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A numerical simulation study of fracture reorientation with a degradable fiber-diverting agent
Journal of Natural Gas Science and Engineering 25 (2015) 215e225
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A numerical simulation study of fracture reorientation with a degradable fiber-diverting agent Daobing Wang a, b, c, Fujian Zhou a, b, *, Wei Ding c, Hongkui Ge a, b, Xinfeng Jia d, Yang Shi c, Xiaoqiong Wang a, b, Xingming Yan e a
Unconventional Natural Gas Research Institute, China University of Petroleum, Beijing 102249, China State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China Beijing Branch, Research Institute of Petroleum Exploration and Development, China National Petroleum Cooperation, Beijing 100083, China d The Department of Chemical and Petroleum Engineering, University of Calgary, Alberta T2N 1N4, Canada e Langfang Branch, Research Institute of Petroleum Exploration and Development, China National Petroleum Cooperation, Hebei 065007, China b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 January 2015 Received in revised form 2 May 2015 Accepted 4 May 2015 Available online
Degradable fiber can temporarily plug a natural fracture or artificial fracture. It has been successfully applied in the stimulated reservoir volume (SRV) fracturing or re-fracturing of unconventional reservoirs. Based on the classical analytical stress field equation, a new mathematical model is established in this paper to model the crack reorientation path after injecting fiber diversion fluid according to the tensile failure criterion. Factors influencing the diverting radius are intensively analyzed through numerical simulation. The results indicate that the horizontal stress difference, fracturing fluid viscosity, and injection time (fracturing fluid volume) have larger effects on the diverting radius than do the formation permeability (1e50 mD) and bottomhole pressure (90e160 MPa). The simulation results are successfully verified, matching well with the experimental data from the true tri-axial fracture reorientation tests in the laboratory. The model is successfully applied to the heterogeneous carbonate reservoirs in northwest China. © 2015 Elsevier B.V. All rights reserved.
Keywords: Fracture reorientation Hydraulic fracturing Mathematical model Fiber Stress field
1. Introduction Hydraulic fracturing techniques are effective means of the economic development of unconventional oil and gas resources. Due to the ultralow matrix permeability, the stimulation reservoir volume (SRV) needs to be expanded by increasing the degree of complexity of the hydraulic fracturing treatment. SRV fracturing can carry out three-dimensional reconstruction of oil and gas layers, form an artificial fracture network to maximize the swept volume of artificial fractures within the reservoir and thereby increase the effective permeability to improve oil and gas production. When objective factors of formation, such as large local stress difference, are not conducive to generating a complex fracture network, a temporary blocking agent can be introduced to form an artificial additional shielding and seal cracks and pre-existing flow channels to divert the artificial fractures (Economides and Nolte,
* Corresponding author. Unconventional Natural Gas Research Institute, China University of Petroleum, Beijing 102249, China. E-mail addresses: [email protected] (D. Wang), [email protected] (F. Zhou). http://dx.doi.org/10.1016/j.jngse.2015.05.002 1875-5100/© 2015 Elsevier B.V. All rights reserved.
2003; Wang and Zhang, 1998; Wang, 2013). Therefore, diverting fracturing can increase the scope and effectiveness of reservoir stimulation. It is important for stimulating low-permeability reservoirs, such as shale gas plays, tight gas reservoirs and deep carbonate reservoirs. Diverting agent materials is crucial to successful diverting fracturing treatment. Smith et al. (1969) simply viewed the conventional diverting agent materials used for fluid diversion. They contain oil or water soluble diverting agents, foam, naphthalene, rock salt, paraformaldehyde and a wax-polymer. Ball-sealer diversion is used in casing perforated wells to divert fracturing fluids by temporarily blocking perforations with rubber-coated balls. Nozaki et al. (2013) developed an empirical correlation for ball-sealer performance according to a laboratory experiment. However, these diverting materials have inherent disadvantages, such as low seal pressure, incomplete degradation and damage to formation. Zhou et al. (2009) proposed a novel fiberassisted diversion acid fracturing (acidizing) technique that integrates fracture reorientation and fluid diversion to divert the artificial fractures and enhance the chance of connecting fractureand-cavern area in deep carbonate reservoirs. The fibers can form
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strong and stable fiber networks that can effectively block natural fractures and perforations. They automatically and completely degrade under the effect of formation temperature when they make contact with hydrocarbons or solvents after the stimulation (Cohen et al., 2010). Mixing cellulose fiber in concrete has small effect on the compressive strength but can improve the tensile strength of concrete, reduce the fracture width and failure area of concrete, and improve crack resistance (Xiao et al., 2014). A new degradable fiber diversion system has been applied in recent years in carbonate reservoirs (Solares et al., 2008; Retnanto et al., 2012). Liu et al. (2013) introduced a fiber diversion technique to improve vertical coverage and deep placement with proppants. The fibers bridge across perforations and natural fractures to form a filter cake (Cohen et al., 2010). Bukovac et al. (2012) presented a mathematical model for fiber accumulation within perforations and explained how parameters such as formation-permeability contrast and fibercake permeability impact the diversion efficiency (Powell et al., 2007; Cohen et al., 2010; Mahdiyar et al., 2007). Allison et al. (2011) used biodegradable particulates to facilitate the temporary diversion and concentration of frac energy, which has enhanced the success of re-fracturing. Re-fracturing is an important technology for stimulating lowpermeability reservoirs when oil or gas production decreases after initial fracturing (Zhang and Chen, 2010a; 2010b). Siebrits et al. (2000) gave evidence of increased output due to re-fracturing in two tight gas wells. Surface tilt-meter measurements showed refracture orientations at oblique angles to the azimuth of the initial fractures. Zhang et al. (2010a, 2010b) proposed a model for a dynamic fracture propagation path during re-stimulation, and numerical simulation results showed that stress difference and initiation angle resulted in changes of the path. Behnia et al. (2015) used the crack tip element and a higher order displacement discontinuity method to study hydraulic fracture extension and reorientation. They came to the conclusion that fracture reorientation is influenced by stress deviation, fluid pressure and fracture inclination. By using a tri-axial fracturing system in the laboratory, Zhou et al. (2008) found that tortuous fractures are along the fracture height direction if the horizontal stress is the maximum principle stress. Rahman and Rahman (2013) proposed a fully coupled poroelastic model of hydraulic fracture-propagation in fractured reservoirs. The results showed that the orientation of the natural fracture has a remarkable effect on the induced hydraulic fracture propagation. Han and Yang (2009) studied the deformations and re-fracture characteristics of rock mass on the basis of uniaxial and tri-axial experiments. The experimental results showed that the refracture of the rock sample depends on mechanical characteristics of the rock and fracture plane. Degradable fiber has an effective temporary plugging effect on natural or artificial hydraulic fractures (Zhou et al., 2014; Solares et al., 2008; Romero et al., 2002). It can be successfully applied in the SRV fracturing or re-fracturing of unconventional reservoirs (Potapenko et al., 2009; Wang et al., 2015). However, the mechanisms of fracture reorientation with a degradable fiber diverting agent and its influence remain poorly understood. Based on the classical analytical stress field equation, a new mathematical model is established in this paper to model the crack reorientation path after injecting fiber diversion fluid according to the tensile failure criterion. Factors influencing the diverting radius are analyzed through numerical solution thereafter. The results indicate that the horizontal stress difference, fracturing fluid viscosity and injection time (fracturing fluid volume) have greater impacts on the diverting radius than do the formation permeability (1e50 mD) and bottomhole pressure (90e160 MPa). The simulation results are verified with experimental data from true tri-axial fracture reorientation tests.
2. Mathematical model 2.1. Physical model A diagram of the physical model is shown in Fig. 1. Its assumed conditions are as follows: (1) A vertical well is located in finite reservoirs; the reservoir is homogeneous and isotropic; rw denotes the borehole radius; re denotes the reservoir outer radius; re >> rw; k denotes matrix permeability; (2) The pressure at the inner or outer boundary is constant; pw denotes the pressure at the borehole; pe denotes the pressure at the outer boundary; (3) Injection fluid is a slightly compressible fluid; its viscosity m and total compression coefficient ct are invariant with pressure; (4) The original stress field is anisotric; sH denotes the maximum horizontal principal stress; sh denotes the minimum horizontal principal stress; (5) The influence of temperature variation on in-situ stress is out of consideration; (6) Because reservoir thickness is far less than drainage radius, plane strain state can be approximately assumed; (7) The fluid-solid coupling effect is neglected, i.e., formation permeability k, porosity f and fluid viscosity are invariant with pressure; (8) sq denotes the tangential or circumferential stress in polar coordinates; sr denotes the radial stress in polar coordinates. The degradable fiber-assisted diverting fracturing process is composed of three stages: (1) in the first fracturing process, fracturing fluid is injected into the formation to produce an artificial fracture; (2) fibers are injected with low viscosity carrying fluid (the mixture fluid is called DCF, and the carrying fluid is often slick water) to temporarily plug the cracks in the first fracturing. Crosslinked gel liquid with high viscosity is often followed by DCF to improve the plugging effect on cracks; due to the low viscosity of DCF, fluid phase are much easier to filtrate into the fracture wall. Then, the fiber filtration cake remains at the fracture mouth or inside the fracture. (3) In the second fracturing process, fracturing fluid is injected into the wellbore. Because of the good effect on old
Fig. 1. Diagram of the physical model.
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225
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fractures, net pressure will remarkably increase in this stage (the effect of fiber is to temporarily plug old fractures and improve net pressure in the fracture; the influence of fiber cake on in-situ stress is out of consideration). Then, a new fracture will initiate and propagate along a different direction. Its fracture pressure in the second fracturing is higher than that in the first fracturing. Repeating steps 2 to 3 can fracture networks. After the stimulation treatment, fibers can be automatically degradable at formation temperature. They change into very low viscous fluids (less than 5 mPa s), which are easier to flow back.
respectively; t denotes the injection time, s; h and c denote elastic coefficients of the rock body, dimensionless unit; L1 is the inverse Laplace transform that is performed by the Stehfest algorithm (Stehfest, 1970); K0 and K1 are the first order modified Bessel function of the first and second kind, respectively. The poroelastic stress field induced by injecting the fracturing fluid for 2 h was obtained by associating Eqs. (1)e(8), as shown in Fig. 2. It shows that the radial stress field and circumferential stress field are changed because of the fluid injection and resultant pore pressure increase.
2.2. Stress field induced by injection fluid
2.3. Stress field induced by open fracture
When fracturing fluids are pumped into a well to create conductive fractures, the wellbore is pressurized. This causes the pore pressure near the borehole to increase, which redistributes the original stress field and leads to a so-called stress field induced by the injection fluid (Detournay and Cheng, 1988). The analytical solution of pore pressure and stress field in a cylindrical coordinate system is shown below:
If a well is fractured to induce a hydraulic fracture, the virgin stress field will be perturbed (Warpinski and Branagan, 1989).
pðr; tÞ ¼ p0 þ ðpw p0 Þgðr; tÞ srr ðr; qÞ ¼
(1)
4 2 sH þ sh r2 sH sh rw rw 1 w 1 þ 3 þ cosð2qÞ 4 2 2 r2 r4 r2 þ pw
2 rw rw þ 2hðpw p0 Þ hðr; tÞ r r2
(2) sqq ðr; qÞ ¼
4 sH þ sh r2 sH sh rw 1þ w 1 þ 3 cosð2qÞ 2 2 r2 r4 i hr r2 w pw w hðr; tÞ þ gðr; tÞ 2hðp p Þ w 0 r r2
(3)
2 s sh r4 rw trq ðr; qÞ ¼ H 13 w sinð2qÞ þ 2 2 r4 r2
(4)
~ðr; sÞ gðr; tÞ ¼ L1 ½g
(5)
~ðr; sÞ ¼ g
K0 ðxÞ sK0 ðbÞ
h i ~ sÞ hðr; tÞ ¼ L1 hðr; ~ sÞ ¼ 1 K1 ðxÞ rw K1 ðbÞ hðr; s bK0 ðbÞ r bK0 ðbÞ
(6)
(7)
(8)
where
b ¼ rw
rffiffiffi rffiffiffi s s ; x¼r ; c c
p0 denotes the initial formation pressure, MPa; pw denotes the flowing bottom hole pressure, MPa; sH denotes the maximum horizontal principle stress, MPa; sh denotes the minimum horizontal principle stress, MPa; sqq denotes the tangential or circumferential stress, MPa; srr denotes the radial stress, MPa; trq denotes the shear stress, MPa; rw denotes the borehole radius, m; r and q denote polar coordinates; r denotes the distance away from the borehole center, m; s denotes the time variable in the Laplace ~ ~ and h, space; g and h denote the inversion Laplace transform of g
Fig. 2. Distribution of poroelastic stress field induced by injecting fluid after 2 h: (a) radial stress field and (b) circumferential stress field.
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Sneddon and Elliot (1946) proposed a model for the stress field around an infinitely long 2D crack (plane strain) in a homogeneous, isotropic, and elastic body. The analytical solution to this model in a Cartesian coordinate system is shown below:
8 > < r q þ q2 Dsx ¼ p pffiffiffiffiffiffiffiffiffi cos q 1 > 2 : r1 r2 9 > = 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q sin ðq1 þ q2 Þ 1 > 2 ; ðr1 r2 Þ3 c2 r
(9)
8 > < r q þ q2 Dsz ¼ p pffiffiffiffiffiffiffiffiffi cos q 1 > 2 : r1 r2 9 > = c2 r 3 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q sin ðq1 þ q2 Þ 1 > 2 ; ðr1 r2 Þ3
Dtxz
8 9 > > < c2 r = 3 ¼ p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q cos ðq1 þ q2 Þ > > 2 : ðr r Þ3 ; 1 2
Dsy ¼ yðDsx þ Dsz Þ
(10)
(11)
(12)
where p is the net pressure in the crack, Pa; h is the crack height, m; y is Poisson's ratio, m/m. Related geometric relations between the Cartesian and cylindrical systems are given by:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > þ z2 r ¼ qxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 2 r1 ¼ x þ ðz cÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : r2 ¼ x2 þ ðz þ cÞ2
(13)
8 x > q ¼ arctan > > > z > > < x q1 ¼ arctan > zc > > x > > > : q2 ¼ arctan zþc
(14)
Negative values of q, q1, and q2 should be replaced, respectively, by p þ q, p þ q1, and p þ q2. Given the artificial fracture height of 50 m, half fracture length of 200 m, and net pressure of 10 MPa, the stress field distribution induced by the fracturing can be determined by the above equations, as shown in Fig. 3. It is found that the closer the stress field location is to fracture, the larger both of the induced stress changes are. In addition, the stress change added to the minimum horizontal direction is greater than that added to the maximum horizontal direction. If the net pressure is large enough or if the horizontal stress contrast is small enough, the minimum and maximum stresses in the virgin stress field can be reversed in the induced stress field, which can be greatly favorable to the fracture re-orientation practice.
Fig. 3. Distribution of stress field induced by open fracture: Stress changes in the (a) maximum and (b) minimum horizontal stress directions, respectively.
denote radial, circumferential, and shear stresses induced by open fracture, respectively, MPa. According to the linear superposition principle, the total stress field after injecting fiber-laden diversion fluid is expressed as:
sr ¼ sr0 þ sr1 þ sr2
(15)
sq ¼ sq0 þ sq1 þ sq2
(16)
trq ¼ trq0 þ trq1 þ trq2
(17)
2.4. Total stress field after injecting fiber-laden diversion fluid It is assumed that sr0, sq0, and trq0 denote radial, circumferential, and shear stresses after drilling the borehole, respectively, MPa; sr1, sq1, and trq1 denote radial, circumferential, and shear stresses induced by fluid injection, respectively, MPa; sr2, sq2, and trq2
where sr0, sq0, and trq0 can be calculated by the elasticity theory. Coordinate transform from the Cartesian to cylindrical coordinate system can be made through the following formulas:
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225 Table 1 Physical properties of a base model. Property
Value
Young's modulus(E) Poisson's ratio(y) Permeability(k) Porosity(f) Total compressibility(ct) Viscosity of fracturing fluid(m) Initial reservoir pressure(p0) Bottomhole pressure(pw) Maximum horizontal principle stress(sH) Minimum horizontal principal stress(sh) Borehole radius(rw) Biot coefficient(a) Reservoir outer radius(re)
35,000 MPa 0.28 50 mD 0.08 1.45 103 MPa1 120 mPa s 75 MPa 100 MPa 99.26 MPa 93.47 MPa 0.1 m 0.8 3000 m
Fig. 4. The artificial fracture propagation trajectory determined by the total tangential stress after injecting fiber-laden diversion fluid: (a) the first fracture in the first fracturing; (b) the second fracture in the second fracturing.
219
sr2 ¼ Dsx cos2 q þ Dsy sin2 q þ Dtxy sin 2 q
(18)
sq2 ¼ Ds2x sin2 q þ Ds2y cos2 q Dtxy sinð2qÞ
(19)
trq ¼ Dsy Dsx cos q sin q þ Dtxy cos 2 q
(20)
3. Simulation results 3.1. Initiation and propagation path of fracture reorientation The initiation and propagation paths of reoriented fractures are simulated and analyzed in this part. The physical properties of a base model are listed in Table 1. Tensile failure occurs when the effective tensile stress across some plane in the sample exceeds a critical limitation (Fjar et al., 2008). The failure criterion, which specifies the stress condition by which tensile failure will occur and identifies the location of the failure surface in principal stress space, is given as: sqT0. Therefore, when the tangential or circumferential stress near the wellbore meets this condition, artificial fracture perpendicular to the tangential stress direction will be generated. In other words, the minimum point of tangential stress near the wellbore will be the first to fail. The total stress field distribution after injecting the fiber-laden diversion fluid is determined by using Eqs. (1)e(8), as shown in Fig. 3. According to the rock tensile fracture criterion, for each radius r, artificial cracks extend along the minimum point of tangential stress (hoop stress). Thereby, the crack initiation and extension path can be determined by the following procedures: First, make contour maps for a tangential stress distribution (In the first fracturing, a lot of fracturing fluid is pumped along the wellbore. Because of the increasing pore pressure, there is stress field induced by the fluid injection. After the first fracturing, an open fracture is formed. The stress field induced by the open fracture exists. According to Eq. (16), the total stress field after injecting the fiber-laden diversion fluid can be obtained.). Then, draw a circle at different radius levels to find the minimum point tangent to the contour (According to the fracture criterion, the minimum point of tangential stress near the wellbore will be the first to fail.). Finally,
Fig. 5. Regressed correlations between horizontal stress difference and diverting radius at different viscosity levels.
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all of the cut-off points are connected into a line. Fig. 4 shows the artificial fracture propagation trajectory (red line) (in the web version). The second fracture extended in a new direction, which is due to the stress field change around the wellbore. However, after a certain distance, the extension direction of the crack came back to the original maximum principal stress direction. The diversion radius is defined as the extension distance of the new fracture from the wellbore to the direction turning point. 3.2. Influencing factors of diverting radius 3.2.1. Horizontal principle stress difference Contour maps of the circumferential stress field are obtained at different levels of the horizontal principle stress difference. According to section 3.1, the artificial fracture propagation trajectories corresponding to the horizontal principle stress difference are acquired. On the basis of the measured data, the horizontal principle
stress difference is correlated to the diverting radius by applying the logarithm regression at three different viscosity levels, respectively (Fig. 5). It is found that at the same level of viscosity of the fracturing fluid, a larger horizontal principle stress difference led to a smaller diverting radius. When the stress difference approached 8 MPa, the diverting radius decreased almost to zero, which indicates that the fracture extended in the original maximum principle stress direction, while the second fracture would not initiate in a new direction. At the same level of horizontal principle stress difference, a larger fracturing fluid viscosity results in a greater diverting radius. 3.2.2. Fracturing fluid viscosity In the same manner as that mentioned in Section 3.1, the fracture reorientation trajectories were obtained at different fracturing fluid viscosity levels. Fig. 6 shows the regressed correlations for the viscosity and diverting radius at different stress difference levels. It also shows that at a similar horizontal stress difference, the diverting radius increases when the fluid viscosity is less than 200 mPa s and remains constant when the viscosity is greater than
Fig. 6. Regressed correlations between viscosity and diverting radius at different stress difference levels. Fig. 8. Regressed correlations between formation permeability and diverting radius at different stress difference levels.
Fig. 7. Regressed correlations between injection time and diverting radius at different stress difference levels.
Fig. 9. Regressed correlations between bottomhole pressure and diverting radius at different stress difference levels.
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225
200 mPa s. Thereby, the optimum viscosity range was determined to be 200300 mPa s. A less viscous fluid would result in a smaller net pressure (Economides and Nolte, 2003; Wang and Zhang, 1998). This will further lead to a minor fracturing induced-stress field and a smaller diverting radius as well. On the contrary, more viscous fluid would cause a larger fracture net pressure, stronger stress field, and a bigger diverting radius (Economides and Nolte, 2003; Wang and Zhang, 1998). When the viscosity exceeded a certain point, the fracture net pressure stayed almost constant and the diverting radius remained unchanged. 3.2.3. Injection time The fracture reorientation trajectories are also obtained at different levels of injection time. Fig. 7 presents the regressed correlations for the injection time and the diverting radius. It can be seen that at the same levels of horizontal stress difference, the longer the injection time is, the larger the diverting radius will be,
100
which indicates that pumping a larger fracturing fluid volume can increase the diverting radius during fracturing treatment. A longer injection led to a larger invasion radius of the fracturing fluid. Therefore, the range of pore pressure changes becomes wider, leading to the spread range of the stress field induced by pore pressure being wider, followed by increasing diverting radius. 3.2.4. Formation permeability The fracture reorientation trajectories were also obtained at different permeability levels, and the correlations between the matrix permeability and diverting radius were regressed and plotted in Fig. 8. It is found that the diverting radius varies between 20 and 25 m when the permeability range is 1e50 mD. This minor change indicated an insignificant effect of permeability on the diverting radius. The common permeability (150 mD) and limited pore pressure change leads to a small stress field induced by the injection fluid. Therefore, the impact of formation permeability on the diverting radius is negligible. 3.2.5. Bottomhole pressure The fracture reorientation trajectories were determined at different levels of bottomhole pressure. Fig. 9 shows the regressed
80
initiation angle (degree)
221
60
y=119.59-21.875x+0.8153x 2 R =0.9457
2
40
20
0 0
1
2
3
4
5
6
7
8
9
horizontal stress difference (MPa)
(a) 1MPa 2MPa 3MPa 4MPa 5MPa 6MPa 7MPa 8MPa
y (m)
40
20
0
0
20
40
60
80
100
x (m)
(b) Fig. 10. Simulation results of the extension path and initiation angle of a diverting fracture at different horizontal principal stress difference levels: (a) initiation angle of diverting fracture and (b) extension path of diverting fracture.
Fig. 11. Schematic diagram of experimental set-up: (a) true tri-axial simulation test device for hydraulic fracturing and (b) 30 30 30 cm3 core sample.
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correlations between the bottomhole pressure and diverting radius. It can be seen that the diverting radius varies between 20 and 25 m when the bottomhole pressure ranges from 90 to 160 MPa. This minor change indicates the small effect of pressure from 90 to 160 MPa on the diverting radius. The pore pressure changes grow larger as the bottomhole pressure increases, resulting in an incremental of the injection-induced stress field (negative). This decreases the tangential stress and increases the fracture net pressure in the meantime, leading to an augmented stress field
induced by artificial fracture increases (positive). These two effects offset each other, together causing the small impact of bottomhole pressure on the diverting radius. 3.3. Initiation angle of fracture reorientation The extension path and initiation angle of the diverting fracture were also simulated at different horizontal principal stress difference levels (Fig. 10). With the increase of principal stress difference,
Fig. 12. Experimental results of fracture reorientation with fiber-diverting agent: (aeb) Sample #1, Ds ¼ 2.5 MPa; (ced) Sample #2, Ds ¼ 5.0 MPa; (eef) Sample #3, Ds ¼ 7.5 MPa. The left figure represents the artificial fracture morphology after the first fracturing; the right figure represents the diverting fracture morphology after the second fracturing.
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225
4. Model verification 4.1. Laboratory experiment The crack initiation and propagation of hydraulic fracture reorientation with a degradable fiber diverting agent was simulated with a true tri-axial simulation system. This system consisted of a large size, true tri-axial experimental aircraft, MTS servo booster pump, data acquisition system, power source, oil and water separator, and another auxiliary device component (Fig. 11a). The limestone outcrop samples had a dimension of 30 30 30 cm3, as shown in Fig. 11b. There is a drilling hole at the center of block sample at a depth of 17 cm. The hole's diameter is 27 mm. A steel tube is tightly cemented on the hole wall, 10 cm in length. The surplus open hole is 7 cm in length. Before injection fluid, vertical principal stress is applied to the top face, and two horizontal principle stresses are applied, respectively, to the leftright and front-back face. The sample was fractured twice: the first fracturing was conducted with a base fluid that was injected into the hole at the center of the sample to induce an artificial fracture. The secondary stimulation was performed with a fiberladen fluid that was injected into the same hole to bridge and plug the first fracture and create a new second fracture (Shi et al., 2013). The fiber-based fracturing fluid was composed of 0.3% guar, 0.025% citric acid, 1.5% fiber and 0.8% cross-linking agent. The fracturing base fluid consisted of 0.3% guar and 0.025% citric acid, and its viscosity was approximately 27 mPa s. The injection rate was in the range of 5e10 mL/min and was held constant during the two fracturings. Red tracer was added into the fracturing fluids to more clearly observe the artificial fracture propagation path (Shi et al., 2013; Zhou et al., 2009 & 2014). The initiation and propagation path of the diverting fracture was physically simulated at three horizontal principle stress difference levels: 2.5, 5, and 7.5 MPa, respectively (Fig. 12). It can be seen that the first fracture extended along the maximum principal stress direction. The initiation angle of the diverting fracture became larger at a greater horizontal principle stress difference. At 7.5 MPa, the secondary fracture extended along the original maximum principal stress direction, which indicated an ineffective diversion of fracture. The real diverting radius can be measured after the second fracturing. The greater the horizontal principle stress difference is, the less the diverting radius is.
4.2. Model verification The correlation for the experimental stress difference and diverting radius was regressed to testify concerning the validation of the model established in Section 2, as shown in Fig. 13. According to the model in section 2, the circumferential stress contours were plotted for the stress-difference range of 0.5e7.5 MPa to determine the diverting radius. It is found that the simulation results match well with the experimental results. After a stress difference exceeding 7.5 MPa, the diverting fracture hardly occurs. The less the stress difference is, the larger the angle between the two cracks would be, which proved the reliability of the model developed in this study.
4.3. Field application A vertical well, denoted as Well A, is an exploratory well located in the Tarim Basin with an open hole completion. The target formation is in the Ordovician at a depth of 6618.5e6700 m. Its main rocks are gray and taupe gray grainstone, oolitic limestone, bioclast limestone, etc. Oil and gas shows do not appear in the drilling and mud-logging process. Drill-stem testing is conducted before acid fracturing treatment. The results demonstrate that the percolation flow capacity is very weak near the wellbore. There is a lower pressure recover coefficient after shutting in the well, with relatively poor reservoir properties. Seismic data interpretation reveals that there exists a fracture-cave cube in the maximum principal stress direction. The distance from the wellbore to the fracture-cave cube is approximately 55 m, as shown Fig. 14a (Wang, 2013). The maximum principal stress direction in the block area is generally approximately NE40 , but the dipole shear wave image logging results show that the maximum principal stress direction of this well is NW300-330 . The angle between them is approximately 85 . If artificial fractures propagate along the direction NE40 , then it will be bad for connect the fracture-cave cube. Thus, degradable fiber-assisted diverting acid fracturing technology is used to enhance the communication chance. If no obvious pump pressure drop occurred during the first fracturing, then degradable fiber will be injected to temporarily block the formed fractures. The fracturing fluid is then pumped into the bottomhole in the second fracturing process, forcing the hydraulic fracture to initiate in a new direction. The horizontal principal stress difference is 3 MPa, which is lower than 7.5 MPa. According to section 3.2.1, fracture diversion will occur in this case. This hydraulic fracturing construction curve is shown in Fig. 14b. In the first fracturing process, the pump pressure gradually increases from 76.8 MPa to 88.4 MPa at a rate of 5 m3/min. This indicates that the artificial fracture cannot communicate the reservoir body during the first fracturing. Therefore, fiber is injected into the formation, carried by slick water fluid. The fiber-carrying fluid is called DCF. When DCF arrives at the hydraulic fracture mouth, the pump pressure increases by 20 MPa. At the beginning of the second fracturing process, the pump pressure is approximately 90 MPa at a rate of 4.5 m3/min. These demonstrate that this degradable fiber
diverting radius Polynomial Fit of diverting radius 100
y = 0.015x - 0.3855x + 3.711x - 15.895x + 27.961x - 30.865x + 110.32 R = 0.9993
80
diverting radius (cm )
the crack initiation angle gradually decreased from 90 to 0 (Fig. 10a) and the diverting radius also diminished (Fig. 10b). Beyond a certain distance, the crack tended to propagate back in the original maximum principal stress direction, which agreed with the previous simulation results (Zhang et al., 2008).
223
60
40
20
0 0
1
2
3
4
5
6
7
8
horizontal stress difference (MPa) Fig. 13. Regressed correlations between stress difference and diverting radius in the physical simulation experiment of fiber-laden diverting fracturing.
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has a good temporary plugging effect on fractures. At the end of the second fracturing, the pump pressure increases by 2 MPa. When acid is pumped into the formation, the pump pressure sharply decreases by more than 20 MPa. Combined with the different fracture pressures during the two stages, fracture diversion occurs in the second fracturing process (Wang, 2013).
According to section 3, the propagation path of the diversion fracture is numerically simulated, as shown in Fig. 14c. Its diverting radius is approximately 62 m, which is very close to the distance between the wellbore and reservoir body (55 m). This again indicates that the model developed in this study is reliable. After the stimulation treatment, natural gas production is 5500 cubic meters per day and water production is 23.92 cubic meters per day. This demonstrates that there is mainly water in the reservoir body. This is a successful field application of this stimulation technology (Wang, 2013). 5. Conclusions This paper developed a new mathematical model for the crack reorientation path after injecting fiber diversion fluid. The following conclusions can be made from this study: 1. Three stress fields (stress field induced by drilling borehole, stress field induced by injection fluid and stress field induced by open fracture) were superposed on the basis of the classical analytical formulation. Then, the total circumferential stress field after injecting fiber diversion fluid was calculated to determine the fracture initiation and propagation path according to the tensile failure criterion. 2. The effects of stress difference, fracturing fluid viscosity, and permeability on the diverting radius were analyzed through numerical simulation. Correlations among them were regressed to determine the conditions for fracture reorientation. Simulation results indicated that the horizontal stress difference, fracturing fluid viscosity and injection time (fracturing fluid volume) have a larger effect on the diverting radius than do the formation permeability (150 mD) and bottomhole pressure (90160 MPa). 3. The simulation results were verified through the true tri-axial fracture reorientation experiment and field application. The limit of stress differences for fracture reorientation were 7.87 and 7.5 MPa from the numerical simulation and laboratory experiment, respectively, which showed a good match. Acknowledgments The authors would like to give our sincere gratitude to the precious financial support from China's Ministry of Science and Technology (973 program, Grant No. 2015CB250903), National Science Foundation of China (Grant No. 51490652 and 41304141), China National Petroleum Cooperation (Scientific Research and Technological Development Project, Grant No. 2010E-2105 and 2014A-4212; Science and Technology Innovation Fund, Grant No. 2013D-5006-0213), as well as Chinese Academy of Sciences (Strategic Leading Science and Technology Project, Grant No. XDB10050203). Abbreviations p0 pw T0 rw re r t k
Fig. 14. Images related with Well A: (a) seismic interpretation results; (b) acid fracturing curve; (c) the numerically simulated propagation path of the diversion fracture.
f m h and c
the initial formation pressure the flowing bottom hole pressure tensile strength of rock borehole radius reservoir outer radius distance away from borehole center the injection time permeability porosity viscosity of fracturing fluid elastic coefficients of rock body
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225
ct L1 K0 K1 P H E
a y sH sh sr sq trq sx sy txy q
total compressibility the inverse Laplace transform that is performed by the Stehfest algorithm the first order modified Bessel function of the first kind the first order modified Bessel function of the second kind the net pressure in the crack the crack height Young's modulus Biot coefficient Poisson's ratio the maximum horizontal principle stress the minimum horizontal principle stress the radial stress in the cylindrical coordinate system the hoop/ circumferential stress in the cylindrical coordinate system the stress along x axis in the cylindrical coordinate system the shear along y axis in the Cartesian coordinate system the shear stress in the Cartesian coordinate system the shear stress in the Cartesian coordinate system angle
References Allison, D., Curry, S., Todd, B., 2011. Restimulation of wells using biodegradable particulates as temporary diverting agents. In: Presented at the Canadian Unconventional Resources Conference, Calgary, Alberta, 15e17 November. CSUG/ SPE-149221-MS. Behnia, M., Goshtasbi, K., Zhang, G., Mirzeinaly Yazdi, S.H., 2015. Numerical modeling of hydraulic fracture propagation and reorientation. Eur. J. Environ. Civ. Eng. 19 (2), 152e167. Bukovac, T., Gurmen Nihat, M.G., Jauregui, J.L., Malik, A.R., Bolarinwa, S., AlGhurairi, F., 2012. Stimulaion strategies to guard against uncertainties of carbonate reservoirs. In: Presented at the SPE Saudi Arabia Section Technical Symposium and Exhibition, L-Khobar, Saudi Arabia, 8e11 April. SPE-160887MS. 2012. Cohen, C.E., Tardy, P.M.J., Lesko, T., Lecerf, B., Pavlova, S., Voropaev, S., Mchaweh, A., 2010. Understanding diversion with a novel fiber laden acid system for matrix acidizing of carbonate formations. In: Presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy, 19e22 September. SPE-134495-MS. Detournay, E., Cheng, A.D., 1988. Poroelastic response of a borehole in a nonhydrostatic stress field. Int. J. Rock Mech. Min. Sci. Geomechanics 25 (3), 171e182. Economides, M.J., Nolte, K.G., 2003. Reservoir Stimulation, Third ed. John Wiley & Sons. Inc, New York. Fjar, E., Holt, R.M., Raaen, A.M., et al., 2008. Petroleum Related Rock Mechanics, , Second ed.vol. 53. Elsevier, pp. 58e65. Han, L., Yang, M., 2009. Re-fracture process and mechanical characteristics of cracked rock samples. Int. J. Rock Mech. Min. Sci. 46 (4), 738e747. Liu, H., Luo, Y., Zhang, N., Yang, D., Dong, W., Qi, D., Gao, Y., 2013. Unlock shale oil reserves using advanced fracturing techniques: a case study in China. In: Presented in IPTC 2013: International Petroleum Technology Conference, 26e28 March. IPTC-16522-MS. Mahdiyar, H., Jamiolahmady, M., Danesh, A., 2007. New mechanical and damage skin factor correlations for hydraulic fractured wells. In: Presented at the European Formation Damage Conference, Scheveningen, Neitherland, 27e29 May. SPE-107634-MS. Nozaki, M., Zhu, D., Hill, A.D., 2013. Experimental and field data analyses of ballsealer diversion. SPE Prod. Operations 28 (03), 286e295. Potapenko, D.I., Tinkham, S.K., Lecerf, B., Fredd, C.N., Samuelson, M.L., Gillard, M.R., Le Calvez, J.H., Daniels, J.L., 2009. Barnett shale refracture stimulations using a novel diversion technique. In: Presented at the SPE Hydraulic Fracturing Technology Conference, Woodlands, Texas, 19e21 January. SPE-119636-MS.
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Powell, A., Bustos, O., Kordziel, W., Olsen, T., Sobernheim, D., Vizurraga, T., 2007. Fiber-Laden fracturing fluid improves production in the Bakken Shale multilateral play. In: Presented at the SPE Rocky Mountain Oil & Gas Technology Symposium, Denver, Colorado, 16e18 April. SPE-107979-MS. Rahman, M.M., Rahman, S.S., 2013. Studies of hydraulic fracture-propagation behavior in presence of natural fractures: fully coupled fractured-reservoir modeling in poroelastic environments. Int. J. Geomechanics 13 (6), 809e826. Retnanto, A., Lynn, C., Orellana, E., 2012. Managing uncertainty of reservoir heterogeneity and optimizing acid placement in thick carbonate reservoirs. In: Presented at the SPE International Production and Operations Conference and Exhibition. Doha, Datar, 14e16 May. SPE-155079-MS. Romero, D.J., Valko, P.P., Economides, M.J., 2002. The optimization of the productivity Index and the fracture geometry of a stimulated well with fracture face and choke skins. In: Presented at the SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, Lousiana, 20e21 Feburary. SPE73758-MS. Shi, Y., Zhou, F., Yang, X., Liu, X., Lian, S., Li, X., 2013. Laboratory study and field application of fiber-based fracture reorientation technology. In: Presented at the International Petroleum Technology Conference, Beijing, 26e28 March. IPTC-16736-MS. Siebrits, E., Elbel, J.L., Hoover, R.S., Diyashev, I.R., Griffin, L.G., Demetrius, S.L., Hill, D.G., 2000. Refracture reorientation enhances gas production in Barnett Shale tight gas wells. In: Presented at the SPE Annual Technical Conference, Dallas, Texas, 1e4 October. SPE-63030-MS. Smith, C.L., Anderson, J.L., Roberts, P.G., 1969. New diverting techniques for acidizing and fracturing. In: Presented at the SPE California Regional Meeting, San Francisco, Ca.lif.,6e7 November. Society of Petroleum Engineers. SPE-2751-MS. Sneddon, I.N., Elliot, H.A., 1946. The opening of a griffith crack under internal pressure. Quart. Appl. Math. 4 (3), 262e267. Solares, J.R., Duenas, J.J., Al-Harbi, M., Al-Sagr, A., Ramanathan, V., Hellman, R., 2008. Field trial of a new non-damaging degradable fiber-diverting full zonal coverage during acid fracturing in a deep gas producer in Saudi Arabia. In: Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 21e24 September. SPE-115525-MS. Stehfest, H., 1970. Algorithm 368: numerical inversion of laplace transforms [D5]. Commun. ACM 13 (1), 47e49. Wang, D.B., 2013. Study on the Mathematical Model about the Fiber Temporary Plugging Diverting Fracturing Crack Initiation and Propagation of the Vertical Wells. Master Thesis. Chengdu University of Technology, Chengdu, China (June 2013). Wang, H., Zhang, S., 1998. Hydraulic Fracturing Numerical Calculation Method, First ed. Petroleum Industry Press, Beijing. Warpinski, N.R., Branagan, P.T., 1989. Altered-stress fracturing. SPE. J. 41 (09), 990e997. Wang, W.D., Su, Y.L., Sheng, G.L., Manuel, Cossio, 2015. A mathematical model considering complex fractures and fractal flow for pressure transient analysis of fractured horizontal wells in unconventional oil reservoirs. J. Nat. Gas Sci. Eng. 23, 139e147. Xiao, K.T., Yan, J.J., Peng, S.S., 2014. Study on the effect of cellulose fiber on the crack resistance of high performance concrete. Adv. Mater. Res. 838, 96e100. Zhang, G.Q., Chen, M., Yao, F., Zhao, Z.F., 2008. Study on optimal re-fracturing timing in anisotropic formation and its influencing factors. Acta Pet. Sin. 29 (6), 885e893. Zhang, G.Q., Chen, M., 2010a. Dynamic fracture propagation in hydraulic re-fracturing. J. Petroleum Sci. Eng. 70 (3), 266e272. Zhang, G.Q., Chen, M., 2010b. The relationship between the production rate and initiation location of new fractures in a refractured well. Petroleum Sci. Technol. 28 (7), 655e666. Zhou, F., Yi, X., Yang, X., Liu, X., Wang, D., 2014. Dynamic filtration experiment of fiber temporary plugging artificial fracture. Drill. Prod. Technol. 37 (4), 83e86. Zhou, F., Liu, Y., Liu, X., Xiong, C., Yang, X., Jia, X., Li, X., Wang, D., Zhang, F., Shi, H., Lian, Y., Tao, S., Qian, C., 2009. Case study: YM204 obtained high petroleum production by acid fracture treatment combining fluid diversion and fracture reorientation. In: Presented at the SPE European Damage Conference, Scheveningen, Neitherland, 27e29 May. SPE-121827-MS. Zhou, J., Chen, M., Jin, Y., Zhang, G.Q., 2008. Analysis of fracture propagation behavior and fracture geometry using a tri-axial fracturing system in naturally fractured reservoirs. Int. J. Rock Mech. Min. Sci. 45 (7), 1143e1152.
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Journal of Natural Gas Science and Engineering journal homepage: www.elsevier.com/locate/jngse
A numerical simulation study of fracture reorientation with a degradable fiber-diverting agent Daobing Wang a, b, c, Fujian Zhou a, b, *, Wei Ding c, Hongkui Ge a, b, Xinfeng Jia d, Yang Shi c, Xiaoqiong Wang a, b, Xingming Yan e a
Unconventional Natural Gas Research Institute, China University of Petroleum, Beijing 102249, China State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China Beijing Branch, Research Institute of Petroleum Exploration and Development, China National Petroleum Cooperation, Beijing 100083, China d The Department of Chemical and Petroleum Engineering, University of Calgary, Alberta T2N 1N4, Canada e Langfang Branch, Research Institute of Petroleum Exploration and Development, China National Petroleum Cooperation, Hebei 065007, China b c
a r t i c l e i n f o
a b s t r a c t
Article history: Received 8 January 2015 Received in revised form 2 May 2015 Accepted 4 May 2015 Available online
Degradable fiber can temporarily plug a natural fracture or artificial fracture. It has been successfully applied in the stimulated reservoir volume (SRV) fracturing or re-fracturing of unconventional reservoirs. Based on the classical analytical stress field equation, a new mathematical model is established in this paper to model the crack reorientation path after injecting fiber diversion fluid according to the tensile failure criterion. Factors influencing the diverting radius are intensively analyzed through numerical simulation. The results indicate that the horizontal stress difference, fracturing fluid viscosity, and injection time (fracturing fluid volume) have larger effects on the diverting radius than do the formation permeability (1e50 mD) and bottomhole pressure (90e160 MPa). The simulation results are successfully verified, matching well with the experimental data from the true tri-axial fracture reorientation tests in the laboratory. The model is successfully applied to the heterogeneous carbonate reservoirs in northwest China. © 2015 Elsevier B.V. All rights reserved.
Keywords: Fracture reorientation Hydraulic fracturing Mathematical model Fiber Stress field
1. Introduction Hydraulic fracturing techniques are effective means of the economic development of unconventional oil and gas resources. Due to the ultralow matrix permeability, the stimulation reservoir volume (SRV) needs to be expanded by increasing the degree of complexity of the hydraulic fracturing treatment. SRV fracturing can carry out three-dimensional reconstruction of oil and gas layers, form an artificial fracture network to maximize the swept volume of artificial fractures within the reservoir and thereby increase the effective permeability to improve oil and gas production. When objective factors of formation, such as large local stress difference, are not conducive to generating a complex fracture network, a temporary blocking agent can be introduced to form an artificial additional shielding and seal cracks and pre-existing flow channels to divert the artificial fractures (Economides and Nolte,
* Corresponding author. Unconventional Natural Gas Research Institute, China University of Petroleum, Beijing 102249, China. E-mail addresses: [email protected] (D. Wang), [email protected] (F. Zhou). http://dx.doi.org/10.1016/j.jngse.2015.05.002 1875-5100/© 2015 Elsevier B.V. All rights reserved.
2003; Wang and Zhang, 1998; Wang, 2013). Therefore, diverting fracturing can increase the scope and effectiveness of reservoir stimulation. It is important for stimulating low-permeability reservoirs, such as shale gas plays, tight gas reservoirs and deep carbonate reservoirs. Diverting agent materials is crucial to successful diverting fracturing treatment. Smith et al. (1969) simply viewed the conventional diverting agent materials used for fluid diversion. They contain oil or water soluble diverting agents, foam, naphthalene, rock salt, paraformaldehyde and a wax-polymer. Ball-sealer diversion is used in casing perforated wells to divert fracturing fluids by temporarily blocking perforations with rubber-coated balls. Nozaki et al. (2013) developed an empirical correlation for ball-sealer performance according to a laboratory experiment. However, these diverting materials have inherent disadvantages, such as low seal pressure, incomplete degradation and damage to formation. Zhou et al. (2009) proposed a novel fiberassisted diversion acid fracturing (acidizing) technique that integrates fracture reorientation and fluid diversion to divert the artificial fractures and enhance the chance of connecting fractureand-cavern area in deep carbonate reservoirs. The fibers can form
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strong and stable fiber networks that can effectively block natural fractures and perforations. They automatically and completely degrade under the effect of formation temperature when they make contact with hydrocarbons or solvents after the stimulation (Cohen et al., 2010). Mixing cellulose fiber in concrete has small effect on the compressive strength but can improve the tensile strength of concrete, reduce the fracture width and failure area of concrete, and improve crack resistance (Xiao et al., 2014). A new degradable fiber diversion system has been applied in recent years in carbonate reservoirs (Solares et al., 2008; Retnanto et al., 2012). Liu et al. (2013) introduced a fiber diversion technique to improve vertical coverage and deep placement with proppants. The fibers bridge across perforations and natural fractures to form a filter cake (Cohen et al., 2010). Bukovac et al. (2012) presented a mathematical model for fiber accumulation within perforations and explained how parameters such as formation-permeability contrast and fibercake permeability impact the diversion efficiency (Powell et al., 2007; Cohen et al., 2010; Mahdiyar et al., 2007). Allison et al. (2011) used biodegradable particulates to facilitate the temporary diversion and concentration of frac energy, which has enhanced the success of re-fracturing. Re-fracturing is an important technology for stimulating lowpermeability reservoirs when oil or gas production decreases after initial fracturing (Zhang and Chen, 2010a; 2010b). Siebrits et al. (2000) gave evidence of increased output due to re-fracturing in two tight gas wells. Surface tilt-meter measurements showed refracture orientations at oblique angles to the azimuth of the initial fractures. Zhang et al. (2010a, 2010b) proposed a model for a dynamic fracture propagation path during re-stimulation, and numerical simulation results showed that stress difference and initiation angle resulted in changes of the path. Behnia et al. (2015) used the crack tip element and a higher order displacement discontinuity method to study hydraulic fracture extension and reorientation. They came to the conclusion that fracture reorientation is influenced by stress deviation, fluid pressure and fracture inclination. By using a tri-axial fracturing system in the laboratory, Zhou et al. (2008) found that tortuous fractures are along the fracture height direction if the horizontal stress is the maximum principle stress. Rahman and Rahman (2013) proposed a fully coupled poroelastic model of hydraulic fracture-propagation in fractured reservoirs. The results showed that the orientation of the natural fracture has a remarkable effect on the induced hydraulic fracture propagation. Han and Yang (2009) studied the deformations and re-fracture characteristics of rock mass on the basis of uniaxial and tri-axial experiments. The experimental results showed that the refracture of the rock sample depends on mechanical characteristics of the rock and fracture plane. Degradable fiber has an effective temporary plugging effect on natural or artificial hydraulic fractures (Zhou et al., 2014; Solares et al., 2008; Romero et al., 2002). It can be successfully applied in the SRV fracturing or re-fracturing of unconventional reservoirs (Potapenko et al., 2009; Wang et al., 2015). However, the mechanisms of fracture reorientation with a degradable fiber diverting agent and its influence remain poorly understood. Based on the classical analytical stress field equation, a new mathematical model is established in this paper to model the crack reorientation path after injecting fiber diversion fluid according to the tensile failure criterion. Factors influencing the diverting radius are analyzed through numerical solution thereafter. The results indicate that the horizontal stress difference, fracturing fluid viscosity and injection time (fracturing fluid volume) have greater impacts on the diverting radius than do the formation permeability (1e50 mD) and bottomhole pressure (90e160 MPa). The simulation results are verified with experimental data from true tri-axial fracture reorientation tests.
2. Mathematical model 2.1. Physical model A diagram of the physical model is shown in Fig. 1. Its assumed conditions are as follows: (1) A vertical well is located in finite reservoirs; the reservoir is homogeneous and isotropic; rw denotes the borehole radius; re denotes the reservoir outer radius; re >> rw; k denotes matrix permeability; (2) The pressure at the inner or outer boundary is constant; pw denotes the pressure at the borehole; pe denotes the pressure at the outer boundary; (3) Injection fluid is a slightly compressible fluid; its viscosity m and total compression coefficient ct are invariant with pressure; (4) The original stress field is anisotric; sH denotes the maximum horizontal principal stress; sh denotes the minimum horizontal principal stress; (5) The influence of temperature variation on in-situ stress is out of consideration; (6) Because reservoir thickness is far less than drainage radius, plane strain state can be approximately assumed; (7) The fluid-solid coupling effect is neglected, i.e., formation permeability k, porosity f and fluid viscosity are invariant with pressure; (8) sq denotes the tangential or circumferential stress in polar coordinates; sr denotes the radial stress in polar coordinates. The degradable fiber-assisted diverting fracturing process is composed of three stages: (1) in the first fracturing process, fracturing fluid is injected into the formation to produce an artificial fracture; (2) fibers are injected with low viscosity carrying fluid (the mixture fluid is called DCF, and the carrying fluid is often slick water) to temporarily plug the cracks in the first fracturing. Crosslinked gel liquid with high viscosity is often followed by DCF to improve the plugging effect on cracks; due to the low viscosity of DCF, fluid phase are much easier to filtrate into the fracture wall. Then, the fiber filtration cake remains at the fracture mouth or inside the fracture. (3) In the second fracturing process, fracturing fluid is injected into the wellbore. Because of the good effect on old
Fig. 1. Diagram of the physical model.
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217
fractures, net pressure will remarkably increase in this stage (the effect of fiber is to temporarily plug old fractures and improve net pressure in the fracture; the influence of fiber cake on in-situ stress is out of consideration). Then, a new fracture will initiate and propagate along a different direction. Its fracture pressure in the second fracturing is higher than that in the first fracturing. Repeating steps 2 to 3 can fracture networks. After the stimulation treatment, fibers can be automatically degradable at formation temperature. They change into very low viscous fluids (less than 5 mPa s), which are easier to flow back.
respectively; t denotes the injection time, s; h and c denote elastic coefficients of the rock body, dimensionless unit; L1 is the inverse Laplace transform that is performed by the Stehfest algorithm (Stehfest, 1970); K0 and K1 are the first order modified Bessel function of the first and second kind, respectively. The poroelastic stress field induced by injecting the fracturing fluid for 2 h was obtained by associating Eqs. (1)e(8), as shown in Fig. 2. It shows that the radial stress field and circumferential stress field are changed because of the fluid injection and resultant pore pressure increase.
2.2. Stress field induced by injection fluid
2.3. Stress field induced by open fracture
When fracturing fluids are pumped into a well to create conductive fractures, the wellbore is pressurized. This causes the pore pressure near the borehole to increase, which redistributes the original stress field and leads to a so-called stress field induced by the injection fluid (Detournay and Cheng, 1988). The analytical solution of pore pressure and stress field in a cylindrical coordinate system is shown below:
If a well is fractured to induce a hydraulic fracture, the virgin stress field will be perturbed (Warpinski and Branagan, 1989).
pðr; tÞ ¼ p0 þ ðpw p0 Þgðr; tÞ srr ðr; qÞ ¼
(1)
4 2 sH þ sh r2 sH sh rw rw 1 w 1 þ 3 þ cosð2qÞ 4 2 2 r2 r4 r2 þ pw
2 rw rw þ 2hðpw p0 Þ hðr; tÞ r r2
(2) sqq ðr; qÞ ¼
4 sH þ sh r2 sH sh rw 1þ w 1 þ 3 cosð2qÞ 2 2 r2 r4 i hr r2 w pw w hðr; tÞ þ gðr; tÞ 2hðp p Þ w 0 r r2
(3)
2 s sh r4 rw trq ðr; qÞ ¼ H 13 w sinð2qÞ þ 2 2 r4 r2
(4)
~ðr; sÞ gðr; tÞ ¼ L1 ½g
(5)
~ðr; sÞ ¼ g
K0 ðxÞ sK0 ðbÞ
h i ~ sÞ hðr; tÞ ¼ L1 hðr; ~ sÞ ¼ 1 K1 ðxÞ rw K1 ðbÞ hðr; s bK0 ðbÞ r bK0 ðbÞ
(6)
(7)
(8)
where
b ¼ rw
rffiffiffi rffiffiffi s s ; x¼r ; c c
p0 denotes the initial formation pressure, MPa; pw denotes the flowing bottom hole pressure, MPa; sH denotes the maximum horizontal principle stress, MPa; sh denotes the minimum horizontal principle stress, MPa; sqq denotes the tangential or circumferential stress, MPa; srr denotes the radial stress, MPa; trq denotes the shear stress, MPa; rw denotes the borehole radius, m; r and q denote polar coordinates; r denotes the distance away from the borehole center, m; s denotes the time variable in the Laplace ~ ~ and h, space; g and h denote the inversion Laplace transform of g
Fig. 2. Distribution of poroelastic stress field induced by injecting fluid after 2 h: (a) radial stress field and (b) circumferential stress field.
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Sneddon and Elliot (1946) proposed a model for the stress field around an infinitely long 2D crack (plane strain) in a homogeneous, isotropic, and elastic body. The analytical solution to this model in a Cartesian coordinate system is shown below:
8 > < r q þ q2 Dsx ¼ p pffiffiffiffiffiffiffiffiffi cos q 1 > 2 : r1 r2 9 > = 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q sin ðq1 þ q2 Þ 1 > 2 ; ðr1 r2 Þ3 c2 r
(9)
8 > < r q þ q2 Dsz ¼ p pffiffiffiffiffiffiffiffiffi cos q 1 > 2 : r1 r2 9 > = c2 r 3 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q sin ðq1 þ q2 Þ 1 > 2 ; ðr1 r2 Þ3
Dtxz
8 9 > > < c2 r = 3 ¼ p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin q cos ðq1 þ q2 Þ > > 2 : ðr r Þ3 ; 1 2
Dsy ¼ yðDsx þ Dsz Þ
(10)
(11)
(12)
where p is the net pressure in the crack, Pa; h is the crack height, m; y is Poisson's ratio, m/m. Related geometric relations between the Cartesian and cylindrical systems are given by:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 > þ z2 r ¼ qxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < 2 r1 ¼ x þ ðz cÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : r2 ¼ x2 þ ðz þ cÞ2
(13)
8 x > q ¼ arctan > > > z > > < x q1 ¼ arctan > zc > > x > > > : q2 ¼ arctan zþc
(14)
Negative values of q, q1, and q2 should be replaced, respectively, by p þ q, p þ q1, and p þ q2. Given the artificial fracture height of 50 m, half fracture length of 200 m, and net pressure of 10 MPa, the stress field distribution induced by the fracturing can be determined by the above equations, as shown in Fig. 3. It is found that the closer the stress field location is to fracture, the larger both of the induced stress changes are. In addition, the stress change added to the minimum horizontal direction is greater than that added to the maximum horizontal direction. If the net pressure is large enough or if the horizontal stress contrast is small enough, the minimum and maximum stresses in the virgin stress field can be reversed in the induced stress field, which can be greatly favorable to the fracture re-orientation practice.
Fig. 3. Distribution of stress field induced by open fracture: Stress changes in the (a) maximum and (b) minimum horizontal stress directions, respectively.
denote radial, circumferential, and shear stresses induced by open fracture, respectively, MPa. According to the linear superposition principle, the total stress field after injecting fiber-laden diversion fluid is expressed as:
sr ¼ sr0 þ sr1 þ sr2
(15)
sq ¼ sq0 þ sq1 þ sq2
(16)
trq ¼ trq0 þ trq1 þ trq2
(17)
2.4. Total stress field after injecting fiber-laden diversion fluid It is assumed that sr0, sq0, and trq0 denote radial, circumferential, and shear stresses after drilling the borehole, respectively, MPa; sr1, sq1, and trq1 denote radial, circumferential, and shear stresses induced by fluid injection, respectively, MPa; sr2, sq2, and trq2
where sr0, sq0, and trq0 can be calculated by the elasticity theory. Coordinate transform from the Cartesian to cylindrical coordinate system can be made through the following formulas:
D. Wang et al. / Journal of Natural Gas Science and Engineering 25 (2015) 215e225 Table 1 Physical properties of a base model. Property
Value
Young's modulus(E) Poisson's ratio(y) Permeability(k) Porosity(f) Total compressibility(ct) Viscosity of fracturing fluid(m) Initial reservoir pressure(p0) Bottomhole pressure(pw) Maximum horizontal principle stress(sH) Minimum horizontal principal stress(sh) Borehole radius(rw) Biot coefficient(a) Reservoir outer radius(re)
35,000 MPa 0.28 50 mD 0.08 1.45 103 MPa1 120 mPa s 75 MPa 100 MPa 99.26 MPa 93.47 MPa 0.1 m 0.8 3000 m
Fig. 4. The artificial fracture propagation trajectory determined by the total tangential stress after injecting fiber-laden diversion fluid: (a) the first fracture in the first fracturing; (b) the second fracture in the second fracturing.
219
sr2 ¼ Dsx cos2 q þ Dsy sin2 q þ Dtxy sin 2 q
(18)
sq2 ¼ Ds2x sin2 q þ Ds2y cos2 q Dtxy sinð2qÞ
(19)
trq ¼ Dsy Dsx cos q sin q þ Dtxy cos 2 q
(20)
3. Simulation results 3.1. Initiation and propagation path of fracture reorientation The initiation and propagation paths of reoriented fractures are simulated and analyzed in this part. The physical properties of a base model are listed in Table 1. Tensile failure occurs when the effective tensile stress across some plane in the sample exceeds a critical limitation (Fjar et al., 2008). The failure criterion, which specifies the stress condition by which tensile failure will occur and identifies the location of the failure surface in principal stress space, is given as: sqT0. Therefore, when the tangential or circumferential stress near the wellbore meets this condition, artificial fracture perpendicular to the tangential stress direction will be generated. In other words, the minimum point of tangential stress near the wellbore will be the first to fail. The total stress field distribution after injecting the fiber-laden diversion fluid is determined by using Eqs. (1)e(8), as shown in Fig. 3. According to the rock tensile fracture criterion, for each radius r, artificial cracks extend along the minimum point of tangential stress (hoop stress). Thereby, the crack initiation and extension path can be determined by the following procedures: First, make contour maps for a tangential stress distribution (In the first fracturing, a lot of fracturing fluid is pumped along the wellbore. Because of the increasing pore pressure, there is stress field induced by the fluid injection. After the first fracturing, an open fracture is formed. The stress field induced by the open fracture exists. According to Eq. (16), the total stress field after injecting the fiber-laden diversion fluid can be obtained.). Then, draw a circle at different radius levels to find the minimum point tangent to the contour (According to the fracture criterion, the minimum point of tangential stress near the wellbore will be the first to fail.). Finally,
Fig. 5. Regressed correlations between horizontal stress difference and diverting radius at different viscosity levels.
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all of the cut-off points are connected into a line. Fig. 4 shows the artificial fracture propagation trajectory (red line) (in the web version). The second fracture extended in a new direction, which is due to the stress field change around the wellbore. However, after a certain distance, the extension direction of the crack came back to the original maximum principal stress direction. The diversion radius is defined as the extension distance of the new fracture from the wellbore to the direction turning point. 3.2. Influencing factors of diverting radius 3.2.1. Horizontal principle stress difference Contour maps of the circumferential stress field are obtained at different levels of the horizontal principle stress difference. According to section 3.1, the artificial fracture propagation trajectories corresponding to the horizontal principle stress difference are acquired. On the basis of the measured data, the horizontal principle
stress difference is correlated to the diverting radius by applying the logarithm regression at three different viscosity levels, respectively (Fig. 5). It is found that at the same level of viscosity of the fracturing fluid, a larger horizontal principle stress difference led to a smaller diverting radius. When the stress difference approached 8 MPa, the diverting radius decreased almost to zero, which indicates that the fracture extended in the original maximum principle stress direction, while the second fracture would not initiate in a new direction. At the same level of horizontal principle stress difference, a larger fracturing fluid viscosity results in a greater diverting radius. 3.2.2. Fracturing fluid viscosity In the same manner as that mentioned in Section 3.1, the fracture reorientation trajectories were obtained at different fracturing fluid viscosity levels. Fig. 6 shows the regressed correlations for the viscosity and diverting radius at different stress difference levels. It also shows that at a similar horizontal stress difference, the diverting radius increases when the fluid viscosity is less than 200 mPa s and remains constant when the viscosity is greater than
Fig. 6. Regressed correlations between viscosity and diverting radius at different stress difference levels. Fig. 8. Regressed correlations between formation permeability and diverting radius at different stress difference levels.
Fig. 7. Regressed correlations between injection time and diverting radius at different stress difference levels.
Fig. 9. Regressed correlations between bottomhole pressure and diverting radius at different stress difference levels.
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200 mPa s. Thereby, the optimum viscosity range was determined to be 200300 mPa s. A less viscous fluid would result in a smaller net pressure (Economides and Nolte, 2003; Wang and Zhang, 1998). This will further lead to a minor fracturing induced-stress field and a smaller diverting radius as well. On the contrary, more viscous fluid would cause a larger fracture net pressure, stronger stress field, and a bigger diverting radius (Economides and Nolte, 2003; Wang and Zhang, 1998). When the viscosity exceeded a certain point, the fracture net pressure stayed almost constant and the diverting radius remained unchanged. 3.2.3. Injection time The fracture reorientation trajectories are also obtained at different levels of injection time. Fig. 7 presents the regressed correlations for the injection time and the diverting radius. It can be seen that at the same levels of horizontal stress difference, the longer the injection time is, the larger the diverting radius will be,
100
which indicates that pumping a larger fracturing fluid volume can increase the diverting radius during fracturing treatment. A longer injection led to a larger invasion radius of the fracturing fluid. Therefore, the range of pore pressure changes becomes wider, leading to the spread range of the stress field induced by pore pressure being wider, followed by increasing diverting radius. 3.2.4. Formation permeability The fracture reorientation trajectories were also obtained at different permeability levels, and the correlations between the matrix permeability and diverting radius were regressed and plotted in Fig. 8. It is found that the diverting radius varies between 20 and 25 m when the permeability range is 1e50 mD. This minor change indicated an insignificant effect of permeability on the diverting radius. The common permeability (150 mD) and limited pore pressure change leads to a small stress field induced by the injection fluid. Therefore, the impact of formation permeability on the diverting radius is negligible. 3.2.5. Bottomhole pressure The fracture reorientation trajectories were determined at different levels of bottomhole pressure. Fig. 9 shows the regressed
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y=119.59-21.875x+0.8153x 2 R =0.9457
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(a) 1MPa 2MPa 3MPa 4MPa 5MPa 6MPa 7MPa 8MPa
y (m)
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(b) Fig. 10. Simulation results of the extension path and initiation angle of a diverting fracture at different horizontal principal stress difference levels: (a) initiation angle of diverting fracture and (b) extension path of diverting fracture.
Fig. 11. Schematic diagram of experimental set-up: (a) true tri-axial simulation test device for hydraulic fracturing and (b) 30 30 30 cm3 core sample.
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correlations between the bottomhole pressure and diverting radius. It can be seen that the diverting radius varies between 20 and 25 m when the bottomhole pressure ranges from 90 to 160 MPa. This minor change indicates the small effect of pressure from 90 to 160 MPa on the diverting radius. The pore pressure changes grow larger as the bottomhole pressure increases, resulting in an incremental of the injection-induced stress field (negative). This decreases the tangential stress and increases the fracture net pressure in the meantime, leading to an augmented stress field
induced by artificial fracture increases (positive). These two effects offset each other, together causing the small impact of bottomhole pressure on the diverting radius. 3.3. Initiation angle of fracture reorientation The extension path and initiation angle of the diverting fracture were also simulated at different horizontal principal stress difference levels (Fig. 10). With the increase of principal stress difference,
Fig. 12. Experimental results of fracture reorientation with fiber-diverting agent: (aeb) Sample #1, Ds ¼ 2.5 MPa; (ced) Sample #2, Ds ¼ 5.0 MPa; (eef) Sample #3, Ds ¼ 7.5 MPa. The left figure represents the artificial fracture morphology after the first fracturing; the right figure represents the diverting fracture morphology after the second fracturing.
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4. Model verification 4.1. Laboratory experiment The crack initiation and propagation of hydraulic fracture reorientation with a degradable fiber diverting agent was simulated with a true tri-axial simulation system. This system consisted of a large size, true tri-axial experimental aircraft, MTS servo booster pump, data acquisition system, power source, oil and water separator, and another auxiliary device component (Fig. 11a). The limestone outcrop samples had a dimension of 30 30 30 cm3, as shown in Fig. 11b. There is a drilling hole at the center of block sample at a depth of 17 cm. The hole's diameter is 27 mm. A steel tube is tightly cemented on the hole wall, 10 cm in length. The surplus open hole is 7 cm in length. Before injection fluid, vertical principal stress is applied to the top face, and two horizontal principle stresses are applied, respectively, to the leftright and front-back face. The sample was fractured twice: the first fracturing was conducted with a base fluid that was injected into the hole at the center of the sample to induce an artificial fracture. The secondary stimulation was performed with a fiberladen fluid that was injected into the same hole to bridge and plug the first fracture and create a new second fracture (Shi et al., 2013). The fiber-based fracturing fluid was composed of 0.3% guar, 0.025% citric acid, 1.5% fiber and 0.8% cross-linking agent. The fracturing base fluid consisted of 0.3% guar and 0.025% citric acid, and its viscosity was approximately 27 mPa s. The injection rate was in the range of 5e10 mL/min and was held constant during the two fracturings. Red tracer was added into the fracturing fluids to more clearly observe the artificial fracture propagation path (Shi et al., 2013; Zhou et al., 2009 & 2014). The initiation and propagation path of the diverting fracture was physically simulated at three horizontal principle stress difference levels: 2.5, 5, and 7.5 MPa, respectively (Fig. 12). It can be seen that the first fracture extended along the maximum principal stress direction. The initiation angle of the diverting fracture became larger at a greater horizontal principle stress difference. At 7.5 MPa, the secondary fracture extended along the original maximum principal stress direction, which indicated an ineffective diversion of fracture. The real diverting radius can be measured after the second fracturing. The greater the horizontal principle stress difference is, the less the diverting radius is.
4.2. Model verification The correlation for the experimental stress difference and diverting radius was regressed to testify concerning the validation of the model established in Section 2, as shown in Fig. 13. According to the model in section 2, the circumferential stress contours were plotted for the stress-difference range of 0.5e7.5 MPa to determine the diverting radius. It is found that the simulation results match well with the experimental results. After a stress difference exceeding 7.5 MPa, the diverting fracture hardly occurs. The less the stress difference is, the larger the angle between the two cracks would be, which proved the reliability of the model developed in this study.
4.3. Field application A vertical well, denoted as Well A, is an exploratory well located in the Tarim Basin with an open hole completion. The target formation is in the Ordovician at a depth of 6618.5e6700 m. Its main rocks are gray and taupe gray grainstone, oolitic limestone, bioclast limestone, etc. Oil and gas shows do not appear in the drilling and mud-logging process. Drill-stem testing is conducted before acid fracturing treatment. The results demonstrate that the percolation flow capacity is very weak near the wellbore. There is a lower pressure recover coefficient after shutting in the well, with relatively poor reservoir properties. Seismic data interpretation reveals that there exists a fracture-cave cube in the maximum principal stress direction. The distance from the wellbore to the fracture-cave cube is approximately 55 m, as shown Fig. 14a (Wang, 2013). The maximum principal stress direction in the block area is generally approximately NE40 , but the dipole shear wave image logging results show that the maximum principal stress direction of this well is NW300-330 . The angle between them is approximately 85 . If artificial fractures propagate along the direction NE40 , then it will be bad for connect the fracture-cave cube. Thus, degradable fiber-assisted diverting acid fracturing technology is used to enhance the communication chance. If no obvious pump pressure drop occurred during the first fracturing, then degradable fiber will be injected to temporarily block the formed fractures. The fracturing fluid is then pumped into the bottomhole in the second fracturing process, forcing the hydraulic fracture to initiate in a new direction. The horizontal principal stress difference is 3 MPa, which is lower than 7.5 MPa. According to section 3.2.1, fracture diversion will occur in this case. This hydraulic fracturing construction curve is shown in Fig. 14b. In the first fracturing process, the pump pressure gradually increases from 76.8 MPa to 88.4 MPa at a rate of 5 m3/min. This indicates that the artificial fracture cannot communicate the reservoir body during the first fracturing. Therefore, fiber is injected into the formation, carried by slick water fluid. The fiber-carrying fluid is called DCF. When DCF arrives at the hydraulic fracture mouth, the pump pressure increases by 20 MPa. At the beginning of the second fracturing process, the pump pressure is approximately 90 MPa at a rate of 4.5 m3/min. These demonstrate that this degradable fiber
diverting radius Polynomial Fit of diverting radius 100
y = 0.015x - 0.3855x + 3.711x - 15.895x + 27.961x - 30.865x + 110.32 R = 0.9993
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the crack initiation angle gradually decreased from 90 to 0 (Fig. 10a) and the diverting radius also diminished (Fig. 10b). Beyond a certain distance, the crack tended to propagate back in the original maximum principal stress direction, which agreed with the previous simulation results (Zhang et al., 2008).
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has a good temporary plugging effect on fractures. At the end of the second fracturing, the pump pressure increases by 2 MPa. When acid is pumped into the formation, the pump pressure sharply decreases by more than 20 MPa. Combined with the different fracture pressures during the two stages, fracture diversion occurs in the second fracturing process (Wang, 2013).
According to section 3, the propagation path of the diversion fracture is numerically simulated, as shown in Fig. 14c. Its diverting radius is approximately 62 m, which is very close to the distance between the wellbore and reservoir body (55 m). This again indicates that the model developed in this study is reliable. After the stimulation treatment, natural gas production is 5500 cubic meters per day and water production is 23.92 cubic meters per day. This demonstrates that there is mainly water in the reservoir body. This is a successful field application of this stimulation technology (Wang, 2013). 5. Conclusions This paper developed a new mathematical model for the crack reorientation path after injecting fiber diversion fluid. The following conclusions can be made from this study: 1. Three stress fields (stress field induced by drilling borehole, stress field induced by injection fluid and stress field induced by open fracture) were superposed on the basis of the classical analytical formulation. Then, the total circumferential stress field after injecting fiber diversion fluid was calculated to determine the fracture initiation and propagation path according to the tensile failure criterion. 2. The effects of stress difference, fracturing fluid viscosity, and permeability on the diverting radius were analyzed through numerical simulation. Correlations among them were regressed to determine the conditions for fracture reorientation. Simulation results indicated that the horizontal stress difference, fracturing fluid viscosity and injection time (fracturing fluid volume) have a larger effect on the diverting radius than do the formation permeability (150 mD) and bottomhole pressure (90160 MPa). 3. The simulation results were verified through the true tri-axial fracture reorientation experiment and field application. The limit of stress differences for fracture reorientation were 7.87 and 7.5 MPa from the numerical simulation and laboratory experiment, respectively, which showed a good match. Acknowledgments The authors would like to give our sincere gratitude to the precious financial support from China's Ministry of Science and Technology (973 program, Grant No. 2015CB250903), National Science Foundation of China (Grant No. 51490652 and 41304141), China National Petroleum Cooperation (Scientific Research and Technological Development Project, Grant No. 2010E-2105 and 2014A-4212; Science and Technology Innovation Fund, Grant No. 2013D-5006-0213), as well as Chinese Academy of Sciences (Strategic Leading Science and Technology Project, Grant No. XDB10050203). Abbreviations p0 pw T0 rw re r t k
Fig. 14. Images related with Well A: (a) seismic interpretation results; (b) acid fracturing curve; (c) the numerically simulated propagation path of the diversion fracture.
f m h and c
the initial formation pressure the flowing bottom hole pressure tensile strength of rock borehole radius reservoir outer radius distance away from borehole center the injection time permeability porosity viscosity of fracturing fluid elastic coefficients of rock body
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ct L1 K0 K1 P H E
a y sH sh sr sq trq sx sy txy q
total compressibility the inverse Laplace transform that is performed by the Stehfest algorithm the first order modified Bessel function of the first kind the first order modified Bessel function of the second kind the net pressure in the crack the crack height Young's modulus Biot coefficient Poisson's ratio the maximum horizontal principle stress the minimum horizontal principle stress the radial stress in the cylindrical coordinate system the hoop/ circumferential stress in the cylindrical coordinate system the stress along x axis in the cylindrical coordinate system the shear along y axis in the Cartesian coordinate system the shear stress in the Cartesian coordinate system the shear stress in the Cartesian coordinate system angle
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