Merge pull request #2989 from wenqing/permeablility_stress

PermeabilityMohrCoulombFailureIndexModel

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/c_PermeabilityMohrCoulombFailureIndexModel.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_cohesion.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::c_

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_fitting_factor.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::b_

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_friction_angle.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::phi_

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_initial_permeability.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::k0_
+Parameter name to be used as initial permeability.

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_maximum_permeability.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::k_max_

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_reference_permeability.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::kr_

Documentation/ProjectFile/properties/property/PermeabilityMohrCoulombFailureIndexModel/t_tensile_strength_parameter.md

+\copydoc MaterialPropertyLib::PermeabilityMohrCoulombFailureIndexModel::t_sigma_max_

Documentation/bibliography/other.bib

@@ -140,3 +140,28 @@
   doi = "10.1007/s10596-013-9341-7",
   publisher={Springer}
 }
+
+@article{wangetalPerm2020,
+  title={Analysis of coupled thermal-hydro-mechanical processes during small
+         scale in situ heater experiment in Callovo-Oxfordian clay rock introducing
+          a failure-index permeability mode},
+  author={Wang, Wenqing and Shao, Hua and  Nagel, Thomas and Fischer,
+           Thomas and Kolditz, Olaf},
+  journal={international journal of rock mechanics and mining sciences},
+  volume={under review},
+  number={},
+  pages={},
+  year={2020},
+}
+
+@article{labuz2012mohr,
+  title={Mohr--Coulomb failure criterion},
+  author={Labuz, Joseph F and Zang, Arno},
+  journal={Rock mechanics and rock engineering},
+  volume={45},
+  number={6},
+  pages={975--979},
+  year={2012},
+  publisher={Springer}
+}
+

MaterialLib/MPL/CreateProperty.cpp

@@ -27,7 +27,7 @@
 namespace
 {
 std::unique_ptr<MaterialPropertyLib::Property> createProperty(
-    int const /*geometry_dimension*/,
+    int const geometry_dimension,
     BaseLib::ConfigTree const& config,
     std::vector<std::unique_ptr<ParameterLib::ParameterBase>> const& parameters,
     ParameterLib::CoordinateSystem const* const local_coordinate_system,
@@ -73,6 +73,13 @@ std::unique_ptr<MaterialPropertyLib::Property> createProperty(
         return createIdealGasLaw(config);
     }
 
+    if (boost::iequals(property_type,
+                       "PermeabilityMohrCoulombFailureIndexModel"))
+    {
+        return createPermeabilityMohrCoulombFailureIndexModel(
+            geometry_dimension, config, parameters, local_coordinate_system);
+    }
+
     if (boost::iequals(property_type, "PermeabilityOrthotropicPowerLaw"))
     {
         return createPermeabilityOrthotropicPowerLaw(config,

MaterialLib/MPL/Properties/CreatePermeabilityMohrCoulombFailureIndexModel.cpp

+/**
+ * \file
+ * \copyright
+ * Copyright (c) 2012-2020, OpenGeoSys Community (http://www.opengeosys.org)
+ *            Distributed under a Modified BSD License.
+ *              See accompanying file LICENSE.txt or
+ *              http://www.opengeosys.org/project/license
+ *
+ * Created on June 5, 2020, 9:07 AM
+ */
+
+#include "CreatePermeabilityMohrCoulombFailureIndexModel.h"
+
+#include <string>
+
+#include "BaseLib/ConfigTree.h"
+#include "Parameter.h"
+#include "ParameterLib/CoordinateSystem.h"
+#include "ParameterLib/Parameter.h"
+#include "ParameterLib/Utils.h"
+#include "PermeabilityMohrCoulombFailureIndexModel.h"
+
+namespace MaterialPropertyLib
+{
+std::unique_ptr<Property> createPermeabilityMohrCoulombFailureIndexModel(
+    int const geometry_dimension,
+    BaseLib::ConfigTree const& config,
+    std::vector<std::unique_ptr<ParameterLib::ParameterBase>> const& parameters,
+    ParameterLib::CoordinateSystem const* const local_coordinate_system)
+{
+    if ((geometry_dimension != 2) && (geometry_dimension != 3))
+    {
+        OGS_FATAL(
+            "The PermeabilityMohrCoulombFailureIndexModel is implemented only "
+            "for 2D or 3D problems");
+    }
+
+    //! \ogs_file_param{properties__property__type}
+    config.checkConfigParameter("type",
+                                "PermeabilityMohrCoulombFailureIndexModel");
+
+    // Second access for storage.
+    //! \ogs_file_param{properties__property__name}
+    auto property_name = config.peekConfigParameter<std::string>("name");
+
+    DBUG("Create PermeabilityMohrCoulombFailureIndexModel property {:s}.",
+         property_name);
+
+    std::string const& parameter_name =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__initial_permeability}
+        config.getConfigParameter<std::string>(
+            "initial_permeability");
+    auto const& parameter_k0 = ParameterLib::findParameter<double>(
+        parameter_name, parameters, 0, nullptr);
+
+    auto const kr =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__reference_permeability}
+        config.getConfigParameter<double>("reference_permeability");
+    auto const b =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__fitting_factor}
+        config.getConfigParameter<double>("fitting_factor");
+    auto const c =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__cohesion}
+        config.getConfigParameter<double>("cohesion");
+    auto const phi =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__friction_angle}
+        config.getConfigParameter<double>("friction_angle");
+    auto const max_k =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__maximum_permeability}
+        config.getConfigParameter<double>("maximum_permeability");
+    auto const t_sigma_max =
+        //! \ogs_file_param{properties__property__PermeabilityMohrCoulombFailureIndexModel__tensile_strength_parameter}
+        config.getConfigParameter<double>("tensile_strength_parameter");
+
+    if (geometry_dimension == 2)
+    {
+        return std::make_unique<PermeabilityMohrCoulombFailureIndexModel<2>>(
+            std::move(property_name), parameter_k0, kr, b, c, phi, max_k,
+            t_sigma_max, local_coordinate_system);
+    }
+
+    return std::make_unique<PermeabilityMohrCoulombFailureIndexModel<3>>(
+        std::move(property_name), parameter_k0, kr, b, c, phi, max_k,
+        t_sigma_max, local_coordinate_system);
+}
+}  // namespace MaterialPropertyLib

MaterialLib/MPL/Properties/CreatePermeabilityMohrCoulombFailureIndexModel.h

+/**
+ * \file
+ * \copyright
+ * Copyright (c) 2012-2020, OpenGeoSys Community (http://www.opengeosys.org)
+ *            Distributed under a Modified BSD License.
+ *              See accompanying file LICENSE.txt or
+ *              http://www.opengeosys.org/project/license
+ *
+ * Created on June 5, 2020, 9:07 AM
+ */
+
+#pragma once
+
+#include <memory>
+#include <vector>
+
+namespace BaseLib
+{
+class ConfigTree;
+}
+
+namespace ParameterLib
+{
+struct CoordinateSystem;
+struct ParameterBase;
+}  // namespace ParameterLib
+
+namespace MaterialPropertyLib
+{
+class Property;
+}
+
+namespace MaterialPropertyLib
+{
+std::unique_ptr<Property> createPermeabilityMohrCoulombFailureIndexModel(
+    int const geometry_dimension,
+    BaseLib::ConfigTree const& config,
+    std::vector<std::unique_ptr<ParameterLib::ParameterBase>> const& parameters,
+    ParameterLib::CoordinateSystem const* const local_coordinate_system);
+}  // namespace MaterialPropertyLib

MaterialLib/MPL/Properties/CreateProperties.h

@@ -26,6 +26,7 @@
 #include "CreateIdealGasLaw.h"
 #include "CreateLinear.h"
 #include "CreateParameter.h"
+#include "CreatePermeabilityMohrCoulombFailureIndexModel.h"
 #include "CreatePermeabilityOrthotropicPowerLaw.h"
 #include "CreatePorosityFromMassBalance.h"
 #include "CreateSaturationDependentSwelling.h"

MaterialLib/MPL/Properties/PermeabilityMohrCoulombFailureIndexModel.cpp

+/**
+ * \file
+ * \copyright
+ * Copyright (c) 2012-2020, OpenGeoSys Community (http://www.opengeosys.org)
+ *            Distributed under a Modified BSD License.
+ *              See accompanying file LICENSE.txt or
+ *              http://www.opengeosys.org/project/license
+ *
+ * Created on June 4, 2020, 10:13 AM
+ */
+
+#include "PermeabilityMohrCoulombFailureIndexModel.h"
+
+#include <boost/math/constants/constants.hpp>
+#include <cmath>
+#include <limits>
+
+#include "BaseLib/Error.h"
+#include "MaterialLib/MPL/Medium.h"
+#include "MaterialLib/MPL/Utils/FormEigenTensor.h"
+#include "MaterialLib/MPL/Utils/GetSymmetricTensor.h"
+#include "MathLib/KelvinVector.h"
+#include "MathLib/MathTools.h"
+#include "ParameterLib/CoordinateSystem.h"
+#include "ParameterLib/Parameter.h"
+
+namespace MaterialPropertyLib
+{
+template <int DisplacementDim>
+PermeabilityMohrCoulombFailureIndexModel<DisplacementDim>::
+    PermeabilityMohrCoulombFailureIndexModel(
+        std::string name, ParameterLib::Parameter<double> const& k0,
+        double const kr, double const b, double const c, double const phi,
+        double const k_max, double const t_sigma_max,
+        ParameterLib::CoordinateSystem const* const local_coordinate_system)
+    : k0_(k0),
+      kr_(kr),
+      b_(b),
+      c_(c),
+      phi_(boost::math::constants::degree<double>() * phi),
+      k_max_(k_max),
+      t_sigma_max_(t_sigma_max),
+      local_coordinate_system_(local_coordinate_system)
+{
+    const double t_sigma_upper = c_ / std::tan(phi_);
+    if (t_sigma_max_ <= 0.0 || t_sigma_max_ > t_sigma_upper ||
+        std::fabs(t_sigma_max_ - t_sigma_upper) <
+            std::numeric_limits<double>::epsilon())
+    {
+        OGS_FATAL(
+            "Tensile strength parameter of {:e} is out of the range (0, "
+            "c/tan(phi)) = (0, {:e})",
+            t_sigma_max_, t_sigma_upper);
+    }
+
+    name_ = std::move(name);
+}
+
+template <int DisplacementDim>
+void PermeabilityMohrCoulombFailureIndexModel<DisplacementDim>::checkScale()
+    const
+{
+    if (!std::holds_alternative<Medium*>(scale_))
+    {
+        OGS_FATAL(
+            "The property 'PermeabilityMohrCoulombFailureIndexModel' is "
+            "implemented on the 'medium' scale only.");
+    }
+}
+
+template <int DisplacementDim>
+PropertyDataType
+PermeabilityMohrCoulombFailureIndexModel<DisplacementDim>::value(
+    VariableArray const& variable_array,
+    ParameterLib::SpatialPosition const& pos, double const t,
+    double const /*dt*/) const
+{
+    auto const& stress_vector = std::get<SymmetricTensor<DisplacementDim>>(
+        variable_array[static_cast<int>(Variable::stress)]);
+
+    auto const& stress_tensor =
+        formEigenTensor<3>(static_cast<PropertyDataType>(stress_vector));
+
+    Eigen::SelfAdjointEigenSolver<Eigen::Matrix<double, 3, 3>>
+        eigenvalue_solver(stress_tensor);
+
+    // Principle stress
+    auto const sigma = eigenvalue_solver.eigenvalues();
+
+    auto k_data = k0_(t, pos);
+
+    double const max_sigma = std::max(std::fabs(sigma[0]), std::fabs(sigma[2]));
+
+    if (max_sigma < std::numeric_limits<double>::epsilon())
+    {
+        return fromVector(k_data);
+    }
+
+    double const sigma_m = 0.5 * (sigma[2] + sigma[0]);
+
+    double const tau_m = 0.5 * std::fabs(sigma[2] - sigma[0]);
+    double f = 0.0;
+    if (sigma_m > t_sigma_max_)
+    {
+        // tensile failure criterion
+        f = sigma_m / t_sigma_max_;
+
+        double const tau_tt =
+            c_ * std::cos(phi_) - t_sigma_max_ * std::sin(phi_);
+
+        f = std::max(f, tau_m / tau_tt);
+    }
+    else
+    {
+        // Mohr Coulomb failure criterion
+        f = tau_m / (c_ * std::cos(phi_) - sigma_m * std::sin(phi_));
+    }
+
+    if (f >= 1.0)
+    {
+        const double exp_value = std::exp(b_ * f);
+        for (auto& k_i : k_data)
+        {
+            k_i = std::min(k_i + kr_ * exp_value, k_max_);
+        }
+    }
+
+    // Local coordinate transformation is only applied for the case that the
+    // initial intrinsic permeability is given with orthotropic assumption.
+    if (local_coordinate_system_ && (k_data.size() == DisplacementDim))
+    {
+        Eigen::Matrix<double, DisplacementDim, DisplacementDim> const e =
+            local_coordinate_system_->transformation<DisplacementDim>(pos);
+        Eigen::Matrix<double, DisplacementDim, DisplacementDim> k =
+            Eigen::Matrix<double, DisplacementDim, DisplacementDim>::Zero();
+
+        for (int i = 0; i < DisplacementDim; ++i)
+        {
+            Eigen::Matrix<double, DisplacementDim, DisplacementDim> const
+                ei_otimes_ei = e.col(i) * e.col(i).transpose();
+
+            k += k_data[i] * ei_otimes_ei;
+        }
+        return k;
+    }
+
+    return fromVector(k_data);
+}
+
+template <int DisplacementDim>
+PropertyDataType
+PermeabilityMohrCoulombFailureIndexModel<DisplacementDim>::dValue(
+    VariableArray const& /*variable_array*/, Variable const /*variable*/,
+    ParameterLib::SpatialPosition const& /*pos*/, double const /*t*/,
+    double const /*dt*/) const
+{
+    OGS_FATAL(
+        "The derivative of the intrinsic permeability k(sigma, ...) with "
+        "respect to stress tensor (sigma) is not implemented because that "
+        "dk/du is normally omitted.");
+}
+template class PermeabilityMohrCoulombFailureIndexModel<2>;
+template class PermeabilityMohrCoulombFailureIndexModel<3>;
+}  // namespace MaterialPropertyLib

MaterialLib/MPL/Properties/PermeabilityMohrCoulombFailureIndexModel.h

+/**
+ * \file
+ * \copyright
+ * Copyright (c) 2012-2020, OpenGeoSys Community (http://www.opengeosys.org)
+ *            Distributed under a Modified BSD License.
+ *              See accompanying file LICENSE.txt or
+ *              http://www.opengeosys.org/project/license
+ *
+ * Created on June 4, 2020, 10:13 AM
+ */
+
+#pragma once
+
+#include "MaterialLib/MPL/Property.h"
+#include "MaterialLib/MPL/VariableType.h"
+
+namespace ParameterLib
+{
+struct CoordinateSystem;
+template <typename T>
+struct Parameter;
+}  // namespace ParameterLib
+
+namespace MaterialPropertyLib
+{
+/**
+ * \brief A failure index dependent permeability model \cite wangetalPerm2020
+ *
+ * \f[ \mathbf{k} =
+ *      \mathbf{k}_0+ H(f-1) k_\text{r} \mathrm{e}^{b f}\mathbf{I}\f]
+ *
+ * where
+ *   \f$\mathbf{k}_0\f$ is the intrinsic permeability
+ *  of the undamaged material,
+ * \f$H\f$ is the Heaviside step function, \f$f\f$ is the failure index,
+ * \f$k_\text{r}\f$ is a reference permeability,
+ * \f$b\f$ is a fitting parameter.
+ * \f$k_\text{r}\f$ and \f$b\f$ can be calibrated by experimental data.
+ *
+ * The failure index \f$f\f$ 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 \cite labuz2012mohr takes
+ * the form
+ *   \f[\tau(\sigma)=c-\sigma \mathrm{tan} \phi\f]
+ *   with \f$\tau\f$ the shear stress, \f$c\f$ the cohesion, \f$\sigma\f$ the
+ * normal stress, and \f$\phi\f$ the internal friction angle.
+ *
+ *  The failure index of the Mohr Coulomb model is calculated by
+ *   \f[
+ *         f_{MC}=\frac{|\tau_m| }{\cos(\phi)\tau(\sigma_m)}
+ *   \f]
+ *   with
+ *   \f$\tau_m=(\sigma_3-\sigma_1)/2\f$
+ *   and \f$\sigma_m=(\sigma_1+\sigma_3)/2\f$,
+ *  where \f$\sigma_1\f$ and \f$\sigma_3\f$ are the minimum and maximum shear
+ * stress, respectively.
+ *
+ * The tensile failure index is calculated by
+ *  \f[
+ *    f_{t} = \sigma_m / \sigma^t_{max}
+ * \f]
+ * with, \f$0 < \sigma^t_{max} < c \tan(\phi) \f$, 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 \f$\sigma_m>
+ * \sigma^t_{max}\f$. The failure index is then calculated by
+ * \f[
+ *   f =
+ *  \begin{cases}
+ *    f_{MC}, & \sigma_{m} \leq \sigma^t_{max}\\
+ *    max(f_{MC}, f_t), & \sigma_{m} > \sigma^t_{max}\\
+ *  \end{cases}
+ * \f]
+ *
+ *  The computed permeability components are restricted with an upper bound,
+ *  i.e. \f$\mathbf{k}:=k_{ij} < k_{max}\f$.
+ *
+ * If \f$\mathbf{k}_0\f$ is orthogonal, i.e input two or three numbers
+ * for its diagonal entries, a coordinate system rotation of \f$\mathbf{k}\f$
+ * is possible if it is needed.
+ *
+ *  Note: the conventional mechanics notations are used, which mean that tensile
+ * stress is positive.
+ *
+ */
+template <int DisplacementDim>
+class PermeabilityMohrCoulombFailureIndexModel final : public Property
+{
+public:
+    PermeabilityMohrCoulombFailureIndexModel(
+        std::string name, ParameterLib::Parameter<double> const& k0,
+        double const kr, double const b, double const c, double const phi,
+        double const k_max, double const t_sigma_max,
+        ParameterLib::CoordinateSystem const* const local_coordinate_system);
+
+    void checkScale() const override;
+
+    PropertyDataType value(VariableArray const& variable_array,
+                           ParameterLib::SpatialPosition const& pos,
+                           double const t, double const dt) const override;
+    PropertyDataType dValue(VariableArray const& variable_array,
+                            Variable const variable,
+                            ParameterLib::SpatialPosition const& pos,
+                            double const t, double const dt) const override;
+
+private:
+    /// Intrinsic permeability for undamaged material. It can be a scalar or
+    /// tensor for anisotropic material.
+    ParameterLib::Parameter<double> const& k0_;
+    /// Reference permeability.
+    double const kr_;
+    /// Fitting parameter.
+    double const b_;
+    /// Cohesion.
+    double const c_;
+    /// Angle of internal friction.
+    double const phi_;
+
+    /// Maximum permeability.
+    double const k_max_;
+
+    /// Tensile strength parameter.
+    double const t_sigma_max_;
+    ParameterLib::CoordinateSystem const* const local_coordinate_system_;
+};
+
+}  // namespace MaterialPropertyLib

Tests/MaterialLib/TestMPLPermeabilityMohrCoulombFailureIndexModel.cpp

+/**
+ * \file
+ * \copyright
+ * Copyright (c) 2012-2020, OpenGeoSys Community (http://www.opengeosys.org)
+ *            Distributed under a Modified BSD License.
+ *              See accompanying file LICENSE.txt or
+ *              http://www.opengeosys.org/project/license
+ *
+ * Created on June 8, 2020, 9:17 AM
+ */
+
+#include <gtest/gtest.h>
+
+#include <Eigen/Eigen>
+#include <boost/math/constants/constants.hpp>
+
+#include "MaterialLib/MPL/Medium.h"
+#include "MaterialLib/MPL/Properties/CreatePermeabilityMohrCoulombFailureIndexModel.h"
+#include "MaterialLib/MPL/Properties/PermeabilityMohrCoulombFailureIndexModel.h"
+#include "MaterialLib/MPL/Utils/FormEigenTensor.h"
+#include "MathLib/KelvinVector.h"
+#include "ParameterLib/ConstantParameter.h"
+#include "TestMPL.h"
+#include "Tests/TestTools.h"
+
+TEST(MaterialPropertyLib, PermeabilityMohrCoulombFailureIndexModel)
+{
+    ParameterLib::ConstantParameter<double> const k0("k0", 1.e-20);
+    double const kr = 1.0e-19;
+    double const b = 3.0;
+    double const c = 1.0e+6;
+    double const phi = 15.0;
+    double const k_max = 1.e-10;
+    double const t_sigma_max = c;
+
+    auto const k_model = MPL::PermeabilityMohrCoulombFailureIndexModel<3>(
+        "k_f", k0, kr, b, c, phi, k_max, t_sigma_max, nullptr);
+
+    const int symmetric_tensor_size = 6;
+    using SymmetricTensor = Eigen::Matrix<double, symmetric_tensor_size, 1>;
+
+    // Under failure, i,e stress beyond the yield.
+    SymmetricTensor stress;
+    stress[0] = -1.36040e+7;
+    stress[1] = -1.78344e+7;
+    stress[2] = -1.36627e+7;
+    stress[3] = -105408;
+    stress[4] = -25377.2;
+    stress[5] = -1.39944e+7;
+
+    MPL::VariableArray vars;
+    vars[static_cast<int>(MPL::Variable::stress)].emplace<SymmetricTensor>(
+        stress);
+
+    ParameterLib::SpatialPosition const pos;
+    double const t = std::numeric_limits<double>::quiet_NaN();
+    double const dt = std::numeric_limits<double>::quiet_NaN();
+
+    auto const k = MPL::formEigenTensor<3>(k_model.value(vars, pos, t, dt));
+
+    double const k_expected = 1.1398264890628033e-15;
+
+    ASSERT_LE(std::fabs(k_expected - k(0, 0)) / k_expected, 1e-10)
+        << "for expected changed permeability " << k_expected
+        << " and for computed changed permeability." << k(0, 0);
+
+    // Stress outside of the yield surface. No change in permeability
+    stress[0] = -1.2e+7;
+    stress[1] = -1.5e+7;
+    stress[2] = -1.2e+7;
+    stress[3] = -1e+5;
+    stress[4] = -2200.2;
+    stress[5] = -8e+5;
+    vars[static_cast<int>(MPL::Variable::stress)] = stress;
+    auto const k_non_f =
+        MPL::formEigenTensor<3>(k_model.value(vars, pos, t, dt));
+
+    auto const k_non_f_expected = k0(t, pos)[0];
+
+    ASSERT_LE(std::fabs(k_non_f_expected - k_non_f(0, 0)) / k_non_f_expected,
+              1e-19)
+        << "for expected untouched permeability" << k_non_f_expected
+        << " and for computed untouched permeability." << k_non_f(0, 0);
+
+    // Stress at the apex. No change in permeability.
+    const double val =
+        2.0 * c / std::tan(phi * boost::math::constants::degree<double>());
+    stress[0] = 0.7 * val;
+    stress[1] = 0.3 * val;
+    stress[2] = 0.3 * val;
+    stress[3] = 0.0;
+    stress[4] = 0.0;
+    stress[5] = 0.0;
+    vars[static_cast<int>(MPL::Variable::stress)] = stress;
+    auto const k_apex_f =
+        MPL::formEigenTensor<3>(k_model.value(vars, pos, t, dt));
+
+    double const k_apex_expacted = 7.2849707418474819e-15;
+    ASSERT_LE(std::fabs(k_apex_expacted - k_apex_f(0, 0)) / k_apex_expacted,
+              1e-19)
+        << "for expected untouched permeability" << k_non_f_expected
+        << " and for computed untouched permeability." << k_apex_f(0, 0);
+}