Equations

Equations and nomenclatures used for results post processing.

This section provides information regarding the equations used to perform the post-processing calculations.

Stress Manipulation

Table 1. Solver supplied results and their Abbreviation list
Abbreviation Name
S1 Max Principal Stress
S2 Intermediate Principal Stress
S3 Min Principal Stress
S11 Normal Stress in global X direction
S22 Normal Stress in global Y direction
S33 Normal Stress in global Z direction
S12 Shear Stress in global XY direction
S13 Shear Stress in global XZ direction
S23 Shear Stress in global YZ direction
Pp PorePressure
VoidR Void Ratio
iPp Initial Pore Pressure
Table 2. Consolidated equations used for deriving Derived Scalars
Name Abbreviation Equation Comments
Vertical stress Sv S33 S33-eff or total.
MeanMaxMin Smxn (S1 + S3) / 2 Mean2D
Differential Stress Sd (S1 - S3)  
MeanStress Smn (S1 + S2 + S3) / 3 Pressure-eff or total; 1st invariant.
Deviatoric S1 dvS1 abs(S1 - Smn)  
Deviatoric S2 dvS2 abs(S2 - Smn)  
Deviatoric S3 dvS3 abs(S3 - Smn)  
vonMises Stress Svm sqrt[{(S1 - S2)/2)^2} + {(S1 - S3)/2)^2} + {(S2 - S3)/2)^2}] 2nd invariant; the simple calc.
Tresca Stress Sta (S1 - S3) / 2 MaxShear: 3rd invariant; the very simple calc.
equivalent Pp Pp iPp + SkB*(Sm - Sm0) ePp;pPp-Calculated when only total stresses are available (i.e. Non poroelastic elements).
Porosity Phi VoidR / (1 + VoidR)  
Table 3. Consolidated equations used for deriving In-Plane from Tensor results
Name Abbreviation Equation Comments
Max Horizontal Stress SHx Lund&Townend Maximum Horizontal Stress Magnitude
Min Horizontal Stress Shn Lund&Townend Minimum Horizontal Stress Magnitude
Max Horizontal Stress Azimuth SHxAzi Lund&Townend Maximum Horizontal Stress Azimuth
Mean Horizontal SHmn (SHx + Shn) / 2  
Max Shear Horizontal MxShrHz (SHx - Shn) / 2  
Table 4. Consolidated equations used for deriving Ratios
Name Abbreviation Equation
Vertical to Maximun Principal Stress Sv:S1 S33 / S1
Vertical to Intermediate Principal Stress Sv:S2 S33 / S2
Vertical to Minimum Principal Stress Sv:S3 S33 / S3
Max to Minimum Principal Stress S1:S3 S1 / S3
Max to Intermediate Principal Stress S1:S2 S1 / S2
Intermediate to Minimum Principal Stress S2:S3 S2 / S1
Vertical to Horizontal Stress Sv:Shn S33 / Shn
Horizontal Stress Anisotropy SHx:Shn SHx / Shn

Planar (Surface) Results

Table 5. Consolidated equations used for deriving Planar (Surface) results
Name Abbreviation Equation Comments
Dip Dip   User Defined; used with DipDir to calculate Normal Vector (NV)
Dip Direction DipDir   User Defined; used with Dip to calculate Normal Vector (NV)
NormalVector Normalized NV   Facet Normal from a shell mesh, surface or user input.
Traction Magnitude on surface Trac Tensor · NV Each Tensor row · the col vector; · = dot product, x = cross product
Normal Stress Sn (Trac · NV) * NV · = dot product, x = cross product.
Shear Stress Tau NV x (Trac x NV) · = dot product, x = cross product.
Traction Vectors on surfaces TracV Tensor · NV · = dot product, x = cross product.
Max Shear vector on surfaces TauV NV x (Trac x NV) · = dot product, x = cross product.

User Entered Results

Table 6. Consolidated equations for User entered results
Name Abbreviation Equation Default (SI) Comments
Coeficient of Internal Friction Mui tan(Phii) 0.6

Could supply either coef or angle and calc other).
Angle of Internal Friction AiF atan(Mui) 30  
Coeficient of Sliding Friction Mus tan(Phis) 0.4  
Angle of Sliding Friction AsF atan(Mus) 21.80140949  
Tensile strength T   0

As a positive (strength) value.
Cohesion Co   0

As a positive (strength) value.
BiotConst Bio   1  
Skemptons Beta SkB   1  
Porosity Phi   0.2

If Non PoroElastic.
BulkDensity RhoB   2000  
GrainDensity RhoG   2200  
Fluid Density RhoF   1000  
Saturation Sat   1  
Oil Saturation OSat   0  
Gas Saturation Gsat   0  
Water Saturation Wsat   1  
Permeability K   1.00E-12  
Oil Perm. OK   0  
Gas Perm. GK   0  
Water Perm. WK   1.00E-12  
Slip Vector SlpV    

As vector components or plunge and plunge direction.