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Geomechanics and fracturing formulas and calculations
Chapter 11
Geomechanics and fracturing formulas and calculations Chapter Outline 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 11.22 11.23 11.24 11.25 11.26 11.27 11.28 11.29 11.30 11.31 11.32 11.33 11.34
Axial stress around vertical wellbore Axis of a deviated borehole from an arbitrary origin Bulk modulus (using Lame) Bulk modulus (using Poisson’s ratio and Lame’s constant) Bulk modulus (using Poisson’s ratio and shear modulus) Change in pore volume due to initial water and rock expansion Cohesive strength of rocks Compressibility of a coalbed methane formation Effect of pore pressure on stress Effective stress on individual grains Failure criteria (Mohr-Coulomb) Formation compressibility by using hydrofrac data Fracture conductivity Fracture gradient (Eaton) Fracture gradient (Holbrook) Fracture gradient (Matthews and Kelly) Fracture gradient (Zoback and Healy) Fracture pressure (Hubert & Willis) Fracture volume (GDK method) Fracture volume (Perkins and Kern method) Fracture width (GDK method) Fracture width (Perkins and Kern method) Hoek and brown criteria for principal stress failure Horizontal effective stress (assuming no lateral strain as per Lorenz and Teufel) Horizontal maximum stress (Bredehoeft) Induced fracture dip Initial effective horizontal stress Isothermal compressibility of limestones (Newman correlation) Least principal stress as function of depth in Gulf of Mexico (Hubbert and Willis) Least principal stress as function of depth in Gulf of Mexico (Matthew and Kelly) Linearized Mohr failure line Linearized Mohr Coulomb criteria M modulus (using shear modulus and bulk modulus) M modulus (using Young’s modulus and Poisson’s ratio)
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11.35 Maximum anisotropic failure stress 11.36 Maximum compression at vertical wellbore 11.37 Maximum normal stress in tangential direction at wellbore wall (hoop stress) 11.38 Maximum plane tangential stress acting on deviated wellbore 11.39 Maximum principal stress failure (Hoek and Brown) 11.40 Maximum principal stress in normal faulting 11.41 Maximum principal stress in reverse faulting 11.42 Maximum principal stress in strike-slip faulting 11.43 Maximum principal stress calculation using breakout width 11.44 Minimum compression at vertical wellbore 11.45 Minimum normal stress in tangential direction at wellbore wall (hoop stress) 11.46 Maximum plane tangential stress acting on deviated wellbore 11.47 Modified lade criterion 11.48 Normal stress in radial direction near wellbore 11.49 Normal stress in rock at failure 11.50 Normal stress in tangential direction at wellbore wall (hoop stress) 11.51 Normal stress in tangential direction near wellbore (hoop stress) 11.52 Pore pressure increase due to fluid activity (Mody & Hale) 11.53 Pore pressure increase due to given fluid activity contrast (Mody and Hale) 11.54 Pore pressure of shale (Flemings) 11.55 Pore pressure of shale (Traugott) 11.56 Porosity irreversible plastic deformation occurs 11.57 Pressure required to induce a tensile fracture (breakdown pressure) 11.58 Pressure to grow fractures (Abe, Mura, et al.) 11.59 Radial stress around vertical wellbore 11.60 Ratio of pore pressure change to original due to depletion 11.61 Rotation of maximum principal stress near wellbore 11.62 Rotation of maximum principal stress near wellbore (Zoback & Day-Lewis) 11.63 Shale compaction 11.64 Shear modulus
Formulas and Calculations for Petroleum Engineering. https://doi.org/10.1016/B978-0-12-816508-9.00011-1 © 2019 Elsevier Inc. All rights reserved.
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Formulas and calculations for petroleum engineering
11.65 11.66 11.67 11.68 11.69 11.70 11.71
Shear modulus from Young’s modulus Shear stress near vertical well Slowness of the formation Storativity of fractures Stress at edge of wellbore breakout Stress component near normal faulting in reservoir Stress components in original coordinate system in depletion drive 11.72 Stress intensity at tip of mode I fracture 11.73 Stress path (induced normal faulting) 11.74 Stress path of reservoir with changes in production
11.1
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11.75 Stress perturbation (Segall and Fitzgerald) 11.76 Subsidence due to uniform pore pressure reduction in free surfaces 11.77 Unconfined compressive strength of rock 11.78 Velocity of bulk compressional waves 11.79 Velocity of compression waves 11.80 Velocity of shear waves 11.81 Vp and Vs calculation (Eberhart-Phillips) 11.82 Vp and vs calculation (geomechanical model) 11.83 Yield strength (Bingham plastic model)
Axial stress around vertical wellbore
Input(s) R: Radius of Wellbore (ft) r: Position in Respect to Centre of Wellbore (ft) Y: Azimuth of Shmax (rad) Shmax: Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Shmin:
Output(s) τ:
Twisting Stress (psi)
Formula(s)
2∗ R2 3∗ R4 ∗ sin ð2∗ YÞ τ ¼ 0:5 ∗ ðShmax Shmin Þ ∗ 1 + r2 r4 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.2
Axis of a deviated borehole from an arbitrary origin
Input(s) Po: Pore Pressure (psi) R: Radius of Wellbore (ft) r: Position in Respect to Centre of Wellbore (ft) Y: Azimuth of Shmax (rad) Thermal Stress (psi) sDt: Maximum Horizontal Stress (psi) Shmax: Minimum Horizontal Stress (psi) Shmin:
Output(s) saa:
Stress (psi)
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Formula(s) 2 R 3 ∗ R4 Po ∗ R2 0:5∗ ðShmax Shmin Þ∗ 1 + ∗ cos ð 2∗ Y Þ sDt saa ¼ 0:5 ∗ ðShmax + Shmin 2 ∗Po Þ∗ 1 + 2 r r4 r2 Reference: Mark D. Zoback., Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.3
Bulk modulus (using Lame)
Input(s) p: Pressure (Pa) DV: Volume (m3) v: Volume (m3)
Output(s) K:
Bulk Modulus (Pa)
Formula(s) K¼
p DV v
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 45.
11.4
Bulk modulus (using Poisson’s ratio and Lame’s constant)
Input(s) l: G:
Lame (dimensionless) Shear Modulus (N/m2)
Output(s) K:
Bulk Modulus (N/m2)
Formula(s) K ¼ l + 2∗
G 3
Reference: PetroWiki.org.
11.5
Bulk modulus (using Poisson’s ratio and shear modulus)
Input(s) l: G:
Lame (dimensionless) Shear Modulus (N/m2)
Output(s) K:
Bulk Modulus (N/m2)
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Formulas and calculations for petroleum engineering
Formula(s) K ¼ l + 2∗
G 3
Reference: PetroWiki.org.
11.6
Change in pore volume due to initial water and rock expansion
Input(s) l: n:
Lame (dimensionless) Poisson (dimensionless)
Output(s) Bulk Modulus (N/m2)
K:
Formula(s) K¼
l ∗ ð 1 + nÞ 3∗n
Reference: PetroWiki.org.
11.7
Cohesive strength of rocks
Input(s) G: n:
Shear Modulus (N/m2) Poisson (dimensionless)
Output(s) K:
Bulk Modulus (N/m2)
Formula(s) K¼
2∗ G ∗ ð1 + nÞ 3 ∗ ð 1 2 ∗ nÞ
Reference: PetroWiki.org.
11.8
Compressibility of a coalbed methane formation
Input(s) Wi: W p: Pi: Pd :
Total Volume of Water in the Reservoir (bbl) Total Volume of Water Removed (bbl) Initial Reservoir Pressure (psi) Desorption Pressure (psi)
Output(s) ct:
Total Compressibility (1/psi)
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Formula(s) ct ¼
Wp 1 ∗ P i Pd Wi
Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:3, Page: 219.
11.9
Effect of pore pressure on stress
Input(s) l: d: x o: x: G: a: P:
Lames First Constant (psi) Kronecker Delta (dimensionless) Initial Strain (dimensionless) Final Strain (dimensionless) Modulus of Shear (psi) Biot (dimensionless) Pressure Applied (psi)
Output(s) S:
Stress (psi)
Formula(s) S ¼ l ∗ xo ∗ d + 2 ∗ G ∗ x a∗ d ∗ P Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 68.
11.10 Effective stress on individual grains Input(s) S: Pp:
Normal Stress (psi) Pore Pressure (psi)
Output(s) sg:
Effective Stress (psi)
Formula(s) sg ¼ S Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 87.
11.11 Failure criteria (Mohr-Coulomb) Input(s) sa: sb : b:
Principle Stress (psi) Second Strongest Stress (psi) Angle Between Fault and Direction of Principle Stress (degrees)
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Formulas and calculations for petroleum engineering
Output(s) τ: s:
Shear Stress (psi) Normal Stress (psi)
Formula(s) τ ¼ 0:5 ∗ ðsa sb Þ ∗ sin ð2 ∗ bÞ s ¼ 0:5 ∗ ðsa + sb Þ + 0:5∗ ðsa sb Þ∗ cos ð2 ∗bÞ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
11.12 Formation compressibility by using hydrofrac data Input(s) Vs: Volume Associated to Conduct a Hydrofrac (bbl) dVs: Change in Volume (bbl) dP: Change in Pressure (psi)
Output(s) b:
Formation Compressibility (psi)
Formula(s) b¼
1 dVs ∗ Vs dP
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 221.
11.13 Fracture conductivity Input(s) kf: wf:
Fracture Permeability (mD) Width of Fracture (ft)
Output(s) FC :
Fracture Conductivity (mD ft)
Formula(s) FC ¼ kf ∗wf Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:1, Page: 93.
11.14 Fracture gradient (Eaton) Input(s) u: S v: Pp :
Poisson (dimensionless) Vertical Stress (psi) Pore Pressure (psi)
Geomechanics and fracturing formulas and calculations Chapter
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) u ∗ Sv Pp + Pp 1u
Shmin ¼
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 282.
11.15 Fracture gradient (Holbrook) Input(s) u: S v: Pp :
Poisson (dimensionless) Vertical Stress (psi) Pore Pressure (psi)
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
u ∗ S v Pp + P p 1u
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 282.
11.16 Fracture gradient (Matthews and Kelly) Input(s) smin: Minimum Principle Horizontal Stress (psi) Vertical Principle Stress (psi) sv : Vertical Stress (psi) S v: Pp : Pore Pressure (psi)
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
smin ∗ Sv Pp + Pp sv
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.17 Fracture gradient (Zoback and Healy) Input(s) m: S v: Sp :
Viscosity (cP) Vertical Stress (psi) Pore Pressure (psi)
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Formulas and calculations for petroleum engineering
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
1 + m2
0:5
+m
2 ∗ Sv Pp + Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.18 Fracture pressure (Hubert & Willis) Input(s) S v: Pp :
Vertical Stress (psi) Pore Pressure (psi)
Output(s) Shmin:
Fracture Pressure (psi)
Formula(s) Shmin ¼ 0:3 ∗ Sv Pp + Pp shmin ¼ 0:3 sv Notes: Fracture Pressure Is assumed to be Equal to Minimum Horizontal Stress. Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.19 Fracture volume (GDK method) Input(s) Q: m: L: H: G:
Flow Rate (B/m) Viscosity (cP) Length (ft) Height (ft) Shear Modulus (psi)
Output(s) Vf:
Fracture Volume (ft3)
Formula(s)
H3 Vf ¼ 0:03561 ∗ m ∗ Q ∗ L ∗ G
0:25
6
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 70.
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11.20 Fracture volume (Perkins and Kern method) Input(s) H: n: m: Q: E: L:
Fracture Height (ft) Poisson (unitless) Viscosity (cP) Flowrate (B/m) Young (psi) Fracture Length (ft)
Output(s) Fracture Volume (ft3)
Vf:
Formula(s) Vf ¼ 0:04 ∗ H ∗
Q 1 n2 ∗ m ∗ E
0:25
5
∗ L4
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 57.
11.21 Fracture width (GDK method) Input(s) Q: G: m: L: H:
Flow Rate (B/m) Shear Modulus (psi) Viscosity (cP) Length (ft) Height (ft)
Output(s) w:
Fracture Width (in.)
Formula(s) 0:25 L2 w ¼ 0:272∗ m ∗ Q ∗ G∗H Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 70.
11.22 Fracture width (Perkins and Kern method) Input(s) n: Q: m: L: E:
Poisson (unitless) Flowrate (B/m) Viscosity (cP) Fracture Length (ft) Young’s Modulus (psi)
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Output(s) Wmax:
Fracture Width (in.)
Formula(s) Wmax ¼ 0:389 ∗
L 1 n ∗Q∗m∗ E 2
0:25
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 58.
11.23 Hoek and Brown criteria for principal stress failure Input(s) Co: Unconfined Compressive Strength of Rock (psi) m: Constant Depending on Property of Rock and Extent to Which It Is Broken (dimensionless) s: Constant Depending on Property of Rock and Extent to Which It Is Broken (dimensionless) Minimum Effective Principal Stress (psi) sc:
Output(s) sa:
Maximum Effective Principal Stress (psi)
Formula(s) 0:5 s sa ¼ sc + Co ∗ m ∗ c + s Co Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 98.
11.24 Horizontal effective stress (assuming no lateral strain as per Lorenz and Teufel) Input(s) n: Poisson Ratio (dimensionless) Sv: Vertical Overburden Stress (psi) a: Biot Coefficient (dimensionless) P: Pore Pressure (psi)
Output(s) Shor:
Horizontal Stress (psi)
Formula(s) Shor ¼
n n ∗ Sv + a ∗ P ∗ 1 1n 1n
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.25 Horizontal maximum stress (Bredehoeft) Input(s) Shmin:
Minimum Horizontal Stress (psi)
Geomechanics and fracturing formulas and calculations Chapter
Pb: Pp :
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Breakdown Pressure at initial Hydrofrac (psi) Pore Pressure (psi)
Output(s) Shmax:
Maximum Horizontal Stress (psi)
Formula(s) Shmax ¼ 3∗ Shmin Pb Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 220.
11.26 Induced fracture dip Input(s) h: d:
Height of Fracture (ft) Diameter of Well (ft)
Output(s) Dip:
Dip (degrees)
Formula(s) Dip ¼ arctan
h d
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 146.
11.27 Initial effective horizontal stress Input(s) n: r: H: a: pp :
Poisson (unitless) Overburden Density (lb/ft3) Formation Depth (ft) Biot Constant (unitless) Reservoir Pressure (psi)
Output(s) sh:
Effective Horizontal Stress (psi)
Formula(s) n H a ∗ pp sh ¼ ∗ r∗ 1n 144 Reference: Boyun, G., William, C., & Ali Ghalambor, G. (2007). Petroleum Production Engineering: A Computer-Assisted Approach Page: 259.
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11.28 Isothermal compressibility of limestones (Newman correlation) Input(s) ø:
Porosity (fraction)
Output(s) Ct:
Compressibility (psi1)
Formula(s) Ct ¼
97:32∗ 106 ð1 + 55:8721 ∗ øÞ1:42869
Notes: Check for 0.02 < Ø < 0.23. Reference: Applied Petroleum Reservoir Engineering, Craft & Hawkins, Page: 11.
11.29 Least principal stress as function of depth in Gulf of Mexico (Hubbert and Willis) Input(s) Sv: Pp:
Vertical Overburden Stress (psi) Pore Pressure in Reservoir (psi)
Output(s) Shmin:
Minimum Principal Stress in Reservoir (psi)
Formula(s) Shmin ¼ 0:3 ∗ ðSv PpÞ + Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 280.
11.30 Least principal stress as function of depth in Gulf of Mexico (Matthew and Kelly) Input(s) Sv: Ki: Pp:
Vertical Overburden Stress (psi) Constant as Function of Depth (dimensionless) Pore Pressure in Reservoir (psi)
Output(s) Shmin:
Minimum Principal Stress in Reservoir (psi)
Formula(s) Shmin ¼ Ki ∗ ðSv PpÞ + Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 280.
Geomechanics and fracturing formulas and calculations Chapter
11.31 Linearized Mohr failure line Input(s) S o: sn: mi:
Stress (psi) Normal Stress (psi) Coefficient of internal Friction (cP)
Output(s) τ:
Shear Stress (psi)
Formula(s) τ ¼ So + s n ∗ m i Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
11.32 Linearized Mohr coulomb criteria Input(s) Co: m: sc:
Unconfined Compressive Strength of Rock (psi) Slope of Failure Line (dimensionless) Minimum Effective Principal Stress (psi)
Output(s) s1:
Maximum Effective Principal Stress (psi)
Formula(s) s1 ¼ ð Co Þ +
m2 + 1
0:5
2 + m ∗ sc
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 93.
11.33 M Modulus (using shear modulus and bulk modulus) Input(s) G: K:
Shear Modulus (N/m2) Bulk Modulus (N/m2)
Output(s) M:
M Modulus (N/m2)
Formula(s) M ¼ K + 4∗
G 3
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 64.
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11.34 M Modulus (using Young’s modulus and Poisson’s ratio) Input(s) n: E:
Poisson Ratio (dimensionless) Young Modulus (N/m2)
Output(s) M:
M Modulus (N/m2)
Formula(s) M ¼ E∗
1n ð1 + nÞ∗ ð1 2 ∗ nÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 64.
11.35 Maximum anisotropic failure stress Input(s) sc: Sw : m: b:
Minimum Principle Stress (psi) Intact Rock Strength (psi) Internal Friction of Weak Bedding (cP) Angle of Weak Plane to Maximum Principle Stress (degrees)
Output(s) s:
Stress (cm/s)
Formula(s) s¼
s c ∗ 2 ∗ ðSw + m ∗ sc Þ ð1 m ∗ cot ðbÞÞ ∗ sin ð2 ∗ bÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 107.
11.36 Maximum compression at vertical wellbore Input(s) sc: Sw: m: b:
Minimum Principle Stress (psi) Intact Rock Strength (psi) Internal Friction of Weak Bedding (cP) Angle of Weak Plane to Maximum Principle Stress (degrees)
Output(s) s:
Stress (cm/s)
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Formula(s) s¼
sc ∗ 2∗ ðSw + m ∗ sc Þ ð1 m∗ cot ðbÞÞ∗ sin ð2∗ bÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 107.
11.37 Maximum normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in Reservoir (psi) Minimum Principal Stress in Reservoir (psi) Shmin: Po: Pore Pressure (psi) Stress induced Due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi)
Output(s) sigth:
Maximum Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) sigth ¼ 3 ∗ Shmax Shmin 2∗ Po dP Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 174.
11.38 Maximum plane tangential stress acting on deviated wellbore Input(s) szz: Stress in Radial Direction (psi) Stress in Axial Direction (psi) saa: τ: Shear Stress (psi)
Output(s) stmax:
Maximum Tangential Stress (psi)
Formula(s) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 stmax ¼ ∗ szz + saa + ðszz saa Þ2 + 4∗ τ2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 stmax ¼ ∗ szz + saa ðszz saa Þ2 + 4 ∗ τ2 2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 239.
11.39 Maximum principal stress failure (Hoek and Brown) Input(s) sb: Second Largest Stress (psi) C: Rock Compressive Strength (lbf) m: Hoek - Brown Constant Dependent on Rock Type (dimensionless) s: Hoek and Brown Constant Dependent on Shape (dimensionless)
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Formulas and calculations for petroleum engineering
Output(s) sa:
Principal Stress (psi)
Formula(s) 0:5 s sa ¼ sb + C ∗ m ∗ b + s C Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 98.
11.40 Maximum principal stress in normal faulting Input(s) sc: Least Principle Stress (psi) Vertical Stress (psi) S v: Pore Pressure (psi) Pp : Minimum Horizontal Stress (psi) Shm:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Sv Pp Shm Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
11.41 Maximum principal stress in reverse faulting Input(s) sc: Least Principle Stress (psi) Pore Pressure (psi) P p: Vertical Stress (psi) S v: Maximum Horizontal Stress (psi) Shmax:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Shmax Pp Sv Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
Geomechanics and fracturing formulas and calculations Chapter
11.42 Maximum principal stress in strike-slip faulting Input(s) sc: Least Principle Stress (psi) Maximum Horizontal Stress (psi) Shmax: Pore Pressure (psi) P p: Minimum Horizontal Stress (psi) Shmin:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Shmax Pp Shmin Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
11.43 Maximum principal stress calculation using breakout width Input(s) Shmin: Minimum Principal Stress in Reservoir (psi) Distance from Wellbore (ft) C o: Pp: Pore Pressure (psi) Stress induced Due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi) Y: Angle from Wellbore Breakout Width (rad)
Output(s) Shmin:
Maximum Principal Stress in Reservoir (psi)
Formula(s) Shmax 5
ððCo Þ + 2∗ Pp + dP + SdtÞ Shmin ∗ ð1 + 2∗ cos ðYÞÞ 1 2 ∗ cos ðYÞ
Reference: Mark D. Zoback., Reservoir Geomechanics, Cambridge University Press, UK, Page: 223.
11.44 Minimum compression at vertical wellbore Input(s) Shmin: Minimum Horizontal Stress (psi) Maximum Horizontal Stress (psi) Shmax: Pore Pressure (psi) P o: P: Pressure Drawdown (psi) Thermal Stress (psi) st:
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Formulas and calculations for petroleum engineering
Output(s) sminaxial:
Minimum Axial Stress (psi)
Formula(s) sminaxial 53 ∗Shmin 2Shmax 22∗ Po 2P2st Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 174.
11.45 Minimum normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in Reservoir (psi) Minimum Principal Stress in Reservoir (psi) Shmin: Po: Pore Pressure (psi) Stress induced due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi)
Output(s) sigth:
Minimum Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) sigth ¼ 3∗ Shmin Shmax 2 ∗ Po dP Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 238.
11.46 Maximum plane tangential stress acting on deviated wellbore Input(s) szz: Radial Stress (psi) Axial Stress (psi) saa: τ: Shear Stress (psi)
Output(s) stmax:
MaximumTangential Stress (psi)
Formula(s) 0:5 2 2 stmax ¼ 0:5 ∗ szz + saa ðszz saa Þ + 4 ∗ τ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 239.
11.47 Modified lade criterion Input(s) S a: Sb :
Principle Stress (psi) Intermediate Stress (psi)
Geomechanics and fracturing formulas and calculations Chapter
S c: Pa : m:
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Minimum Stress (psi) Pressure (psi) Material Strength Constant (dimensionless)
Output(s) Ia: Ic: :
First Invariant of Stress Tensor (psi) Third Invariant of Stress Tensor (psi3) Lades Coefficient (dimensionless)
Formula(s) Ia ¼ Sa + Sb + Sc I c ¼ S a ∗ Sb ∗ Sc 3 m Ia Ia 27 ∗ Z¼ Pa I3c Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 99.
11.48 Normal stress in radial direction near wellbore Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) Po: Pore Pressure (psi) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
Output(s) sigrr :
Normal Stress in Radial Direction Near Wellbore (psi)
Formula(s) R2 R2 R4 3:142 R2 + Po ∗ 2 sigrr ¼ 0:5 ∗ ðShmax + Shmin 2∗ PoÞ ∗ 1 2 + 0:5 ∗ ðShmax ShminÞ ∗ 1 4∗ 2 + 3∗ 4 ∗ cos 2∗ y ∗ r r r r 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 100.
11.49 Normal stress in rock at failure Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) Po: Pore Pressure (psi) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
462
Formulas and calculations for petroleum engineering
Output(s) sigrr :
Normal Stress in Radial Direction Near Wellbore (psi)
Formula(s) R2 R2 R4 3:142 R2 + Po∗ 2 sigrr ¼ 0:5 ∗ ðShmax + Shmin 2 ∗PoÞ∗ 1 2 + 0:5∗ ðShmax ShminÞ∗ 1 4 ∗ 2 + 3 ∗ 4 ∗ cos 2 ∗ y∗ r r r r 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 101.
11.50 Normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Shmin: Minimum Principal Stress in reservoir (psi) Po: Pore Pressure (psi) Sdt: Stress induced due to temperature (psi) dP: Difference between Wellbore Pressure and Mud Weight (psi) y: Angle from Shmax at which stress is measured (degrees)
Output(s) sigth :
Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) 3:142 2∗ Po dP Sdt sigth ¼ ðShmax + ShminÞ 2∗ ðShmax ShminÞ∗ cos 2 ∗ y∗ 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 102.
11.51 Normal stress in tangential direction near wellbore (hoop stress) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Shmin: Minimum Principal Stress in reservoir (psi) Po: Pore Pressure (psi) Sdt: Stress induced due to temperature (psi) dP: Difference between Wellbore Pressure and Mud Weight (psi) y: Angle from Shmax at which stress is measured (degrees) r: Distance from wellbore (ft) R: Radius of wellbore (ft)
Output(s) Sgth:
Normal Stress in Tangential Direction Near Wellbore (Hoop Stress) (psi)
Geomechanics and fracturing formulas and calculations Chapter
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Formula(s) R2 R4 3:142 R2 Po ∗ 2 Sgth ¼ 0:5∗ ðShmax + Shmin 2 ∗ PoÞ∗ 1 + 2 0:5 ∗ ðShmax ShminÞ ∗ 1 + 3∗ 4 ∗ cos 2∗ y ∗ r r r 180 Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 104.
11.52 Pore pressure increase due to fluid activity (Mody & Hale) Input(s) Em : Membrane Efficiency (dimensionless) R: Gas Constant (psi/mol K) T: Temperature (K) V: Volume (ft3) Pore Fluid Activity (dimensionless) A p: Am : Mud Activity (dimensionless)
Output(s) dP:
Pore Pressure Increase (psi)
Formula(s) Ap R ∗T ∗ ln dP ¼ Em ∗ Am V
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 321.
11.53 Pore pressure increase due to given fluid activity contrast (Mody and Hale) Input(s) E m: Membrane Efficiency (dimensionless) R: Gas Constant (dimensionless) T: Temperature (K) V: Molar Volume of Water (L/mol) Pore Fluid Activity (dimensionless) Ap: Mud Activity (dimensionless) Am:
Output(s) dP:
Pore Pressure Increase (psi)
Formula(s)
T Ap ∗ log dP ¼ Em ∗ R ∗ V Am Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 105.
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Formulas and calculations for petroleum engineering
11.54 Pore pressure of shale (Flemings) Input(s) S v: bc: ø o: t: ø:
Sonic Velocity (ft/s) Compressibility (1/psi) Initial Porosity (fraction) Sonic Travel Time (s) Porosity from Sonic Log (fraction)
Output(s) Pp:
Pore Pressure (psi)
Formula(s) Pp 5Sv 2
1 ø ∗ ln o ø bc
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 48.
11.55 Pore pressure of shale (Traugott) Input(s) z: Depth (ft) S v: Sonic Velocity (ft/s) Resistivity of Shale (ohm ft) Ro : Expected Resistivity (ohm ft) Rn : Hydrostatic Pore Pressure (psi) ( Pp)hydro:
Output(s) Ppsh:
Pore Pressure of Shale (psi)
Formula(s) 0 0 hydro 11 1 1:2 ! P Ro B S B p CC B S C Ppsh ¼ z∗ @ v @ v @ AA ∗ A z z z Rn 0
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 47.
11.56 Porosity irreversible plastic deformation occurs Input(s) m: S v: SH : Sh : Pp :
Viscosity (cP) Vertical Stress (psi) Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Change in Pore Pressure (psi)
Geomechanics and fracturing formulas and calculations Chapter
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Output(s) M: p:
Mobility (dimensionless) Porosity (dimensionless)
Formula(s) M¼
6∗m 2 3 ∗ ððm Þ + 1Þ0:5 m
! 2 1 9 9 2 2 2 ∗ 9 ∗ Pp + 1+ 2 ∗ Sv + SH + Sh + 2 2 ∗ ðSv ∗ SH +Sv ∗ Sh +SH ∗ Sh Þ6∗ Pp ∗ ðSv + SH + Sh Þ p¼ 3 ∗ ðSv ∗ SH + Sh Þ 9 ∗ Pp M M Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 400.
11.57 Pressure required to induce a tensile fracture (breakdown pressure) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: Pore Pressure (psi) Pp: Minimum Hoop Stress for formation at which crack initiates (psi) To :
Output(s) Pb:
Breakdown Pressure (psi)
Formula(s) Pb ¼ 3 ∗ Shmin Shmax Pp + To Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 56.
11.58 Pressure to grow fractures (Abe, Mura, et al.) Input(s) S c: Pp : cf: ci:
Minimum Principle Stress (psi) Pore Pressure (psi) Radius of Fracture (in.) Radius of Invaded Zone (in.)
Output(s) Pgrow:
Growth Pressure (psi)
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Formulas and calculations for petroleum engineering
Formula(s) 0
1 2 !0:5 P c p B1 C ∗ 1 f B C Sc ci B C B 0 Pgrow ¼ Sc ∗ B !0:5 1 C C 2 B C @ 1 @ 1 cf A A ci Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 326.
11.59 Radial stress around vertical wellbore Input(s) Shmax: Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Shmin: Pore Pressure (psi) Po : R: Radius of Wellbore (ft) r: Relative Position to Centre (ft) y: Azimuth of Shmax (rad)
Output(s) srr:
Stress (psi)
Formula(s) 2 R 3 ∗ R4 4∗ R2 Po ∗ R2 + 0:5∗ ðShmax Shmin Þ ∗ 1 + ∗ cos ð 2 ∗ y Þ + srr ¼ 0:5 ∗ ðShmax + Shmin 2∗ Po Þ∗ 1 2 r r4 r2 r2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.60 Ratio of pore pressure change to original due to depletion Input(s) P: Change in Pore Pressure (psi) Maximum Horizontal Stress (psi) Shmax: Minimum Horizontal Stress (psi) Shmin:
Output(s) q:
Pore Pressure Ratio (fraction)
Formula(s) q¼
Pp Shmax Shmin
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
Geomechanics and fracturing formulas and calculations Chapter
11
11.61 Rotation of maximum principal stress near wellbore Input(s) A: Stress Field Direction (dimensionless) Change in Pore Pressure (psi) Pp: y: Fault Orientation (degrees) Maximum Principle Stress (psi) Shmax: Shmin: Minimum Principle Stress (psi)
Output(s) g:
Rotation (degrees)
Formula(s) ! A ∗ Pp ∗ sin ð2∗ yÞ g ¼ 0:5∗ atan Shmax Shmin + A ∗ Pp ∗ cos ð2 ∗ yÞ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.62 Rotation of maximum principal stress near wellbore (Zoback & Day-Lewis) Input(s) A: D: q:
Constant a Value (dimensionless) Fault Orientation (degrees) Ratio of Pore Pressure to Differential Stress (dimensionless)
Output(s) g:
Stress Rotation (radian)
Formula(s)
sin ð2 ∗ DÞ g ¼ 0:5 ∗ atan A ∗ q ∗ 1 + A ∗ q ∗ cos ð2 ∗ DÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.63 Shale compaction Input(s) Ø: b: sv:
Porosity (fraction) Second Empirical Constant from Porosity vs Vertical Stress Graph (1/MPa) Vertical Effective Stress (MPa)
Output(s) Øe:
Shale Compaction (fraction)
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Formulas and calculations for petroleum engineering
Formula(s) Øe ¼ Ø ∗ eb ∗ sv Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 46.
11.64 Shear modulus Input(s) F: A: y:
Force (N) Area (m2) Deformation Angle (degrees)
Output(s) G:
Shear Modulus (Pa)
Formula(s) G¼
F=A y
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 45.
11.65 Shear modulus from Young’s modulus Input(s) n: E:
Poisson’s Ratio (dimensionless) Young’s Modulus (N/m2)
Output(s) G:
Modulus of Rigidity (psi)
Formula(s) G¼
E 2∗ ð1 + nÞ
Reference: Samuel. E Robello. 501 Solved Problems and Calculations for Drilling Operations. Sigma Quadrant. 2015. Houston, Texas, Page: 356.
11.66 Shear stress near vertical well Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
Geomechanics and fracturing formulas and calculations Chapter
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Output(s) sigrth:
Shear Stress near Vertical Well (psi)
Formula(s)
R2 R4 3:142 sigrth ¼ 0:5 ∗ ðShmax + Shmin Þ ∗ 1 + 2∗ 2 3 ∗ 4 ∗ sin 2 ∗y ∗ r r 180 Reference: petrowiki.org.
11.67 Slowness of the formation Input(s) d h: Dtm: dt: Ls: lc: tl:
Borehole Diameter (in.) Interval Travel Time (ms/ft) Tool Diameter (in.) Spacing of the Tool (ft) Eccentricity of the Tool (ft) Time Between Initiation of the Pulse and First Arrival Acoustic Energy at the Receiver (ms/ft)
Output(s) Dt: tm:
Slowness of Formation Observed by Sonic Log (ms/ft) Mud Path Correction Time (ms/ft)
Formula(s) Dt ¼
tl tm Ls
Dt tm ¼ ðDtm Þ∗ ðdh ðdt + 2 ∗ lc ÞÞ ∗ 1 Dtm
2 !0:5
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 10, Page: 189.
11.68 Storativity of fractures Input(s) øf: ø m: h f: h m: ctf: ctm:
Porosity of Fracture (fraction) Porosity of Matrix (fraction) Fracture Thickness (ft) Matrix Thickness (ft) Total Fracture Compressibility (1/psi) Total Matrix Compressibility (1/psi)
Output(s) o:
Storativity of Fracture (dimensionless)
470
Formulas and calculations for petroleum engineering
Formula(s) o¼
øf ∗ hf ∗ctf øf ∗ hf ∗ ctf + øm ∗ hm ∗ ctm
Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:1, Page: 82.
11.69 Stress at edge of wellbore breakout Input(s) Co: Wellbore Strength (psi) Pore Pressure (psi) Pp: P: Drawdown Differential in Pressure (psi) Thermally Induced Stress (psi) st: Wellbore Breakout (degrees) wb: Minimum Horizontal Stress (psi) Shmin:
Output(s) y b: Breakout Angle (degrees) Maximum Principle Stress (psi) Shmax:
Formula(s) Shmax ¼
2yb ¼ p wb ðCo Þ + 2 ∗ Pp + P + st Shmin ∗ ð1 + 2∗ cos ðyb ÞÞ 1 2∗ cos ðyb Þ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 223.
11.70 Stress component near normal faulting in reservoir Input(s) a: Biot (dimensionless) n: Poisson (dimensionless) Maximum Principal Stress (psi) Shmax: Minimum Principal Stress (psi) Shmin: dP: Change in Pore Pressure (psi) y: Fault Orientation (degrees)
Output(s) A: Constant a Value (dimensionless) Stress in X Direction (psi) Sx: Stress in Y Direction (psi) Sy : Normal Stress in Y Direction (psi) Txy:
Geomechanics and fracturing formulas and calculations Chapter
Formula(s) 1 2∗n 1n dP p Sx ¼ Shmax A ∗ dP A ∗ ∗ 1 cos 2 ∗y ∗ 2 180 dP p Sy ¼ Shmin A ∗ dP A ∗ ∗ 1 + cos 2 ∗ y∗ 2 180 dP p Txy ¼ A ∗ ∗ sin 2 ∗ y ∗ 2 180 A ¼ a∗
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.71 Stress components in original coordinate system in depletion drive Input(s) Shmax: Maximum Stress in Horizontal Direction (psi) Minimum Stress in Horizontal Direction (psi) Shmin: Change in Pore Pressure due to Depletion (psi) dPp: A: Stress Path (dimensionless) D: Fault Orientation (degrees)
Output(s) Sx: Sy : τ:
Stress in X-direction (psi) Stress in Y-direction (psi) Stress Around Wellbore (psi)
Formula(s) A ∗ dPp ∗ ð1 cos ð2∗ DÞÞ 2 A ∗ dPp Sy ¼ Shmin A ∗ Pp ∗ ð1 + cos ð2 ∗ DÞÞ 2 A ∗ dPp τ¼ ∗ sin ð2∗ DÞ 2
Sx ¼ Shmax A ∗Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.72 Stress intensity at tip of mode I fracture Input(s) Pf: Sc: L:
Fracture Pressure (psi) Minimum Principle Stress (psi) Length of Fracture (ft)
Output(s) K:
Stress Intensity (psi ft)
11
471
472
Formulas and calculations for petroleum engineering
Formula(s) K ¼ ðPf Sc Þ∗ p ∗ L0:5 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 122.
11.73 Stress path (induced normal faulting) Input(s) m:
Friction Coefficient (dimensionless)
Output(s) A:
Stress Path (dimensionless)
Formula(s) A ¼ 1
1
2 ðm2 + 1Þ0:5 + m
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 385.
11.74 Stress path of reservoir with changes in production Input(s) a: u:
Biots Coefficient (dimensionless) Poisson (dimensionless)
Output(s) A:
Stress Path (dimensionless)
Formula(s) A¼
a ∗ ð1 2 ∗ uÞ 1u
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.75 Stress perturbation (Segall and Fitzgerald) Input(s) u: H: R: a:
Poisson (dimensionless) Height of Reservoir (ft) Half the Lateral Extent (ft) Constant of Stress Propagation (dimensionless)
Output(s) M:
Stress Perturbation (dimensionless)
Geomechanics and fracturing formulas and calculations Chapter
11
Formula(s) M ¼ a∗
ð1 2 ∗uÞ ∗ p ∗ H ð1 uÞ ∗4 ∗ 2 ∗ R
Reference: Mark D. Zoback, Reservoir Geomechanics. Cambridge University Press. Cambridge, UK, Page: 112.
11.76 Subsidence due to uniform pore pressure reduction in free surfaces Input(s) cm: Formation Compaction per Unit Change in Pore Pressure Reduction (ft3/psi) u: Poisson’s (dimensionless) r: Radius of Area Involved (ft) D: Depth of Formation in Consideration (ft) Pore Pressure Change (psi) DPp: V: Volume of Reservoir (ft3)
Output(s) u z: ur:
Subsidence in Z Direction (ft) Subsidence Along R (ft)
Formula(s) ! cm ∗ ð1 uÞ∗ D ∗ DPp ∗ V uz ¼ ð1Þ∗ 1:5 p∗ ðr2 Þ + D2 ! cm ∗ ð1 uÞ ∗r ∗ DPp ∗ V ur ¼ 1:5 p ∗ ðr2 Þ + D2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 412.
11.77 Unconfined compressive strength of rock Input(s) S o: m:
Cohesive Strength (psi) Slope of Failure Line (dimensionless)
Output(s) Co:
Unconfined Compressive Strength of Rock (psi)
Formula(s) Co ¼ 2 ∗So ∗
m2 + 1
0:5
+m
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
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Formulas and calculations for petroleum engineering
11.78 Velocity of bulk compressional waves Input(s) E: r: m:
Young’s Modulus (lbf/ft2) Density (lbm/ft3) Poisson’s Ratio (dimensionless)
Output(s) Vb:
Velocity of Bulk Compressional Waves (ft/s)
Formula(s)
E 1m Vb ¼ r ∗ ð1 + mÞ ∗ ð1 2 ∗ mÞ
0:5
Reference: Core Laboratories. 2005. Formation Evaluation and Petrophysics, Page: 23.
11.79 Velocity of compression waves Input(s) K: G: r:
Bulk Modulus (Pa) Shear Modulus (Pa) Density (kg/m3)
Output(s) Vp:
Velocity of Compression Waves (m/s)
Formula(s) 4 G 0:5 Vp ¼ K+ 3 ∗r Reference: Bassiouni, Z. 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 46.
11.80 Velocity of shear waves Input(s) G: r:
Shear Modulus (Pa) Density (kg/m3)
Output(s) Vs :
Velocity of Shear Waves (m/s)
Geomechanics and fracturing formulas and calculations Chapter
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Formula(s) 0:5 G Vs ¼ r Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 46.
11.81 Vp and Vs calculation (Eberhart-Phillips) Input(s) ø: C: s:
Porosity (fraction) Clay Content (fraction) Effective Stress (psi)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) Vp ¼ 5:77 6:94 ∗ ø 1:73 ∗ C0:5 + 0:446 ∗ s ð1Þ ∗eð1Þ ∗ 16:7 ∗ s Vs ¼ 3:7 4:94 ∗ ø 1:57 ∗ C0:5 + 0:361 ∗ s ð1Þ∗ eð1Þ ∗ 16:7 ∗ s Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 53.
11.82 Vp and Vs calculation (geomechanical model) Input(s) K: G: r:
Bulk Modulus (psi) Shear Modulus (psi) Density (ppg)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) 1 4 G 0:5 K+ ∗ B 3 C Vp ¼ @ A r 0
0:5 G Vs ¼ r Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 63.
476
Formulas and calculations for petroleum engineering
11.83 Yield strength (Bingham plastic model) Input(s) K: G: r:
Bulk Modulus (psi) Shear Modulus (psi) Density (ppg)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) 1 4 G 0:5 K+ ∗ B 3 C Vp ¼ @ A r 0
Vs ¼
0:5 G r
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, page 63.
Geomechanics and fracturing formulas and calculations Chapter Outline 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 11.22 11.23 11.24 11.25 11.26 11.27 11.28 11.29 11.30 11.31 11.32 11.33 11.34
Axial stress around vertical wellbore Axis of a deviated borehole from an arbitrary origin Bulk modulus (using Lame) Bulk modulus (using Poisson’s ratio and Lame’s constant) Bulk modulus (using Poisson’s ratio and shear modulus) Change in pore volume due to initial water and rock expansion Cohesive strength of rocks Compressibility of a coalbed methane formation Effect of pore pressure on stress Effective stress on individual grains Failure criteria (Mohr-Coulomb) Formation compressibility by using hydrofrac data Fracture conductivity Fracture gradient (Eaton) Fracture gradient (Holbrook) Fracture gradient (Matthews and Kelly) Fracture gradient (Zoback and Healy) Fracture pressure (Hubert & Willis) Fracture volume (GDK method) Fracture volume (Perkins and Kern method) Fracture width (GDK method) Fracture width (Perkins and Kern method) Hoek and brown criteria for principal stress failure Horizontal effective stress (assuming no lateral strain as per Lorenz and Teufel) Horizontal maximum stress (Bredehoeft) Induced fracture dip Initial effective horizontal stress Isothermal compressibility of limestones (Newman correlation) Least principal stress as function of depth in Gulf of Mexico (Hubbert and Willis) Least principal stress as function of depth in Gulf of Mexico (Matthew and Kelly) Linearized Mohr failure line Linearized Mohr Coulomb criteria M modulus (using shear modulus and bulk modulus) M modulus (using Young’s modulus and Poisson’s ratio)
444 444 445 445 445 446 446 446 447 447 447 448 448 448 449 449 449 450 450 451 451 451 452 452 452 453 453 454 454 454 455 455 455 456
11.35 Maximum anisotropic failure stress 11.36 Maximum compression at vertical wellbore 11.37 Maximum normal stress in tangential direction at wellbore wall (hoop stress) 11.38 Maximum plane tangential stress acting on deviated wellbore 11.39 Maximum principal stress failure (Hoek and Brown) 11.40 Maximum principal stress in normal faulting 11.41 Maximum principal stress in reverse faulting 11.42 Maximum principal stress in strike-slip faulting 11.43 Maximum principal stress calculation using breakout width 11.44 Minimum compression at vertical wellbore 11.45 Minimum normal stress in tangential direction at wellbore wall (hoop stress) 11.46 Maximum plane tangential stress acting on deviated wellbore 11.47 Modified lade criterion 11.48 Normal stress in radial direction near wellbore 11.49 Normal stress in rock at failure 11.50 Normal stress in tangential direction at wellbore wall (hoop stress) 11.51 Normal stress in tangential direction near wellbore (hoop stress) 11.52 Pore pressure increase due to fluid activity (Mody & Hale) 11.53 Pore pressure increase due to given fluid activity contrast (Mody and Hale) 11.54 Pore pressure of shale (Flemings) 11.55 Pore pressure of shale (Traugott) 11.56 Porosity irreversible plastic deformation occurs 11.57 Pressure required to induce a tensile fracture (breakdown pressure) 11.58 Pressure to grow fractures (Abe, Mura, et al.) 11.59 Radial stress around vertical wellbore 11.60 Ratio of pore pressure change to original due to depletion 11.61 Rotation of maximum principal stress near wellbore 11.62 Rotation of maximum principal stress near wellbore (Zoback & Day-Lewis) 11.63 Shale compaction 11.64 Shear modulus
Formulas and Calculations for Petroleum Engineering. https://doi.org/10.1016/B978-0-12-816508-9.00011-1 © 2019 Elsevier Inc. All rights reserved.
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443
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Formulas and calculations for petroleum engineering
11.65 11.66 11.67 11.68 11.69 11.70 11.71
Shear modulus from Young’s modulus Shear stress near vertical well Slowness of the formation Storativity of fractures Stress at edge of wellbore breakout Stress component near normal faulting in reservoir Stress components in original coordinate system in depletion drive 11.72 Stress intensity at tip of mode I fracture 11.73 Stress path (induced normal faulting) 11.74 Stress path of reservoir with changes in production
11.1
468 468 469 469 470 470 471 471 472 472
11.75 Stress perturbation (Segall and Fitzgerald) 11.76 Subsidence due to uniform pore pressure reduction in free surfaces 11.77 Unconfined compressive strength of rock 11.78 Velocity of bulk compressional waves 11.79 Velocity of compression waves 11.80 Velocity of shear waves 11.81 Vp and Vs calculation (Eberhart-Phillips) 11.82 Vp and vs calculation (geomechanical model) 11.83 Yield strength (Bingham plastic model)
Axial stress around vertical wellbore
Input(s) R: Radius of Wellbore (ft) r: Position in Respect to Centre of Wellbore (ft) Y: Azimuth of Shmax (rad) Shmax: Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Shmin:
Output(s) τ:
Twisting Stress (psi)
Formula(s)
2∗ R2 3∗ R4 ∗ sin ð2∗ YÞ τ ¼ 0:5 ∗ ðShmax Shmin Þ ∗ 1 + r2 r4 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.2
Axis of a deviated borehole from an arbitrary origin
Input(s) Po: Pore Pressure (psi) R: Radius of Wellbore (ft) r: Position in Respect to Centre of Wellbore (ft) Y: Azimuth of Shmax (rad) Thermal Stress (psi) sDt: Maximum Horizontal Stress (psi) Shmax: Minimum Horizontal Stress (psi) Shmin:
Output(s) saa:
Stress (psi)
472 473 473 474 474 474 475 475 476
Geomechanics and fracturing formulas and calculations Chapter
11
445
Formula(s) 2 R 3 ∗ R4 Po ∗ R2 0:5∗ ðShmax Shmin Þ∗ 1 + ∗ cos ð 2∗ Y Þ sDt saa ¼ 0:5 ∗ ðShmax + Shmin 2 ∗Po Þ∗ 1 + 2 r r4 r2 Reference: Mark D. Zoback., Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.3
Bulk modulus (using Lame)
Input(s) p: Pressure (Pa) DV: Volume (m3) v: Volume (m3)
Output(s) K:
Bulk Modulus (Pa)
Formula(s) K¼
p DV v
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 45.
11.4
Bulk modulus (using Poisson’s ratio and Lame’s constant)
Input(s) l: G:
Lame (dimensionless) Shear Modulus (N/m2)
Output(s) K:
Bulk Modulus (N/m2)
Formula(s) K ¼ l + 2∗
G 3
Reference: PetroWiki.org.
11.5
Bulk modulus (using Poisson’s ratio and shear modulus)
Input(s) l: G:
Lame (dimensionless) Shear Modulus (N/m2)
Output(s) K:
Bulk Modulus (N/m2)
446
Formulas and calculations for petroleum engineering
Formula(s) K ¼ l + 2∗
G 3
Reference: PetroWiki.org.
11.6
Change in pore volume due to initial water and rock expansion
Input(s) l: n:
Lame (dimensionless) Poisson (dimensionless)
Output(s) Bulk Modulus (N/m2)
K:
Formula(s) K¼
l ∗ ð 1 + nÞ 3∗n
Reference: PetroWiki.org.
11.7
Cohesive strength of rocks
Input(s) G: n:
Shear Modulus (N/m2) Poisson (dimensionless)
Output(s) K:
Bulk Modulus (N/m2)
Formula(s) K¼
2∗ G ∗ ð1 + nÞ 3 ∗ ð 1 2 ∗ nÞ
Reference: PetroWiki.org.
11.8
Compressibility of a coalbed methane formation
Input(s) Wi: W p: Pi: Pd :
Total Volume of Water in the Reservoir (bbl) Total Volume of Water Removed (bbl) Initial Reservoir Pressure (psi) Desorption Pressure (psi)
Output(s) ct:
Total Compressibility (1/psi)
Geomechanics and fracturing formulas and calculations Chapter
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Formula(s) ct ¼
Wp 1 ∗ P i Pd Wi
Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:3, Page: 219.
11.9
Effect of pore pressure on stress
Input(s) l: d: x o: x: G: a: P:
Lames First Constant (psi) Kronecker Delta (dimensionless) Initial Strain (dimensionless) Final Strain (dimensionless) Modulus of Shear (psi) Biot (dimensionless) Pressure Applied (psi)
Output(s) S:
Stress (psi)
Formula(s) S ¼ l ∗ xo ∗ d + 2 ∗ G ∗ x a∗ d ∗ P Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 68.
11.10 Effective stress on individual grains Input(s) S: Pp:
Normal Stress (psi) Pore Pressure (psi)
Output(s) sg:
Effective Stress (psi)
Formula(s) sg ¼ S Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 87.
11.11 Failure criteria (Mohr-Coulomb) Input(s) sa: sb : b:
Principle Stress (psi) Second Strongest Stress (psi) Angle Between Fault and Direction of Principle Stress (degrees)
448
Formulas and calculations for petroleum engineering
Output(s) τ: s:
Shear Stress (psi) Normal Stress (psi)
Formula(s) τ ¼ 0:5 ∗ ðsa sb Þ ∗ sin ð2 ∗ bÞ s ¼ 0:5 ∗ ðsa + sb Þ + 0:5∗ ðsa sb Þ∗ cos ð2 ∗bÞ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
11.12 Formation compressibility by using hydrofrac data Input(s) Vs: Volume Associated to Conduct a Hydrofrac (bbl) dVs: Change in Volume (bbl) dP: Change in Pressure (psi)
Output(s) b:
Formation Compressibility (psi)
Formula(s) b¼
1 dVs ∗ Vs dP
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 221.
11.13 Fracture conductivity Input(s) kf: wf:
Fracture Permeability (mD) Width of Fracture (ft)
Output(s) FC :
Fracture Conductivity (mD ft)
Formula(s) FC ¼ kf ∗wf Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:1, Page: 93.
11.14 Fracture gradient (Eaton) Input(s) u: S v: Pp :
Poisson (dimensionless) Vertical Stress (psi) Pore Pressure (psi)
Geomechanics and fracturing formulas and calculations Chapter
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) u ∗ Sv Pp + Pp 1u
Shmin ¼
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 282.
11.15 Fracture gradient (Holbrook) Input(s) u: S v: Pp :
Poisson (dimensionless) Vertical Stress (psi) Pore Pressure (psi)
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
u ∗ S v Pp + P p 1u
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 282.
11.16 Fracture gradient (Matthews and Kelly) Input(s) smin: Minimum Principle Horizontal Stress (psi) Vertical Principle Stress (psi) sv : Vertical Stress (psi) S v: Pp : Pore Pressure (psi)
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
smin ∗ Sv Pp + Pp sv
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.17 Fracture gradient (Zoback and Healy) Input(s) m: S v: Sp :
Viscosity (cP) Vertical Stress (psi) Pore Pressure (psi)
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Formulas and calculations for petroleum engineering
Output(s) Shmin:
Minimum Horizontal Stress (psi)
Formula(s) Shmin ¼
1 + m2
0:5
+m
2 ∗ Sv Pp + Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.18 Fracture pressure (Hubert & Willis) Input(s) S v: Pp :
Vertical Stress (psi) Pore Pressure (psi)
Output(s) Shmin:
Fracture Pressure (psi)
Formula(s) Shmin ¼ 0:3 ∗ Sv Pp + Pp shmin ¼ 0:3 sv Notes: Fracture Pressure Is assumed to be Equal to Minimum Horizontal Stress. Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 281.
11.19 Fracture volume (GDK method) Input(s) Q: m: L: H: G:
Flow Rate (B/m) Viscosity (cP) Length (ft) Height (ft) Shear Modulus (psi)
Output(s) Vf:
Fracture Volume (ft3)
Formula(s)
H3 Vf ¼ 0:03561 ∗ m ∗ Q ∗ L ∗ G
0:25
6
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 70.
Geomechanics and fracturing formulas and calculations Chapter
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11.20 Fracture volume (Perkins and Kern method) Input(s) H: n: m: Q: E: L:
Fracture Height (ft) Poisson (unitless) Viscosity (cP) Flowrate (B/m) Young (psi) Fracture Length (ft)
Output(s) Fracture Volume (ft3)
Vf:
Formula(s) Vf ¼ 0:04 ∗ H ∗
Q 1 n2 ∗ m ∗ E
0:25
5
∗ L4
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 57.
11.21 Fracture width (GDK method) Input(s) Q: G: m: L: H:
Flow Rate (B/m) Shear Modulus (psi) Viscosity (cP) Length (ft) Height (ft)
Output(s) w:
Fracture Width (in.)
Formula(s) 0:25 L2 w ¼ 0:272∗ m ∗ Q ∗ G∗H Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 70.
11.22 Fracture width (Perkins and Kern method) Input(s) n: Q: m: L: E:
Poisson (unitless) Flowrate (B/m) Viscosity (cP) Fracture Length (ft) Young’s Modulus (psi)
452
Formulas and calculations for petroleum engineering
Output(s) Wmax:
Fracture Width (in.)
Formula(s) Wmax ¼ 0:389 ∗
L 1 n ∗Q∗m∗ E 2
0:25
Reference: Daneshy, A. 2013. Fundamentals of Hydraulic Fracturing, Daneshy Consultants International, Page: 58.
11.23 Hoek and Brown criteria for principal stress failure Input(s) Co: Unconfined Compressive Strength of Rock (psi) m: Constant Depending on Property of Rock and Extent to Which It Is Broken (dimensionless) s: Constant Depending on Property of Rock and Extent to Which It Is Broken (dimensionless) Minimum Effective Principal Stress (psi) sc:
Output(s) sa:
Maximum Effective Principal Stress (psi)
Formula(s) 0:5 s sa ¼ sc + Co ∗ m ∗ c + s Co Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 98.
11.24 Horizontal effective stress (assuming no lateral strain as per Lorenz and Teufel) Input(s) n: Poisson Ratio (dimensionless) Sv: Vertical Overburden Stress (psi) a: Biot Coefficient (dimensionless) P: Pore Pressure (psi)
Output(s) Shor:
Horizontal Stress (psi)
Formula(s) Shor ¼
n n ∗ Sv + a ∗ P ∗ 1 1n 1n
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.25 Horizontal maximum stress (Bredehoeft) Input(s) Shmin:
Minimum Horizontal Stress (psi)
Geomechanics and fracturing formulas and calculations Chapter
Pb: Pp :
11
453
Breakdown Pressure at initial Hydrofrac (psi) Pore Pressure (psi)
Output(s) Shmax:
Maximum Horizontal Stress (psi)
Formula(s) Shmax ¼ 3∗ Shmin Pb Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 220.
11.26 Induced fracture dip Input(s) h: d:
Height of Fracture (ft) Diameter of Well (ft)
Output(s) Dip:
Dip (degrees)
Formula(s) Dip ¼ arctan
h d
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 146.
11.27 Initial effective horizontal stress Input(s) n: r: H: a: pp :
Poisson (unitless) Overburden Density (lb/ft3) Formation Depth (ft) Biot Constant (unitless) Reservoir Pressure (psi)
Output(s) sh:
Effective Horizontal Stress (psi)
Formula(s) n H a ∗ pp sh ¼ ∗ r∗ 1n 144 Reference: Boyun, G., William, C., & Ali Ghalambor, G. (2007). Petroleum Production Engineering: A Computer-Assisted Approach Page: 259.
454
Formulas and calculations for petroleum engineering
11.28 Isothermal compressibility of limestones (Newman correlation) Input(s) ø:
Porosity (fraction)
Output(s) Ct:
Compressibility (psi1)
Formula(s) Ct ¼
97:32∗ 106 ð1 + 55:8721 ∗ øÞ1:42869
Notes: Check for 0.02 < Ø < 0.23. Reference: Applied Petroleum Reservoir Engineering, Craft & Hawkins, Page: 11.
11.29 Least principal stress as function of depth in Gulf of Mexico (Hubbert and Willis) Input(s) Sv: Pp:
Vertical Overburden Stress (psi) Pore Pressure in Reservoir (psi)
Output(s) Shmin:
Minimum Principal Stress in Reservoir (psi)
Formula(s) Shmin ¼ 0:3 ∗ ðSv PpÞ + Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 280.
11.30 Least principal stress as function of depth in Gulf of Mexico (Matthew and Kelly) Input(s) Sv: Ki: Pp:
Vertical Overburden Stress (psi) Constant as Function of Depth (dimensionless) Pore Pressure in Reservoir (psi)
Output(s) Shmin:
Minimum Principal Stress in Reservoir (psi)
Formula(s) Shmin ¼ Ki ∗ ðSv PpÞ + Pp Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 280.
Geomechanics and fracturing formulas and calculations Chapter
11.31 Linearized Mohr failure line Input(s) S o: sn: mi:
Stress (psi) Normal Stress (psi) Coefficient of internal Friction (cP)
Output(s) τ:
Shear Stress (psi)
Formula(s) τ ¼ So + s n ∗ m i Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
11.32 Linearized Mohr coulomb criteria Input(s) Co: m: sc:
Unconfined Compressive Strength of Rock (psi) Slope of Failure Line (dimensionless) Minimum Effective Principal Stress (psi)
Output(s) s1:
Maximum Effective Principal Stress (psi)
Formula(s) s1 ¼ ð Co Þ +
m2 + 1
0:5
2 + m ∗ sc
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 93.
11.33 M Modulus (using shear modulus and bulk modulus) Input(s) G: K:
Shear Modulus (N/m2) Bulk Modulus (N/m2)
Output(s) M:
M Modulus (N/m2)
Formula(s) M ¼ K + 4∗
G 3
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 64.
11
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Formulas and calculations for petroleum engineering
11.34 M Modulus (using Young’s modulus and Poisson’s ratio) Input(s) n: E:
Poisson Ratio (dimensionless) Young Modulus (N/m2)
Output(s) M:
M Modulus (N/m2)
Formula(s) M ¼ E∗
1n ð1 + nÞ∗ ð1 2 ∗ nÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 64.
11.35 Maximum anisotropic failure stress Input(s) sc: Sw : m: b:
Minimum Principle Stress (psi) Intact Rock Strength (psi) Internal Friction of Weak Bedding (cP) Angle of Weak Plane to Maximum Principle Stress (degrees)
Output(s) s:
Stress (cm/s)
Formula(s) s¼
s c ∗ 2 ∗ ðSw + m ∗ sc Þ ð1 m ∗ cot ðbÞÞ ∗ sin ð2 ∗ bÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 107.
11.36 Maximum compression at vertical wellbore Input(s) sc: Sw: m: b:
Minimum Principle Stress (psi) Intact Rock Strength (psi) Internal Friction of Weak Bedding (cP) Angle of Weak Plane to Maximum Principle Stress (degrees)
Output(s) s:
Stress (cm/s)
Geomechanics and fracturing formulas and calculations Chapter
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Formula(s) s¼
sc ∗ 2∗ ðSw + m ∗ sc Þ ð1 m∗ cot ðbÞÞ∗ sin ð2∗ bÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 107.
11.37 Maximum normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in Reservoir (psi) Minimum Principal Stress in Reservoir (psi) Shmin: Po: Pore Pressure (psi) Stress induced Due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi)
Output(s) sigth:
Maximum Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) sigth ¼ 3 ∗ Shmax Shmin 2∗ Po dP Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 174.
11.38 Maximum plane tangential stress acting on deviated wellbore Input(s) szz: Stress in Radial Direction (psi) Stress in Axial Direction (psi) saa: τ: Shear Stress (psi)
Output(s) stmax:
Maximum Tangential Stress (psi)
Formula(s) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 stmax ¼ ∗ szz + saa + ðszz saa Þ2 + 4∗ τ2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 stmax ¼ ∗ szz + saa ðszz saa Þ2 + 4 ∗ τ2 2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 239.
11.39 Maximum principal stress failure (Hoek and Brown) Input(s) sb: Second Largest Stress (psi) C: Rock Compressive Strength (lbf) m: Hoek - Brown Constant Dependent on Rock Type (dimensionless) s: Hoek and Brown Constant Dependent on Shape (dimensionless)
458
Formulas and calculations for petroleum engineering
Output(s) sa:
Principal Stress (psi)
Formula(s) 0:5 s sa ¼ sb + C ∗ m ∗ b + s C Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 98.
11.40 Maximum principal stress in normal faulting Input(s) sc: Least Principle Stress (psi) Vertical Stress (psi) S v: Pore Pressure (psi) Pp : Minimum Horizontal Stress (psi) Shm:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Sv Pp Shm Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
11.41 Maximum principal stress in reverse faulting Input(s) sc: Least Principle Stress (psi) Pore Pressure (psi) P p: Vertical Stress (psi) S v: Maximum Horizontal Stress (psi) Shmax:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Shmax Pp Sv Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
Geomechanics and fracturing formulas and calculations Chapter
11.42 Maximum principal stress in strike-slip faulting Input(s) sc: Least Principle Stress (psi) Maximum Horizontal Stress (psi) Shmax: Pore Pressure (psi) P p: Minimum Horizontal Stress (psi) Shmin:
Output(s) s:
Maximum Principle Stress (psi)
Formula(s) s¼
sc ∗ Shmax Pp Shmin Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 133.
11.43 Maximum principal stress calculation using breakout width Input(s) Shmin: Minimum Principal Stress in Reservoir (psi) Distance from Wellbore (ft) C o: Pp: Pore Pressure (psi) Stress induced Due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi) Y: Angle from Wellbore Breakout Width (rad)
Output(s) Shmin:
Maximum Principal Stress in Reservoir (psi)
Formula(s) Shmax 5
ððCo Þ + 2∗ Pp + dP + SdtÞ Shmin ∗ ð1 + 2∗ cos ðYÞÞ 1 2 ∗ cos ðYÞ
Reference: Mark D. Zoback., Reservoir Geomechanics, Cambridge University Press, UK, Page: 223.
11.44 Minimum compression at vertical wellbore Input(s) Shmin: Minimum Horizontal Stress (psi) Maximum Horizontal Stress (psi) Shmax: Pore Pressure (psi) P o: P: Pressure Drawdown (psi) Thermal Stress (psi) st:
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Formulas and calculations for petroleum engineering
Output(s) sminaxial:
Minimum Axial Stress (psi)
Formula(s) sminaxial 53 ∗Shmin 2Shmax 22∗ Po 2P2st Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 174.
11.45 Minimum normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in Reservoir (psi) Minimum Principal Stress in Reservoir (psi) Shmin: Po: Pore Pressure (psi) Stress induced due to Temperature (psi) Sdt: dP: Difference Between Wellbore Pressure and Mud Weight (psi)
Output(s) sigth:
Minimum Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) sigth ¼ 3∗ Shmin Shmax 2 ∗ Po dP Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 238.
11.46 Maximum plane tangential stress acting on deviated wellbore Input(s) szz: Radial Stress (psi) Axial Stress (psi) saa: τ: Shear Stress (psi)
Output(s) stmax:
MaximumTangential Stress (psi)
Formula(s) 0:5 2 2 stmax ¼ 0:5 ∗ szz + saa ðszz saa Þ + 4 ∗ τ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 239.
11.47 Modified lade criterion Input(s) S a: Sb :
Principle Stress (psi) Intermediate Stress (psi)
Geomechanics and fracturing formulas and calculations Chapter
S c: Pa : m:
11
461
Minimum Stress (psi) Pressure (psi) Material Strength Constant (dimensionless)
Output(s) Ia: Ic: :
First Invariant of Stress Tensor (psi) Third Invariant of Stress Tensor (psi3) Lades Coefficient (dimensionless)
Formula(s) Ia ¼ Sa + Sb + Sc I c ¼ S a ∗ Sb ∗ Sc 3 m Ia Ia 27 ∗ Z¼ Pa I3c Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 99.
11.48 Normal stress in radial direction near wellbore Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) Po: Pore Pressure (psi) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
Output(s) sigrr :
Normal Stress in Radial Direction Near Wellbore (psi)
Formula(s) R2 R2 R4 3:142 R2 + Po ∗ 2 sigrr ¼ 0:5 ∗ ðShmax + Shmin 2∗ PoÞ ∗ 1 2 + 0:5 ∗ ðShmax ShminÞ ∗ 1 4∗ 2 + 3∗ 4 ∗ cos 2∗ y ∗ r r r r 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 100.
11.49 Normal stress in rock at failure Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) Po: Pore Pressure (psi) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
462
Formulas and calculations for petroleum engineering
Output(s) sigrr :
Normal Stress in Radial Direction Near Wellbore (psi)
Formula(s) R2 R2 R4 3:142 R2 + Po∗ 2 sigrr ¼ 0:5 ∗ ðShmax + Shmin 2 ∗PoÞ∗ 1 2 + 0:5∗ ðShmax ShminÞ∗ 1 4 ∗ 2 + 3 ∗ 4 ∗ cos 2 ∗ y∗ r r r r 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 101.
11.50 Normal stress in tangential direction at wellbore wall (hoop stress) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Shmin: Minimum Principal Stress in reservoir (psi) Po: Pore Pressure (psi) Sdt: Stress induced due to temperature (psi) dP: Difference between Wellbore Pressure and Mud Weight (psi) y: Angle from Shmax at which stress is measured (degrees)
Output(s) sigth :
Hoop Stress in Tangential Direction at Wellbore Wall (psi)
Formula(s) 3:142 2∗ Po dP Sdt sigth ¼ ðShmax + ShminÞ 2∗ ðShmax ShminÞ∗ cos 2 ∗ y∗ 180 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 102.
11.51 Normal stress in tangential direction near wellbore (hoop stress) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Shmin: Minimum Principal Stress in reservoir (psi) Po: Pore Pressure (psi) Sdt: Stress induced due to temperature (psi) dP: Difference between Wellbore Pressure and Mud Weight (psi) y: Angle from Shmax at which stress is measured (degrees) r: Distance from wellbore (ft) R: Radius of wellbore (ft)
Output(s) Sgth:
Normal Stress in Tangential Direction Near Wellbore (Hoop Stress) (psi)
Geomechanics and fracturing formulas and calculations Chapter
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Formula(s) R2 R4 3:142 R2 Po ∗ 2 Sgth ¼ 0:5∗ ðShmax + Shmin 2 ∗ PoÞ∗ 1 + 2 0:5 ∗ ðShmax ShminÞ ∗ 1 + 3∗ 4 ∗ cos 2∗ y ∗ r r r 180 Sdt Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 104.
11.52 Pore pressure increase due to fluid activity (Mody & Hale) Input(s) Em : Membrane Efficiency (dimensionless) R: Gas Constant (psi/mol K) T: Temperature (K) V: Volume (ft3) Pore Fluid Activity (dimensionless) A p: Am : Mud Activity (dimensionless)
Output(s) dP:
Pore Pressure Increase (psi)
Formula(s) Ap R ∗T ∗ ln dP ¼ Em ∗ Am V
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 321.
11.53 Pore pressure increase due to given fluid activity contrast (Mody and Hale) Input(s) E m: Membrane Efficiency (dimensionless) R: Gas Constant (dimensionless) T: Temperature (K) V: Molar Volume of Water (L/mol) Pore Fluid Activity (dimensionless) Ap: Mud Activity (dimensionless) Am:
Output(s) dP:
Pore Pressure Increase (psi)
Formula(s)
T Ap ∗ log dP ¼ Em ∗ R ∗ V Am Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 105.
464
Formulas and calculations for petroleum engineering
11.54 Pore pressure of shale (Flemings) Input(s) S v: bc: ø o: t: ø:
Sonic Velocity (ft/s) Compressibility (1/psi) Initial Porosity (fraction) Sonic Travel Time (s) Porosity from Sonic Log (fraction)
Output(s) Pp:
Pore Pressure (psi)
Formula(s) Pp 5Sv 2
1 ø ∗ ln o ø bc
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 48.
11.55 Pore pressure of shale (Traugott) Input(s) z: Depth (ft) S v: Sonic Velocity (ft/s) Resistivity of Shale (ohm ft) Ro : Expected Resistivity (ohm ft) Rn : Hydrostatic Pore Pressure (psi) ( Pp)hydro:
Output(s) Ppsh:
Pore Pressure of Shale (psi)
Formula(s) 0 0 hydro 11 1 1:2 ! P Ro B S B p CC B S C Ppsh ¼ z∗ @ v @ v @ AA ∗ A z z z Rn 0
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 47.
11.56 Porosity irreversible plastic deformation occurs Input(s) m: S v: SH : Sh : Pp :
Viscosity (cP) Vertical Stress (psi) Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Change in Pore Pressure (psi)
Geomechanics and fracturing formulas and calculations Chapter
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465
Output(s) M: p:
Mobility (dimensionless) Porosity (dimensionless)
Formula(s) M¼
6∗m 2 3 ∗ ððm Þ + 1Þ0:5 m
! 2 1 9 9 2 2 2 ∗ 9 ∗ Pp + 1+ 2 ∗ Sv + SH + Sh + 2 2 ∗ ðSv ∗ SH +Sv ∗ Sh +SH ∗ Sh Þ6∗ Pp ∗ ðSv + SH + Sh Þ p¼ 3 ∗ ðSv ∗ SH + Sh Þ 9 ∗ Pp M M Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 400.
11.57 Pressure required to induce a tensile fracture (breakdown pressure) Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: Pore Pressure (psi) Pp: Minimum Hoop Stress for formation at which crack initiates (psi) To :
Output(s) Pb:
Breakdown Pressure (psi)
Formula(s) Pb ¼ 3 ∗ Shmin Shmax Pp + To Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 56.
11.58 Pressure to grow fractures (Abe, Mura, et al.) Input(s) S c: Pp : cf: ci:
Minimum Principle Stress (psi) Pore Pressure (psi) Radius of Fracture (in.) Radius of Invaded Zone (in.)
Output(s) Pgrow:
Growth Pressure (psi)
466
Formulas and calculations for petroleum engineering
Formula(s) 0
1 2 !0:5 P c p B1 C ∗ 1 f B C Sc ci B C B 0 Pgrow ¼ Sc ∗ B !0:5 1 C C 2 B C @ 1 @ 1 cf A A ci Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 326.
11.59 Radial stress around vertical wellbore Input(s) Shmax: Maximum Horizontal Stress (psi) Minimum Horizontal Stress (psi) Shmin: Pore Pressure (psi) Po : R: Radius of Wellbore (ft) r: Relative Position to Centre (ft) y: Azimuth of Shmax (rad)
Output(s) srr:
Stress (psi)
Formula(s) 2 R 3 ∗ R4 4∗ R2 Po ∗ R2 + 0:5∗ ðShmax Shmin Þ ∗ 1 + ∗ cos ð 2 ∗ y Þ + srr ¼ 0:5 ∗ ðShmax + Shmin 2∗ Po Þ∗ 1 2 r r4 r2 r2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 170.
11.60 Ratio of pore pressure change to original due to depletion Input(s) P: Change in Pore Pressure (psi) Maximum Horizontal Stress (psi) Shmax: Minimum Horizontal Stress (psi) Shmin:
Output(s) q:
Pore Pressure Ratio (fraction)
Formula(s) q¼
Pp Shmax Shmin
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
Geomechanics and fracturing formulas and calculations Chapter
11
11.61 Rotation of maximum principal stress near wellbore Input(s) A: Stress Field Direction (dimensionless) Change in Pore Pressure (psi) Pp: y: Fault Orientation (degrees) Maximum Principle Stress (psi) Shmax: Shmin: Minimum Principle Stress (psi)
Output(s) g:
Rotation (degrees)
Formula(s) ! A ∗ Pp ∗ sin ð2∗ yÞ g ¼ 0:5∗ atan Shmax Shmin + A ∗ Pp ∗ cos ð2 ∗ yÞ Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.62 Rotation of maximum principal stress near wellbore (Zoback & Day-Lewis) Input(s) A: D: q:
Constant a Value (dimensionless) Fault Orientation (degrees) Ratio of Pore Pressure to Differential Stress (dimensionless)
Output(s) g:
Stress Rotation (radian)
Formula(s)
sin ð2 ∗ DÞ g ¼ 0:5 ∗ atan A ∗ q ∗ 1 + A ∗ q ∗ cos ð2 ∗ DÞ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.63 Shale compaction Input(s) Ø: b: sv:
Porosity (fraction) Second Empirical Constant from Porosity vs Vertical Stress Graph (1/MPa) Vertical Effective Stress (MPa)
Output(s) Øe:
Shale Compaction (fraction)
467
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Formulas and calculations for petroleum engineering
Formula(s) Øe ¼ Ø ∗ eb ∗ sv Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 46.
11.64 Shear modulus Input(s) F: A: y:
Force (N) Area (m2) Deformation Angle (degrees)
Output(s) G:
Shear Modulus (Pa)
Formula(s) G¼
F=A y
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 45.
11.65 Shear modulus from Young’s modulus Input(s) n: E:
Poisson’s Ratio (dimensionless) Young’s Modulus (N/m2)
Output(s) G:
Modulus of Rigidity (psi)
Formula(s) G¼
E 2∗ ð1 + nÞ
Reference: Samuel. E Robello. 501 Solved Problems and Calculations for Drilling Operations. Sigma Quadrant. 2015. Houston, Texas, Page: 356.
11.66 Shear stress near vertical well Input(s) Shmax: Maximum Principal Stress in reservoir (psi) Minimum Principal Stress in reservoir (psi) Shmin: r: Distance from wellbore (ft) R: Radius of wellbore (ft) y: Angle from Shmax at which stress is measured (degrees)
Geomechanics and fracturing formulas and calculations Chapter
11
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Output(s) sigrth:
Shear Stress near Vertical Well (psi)
Formula(s)
R2 R4 3:142 sigrth ¼ 0:5 ∗ ðShmax + Shmin Þ ∗ 1 + 2∗ 2 3 ∗ 4 ∗ sin 2 ∗y ∗ r r 180 Reference: petrowiki.org.
11.67 Slowness of the formation Input(s) d h: Dtm: dt: Ls: lc: tl:
Borehole Diameter (in.) Interval Travel Time (ms/ft) Tool Diameter (in.) Spacing of the Tool (ft) Eccentricity of the Tool (ft) Time Between Initiation of the Pulse and First Arrival Acoustic Energy at the Receiver (ms/ft)
Output(s) Dt: tm:
Slowness of Formation Observed by Sonic Log (ms/ft) Mud Path Correction Time (ms/ft)
Formula(s) Dt ¼
tl tm Ls
Dt tm ¼ ðDtm Þ∗ ðdh ðdt + 2 ∗ lc ÞÞ ∗ 1 Dtm
2 !0:5
Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 10, Page: 189.
11.68 Storativity of fractures Input(s) øf: ø m: h f: h m: ctf: ctm:
Porosity of Fracture (fraction) Porosity of Matrix (fraction) Fracture Thickness (ft) Matrix Thickness (ft) Total Fracture Compressibility (1/psi) Total Matrix Compressibility (1/psi)
Output(s) o:
Storativity of Fracture (dimensionless)
470
Formulas and calculations for petroleum engineering
Formula(s) o¼
øf ∗ hf ∗ctf øf ∗ hf ∗ ctf + øm ∗ hm ∗ ctm
Reference: Ahmed, T., McKinney, P.D. 2005. Advanced Reservoir Engineering, Gulf Publishing of Elsevier, Chapter:1, Page: 82.
11.69 Stress at edge of wellbore breakout Input(s) Co: Wellbore Strength (psi) Pore Pressure (psi) Pp: P: Drawdown Differential in Pressure (psi) Thermally Induced Stress (psi) st: Wellbore Breakout (degrees) wb: Minimum Horizontal Stress (psi) Shmin:
Output(s) y b: Breakout Angle (degrees) Maximum Principle Stress (psi) Shmax:
Formula(s) Shmax ¼
2yb ¼ p wb ðCo Þ + 2 ∗ Pp + P + st Shmin ∗ ð1 + 2∗ cos ðyb ÞÞ 1 2∗ cos ðyb Þ
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 223.
11.70 Stress component near normal faulting in reservoir Input(s) a: Biot (dimensionless) n: Poisson (dimensionless) Maximum Principal Stress (psi) Shmax: Minimum Principal Stress (psi) Shmin: dP: Change in Pore Pressure (psi) y: Fault Orientation (degrees)
Output(s) A: Constant a Value (dimensionless) Stress in X Direction (psi) Sx: Stress in Y Direction (psi) Sy : Normal Stress in Y Direction (psi) Txy:
Geomechanics and fracturing formulas and calculations Chapter
Formula(s) 1 2∗n 1n dP p Sx ¼ Shmax A ∗ dP A ∗ ∗ 1 cos 2 ∗y ∗ 2 180 dP p Sy ¼ Shmin A ∗ dP A ∗ ∗ 1 + cos 2 ∗ y∗ 2 180 dP p Txy ¼ A ∗ ∗ sin 2 ∗ y ∗ 2 180 A ¼ a∗
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.71 Stress components in original coordinate system in depletion drive Input(s) Shmax: Maximum Stress in Horizontal Direction (psi) Minimum Stress in Horizontal Direction (psi) Shmin: Change in Pore Pressure due to Depletion (psi) dPp: A: Stress Path (dimensionless) D: Fault Orientation (degrees)
Output(s) Sx: Sy : τ:
Stress in X-direction (psi) Stress in Y-direction (psi) Stress Around Wellbore (psi)
Formula(s) A ∗ dPp ∗ ð1 cos ð2∗ DÞÞ 2 A ∗ dPp Sy ¼ Shmin A ∗ Pp ∗ ð1 + cos ð2 ∗ DÞÞ 2 A ∗ dPp τ¼ ∗ sin ð2∗ DÞ 2
Sx ¼ Shmax A ∗Pp
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 393.
11.72 Stress intensity at tip of mode I fracture Input(s) Pf: Sc: L:
Fracture Pressure (psi) Minimum Principle Stress (psi) Length of Fracture (ft)
Output(s) K:
Stress Intensity (psi ft)
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472
Formulas and calculations for petroleum engineering
Formula(s) K ¼ ðPf Sc Þ∗ p ∗ L0:5 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 122.
11.73 Stress path (induced normal faulting) Input(s) m:
Friction Coefficient (dimensionless)
Output(s) A:
Stress Path (dimensionless)
Formula(s) A ¼ 1
1
2 ðm2 + 1Þ0:5 + m
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 385.
11.74 Stress path of reservoir with changes in production Input(s) a: u:
Biots Coefficient (dimensionless) Poisson (dimensionless)
Output(s) A:
Stress Path (dimensionless)
Formula(s) A¼
a ∗ ð1 2 ∗ uÞ 1u
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 381.
11.75 Stress perturbation (Segall and Fitzgerald) Input(s) u: H: R: a:
Poisson (dimensionless) Height of Reservoir (ft) Half the Lateral Extent (ft) Constant of Stress Propagation (dimensionless)
Output(s) M:
Stress Perturbation (dimensionless)
Geomechanics and fracturing formulas and calculations Chapter
11
Formula(s) M ¼ a∗
ð1 2 ∗uÞ ∗ p ∗ H ð1 uÞ ∗4 ∗ 2 ∗ R
Reference: Mark D. Zoback, Reservoir Geomechanics. Cambridge University Press. Cambridge, UK, Page: 112.
11.76 Subsidence due to uniform pore pressure reduction in free surfaces Input(s) cm: Formation Compaction per Unit Change in Pore Pressure Reduction (ft3/psi) u: Poisson’s (dimensionless) r: Radius of Area Involved (ft) D: Depth of Formation in Consideration (ft) Pore Pressure Change (psi) DPp: V: Volume of Reservoir (ft3)
Output(s) u z: ur:
Subsidence in Z Direction (ft) Subsidence Along R (ft)
Formula(s) ! cm ∗ ð1 uÞ∗ D ∗ DPp ∗ V uz ¼ ð1Þ∗ 1:5 p∗ ðr2 Þ + D2 ! cm ∗ ð1 uÞ ∗r ∗ DPp ∗ V ur ¼ 1:5 p ∗ ðr2 Þ + D2 Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 412.
11.77 Unconfined compressive strength of rock Input(s) S o: m:
Cohesive Strength (psi) Slope of Failure Line (dimensionless)
Output(s) Co:
Unconfined Compressive Strength of Rock (psi)
Formula(s) Co ¼ 2 ∗So ∗
m2 + 1
0:5
+m
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 89.
473
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Formulas and calculations for petroleum engineering
11.78 Velocity of bulk compressional waves Input(s) E: r: m:
Young’s Modulus (lbf/ft2) Density (lbm/ft3) Poisson’s Ratio (dimensionless)
Output(s) Vb:
Velocity of Bulk Compressional Waves (ft/s)
Formula(s)
E 1m Vb ¼ r ∗ ð1 + mÞ ∗ ð1 2 ∗ mÞ
0:5
Reference: Core Laboratories. 2005. Formation Evaluation and Petrophysics, Page: 23.
11.79 Velocity of compression waves Input(s) K: G: r:
Bulk Modulus (Pa) Shear Modulus (Pa) Density (kg/m3)
Output(s) Vp:
Velocity of Compression Waves (m/s)
Formula(s) 4 G 0:5 Vp ¼ K+ 3 ∗r Reference: Bassiouni, Z. 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 46.
11.80 Velocity of shear waves Input(s) G: r:
Shear Modulus (Pa) Density (kg/m3)
Output(s) Vs :
Velocity of Shear Waves (m/s)
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Formula(s) 0:5 G Vs ¼ r Reference: Bassiouni, Z., 1994, Theory, Measurement, and Interpretation of Well Logs. SPE Textbook Series Vol. 4. Chapter 3, Page: 46.
11.81 Vp and Vs calculation (Eberhart-Phillips) Input(s) ø: C: s:
Porosity (fraction) Clay Content (fraction) Effective Stress (psi)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) Vp ¼ 5:77 6:94 ∗ ø 1:73 ∗ C0:5 + 0:446 ∗ s ð1Þ ∗eð1Þ ∗ 16:7 ∗ s Vs ¼ 3:7 4:94 ∗ ø 1:57 ∗ C0:5 + 0:361 ∗ s ð1Þ∗ eð1Þ ∗ 16:7 ∗ s Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 53.
11.82 Vp and Vs calculation (geomechanical model) Input(s) K: G: r:
Bulk Modulus (psi) Shear Modulus (psi) Density (ppg)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) 1 4 G 0:5 K+ ∗ B 3 C Vp ¼ @ A r 0
0:5 G Vs ¼ r Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, Page: 63.
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Formulas and calculations for petroleum engineering
11.83 Yield strength (Bingham plastic model) Input(s) K: G: r:
Bulk Modulus (psi) Shear Modulus (psi) Density (ppg)
Output(s) Vp: Vs :
Velocity of Compressional Waves (ft/s) Shear Waves (ft/s)
Formula(s) 1 4 G 0:5 K+ ∗ B 3 C Vp ¼ @ A r 0
Vs ¼
0:5 G r
Reference: Mark D. Zoback, Reservoir Geomechanics, Cambridge University Press, UK, page 63.