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A three-dimensional solution of hydraulic fracture width for wellbore strengthening applications

A three-dimensional solution of hydraulic fracture width for wellbore strengthening applications Determining the width of an induced hydraulic fracture is the first step for applying wellbore strengthening and hydrau- lic fracturing techniques. However, current 2-D analytical solutions obtained from the plane strain assumption may have large uncertainties when the fracture height is small. To solve this problem, a 3-D finite element method (FEM) is used to model wellbore strengthening and calculate the fracture width. Comparisons show that the 2-D plane strain solution is the asymptote of the 3-D FEM solution. Therefore, the 2-D solution may overestimate the fracture width. This indicates that the 2-D solution may not be applicable in 3-D conditions. Based on the FEM modeling, a new 3-D semi-analytical solution for determining the fracture width is proposed, which accounts for the effects of 3-D fracture dimensions, stress anisotropy and borehole inclination. Compared to the 2-D solution, this new 3-D semi-analytical solution predicts a smaller fracture width. This implies that the 2-D-based old design for wellbore strengthening may overestimate the fracture width, which can be reduced using the proposed 3-D solution. It also allows an easy way to calculate the fracture width in complex geometrical and geological conditions. This solution has been verified against 3-D finite element calculations for field applications. Keywords Hydraulic fracture · Fracture width · Wellbore strengthening · Fracture propagation · 3-D modeling 1 Introduction methods (Alberty and McLean 2004; van Oort et al. 2009; Zhang et al. 2016). Drilling operations encounter many challenges when the Numerical modeling indicates that after creating a bi-wing mud weight windows are narrow. For example, severe mud fracture in a wellbore, the tangential stress increases mark- losses or lost circulation is one of the major challenges for edly around the fracture and the wellbore (Fig. 1). When the drilling in a depleted reservoir because of its low fracture fracture is plugged, the wellbore is strengthened which makes gradient. Therefore, wellbore strengthening is required for it difficult to create new fractures around the wellbore. Labo- increasing the fracture gradient of the rock, widening the ratory experiments verie fi d that wellbore strengthening can mud weight window and consequently enhancing the well greatly increase the fracture gradient. For instance, laboratory integrity and mitigating mud losses. Wellbore strengthening experiments in Roubidoux sandstone cores show that the for- techniques have been successfully used to increase formation mation breakdown pressures were increased by 39%–65% and fracture gradient, reduce mud losses and access resources fracture initiation pressures were increased by 15%–36% when that may have been undrillable using conventional drilling oil-based mud with nanoparticles combined with graphite was used to seal the fractures (Contreras et al. 2014). Methods for wellbore strengthening and enhancing fracture gradients have Edited by Yan-Hua Sun been proposed, and great outcomes have been achieved in drilling to reduce drilling risks of mud losses (van Oort et al. * Jincai Zhang zhangjincai@yahoo.com 2009; Zhang and Yin 2017). The drilling industry converges mainly on two wellbore strengthening methods: i.e., the stress * Shangxian Yin yinshx03@126.com cage method (Alberty and McLean 2004) and tip resistance by the development of an immobile mass (Dupriest 2005). Sinopec Tech Houston, Houston, USA Dupriest (2005) pointed out that regardless of the type of treat- North China Institute of Science and Technology, ment used, integrity is increased by widening the fracture to Yanjiao 065201, Hebei, China Vol:.(1234567890) 1 3 Petroleum Science (2019) 16:808–815 809 Surface: Tangential stress difference, psi 1.47×10 ×10 1.2 5 1.0 Fig. 2 Wellbore and fracture growth model 0.8 0.6 Poisson’s ratio; p is the internal pressure in the crack; L is -5 0.4 the fracture half-length or the length in one side of the frac- 0.2 -10 ture; and x is the distance from the crack center. Perkins and Kern (1961) and Geertsma and de Klerk -15 -1.47×10 (1969) applied Sneddon and Elliott’s solution to the oil and -10 0 10 gas industry for hydraulic fracturing applications. They Distance, in treated the borehole as a circular crack with an internal pres- sure inside the borehole and an isotropic far-field horizontal Fig. 1 Increased area of the tangential stress induced by creating a stress exerted to the wellbore. Alberty and McLean (2004) bi-wing fracture in a wellbore modeled by the finite element method proposed a 2-D solution to determine the fracture width in (modeling parameters can be found in Table 1) the stress cage method for wellbore strengthening based on the solution of Sneddon and Elliott (1946). They assumed increase its closing stress. The stress cage wellbore strengthen- that the borehole is a circular crack. Using the linear model ing method has been applied to several hundred wells (Aston of fracture propagation (Sneddon and Elliott 1946; Geertsma et al. 2004, 2007), making those conventionally undrillable and de Klerk 1969), the closed-form solution of the fracture wells (because of their narrow drilling margins) able to be width in the plane strain condition for a vertical well is writ- drilled successfully. For a successful wellbore strengthening ten in the following equation (Alberty and McLean 2004): application, the fracture width is one of the most important 2 √ 4(1 −  ) 2 2 parameters, because the particle concentration required for w(x)= (p − S ) (L + R) − x (2) w h plugging the fracture in wellbore strengthening is dependent strongly on the fracture width as laboratory experiments have where p is the internal mud pressure in the wellbore; S is w h verified (Guo et al. 2014). the horizontal stress with assumption of two equal horizontal The methods for the fracture width calculations are summa- stresses; and R is the wellbore radius (Fig. 2). rized in this paper. Then, a 3-D finite element method (FEM), This model is limited to a vertical borehole with two Comsol Multiphysics software, is used to model wellbore equal horizontal stresses. As the difference between the strengthening and calculate the fracture width. A comparison minimum and maximum horizontal stresses increases, the of 3-D FEM results and the 2-D plane strain solution shows aperture of the fracture becomes larger than that predicted that the 2-D model may overestimate the fracture width. by Eq. 2. Therefore, a 3-D solution for the fracture width is needed, which is the motivation for this paper. 2.2 Semi‑analytical solution accounting for stress anisotropy 2 2‑D solutions of the fracture widths near the wellbores Many of the wells being drilled are deviated, and the three far-field stresses are very different. In this case, the analyti- 2.1 Existing analytical solutions for the fracture cal solution offered by Alberty and McLean (2004) in Eq. (2 ) widths would underestimate the fracture width due to the stress ani- sotropy resulting from the borehole deviation. Therefore, a The 2-D plane strain solution of the fracture width for a circu- new method is needed to handle stress anisotropy for calcu- lar crack was first proposed by Sneddon and Elliott (1946) in lating the fracture width. Some research has been conducted the following equation: considering the stress anisotropy (Guo et al. 2011; Marita and Fuh 2012; Shahri et al. 2015; Mehrabian and Jamison 2015; 2 √ 4(1 −  ) 2 2 w(x)= p L − x (1) Zhong et al. 2017). However, there is no commonly used analytical method to handle the stress anisotropy; therefore, where w(x) is the fracture width or aperture at a distance a numerical method is required to model the fracture width. x from the center of the crack; E is Young’s modulus; ν is The numerical solution is normally performed using complex 1 3 810 Petroleum Science (2019) 16:808–815 S S h h p p w w S S S S S -S S -S H H h h H h H h S S h h 0max p p 0min e 0 p =p -S e w h p =c(S -S ) 0 H h Fig. 3 Schematic wellbore and fracture model under anisotropic far-field stresses, which can be superposed by two separate models commercial software, which requires special training, licens- ing for each potential user and the manual transfer of data from company databases to the commercial FEM software. This requires specialists to conduct the analysis and increases costs due to software licensing, as well as the potential for human-generated errors in manually moving data between the different programs. An analytical solution for the presence of -100 stress anisotropy would be a practical solution to address these -200 FEM (Large anisotropy) issues. Applying the superposition principle (Fig. 3), a new Semi-analytical solution -300 2-D semi-analytical solution for the fracture width in a vertical FEM (Small anisotropy) Semi-analytical solution borehole accounting for in situ stress anisotropy was proposed -400 (Zhang et al. 2016) in the following equation: Distance from the wellbore wall, in 2 √ 4(1 −  ) 2 2 w(x)= [p − S + c(S − S )] (L + R) − x w h H h E Fig. 4 Comparisons of the semi-analytical solution calculated from Eqs. 3 and 4 and the 2-D FEM solution in small and large horizontal (3) stress anisotropies where E is Young’s modulus of the rock (for practical appli- cations, the dynamic modulus is suggested to be used) and c is the stress anisotropy factor. where the units of R, L and x are in inches. Although c has a The derivation of Eq. 3 can be found in the appendix in unit, when the units are consistent, it is a constant; therefore, Zhang et al. (2016). In the derivation, the stress anisotropy fac- c is an empirical parameter. tor (c) was considered to account for the impact of the horizon- In metric unit, if the length and radius are expressed in tal stress difference. It was obtained from the FEM numerical meters, c can be rewritten in the following form: modeling based on more than a hundred case applications. The 1∕2 stress anisotropy factor can be expressed in the following form, 0.137R c = (5) 1∕1.3 if the length and radius are expressed in inches: [L + 3(x − R)] 1∕2 0.368R where the units of R, L and x are in meters. c = (4) 1∕1.3 [L + 3(x − R)] Figure 4 displays the fracture shapes calculated from the semi-analytical solution and from the 2-D FEM numerical model for a small anisotropy case (S –S = 100 psi) and a H h 1 3 Fracture half width, µm Petroleum Science (2019) 16:808–815 811 Table 1 In situ stresses and rock properties used in the FEM analysis Minimum horizon- Maximum hori- Vertical stress Downhole mud Young’s modulus Poisson’s ratio of Hole diameter d, in tal stress S , psi zontal stress S , S , psi pressure p , psi of the rock E, the rock ν h H V w psi Mpsi 5953 6953 7863 6480 1.09 0.225 8.5 large anisotropy case (S –S = 1000 psi). The in situ stresses stress directions (e.g., the vertical and maximum horizontal H h and rock properties used in the analysis are listed in Table 1 stress directions for a vertical hydraulic fracture). In this with varied S , i.e., S = 6053 psi in the small anisotropy case, a 2-D plane strain solution may not work properly, H H case and S = 6953 psi in the large anisotropy case. Figure 4 and a 3-D solution is needed for a better description of near- shows that the results in the two solutions match very well. wellbore fractures. In a wellbore cross section orthogonal to the borehole axis in an inclined borehole, the maximum and minimum far-field principal stresses (S , S ) in the cross section 3 3‑D solution of the fracture width max min can be calculated from the in  situ stresses (S , S , S ). for near‑wellbore fractures h H V Inserting S and S into Eq. 3, the fracture width can max min be obtained for the inclined borehole. In this case, Eq.  3 3.1 3‑D FEM modeling of fracture widths becomes: Numerical methods have been used to model stresses, defor- 4(1 −  ) 2 2 mations and stabilities of boreholes caused by drilling (Feng w(x)= [p − S + c(S − S )] (L + R) − x w max min min et al. 2015; Feng and Gray 2018; Zhang et al. 2003, 2007). (6) In this study, the FEM is used to model stresses and defor- where S and S are the maximum and minimum far- max min mations of near-wellbore hydraulic fractures for wellbore field stresses in the wellbore cross section perpendicular to strengthening applications. Prior to the numerical study, we the borehole axis. In an inclined borehole, S and S can max min validated this numerical method in 3-D wellbores against be estimated using the following equations (Zhang 2013): Kirsch’s analytical solution (Bradley 1979) of wellbore 0 2 2 2 2 stresses and displacements (Zhang et al. 2016). After the =(S cos  + S sin ) cos i + S sin i H h V validation, we built a 3-D elliptical fracture for the finite 0 2 2 = S sin  + S cos H h element modeling (Fig. 5a) to examine the stress distribu- (7) 0 0 tion, fracture deformation and fracture aperture changes in a S = max( ,  ) max x y wellbore. A Gulf of Mexico well from Alberty and McLean 0 0 S = min( ,  ) min x y (2004) is used in the modeling. But here a higher maxi- mum horizontal stress is assumed to examine in situ stress anisotropy. The in situ stresses and rock properties in the where i is the borehole inclination relative to the vertical studied well are listed in Table 1. A 3-D model, similar to direction; α is the borehole azimuth relative to the maximum the PKN elliptical fracture (the PKN is a hydraulic fractur- horizontal stress direction; S is the overburden stress; and 0 0 ing model developed by Perkins and Kern 1961 and then ,  are the intermediate stress values exerted on the well- x y refined by Nordgren 1972), is studied. The 3-D FEM sym- bore cross section in the x- and y-directions, respectively. metrical meshes (the symmetrical plane is the bottom plane of Fig. 5b, i.e., the central horizontal plane of Fig. 5a) shown 2.3 Limitations of the 2‑D plane strain solution in Fig. 5b are used to simulate a borehole with a PKN-type fracture. Using rock mechanics sign convention, the com- The 2-D semi-analytical solution (Eq. 3) has been applied pressive stress is positive, and the tensile stress is negative for wellbore strengthening in several drilling projects in the in this study. Gulf of Mexico and the North Sea. In the 2-D plane strain In the 3-D modeling, we model the fracture width vari- solution, the fracture is assumed very long in one direc- ations with varied fracture heights for each case with a tion compared to other directions. This assumption may be fixed fracture length in order to compare with the 2-D plane appropriate for a long fracture, but it might not be suitable strain solution. The purpose of the modeling is to verify for a near-wellbore fracture or a fracture in its early propa- whether the 2-D plane strain solution is applicable to the gating stage. When a hydraulic fracture is in its early propa- 3-D conditions. In the 3-D FEM modeling, we use Table 1 gation stage, it mainly propagates in the two major principal as the inputs and Fig. 5 as the fracture model. The fracture 1 3 812 Petroleum Science (2019) 16:808–815 Fig. 5 a Schematic 3-D PKN (a) (b) elliptical fracture studied in the 3-D FEM modeling. b FEM half-model is symmetrical on the bottom of the z-plane (i.e., 200 Fracture tip the central horizontal plane of Fig. 5a) and the maximum hori- zontal, minimum horizontal and w(x) vertical stresses are in the x-, y- and z-directions, respectively 100 200 -100 -100 z -200-200 V y x (a) (b) 300 300 250 250 200 200 150 150 100 100 50 50 3D: L=6 in 3D: L=6 in 2D: L=6 in 2D: L=6 in 0 0 04080 120 160 200 240 280 320 360 0102030405060 Fracture height, in H/L Fig. 6 Fracture width variation obtained from 3-D FEM modeling compared to 2-D plane strain FEM result (fracture length L = 6 in.). a Fracture half-width versus the fracture height (H). b Fracture half-width versus the ratio of fracture height to fracture length (H/L). Notice that the frac- ture width and distance from the wellbore wall are plotted in different units for the conventional notation widths have the maximum values in the central plane of the borehole (i.e., the dashed horizontal plane in Fig. 5a). The following analysis focuses on the fracture behavior in this horizon. Figure 6 shows that the fracture half-width, w(x)/2, increases as the fracture height increases (in this case, frac- ture length L = 6 inches). When the fracture height is small (H < 40 inches or H/L < 6), the fracture half-width increases significantly as the fracture height increases. When the frac- ture height is relatively large (H > 60 inches in Fig. 6a or H/L > 10 in Fig. 6b), the fracture width increases slowly and 3D: L=15 in approaches the 2-D plane strain FEM solution (the dashed 2D: L=15 in line in Fig. 6). The same trend is obtained for the cases with 0 2468 10 12 14 16 18 20 22 24 large fracture lengths, as shown in Figs. 7 and 8. It demon- H/L strates that the 2-D plane strain FEM solution is the asymp- tote of the 3-D solution. This implies that the 2-D plane Fig. 7 Fracture width variation versus the ratio of fracture height strain solution might not be applicable in 3-D conditions, to fracture length (H/L) from 3-D FEM modeling compared to 2-D plane strain FEM result (fracture length L = 15 in.) particularly when the fracture height or H/L is small. 1 3 Fracture half width, µm Fracture half width, μm Fracture half width, μm Petroleum Science (2019) 16:808–815 813 3.2 3‑D semi‑analytical solution of the fracture width Fifteen application cases of 3-D FEM near-wellbore frac- tures are modeled, and the fracture widths with different L/H (ratio of fracture length to fracture height) values are calculated in each case. The corresponding 2-D FEM plane 400 strain solution is also obtained for each case. The fracture widths in 2-D and 3-D conditions and L/H in these cases are plotted in Fig. 9. It can be observed that the fracture widths 3D: L=40 in 2D: L=40 in from the 2-D to 3-D solutions are strongly dependent on the ratio of fracture length to fracture height. The following relationship of the 2-D and 3-D solutions is obtained from H/L the FEM modeling results: Fig. 8 Fracture width variation versus the ratio of fracture height 1.2L 2D = 1 + to fracture length (H/L) from 3-D FEM modeling compared to 2-D (8) w H 3D plane strain FEM result (fracture length L = 40 in.) where w and w are the fracture widths in 2-D and 3-D 2D 3D cases, respectively. Or, 2D w = (9) 3D 1.2L where k = 1 + (10) Equations  8–10 are obtained from the fracture initial Low L=6 in Deep L=6 in propagation stage; therefore, they are mainly applicable for Deep L=10 in Deep L=20 in near-wellbore fractures. Deep L=30 in Deep L=40 in Deep L=60 in Low L=6 in We verify Eq.  8 by calculating the 2-D and 3-D FEM L=4 in L=6 in solutions at different fracture length conditions using the L=10 in L=15 in L=20 in L=30 in same input parameters shown in Table  1. We then plot L=40 in Y/X=1.2 w ∕w − 1 and 1.2L∕H versus the fracture heights in 2D 3D 012345 6 Fig. 10. It indicates that the FEM results follow the relation H/L of Eq. 8 very well. We also investigate a deep well to exam- ine the 3-D semi-analytical solution. The in situ stresses and Fig. 9 Fracture widths in 2-D and 3-D FEM solutions related to the rock properties are listed in Table 2. Figure 11 shows the 3-D ratio of fracture length to fracture height for 15 application cases FEM numerical results compared to the results calculated 7 9 (a) (b) W /W -1 W /W -1 2D 3D 2D 3D 6 1.2L/H 1.2L/H 0 0 04080 120 160 200 240 280 320 360 04080 120 160 200 240 280 320 360 Fracture height, in Fracture height, in Fig. 10 FEM results in 3-D and 2-D cases versus L/H. Left: L = 6 inches; right: L = 40 in. 1 3 W /W -1 2D 3D Fracture half width, μm W /W -1 or 1.2L/H 2D 3D W /W -1 or 1.2L/H 2D 3D 814 Petroleum Science (2019) 16:808–815 Table 2 In situ stresses and rock properties used in a deep well Minimum Maximum Vertical stress Downhole mud Young’s modu- Poisson’s ratio of Hole diameter Depth D, ft horizontal stress horizontal stress S , psi pressure p , psi lus of the rock E, the rock ν d, in V w S , psi S , psi Mpsi h H 18,260 18,680 21,680 19,120 3.38 0.15 6.125 25,000 In fact, fracture propagation is markedly dependent on the principal stress magnitudes exerted in the fracture propaga- tion directions. For a vertical hydraulic fracture, the fracture 100 propagation length and height in the horizontal and verti- 50 cal directions in an isotropic and homogeneous formation 0 should be proportional to the far-field stress magnitudes -50 in the corresponding directions; i.e., 2L/H is directly pro- portional to S /S . Therefore, if the fracture propagation is -100 H V FEM 3D: H=360 in -150 dependent on the principal far-field stress magnitudes, then FEM 3D: H=20 in k value in Eq. 10 for a vertical fracture can be rewritten in -200 Proposed 3D Eq., H=20 in Proposed 3D Eq., H=360 in the following form: -250 02468 10 12 0.6S Distance from the wellbore wall, in H k = 1 + (12) Fig. 11 3-D FEM numerical results compared to the proposed 3-D Therefore, for a vertical fracture in a vertical well, Eq. 8 equation (Eq.  8) for a deep well at a depth of 25,000 ft. The w in 2D Eq. 8 is obtained from the 2-D semi-analytical solution (Eq. 3) can be expressed in the following form (Zhang 2019): 2 √ 4(1 −  ) 2 2 w (x)= [p − S + c(S − S )] (L + R) − x 3D w h H h (1 + 0.6S ∕S )E H V from the proposed 3-D fracture width (Eq. 8), in which the (13) w is obtained from the 2-D semi-analytical solution (Eq. 3) 2D This 3-D semi-analytical solution allows an easy way to for a deep well at a depth of 25,000 ft for L = 10 inches and compute the fracture width in complex conditions for well- R = 6.125 inches. Again, the match is very good. bore strengthening, but it has some assumptions and condi- It is commonly assumed that a near-wellbore vertical tions for applications. The assumptions used to obtain the hydraulic fracture propagates uniformly in the vertical and solution include: horizontal directions. Therefore, the fracture length (L) and height (H) for the uniform propagation should satisfy the (1) The studied hydraulic fracture is a near-wellbore verti- condition of H = 2L. Inserting H = 2L into Eq. 10, we obtain cal fracture in an isotropic and homogeneous elastic k = 1.6. For a 3-D radially propagating circular crack or rock; penny-shaped crack, Sneddon’s solution (Sneddon 1946) is (2) The fracture length is short; normally, the half-length equivalent to k = π/2 = 1.57, which is similar to k = 1.6 in the of the fracture is L < 50 inches. linear fracture. The solution of w can be obtained from the 2D (3) The 2-D semi-analytical solution is used to derive the semi-analytical solution in Eq. 3. Therefore, the 3-D semi- 3-D semi-analytical solution, in which the 2-D solution analytical solution of the fracture width can be obtained by is assumed in the plane strain condition. replacing w in Eq. 8 by Eq. 3 and inserting H = 2L. Hence, 2D the 3-D semi-analytical solution of fracture width can be expressed in the following equation: 4 Conclusions 2 √ 4(1 −  ) 2 2 w (x)= [p − S + c(S − S )] (L + R) − x 3D w max min min 1.6E Three-dimensional finite element modeling is applied to (11) model near-wellbore hydraulic fracture behavior and to where w (x) is the fracture width and L is the fracture 3D determine the width of the induced hydraulic fracture. Com- length in one side of the wellbore. Compared to the 2-D parisons demonstrate that the 2-D plane strain solution is solution in Eq. 3, the proposed 3-D solution of Eq. 11 pre- the asymptote of the 3-D solution. This implies that the 2-D dicts a smaller fracture width. solution might not be applicable in 3-D conditions, because it overestimates the fracture width when H/L is small. 1 3 Fracture half width, µm Petroleum Science (2019) 16:808–815 815 Feng Y, Gray KE. 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SPE Drill Complet. 2015;30(1):38–51. https ://doi. ://doi.org/10.1016/j.jngse .2017.01.009. org/10.2118/16800 1-PA. 1 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Petroleum Science Springer Journals

A three-dimensional solution of hydraulic fracture width for wellbore strengthening applications

Zhang, Jincai; Yin, Shangxian
Petroleum Science , Volume 16 (4) – May 3, 2019
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Springer Journals
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Copyright © 2019 by The Author(s)
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Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Policy, Economics and Management
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1672-5107
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1995-8226
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10.1007/s12182-019-0317-7
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Abstract

Determining the width of an induced hydraulic fracture is the first step for applying wellbore strengthening and hydrau- lic fracturing techniques. However, current 2-D analytical solutions obtained from the plane strain assumption may have large uncertainties when the fracture height is small. To solve this problem, a 3-D finite element method (FEM) is used to model wellbore strengthening and calculate the fracture width. Comparisons show that the 2-D plane strain solution is the asymptote of the 3-D FEM solution. Therefore, the 2-D solution may overestimate the fracture width. This indicates that the 2-D solution may not be applicable in 3-D conditions. Based on the FEM modeling, a new 3-D semi-analytical solution for determining the fracture width is proposed, which accounts for the effects of 3-D fracture dimensions, stress anisotropy and borehole inclination. Compared to the 2-D solution, this new 3-D semi-analytical solution predicts a smaller fracture width. This implies that the 2-D-based old design for wellbore strengthening may overestimate the fracture width, which can be reduced using the proposed 3-D solution. It also allows an easy way to calculate the fracture width in complex geometrical and geological conditions. This solution has been verified against 3-D finite element calculations for field applications. Keywords Hydraulic fracture · Fracture width · Wellbore strengthening · Fracture propagation · 3-D modeling 1 Introduction methods (Alberty and McLean 2004; van Oort et al. 2009; Zhang et al. 2016). Drilling operations encounter many challenges when the Numerical modeling indicates that after creating a bi-wing mud weight windows are narrow. For example, severe mud fracture in a wellbore, the tangential stress increases mark- losses or lost circulation is one of the major challenges for edly around the fracture and the wellbore (Fig. 1). When the drilling in a depleted reservoir because of its low fracture fracture is plugged, the wellbore is strengthened which makes gradient. Therefore, wellbore strengthening is required for it difficult to create new fractures around the wellbore. Labo- increasing the fracture gradient of the rock, widening the ratory experiments verie fi d that wellbore strengthening can mud weight window and consequently enhancing the well greatly increase the fracture gradient. For instance, laboratory integrity and mitigating mud losses. Wellbore strengthening experiments in Roubidoux sandstone cores show that the for- techniques have been successfully used to increase formation mation breakdown pressures were increased by 39%–65% and fracture gradient, reduce mud losses and access resources fracture initiation pressures were increased by 15%–36% when that may have been undrillable using conventional drilling oil-based mud with nanoparticles combined with graphite was used to seal the fractures (Contreras et al. 2014). Methods for wellbore strengthening and enhancing fracture gradients have Edited by Yan-Hua Sun been proposed, and great outcomes have been achieved in drilling to reduce drilling risks of mud losses (van Oort et al. * Jincai Zhang zhangjincai@yahoo.com 2009; Zhang and Yin 2017). The drilling industry converges mainly on two wellbore strengthening methods: i.e., the stress * Shangxian Yin yinshx03@126.com cage method (Alberty and McLean 2004) and tip resistance by the development of an immobile mass (Dupriest 2005). Sinopec Tech Houston, Houston, USA Dupriest (2005) pointed out that regardless of the type of treat- North China Institute of Science and Technology, ment used, integrity is increased by widening the fracture to Yanjiao 065201, Hebei, China Vol:.(1234567890) 1 3 Petroleum Science (2019) 16:808–815 809 Surface: Tangential stress difference, psi 1.47×10 ×10 1.2 5 1.0 Fig. 2 Wellbore and fracture growth model 0.8 0.6 Poisson’s ratio; p is the internal pressure in the crack; L is -5 0.4 the fracture half-length or the length in one side of the frac- 0.2 -10 ture; and x is the distance from the crack center. Perkins and Kern (1961) and Geertsma and de Klerk -15 -1.47×10 (1969) applied Sneddon and Elliott’s solution to the oil and -10 0 10 gas industry for hydraulic fracturing applications. They Distance, in treated the borehole as a circular crack with an internal pres- sure inside the borehole and an isotropic far-field horizontal Fig. 1 Increased area of the tangential stress induced by creating a stress exerted to the wellbore. Alberty and McLean (2004) bi-wing fracture in a wellbore modeled by the finite element method proposed a 2-D solution to determine the fracture width in (modeling parameters can be found in Table 1) the stress cage method for wellbore strengthening based on the solution of Sneddon and Elliott (1946). They assumed increase its closing stress. The stress cage wellbore strengthen- that the borehole is a circular crack. Using the linear model ing method has been applied to several hundred wells (Aston of fracture propagation (Sneddon and Elliott 1946; Geertsma et al. 2004, 2007), making those conventionally undrillable and de Klerk 1969), the closed-form solution of the fracture wells (because of their narrow drilling margins) able to be width in the plane strain condition for a vertical well is writ- drilled successfully. For a successful wellbore strengthening ten in the following equation (Alberty and McLean 2004): application, the fracture width is one of the most important 2 √ 4(1 −  ) 2 2 parameters, because the particle concentration required for w(x)= (p − S ) (L + R) − x (2) w h plugging the fracture in wellbore strengthening is dependent strongly on the fracture width as laboratory experiments have where p is the internal mud pressure in the wellbore; S is w h verified (Guo et al. 2014). the horizontal stress with assumption of two equal horizontal The methods for the fracture width calculations are summa- stresses; and R is the wellbore radius (Fig. 2). rized in this paper. Then, a 3-D finite element method (FEM), This model is limited to a vertical borehole with two Comsol Multiphysics software, is used to model wellbore equal horizontal stresses. As the difference between the strengthening and calculate the fracture width. A comparison minimum and maximum horizontal stresses increases, the of 3-D FEM results and the 2-D plane strain solution shows aperture of the fracture becomes larger than that predicted that the 2-D model may overestimate the fracture width. by Eq. 2. Therefore, a 3-D solution for the fracture width is needed, which is the motivation for this paper. 2.2 Semi‑analytical solution accounting for stress anisotropy 2 2‑D solutions of the fracture widths near the wellbores Many of the wells being drilled are deviated, and the three far-field stresses are very different. In this case, the analyti- 2.1 Existing analytical solutions for the fracture cal solution offered by Alberty and McLean (2004) in Eq. (2 ) widths would underestimate the fracture width due to the stress ani- sotropy resulting from the borehole deviation. Therefore, a The 2-D plane strain solution of the fracture width for a circu- new method is needed to handle stress anisotropy for calcu- lar crack was first proposed by Sneddon and Elliott (1946) in lating the fracture width. Some research has been conducted the following equation: considering the stress anisotropy (Guo et al. 2011; Marita and Fuh 2012; Shahri et al. 2015; Mehrabian and Jamison 2015; 2 √ 4(1 −  ) 2 2 w(x)= p L − x (1) Zhong et al. 2017). However, there is no commonly used analytical method to handle the stress anisotropy; therefore, where w(x) is the fracture width or aperture at a distance a numerical method is required to model the fracture width. x from the center of the crack; E is Young’s modulus; ν is The numerical solution is normally performed using complex 1 3 810 Petroleum Science (2019) 16:808–815 S S h h p p w w S S S S S -S S -S H H h h H h H h S S h h 0max p p 0min e 0 p =p -S e w h p =c(S -S ) 0 H h Fig. 3 Schematic wellbore and fracture model under anisotropic far-field stresses, which can be superposed by two separate models commercial software, which requires special training, licens- ing for each potential user and the manual transfer of data from company databases to the commercial FEM software. This requires specialists to conduct the analysis and increases costs due to software licensing, as well as the potential for human-generated errors in manually moving data between the different programs. An analytical solution for the presence of -100 stress anisotropy would be a practical solution to address these -200 FEM (Large anisotropy) issues. Applying the superposition principle (Fig. 3), a new Semi-analytical solution -300 2-D semi-analytical solution for the fracture width in a vertical FEM (Small anisotropy) Semi-analytical solution borehole accounting for in situ stress anisotropy was proposed -400 (Zhang et al. 2016) in the following equation: Distance from the wellbore wall, in 2 √ 4(1 −  ) 2 2 w(x)= [p − S + c(S − S )] (L + R) − x w h H h E Fig. 4 Comparisons of the semi-analytical solution calculated from Eqs. 3 and 4 and the 2-D FEM solution in small and large horizontal (3) stress anisotropies where E is Young’s modulus of the rock (for practical appli- cations, the dynamic modulus is suggested to be used) and c is the stress anisotropy factor. where the units of R, L and x are in inches. Although c has a The derivation of Eq. 3 can be found in the appendix in unit, when the units are consistent, it is a constant; therefore, Zhang et al. (2016). In the derivation, the stress anisotropy fac- c is an empirical parameter. tor (c) was considered to account for the impact of the horizon- In metric unit, if the length and radius are expressed in tal stress difference. It was obtained from the FEM numerical meters, c can be rewritten in the following form: modeling based on more than a hundred case applications. The 1∕2 stress anisotropy factor can be expressed in the following form, 0.137R c = (5) 1∕1.3 if the length and radius are expressed in inches: [L + 3(x − R)] 1∕2 0.368R where the units of R, L and x are in meters. c = (4) 1∕1.3 [L + 3(x − R)] Figure 4 displays the fracture shapes calculated from the semi-analytical solution and from the 2-D FEM numerical model for a small anisotropy case (S –S = 100 psi) and a H h 1 3 Fracture half width, µm Petroleum Science (2019) 16:808–815 811 Table 1 In situ stresses and rock properties used in the FEM analysis Minimum horizon- Maximum hori- Vertical stress Downhole mud Young’s modulus Poisson’s ratio of Hole diameter d, in tal stress S , psi zontal stress S , S , psi pressure p , psi of the rock E, the rock ν h H V w psi Mpsi 5953 6953 7863 6480 1.09 0.225 8.5 large anisotropy case (S –S = 1000 psi). The in situ stresses stress directions (e.g., the vertical and maximum horizontal H h and rock properties used in the analysis are listed in Table 1 stress directions for a vertical hydraulic fracture). In this with varied S , i.e., S = 6053 psi in the small anisotropy case, a 2-D plane strain solution may not work properly, H H case and S = 6953 psi in the large anisotropy case. Figure 4 and a 3-D solution is needed for a better description of near- shows that the results in the two solutions match very well. wellbore fractures. In a wellbore cross section orthogonal to the borehole axis in an inclined borehole, the maximum and minimum far-field principal stresses (S , S ) in the cross section 3 3‑D solution of the fracture width max min can be calculated from the in  situ stresses (S , S , S ). for near‑wellbore fractures h H V Inserting S and S into Eq. 3, the fracture width can max min be obtained for the inclined borehole. In this case, Eq.  3 3.1 3‑D FEM modeling of fracture widths becomes: Numerical methods have been used to model stresses, defor- 4(1 −  ) 2 2 mations and stabilities of boreholes caused by drilling (Feng w(x)= [p − S + c(S − S )] (L + R) − x w max min min et al. 2015; Feng and Gray 2018; Zhang et al. 2003, 2007). (6) In this study, the FEM is used to model stresses and defor- where S and S are the maximum and minimum far- max min mations of near-wellbore hydraulic fractures for wellbore field stresses in the wellbore cross section perpendicular to strengthening applications. Prior to the numerical study, we the borehole axis. In an inclined borehole, S and S can max min validated this numerical method in 3-D wellbores against be estimated using the following equations (Zhang 2013): Kirsch’s analytical solution (Bradley 1979) of wellbore 0 2 2 2 2 stresses and displacements (Zhang et al. 2016). After the =(S cos  + S sin ) cos i + S sin i H h V validation, we built a 3-D elliptical fracture for the finite 0 2 2 = S sin  + S cos H h element modeling (Fig. 5a) to examine the stress distribu- (7) 0 0 tion, fracture deformation and fracture aperture changes in a S = max( ,  ) max x y wellbore. A Gulf of Mexico well from Alberty and McLean 0 0 S = min( ,  ) min x y (2004) is used in the modeling. But here a higher maxi- mum horizontal stress is assumed to examine in situ stress anisotropy. The in situ stresses and rock properties in the where i is the borehole inclination relative to the vertical studied well are listed in Table 1. A 3-D model, similar to direction; α is the borehole azimuth relative to the maximum the PKN elliptical fracture (the PKN is a hydraulic fractur- horizontal stress direction; S is the overburden stress; and 0 0 ing model developed by Perkins and Kern 1961 and then ,  are the intermediate stress values exerted on the well- x y refined by Nordgren 1972), is studied. The 3-D FEM sym- bore cross section in the x- and y-directions, respectively. metrical meshes (the symmetrical plane is the bottom plane of Fig. 5b, i.e., the central horizontal plane of Fig. 5a) shown 2.3 Limitations of the 2‑D plane strain solution in Fig. 5b are used to simulate a borehole with a PKN-type fracture. Using rock mechanics sign convention, the com- The 2-D semi-analytical solution (Eq. 3) has been applied pressive stress is positive, and the tensile stress is negative for wellbore strengthening in several drilling projects in the in this study. Gulf of Mexico and the North Sea. In the 2-D plane strain In the 3-D modeling, we model the fracture width vari- solution, the fracture is assumed very long in one direc- ations with varied fracture heights for each case with a tion compared to other directions. This assumption may be fixed fracture length in order to compare with the 2-D plane appropriate for a long fracture, but it might not be suitable strain solution. The purpose of the modeling is to verify for a near-wellbore fracture or a fracture in its early propa- whether the 2-D plane strain solution is applicable to the gating stage. When a hydraulic fracture is in its early propa- 3-D conditions. In the 3-D FEM modeling, we use Table 1 gation stage, it mainly propagates in the two major principal as the inputs and Fig. 5 as the fracture model. The fracture 1 3 812 Petroleum Science (2019) 16:808–815 Fig. 5 a Schematic 3-D PKN (a) (b) elliptical fracture studied in the 3-D FEM modeling. b FEM half-model is symmetrical on the bottom of the z-plane (i.e., 200 Fracture tip the central horizontal plane of Fig. 5a) and the maximum hori- zontal, minimum horizontal and w(x) vertical stresses are in the x-, y- and z-directions, respectively 100 200 -100 -100 z -200-200 V y x (a) (b) 300 300 250 250 200 200 150 150 100 100 50 50 3D: L=6 in 3D: L=6 in 2D: L=6 in 2D: L=6 in 0 0 04080 120 160 200 240 280 320 360 0102030405060 Fracture height, in H/L Fig. 6 Fracture width variation obtained from 3-D FEM modeling compared to 2-D plane strain FEM result (fracture length L = 6 in.). a Fracture half-width versus the fracture height (H). b Fracture half-width versus the ratio of fracture height to fracture length (H/L). Notice that the frac- ture width and distance from the wellbore wall are plotted in different units for the conventional notation widths have the maximum values in the central plane of the borehole (i.e., the dashed horizontal plane in Fig. 5a). The following analysis focuses on the fracture behavior in this horizon. Figure 6 shows that the fracture half-width, w(x)/2, increases as the fracture height increases (in this case, frac- ture length L = 6 inches). When the fracture height is small (H < 40 inches or H/L < 6), the fracture half-width increases significantly as the fracture height increases. When the frac- ture height is relatively large (H > 60 inches in Fig. 6a or H/L > 10 in Fig. 6b), the fracture width increases slowly and 3D: L=15 in approaches the 2-D plane strain FEM solution (the dashed 2D: L=15 in line in Fig. 6). The same trend is obtained for the cases with 0 2468 10 12 14 16 18 20 22 24 large fracture lengths, as shown in Figs. 7 and 8. It demon- H/L strates that the 2-D plane strain FEM solution is the asymp- tote of the 3-D solution. This implies that the 2-D plane Fig. 7 Fracture width variation versus the ratio of fracture height strain solution might not be applicable in 3-D conditions, to fracture length (H/L) from 3-D FEM modeling compared to 2-D plane strain FEM result (fracture length L = 15 in.) particularly when the fracture height or H/L is small. 1 3 Fracture half width, µm Fracture half width, μm Fracture half width, μm Petroleum Science (2019) 16:808–815 813 3.2 3‑D semi‑analytical solution of the fracture width Fifteen application cases of 3-D FEM near-wellbore frac- tures are modeled, and the fracture widths with different L/H (ratio of fracture length to fracture height) values are calculated in each case. The corresponding 2-D FEM plane 400 strain solution is also obtained for each case. The fracture widths in 2-D and 3-D conditions and L/H in these cases are plotted in Fig. 9. It can be observed that the fracture widths 3D: L=40 in 2D: L=40 in from the 2-D to 3-D solutions are strongly dependent on the ratio of fracture length to fracture height. The following relationship of the 2-D and 3-D solutions is obtained from H/L the FEM modeling results: Fig. 8 Fracture width variation versus the ratio of fracture height 1.2L 2D = 1 + to fracture length (H/L) from 3-D FEM modeling compared to 2-D (8) w H 3D plane strain FEM result (fracture length L = 40 in.) where w and w are the fracture widths in 2-D and 3-D 2D 3D cases, respectively. Or, 2D w = (9) 3D 1.2L where k = 1 + (10) Equations  8–10 are obtained from the fracture initial Low L=6 in Deep L=6 in propagation stage; therefore, they are mainly applicable for Deep L=10 in Deep L=20 in near-wellbore fractures. Deep L=30 in Deep L=40 in Deep L=60 in Low L=6 in We verify Eq.  8 by calculating the 2-D and 3-D FEM L=4 in L=6 in solutions at different fracture length conditions using the L=10 in L=15 in L=20 in L=30 in same input parameters shown in Table  1. We then plot L=40 in Y/X=1.2 w ∕w − 1 and 1.2L∕H versus the fracture heights in 2D 3D 012345 6 Fig. 10. It indicates that the FEM results follow the relation H/L of Eq. 8 very well. We also investigate a deep well to exam- ine the 3-D semi-analytical solution. The in situ stresses and Fig. 9 Fracture widths in 2-D and 3-D FEM solutions related to the rock properties are listed in Table 2. Figure 11 shows the 3-D ratio of fracture length to fracture height for 15 application cases FEM numerical results compared to the results calculated 7 9 (a) (b) W /W -1 W /W -1 2D 3D 2D 3D 6 1.2L/H 1.2L/H 0 0 04080 120 160 200 240 280 320 360 04080 120 160 200 240 280 320 360 Fracture height, in Fracture height, in Fig. 10 FEM results in 3-D and 2-D cases versus L/H. Left: L = 6 inches; right: L = 40 in. 1 3 W /W -1 2D 3D Fracture half width, μm W /W -1 or 1.2L/H 2D 3D W /W -1 or 1.2L/H 2D 3D 814 Petroleum Science (2019) 16:808–815 Table 2 In situ stresses and rock properties used in a deep well Minimum Maximum Vertical stress Downhole mud Young’s modu- Poisson’s ratio of Hole diameter Depth D, ft horizontal stress horizontal stress S , psi pressure p , psi lus of the rock E, the rock ν d, in V w S , psi S , psi Mpsi h H 18,260 18,680 21,680 19,120 3.38 0.15 6.125 25,000 In fact, fracture propagation is markedly dependent on the principal stress magnitudes exerted in the fracture propaga- tion directions. For a vertical hydraulic fracture, the fracture 100 propagation length and height in the horizontal and verti- 50 cal directions in an isotropic and homogeneous formation 0 should be proportional to the far-field stress magnitudes -50 in the corresponding directions; i.e., 2L/H is directly pro- portional to S /S . Therefore, if the fracture propagation is -100 H V FEM 3D: H=360 in -150 dependent on the principal far-field stress magnitudes, then FEM 3D: H=20 in k value in Eq. 10 for a vertical fracture can be rewritten in -200 Proposed 3D Eq., H=20 in Proposed 3D Eq., H=360 in the following form: -250 02468 10 12 0.6S Distance from the wellbore wall, in H k = 1 + (12) Fig. 11 3-D FEM numerical results compared to the proposed 3-D Therefore, for a vertical fracture in a vertical well, Eq. 8 equation (Eq.  8) for a deep well at a depth of 25,000 ft. The w in 2D Eq. 8 is obtained from the 2-D semi-analytical solution (Eq. 3) can be expressed in the following form (Zhang 2019): 2 √ 4(1 −  ) 2 2 w (x)= [p − S + c(S − S )] (L + R) − x 3D w h H h (1 + 0.6S ∕S )E H V from the proposed 3-D fracture width (Eq. 8), in which the (13) w is obtained from the 2-D semi-analytical solution (Eq. 3) 2D This 3-D semi-analytical solution allows an easy way to for a deep well at a depth of 25,000 ft for L = 10 inches and compute the fracture width in complex conditions for well- R = 6.125 inches. Again, the match is very good. bore strengthening, but it has some assumptions and condi- It is commonly assumed that a near-wellbore vertical tions for applications. The assumptions used to obtain the hydraulic fracture propagates uniformly in the vertical and solution include: horizontal directions. Therefore, the fracture length (L) and height (H) for the uniform propagation should satisfy the (1) The studied hydraulic fracture is a near-wellbore verti- condition of H = 2L. Inserting H = 2L into Eq. 10, we obtain cal fracture in an isotropic and homogeneous elastic k = 1.6. For a 3-D radially propagating circular crack or rock; penny-shaped crack, Sneddon’s solution (Sneddon 1946) is (2) The fracture length is short; normally, the half-length equivalent to k = π/2 = 1.57, which is similar to k = 1.6 in the of the fracture is L < 50 inches. linear fracture. The solution of w can be obtained from the 2D (3) The 2-D semi-analytical solution is used to derive the semi-analytical solution in Eq. 3. Therefore, the 3-D semi- 3-D semi-analytical solution, in which the 2-D solution analytical solution of the fracture width can be obtained by is assumed in the plane strain condition. replacing w in Eq. 8 by Eq. 3 and inserting H = 2L. Hence, 2D the 3-D semi-analytical solution of fracture width can be expressed in the following equation: 4 Conclusions 2 √ 4(1 −  ) 2 2 w (x)= [p − S + c(S − S )] (L + R) − x 3D w max min min 1.6E Three-dimensional finite element modeling is applied to (11) model near-wellbore hydraulic fracture behavior and to where w (x) is the fracture width and L is the fracture 3D determine the width of the induced hydraulic fracture. Com- length in one side of the wellbore. Compared to the 2-D parisons demonstrate that the 2-D plane strain solution is solution in Eq. 3, the proposed 3-D solution of Eq. 11 pre- the asymptote of the 3-D solution. This implies that the 2-D dicts a smaller fracture width. solution might not be applicable in 3-D conditions, because it overestimates the fracture width when H/L is small. 1 3 Fracture half width, µm Petroleum Science (2019) 16:808–815 815 Feng Y, Gray KE. 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Published: May 3, 2019

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