs to set up pore pressure boundary condition around the model. It is assumed that the geometry of model is centrosymmetric rectangle and the wellbore is cylindrical shape. According to symmetry principle, the actual stress state of wellbore and fracture can be analyzed by utilizing half of the model. To satisfy the assumed geometry of model, the profile of model is set up as symmetric boundary condition. The top node of cohesive element is defined as displacement boundary condition which can ensure the position of node
where t n , t s and t t are the stress components estimated by tractionseparation criterion before damage; tn , ts and tt are the actual stresses of normal and two shear directions; D is damage variable. It was not a simple linear damage evolution when used B-K criterion to predict fracture propagation. So, exponential damage evolution is used to describe damage variable. It is given by df
D ¼ ∫ dmmax m
Teff dd GC G0
(11)
where qt and qb are the filtration rates of the top and bottom surfaces, respectively; ct and cb are the leak-off coefficients of mud cake of the top and bottom surfaces, respectively; pi is mid surface pressure.
It is assumed that the propagation of fracture is a mixed-mode damage evolution progress which consists of normal and tangential direction propagation, and the B-K criterion (Benzeggagh and Kenane, 1996) is most widely used law to predict fracture propagation. The form is used when critical fracture energies are same during deformation purely along the first and the second shear directions. It is expressed as:
(10)
where q is volume flow rate density vector; d is the gap opening; μ is the fluid viscosity; ∇p is the pressure gradient along the cohesive element. The normal flow is used to simulate filtration of drilling fluid, it is defined as:
where tn0 is normal critical stress (rock tensile strength); ts0 and tt0 are critical shear stresses; The symbol 〈〉 represents a pure compressive deformation or stress state doesn't initiate damage.
GCn þ GCs GCn
d3 ∇p 12μ
(9)
f where dmax m is maximum displacement; dm is fracture displacement; Teff is
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Fig. 3. Mechanism of fracture propagation.
4.3. Loading conditions Fig. 6 shows the main stresses of model are overburden pressure in vertical direction, maximum and minimum principal stress in horizontal direction and drilling fluid column pressure perpendicular to wellbore direction. The injection pressure is set up in the middle of the wellbore to simulate lost circulation. It is generated by concentrated pore fluid. The injection pressure will drops to zero in sealing stage and it will decreases linearly from lost circulation to complete plugging. Table 1 shows the basic parameters of wellbore strengthening which come from previous studies (Onyia, 1994; Morita et al., 1996) and laboratory test of sandstone formation. Fig. 4. The fluid flow in cohesive element.
5. Simulation results will not move in X direction and it also ensures the displacement precision of fracture aperture. In order to avoid the movement of model in Y direction, the bottom of model will be set as displacement boundary condition which can ensure the application of overburden pressure.
5.1. The change of damage factors of cohesive element According to theory of damage mechanics, the damage factors
Fig. 5. The boundary conditions of model. 261
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Fig. 6. The loading conditions of model.
distribution of cohesive element are shown as Fig. 7. The fracture state is simulated by using the method of stiffness degradation from the fracture initiation to sealing process. Fig. 7 (a) shows the damage factors of whole cohesive element at the stage of initiation. As shown in Fig. 7 (a), the damage factors of whole cohesive element was 0 before the initiation of fracture. With the opening of fracture, the drilling fluid begins to leak into the formation. The damage factors around lost circulation point turn to 1. It shows that the damage is generated around the initiation point and the stiffness of cohesive element is gradually degenerating under the pressure of drilling fluid column. When damage factors turned to 1, the cohesive element was completely cracked and the fracture was generated. Fig. 7 (b) shows the damage factors of whole cohesive element at the stage of propagation. As shown in Fig. 7 (b), the fracture was gradually extended to formation after lost circulation happened. The damage factors of red color region is 0, it shows that the cohesive element is completely deboned and the damage factors of blue color region is 1, it shows that the fracture doesn't generate at this region. The narrow transition zone between red and blue region represents the damage condition of fracture tip which means the formation rock still have some tensile strength and the phenomenon of complete cracking doesn't appear. Fig. 7 (c) shows the damage factors of whole cohesive element at the stage of complete sealing. As shown in Fig. 7 (c), although the fracture behind bridging location tended to be closed after sealing, the fracture surface was still at the red region, which represented the fracture would propagate again under the pressure of drilling fluid column after the plugging particle failed.
Table 1 Basic parameters of 3D model. Parameter
Values
Units
Model dimension (length width height) Borehole diameter/D Young's modulus/E Poisson ratio/υ Formation porosity/φ Overburden pressure/συ Minimum horizontal principal stress/Sh Maximum horizontal principal stress/SH Drilling fluid column pressure/Pw Formation pore pressure/Po Permeability/k Specific gravity of drilling fluid/γ Lost circulation rate/v Filtration coefficient of mud Cake/C
0.10795 0.10795 0.10795
mmm
0.010795 12 0.25 0.47 20.70 10.10
m GPa – – MPa MPa
15.15
MPa
12.56
MPa
6.00 0.3, 3, 30, 300 15 680
MPa 103μm2 N/m3
1, 2, 3, 4, 5 5.879 1010, 5.879 109, 5.879 108, 5.879 107
105 m3/s m3/(KPa⋅s)
5.2. The effect of lost circulation rate It is assumed that the initial lost circulation rate is 1 105 m3/s. The hoop stress of wellbore, stress of fracture and geometry of fracture will be researched under initial lost circulation rate of 1–5 times. As illustrated in Fig. 8, the analysis plane will be defined as horizontal plane which contains lost circulation point. According to symmetric principle, half of this plane will be selected as object. Because the stress and displacement of this plane are the critical value of whole model. 5.2.1. Hoop stress of wellbore Fig. 9 shows the hoop stress of wellbore at different lost circulation rate before and after sealing stage. Fig. 9 (a) shows that the hoop stress of wellbore is about equal at lost circulation location (0 point) before sealing. With the increasing of lost circulation rate, the hoop compressive stress of wellbore is gradually increasing in vertical to fracture direction (90 point). The large stress difference between two points implies the fracture is easy to open. Fig. 9 (b) shows that the hoop compressive stress of wellbore increases at fracture location (0 point) after bridging which represents the bearing capacity is improved by sealing, but the increment of compressive stress flatten out after the location of 20 . With the increasing of lost circulation rate, the hoop compressive stress of wellbore is significantly greater than the stress before bridging. When the lost
Fig. 7. Damage factors distribution of cohesive element. 262
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after bridging stage. It is shown in Fig. 10 (a), the stress of fracture mouth is less than stress of fracture tip which means fracture will propagates by using stress of fracture tip before sealing stage. With the increasing of lost circulation rate, the stress of fracture tip will increases along with it. That is to say, the lost circulation rate can provide extended stress for fracture tip. It is shown in Fig. 10 (b), due to the sealing of plugging material, the stress of fracture mouth is increased significantly. The stress of fracture behind the bridging location is reduced, because there is no continuous fluid across the plugging material to keep compressive stress. With the increasing of lost circulation rate, the compressive stress behind the bridging location decreases firstly and then increases. It shows that the fracture propagation will be limited under a higher lost circulation rate.
5.2.2. Hoop stress of fracture Fig. 10 shows the distribution of hoop stress of fracture before and
5.2.3. Fracture geometry Fig. 11 shows the fracture geometry of different lost circulation rates before and after sealing. As shown in Fig. 11 (a), the displacement of fracture surface represents half of fracture width. It shows a linearly increasing trend with the increasing of lost circulation rate. The increasing of lost circulation rate enhances the effective stress of fracture surface which results in the widening of fracture. As shown in Fig. 11 (b), the width of fracture behind bridging location reduces rapidly after sealing, and fracture mouth has little change in width, because the velocity boundary condition is set at fracture to simulate sealing process. The plugging material is assumed as rigid material, thus the geometry of fracture mouth doesn't change so much.
Fig. 9. Circumferential stress distribution of wellbore at different leakage rate.
Fig. 10. Circumferential stress distribution of crack at different leakage rate.
Fig. 8. The selection of analysis plane.
circulation rate speeds up, the total stress acting on wellbore is increased and so is the effective stress of rock matrixes. Thus, the hoop stress of wellbore is enhanced after sealing.
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Fig. 11. Fracture geometry at different leakage rate. Fig. 12. Circumferential stress distribution of wellbore at different permeability.
5.3.2. Hoop stress of fracture Fig. 13 illustrates the hoop stress distribution of crack at different permeability. As shown in Fig. 13 (a) the compressive stress reaches maximum near the fracture mouth. The variation range of compressive stress is wide near the fracture mouth. The results show that the stress to support fracture initiation is higher than the stress of keeping fracture propagation. With the increasing of formation permeability, the compressive stress will increases firstly and then decreases, but compressive stress of fracture tip is obviously higher in low permeability formations. Fig. 13 (b) shows that the compressive stress of fracture tip is decreased after sealing. This indicates that the fracture tip is gradually closed under the action of sealing. Due to the action of sealing material, there is no continuous filtration of drilling in fracture, while the compressive stress of fracture tip is also higher in low permeability formations.
5.3. The effect of permeability In order to investigate the effect of formation permeability on wellbore strengthening result, permeability of 0.3–300 mD is used to observe the stress distribution of wellbore and fracture and the geometry of crack before and after sealing stage. 5.3.1. Hoop stress of wellbore Fig. 12 shows the distribution of wellbore hoop stress at different permeability before and after sealing. Fig. 12 (a) illustrates that the fracture is opened by tension stress of perpendicular to fracture plane. With the increasing of formation permeability, the compressive stress becomes larger at lost circulation location and the value changes little when permeability is low. These results demonstrate that the opening tendency of fracture is enhanced at high permeability condition. While the compressive stress will increases firstly and then decreases at 90 point. Fig. 12 (b) shows that with the increasing of formation permeability, the compressive stress of fracture mouth is gradually enhanced. This means wellbore bearing capacity is significantly enhanced in high permeability formations, the wellbore strengthening techniques are more likely to succeed in high permeability formations.
5.3.3. Fracture geometry Fig. 14 shows geometry of fracture at different permeability. As shown in Fig. 14 (a), with the increasing of formation permeability, the opening of crack is decreased. It is generally considered, drilling fluid will loses a quantity of energy while filtrating in a high permeability formation, which is unfavorable to propagation. According to principle of
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Fig. 13. Circumferential stress distribution of crack at different permeability.
Fig. 14. Fracture geometry at different permeability.
effective stress (Terzaghi et al., 1995), drilling fluid pressure can balance rapidly with pore pressure in high permeability formations to increase the pore pressure of formation, and it effectively reduces the effective stress of rock matrix and aperture of fracture. Fig. 14 (b) illustrates that the displacement of fracture surface behind the bridging location is significantly decreased after sealing in high permeability formations. With the increasing of formation permeability, the decreasing amplitude of fracture width has slightly different in high permeability formations.
of formation changes little with a lower filtration coefficient, thus, the effective stress of wellbore will increases. Fig. 15 (b) shows that the wellbore compressive stress increases significantly after sealing. The wellbore stress increases rapidly at 20 , which indicates the distribution of wellbore hoop stress is changed by bridging and the risk point of wellbore is transferred. The results of sealing are conducive to wellbore stability. With the increasing of filtration coefficient, the hoop stress of wellbore is gradually decreased after sealing.
5.4. The effect of filtration coefficient
5.4.2. Hoop stress of fracture Fig. 16 shows the hoop stress distribution of crack at different filtration coefficient. As shown in Fig. 16 (a), the compressive stress of fracture mouth is higher, the stress is gradually decreased from fracture mouth to fracture. With the increasing of filtration coefficient, the compressive stress of fracture is decreased, but the amplitude of it is reduced when the filtration coefficient ups to a certain value. Fig. 16 (b) shows that the stress changes rapidly in the fracture mouth as a result of sealing in the fracture mouth. The result indicates that the fracture before bridging location can bear higher compressive, and filtration coefficient has little effect on the fracture surface of behind bridging location.
It is assumed that drilling fluid can generate mud cake on fracture surface and the permeability of mud cake is homogeneous after lost circulation. Thus the fluid loss performance of mud cake will be simulated by changing filtration coefficient of cohesive element. 5.4.1. Hoop stress of wellbore Fig. 15 shows the distribution of wellbore hoop stress with different filtration coefficient. As shown in Fig. 15 (a), with the increasing of filtration coefficient, the compressive stress of wellbore is gradually decreased before sealing. When the value of filtration coefficient grows to a certain value, the effect of filtration coefficient will becomes small. The fracture resembles to a relief valve in lost circulation process. This is similar with the mechanism of formation permeability, the pore pressure
5.4.3. Fracture geometry Fig. 17 shows the fracture geometry at different filtration coefficient.
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Fig. 16. Circumferential stress distribution of crack with different filtration coefficient. Fig. 15. Circumferential stress distribution of wellbore at different filtration coefficient.
sealing the fracture, the leak off test is performed by using designer drilling fluid. The breaking pressure reaches 14 MPa in subsequent tests. It means the bearing capacity of wellbore is enhanced by wellbore strengthening treatment. While, it's reported from Aston et al. (2004), the leak off test is failed to reach the bearing capacity of base drilling fluid due to the low permeability, and it is not curved in Fig. 18. Fig. 19 shows the field results of wellbore strengthening in low permeability formations from Dupriest (2005). As shown in Fig. 19, the success rate of wellbore strengthening treatment in high permeability formations in 2003 is almost 100%, and most of the squeeze pressure of high permeability formations is relatively low, according to the data of construction pressure in FCS (fracture closure stress) method. It indicates that the wellbore strengthening treatment is more difficult to perform in low permeability formations. Fig. 20 illustrates the hesitation squeeze treatment in low permeability formations from Dupriest (2005). As shown in Fig. 20, the fracture opening pressure is 2.06 MPa in initial squeeze. In order to enhance the bearing capacity of wellbore, about 3 m3 pills are squeezed into the formations. While, the pressure is changed very little. This is because the lost circulation rate is too low in this formations. It indicates that the LCM (lost circulation materials) can't be filled into the fracture in a short time, and the bearing capacity of wellbore has little change with less effective pills. Thus, the pressure is held for 2 h to fill pills adequately. As shown in Fig. 20, the pressure is increased so much in the later hesitations.
As shown in Fig. 17 (a), with the increasing of filtration coefficient, the displacement of fracture surface is decreased and the amplitude of it is reduced. Fig. 17 (b) shows that with the increasing of filtration coefficient, the geometry of fracture becomes narrow and short. The result indicates that the drilling fluid is difficult to permeate into formation with a lower filtration coefficient. It causes the increasing of fracture pressure and the instability of fracture pressure system. When the bearing capacity of mud cake is insufficient, the fracture will releases energy by propagation. When the filtration coefficient reduces to a certain value, the effect of mud cake is equal to the result of complete sealing, thus the filtration coefficient has little effect on propagation, and the aperture of fracture is roughly same. 5.5. Analysis of field results Fig. 18 shows the effect of wellbore strengthening by two leak off tests in a shale formation from Aston et al. (2004). There are two kinds of drilling fluids in leak off tests, the bridging particles are used in designer drilling fluid. As illustrated in Fig. 18, the breaking fracture achieves 8.27 MPa by using base drilling fluid. With the reducing of pumping pressure, the surface pressure is stabilized at about 6Mpa, and it represents the propagation pressure of fracture. That indicates the propagation pressure of fracture is lower than opening pressure of fracture. After 266
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Fig. 19. The worldwide results of wellbore strengthening in 2003 (reproduced from Dupriest, 2005).
Fig. 17. Fracture geometry at different filtration coefficient. Fig. 20. The field results of wellbore strengthening in low-permeability formations (reproduced from Dupriest, 2005).
6. Conclusions 1) During the propagation process, the damage region of cohesive element is narrow. Although the fracture tip is closed under the in-situ stress, cohesive element still keeps the state of complete fracture. The bearing tensile capacity of fracture tip is not well and the fracture will propagates again under the pressure of drilling fluid column after the failure of plugging particle. 2) With the increasing of lost circulation rate, more plugging material will be carried in unit time which can seal the fracture rapidly. The bearing pressure capacity of wellbore is also improved after bridging. Still, the high velocity of lost circulation also makes the fracture aperture become width. 3) According to principle of effective stress, drilling fluid pressure can balance dramatically with pore pressure in high permeability formation to increase the pore pressure of formation which effectively reduces the effective stress of rock matrix and aperture of fracture. 4) The drilling fluid is difficult to permeate into formation with a lower filtration coefficient which causes the wellbore strengthening and fracture propagating. When the filtration coefficient reduces to a certain value, the effect of mud cake is equal to the result of complete
Fig. 18. Extended leak off test before and after wellbore strengthening (reproduced from Aston et al., 2004). 267
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sealing, thus filtration coefficient has little effect on propagation, and the aperture of fracture is roughly same.
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Acknowledgments This work was supported by the National Key Basic Research Special Foundation of China (2015CB251205), Application and basic research Project of Qingdao (15-9-1-43-jch), and the Fundamental Research Funds for the Central Universities (16CX02023A). Thanks for their supports for this article. References Alberty, M.W., McLean, M.R., 2004. A Physical Model for Stress Cages. Society of Petroleum Engineers. http://dx.doi.org/10.2118/90493-MS. Alberty, M.W., McLean, M.R., 2001. Fracture Gradients in Depleted Reservoirs - Drilling Wells in Late Reservoir Life. Society of Petroleum Engineers. http://dx.doi.org/ 10.2118/67740-MS. Aston, M.S., Alberty, M.W., Duncum, S.D., Bruton, J.R., Friedheim, J.E., Sanders, M.W., 2007. A New Treatment for Wellbore Strengthening in Shale. Society of Petroleum Engineers. http://dx.doi.org/10.2118/110713-MS. Aston, M.S., Alberty, M.W., Mclean, M.R., et al., 2004. Drilling fluids for wellbore strengthening. http://dx.doi.org/10.2118/87130-MS. Benzeggagh, M.L., Kenane, M., 1996. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56 (4), 439–449. Camanho, P.P., Davila, C.G., 2002. Mixed-mode Decohesion Finite Elements for the Simulation of Delamination in Composite Materials, pp. 1–42. NASA/TM2002e211737. Cook, J., Growcock, F., Guo, Q., Hodder, M., Van Oort, E., 2011. Stabilizing the wellbore to prevent lost circulation. Oilfield Rev. 23 (4), 26–35. Dupriest, F.E., 2005. Fracture Closure Stress (FCS) and Lost Returns Practices. Society of Petroleum Engineers. http://dx.doi.org/10.2118/92192-MS. Feng, Y., Gray, K.E., 2016. A parametric study for wellbore strengthening. J. Nat. Gas Sci. Eng. 30, 350–363.
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Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol
Numerical simulation of 3D fracture propagation in wellbore strengthening conditions Jia Li, Zhengsong Qiu *, Dingding Song, Hanyi Zhong, Zhichuan Tang College of Petroleum Engineering, China University of Petroleum, Qingdao 266580, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Wellbore strengthening Numerical simulation Fluid-solid coupling Finite-element Lost circulation
The principle of damage mechanics was adopted in order to solve the lost circulation problem when drilled low bearing capacity formation. According to quadratic nominal stress criterion and B-K criterion, cohesive element with zero thickness was inserted into fracture to simulate fracture initiation and propagation. A 3D fluid-solid coupling finite element model was established to analyze circumferential stress distribution of wellbore, hoop stress distribution of crack and geometry of fracture. Results show that closed fracture surface is still in the stage of complete fracture under the action of crustal stress and crack tip reopens after plugging particle failed. The fracture is difficult to widen for filling more plugging materials in a short time while the lost circulation rate and permeability are low. The wellbore strengthening treatment will makes the fracture become short and narrow under a larger filtration coefficient, but mud cake quality has an insignificant influence on fracture geometry when filtration coefficient increases to a certain value. The simulation results were supported by providing some cases of real wellbore condition.
1. Introduction During the drilling process, drilling fluid can balance formation pressure to prevent well collapse or lost circulation. In the meantime, it may fractures the formation with excessive pressure in low pressure drain well, which can lead to lost circulation (Alberty and McLean, 2001). If taking the method of reducing density of drilling fluid to prevent loss of circulation, it may results in overflow or blowout risks and it can also generate underground complex accidents of wellbore instability in sloughing formation. Lost circulation signifies a serious problem which can cause significant economic losses and increase nonproductive time (Wang et al., 2005). It is shown that approximately 2–4 billion dollars were wasted annually on handling lost circulation from some recent engineering accidents (Cook et al., 2011). With the development of lost circulation protection and sealing technology, the method of improving ground bearing capacity and widening safe drilling fluid density window by using proper drilling fluid technology has become a hot spot in global petroleum industry. For this reason, a large number of studies have been performed over the past decades (Morita et al., 1990; Alberty and McLean, 2004; Dupriest, 2005; Van Oort et al., 2011). The technological core consists of physical, chemical or mechanical methods which can enhance rock strength or strengthen wellbore stress status to improve ground bearing capacity.
* Corresponding author. E-mail address: [email protected] (Z. Qiu). http://dx.doi.org/10.1016/j.petrol.2017.06.010 Received 12 March 2017; Received in revised form 26 May 2017; Accepted 2 June 2017 Available online 4 June 2017 0920-4105/© 2017 Elsevier B.V. All rights reserved.
According to the principle of the methods, three types are included as follows: The first one is using temporary shielding and plugging technology to seal leak channel (You et al., 2013). The second one is using chemical reinforcement to enhance rock strength (Aston et al., 2007). The third one is utilizing wellbore strengthening method to improve formation fracture pressure or fracture reopening pressure (Van Oort et al., 2011). More recently, wellbore strengthening becomes the most commonly used method to expand safe drilling fluid density window which reduces the occurrence of lost circulation effectively. In order to optimize the design of wellbore strengthening technology, a series of numerical researches were performed. Salehi and Nygaard (2010) built a 3D finiteelement model to simulate fracture opening and growth under different rock permeability. The results show that cracks are widened more in higher permeability. However, the influence of bridging is not considered for the fracture geometry. Feng and Gray (2016) used a poroelastic finite-element numerical model to investigate the effect of several parametrics, and obtained the best bridge location and other useful field implications, but the simulation couldn't provide dynamic change process of fracture behavior during wellbore strengthening operations. Furthermore, most of wellbore strengthening models are based on 2D finite-element and the influence of overburden pressure is neglected, and the length of pre-existing fracture is default value in
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previous studies (Alberty and McLean, 2004; Wang et al., 2009, 2007a, 2007b), it can't be determined by the actual stress state of rock. Based on theory of damage mechanics, a 3D finite-element method was established. Cohesive element with zero thickness was inserted into fracture to simulate fracture initiation and propagation, and fluid flow model of cohesive element was utilized to simulate filtration of drilling fluid in fracture. So, this model has a much higher accuracy than previous 2D model in estimate fracture and wellbore behavior and it has significance of theoretical guidance in wellbore strengthening operations. 2. Fluid-solid coupling equation Fig. 1 shows the principle of effective stress. The concept of effective stress was proposed by Terzaghi et al. (1995). It is assumed that the internal stress of fluid-saturated porous medium consists of pore pressure and effective pressure. The normal stress acts on periphery of pore is called pore pressure and the stress which is transferred by interface of rock particles is described as effective pressure. When external load of porous medium changes, the rock matrixes will be deformed with external force which can lead to the change of porosity and permeability. In the meantime, due to the deformation of rock matrixes, the flowing state and pressure of pore fluid also change along with it. According to effective stress principle, the mechanical equilibrium equation of poroelastic medium can be written as follow (Roe and Siegmund, 2003):
∫ V ðσ pw IÞδεdV ¼ ∫ S tδvdS þ ∫ V fδvdV þ ∫ V φρw gδvdV
Fig. 2. Diagram of cohesive element.
where J is volume change of porous medium; t is computation time; nw is void ratio; x is space vector. 3. Cohesive element model Fig. 2 shows the spatial representation of three-dimensional cohesive element. The cohesive element consists of 12 nodes and it can be divided into top face, mid surface and bottom face which is formed by 4 nodes respectively. In addition, the relative motion of the bottom and top faces along the thickness direction represent opening or closing of the fracture. The relative movement of the bottom and top faces along the direction of perpendicular to thickness represents shear behavior of the cohesive element. The membrane strains is generated from tensile and shear motion of mid surface in the cohesive element, but the cohesive elements do not generate any stresses in a purely membrane response.
(1)
where σ is effective stress matrix; pw is pore pressure; δε is virtual strain rate matrix; V is integral region; t is surface force matrix; δv is virtual rate matrix; S is surface of integral region; f is volume force matrix; ρw is density of pore fluid. The fluid flow in rock pore obeys Darcy's law, so that the continuity equation of fluid is expressed as follow (Yao et al., 2010):
1 ∂ ∂ ðJρw nw Þ ¼ ðρw nw vw Þ J ∂t ∂x
3.1. Fracture damage initiation Before the damage of cohesive element, the stress and strain of cohesive element are assumed to satisfy linear elastic response, it can be described as follow (SIMULIA, 2012):
(2)
8 9 8 < tn = < Knn t ¼ ts ¼ Kε ¼ Kns : ; : tt Knt
where vw is flow rate, which is calculated by using Eq. (3):
1 ∂pw ρw g vw ¼ k⋅ nw gρw ∂x
(3)
Fig. 1. The schematic of effective stress theory. 259
Kns Kss Kst
98 9 Knt =< εn = Kst ε ;: s ; Ktt εt
(4)
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effective traction; d is effective displacement; G0 is elastic energy at the stage of damage initiation. Fig. 3 shows that the cohesive element is gradually damaging from bonding state under the external loading. It finally achieved the breaking state and accomplished interface debond.
where K is element stiffness matrix; t is stress vector; tn , ts and tt represent the normal and two shear tractions; ε is strain vector; εn , εs and εt represent the normal and two shear strains. The ε can be expressed as follow:
εn ¼
dn ds dt ; εs ¼ ; εt ¼ To To To
(5)
3.4. Fracture fluid flow equation
where dn , ds and dt represent the separation of normal and two shear directions; To is the original thickness of the cohesive element. When utilized cohesive element to simulate fracture before the damage of element, the real thickness of it was 0. It will generates singularity when the strain is calculated by using geometry thickness. In order to eliminate singularity, the stress and strain will be described by using original thickness. The original thickness is assumed as 1. This paper utilizes a quadratic interaction function involving the nominal stress ratio to determine whether the fracture will be opened. When the sum of square of critical stress ratio is equal to 1, the damage is assumed to be complete. This criterion (Camanho and Davila, 2002) can be represented as
〈tn 〉 tn0
2
2 2 ts tt þ 0 þ 0 ¼1 ts tt
As shown in Fig. 4, the flow of cohesive element consists of two directions. One is tangential direction, which is parallel to propagation direction, and it is used to simulate flow of drilling fluid in fracture during lost circulation. The other is normal direction, which is perpendicular to the upper and lower fracture surface, and it is used to model filtration of drilling fluid. The tangential flow is assumed as Newtonian fluid. The constitutive relation is expressed as follow (Zimmerman et al., 1991):
qd ¼
(6)
qt ¼ ct ðpi pt Þ qb ¼ cb ðpi pb Þ
3.2. Fracture damage mode
Gs þ Gt Gn þ Gs þ Gt
4. Wellbore strengthening model 4.1. Analysis steps
η ¼ GC
The simulation is divided into three analysis steps. The first step is insitu stress equilibrium. The internal stress field and gravity are applied to model by using geostatic step of ABAQUS which can obtain unstrained model under initial stress field. This step will brings the displacement of model up to a much higher precision. The second step is to simulate lost circulation. The lost circulation will be simulated by injecting fluid into cohesive element. The initiation and propagation of fracture will be simulated by actual stress state of rock. The third step is to simulate wellbore strengthening. The research shows that the best place to bridge the fracture is the fracture mouth near wellbore wall (Feng and Gray, 2016). Through defining velocity boundary condition in fracture mouth near wellbore wall, the sealing progress will be simulated to accomplish the goal of wellbore strengthening.
(7)
where GCn , GCs and GCt represent the fracture energy of normal and two shear directions; Gn , Gs and Gt represent work of stresses from normal and two shear directions which are acted on displacement; η is a material parameter. 3.3. Fracture damage evolution The stiffness degradation criterion is used to simulate fracture element damage evolution. This criterion can be represented as (Camanho and Davila, 2002):
ð1 DÞt n ; tn ; ts ¼ ð1 DÞts tt ¼ ð1 DÞt t tn ¼
tn 0 tn < 0
4.2. Initial and boundary conditions
(8) Due to the boundary of model default to impermeable, the special boundary condition needs to be defined when seepage free surface encounters the free draining profile, so the initial pore pressure and porosity and saturation will be defined by using predefined field. As shown in the Fig. 5, in order to satisfy mechanical equilibrium equation of fluid-solid coupling model, it also needs to set up pore pressure boundary condition around the model. It is assumed that the geometry of model is centrosymmetric rectangle and the wellbore is cylindrical shape. According to symmetry principle, the actual stress state of wellbore and fracture can be analyzed by utilizing half of the model. To satisfy the assumed geometry of model, the profile of model is set up as symmetric boundary condition. The top node of cohesive element is defined as displacement boundary condition which can ensure the position of node
where t n , t s and t t are the stress components estimated by tractionseparation criterion before damage; tn , ts and tt are the actual stresses of normal and two shear directions; D is damage variable. It was not a simple linear damage evolution when used B-K criterion to predict fracture propagation. So, exponential damage evolution is used to describe damage variable. It is given by df
D ¼ ∫ dmmax m
Teff dd GC G0
(11)
where qt and qb are the filtration rates of the top and bottom surfaces, respectively; ct and cb are the leak-off coefficients of mud cake of the top and bottom surfaces, respectively; pi is mid surface pressure.
It is assumed that the propagation of fracture is a mixed-mode damage evolution progress which consists of normal and tangential direction propagation, and the B-K criterion (Benzeggagh and Kenane, 1996) is most widely used law to predict fracture propagation. The form is used when critical fracture energies are same during deformation purely along the first and the second shear directions. It is expressed as:
(10)
where q is volume flow rate density vector; d is the gap opening; μ is the fluid viscosity; ∇p is the pressure gradient along the cohesive element. The normal flow is used to simulate filtration of drilling fluid, it is defined as:
where tn0 is normal critical stress (rock tensile strength); ts0 and tt0 are critical shear stresses; The symbol 〈〉 represents a pure compressive deformation or stress state doesn't initiate damage.
GCn þ GCs GCn
d3 ∇p 12μ
(9)
f where dmax m is maximum displacement; dm is fracture displacement; Teff is
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Fig. 3. Mechanism of fracture propagation.
4.3. Loading conditions Fig. 6 shows the main stresses of model are overburden pressure in vertical direction, maximum and minimum principal stress in horizontal direction and drilling fluid column pressure perpendicular to wellbore direction. The injection pressure is set up in the middle of the wellbore to simulate lost circulation. It is generated by concentrated pore fluid. The injection pressure will drops to zero in sealing stage and it will decreases linearly from lost circulation to complete plugging. Table 1 shows the basic parameters of wellbore strengthening which come from previous studies (Onyia, 1994; Morita et al., 1996) and laboratory test of sandstone formation. Fig. 4. The fluid flow in cohesive element.
5. Simulation results will not move in X direction and it also ensures the displacement precision of fracture aperture. In order to avoid the movement of model in Y direction, the bottom of model will be set as displacement boundary condition which can ensure the application of overburden pressure.
5.1. The change of damage factors of cohesive element According to theory of damage mechanics, the damage factors
Fig. 5. The boundary conditions of model. 261
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Fig. 6. The loading conditions of model.
distribution of cohesive element are shown as Fig. 7. The fracture state is simulated by using the method of stiffness degradation from the fracture initiation to sealing process. Fig. 7 (a) shows the damage factors of whole cohesive element at the stage of initiation. As shown in Fig. 7 (a), the damage factors of whole cohesive element was 0 before the initiation of fracture. With the opening of fracture, the drilling fluid begins to leak into the formation. The damage factors around lost circulation point turn to 1. It shows that the damage is generated around the initiation point and the stiffness of cohesive element is gradually degenerating under the pressure of drilling fluid column. When damage factors turned to 1, the cohesive element was completely cracked and the fracture was generated. Fig. 7 (b) shows the damage factors of whole cohesive element at the stage of propagation. As shown in Fig. 7 (b), the fracture was gradually extended to formation after lost circulation happened. The damage factors of red color region is 0, it shows that the cohesive element is completely deboned and the damage factors of blue color region is 1, it shows that the fracture doesn't generate at this region. The narrow transition zone between red and blue region represents the damage condition of fracture tip which means the formation rock still have some tensile strength and the phenomenon of complete cracking doesn't appear. Fig. 7 (c) shows the damage factors of whole cohesive element at the stage of complete sealing. As shown in Fig. 7 (c), although the fracture behind bridging location tended to be closed after sealing, the fracture surface was still at the red region, which represented the fracture would propagate again under the pressure of drilling fluid column after the plugging particle failed.
Table 1 Basic parameters of 3D model. Parameter
Values
Units
Model dimension (length width height) Borehole diameter/D Young's modulus/E Poisson ratio/υ Formation porosity/φ Overburden pressure/συ Minimum horizontal principal stress/Sh Maximum horizontal principal stress/SH Drilling fluid column pressure/Pw Formation pore pressure/Po Permeability/k Specific gravity of drilling fluid/γ Lost circulation rate/v Filtration coefficient of mud Cake/C
0.10795 0.10795 0.10795
mmm
0.010795 12 0.25 0.47 20.70 10.10
m GPa – – MPa MPa
15.15
MPa
12.56
MPa
6.00 0.3, 3, 30, 300 15 680
MPa 103μm2 N/m3
1, 2, 3, 4, 5 5.879 1010, 5.879 109, 5.879 108, 5.879 107
105 m3/s m3/(KPa⋅s)
5.2. The effect of lost circulation rate It is assumed that the initial lost circulation rate is 1 105 m3/s. The hoop stress of wellbore, stress of fracture and geometry of fracture will be researched under initial lost circulation rate of 1–5 times. As illustrated in Fig. 8, the analysis plane will be defined as horizontal plane which contains lost circulation point. According to symmetric principle, half of this plane will be selected as object. Because the stress and displacement of this plane are the critical value of whole model. 5.2.1. Hoop stress of wellbore Fig. 9 shows the hoop stress of wellbore at different lost circulation rate before and after sealing stage. Fig. 9 (a) shows that the hoop stress of wellbore is about equal at lost circulation location (0 point) before sealing. With the increasing of lost circulation rate, the hoop compressive stress of wellbore is gradually increasing in vertical to fracture direction (90 point). The large stress difference between two points implies the fracture is easy to open. Fig. 9 (b) shows that the hoop compressive stress of wellbore increases at fracture location (0 point) after bridging which represents the bearing capacity is improved by sealing, but the increment of compressive stress flatten out after the location of 20 . With the increasing of lost circulation rate, the hoop compressive stress of wellbore is significantly greater than the stress before bridging. When the lost
Fig. 7. Damage factors distribution of cohesive element. 262
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after bridging stage. It is shown in Fig. 10 (a), the stress of fracture mouth is less than stress of fracture tip which means fracture will propagates by using stress of fracture tip before sealing stage. With the increasing of lost circulation rate, the stress of fracture tip will increases along with it. That is to say, the lost circulation rate can provide extended stress for fracture tip. It is shown in Fig. 10 (b), due to the sealing of plugging material, the stress of fracture mouth is increased significantly. The stress of fracture behind the bridging location is reduced, because there is no continuous fluid across the plugging material to keep compressive stress. With the increasing of lost circulation rate, the compressive stress behind the bridging location decreases firstly and then increases. It shows that the fracture propagation will be limited under a higher lost circulation rate.
5.2.2. Hoop stress of fracture Fig. 10 shows the distribution of hoop stress of fracture before and
5.2.3. Fracture geometry Fig. 11 shows the fracture geometry of different lost circulation rates before and after sealing. As shown in Fig. 11 (a), the displacement of fracture surface represents half of fracture width. It shows a linearly increasing trend with the increasing of lost circulation rate. The increasing of lost circulation rate enhances the effective stress of fracture surface which results in the widening of fracture. As shown in Fig. 11 (b), the width of fracture behind bridging location reduces rapidly after sealing, and fracture mouth has little change in width, because the velocity boundary condition is set at fracture to simulate sealing process. The plugging material is assumed as rigid material, thus the geometry of fracture mouth doesn't change so much.
Fig. 9. Circumferential stress distribution of wellbore at different leakage rate.
Fig. 10. Circumferential stress distribution of crack at different leakage rate.
Fig. 8. The selection of analysis plane.
circulation rate speeds up, the total stress acting on wellbore is increased and so is the effective stress of rock matrixes. Thus, the hoop stress of wellbore is enhanced after sealing.
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Fig. 11. Fracture geometry at different leakage rate. Fig. 12. Circumferential stress distribution of wellbore at different permeability.
5.3.2. Hoop stress of fracture Fig. 13 illustrates the hoop stress distribution of crack at different permeability. As shown in Fig. 13 (a) the compressive stress reaches maximum near the fracture mouth. The variation range of compressive stress is wide near the fracture mouth. The results show that the stress to support fracture initiation is higher than the stress of keeping fracture propagation. With the increasing of formation permeability, the compressive stress will increases firstly and then decreases, but compressive stress of fracture tip is obviously higher in low permeability formations. Fig. 13 (b) shows that the compressive stress of fracture tip is decreased after sealing. This indicates that the fracture tip is gradually closed under the action of sealing. Due to the action of sealing material, there is no continuous filtration of drilling in fracture, while the compressive stress of fracture tip is also higher in low permeability formations.
5.3. The effect of permeability In order to investigate the effect of formation permeability on wellbore strengthening result, permeability of 0.3–300 mD is used to observe the stress distribution of wellbore and fracture and the geometry of crack before and after sealing stage. 5.3.1. Hoop stress of wellbore Fig. 12 shows the distribution of wellbore hoop stress at different permeability before and after sealing. Fig. 12 (a) illustrates that the fracture is opened by tension stress of perpendicular to fracture plane. With the increasing of formation permeability, the compressive stress becomes larger at lost circulation location and the value changes little when permeability is low. These results demonstrate that the opening tendency of fracture is enhanced at high permeability condition. While the compressive stress will increases firstly and then decreases at 90 point. Fig. 12 (b) shows that with the increasing of formation permeability, the compressive stress of fracture mouth is gradually enhanced. This means wellbore bearing capacity is significantly enhanced in high permeability formations, the wellbore strengthening techniques are more likely to succeed in high permeability formations.
5.3.3. Fracture geometry Fig. 14 shows geometry of fracture at different permeability. As shown in Fig. 14 (a), with the increasing of formation permeability, the opening of crack is decreased. It is generally considered, drilling fluid will loses a quantity of energy while filtrating in a high permeability formation, which is unfavorable to propagation. According to principle of
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Fig. 13. Circumferential stress distribution of crack at different permeability.
Fig. 14. Fracture geometry at different permeability.
effective stress (Terzaghi et al., 1995), drilling fluid pressure can balance rapidly with pore pressure in high permeability formations to increase the pore pressure of formation, and it effectively reduces the effective stress of rock matrix and aperture of fracture. Fig. 14 (b) illustrates that the displacement of fracture surface behind the bridging location is significantly decreased after sealing in high permeability formations. With the increasing of formation permeability, the decreasing amplitude of fracture width has slightly different in high permeability formations.
of formation changes little with a lower filtration coefficient, thus, the effective stress of wellbore will increases. Fig. 15 (b) shows that the wellbore compressive stress increases significantly after sealing. The wellbore stress increases rapidly at 20 , which indicates the distribution of wellbore hoop stress is changed by bridging and the risk point of wellbore is transferred. The results of sealing are conducive to wellbore stability. With the increasing of filtration coefficient, the hoop stress of wellbore is gradually decreased after sealing.
5.4. The effect of filtration coefficient
5.4.2. Hoop stress of fracture Fig. 16 shows the hoop stress distribution of crack at different filtration coefficient. As shown in Fig. 16 (a), the compressive stress of fracture mouth is higher, the stress is gradually decreased from fracture mouth to fracture. With the increasing of filtration coefficient, the compressive stress of fracture is decreased, but the amplitude of it is reduced when the filtration coefficient ups to a certain value. Fig. 16 (b) shows that the stress changes rapidly in the fracture mouth as a result of sealing in the fracture mouth. The result indicates that the fracture before bridging location can bear higher compressive, and filtration coefficient has little effect on the fracture surface of behind bridging location.
It is assumed that drilling fluid can generate mud cake on fracture surface and the permeability of mud cake is homogeneous after lost circulation. Thus the fluid loss performance of mud cake will be simulated by changing filtration coefficient of cohesive element. 5.4.1. Hoop stress of wellbore Fig. 15 shows the distribution of wellbore hoop stress with different filtration coefficient. As shown in Fig. 15 (a), with the increasing of filtration coefficient, the compressive stress of wellbore is gradually decreased before sealing. When the value of filtration coefficient grows to a certain value, the effect of filtration coefficient will becomes small. The fracture resembles to a relief valve in lost circulation process. This is similar with the mechanism of formation permeability, the pore pressure
5.4.3. Fracture geometry Fig. 17 shows the fracture geometry at different filtration coefficient.
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Fig. 16. Circumferential stress distribution of crack with different filtration coefficient. Fig. 15. Circumferential stress distribution of wellbore at different filtration coefficient.
sealing the fracture, the leak off test is performed by using designer drilling fluid. The breaking pressure reaches 14 MPa in subsequent tests. It means the bearing capacity of wellbore is enhanced by wellbore strengthening treatment. While, it's reported from Aston et al. (2004), the leak off test is failed to reach the bearing capacity of base drilling fluid due to the low permeability, and it is not curved in Fig. 18. Fig. 19 shows the field results of wellbore strengthening in low permeability formations from Dupriest (2005). As shown in Fig. 19, the success rate of wellbore strengthening treatment in high permeability formations in 2003 is almost 100%, and most of the squeeze pressure of high permeability formations is relatively low, according to the data of construction pressure in FCS (fracture closure stress) method. It indicates that the wellbore strengthening treatment is more difficult to perform in low permeability formations. Fig. 20 illustrates the hesitation squeeze treatment in low permeability formations from Dupriest (2005). As shown in Fig. 20, the fracture opening pressure is 2.06 MPa in initial squeeze. In order to enhance the bearing capacity of wellbore, about 3 m3 pills are squeezed into the formations. While, the pressure is changed very little. This is because the lost circulation rate is too low in this formations. It indicates that the LCM (lost circulation materials) can't be filled into the fracture in a short time, and the bearing capacity of wellbore has little change with less effective pills. Thus, the pressure is held for 2 h to fill pills adequately. As shown in Fig. 20, the pressure is increased so much in the later hesitations.
As shown in Fig. 17 (a), with the increasing of filtration coefficient, the displacement of fracture surface is decreased and the amplitude of it is reduced. Fig. 17 (b) shows that with the increasing of filtration coefficient, the geometry of fracture becomes narrow and short. The result indicates that the drilling fluid is difficult to permeate into formation with a lower filtration coefficient. It causes the increasing of fracture pressure and the instability of fracture pressure system. When the bearing capacity of mud cake is insufficient, the fracture will releases energy by propagation. When the filtration coefficient reduces to a certain value, the effect of mud cake is equal to the result of complete sealing, thus the filtration coefficient has little effect on propagation, and the aperture of fracture is roughly same. 5.5. Analysis of field results Fig. 18 shows the effect of wellbore strengthening by two leak off tests in a shale formation from Aston et al. (2004). There are two kinds of drilling fluids in leak off tests, the bridging particles are used in designer drilling fluid. As illustrated in Fig. 18, the breaking fracture achieves 8.27 MPa by using base drilling fluid. With the reducing of pumping pressure, the surface pressure is stabilized at about 6Mpa, and it represents the propagation pressure of fracture. That indicates the propagation pressure of fracture is lower than opening pressure of fracture. After 266
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Fig. 19. The worldwide results of wellbore strengthening in 2003 (reproduced from Dupriest, 2005).
Fig. 17. Fracture geometry at different filtration coefficient. Fig. 20. The field results of wellbore strengthening in low-permeability formations (reproduced from Dupriest, 2005).
6. Conclusions 1) During the propagation process, the damage region of cohesive element is narrow. Although the fracture tip is closed under the in-situ stress, cohesive element still keeps the state of complete fracture. The bearing tensile capacity of fracture tip is not well and the fracture will propagates again under the pressure of drilling fluid column after the failure of plugging particle. 2) With the increasing of lost circulation rate, more plugging material will be carried in unit time which can seal the fracture rapidly. The bearing pressure capacity of wellbore is also improved after bridging. Still, the high velocity of lost circulation also makes the fracture aperture become width. 3) According to principle of effective stress, drilling fluid pressure can balance dramatically with pore pressure in high permeability formation to increase the pore pressure of formation which effectively reduces the effective stress of rock matrix and aperture of fracture. 4) The drilling fluid is difficult to permeate into formation with a lower filtration coefficient which causes the wellbore strengthening and fracture propagating. When the filtration coefficient reduces to a certain value, the effect of mud cake is equal to the result of complete
Fig. 18. Extended leak off test before and after wellbore strengthening (reproduced from Aston et al., 2004). 267
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sealing, thus filtration coefficient has little effect on propagation, and the aperture of fracture is roughly same.
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Acknowledgments This work was supported by the National Key Basic Research Special Foundation of China (2015CB251205), Application and basic research Project of Qingdao (15-9-1-43-jch), and the Fundamental Research Funds for the Central Universities (16CX02023A). Thanks for their supports for this article. References Alberty, M.W., McLean, M.R., 2004. A Physical Model for Stress Cages. Society of Petroleum Engineers. http://dx.doi.org/10.2118/90493-MS. Alberty, M.W., McLean, M.R., 2001. Fracture Gradients in Depleted Reservoirs - Drilling Wells in Late Reservoir Life. Society of Petroleum Engineers. http://dx.doi.org/ 10.2118/67740-MS. Aston, M.S., Alberty, M.W., Duncum, S.D., Bruton, J.R., Friedheim, J.E., Sanders, M.W., 2007. A New Treatment for Wellbore Strengthening in Shale. Society of Petroleum Engineers. http://dx.doi.org/10.2118/110713-MS. Aston, M.S., Alberty, M.W., Mclean, M.R., et al., 2004. Drilling fluids for wellbore strengthening. http://dx.doi.org/10.2118/87130-MS. Benzeggagh, M.L., Kenane, M., 1996. Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Compos. Sci. Technol. 56 (4), 439–449. Camanho, P.P., Davila, C.G., 2002. Mixed-mode Decohesion Finite Elements for the Simulation of Delamination in Composite Materials, pp. 1–42. NASA/TM2002e211737. Cook, J., Growcock, F., Guo, Q., Hodder, M., Van Oort, E., 2011. Stabilizing the wellbore to prevent lost circulation. Oilfield Rev. 23 (4), 26–35. Dupriest, F.E., 2005. Fracture Closure Stress (FCS) and Lost Returns Practices. Society of Petroleum Engineers. http://dx.doi.org/10.2118/92192-MS. Feng, Y., Gray, K.E., 2016. A parametric study for wellbore strengthening. J. Nat. Gas Sci. Eng. 30, 350–363.
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