Permeability in EDZ

Theoretical Background

This implementation report presents the theoretical basis of the failure index dependent permeabilitymodel for the excavation damaged zone (EDZ) by taking into account HM response in rock due to excavation [2] as well as the implementation in OGS-6. The type of that permeability model in OGS-6 is PermeabilityMohrCoulombFailureIndexModel.

The failure index dependent permeability model according to [2] is defined as

where $k_0$ is the intrinsic permeability of the undamaged material, $H$ is the Heaviside step function, $f$ is the failure index, $k_r$ is a reference permeability, $b$ is a fitting parameter. $k_r$ and $b$ can be calibrated by experimental data.

The failure index $f$ is calculated from the Mohr-Coulomb failure criterion comparing an acting shearstress. With the conventional mechanics notations, which mean that tensile stress is positive, the Mohr-Coulomb failure criterion [1] takes the form

with $\tau$ the shear strength, $c$ the cohesion, $\sigma$ the normal stress, and $\phi$ the internal friction angle. We further introduce the maximum shear stress $\tau_{\mathrm{m}}=\left(\sigma_3-\sigma_1\right) / 2$ and the mean stress $\sigma_{\mathrm{m}}=\left(\sigma_1+\sigma_3\right) / 2$, where $\sigma_1$ and $\sigma_3$ are the minimum and maximum shear stress, respectively. A second criterion is implemented similar to a tension cut-off. Let $\sigma_{\mathrm{m}}^{\max } \in(0, c / \tan \phi)$ be a limit value related (but not equivalent) to tensile strength of the material. Then, the failure index is determined by

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

The material properties for the test example are given in Table 1.

Table 1: Material properties

The parameters of the EDZ permeability with the Mohr Coulumb failure index are

This geometry of this example is a square of [0,50] $\times$ [-25,25] $m²$ with a half circle hole with a radius of 2.3 m and a center at (0,0).

The initial pore pressure is 4.7 MPa. The initial displacement and stress components are all zero.

Table 2: Boundary

import os
from pathlib import Path

import matplotlib.pyplot as plt
import ogstools as ot
from ogstools.meshlib import MeshSeries

# creating output path if it doesn't exist already.
out_dir = Path(os.environ.get("OGS_TESTRUNNER_OUT_DIR", "_out"))
out_dir.mkdir(parents=True, exist_ok=True)

model = ot.Project(
    input_file="quad_with_half_hole.prj", output_file="quad_with_half_hole.prj"
)

model.write_input()

model.run_model(logfile=Path(out_dir) / "log.txt")

Project file written to output.
Simulation: quad_with_half_hole.prj
Status: finished successfully.
Execution took 0.506908655166626 s

The computed permeability is shown in Fig. 1, in which one can see that the permeability near the hole is increased with a reasonable distribution pattern. This implies that the permeability model can describe the permeability change in EDZ.

ms = MeshSeries("quad_with_half_hole.pvd").mesh(1)

fig_perm, ax = plt.subplots(figsize=(8, 6))
ms.plot_contourf("permeability", fig=fig_perm, ax=ax, fontsize=15)
ax.set_title("Figure 1: Calculated permeability distribution in [m$^2$].")
plt.tight_layout()
plt.show()

The distributions of the horizontal stress, $\sigma_{xx}$ , and the pore pressure are illustrated in Fig. 2.

pressure = ot.variables.pressure.replace(data_name="pressure_interpolated")
fig_stress, ax = plt.subplots(nrows=1, ncols=2, figsize=(20, 8))

ms.plot_contourf(ot.variables.stress["xx"], fig=fig_stress, ax=ax[0], fontsize=15)
ms.plot_contourf(pressure, fig=fig_stress, ax=ax[1], fontsize=15)
plt.suptitle(
    "Figure 2: Calculated distributions of horizontal stress (left) and pore pressure (right), in [MPa] respectively",
    fontsize=20,
)
plt.tight_layout()
plt.show()

References

[1] J.F. Labuz and A. Zang. Mohr–Coulomb failure criterion. Rock Mechanics and Rock Engineering,45(6):975–979, 2012.

[2] W.Q. Wang, H. Shao, Th. Nagel, and O. Kolditz. Analysis of coupled thermal-hydro-mechanical processes during small scale in-situ heater experiment in Callovo-Oxfordian clay rock introducing a failure-index permeability model. International Journal of Rock Mechanics and Mining Sciences, revised manuscript under review, 2020.