Processing

The processing part of geos-geomechanics package contains functions tools to compute geomechanical properties.

geos.geomechanics.processing.geomechanicsCalculatorFunctions module

Functions to compute additional geomechanical properties.

geos.geomechanics.processing.geomechanicsCalculatorFunctions.biotCoefficient(Kgrain, bulkModulus)

Compute Biot coefficient.

\[b = 1-\frac{K}{K_{grain}}\]

Parameters:

• Kgrain (float) – grain bulk modulus (\(K_{grain}\) - Pa)

• bulkModulus (npt.NDArray[np.float64]) – default bulk modulus (K - Pa)

Returns:

biot coefficient (b)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.bulkModulus(youngModulus, poissonRatio)

Compute bulk Modulus from young modulus and poisson ratio.

\[K = \frac{E}{3(1-2\nu)}\]

Parameters:

• youngModulus (npt.NDArray[np.float64]) – Young modulus (E - Pa)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

Bulk modulus (K - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.compressibility(poissonRatio, bulkModulus, biotCoefficient, porosity)

Compute compressibility from elastic moduli, biot coefficient and porosity.

Compressibility formula is:

\[C = \frac{1-2\nu}{\phi K}\left(\frac{b²(1+\nu)}{1-\nu} + 3(b-\phi)(1-b)\right)\]

Parameters:

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\)).

• bulkModulus (npt.NDArray[np.float64]) – Bulk Modulus (K - Pa)

• biotCoefficient (npt.NDArray[np.float64]) – Biot coefficient (b).

• porosity (npt.NDArray[np.float64]) – Rock porosity (\(\phi\)).

Returns:

Compressibility array (C - Pa^-1).

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.compressibilityOed(shearModulus, bulkModulus, porosity)

Compute compressibility from elastic moduli and porosity.

Compressibility formula is:

\[C = \frac{1}{\phi}.\frac{3}{3K+4G}\]

Parameters:

• shearModulus (npt.NDArray[np.float64]) – shear modulus (G - Pa).

• bulkModulus (npt.NDArray[np.float64]) – Bulk Modulus (K - Pa).

• porosity (npt.NDArray[np.float64]) – Rock porosity (\(\phi\)).

Returns:

Oedometric Compressibility (C - Pa^-1).

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.compressibilityReal(deltaPressure, porosity, porosityInitial)

Compute compressibility from elastic moduli and porosity.

Compressibility formula is:

\[C = \frac{\phi-\phi_0}{\Delta P.\phi_0}\]

Parameters:

• deltaPressure (npt.NDArray[np.float64]) – Pressure deviation (\(\Delta P\) - Pa).

• porosity (npt.NDArray[np.float64]) – Rock porosity (\(\phi\)).

• porosityInitial (npt.NDArray[np.float64]) – initial porosity (\(\phi_0\)).

Returns:

Real compressibility (C - Pa^-1).

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.computeNormalShearStress(stressTensor, directionVector)

Compute normal and shear stress according to stress tensor and direction.

Parameters:

• stressTensor (npt.NDArray[np.float64]) – 3x3 stress tensor

• directionVector (npt.NDArray[np.float64]) – direction vector

Returns:

normal and shear stresses.

Return type:

tuple[float, float]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.computeStressPrincipalComponents(stressTensor)

Compute stress principal components.

Parameters:

stressTensor (npt.NDArray[np.float64]) – stress tensor.

Returns:

Principal components sorted in ascending

order.

Return type:

tuple[float, float, float]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.computeStressPrincipalComponentsFromStressVector(stressVector)

Compute stress principal components from stress vector.

Parameters:

stressVector (npt.NDArray[np.float64]) – stress vector.

Returns:

Principal components sorted in ascending

order.

Return type:

tuple[float, float, float]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.criticalPorePressure(stressVector, rockCohesion, frictionAngle=0.0)

Compute the critical pore pressure.

Fracturing can occur in areas where Critical pore pressure is greater than the pressure. (see Khan, S., Khulief, Y., Juanes, R., Bashmal, S., Usman, M., & Al-Shuhail, A. (2024). Geomechanical Modeling of CO2 Sequestration: A Review Focused on CO2 Injection and Monitoring. Journal of Environmental Chemical Engineering, 112847. https://doi.org/10.1016/j.jece.2024.112847)

\[P_{Cr}=\frac{c.\cos(\alpha)}{1-\sin(\alpha)} + \frac{3\sigma_3-\sigma_1}{2}\]

Parameters:

• stressVector (npt.NDArray[np.float64]) – stress vector (\(\sigma\) - Pa).

• rockCohesion (float) – rock cohesion (c - Pa)

• frictionAngle (float, optional) –

  friction angle (\(\alpha\) - rad).

  Defaults to 0 rad.

Returns:

critical pore pressure (\(P_{Cr}\) - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.criticalPorePressureThreshold(pressure, criticalPorePressure)

Compute the critical pore pressure threshold.

Defined as the ratio between pressure and critical pore pressure. Fracturing can occur in areas where critical pore pressure threshold is >1.

\[P_{Th}=\frac{P}{P_{Cr}}\]

Parameters:

• pressure (npt.NDArray[np.float64]) – pressure (P - Pa)

• criticalPorePressure (npt.NDArray[np.float64]) – Critical pore pressure (\(P_{Cr}\) - Pa)

Returns:

Critical pore pressure threshold (\(P_{Th}\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.criticalTotalStressRatio(pressure, verticalStress)

Compute critical total stress ratio.

Corresponds to the fracture index from Lemgruber-Traby et al (2024). Fracturing can occur in areas where K > Total stress ratio. (see Lemgruber-Traby, A., Cacas, M. C., Bonte, D., Rudkiewicz, J. L., Gout, C., & Cornu, T. (2024). Basin modelling workflow applied to the screening of deep aquifers for potential CO2 storage. Geoenergy, geoenergy2024-010. https://doi.org/10.1144/geoenergy2024-010)

\[\sigma_{Ĉr}=\frac{P}{\sigma_v}\]

Parameters:

• pressure (npt.NDArray[np.float64]) – Pressure (P - Pa)

• verticalStress (npt.NDArray[np.float64]) – Vertical total stress (\(\sigma_v\) - Pa) using Geos convention

Returns:

critical total stress ratio

(\(\sigma_{Cr}\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.deviatoricStressPathOed(poissonRatio)

Compute the Deviatoric Stress Path parameter in oedometric conditions.

This parameter corresponds to the ratio between horizontal and vertical stresses in oedometric conditions.

\[DSP=\frac{\nu}{1-\nu}\]

Parameters:

poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

Deviatoric Stress Path parameter in

oedometric conditions (DSP)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.elasticStrainFromBulkShear(deltaEffectiveStress, bulkModulus, shearModulus)

Compute elastic strain from Bulk and Shear moduli.

See documentation on https://dnicolasespinoza.github.io/node5.html.

\[ \begin{align}\begin{aligned}\epsilon=\Delta\sigma_{eff}.C^{-1}\\\begin{split}C=\begin{pmatrix} K+\frac{4}{3}G & K-\frac{2}{3}G & K-\frac{2}{3}G & 0 & 0 & 0\\ K-\frac{2}{3}G & K+\frac{4}{3}G & K-\frac{2}{3}G & 0 & 0 & 0\\ K-\frac{2}{3}G & K-\frac{2}{3}G & K+\frac{4}{3}G & 0 & 0 & 0\\ 0 & 0 & 0 & \nu & 0 & 0\\ 0 & 0 & 0 & 0 & \nu & 0\\ 0 & 0 & 0 & 0 & 0 & \nu\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]

where C is stiffness tensor.

Parameters:

• deltaEffectiveStress (npt.NDArray[np.float64]) – effective stress variation (\(\Delta\sigma_{eff}\) - Pa) [S11, S22, S33, S23, S13, S12]

• bulkModulus (npt.NDArray[np.float64]) – Bulk modulus (K - Pa)

• shearModulus (npt.NDArray[np.float64]) – Shear modulus (G - Pa)

Returns:

elastic strain (\(\epsilon\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.elasticStrainFromYoungPoisson(deltaEffectiveStress, youngModulus, poissonRatio)

Compute elastic strain from Young modulus and Poisson ratio.

See documentation on https://dnicolasespinoza.github.io/node5.html.

\[ \begin{align}\begin{aligned}\epsilon=\Delta\sigma_{eff}.C^{-1}\\\begin{split}C=\begin{pmatrix} \lambda+2G & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda+2G & \lambda & 0 & 0 & 0\\ \lambda & \lambda & \lambda+2G & 0 & 0 & 0\\ 0 & 0 & 0 & \nu & 0 & 0\\ 0 & 0 & 0 & 0 & \nu & 0\\ 0 & 0 & 0 & 0 & 0 & \nu\\ \end{pmatrix}\end{split}\end{aligned}\end{align} \]

where C is stiffness tensor, \(\nu\) is shear modulus, \(\lambda\) is lambda coefficient.

Parameters:

• deltaEffectiveStress (npt.NDArray[np.float64]) – effective stress variation (\(\Delta\sigma_{eff}\) - Pa) [S11, S22, S33, S23, S13, S12]

• youngModulus (npt.NDArray[np.float64]) – Young modulus (E - Pa)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

elastic strain (\(\epsilon\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.lambdaCoefficient(youngModulus, poissonRatio)

Compute lambda coefficient from young modulus and Poisson ratio.

\[\lambda = \frac{E*\nu}{(1+\nu)(1-2\nu)}\]

Parameters:

• youngModulus (npt.NDArray[np.float64]) – Young modulus (E - Pa)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

lambda coefficient (\(\lambda\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.lithostaticStress(depth, density, gravity)

Compute the lithostatic stress.

Parameters:

• depth (npt.NDArray[np.float64]) – depth from surface - m)

• density (npt.NDArray[np.float64]) – density of the overburden (kg/m³)

• gravity (float) – gravity (m²/s)

Returns:

lithostatic stress (Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.oedometricModulus(Edef, poissonRatio)

Compute Oedometric modulus.

\[M_{oed} = \frac{E_{def}}{1-2\frac{\nu^2}{1-\nu}}\]

Parameters:

• Edef (npt.NDArray[np.float64]) – Deformation modulus (\(E_{def}\) - Pa)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

Oedometric modulus (\(M_{oed}\) - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.poissonRatio(bulkModulus, shearModulus)

Compute Poisson’s ratio.

\[\nu = \frac{3K-2G}{2(3K+G)}\]

Parameters:

• bulkModulus (npt.NDArray[np.float64]) – Bulk modulus (K - Pa)

• shearModulus (npt.NDArray[np.float64]) – Shear modulus (G - Pa)

Returns:

Poisson’s ratio (\(\nu\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.reservoirStressPathOed(biotCoefficient, poissonRatio)

Compute reservoir stress path in oedometric conditions.

\[RSP_{oed}=b\frac{1-2\nu}{1-\nu}\]

Parameters:

• biotCoefficient (npt.NDArray[np.float64]) – biot coefficient (b)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

reservoir stress path (\(RSP_{oed}\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.reservoirStressPathReal(deltaStress, deltaPressure)

Compute real reservoir stress path.

\[RSP_{real}=\frac{\Delta\sigma}{\Delta P}\]

Parameters:

• deltaStress (npt.NDArray[np.float64]) – stress difference from start (\(\Delta\sigma\) - Pa)

• deltaPressure (npt.NDArray[np.float64]) – pressure difference from start (\(\Delta P\) - Pa)

Returns:

reservoir stress path (\(RSP_{real}\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.shearCapacityUtilization(traction, rockCohesion, frictionAngle)

Compute shear capacity utilization (SCU).

\[SCU = \frac{abs(\tau_1)}{\tau_{max}}\]

where tau_{max} is the Mohr-Coulomb failure threshold.

Parameters:

• traction (npt.NDArray[np.float64]) – traction vector (\((\sigma, \tau_1, \tau2)\) - Pa)

• rockCohesion (float) – rock cohesion (c - Pa).

• frictionAngle (float) – friction angle (\(\alpha\) - rad).

Returns:

SCU

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.shearModulus(youngModulus, poissonRatio)

Compute shear Modulus from young modulus and poisson ratio.

\[G = \frac{E}{2(1+\nu)}\]

Parameters:

• youngModulus (npt.NDArray[np.float64]) – Young modulus (E - Pa)

• poissonRatio (npt.NDArray[np.float64]) – Poisson’s ratio (\(\nu\))

Returns:

Shear modulus (G - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.specificGravity(density, specificDensity)

Compute the specific gravity.

\[SG = \frac{\rho}{\rho_f}\]

Parameters:

• density (npt.NDArray[np.float64]) – density (\(\rho\) - kg/m³)

• specificDensity (float) – fluid density (\(\rho_f\) - kg/m³)

Returns:

specific gravity (SG - no unit)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.stressRatio(horizontalStress, verticalStress)

Compute horizontal to vertical stress ratio.

\[r = \frac{\sigma_h}{\sigma_v}\]

Parameters:

• horizontalStress (npt.NDArray[np.float64]) – horizontal stress (\(\sigma_h\) - Pa)

• verticalStress (npt.NDArray[np.float64]) – vertical stress (\(\sigma_v\) - Pa)

Returns:

stress ratio (\(\sigma\) - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.totalStress(effectiveStress, biot, pressure)

Compute total stress from effective stress, pressure, and Biot coeff.

\[\sigma_{tot} = \sigma_{eff}-bP\]

Parameters:

• effectiveStress (npt.NDArray[np.float64]) – effective stress (\(\sigma_{eff}\) - Pa) using Geos convention

• biot (npt.NDArray[np.float64]) – Biot coefficient (b)

• pressure (npt.NDArray[np.float64]) – Pore pressure (P - Pa)

Returns:

total stress (\(\sigma_{tot}\) - Pa)

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.totalStressRatioThreshold(pressure, horizontalStress)

Compute total stress ratio threshold.

Corresponds to the fracture threshold from Lemgruber-Traby et al (2024). Fracturing can occur in areas where FT > 1. Equals FractureIndex / totalStressRatio. (see Lemgruber-Traby, A., Cacas, M. C., Bonte, D., Rudkiewicz, J. L., Gout, C., & Cornu, T. (2024). Basin modelling workflow applied to the screening of deep aquifers for potential CO2 storage. Geoenergy, geoenergy2024-010. https://doi.org/10.1144/geoenergy2024-010)

\[\sigma_{Th}=\frac{P}{\sigma_h}\]

Parameters:

• pressure (npt.NDArray[np.float64]) – Pressure (P - Pa)

• horizontalStress (npt.NDArray[np.float64]) – minimal horizontal total stress (\(\sigma_h\) - Pa) using Geos convention

Returns:

fracture threshold (\(\sigma_{Th}\))

Return type:

npt.NDArray[np.float64]

geos.geomechanics.processing.geomechanicsCalculatorFunctions.youngModulus(bulkModulus, shearModulus)

Compute Young modulus.

\[E = \frac{9K.G}{3K+G}\]

Parameters:

• bulkModulus (npt.NDArray[np.float64]) – Bulk modulus (K - Pa)

• shearModulus (npt.NDArray[np.float64]) – Shear modulus (G - Pa)

Returns:

Young modulus (E - Pa)

Return type:

npt.NDArray[np.float64]