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Impact of elliptical boreholes on in situ stress estimation from leak-off test data

Impact of elliptical boreholes on in situ stress estimation from leak-off test data We developed an inversion technique to determine in situ stresses for elliptical boreholes of arbitrary trajectory. In this approach, borehole geometry, drilling-induced fracture information, and other available leak-off test data were used to construct a mathematical model, which was in turn applied to finding the inverse of an overdetermined system of equations. The method has been demonstrated by a case study in the Appalachian Basin, USA. The calculated horizontal stresses are in reasonable agreement with the reported regional stress study of the area, although there are no field measurement data of the studied well for direct calibration. The results also indicate that 2% of axis difference in the elliptical borehole geometry can cause a 5% difference in minimum horizontal stress calculation and a 10% difference in maximum horizontal stress calculation. Keywords Inversion  Leak-off test data  Elliptical borehole  In situ stress 1 Introduction the hole, which are described by the Kirsch equation (Kirsch 1898). In an oilfield subjected to injection or pro- In situ stresses are generated or controlled by a series of duction activities, stresses vary due to volumetric strain geological events such as sedimentation and tectonic changes. This has been found in conventional resource movements. Far-field stresses imposed on the basin reservoir deformation and casing stability analysis boundary are transferred across the basin (Luo and Dus- (Geertsma 1973; Safai and Pinder 1980; Segall and seault 1998). Three orthogonal in situ stresses are normally Fitzgerald 1998; Du and Olson 2001; Soltanzadeh and assumed for the convenience of description and study: Hawkes 2008; Han and Dusseault 2008), caprock integrity vertical stress (r ), maximum horizontal stress (r ), and analysis (Han et al. 2012; Rahmati et al. 2014), and com- V H minimum horizontal stress (r ). Generally, three stress pletion of unconventional resources (Nagel et al. 2013; Han regimes are defined according to the relative magnitude of et al. 2014). In petroleum engineering, especially in the three principal stresses: normal faulting stress regime development of unconventional resources, effective deter- (r [ r [ r ), strike-slip faulting stress regime mination of in situ stress is critical to ensure successful V H h (r [ r [ r ), and thrust faulting stress regime drilling, quality completion, and reservoir containment H V h (r [ r [ r ) (Anderson 1905). In situ stresses vary due analysis. H h V to some changes in the environment. For example, during Vertical in situ stress is usually assumed to be equal to drilling of a circular hole, stresses will redistribute around the weight of the overlying layers per unit area and can be computed by integrating the bulk density log data. The direction of maximum or minimum horizontal in situ stress Edited by Yan-Hua Sun can either be observed from image logs or caliper logs, or be estimated from basin-scale observation of geological & Shunde Yin events. The magnitude of minimum in situ stress can be shunde.yin@uwaterloo.ca measured by leak-off tests (LOT) or hydraulic fracturing Department of Petroleum Engineering, University of tests (Haimson and Fairhurst 1967; Haimson 1974). Small- Wyoming, Laramie, WY 82071, USA scale hydraulic fracturing tests are called mini-frac tests. Petroleum Engineering Department, University of Stavanger, There are many types of mini-frac tests, such as 4036 Stavanger, Norway 123 Petroleum Science (2018) 15:794–800 795 Halliburton’s Diagnostic Fracture Injection Test (DFIT) σ σ North and Schlumberger’s Modular Formation Dynamics Tester σ (MDT). However, there is no tool for direct measurement Vertical of maximum horizontal in situ stress. The determination of West maximum horizontal in situ stress magnitude is often achieved by calculation from borehole breakouts, mini- East mum horizontal stress, and rock mechanical properties such as cohesion, friction angle, and unconfined compres- sive strength (UCS) (Zoback et al. 1985; Peska and Zoback South 1995). There are many other studies of in situ stress γ x determination. Ervin and Bell used breakdown pressure or leak-off pressure from formation leak-off tests to calculate the maximum horizontal stress estimation (Ervin and Bell Well of arbitraty inclination 1987). Cornet and Valette (1984) developed a method and deviation based on normal stress measurements and fast flow rate reopening tests to calculate in situ stresses. Aadnoy (1990) Fig. 1 Sketch map showing well geometry, coordination system, and in situ stress directions developed a method to determine direction and magnitudes of horizontal in situ stresses based on the formation 2 2 2 2 breakdown pressure of a circular borehole of arbitrary r ¼fr cos ð/  bÞþ r sin ð/  bÞg cos c þ r sin c x H h v trajectory. The method has been successfully applied to a ð1Þ set of North Sea data. All the available methods for max- 2 2 r ¼ r sin ð/  bÞþ r cos ð/  bÞð2Þ y H h imum horizontal stress calculation are based on a circular borehole shape. There are currently no reports found on the where r and r are two orthogonal stresses along the cross x y calculation of in situ stresses from elliptical boreholes. section of the arbitrary well trajectory (r is the normal In most basins, the scenario of perfect circular boreholes in situ stress in the x direction; r is the normal in situ is not common. The drilled boreholes have some degree of stress in the y direction); r is the vertical stress; r is the v H elliptical geometry rather than circular geometry due to maximum horizontal stress; r is the minimum horizontal factors of borehole elastic deformation and/or breakouts. It stress; c is the borehole inclination; / is the borehole azi- is therefore necessary to investigate in situ stress inversion muth; b is the angle between r and the north. from a borehole of elliptical geometry and to compare the Suppose r [ r , the fracture pressure of a circular x y difference between the results of elliptical borehole-based borehole can be derived from the Kirsch equations. inversion and circular borehole-based inversion. To this P ¼ P ¼ 3r  r  P þ T ð3Þ f w y x p 0 end, we developed an inversion technique to calculate in situ stress based on the information of borehole mud where P is the pore pressure; T is the rock tensile p 0 pressure that created drilling-induced fracture for any strength; P is the borehole pressure at fracture; P is the f w elliptical borehole of arbitrary well trajectory. borehole pressure. In the following sections, the theory of the proposed For an arbitrary borehole, r is not necessarily always approach is introduced, and case studies are demonstrated greater than r ,if r \ r , the fracture pressure of a cir- y x y from the Appalachian Basin. The results indicate that even cular borehole becomes: a small amount of borehole ellipticity has considerable P ¼ P ¼ 3r  r  P þ T ð4Þ f w x y p 0 influence on the magnitude of estimated horizontal in situ stresses. However, in real drilling practice, even when a borehole is drilled as a circular hole, elliptical boreholes are often observed because of the heterogeneous stress concentration 2 Mathematical model around the borehole. This will happen as either a defor- mation of the borehole or as wellbore breakouts. Lekhnit- To develop stress equations around elliptical boreholes, we skii (1968) investigated the tangential stresses on the short started with a circular borehole assumption. For an arbi- or long axis of the elliptical hole in plates under tension. trary well trajectory with a given inclination and azimuth Similarly, the elliptical borehole geometry under com- as shown in Fig. 1, the two stresses normal to the circular pression is illustrated in Fig. 2. The two orthogonal com- well bore can be written as: pression stresses r and r apply along the cross section of x y an inclined wellbore. By adapting the work of Lekhnitskii, 123 796 Petroleum Science (2018) 15:794–800 σ σ x x (a) (b) Breakout Breakout σ σ σ B σ y y y y Fracture a Fracture point point σ σ x x Fig. 2 Cross sections of elliptical borehole under the two orthogonal compression stresses r and r . a r [ r , b r \r x y x y x y the tangential stresses at the short or long axis of the It is observed from Eqs. (10) and (12) that the equations elliptical hole under compression can be written as Eqs. (5) are linear. Well deviation c and azimuth / are constants or (6) in the case of r [ r . x y that depend on the well geometry. For an elliptical bore- r ¼ð2c þ 1Þr  r ð2c  1ÞP ð5Þ hole, the axis ratio c and the direction of maximum hori- B y x w zontal stress b will be known. The two unknown factors r ¼ð2=c þ 1Þr  r ð2=c  1ÞP ð6Þ A x y w r =r and r =r are separated on the right side of the H v h v In the case of r \ r , the two equations become: equations. If there are multiple wells or multiple points in a x y single well that have fractured wellbores and share the r ¼ð2c þ 1Þr  r ð2c  1ÞP ð7Þ B x y w same stresses state, a system of equations can be con- r ¼ð2=c þ 1Þr  r ð2=c  1ÞP ð8Þ A y x w structed in the following matrix form: where r is the tangential stress at the long axis point A of A ½P¼ ½A½rð13Þ the elliptical borehole; r is the tangential at the short axis where all parameters on the left-hand side of Eq. (10) point B of the elliptical borehole; c is the ratio of short axis or (12) can be lumped into the matrix [P]; the constant on b over long axis a. the right-hand side can be included into matrix [A]; the When a fracture was induced at point B,if r [ r , x y stresses matrix can be solved by inverse operation of P ¼fð2c þ 1Þr  r  P þ T g=ð2c  1Þð9Þ f y x p 0 Eq. (13): Combining Eq. (9) with Eqs. (1) and (2), we obtain ½r¼ ½An½Pð14Þ fð2c  1ÞP þ P  T g=r þ sin ðcÞ f p 0 v The error is defined as the following: 2 2 2 ¼fð2c þ 1Þsin ð/  bÞ cos ð/  bÞcos ðcÞgðr =r Þ H v ½e¼ ½A½r½Pð15Þ 2 2 2 þfð2c þ 1Þcos ð/  bÞ sin ð/  bÞcos ðcÞgðr =r Þ h v In actual calculation, this error will be minimized by a ð10Þ least-squares method. It is necessary to ensure that the determinant of matrix [A] is non-singular. It should also be if r \ r , x y noted that at least two fracture measurements are needed; P ¼fð2c þ 1Þr  r  P þ T g=ð2c  1Þð11Þ f x y p 0 the more observations than unknowns the better, to have an over-constrained system of equations. For each inversion, Combining Eq. (11) with Eqs. (1) and (2), we obtain the magnitude of the two calculated stresses r and r x y fð2c  1ÞP þ P  T g=r ð2c þ 1Þsin ðcÞ f p 0 v needs to be checked and verified. If r [ r ,Eq. (10) x y 2 2 2 ¼fð2c þ 1Þcos ð/  bÞcos ðcÞ sin ð/  bÞgðr =r Þ H v needs to be used, otherwise Eq. (12) needs to be used. 2 2 2 If the lengths of the two axes are very close, the c value þfð2c þ 1Þsin ð/  bÞcos ðcÞ cos ð/  bÞgðr =r Þ h v will be close to 1. This will be a circular borehole case. In ð12Þ such a case, the calculation will be repeated for all 0  b  180 , the square error is calculated and plotted as 123 Petroleum Science (2018) 15:794–800 797 a function of b, and the minimum value of the squared were reported in several sections in Middlesex and Hun- error gives the direction of maximum horizontal stress and tersville Formations in the image log of the MIP 3H pilot the ratios of the two horizontal stresses over vertical stress hole. In this study we chose fractured sections in the for this angle b. Normally, there will be two b values that Middlesex Formation, which have shown obvious ovality are 90 different and both have minimum squared error. from caliper logs. Figure 3 shows the image log that has Choose the one that corresponds to r [ r , but discard drilling-induced fractures in a section of the Middlesex H h the other one. In matrix [P], the tensile strength T is often Formation. Figure 4 shows the caliper logs at the corre- set to zero for circular borehole-based inversion. sponding depth section of the Middlesex Formation. In the case of elliptical boreholes, well deviation c, well Four-arm caliper logs give the ovality of the sections azimuth /, maximum horizontal stress direction b, and axis that have drilling-induced fractures. In the Middlesex ratio c are all known. In the inversion of two horizontal Formation the ratio of the two axes is around 0.978–0.984. in situ stress magnitudes, a set of T values can be used for The vertical normal stress is assumed to be equal to the calculation. The T value corresponding to the minimum weight of the overlying rock and can be computed by squared error will be considered as the best estimation of integrating the bulk density log data. There is no direct rock tensile strength. This is additional information to the measurement for pore pressure in this well. Eaton’s method estimated stresses in the elliptical borehole-based inver- was applied for pore pressure estimation using acoustic sions. The ratios of the two horizontal stresses at the slowness logging. Pore pressure was calculated from ver- minimum squared error will be the estimated stress ratios. tical stress (calculated from the density log) and the 9.25 Longer diameter 9.20 3 Application of the method Shorter diameter 9.15 9.10 The method has been demonstrated by a field study in West 9.05 Virginia, in the southern part of the Appalachian Basin, 9.00 USA. Drilling data and borehole geometry information of 8.95 the MIP 3H pilot hole were used for both circular borehole- 8.90 based and elliptical borehole-based in situ stress inversions. 8.85 The input data for the inversion calculation of horizontal 8.80 stresses include mud weight pressure at fracture, pore 6798 6799 6800 6801 6802 6803 6804 6805 6806 6807 6808 6809 pressure, vertical stress, borehole deviation, and borehole Dept, ft azimuth. For the cases of elliptical borehole-based in situ stress inversion, axis ratio and induced fracture azimuth are Fig. 4 Four-arm caliper log showing the longer and shorter diameters at corresponding depth in the Middlesex Formation included in the input parameters. Drilling-induced fractures Fig. 3 Section of image log of MIP 3H pilot hole showing induced fractures in the Middlesex Formation Axis diameter, in 798 Petroleum Science (2018) 15:794–800 Table 1 Data input for stress Data set P , psi/ft P , psi/ft r , psi/ft c,  /,  b,  c f p v inversion in the Middlesex Formation 1 0.6325 0.4264 1.0239 1.0121 296.7038 228 0.9844 2 0.6325 0.6767 1.0262 1.0701 274.3615 239 0.9799 3 0.6325 0.6687 1.0262 1.1115 271.6611 239 0.9805 4 0.6325 0.6445 1.0262 1.1276 271.0305 239 0.9841 5 0.6325 0.6367 1.0262 1.142 270.5145 239 0.9832 6 0.6325 0.6333 1.0262 1.1583 269.7962 239 0.9813 7 0.6325 0.6142 1.0262 1.1743 269.3157 239 0.9810 8 0.6325 0.6062 1.0262 1.1904 268.8706 239 0.9779 Inversed stresses from drilling induced fractures Calculated stress inversion from elliptical borehole 0.8 1.0 σσ / H v σσ / h v 0.9 0.7 Errors σσ / 0.8 H v σσ h / v 0.6 Errors 0.7 0.5 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Tensile strength, psi/ft Direction of σ , degrees Fig. 6 Elliptical borehole stress inversion results for the Middlesex Fig. 5 Circular borehole stress inversion results for the Middlesex Formation Formation direction are 0.728 and 0.553, respectively, which are 0.75 difference between actual logged data and the normal trend and 0.57 psi/ft in magnitude at corresponding depth in the line of the acoustic slowness. Middlesex Formation. The calculated maximum horizontal In the drilling process, the mud is circulated in and out stress direction of 229 (or 49) is in reasonable agreement of the well bore by pumping devices. Therefore, the mud with the field observations of the well, which were reported pressure will be higher than the static mud weight. This is as 228 and 239 in the maximum horizontal stress direc- called equivalent circulating density (ECD) of mud weight. tion at the corresponding depth in the Middlesex The value will be normally 3% higher than the static mud Formation. weight. The value of 1.03 times of mud weight was used in In order to compare with the circular borehole-based our calculation. inversion, elliptical borehole-based inversion was run. Table 1 lists the data prepared for the stress inversion. Figure 6 shows the inversion results using the elliptical There are eight datasets in the Middlesex Formation. Both equation for the Middlesex Formation. The tensile strength circular and elliptical equations were applied to the T was assumed a series of small values. In this case, the calculation. squared errors of all runs of the assumed strength T values Figure 5 shows the inversion results for the Middlesex are 0.002. It was not able to differentiate the satisfied T Formation using the circular borehole equation. In the value having smallest error. In comparison with the circular calculation, b was scanned from 0 to 180. Two minimum borehole inversion, in the elliptical borehole for T ¼ 0, the squared error values of 0.0005 were found at 49 and 139. stresses ratios r =r and r [ r are 0.925 and 0.602, H v h v Because at 139 the maximum horizontal stress r is respectively, which are 0.95 and 0.62 psi/ft in magnitude of wrongly smaller than the minimum horizontal stress r , corresponding depth in the Middlesex Formation. 49 (or 229) was chosen as the direction of the maximum horizontal stress. The stress ratios r =r and r =r at this H v h v Stress ratio and error Stress ratio and error Petroleum Science (2018) 15:794–800 799 Table 2 Comparison of inversion results between circular borehole and elliptical borehole in the Middlesex Formation Circular hole Elliptical hole Field observation r , psi/ r , psi/ Direction of r from r , psi/ r , psi/ Direction of r from r , psi/ft r , psi/ft Direction of r from H h H H h H H h H ft ft north,  ft ft north,  north, 0.75 0.57 49 (229) 0.95 0.62 Same as field Up to Up to Measured value 228 and a a observation 1.04 0.73 239 Values calculated using vertical stress gradient of the case study and the stress ratio from the Appalachian Stress Study report 4 Discussion 5 Conclusion Table 2 shows the comparison of the inversion results We can use inverse analysis to estimates of in situ hori- between circular borehole-based calculation and elliptical zontal stresses in an elliptical borehole from leak-off test borehole-based calculation. Large differences exist data; case studies show that even a small amount of around between these two calculations. Tensile strengths are 2% axis difference in an elliptical borehole will cause assumed to be zero in these two calculations for the con- differences of 5%–10% in the estimation of horizontal venience of comparison. All the input parameters are same stresses. Inversion using elliptical borehole equations gives except the axis ratio, which is unit value for a circular better in situ stress estimation than those from circular borehole-based calculation and 0.978–0.984 for the ellip- borehole equations. tical borehole-based calculation. Acknowledgements The support of the United States Department of The results indicate that a 2% axis difference in an Energy (DE-FE0026825, UCFER-University Coalition for Fossil elliptical borehole will cause a 5% difference in the min- Energy Research) is greatly acknowledged. Authors are grateful to the imum horizontal stress calculation and a 10% difference in US Department of Energy and West Virginia University for supplying the maximum horizontal stress calculation. Although there the field data. Special thanks are given to Professor Mark Zoback at Stanford University who has provided constructive suggestions. is no measurement information about horizontal in situ stress magnitudes for this formation, the basin wide stress Open Access This article is distributed under the terms of the Creative study indicated r =r value of up to 0.7 in the corre- h v Commons Attribution 4.0 International License (http://creative sponding depth at this well location (Evans 1989). Evans commons.org/licenses/by/4.0/), which permits unrestricted use, dis- tribution, and reproduction in any medium, provided you give also stated in the Appalachian Stress Study report that the appropriate credit to the original author(s) and the source, provide a magnitude of r varies from high in the northern part of link to the Creative Commons license, and indicate if changes were the basin to low values in the south; the stress state in the made. Devonian shale, of which Middlesex Formation is a part, is either strike-slip or normal fault regime due to the pinch- References out of the underlying salt (Evans 1989). The location of the study well is around the pinch-out area (Pierce and Rich Aadnoy BS. Inversion technique to determine the in situ stress field 1962). Therefore, a unit value of r =r stress ratio should H v from fracturing data. J Pet Sci Eng. 1990;4:127–41. https://doi. be an upper limit. However, the estimated r =r stress H v org/10.1016/0920-4105(90)90021-T. ratio of 0.728 from circular borehole-based calculation Anderson EM. The dynamics of faulting. Trans Edinb Geol Soc. 1905;83:387–402. deviates substantially from that value. With elliptical Cornet FH, Valette B. In situ stress determination from hydraulic borehole inversion, the inversion results of stresses ratios injection test data. J Geophys Res. 1984;89(13):11527–37. r =r and r =r are 0.925 and 0.602, respectively, which H v h v https://doi.org/10.1144/transed.8.3.387. are closer to the ratios reported in the Appalachian Stress Du J, Olson JE. A poroelastic reservoir model for predicting subsidence and mapping subsurface pressure fronts. J Pet Sci Study report. It can be seen that inversion using elliptical Eng. 2001;30(3–4):181–97. https://doi.org/10.1016/S0920- borehole equations gives better in situ stress estimation 4105(01)00131-0. than that by circular boreholes stress equations. Ervine WB, Bell JS. Subsurface in situ stress magnitudes from oil- Since the inverted stress magnitudes are sensitive to the well drilling records: an example from the Venture Area, borehole diameter ratio, the accuracy of the four-arm offshore eastern Canada. Can J Earth Sci. 1987;24:1748–59. https://doi.org/10.1139/e87-167. calipers is important. If the differences are too small to be Evans KF. Appalachian stress study: 3. Regional scale stress picked up by the four-arm calipers, the results will be same variations and their relation to structure and contemporary as the circular borehole calculations. tectonics. J Geophys Res. 1989;94(B12):17619–45. https://doi. org/10.1029/JB094iB12p17619. 123 800 Petroleum Science (2018) 15:794–800 Geertsma J. Land subsidence above compacting oil and gas Nagel N, Zhang, F, Sanchez-Nagel M, Lee B, Agharazi A. Stress reservoirs. J Pet Technol. 1973;25:734–44. https://doi.org/10. shadow evaluations for completion design in unconventional 2118/3730-PA. plays. 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In: SPE/ISRM rock mechanics in org/10.1029/JB090iB07p05523. petroleum engineering conference, Trondheim, Norway, 8–10 July; 1998. http://dx.doi.org/10.2118/47321-MS. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Petroleum Science Springer Journals

Impact of elliptical boreholes on in situ stress estimation from leak-off test data

Han, Hong; Yin, Shunde; Aadnoy, Bernt
Petroleum Science , Volume 15 (4) – Jul 17, 2018
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Springer Journals
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Copyright © 2018 by The Author(s)
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Earth Sciences; Mineral Resources; Industrial Chemistry/Chemical Engineering; Industrial and Production Engineering; Energy Economics
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Abstract

We developed an inversion technique to determine in situ stresses for elliptical boreholes of arbitrary trajectory. In this approach, borehole geometry, drilling-induced fracture information, and other available leak-off test data were used to construct a mathematical model, which was in turn applied to finding the inverse of an overdetermined system of equations. The method has been demonstrated by a case study in the Appalachian Basin, USA. The calculated horizontal stresses are in reasonable agreement with the reported regional stress study of the area, although there are no field measurement data of the studied well for direct calibration. The results also indicate that 2% of axis difference in the elliptical borehole geometry can cause a 5% difference in minimum horizontal stress calculation and a 10% difference in maximum horizontal stress calculation. Keywords Inversion  Leak-off test data  Elliptical borehole  In situ stress 1 Introduction the hole, which are described by the Kirsch equation (Kirsch 1898). In an oilfield subjected to injection or pro- In situ stresses are generated or controlled by a series of duction activities, stresses vary due to volumetric strain geological events such as sedimentation and tectonic changes. This has been found in conventional resource movements. Far-field stresses imposed on the basin reservoir deformation and casing stability analysis boundary are transferred across the basin (Luo and Dus- (Geertsma 1973; Safai and Pinder 1980; Segall and seault 1998). Three orthogonal in situ stresses are normally Fitzgerald 1998; Du and Olson 2001; Soltanzadeh and assumed for the convenience of description and study: Hawkes 2008; Han and Dusseault 2008), caprock integrity vertical stress (r ), maximum horizontal stress (r ), and analysis (Han et al. 2012; Rahmati et al. 2014), and com- V H minimum horizontal stress (r ). Generally, three stress pletion of unconventional resources (Nagel et al. 2013; Han regimes are defined according to the relative magnitude of et al. 2014). In petroleum engineering, especially in the three principal stresses: normal faulting stress regime development of unconventional resources, effective deter- (r [ r [ r ), strike-slip faulting stress regime mination of in situ stress is critical to ensure successful V H h (r [ r [ r ), and thrust faulting stress regime drilling, quality completion, and reservoir containment H V h (r [ r [ r ) (Anderson 1905). In situ stresses vary due analysis. H h V to some changes in the environment. For example, during Vertical in situ stress is usually assumed to be equal to drilling of a circular hole, stresses will redistribute around the weight of the overlying layers per unit area and can be computed by integrating the bulk density log data. The direction of maximum or minimum horizontal in situ stress Edited by Yan-Hua Sun can either be observed from image logs or caliper logs, or be estimated from basin-scale observation of geological & Shunde Yin events. The magnitude of minimum in situ stress can be shunde.yin@uwaterloo.ca measured by leak-off tests (LOT) or hydraulic fracturing Department of Petroleum Engineering, University of tests (Haimson and Fairhurst 1967; Haimson 1974). Small- Wyoming, Laramie, WY 82071, USA scale hydraulic fracturing tests are called mini-frac tests. Petroleum Engineering Department, University of Stavanger, There are many types of mini-frac tests, such as 4036 Stavanger, Norway 123 Petroleum Science (2018) 15:794–800 795 Halliburton’s Diagnostic Fracture Injection Test (DFIT) σ σ North and Schlumberger’s Modular Formation Dynamics Tester σ (MDT). However, there is no tool for direct measurement Vertical of maximum horizontal in situ stress. The determination of West maximum horizontal in situ stress magnitude is often achieved by calculation from borehole breakouts, mini- East mum horizontal stress, and rock mechanical properties such as cohesion, friction angle, and unconfined compres- sive strength (UCS) (Zoback et al. 1985; Peska and Zoback South 1995). There are many other studies of in situ stress γ x determination. Ervin and Bell used breakdown pressure or leak-off pressure from formation leak-off tests to calculate the maximum horizontal stress estimation (Ervin and Bell Well of arbitraty inclination 1987). Cornet and Valette (1984) developed a method and deviation based on normal stress measurements and fast flow rate reopening tests to calculate in situ stresses. Aadnoy (1990) Fig. 1 Sketch map showing well geometry, coordination system, and in situ stress directions developed a method to determine direction and magnitudes of horizontal in situ stresses based on the formation 2 2 2 2 breakdown pressure of a circular borehole of arbitrary r ¼fr cos ð/  bÞþ r sin ð/  bÞg cos c þ r sin c x H h v trajectory. The method has been successfully applied to a ð1Þ set of North Sea data. All the available methods for max- 2 2 r ¼ r sin ð/  bÞþ r cos ð/  bÞð2Þ y H h imum horizontal stress calculation are based on a circular borehole shape. There are currently no reports found on the where r and r are two orthogonal stresses along the cross x y calculation of in situ stresses from elliptical boreholes. section of the arbitrary well trajectory (r is the normal In most basins, the scenario of perfect circular boreholes in situ stress in the x direction; r is the normal in situ is not common. The drilled boreholes have some degree of stress in the y direction); r is the vertical stress; r is the v H elliptical geometry rather than circular geometry due to maximum horizontal stress; r is the minimum horizontal factors of borehole elastic deformation and/or breakouts. It stress; c is the borehole inclination; / is the borehole azi- is therefore necessary to investigate in situ stress inversion muth; b is the angle between r and the north. from a borehole of elliptical geometry and to compare the Suppose r [ r , the fracture pressure of a circular x y difference between the results of elliptical borehole-based borehole can be derived from the Kirsch equations. inversion and circular borehole-based inversion. To this P ¼ P ¼ 3r  r  P þ T ð3Þ f w y x p 0 end, we developed an inversion technique to calculate in situ stress based on the information of borehole mud where P is the pore pressure; T is the rock tensile p 0 pressure that created drilling-induced fracture for any strength; P is the borehole pressure at fracture; P is the f w elliptical borehole of arbitrary well trajectory. borehole pressure. In the following sections, the theory of the proposed For an arbitrary borehole, r is not necessarily always approach is introduced, and case studies are demonstrated greater than r ,if r \ r , the fracture pressure of a cir- y x y from the Appalachian Basin. The results indicate that even cular borehole becomes: a small amount of borehole ellipticity has considerable P ¼ P ¼ 3r  r  P þ T ð4Þ f w x y p 0 influence on the magnitude of estimated horizontal in situ stresses. However, in real drilling practice, even when a borehole is drilled as a circular hole, elliptical boreholes are often observed because of the heterogeneous stress concentration 2 Mathematical model around the borehole. This will happen as either a defor- mation of the borehole or as wellbore breakouts. Lekhnit- To develop stress equations around elliptical boreholes, we skii (1968) investigated the tangential stresses on the short started with a circular borehole assumption. For an arbi- or long axis of the elliptical hole in plates under tension. trary well trajectory with a given inclination and azimuth Similarly, the elliptical borehole geometry under com- as shown in Fig. 1, the two stresses normal to the circular pression is illustrated in Fig. 2. The two orthogonal com- well bore can be written as: pression stresses r and r apply along the cross section of x y an inclined wellbore. By adapting the work of Lekhnitskii, 123 796 Petroleum Science (2018) 15:794–800 σ σ x x (a) (b) Breakout Breakout σ σ σ B σ y y y y Fracture a Fracture point point σ σ x x Fig. 2 Cross sections of elliptical borehole under the two orthogonal compression stresses r and r . a r [ r , b r \r x y x y x y the tangential stresses at the short or long axis of the It is observed from Eqs. (10) and (12) that the equations elliptical hole under compression can be written as Eqs. (5) are linear. Well deviation c and azimuth / are constants or (6) in the case of r [ r . x y that depend on the well geometry. For an elliptical bore- r ¼ð2c þ 1Þr  r ð2c  1ÞP ð5Þ hole, the axis ratio c and the direction of maximum hori- B y x w zontal stress b will be known. The two unknown factors r ¼ð2=c þ 1Þr  r ð2=c  1ÞP ð6Þ A x y w r =r and r =r are separated on the right side of the H v h v In the case of r \ r , the two equations become: equations. If there are multiple wells or multiple points in a x y single well that have fractured wellbores and share the r ¼ð2c þ 1Þr  r ð2c  1ÞP ð7Þ B x y w same stresses state, a system of equations can be con- r ¼ð2=c þ 1Þr  r ð2=c  1ÞP ð8Þ A y x w structed in the following matrix form: where r is the tangential stress at the long axis point A of A ½P¼ ½A½rð13Þ the elliptical borehole; r is the tangential at the short axis where all parameters on the left-hand side of Eq. (10) point B of the elliptical borehole; c is the ratio of short axis or (12) can be lumped into the matrix [P]; the constant on b over long axis a. the right-hand side can be included into matrix [A]; the When a fracture was induced at point B,if r [ r , x y stresses matrix can be solved by inverse operation of P ¼fð2c þ 1Þr  r  P þ T g=ð2c  1Þð9Þ f y x p 0 Eq. (13): Combining Eq. (9) with Eqs. (1) and (2), we obtain ½r¼ ½An½Pð14Þ fð2c  1ÞP þ P  T g=r þ sin ðcÞ f p 0 v The error is defined as the following: 2 2 2 ¼fð2c þ 1Þsin ð/  bÞ cos ð/  bÞcos ðcÞgðr =r Þ H v ½e¼ ½A½r½Pð15Þ 2 2 2 þfð2c þ 1Þcos ð/  bÞ sin ð/  bÞcos ðcÞgðr =r Þ h v In actual calculation, this error will be minimized by a ð10Þ least-squares method. It is necessary to ensure that the determinant of matrix [A] is non-singular. It should also be if r \ r , x y noted that at least two fracture measurements are needed; P ¼fð2c þ 1Þr  r  P þ T g=ð2c  1Þð11Þ f x y p 0 the more observations than unknowns the better, to have an over-constrained system of equations. For each inversion, Combining Eq. (11) with Eqs. (1) and (2), we obtain the magnitude of the two calculated stresses r and r x y fð2c  1ÞP þ P  T g=r ð2c þ 1Þsin ðcÞ f p 0 v needs to be checked and verified. If r [ r ,Eq. (10) x y 2 2 2 ¼fð2c þ 1Þcos ð/  bÞcos ðcÞ sin ð/  bÞgðr =r Þ H v needs to be used, otherwise Eq. (12) needs to be used. 2 2 2 If the lengths of the two axes are very close, the c value þfð2c þ 1Þsin ð/  bÞcos ðcÞ cos ð/  bÞgðr =r Þ h v will be close to 1. This will be a circular borehole case. In ð12Þ such a case, the calculation will be repeated for all 0  b  180 , the square error is calculated and plotted as 123 Petroleum Science (2018) 15:794–800 797 a function of b, and the minimum value of the squared were reported in several sections in Middlesex and Hun- error gives the direction of maximum horizontal stress and tersville Formations in the image log of the MIP 3H pilot the ratios of the two horizontal stresses over vertical stress hole. In this study we chose fractured sections in the for this angle b. Normally, there will be two b values that Middlesex Formation, which have shown obvious ovality are 90 different and both have minimum squared error. from caliper logs. Figure 3 shows the image log that has Choose the one that corresponds to r [ r , but discard drilling-induced fractures in a section of the Middlesex H h the other one. In matrix [P], the tensile strength T is often Formation. Figure 4 shows the caliper logs at the corre- set to zero for circular borehole-based inversion. sponding depth section of the Middlesex Formation. In the case of elliptical boreholes, well deviation c, well Four-arm caliper logs give the ovality of the sections azimuth /, maximum horizontal stress direction b, and axis that have drilling-induced fractures. In the Middlesex ratio c are all known. In the inversion of two horizontal Formation the ratio of the two axes is around 0.978–0.984. in situ stress magnitudes, a set of T values can be used for The vertical normal stress is assumed to be equal to the calculation. The T value corresponding to the minimum weight of the overlying rock and can be computed by squared error will be considered as the best estimation of integrating the bulk density log data. There is no direct rock tensile strength. This is additional information to the measurement for pore pressure in this well. Eaton’s method estimated stresses in the elliptical borehole-based inver- was applied for pore pressure estimation using acoustic sions. The ratios of the two horizontal stresses at the slowness logging. Pore pressure was calculated from ver- minimum squared error will be the estimated stress ratios. tical stress (calculated from the density log) and the 9.25 Longer diameter 9.20 3 Application of the method Shorter diameter 9.15 9.10 The method has been demonstrated by a field study in West 9.05 Virginia, in the southern part of the Appalachian Basin, 9.00 USA. Drilling data and borehole geometry information of 8.95 the MIP 3H pilot hole were used for both circular borehole- 8.90 based and elliptical borehole-based in situ stress inversions. 8.85 The input data for the inversion calculation of horizontal 8.80 stresses include mud weight pressure at fracture, pore 6798 6799 6800 6801 6802 6803 6804 6805 6806 6807 6808 6809 pressure, vertical stress, borehole deviation, and borehole Dept, ft azimuth. For the cases of elliptical borehole-based in situ stress inversion, axis ratio and induced fracture azimuth are Fig. 4 Four-arm caliper log showing the longer and shorter diameters at corresponding depth in the Middlesex Formation included in the input parameters. Drilling-induced fractures Fig. 3 Section of image log of MIP 3H pilot hole showing induced fractures in the Middlesex Formation Axis diameter, in 798 Petroleum Science (2018) 15:794–800 Table 1 Data input for stress Data set P , psi/ft P , psi/ft r , psi/ft c,  /,  b,  c f p v inversion in the Middlesex Formation 1 0.6325 0.4264 1.0239 1.0121 296.7038 228 0.9844 2 0.6325 0.6767 1.0262 1.0701 274.3615 239 0.9799 3 0.6325 0.6687 1.0262 1.1115 271.6611 239 0.9805 4 0.6325 0.6445 1.0262 1.1276 271.0305 239 0.9841 5 0.6325 0.6367 1.0262 1.142 270.5145 239 0.9832 6 0.6325 0.6333 1.0262 1.1583 269.7962 239 0.9813 7 0.6325 0.6142 1.0262 1.1743 269.3157 239 0.9810 8 0.6325 0.6062 1.0262 1.1904 268.8706 239 0.9779 Inversed stresses from drilling induced fractures Calculated stress inversion from elliptical borehole 0.8 1.0 σσ / H v σσ / h v 0.9 0.7 Errors σσ / 0.8 H v σσ h / v 0.6 Errors 0.7 0.5 0.6 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 Tensile strength, psi/ft Direction of σ , degrees Fig. 6 Elliptical borehole stress inversion results for the Middlesex Fig. 5 Circular borehole stress inversion results for the Middlesex Formation Formation direction are 0.728 and 0.553, respectively, which are 0.75 difference between actual logged data and the normal trend and 0.57 psi/ft in magnitude at corresponding depth in the line of the acoustic slowness. Middlesex Formation. The calculated maximum horizontal In the drilling process, the mud is circulated in and out stress direction of 229 (or 49) is in reasonable agreement of the well bore by pumping devices. Therefore, the mud with the field observations of the well, which were reported pressure will be higher than the static mud weight. This is as 228 and 239 in the maximum horizontal stress direc- called equivalent circulating density (ECD) of mud weight. tion at the corresponding depth in the Middlesex The value will be normally 3% higher than the static mud Formation. weight. The value of 1.03 times of mud weight was used in In order to compare with the circular borehole-based our calculation. inversion, elliptical borehole-based inversion was run. Table 1 lists the data prepared for the stress inversion. Figure 6 shows the inversion results using the elliptical There are eight datasets in the Middlesex Formation. Both equation for the Middlesex Formation. The tensile strength circular and elliptical equations were applied to the T was assumed a series of small values. In this case, the calculation. squared errors of all runs of the assumed strength T values Figure 5 shows the inversion results for the Middlesex are 0.002. It was not able to differentiate the satisfied T Formation using the circular borehole equation. In the value having smallest error. In comparison with the circular calculation, b was scanned from 0 to 180. Two minimum borehole inversion, in the elliptical borehole for T ¼ 0, the squared error values of 0.0005 were found at 49 and 139. stresses ratios r =r and r [ r are 0.925 and 0.602, H v h v Because at 139 the maximum horizontal stress r is respectively, which are 0.95 and 0.62 psi/ft in magnitude of wrongly smaller than the minimum horizontal stress r , corresponding depth in the Middlesex Formation. 49 (or 229) was chosen as the direction of the maximum horizontal stress. The stress ratios r =r and r =r at this H v h v Stress ratio and error Stress ratio and error Petroleum Science (2018) 15:794–800 799 Table 2 Comparison of inversion results between circular borehole and elliptical borehole in the Middlesex Formation Circular hole Elliptical hole Field observation r , psi/ r , psi/ Direction of r from r , psi/ r , psi/ Direction of r from r , psi/ft r , psi/ft Direction of r from H h H H h H H h H ft ft north,  ft ft north,  north, 0.75 0.57 49 (229) 0.95 0.62 Same as field Up to Up to Measured value 228 and a a observation 1.04 0.73 239 Values calculated using vertical stress gradient of the case study and the stress ratio from the Appalachian Stress Study report 4 Discussion 5 Conclusion Table 2 shows the comparison of the inversion results We can use inverse analysis to estimates of in situ hori- between circular borehole-based calculation and elliptical zontal stresses in an elliptical borehole from leak-off test borehole-based calculation. Large differences exist data; case studies show that even a small amount of around between these two calculations. Tensile strengths are 2% axis difference in an elliptical borehole will cause assumed to be zero in these two calculations for the con- differences of 5%–10% in the estimation of horizontal venience of comparison. All the input parameters are same stresses. Inversion using elliptical borehole equations gives except the axis ratio, which is unit value for a circular better in situ stress estimation than those from circular borehole-based calculation and 0.978–0.984 for the ellip- borehole equations. tical borehole-based calculation. Acknowledgements The support of the United States Department of The results indicate that a 2% axis difference in an Energy (DE-FE0026825, UCFER-University Coalition for Fossil elliptical borehole will cause a 5% difference in the min- Energy Research) is greatly acknowledged. Authors are grateful to the imum horizontal stress calculation and a 10% difference in US Department of Energy and West Virginia University for supplying the maximum horizontal stress calculation. Although there the field data. Special thanks are given to Professor Mark Zoback at Stanford University who has provided constructive suggestions. is no measurement information about horizontal in situ stress magnitudes for this formation, the basin wide stress Open Access This article is distributed under the terms of the Creative study indicated r =r value of up to 0.7 in the corre- h v Commons Attribution 4.0 International License (http://creative sponding depth at this well location (Evans 1989). Evans commons.org/licenses/by/4.0/), which permits unrestricted use, dis- tribution, and reproduction in any medium, provided you give also stated in the Appalachian Stress Study report that the appropriate credit to the original author(s) and the source, provide a magnitude of r varies from high in the northern part of link to the Creative Commons license, and indicate if changes were the basin to low values in the south; the stress state in the made. Devonian shale, of which Middlesex Formation is a part, is either strike-slip or normal fault regime due to the pinch- References out of the underlying salt (Evans 1989). The location of the study well is around the pinch-out area (Pierce and Rich Aadnoy BS. Inversion technique to determine the in situ stress field 1962). Therefore, a unit value of r =r stress ratio should H v from fracturing data. J Pet Sci Eng. 1990;4:127–41. https://doi. be an upper limit. 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