[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.

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