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In-Situ Stress and Rock Mechanics Equations

Browse 72 in-situ stress and rock mechanics petroleum engineering equations with formulas, inputs, outputs, units, and sources.

In-Situ Stress and Rock Mechanics equations group related upstream petroleum engineering formulas by workflow so engineers can find the right calculation faster.

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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Bulk Modulus from Lame Constant and Poisson Ratio

K=λ(1+ν)3νK=\frac{\lambda(1+\nu)}{3\nu}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Bulk Modulus from Lame Constant and Shear Modulus

K=λ+2G3K=\lambda+\frac{2G}{3}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Bulk Modulus from Pressure and Volume Strain

K=pΔV/VK=\frac{p}{\Delta V/V}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Bulk Modulus from Shear Modulus and Poisson Ratio

K=2G(1+ν)3(1−2ν)K=\frac{2G(1+\nu)}{3(1-2\nu)}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Bulk Modulus from Young's Modulus and Poisson Ratio

K=E3(1−2ν)K = \frac{E}{3(1 - 2\nu)}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Classic Breakdown Pressure for a Vertical Wellbore

Pb=3Shmin−SHmax−Pp+ToP_b=3S_{hmin}-S_{Hmax}-P_p+T_o
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Compressional and Shear Wave Velocities from Elastic Moduli

Vp=K+4G/3ρV_p=\sqrt{\frac{K+4G/3}{\rho}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Drilling-Induced Tensile Fracture Margin

MT=σθθ+TsM_T=\sigma_{\theta\theta}+T_s
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Eaton Minimum Horizontal Stress Fracture Pressure

Shmin=ν1−ν(Sv−Pp)+PpS_{hmin}=\frac{\nu}{1-\nu}(S_v-P_p)+P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Eaton Pore Pressure Gradient from Corrected D-Exponent

Gpp=Gob−(Gob−Gnp)(dcodcn)1.2G_{pp}=G_{ob}-(G_{ob}-G_{np})\left(\frac{d_{co}}{d_{cn}}\right)^{1.2}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Eaton Pore Pressure Gradient from Resistivity Ratio

Gpp=Gob−(Gob−Gnp)(RoRn)1.2G_{pp}=G_{ob}-(G_{ob}-G_{np})\left(\frac{R_o}{R_n}\right)^{1.2}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Eaton Pore Pressure Gradient from Sonic Transit Time

Gpp=Gob−(Gob−Gnp)(ΔtnΔto)3G_{pp}=G_{ob}-(G_{ob}-G_{np})\left(\frac{\Delta t_n}{\Delta t_o}\right)^3
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Effect of Pore Pressure on Stress

S=λξoδ+2Gξ−αδPS = \lambda \xi_o \delta + 2G\xi - \alpha\delta P
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Effective Stress on Individual Grains

σg=S−Pp\sigma_g = S - P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Failure Criteria - Mohr-Coulomb Stress Components

τ=0.5(σa−σb)sin⁡(2β)\tau=0.5(\sigma_a-\sigma_b)\sin(2\beta)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Formation Compressibility from Hydrofrac Data

β=1VsdVsdP\beta=\frac{1}{V_s}\frac{dV_s}{dP}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Fracture Gradient - Zoback and Healy

Shmin=(1+μf2+μf)−2(Sv−Pp)+PpS_{hmin} = (\sqrt{1 + \mu_f^2} + \mu_f)^{-2}(S_v - P_p) + P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Fracture Pressure - Hubbert and Willis

Shmin=0.3(Sv−Pp)+PpS_{hmin}=0.3(S_v-P_p)+P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Hoek and Brown Criteria for Principal Stress Failure

σa=σc+Comσc/Co+s\sigma_a=\sigma_c+C_o\sqrt{m\sigma_c/C_o+s}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Horizontal Effective Stress - Lorenz and Teufel

Shor=ν1−νSv+αP(1−ν1−ν)S_{hor}=\frac{\nu}{1-\nu}S_v+\alpha P\left(1-\frac{\nu}{1-\nu}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Horizontal Maximum Stress - Bredehoeft

Shmax=3Shmin−Pb−PpS_{hmax}=3S_{hmin}-P_b-P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Induced Fracture Dip

Dip=tan⁡−1(hd)\mathrm{Dip}=\tan^{-1}\left(\frac{h}{d}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Initial Effective Horizontal Stress

σh=ν1−ν(ρbH144−αPp)\sigma_h=\frac{\nu}{1-\nu}\left(\frac{\rho_bH}{144}-\alpha P_p\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Isothermal Compressibility of Limestones - Newman Correlation

Ct=97.32×10−6(1+55.8721ϕ)1.42869C_t=\frac{97.32\times10^{-6}}{(1+55.8721\phi)^{1.42869}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Least Principal Stress - Matthews and Kelly

Shmin=Ki(Sv−Pp)+PpS_{hmin}=K_i(S_v-P_p)+P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Linearized Mohr Failure Line

τ=So+σnμi\tau=S_o+\sigma_n\mu_i
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Linearized Mohr-Coulomb Criteria

σ1=Co+(μ2+1+μ)2σc\sigma_1=C_o+(\sqrt{\mu^2+1}+\mu)^2\sigma_c
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

M Modulus from Shear and Bulk Modulus

M=K+4G3M=K+\frac{4G}{3}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

M Modulus from Young's Modulus and Poisson Ratio

M=E1−ν(1+ν)(1−2ν)M=E\frac{1-\nu}{(1+\nu)(1-2\nu)}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Matthews-Kelly Fracture Gradient from Stress Ratio

Shmin=(σminσv)(Sv−Pp)+PpS_{hmin}=\left(\frac{\sigma_{min}}{\sigma_v}\right)(S_v-P_p)+P_p
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Maximum Hoop Stress at the Wellbore Wall

σθ,max=3Shmax−Shmin−2Po−ΔP−Sdt\sigma_{\theta,max}=3S_{hmax}-S_{hmin}-2P_o-\Delta P-S_{dt}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Maximum Horizontal Stress from Wellbore Breakout Width

Shmax=Co+2Pp+P+σt−Shmin(1+2cos⁡θb)1−2cos⁡θb,θb=(π−wb)/2S_{hmax}=\frac{C_o+2P_p+P+\sigma_t-S_{hmin}(1+2\cos\theta_b)}{1-2\cos\theta_b},\quad \theta_b=(\pi-w_b)/2
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Maximum Principal Stress in Normal Faulting

σmax=σc(Sv−Pp)Shmin−Pp\sigma_{max}=\frac{\sigma_c(S_v-P_p)}{S_{hmin}-P_p}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Maximum Principal Stress in Reverse Faulting

σmax=σc(Shmax−Pp)Sv−Pp\sigma_{max}=\frac{\sigma_c(S_{hmax}-P_p)}{S_v-P_p}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Maximum Principal Stress in Strike-Slip Faulting

σmax=σc(Shmax−Pp)Shmin−Pp\sigma_{max}=\frac{\sigma_c(S_{hmax}-P_p)}{S_{hmin}-P_p}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Minimum Axial Stress at a Vertical Wellbore

σminaxial=3Shmin−Shmax−2Po−P−σt\sigma_{minaxial}=3S_{hmin}-S_{hmax}-2P_o-P-\sigma_t
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Minimum Hoop Stress at the Wellbore Wall

σθ,min=3Shmin−Shmax−2Po−ΔP−Sdt\sigma_{\theta,min}=3S_{hmin}-S_{hmax}-2P_o-\Delta P-S_{dt}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Minimum Mud Pressure to Avoid Wellbore Breakout

PWshear=Pp+3σHmax−σhmin−UCS1+qP_{Wshear}=P_p+\frac{3\sigma_{Hmax}-\sigma_{hmin}-UCS}{1+q}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Mode I Fracture Stress Intensity

KI=(Pf−Sc)πL0.5K_I=(P_f-S_c)\pi L^{0.5}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Modified Eaton Pore Pressure Gradient from Specific Energy

Gpp=Gob−(Gob−Gnp)(SPEoSPEn)mG_{pp}=G_{ob}-(G_{ob}-G_{np})\left(\frac{SPE_o}{SPE_n}\right)^m
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Modified Lade Failure Criterion

η=(I13I3−27)(I1Pa)m\eta=\left(\frac{I_1^3}{I_3}-27\right)\left(\frac{I_1}{P_a}\right)^m
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Mody-Hale Osmotic Pore-Pressure Increase from Molar Volume

ΔP=Em(RψLTVw)ln⁡(ApAm)\Delta P=E_m\left(\frac{R_{\psi L}T}{V_w}\right)\ln\left(\frac{A_p}{A_m}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Mud Pressure for Target Breakout Width

PWBO=Pp+(σHmax+σhmin)−2(σHmax−σhmin)cos⁡(π−wBO)−UCS1+qP_{WBO}=P_p+\frac{(\sigma_{Hmax}+\sigma_{hmin})-2(\sigma_{Hmax}-\sigma_{hmin})\cos(\pi-w_{BO})-UCS}{1+q}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Normal Hoop Stress at the Wellbore Wall

σθ=Shmax+Shmin−2(Shmax−Shmin)cos⁡(2θ)−2Po−ΔP−Sdt\sigma_\theta=S_{hmax}+S_{hmin}-2(S_{hmax}-S_{hmin})\cos(2\theta)-2P_o-\Delta P-S_{dt}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Normal Radial Stress Near a Wellbore

σrr=12(Shmax+Shmin−2Po)(1−R2r2)+12(Shmax−Shmin)(1−4R2r2+3R4r4)cos⁡(2θ)+PoR2r2\sigma_{rr}=\frac{1}{2}(S_{hmax}+S_{hmin}-2P_o)\left(1-\frac{R^2}{r^2}\right)+\frac{1}{2}(S_{hmax}-S_{hmin})\left(1-4\frac{R^2}{r^2}+3\frac{R^4}{r^4}\right)\cos(2\theta)+P_o\frac{R^2}{r^2}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Normal Tangential Hoop Stress Near a Wellbore

Sgth=12(Shmax+Shmin−2Po)(1+R2r2)−12(Shmax−Shmin)(1+3R4r4)cos⁡(2θ)−PoR2r2−SdtSg_{th}=\frac{1}{2}(S_{hmax}+S_{hmin}-2P_o)\left(1+\frac{R^2}{r^2}\right)-\frac{1}{2}(S_{hmax}-S_{hmin})\left(1+\frac{3R^4}{r^4}\right)\cos(2\theta)-P_o\frac{R^2}{r^2}-S_{dt}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Pore Pressure Increase Due to Fluid Activity Mody and Hale

δP=Em(RTV)ln⁡(ApAm)\delta P=E_m\left(\frac{RT}{V}\right)\ln\left(\frac{A_p}{A_m}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Pore Pressure of Shale Flemings

Pp=Sv−1βcln⁡(ϕoϕ)P_p=S_v-\frac{1}{\beta_c}\ln\left(\frac{\phi_o}{\phi}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Pore Pressure of Shale Traugott

Ppsh=z[Svz−(Svz−Phydroz)(RoRn)1.2]P_{psh}=z\left[\frac{S_v}{z}-\left(\frac{S_v}{z}-\frac{P_{hydro}}{z}\right)\left(\frac{R_o}{R_n}\right)^{1.2}\right]
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Pressure to Grow Fractures - Abe, Mura et al.

Pgrow=Sc1−(Pp/Sc)1−(cf/ci)21−1−(cf/ci)2P_{grow}=S_c\frac{1-(P_p/S_c)\sqrt{1-(c_f/c_i)^2}}{1-\sqrt{1-(c_f/c_i)^2}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Radial Stress Around a Vertical Wellbore

σrr=0.5(Shmax+Shmin−2Po)(1−R2/r2)+0.5(Shmax−Shmin)(1+3R4/r4−4R2/r2)cos⁡(2θ)+PoR2/r2\sigma_{rr}=0.5(S_{hmax}+S_{hmin}-2P_o)(1-R^2/r^2)+0.5(S_{hmax}-S_{hmin})(1+3R^4/r^4-4R^2/r^2)\cos(2\theta)+P_oR^2/r^2
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Ratio of Pore Pressure Change to Original Due to Depletion

q=ΔPpShmax−Shminq=\frac{\Delta P_p}{S_{hmax}-S_{hmin}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Reservoir Stress Path from Biot Coefficient and Poisson Ratio

A=α(1−2ν)1−νA=\frac{\alpha(1-2\nu)}{1-\nu}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Rotation of Maximum Principal Stress Near Wellbore

γ=12tan⁡−1(AΔPpsin⁡2θShmax−Shmin+AΔPpcos⁡2θ)\gamma=\frac{1}{2}\tan^{-1}\left(\frac{A\Delta P_p\sin 2\theta}{S_{hmax}-S_{hmin}+A\Delta P_p\cos 2\theta}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Rotation of Maximum Principal Stress Zoback Day Lewis

γ=12tan⁡−1(Aqsin⁡2Δ1+Aqcos⁡2Δ)\gamma=\frac{1}{2}\tan^{-1}\left(\frac{Aq\sin 2\Delta}{1+Aq\cos 2\Delta}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Shale Compaction

ϕe=ϕe−βsσv\phi_e=\phi e^{-\beta_s\sigma_v}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Shear Modulus from Force Area and Deformation Angle

G=F/AθG=\frac{F/A}{\theta}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Shear Modulus from Young's Modulus and Poisson Ratio

G=E2(1+ν)G = \frac{E}{2(1 + \nu)}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Shear Wave Velocity from Shear Modulus and Density

Vs=(Gρ)0.5V_s=\left(\frac{G}{\rho}\right)^{0.5}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Stress Components in Original Coordinate System in Depletion Drive

Sx=Shmax−APp−AΔPp2(1−cos⁡2Δ)S_x=S_{hmax}-AP_p-\frac{A\Delta P_p}{2}(1-\cos 2\Delta)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Stress Components Near Normal Faulting in Reservoir

Sx=Shmax−AdP−AdP2(1−cos⁡2θ)S_x=S_{hmax}-AdP-\frac{AdP}{2}(1-\cos 2\theta)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Stress Path for Induced Normal Faulting

A=1−1(μ2+1+μ)2A=1-\frac{1}{(\sqrt{\mu^2+1}+\mu)^2}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Stress Perturbation Segall and Fitzgerald

M=α(1−2ν)πH(1−ν)8RM=\alpha\frac{(1-2\nu)\pi H}{(1-\nu)8R}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Subsidence Due to Uniform Pore Pressure Reduction in Free Surfaces

uz=−cm(1−ν)DΔPpVπ(r2+D2)1.5u_z=-\frac{c_m(1-\nu)D\Delta P_pV}{\pi(r^2+D^2)^{1.5}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Tangential Stress Extremes at a Deviated Wellbore

σtmax=12(σzz+σaa+(σzz−σaa)2+4τ2)\sigma_{tmax}=\frac{1}{2}\left(\sigma_{zz}+\sigma_{aa}+\sqrt{(\sigma_{zz}-\sigma_{aa})^2+4\tau^2}\right)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Tensile Breakdown Pressure with Thermal Stress

Pb=Pp−σHmax+3σhmin+Ts+σΔTP_b=P_p-\sigma_{Hmax}+3\sigma_{hmin}+T_s+\sigma^{\Delta T}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Thermal Hoop Stress from Wellbore Temperature Change

σΔT=αTE1−νΔT\sigma^{\Delta T}=\frac{\alpha_T E}{1-\nu}\Delta T
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Twisting Stress Around a Vertical Wellbore

τ=12(Shmax−Shmin)(1+2R2r2−3R4r4)sin⁡(2θ)\tau=\frac{1}{2}(S_{hmax}-S_{hmin})\left(1+\frac{2R^2}{r^2}-\frac{3R^4}{r^4}\right)\sin(2\theta)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Unconfined Compressive Strength of Rock

Co=2So(μ2+1+μ)C_o=2S_o(\sqrt{\mu^2+1}+\mu)
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Velocity of Bulk Compressional Waves

Vb=Eρ1−ν(1+ν)(1−2ν)V_b=\sqrt{\frac{E}{\rho}\frac{1-\nu}{(1+\nu)(1-2\nu)}}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Weak-Plane Anisotropic Failure Stress

σ=2σc(Sw+μσc)[1−μcot⁡(β)]sin⁡(2β)\sigma=\frac{2\sigma_c(S_w+\mu\sigma_c)}{[1-\mu\cot(\beta)]\sin(2\beta)}
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Geomechanics and FracturingIn-Situ Stress and Rock Mechanics

Wellbore Breakout Width from Mud Pressure

wBO=π−arccos⁡(σHmax+σhmin−UCS−(1+q)(PW−Pp)2(σHmax−σhmin))w_{BO}=\pi-\arccos\left(\frac{\sigma_{Hmax}+\sigma_{hmin}-UCS-(1+q)(P_W-P_p)}{2(\sigma_{Hmax}-\sigma_{hmin})}\right)
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