A failure index dependent permeability model [44].

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

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.

Child parameters, attributes and cases

Additional info

Used in the following test data files