Stress Tensors

Several tensors need the Geomechanics licence feature

Every licence has the Stress Tensors branch and one stress tensor you can edit, which every stress analysis uses — what the old Project Properties stress field did. Adding more tensors (New Stress Tensor, Duplicate), Set as Active, choosing a tensor for a DFN and choosing one per slip-tendency run need the Geomechanics feature (GEOMECH). Without it those commands say so instead of running. A project made with the feature still opens without it: all its tensors are kept and can be edited, but the analyses use the active one. See Feature availability.

The Stress Tensors branch holds the project's in-situ stress states. Each tensor is a set of three principal stresses, with a magnitude and an orientation for each, plus the Coulomb failure parameters. A project can hold as many as you need: a regional state and a local one, a measured state and an interpreted alternative, or the stress at several stages of a basin's history.

One tensor is active, shown in bold. It is the stress the stress-dependent tools use unless told otherwise:

• slip potential on meshes — where you can also pick any other tensor from the command's submenu (Triangular Meshes);
• the stereonet's Stress Axes markers (Stereonet User Guide) and the Charts panel's Mohr Diagram, which can also plot any other tensor (Charts);
• slip tendency on orientation measurements — where you can also pick any other tensor from the command's submenu (Orientation);
• the stress attributes, and closure under stress, of a Discrete Fracture Network — each DFN can instead be given a tensor of its own under Stress Parameters > Stress Tensor;
• the stereonet's stress-attribute field and kinematic analysis (Stereonet User Guide);
• the default friction angle of Slope Stability.

To compare stress states, make another tensor active, or give each DFN its own, and rerun the analysis. An open stereonet redraws its stress field on its own. Renaming a tensor keeps the DFNs that chose it; deleting one sends them back to the active tensor.

Earlier versions

In VRGS 3.4.7 and earlier a project had a single stress field, set in Project Properties. It now appears here as the tensor Regional Stress, with the values it had. A new project starts with the same tensor, set to the published-data default.

Tensor properties

Select a tensor to edit it in the Properties panel.

Setting	What it does
Name	The tensor's name. Names are unique within a project.
Status	Active or Inactive. Read-only: use Set as Active to change it.
Validation	Valid, the warnings the tensor carries, or why the last change was not applied.
Sigma 1	Major principal stress magnitude, in MPa, compression positive.
Sigma 1 Orientation	Plunge and trend of Sigma 1.
Sigma 2	Intermediate principal stress magnitude.
Sigma 2 Orientation	Derived from Sigma 1 and the Sigma 3 trend, so read-only.
Sigma 3	Minor principal stress magnitude.
Sigma 3 Orientation	Set the trend; the plunge is derived so that Sigma 3 is perpendicular to Sigma 1.

Under Failure Parameters:

Setting	What it does
Friction Coefficient	Coulomb friction coefficient μ, typically 0.6 for rock. New DFNs and the stereonet's kinematic analysis take it, and Slope Stability starts its friction angle from it (the angle whose tangent is μ: about 31° for 0.6).
Pore Pressure	Pore fluid pressure Pf, in MPa, for the effective-stress and failure terms.
Cohesion	Cohesive strength C0, in MPa, in the Coulomb failure criterion.

The three axes are always kept perpendicular: you set Sigma 1 fully and the Sigma 3 trend, and VRGS works out the rest. A trend outside 0–360° is wrapped into that range.

Variation with depth

A tensor's magnitudes and pore pressure can change with depth, so a tall slope or a deep mine is not given one stress from crest to toe. Only the rows that apply to your choices are shown.

Property	Description
Stress	Uniform (default): the magnitudes above everywhere. Gravitational (Sv = unit weight × depth): the vertical stress grows with depth below ground; the steepest axis takes it, and the other two take SHmax (the larger of them above) and Shmin from Horizontal Stress. The axes keep their orientations; the magnitudes are re-sorted at each point so Sigma 1 ≥ Sigma 2 ≥ Sigma 3.
Ground Surface	The topographic mesh depth is measured below, straight down. A point the mesh does not cover gets no value. (fixed elevation) measures depth below Ground Elevation instead.
Ground Elevation (m)	The ground's elevation with no ground surface.
Unit Weight (kN/m³)	Of the overburden; 27 kN/m³ gives 0.027 MPa per metre.
Horizontal Stress	k Ratios: SHmax and Shmin as multiples of Sv. Sheorey (1994): k = 0.25 + 7 Eh (0.001 + 1/z), the mean horizontal stress k·Sv, split in the ratio SHmax / Shmin. Gradients: SHmax and Shmin = value at surface + gradient × depth.
SHmax / Sv, Shmin / Sv	(k Ratios)
Eh (GPa), SHmax / Shmin	(Sheorey) Eh is the average horizontal deformation modulus of the upper crust.
SHmax at Surface, SHmax Gradient, Shmin at Surface, Shmin Gradient	(Gradients) MPa and MPa/m.
Pore Pressure	Constant (Failure Parameters): the Pore Pressure above, everywhere. Hydrostatic below a water level: the unit weight of water × the depth below a water level at Water Level Depth below ground. Below a water table surface: the same below a phreatic surface mesh, Water Table Surface; above it, and where it does not reach, 0.
Reference Depth (m)	Where analyses with no position take the stress: the stereonet's tendency field and stress axes, the Mohr diagram when its planes have no position, and apertures closed where no fracture is placed. Default 100 m.
Water Level Depth at the Reference (m)	For a water table model: the table's depth below ground at the reference depth, since the reference has no position for the mesh to give it. Default 0 (the table at the ground).
At the Reference Depth	Read-only: σ1, σ2, σ3 and Pf there. A water table counts as a water level at the depth above.

Where the stress is evaluated:

• Slip Potential (Stress Tensor) on a mesh: at each vertex. With a depth model it also writes a Pore Pressure layer, and normalises Ts and Td by the maxima at each vertex's depth.
• Stress Attributes (Slip Tendency) on orientation measurements: at each measurement's position.
• A DFN's stress attributes and closure under stress: at each fracture's centre.
• The Mohr diagram: at the centroid of the plotted planes; the subtitle gives the depth.
• The stereonet: at the reference depth.

A point below no ground (off the ground surface) gets no value, and the messages panel counts them. Depths shallower than 1 m are taken as 1 m, so the ratios stay finite at the surface. If a surface the model names is missing from the project, the analysis says so and uses the tensor as uniform.

Validation

Every tensor is checked when it is added, and again after every change. A change that would make the tensor invalid is not applied: the values go back to the last valid ones, the Validation row says why, and the reason is written to the messages panel.

A tensor is rejected if:

• any value is not a number;
• Sigma 1 is not greater than zero, or Sigma 3 is negative (magnitudes are compression positive);
• the magnitudes are out of order: Sigma 1 ≥ Sigma 2 ≥ Sigma 3 is required;
• Sigma 1 equals Sigma 3 — an isotropic stress has no principal directions, so slip and dilation tendency are undefined;
• a plunge is outside 0–90°;
• the axes are not perpendicular;
• the friction coefficient is not greater than zero, or pore pressure or cohesion is negative.

Two states are accepted with a warning, because they are possible but usually a mistake:

• a friction coefficient above 1.2;
• a pore pressure at or above Sigma 3, which makes the effective minimum stress tensile.

Sigma 1 = Sigma 2, or Sigma 2 = Sigma 3, is allowed.

A tensor saved in the project file is checked when the project opens. One that fails — a file edited by hand, or an older project whose stress field was never checked — is kept, so nothing is lost, but it is reported in the messages panel and cannot be made active until it is corrected. If no tensor in the project is valid, the first one stays active, so the analyses run as they did before.

Context menus

Right-click the Stress Tensors branch header:

Command	What it does
New Stress Tensor	Adds a tensor named Stress Tensor (numbered if the name is taken), starting from the active tensor's values (and its depth model), and selects it.
Invert from Fault Slip (CSV)...	Inverts fault-slip data for the stress axes and ratio, as a new tensor. See Inverting fault-slip data.

Right-click a tensor:

Command	What it does
New Stress Tensor	As on the branch header.
Duplicate	Adds a copy of this tensor, with a number after its name.
Set from Regime...	Sets this tensor's axes the Andersonian way: the faulting regime (normal: σ1 vertical; strike-slip: σ2 vertical; reverse: σ3 vertical), the SHmax azimuth, the stress ratio φ = (σ2 − σ3)/(σ1 − σ3) and the σ1 and σ3 magnitudes. It opens with the tensor's own values; its failure parameters and depth model are kept.
Invert from Fault Slip (CSV)...	As on the branch header.
Set as Active	Makes this the tensor the analyses use. Unavailable for the tensor that is already active, and refused for one that is not valid.
Rename	Edits the name in the tree. A name another tensor has is refused.
Delete	Removes the tensor. Deleting the active one makes the first remaining tensor active. The last tensor cannot be deleted: there is always one for the analyses to use.

The tensors, with their depth models and inversion records, are saved in the project database, and in ProjectPrefs.json for builds that read only that.

Inverting fault-slip data

Invert from Fault Slip (CSV)..., on the branch header and on every tensor, reads a CSV file of faults — the dip direction, dip and rake of each, and optionally a weight. Invert Stress from Selected Faults... on the Orientation menu takes fault planes and the slickenline lineations measured on them instead. Either finds the principal stress axes and the stress ratio φ = (σ2 − σ3) / (σ1 − σ3) that best explain the slip — by the least angular misfit (Angelier 1979) or Michael's (1984) linear method — with a bootstrap for the confidence. It needs at least four faults and the Geomechanics module.

The messages panel gives the axes, φ with its 95% interval, the spread of the bootstrap axes and the mean misfit between observed and predicted slip. A new tensor, Inverted (N faults), is added and selected but not made active; its page has an Inversion group, and the stereonet's Stress Axes draws its bootstrap axes. An inversion resolves no magnitudes: the new tensor takes the magnitudes of σ1 and σ3, the failure parameters and the depth model from the active tensor, with σ2 = σ3 + φ (σ1 − σ3).

The file format, the pairing of measurements and the methods are on the Stress Inversion page.

References

• Sheorey, P. R. (1994). A theory for in situ stresses in isotropic and transversely isotropic rock. International Journal of Rock Mechanics and Mining Sciences, 31, 23–34.