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Extending behavior of hydraulic fracture when reaching formation interface
Journal of Petroleum Science and Engineering 74 (2010) 26–30
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l
Extending behavior of hydraulic fracture when reaching formation interface Haifeng Zhao ⁎, Mian Chen MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China
a r t i c l e
i n f o
Article history: Received 2 April 2009 Accepted 9 August 2010 Keywords: rock fracture mechanics hydraulic fracturing earth stress fracture toughness formation interface
a b s t r a c t Stopping extension, extending along formation interface and directly penetrating into bounding layer are three possible reactions of fracture extension when hydraulic fractures reach formation interfaces. The three types of extending behavior are analyzed using rock fracture mechanics and three respective judging criterions are presented. Layered earth stress, layered rock mechanics parameters, formation interface effect, reservoir thickness and operating parameters are taken into consideration. This study indicates that there exists a critical fracture length when fracture extension stops. Hydraulic fractures will extend along formation interfaces or penetrate into bounding layers when fracture lengths are larger than the critical fracture length. The method to calculate critical fracture length is introduced. In the case of direct penetration into the bounding layer, the method to estimate hydraulic fracture height is presented based on superposition of stress intensity factors. A case study is also included to show the operability of our theory. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Hydraulic fracture treatment to enhance recovery rate in lowpermeability reservoirs has become standard practice in Petroleum Engineering. A pair of fractures usually grows outward in a vertical plane and propagates above and below the packers as well as laterally away from the wellbore. The vertical extending limit of a fracture (i.e. fracture height) is an important issue in hydraulic fracturing design. To some degree, fracture height affects fracturing efficiency and plays a key role in determining the success of hydraulic fracturing (Biot et al., 1983; Chen and Chen, 1994; Chen et al., 1997; Hudson et al., 2003; Adachi and Detournay, 2008). Therefore before fracturing operation, it is essential to know whether the fractures will break into bounding layers and predict the vertical extending limit of the fractures in order to establish operational parameter. It is easy to predict fracture height if the formation is isotropic and earth stress is uniform. In this case the fracture is circular and fracture height equals fracture length. We can use delivery capacity, filter loss and fracture volume to predict fracture shape (Yi and Zhu, 2005). However, in most cases hydraulic fracturing is conducted where the mechanical properties of the reservoir rock are different from the bounding layers and the earth stress is not uniform. This requires an improved understanding of the extending behavior of hydraulic fractures reaching formation interfaces. The extending behavior of hydraulic fractures reaching formation interfaces is controlled by formation condition and work condition, and little quantitative research has been done in this field. The factors affecting the fracture extending behavior when reaching formation
⁎ Corresponding author. Tel.: +86 1089732165. E-mail address: [email protected] (H. Zhao). 0920-4105/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2010.08.003
interface include: layered difference of earth stress, elastic parameters and fracture toughness, interfacial strength between pay formation and bounding layers, pressure distribution of fracturing fluid, rheological property and viscosity of fracturing fluid (Warpinski et al., 1999; Fan, 2003; Hossain and Rahman, 2008). Moreover, anisotropy of formation, natural fractures, filtration characteristics of fracturing fluid and injection rate also affect the fracture extending behavior. Biot et al. (1983) assumed that hydraulic fracture is under plane strain condition along fracture width direction, the fracture width profile is sinusoidal and the formation interface is well consolidated. Without considering earth stress difference and filtration, the elastic equations are simplified to Laplace equation and a simple criterion is introduced p2 = p1
sffiffiffiffiffiffiffiffiffiffi S2 G2 S1 G1
ð1Þ
where p1 and p2 are extending pressure in pay formation and bounding layer respectively, G1 and G2 are shear modulus of pay formation and bounding layer respectively, and S1 and S2 are interface energy density of pay formation and bounding layer respectively. If S2G2 b S1G1, the fracture will propagate into bounding layer when reaching the formation interface. Otherwise, the fracture will stop extending. Biot's theory does not consider earth stress difference and interfacial effect. In practice hydraulic fractures break through formation interfaces and enter bounding layers even if G2 is 20 times larger than G1 (Warpinski et al., 1999), which indicates that the fracture behavior is not completely controlled by shear modulus difference. When burial depth is large and strata dip is small, overburden rock applies large normal stress on formation interface
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
and hydraulic fracture cannot extend along formation interface. But when strata dip is large or burial depth is small, we must consider interfacial effect. Chen et al. (1997) assumed the hydraulic fractures will certainly enter bounding layers more or less when reaching formation interfaces. They built a numerical model using elastic theory of rock mechanics to simulate vertical extension of fracture in layered media. With the numerical model, they studied the effect of earth stress profile, fracture toughness difference and fracture fluid density on fracture height. Another researcher Chen and Chen (1994) used complex variable method to calculate stress intensity factor of fracture in layered media. They resolved stress intensity factor into singular integral equations and used Lobatto–Chebyshev orthogonal polynomials to get numerical results. In this paper, the authors use rock fracture mechanics to describe the reactions of fracture growth when hydraulic fracture reaches formation interface. A homologous judgment criterion is introduced. In the model we take into account the effect of layered earth stress, layered rock mechanics parameters, formation interface, reservoir thickness and work parameters. In the case of fracturing through bounding layers, the method to calculate fracture height in bounding layers is presented.
Also by stress analysis of pay formation we get the normal stress and shear stress on interface I: 8 < σ = σ cos2 θ + σ sin2 θ 1 21 1 23 1 : τ1 = ðσ 21 + σ 23 Þ sinθ1 cosθ1
ð3Þ
Therefore the layered earth stress σij should meet the following condition: 8 2 > > 2 < σ −σ i1 i + 1;1 cos θi = σ i + 1;3 −σ i3 sin θi > > : ðσ + σ Þ sinθ cosθ = σ i1 i3 i i i + 1;1 + σ i + 1;3 sinθi cosθi
ð4Þ
where i = 1, 2. 2.2. Fundamental assumptions In order to appropriately simplify the calculation, we have the following assumptions.
2. Layered earth stress and fundamental assumptions 2.1. Layered earth stress Extension of hydraulic fracture in layered formations is shown in Fig. 1. In Fig. 1, σ11 is the overburden pressure of upper bounding layer, MPa; σ13 is minimal horizontal earth stress, MPa; the index i of σij indicates upper bounding layer (when i = 1), pay formation (when i = 2) or lower bounding layer (when i = 3), j indicates minimal horizontal earth stress (when j = 3) or overburden pressure (when j = 1); θ1 and θ2 are strata dips of upper and lower bounding layers, deg and h1(h2) is distance between fracture center and upper (lower) bounding layer, m. For the upper bounding layer, by stress analysis we get the normal stress and shear stress on interface I (interface between upper bounding layer and pay formation): 8 < σ = σ cos2 θ + σ sin2 θ 1 11 1 13 1 : τ1 = ðσ 11 + σ 13 Þ sinθ1 cosθ1
27
ð2Þ
(1) Rock fracturing can be characterized by linear elastic fracture mechanics. This assumption indicates rock is considered as elastic material and fracture propagation criterion is stress intensity factor equals fracture toughness. For most sandstone reservoirs, this assumption is sufficiently precise. If the reservoir is plastic material, such as shale gas, this assumption may be invalid. (2) Hydraulic fracture shape is pseudo 3-dimensional, it is elliptic along the plane normal to minimal horizontal earth stress. In homogeneous reservoir, hydraulic fracture shape is pseudo 3dimensional. Furthermore, it is difficult to get analytical solution for the stress intensity factor if the fracture shape is considered as true 3-dimensional. (3) The static and dynamic pressure drop of fracturing fluid along the direction of fracture height is ignored. In order to capture the fundamentals of the problem, we do not consider the effect of gravity and viscous force.
Fig. 1. Configuration of hydraulic fracture in layered formations.
28
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
3.2. Fracture extending along formation interface
3. Fracture propagation reaching formation interface and judgment criterion 3.1. Fracture stops extending When reaching formation interface, hydraulic fracture stops extending along the direction of height and only can extend along the direction of length. As the length and height of hydraulic fracture are much larger than width, the inside hydraulic pressure can be considered as along the direction of Z-axis. By using the method of complex function (Fan, 2003), we get the I-form stress intensity factor at arbitrary point on the frontal side of elliptic fracture, rffiffiffiffiffiffi 1 = 4 πh p−σ 23 2 2 2 2 K= h cos θ + l sin θ l EðkÞ
ð5Þ
where h is half fracture height, m; l is half fracture length, m; p is hydraulic pressure in the fracture, MPa; k2 = (l2 − h2) / l2, E is the second complete elliptic integral function and θ is angular variable of ellipse parametric equation. When hydraulic fracture reaches upper interface, l = h = h1. h1 can be considered as the distance between perforation center and interface I. In this case the hydraulic fracture extending condition on point D is as follows: rffiffiffiffiffi h1 KD = 2 ðp−σ 23 Þ = K2c π
ð6Þ
where K2c is fracture toughness of pay formation, MPa m1/2. From Eq. (6) we can get critical hydraulic pressure: p1 =
K2c 2
rffiffiffiffiffi π + σ 23 h1
ð7Þ
Fracture length increases with the proceeding of fracturing work, the fracture shape changes from circle to ellipse. The stress intensity factor on point D decreases under the same hydraulic pressure, as shown in Fig. 2. In Fig. 2, the dimensionless number 1 is the ratio (KD)l N h1 / (KD)l = h1, dimensionless number 2 is critical pressure (the minimal hydraulic pressure required to extend the fracture) ratio (p − σ23)l N h1 / (p1 − σ23) and fracture length n is also in dimensionless form (n = l / h1). In other words, the critical hydraulic pressure of fracture extending along length direction increases. It is possible that there exists a critical fracture length. When fracture length is larger than the critical fracture length, hydraulic fracture will break through formation interface or extend along formation interface.
Fig. 2. Variation of stress intensity factor and critical pressure vs. fracture length.
If the tensile strength of formation interface is weak and the normal stress on formation interface is small, which may occur in shallow formations or large dip angle formations, hydraulic fracture may extend along formation interface. Then the tension failure criterion should be adopted to determine fracture extension. The effective normal stress on interface I is as follows: 2
2
σ 1 = σ 11 cos θ1 + σ 13 sin θ1 −p
ð8Þ
Critical condition satisfies σ1 = −St1, where St1 is the tensile strength of interface I, MPa. The critical hydraulic pressure is 2
2
p2 = σ 11 cos θ1 + σ 13 sin θ1 + St1
ð9Þ
3.3. Fracturing through bounding layer If hydraulic fracture directly breaks through formation interface to bounding layer, the formation interface can be considered to be well consolidated. When hydraulic fracture meets the upper interface, l = h = h1, the hydraulic fracture extending condition on point A is rffiffiffiffiffi h1 ðp−σ 23 Þ = K1c π
KA = 2
ð10Þ
where KA is stress intensity factor on point A, MPa m1/2 and K1c is fracture toughness of the upper bounding layer, MPa m1/2. From Eq. (10) we can get the critical hydraulic pressure p3 =
K1c 2
rffiffiffiffiffi π + σ 23 h1
ð11Þ
Hydraulic fracture height in bounding layer is closely related to earth stress difference between pay formation and bounding layer (Brudy et al., 1997). Stress intensity factor on point A can be calculated by three section superpositions after point A enters the upper bounding layer, as shown in Fig. 3. The center of coordinates system y′z′ is fracture center. The relationship between coordinate system y′z′ and yz (which center is perforation center) is: ′
y = y + d= 2 z = z′
Fig. 3. Three sections for calculating stress intensity factor.
ð12Þ
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
29
where d is the fracture penetrating distance into the upper bounding layer, m; h is half fracture height and h = h1 + d/2, m. The contribution of I, II and III sections to the stress intensity factor on point A can be solved with analytic solution (Fan, 2003). By calculation and superimposition we get the stress intensity factor on point A KA =
rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h π −1 + arcsinð1−d = hÞ− 2d = h−ðd =hÞ2 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp−σ 13 Þ h π − arcsinð1−d = hÞ + 2d = h−ðd=hÞ2 + π−1 π 2 2ðp−σ 23 Þ π−1
ð13Þ Critical condition is KA = K1c
ð14Þ
From Eqs. (13) and (14), we can calculate penetrating distance d under hydraulic pressure p. The problem is more complicated if hydraulic fracture enters both upper and lower bounding layers, as shown in Fig. 4. The stress intensity factor should be calculated by four section superpositions. By calculation and superimposition, we can get the stress intensity factor on points A and B, KA =
rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h π − arcsinð1−d2 = hÞ− 2d2 = h−ðd2 =hÞ2 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d2 = hÞ + 2d2 = h−ðd2 = hÞ2 −1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d1 = hÞ− 2d1 = h−ðd1 =hÞ2 + 1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 13 Þ h π − arcsinð1−d1 = hÞ + 2d1 = h−ðd1 =hÞ2 + π−1 π 2 2ð p−σ 33 Þ π−1
ð15Þ rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp−σ 33 Þ h π − arcsinð1−d2 = hÞ + 2d2 = h−ðd2 =hÞ2 KB = π−1 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d2 = hÞ− 2d2 = h−ðd2 =hÞ2 + 1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d1 = hÞ + 2d1 = h−ðd1 =hÞ2 −1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 13 Þ h π − arcsinð1−d1 = hÞ− 2d1 = h−ðd1 =hÞ2 + π−1 π 2 ð16Þ Critical condition is KB = K3c KA = K1c
ð17Þ
where K3c is fracture toughness of the lower bounding layer (MPa m1/2), d1 and d2 are penetrating distances in upper and lower bounding layers respectively (m) and h is half fracture height, h = (d2 + d1 + h1 + h2)/2. From Eqs. (15)–(17), we can calculate the penetrating distances d1 and d2 under hydraulic pressure p. 3.4. Judging criterion (1) If min(p1, p2, p3) = p1, hydraulic fracture stops extending along height direction when reaching formation interface and only can extend along length direction. When fracture length reaches a critical value, the fracture will extend along formation interface or penetrate into bounding layers. From
Fig. 4. Four sections for calculating stress intensity factor.
Eqs. (7) and (11), the critical fracture length satisfies the following conditions, ðpc −σ 23 Þl = nh1 ðp3 −σ 23 Þ ðpc −σ 23 Þl = nh1 ðp1 −σ 23 Þ
2EðkÞ π pffiffiffi 2 nEðkÞ = π =
ð18Þ
That means when fracture length l1 =
p3 −σ 23 2 h1 p1 −σ 23
ð19Þ
The hydraulic fracture will enter bounding layer. From Eqs. (9) and (11), another critical fracture length satisfies the following condition: ðpc −σ 23 Þl = nh1 ðp2 −σ 23 Þ ðpc −σ 23 Þl = nh1 ðp1 −σ 23 Þ
=1 pffiffiffi 2 nEðkÞ = π
ð20Þ
That means when fracture length satisfies pffiffiffi 2 nEðkÞ p −σ 23 = 2 π p1 −σ 23
ð21Þ
the fracture will extend along formation interface. Eq. (21) can be solved in Fig. 2. (2) If min(p1, p2, p3) = p2, hydraulic fracture will extend along formation interface, which means the fracture will deflect along height direction. (3) If min(p1, p2, p3) = p3, hydraulic fracture directly breaks through formation interface to bounding layer. The penetrating distance is related to stress difference, fracture toughness difference and reservoir thickness. 4. Case study Dip angles θ1 = θ2 = 0°, form Eq. (4) we know that overburden pressure is continuous on interfaces I and II. Overburden pressure is 46.0 MPa on interface I and 46.7 MPa on interface II. σ23 = 30 MPa, σ13 − σ23 = 5 MPa, σ33 − σ23 = 7 MPa, distances from perforation to upper and lower interfaces are h1 = 3 m, h2 = 7 m, K2c = 1 MPa m1/2,
30
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
K1c = K3c = 2 MPa m1/2, and interface tensile strength is St1 = St2 = 0.5 MPa. The critical pressures are: p1 = 30.5 MPa, p2 = 46.5 MPa and p3 = 31 MPa. Thus by judging criterion the fracture will stop extending when reaching formation interface I. Form Eqs. (19) and (21), the critical half fracture length is l = 4 h1 = 12 m N h2. That means before fracture length reaches the critical value, the fracture has already met interface II. Before hydraulic pressure reaches 31 Mpa, the fracture has already extended into upper and lower bounding layers. From Eqs. (15)–(17) the relationship between penetrating depths d1 and d2 and hydraulic pressure can be calculated, as shown in Fig. 5. 5. Conclusions and discussions (1) We use rock fracture mechanics to analyze the three types of possible extending behavior when hydraulic fracture reaches formation interface: stopping extension, extending along formation interface and penetrating into bounding layers. The judgment criterion is established. (2) If extension along height direction stops when fracture reaches formation interface, the fracture will extend along length direction. The fracture will extend along formation interface or penetrate into bounding layers when fracture length reaches a critical value. Form Eqs. (19) and (21), we find that selection of perforation positions is closely related to the critical fracture length. Perforation positions should be optimized according to actual formations. Pay formation center is not always the most appropriate selection. (3) If the fracture breaks through the interface and penetrates into bounding layers, we present a method to calculate penetrating depth in bounding layers. The single-sided penetration and two-sided penetration should be treated differently. Form the calculated results we find that earth stress and hydraulic pressure are significant factors determining penetrating depth. The lower the earth stress in bounding layer is, the faster penetration happens. When hydraulic pressure reaches earth stress in bounding layer, penetration depth will sharply increase. (4) In this paper, we consider pay formation and bounding layers as isotropic and intact media. Anisotropy and natural damage (including natural fractures, pores and joints etc.) of formations are neglected. But the influence of these factors can be removed to a certain degree when we acquire rock mechanic para-
Fig. 5. Vertical penetrating distances d1 and d2 vs. hydraulic pressure in fracture.
meters. For example, the measurement of fracture toughness may have been conducted using cores with natural fractures.
References Adachi, J.I., Detournay, E., 2008. Plane strain propagation of a hydraulic fracture in a permeable rock. Eng. Fract. Mech. 75 (16), 4666–4694. Biot, M.A., Medlin, W.L., Masse, L., 1983. Fracture penetration through an interface [R]. SPE 10372, 857–869. Brudy, M., Zoback, M.D., Fuchs, K., Rummel, F., Baumgartner, J., 1997. Estimation of the complete stress tensor to 8 km depth in the KTB scientific drill holes: implications for crustal strength [J]. J. Geophys. Res. 102 (B8), 18453–18475. Chen, M., Chen, Z.X., 1994. Study on the Extension of Hydraulic Fracture in Payered Formations: the Fifth National Conference on Numerical Methods and Theoretical Methods in Rock and Soil Mechanics, Chongqi, 1994 [C]. Chongqi University Press, Chongqi. Chen, Z.X., Chen, M., Huang, R.Z., Shen, Z.H., 1997. Vertical growth of hydraulic fracture in payered formations [J]. J. China Univ. Petrol.: Ed. Nat. Sci. 21 (4), 23–26. Fan, T.Y., 2003. Fracture Theoretical Basis [M]. Science Press, Beijing, pp. 170–177. Hossain, M.M., Rahman, M.K., 2008. Numerical simulation of complex fracture growth during tight reservoir stimulation by hydraulic fracturing. J. Petrol. Sci. Eng. 60 (2), 86–104. Hudson, J.A., Cornet, F.H., Chistiansson, R., 2003. ISRM suggested method for rock stress estimation—part 1: strategy for rock stress estimation [J]. Int. J. Rock Mech. Min. Sci. 40 (7/8), 991–998. Warpinski, N.R., Schmidt, R.A., Northrop, D.A., 1999. Earth stresses: the predominant influence on hydraulic fracture containment [R]. SPE 8932, 653–664. Yi, S.M., Zhu, Z.D., 2005. Primary Damage Mechanics of Fractured Rock [M]. Science Press, Beijing, pp. 84–93.
Contents lists available at ScienceDirect
Journal of Petroleum Science and Engineering j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p e t r o l
Extending behavior of hydraulic fracture when reaching formation interface Haifeng Zhao ⁎, Mian Chen MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China
a r t i c l e
i n f o
Article history: Received 2 April 2009 Accepted 9 August 2010 Keywords: rock fracture mechanics hydraulic fracturing earth stress fracture toughness formation interface
a b s t r a c t Stopping extension, extending along formation interface and directly penetrating into bounding layer are three possible reactions of fracture extension when hydraulic fractures reach formation interfaces. The three types of extending behavior are analyzed using rock fracture mechanics and three respective judging criterions are presented. Layered earth stress, layered rock mechanics parameters, formation interface effect, reservoir thickness and operating parameters are taken into consideration. This study indicates that there exists a critical fracture length when fracture extension stops. Hydraulic fractures will extend along formation interfaces or penetrate into bounding layers when fracture lengths are larger than the critical fracture length. The method to calculate critical fracture length is introduced. In the case of direct penetration into the bounding layer, the method to estimate hydraulic fracture height is presented based on superposition of stress intensity factors. A case study is also included to show the operability of our theory. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Hydraulic fracture treatment to enhance recovery rate in lowpermeability reservoirs has become standard practice in Petroleum Engineering. A pair of fractures usually grows outward in a vertical plane and propagates above and below the packers as well as laterally away from the wellbore. The vertical extending limit of a fracture (i.e. fracture height) is an important issue in hydraulic fracturing design. To some degree, fracture height affects fracturing efficiency and plays a key role in determining the success of hydraulic fracturing (Biot et al., 1983; Chen and Chen, 1994; Chen et al., 1997; Hudson et al., 2003; Adachi and Detournay, 2008). Therefore before fracturing operation, it is essential to know whether the fractures will break into bounding layers and predict the vertical extending limit of the fractures in order to establish operational parameter. It is easy to predict fracture height if the formation is isotropic and earth stress is uniform. In this case the fracture is circular and fracture height equals fracture length. We can use delivery capacity, filter loss and fracture volume to predict fracture shape (Yi and Zhu, 2005). However, in most cases hydraulic fracturing is conducted where the mechanical properties of the reservoir rock are different from the bounding layers and the earth stress is not uniform. This requires an improved understanding of the extending behavior of hydraulic fractures reaching formation interfaces. The extending behavior of hydraulic fractures reaching formation interfaces is controlled by formation condition and work condition, and little quantitative research has been done in this field. The factors affecting the fracture extending behavior when reaching formation
⁎ Corresponding author. Tel.: +86 1089732165. E-mail address: [email protected] (H. Zhao). 0920-4105/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.petrol.2010.08.003
interface include: layered difference of earth stress, elastic parameters and fracture toughness, interfacial strength between pay formation and bounding layers, pressure distribution of fracturing fluid, rheological property and viscosity of fracturing fluid (Warpinski et al., 1999; Fan, 2003; Hossain and Rahman, 2008). Moreover, anisotropy of formation, natural fractures, filtration characteristics of fracturing fluid and injection rate also affect the fracture extending behavior. Biot et al. (1983) assumed that hydraulic fracture is under plane strain condition along fracture width direction, the fracture width profile is sinusoidal and the formation interface is well consolidated. Without considering earth stress difference and filtration, the elastic equations are simplified to Laplace equation and a simple criterion is introduced p2 = p1
sffiffiffiffiffiffiffiffiffiffi S2 G2 S1 G1
ð1Þ
where p1 and p2 are extending pressure in pay formation and bounding layer respectively, G1 and G2 are shear modulus of pay formation and bounding layer respectively, and S1 and S2 are interface energy density of pay formation and bounding layer respectively. If S2G2 b S1G1, the fracture will propagate into bounding layer when reaching the formation interface. Otherwise, the fracture will stop extending. Biot's theory does not consider earth stress difference and interfacial effect. In practice hydraulic fractures break through formation interfaces and enter bounding layers even if G2 is 20 times larger than G1 (Warpinski et al., 1999), which indicates that the fracture behavior is not completely controlled by shear modulus difference. When burial depth is large and strata dip is small, overburden rock applies large normal stress on formation interface
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
and hydraulic fracture cannot extend along formation interface. But when strata dip is large or burial depth is small, we must consider interfacial effect. Chen et al. (1997) assumed the hydraulic fractures will certainly enter bounding layers more or less when reaching formation interfaces. They built a numerical model using elastic theory of rock mechanics to simulate vertical extension of fracture in layered media. With the numerical model, they studied the effect of earth stress profile, fracture toughness difference and fracture fluid density on fracture height. Another researcher Chen and Chen (1994) used complex variable method to calculate stress intensity factor of fracture in layered media. They resolved stress intensity factor into singular integral equations and used Lobatto–Chebyshev orthogonal polynomials to get numerical results. In this paper, the authors use rock fracture mechanics to describe the reactions of fracture growth when hydraulic fracture reaches formation interface. A homologous judgment criterion is introduced. In the model we take into account the effect of layered earth stress, layered rock mechanics parameters, formation interface, reservoir thickness and work parameters. In the case of fracturing through bounding layers, the method to calculate fracture height in bounding layers is presented.
Also by stress analysis of pay formation we get the normal stress and shear stress on interface I: 8 < σ = σ cos2 θ + σ sin2 θ 1 21 1 23 1 : τ1 = ðσ 21 + σ 23 Þ sinθ1 cosθ1
ð3Þ
Therefore the layered earth stress σij should meet the following condition: 8 2 > > 2 < σ −σ i1 i + 1;1 cos θi = σ i + 1;3 −σ i3 sin θi > > : ðσ + σ Þ sinθ cosθ = σ i1 i3 i i i + 1;1 + σ i + 1;3 sinθi cosθi
ð4Þ
where i = 1, 2. 2.2. Fundamental assumptions In order to appropriately simplify the calculation, we have the following assumptions.
2. Layered earth stress and fundamental assumptions 2.1. Layered earth stress Extension of hydraulic fracture in layered formations is shown in Fig. 1. In Fig. 1, σ11 is the overburden pressure of upper bounding layer, MPa; σ13 is minimal horizontal earth stress, MPa; the index i of σij indicates upper bounding layer (when i = 1), pay formation (when i = 2) or lower bounding layer (when i = 3), j indicates minimal horizontal earth stress (when j = 3) or overburden pressure (when j = 1); θ1 and θ2 are strata dips of upper and lower bounding layers, deg and h1(h2) is distance between fracture center and upper (lower) bounding layer, m. For the upper bounding layer, by stress analysis we get the normal stress and shear stress on interface I (interface between upper bounding layer and pay formation): 8 < σ = σ cos2 θ + σ sin2 θ 1 11 1 13 1 : τ1 = ðσ 11 + σ 13 Þ sinθ1 cosθ1
27
ð2Þ
(1) Rock fracturing can be characterized by linear elastic fracture mechanics. This assumption indicates rock is considered as elastic material and fracture propagation criterion is stress intensity factor equals fracture toughness. For most sandstone reservoirs, this assumption is sufficiently precise. If the reservoir is plastic material, such as shale gas, this assumption may be invalid. (2) Hydraulic fracture shape is pseudo 3-dimensional, it is elliptic along the plane normal to minimal horizontal earth stress. In homogeneous reservoir, hydraulic fracture shape is pseudo 3dimensional. Furthermore, it is difficult to get analytical solution for the stress intensity factor if the fracture shape is considered as true 3-dimensional. (3) The static and dynamic pressure drop of fracturing fluid along the direction of fracture height is ignored. In order to capture the fundamentals of the problem, we do not consider the effect of gravity and viscous force.
Fig. 1. Configuration of hydraulic fracture in layered formations.
28
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
3.2. Fracture extending along formation interface
3. Fracture propagation reaching formation interface and judgment criterion 3.1. Fracture stops extending When reaching formation interface, hydraulic fracture stops extending along the direction of height and only can extend along the direction of length. As the length and height of hydraulic fracture are much larger than width, the inside hydraulic pressure can be considered as along the direction of Z-axis. By using the method of complex function (Fan, 2003), we get the I-form stress intensity factor at arbitrary point on the frontal side of elliptic fracture, rffiffiffiffiffiffi 1 = 4 πh p−σ 23 2 2 2 2 K= h cos θ + l sin θ l EðkÞ
ð5Þ
where h is half fracture height, m; l is half fracture length, m; p is hydraulic pressure in the fracture, MPa; k2 = (l2 − h2) / l2, E is the second complete elliptic integral function and θ is angular variable of ellipse parametric equation. When hydraulic fracture reaches upper interface, l = h = h1. h1 can be considered as the distance between perforation center and interface I. In this case the hydraulic fracture extending condition on point D is as follows: rffiffiffiffiffi h1 KD = 2 ðp−σ 23 Þ = K2c π
ð6Þ
where K2c is fracture toughness of pay formation, MPa m1/2. From Eq. (6) we can get critical hydraulic pressure: p1 =
K2c 2
rffiffiffiffiffi π + σ 23 h1
ð7Þ
Fracture length increases with the proceeding of fracturing work, the fracture shape changes from circle to ellipse. The stress intensity factor on point D decreases under the same hydraulic pressure, as shown in Fig. 2. In Fig. 2, the dimensionless number 1 is the ratio (KD)l N h1 / (KD)l = h1, dimensionless number 2 is critical pressure (the minimal hydraulic pressure required to extend the fracture) ratio (p − σ23)l N h1 / (p1 − σ23) and fracture length n is also in dimensionless form (n = l / h1). In other words, the critical hydraulic pressure of fracture extending along length direction increases. It is possible that there exists a critical fracture length. When fracture length is larger than the critical fracture length, hydraulic fracture will break through formation interface or extend along formation interface.
Fig. 2. Variation of stress intensity factor and critical pressure vs. fracture length.
If the tensile strength of formation interface is weak and the normal stress on formation interface is small, which may occur in shallow formations or large dip angle formations, hydraulic fracture may extend along formation interface. Then the tension failure criterion should be adopted to determine fracture extension. The effective normal stress on interface I is as follows: 2
2
σ 1 = σ 11 cos θ1 + σ 13 sin θ1 −p
ð8Þ
Critical condition satisfies σ1 = −St1, where St1 is the tensile strength of interface I, MPa. The critical hydraulic pressure is 2
2
p2 = σ 11 cos θ1 + σ 13 sin θ1 + St1
ð9Þ
3.3. Fracturing through bounding layer If hydraulic fracture directly breaks through formation interface to bounding layer, the formation interface can be considered to be well consolidated. When hydraulic fracture meets the upper interface, l = h = h1, the hydraulic fracture extending condition on point A is rffiffiffiffiffi h1 ðp−σ 23 Þ = K1c π
KA = 2
ð10Þ
where KA is stress intensity factor on point A, MPa m1/2 and K1c is fracture toughness of the upper bounding layer, MPa m1/2. From Eq. (10) we can get the critical hydraulic pressure p3 =
K1c 2
rffiffiffiffiffi π + σ 23 h1
ð11Þ
Hydraulic fracture height in bounding layer is closely related to earth stress difference between pay formation and bounding layer (Brudy et al., 1997). Stress intensity factor on point A can be calculated by three section superpositions after point A enters the upper bounding layer, as shown in Fig. 3. The center of coordinates system y′z′ is fracture center. The relationship between coordinate system y′z′ and yz (which center is perforation center) is: ′
y = y + d= 2 z = z′
Fig. 3. Three sections for calculating stress intensity factor.
ð12Þ
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
29
where d is the fracture penetrating distance into the upper bounding layer, m; h is half fracture height and h = h1 + d/2, m. The contribution of I, II and III sections to the stress intensity factor on point A can be solved with analytic solution (Fan, 2003). By calculation and superimposition we get the stress intensity factor on point A KA =
rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h π −1 + arcsinð1−d = hÞ− 2d = h−ðd =hÞ2 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp−σ 13 Þ h π − arcsinð1−d = hÞ + 2d = h−ðd=hÞ2 + π−1 π 2 2ðp−σ 23 Þ π−1
ð13Þ Critical condition is KA = K1c
ð14Þ
From Eqs. (13) and (14), we can calculate penetrating distance d under hydraulic pressure p. The problem is more complicated if hydraulic fracture enters both upper and lower bounding layers, as shown in Fig. 4. The stress intensity factor should be calculated by four section superpositions. By calculation and superimposition, we can get the stress intensity factor on points A and B, KA =
rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h π − arcsinð1−d2 = hÞ− 2d2 = h−ðd2 =hÞ2 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d2 = hÞ + 2d2 = h−ðd2 = hÞ2 −1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d1 = hÞ− 2d1 = h−ðd1 =hÞ2 + 1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 13 Þ h π − arcsinð1−d1 = hÞ + 2d1 = h−ðd1 =hÞ2 + π−1 π 2 2ð p−σ 33 Þ π−1
ð15Þ rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðp−σ 33 Þ h π − arcsinð1−d2 = hÞ + 2d2 = h−ðd2 =hÞ2 KB = π−1 π 2 rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d2 = hÞ− 2d2 = h−ðd2 =hÞ2 + 1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 23 Þ h arcsinð1−d1 = hÞ + 2d1 = h−ðd1 =hÞ2 −1 + π−1 π rffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð p−σ 13 Þ h π − arcsinð1−d1 = hÞ− 2d1 = h−ðd1 =hÞ2 + π−1 π 2 ð16Þ Critical condition is KB = K3c KA = K1c
ð17Þ
where K3c is fracture toughness of the lower bounding layer (MPa m1/2), d1 and d2 are penetrating distances in upper and lower bounding layers respectively (m) and h is half fracture height, h = (d2 + d1 + h1 + h2)/2. From Eqs. (15)–(17), we can calculate the penetrating distances d1 and d2 under hydraulic pressure p. 3.4. Judging criterion (1) If min(p1, p2, p3) = p1, hydraulic fracture stops extending along height direction when reaching formation interface and only can extend along length direction. When fracture length reaches a critical value, the fracture will extend along formation interface or penetrate into bounding layers. From
Fig. 4. Four sections for calculating stress intensity factor.
Eqs. (7) and (11), the critical fracture length satisfies the following conditions, ðpc −σ 23 Þl = nh1 ðp3 −σ 23 Þ ðpc −σ 23 Þl = nh1 ðp1 −σ 23 Þ
2EðkÞ π pffiffiffi 2 nEðkÞ = π =
ð18Þ
That means when fracture length l1 =
p3 −σ 23 2 h1 p1 −σ 23
ð19Þ
The hydraulic fracture will enter bounding layer. From Eqs. (9) and (11), another critical fracture length satisfies the following condition: ðpc −σ 23 Þl = nh1 ðp2 −σ 23 Þ ðpc −σ 23 Þl = nh1 ðp1 −σ 23 Þ
=1 pffiffiffi 2 nEðkÞ = π
ð20Þ
That means when fracture length satisfies pffiffiffi 2 nEðkÞ p −σ 23 = 2 π p1 −σ 23
ð21Þ
the fracture will extend along formation interface. Eq. (21) can be solved in Fig. 2. (2) If min(p1, p2, p3) = p2, hydraulic fracture will extend along formation interface, which means the fracture will deflect along height direction. (3) If min(p1, p2, p3) = p3, hydraulic fracture directly breaks through formation interface to bounding layer. The penetrating distance is related to stress difference, fracture toughness difference and reservoir thickness. 4. Case study Dip angles θ1 = θ2 = 0°, form Eq. (4) we know that overburden pressure is continuous on interfaces I and II. Overburden pressure is 46.0 MPa on interface I and 46.7 MPa on interface II. σ23 = 30 MPa, σ13 − σ23 = 5 MPa, σ33 − σ23 = 7 MPa, distances from perforation to upper and lower interfaces are h1 = 3 m, h2 = 7 m, K2c = 1 MPa m1/2,
30
H. Zhao, M. Chen / Journal of Petroleum Science and Engineering 74 (2010) 26–30
K1c = K3c = 2 MPa m1/2, and interface tensile strength is St1 = St2 = 0.5 MPa. The critical pressures are: p1 = 30.5 MPa, p2 = 46.5 MPa and p3 = 31 MPa. Thus by judging criterion the fracture will stop extending when reaching formation interface I. Form Eqs. (19) and (21), the critical half fracture length is l = 4 h1 = 12 m N h2. That means before fracture length reaches the critical value, the fracture has already met interface II. Before hydraulic pressure reaches 31 Mpa, the fracture has already extended into upper and lower bounding layers. From Eqs. (15)–(17) the relationship between penetrating depths d1 and d2 and hydraulic pressure can be calculated, as shown in Fig. 5. 5. Conclusions and discussions (1) We use rock fracture mechanics to analyze the three types of possible extending behavior when hydraulic fracture reaches formation interface: stopping extension, extending along formation interface and penetrating into bounding layers. The judgment criterion is established. (2) If extension along height direction stops when fracture reaches formation interface, the fracture will extend along length direction. The fracture will extend along formation interface or penetrate into bounding layers when fracture length reaches a critical value. Form Eqs. (19) and (21), we find that selection of perforation positions is closely related to the critical fracture length. Perforation positions should be optimized according to actual formations. Pay formation center is not always the most appropriate selection. (3) If the fracture breaks through the interface and penetrates into bounding layers, we present a method to calculate penetrating depth in bounding layers. The single-sided penetration and two-sided penetration should be treated differently. Form the calculated results we find that earth stress and hydraulic pressure are significant factors determining penetrating depth. The lower the earth stress in bounding layer is, the faster penetration happens. When hydraulic pressure reaches earth stress in bounding layer, penetration depth will sharply increase. (4) In this paper, we consider pay formation and bounding layers as isotropic and intact media. Anisotropy and natural damage (including natural fractures, pores and joints etc.) of formations are neglected. But the influence of these factors can be removed to a certain degree when we acquire rock mechanic para-
Fig. 5. Vertical penetrating distances d1 and d2 vs. hydraulic pressure in fracture.
meters. For example, the measurement of fracture toughness may have been conducted using cores with natural fractures.
References Adachi, J.I., Detournay, E., 2008. Plane strain propagation of a hydraulic fracture in a permeable rock. Eng. Fract. Mech. 75 (16), 4666–4694. Biot, M.A., Medlin, W.L., Masse, L., 1983. Fracture penetration through an interface [R]. SPE 10372, 857–869. Brudy, M., Zoback, M.D., Fuchs, K., Rummel, F., Baumgartner, J., 1997. Estimation of the complete stress tensor to 8 km depth in the KTB scientific drill holes: implications for crustal strength [J]. J. Geophys. Res. 102 (B8), 18453–18475. Chen, M., Chen, Z.X., 1994. Study on the Extension of Hydraulic Fracture in Payered Formations: the Fifth National Conference on Numerical Methods and Theoretical Methods in Rock and Soil Mechanics, Chongqi, 1994 [C]. Chongqi University Press, Chongqi. Chen, Z.X., Chen, M., Huang, R.Z., Shen, Z.H., 1997. Vertical growth of hydraulic fracture in payered formations [J]. J. China Univ. Petrol.: Ed. Nat. Sci. 21 (4), 23–26. Fan, T.Y., 2003. Fracture Theoretical Basis [M]. Science Press, Beijing, pp. 170–177. Hossain, M.M., Rahman, M.K., 2008. Numerical simulation of complex fracture growth during tight reservoir stimulation by hydraulic fracturing. J. Petrol. Sci. Eng. 60 (2), 86–104. Hudson, J.A., Cornet, F.H., Chistiansson, R., 2003. ISRM suggested method for rock stress estimation—part 1: strategy for rock stress estimation [J]. Int. J. Rock Mech. Min. Sci. 40 (7/8), 991–998. Warpinski, N.R., Schmidt, R.A., Northrop, D.A., 1999. Earth stresses: the predominant influence on hydraulic fracture containment [R]. SPE 8932, 653–664. Yi, S.M., Zhu, Z.D., 2005. Primary Damage Mechanics of Fractured Rock [M]. Science Press, Beijing, pp. 84–93.