3D Biot coefficient (Pore pressure effectiveness)


Pore pressure and pore pressure effectiveness

The pore pressure acts as an internal loading against the total stress. Its effect is generally limited by the rock structure. This usually results in a mechanically effective pore pressure (MOPP) less than the fluid pressure in the pores (see the section: Evaluation → 3D in situ stresses). Therefore, the effective stress (relevant for deformation, instability and fracturing) is commonly greater than that obtained from a consideration of the complete pore (fluid) pressure.

These effects are quantified by the direction dependent pore pressure effectiveness (Biot coefficient in 3D). The magnitude of this depends on the rock and the loading situation and may have a value between 0 (no pore pressure effect) and 1.

Calculation of pore pressure effectiveness

The pore pressure effectiveness ω is a deformation value, calculated on the basis of the dynamic elastic parameters of the rock and of its matrix (solid).

Generally relationship ω from Cheng1

Stiffness tensor of the rock

Compliance tensor of the rock

Isotropic solution (Biot coefficient n)

Rock compressibility

Solid compressibility

1 see: Cheng, A. H.-D.: Material Coefficients of Anisotropic Poroelasticity. Int. J. Rock Mech. Min. Sci., Vol 34, No 2, S. 199-205, 1997.

These parameters are calculated for the compaction conditions reached at each relevant loading level. The elastic stiffness tensor of the rock and the compliance tensor of the solid are determined in RACOS® with special analyses.

Rock properties and pore pressure effectiveness

The pore pressure effectiveness becomes greater as the rock compressibility increases (the rock stiffness decreases) in comparison with the compressibility of the solids. For this reason rocks with higher porosity (generally also with higher permeability), because of their greater rock compressibility, also have larger pore pressure effectiveness than rocks with similar solids compressibility.
In contrast, rocks with a very low porosity and/or a strongly deformable solids matrix have rather low pore pressure effectivenesses.

Correlation of pore pressure effectiveness with the:

rock porosity

rock permeability

The minimum principal component of pore pressure effectiveness is generally in the direction of the maximum permeability. This enables a fast 3D characterization to be made of the preferred flow direction.

Rock strain and pore pressure effectiveness


Geographic orientation

When the principal directions of the rock strain (large symbols in the geographic orientation plot) and the pore pressure effectiveness (small symbols) are identical, the maximum pore pressure effectiveness is found to be in the direction of the maximum deformation.

Discrepancies between the principal directions of the pore pressure effectiveness and those of the rock deformation result when the principal directions of the compressibility of the solids differ from those of the rock.

Effect of compaction/loosening on pore pressure effectiveness

The initial values of pore pressure effectiveness result from the rock properties and the initial loading. A change of the effective in situ loads may lead to compaction/loosening of the rock mass and thereby to the modification of the pore pressure effectivenesses.


Compaction during deformation is shown by increasing compressional wave velocity (vp), decreasing permeability (k) and reduction of the pore pressure effectiveness (x in the figure below).

On loosening resulting from higher (critical) loadings these tendencies reverse.



The pore pressure effectiveness (x) can be determined directly in conventional laboratory tests from comparisons of the deformation and failure parameters of a rock under different internal and external loadings2. This requires a large number of samples and tests.

Using the modern RACOS® analysis procedure only a few samples are necessary to determine the 3D pore pressure effectivenesses for any compaction condition.

2 see: Braun, R.: Zum mechanischen Verhalten poröser Gesteine bei innerer und äußerer Beanspruchung“. PhD Thesis, TU Bergakademie Freiberg, 1982

Pore pressure effectiveness, MOPP and effective stress

From compaction/loosening of the rock/rock mass result the modifications of the pore pressure effectivenesses described above and so too variations of the mechanically operating pore pressures (MOPP). These modify the effective in situ loadings and the level of stability.

Compaction and MOPP-
On reduction of the pore pressure and/or an increase in the external loading, but without reaching any critical loading condition, rock compaction occurs and with this a reduction of the pore pressure effectiveness. From this reduction there results a decrease of the mechanically operating pore pressure (MOPP-) and thus a further increase in the effective stress.

Loosening and MOPP+
Levels of shear stress approaching critical values cause micro and macro fractures. These result in the pore pressure effectiveness increasing to a magnitude of 1. This in turn results in an increase in the mechanically operating pore pressure (MOPP+). This reduces the level of effective stress to approach more closely to a critical loading condition (see figure below).

The increases in pore pressure effectiveness which result from loosening, and thus also the corresponding increases in mechanically operating pore pressure, are greatest in rocks with small values of initial pore pressure effectiveness (low porosity etc.). MOPP+ effects are also stronger when the pore pressure is high.

Example of MOPP+
There follows an example of the effect of MOPP+, this resulting from pore pressure increases (caused by injection, fracs or similar), on the failure/stability situation.


Initial pore pressure & effective stress state


Pore pressure↑→effective stress↓→looseningMOPP+


Effective stress↓↓→additional looseningMOPP+


Initial pore pressure but with residual MOPP+ effects

Loosening and the resulting increase of the pore pressure effectiveness is irreversible.