OGS
[case] PermeabilityMohrCoulombFailureIndexModel

A failure index dependent permeability model [42].

$\mathbf{k} = \mathbf{k}_0+ H(f-1) k_\text{r} \mathrm{e}^{b f}\mathbf{I}$

where $$\mathbf{k}_0$$ is the intrinsic permeability of the undamaged material, $$H$$ is the Heaviside step function, $$f$$ is the failure index, $$k_\text{r}$$ is a reference permeability, $$b$$ is a fitting parameter. $$k_\text{r}$$ and $$b$$ can be calibrated by experimental data.

The failure index $$f$$ is calculated from the Mohr Coulomb failure criterion comparing an acting shear stress for the shear dominated failure. The tensile failure is governed by an input parameter of tensile_strength_parameter .

The Mohr Coulomb failure criterion [25] takes the form

$\tau(\sigma)=c-\sigma \mathrm{tan} \phi$

with $$\tau$$ the shear stress, $$c$$ the cohesion, $$\sigma$$ the normal stress, and $$\phi$$ the internal friction angle.

The failure index of the Mohr Coulomb model is calculated by

$f_{MC}=\frac{|\tau_m| }{\cos(\phi)\tau(\sigma_m)}$

with $$\tau_m=(\sigma_3-\sigma_1)/2$$ and $$\sigma_m=(\sigma_1+\sigma_3)/2$$, where $$\sigma_1$$ and $$\sigma_3$$ are the minimum and maximum shear stress, respectively.

The tensile failure index is calculated by

$f_{t} = \sigma_m / \sigma^t_{max}$

with, $$0 < \sigma^t_{max} < c \tan(\phi)$$, a parameter of tensile strength for the cutting of the apex of the Mohr Coulomb model.

The tensile stress status is determined by a condition of $$\sigma_m> \sigma^t_{max}$$. The failure index is then calculated by

$f = \begin{cases} f_{MC}, & \sigma_{m} \leq \sigma^t_{max}\\ max(f_{MC}, f_t), & \sigma_{m} > \sigma^t_{max}\\ \end{cases}$

The computed permeability components are restricted with an upper bound, i.e. $$\mathbf{k}:=k_{ij} < k_{max}$$.

If $$\mathbf{k}_0$$ is orthogonal, i.e input two or three numbers for its diagonal entries, a coordinate system rotation of $$\mathbf{k}$$ is possible if it is needed.

Note: the conventional mechanics notations are used, which mean that tensile stress is positive.