1 Introduction

The characterisation of gas transport through claystone is of great importance for the safety assessment of a repository for radioactive materials, as claystone is considered a possible host rock for such a repository (de La Vaissière et al. 2015). In conventional continuum models, multiphase flow in porous media is often modelled with standard two-phase flow models. However, in addition to standard two-phase flow, complex hydraulic-mechanical coupled constitutive equations are necessary to describe gas flow in claystone, as these can better represent the observed processes (Seyedi et al. 2021). Gas flow in claystone during in-situ investigations with high-gas pressures is dependent on an EDZ, dilatancy-induced permeability alteration and hydraulic fracturing of the rock (de La Vaissière et al. 2015). In continuum mechanics, there are numerous well-established empirical solutions for those processes, although no general theory or concept can describe all processes related to gas transport through claystone.

Approaches to numerically simulate the EDZ have been developed by Jung et al. (2023), Manica et al. (2017), Pardoen and Collin (2017), Souley et al. (2017), Vu et al. (2020) and Wang et al. (2021), among others. The developed approaches employ sophisticated hydro-mechanical models with strength hardening and softening (Manica et al. 2017), strain localisation (Pardoen and Collin 2017; Vu et al. 2020), elasto-viscoplasticity (Manica et al. 2017; Souley et al. 2017) and damage (Manica et al. 2017; Souley et al. 2017; Wang et al. 2021). Empirical or semi-empirical hydro-mechanical (HM) coupled models developed in the framework of Biot's consolidation theory simulating dilatancy-controlled gas flow have also been proposed by several authors (Gerard et al 2014; Nguyen and Le, 2015; Olivella and Alonso 2008; Radeisen et al. 2023a; Senger et al 2014; Senger et al 2018; Xu et al 2013; Yang et al 2020).

In this paper, two constitutive sets of equations are combined, which are used to model gas flow in the EDZ and the dilatancy-dependent gas flow. The two constitutive equation sets are based on different dominant failure modes, the shear failure mode during excavation and the tensile failure mode during gas-injection testing. The developed approach is applied in a numerical simulation of the PGZ experiment at the Meuse/Haute-Marne Underground Research Laboratory (MHM URL) in France (Delay et al. 2011). The PGZ experiment was conducted by the French National Radioactive Waste Management Agency (ANDRA) to investigate the effects of stress alignment and permeability increase induced by drilling/excavation and the effect of gas injection on the barrier integrity of the Callovo-Oxfordian (COx) argillaceous rock formation (Plúa et al. 2021). The simulated-gas-injection test was carried out in December 2020 in boreholes PGZ1002 and PGZ1003, which are aligned parallel to the minor and major horizontal stress, respectively. The COx is a well-examined and characterised argillaceous rock formation, so that the values of the material parameters can be well determined and the results can be validated more accurately (Belmokhtar et al. 2016; Plúa et al. 2021; Seyedi et al. 2017). In-situ measurements are available for the COx with regard to the dimension and characteristics of the EDZ (Armand et al. 2014; de La Vaissière et al. 2015). Measurements of the gas pressure and water pressure in the injection filter are available for the gas-injection test in PGZ1003, provided by the DECOVALEX-2023 project (Birkholzer and Bond 2022) and used for a comparison with the numerical results.

The numerical application is conducted with the open-source code OpenGeoSys (OGS). OGS is based on an object-oriented finite element method (FEM) and has been developed since the mid-1980s as an open-source initiative for numerical simulations of thermo-hydro-mechano-chemical processes in porous and fractured media (Kolditz et al. 2012). Similiar hydro-mechanically coupled simulations were performed in recent years with the numerical code on bentonites (Radeisen et al. 2023b, Radeisen et al. 2024).  The paper first presents the numerical fundamentals including balance equations and the approach for combining the permeability constitutive models. Subsequently, the experiment is presented and the phases of the simulation are described. Finally, the results are presented and a short conclusion is given.

2 Numerical Methods

2.1 Balance Equations

The two-phase hydraulic-mechanical (H2M) coupled processes are calculated by a staggered scheme. The processes are based on the balance equations. The coupled processes satisfy the following equations at all times. The momentum balance equation, derived from the effective stress law can be expressed as follows (Wang et al. 2011):

$$\nabla \cdot \left[{{\varvec{\upsigma}}}^{\mathrm{^{\prime}}}-\alpha \left({p}_{{\text{g}}}-{S}_{{\text{w}}}{p}_{{\text{c}}}\right)\mathbf{I}\right]+\rho \mathbf{g}=0$$
(1)

where \(\nabla\) is the Nabla operator, \({\mathbf{\sigma^{\prime}}}\) is the effective stress, α is the Biot coefficient, pg is the pressure of the gaseous phase (non-wetting phase), Sw is the saturation of the wetting fluid, pc is the capillary pressure (suction), \(\mathbf{I}\) is the identity tensor, ρ is the bulk density and \(\mathbf{g}\) is the gravitational acceleration. The wetting fluid is assumed to behave as an immiscible Newtonian fluid. Since phase transitions and dissolutions are of little significance in simulations with high-gas injection rates, phase immiscibility is adopted as a simplification here. Assuming an isothermal two-phase flow process, the balance equation for the wetting phase can be expressed as follows:

$$\phi {\rho }_{{\text{w}}}\frac{\partial {S}_{{\text{w}}}}{\partial {p}_{{\text{c}}}}{\dot{p}}_{{\text{c}}}+\alpha {\rho }_{{\text{w}}}{S}_{{\text{w}}}\nabla \cdot \dot{\mathbf{u}}+\nabla \cdot \left[{\rho }_{{\text{w}}}\frac{\mathbf{k}{k}_{{\text{w}}}^{{\text{r}}}}{{\eta }_{{\text{w}}}}\left(-\nabla {p}_{{\text{g}}}+\nabla {p}_{{\text{c}}}+{\rho }_{{\text{w}}}\mathbf{g}\right)\right]={Q}_{{\text{w}}}$$
(2)

where ϕ is the porosity, ρw is the density of the wetting fluid, \({\dot{p}}_{{\text{c}}}={{\text{d}}p}_{c}/{\text{d}}t\) is the temporal derivative of the capillary pressure, \(\dot{\mathbf{u}}={\text{d}}\mathbf{u}/{\text{d}}t\) is the temporal derivative of the solid displacement vector, k is the intrinsic permeability, krw is the relative permeability of the wetting fluid, ηw is the viscosity of the wetting fluid and Qw represents the source and sink term of the wetting fluid. Similarly, the mass balance equation of the gas phase can be expressed as follows:

$$-\phi {\rho }_{{\text{g}}}\frac{\partial {S}_{{\text{w}}}}{\partial {p}_{{\text{c}}}}{\dot{p}}_{{\text{c}}}+\alpha {\rho }_{{\text{g}}}{(1-S}_{{\text{w}}})\nabla \cdot \dot{\mathbf{u}}+\phi (1-{S}_{{\text{w}}})\left(\frac{\partial {\rho }_{{\text{g}}}}{\partial {p}_{{\text{g}}}}{\dot{p}}_{{\text{g}}}+\frac{\partial {\rho }_{{\text{g}}}}{\partial {p}_{{\text{c}}}}{\dot{p}}_{{\text{c}}}\right)+\nabla \cdot \left[{\rho }_{{\text{g}}}\frac{\mathbf{k}{k}_{{\text{g}}}^{{\text{r}}}}{{\eta }_{{\text{g}}}}\left(-\nabla {p}_{{\text{g}}}+{\rho }_{{\text{g}}}\mathbf{g}\right)\right]={Q}_{{\text{g}}}$$
(3)

where ρg is the density of the non-wetting fluid,\({\dot{p}}_{{\text{g}}}={\text{d}}{p}_{g}/{\text{d}}t\) is the temporal derivative of the non-wetting fluid pressure, krg is the relative permeability of the non-wetting fluid, ηg is the viscosity of the non-wetting fluid and Qg represents the source and sink term of the non-wetting fluid. All balance equations are formulated with the primary variables u, pg and pc as implemented in the H2M process in OGS. Detailed derivations of the balance equations can be found in (Kolditz et al. 2012).

2.2 Hydraulic Constitutive Equations

In the H2M process of OGS, the capillary pressure and the phase pressure of the non-wetting fluid are primary variables. The pressure of the liquid phase is consequently defined as:

$${p}_{{\text{w}}}={p}_{{\text{g}}}-{p}_{{\text{c}}}$$
(4)

Equation (4) can be used to calculate the pore pressure (pp), based on \({S}_{{\text{w}}}+{S}_{{\text{g}}}=1\) as:

$${p}_{{\text{p}}}={S}_{{\text{g}}}{p}_{{\text{g}}}+{S}_{{\text{w}}}{p}_{{\text{w}}}={p}_{{\text{g}}}-{S}_{{\text{w}}}{p}_{{\text{c}}}$$
(5)

The saturation can be calculated with the van Genuchten equation (van Genuchten 1980):

$$S_{{\text{w}}} = \left\{ {\begin{array}{*{20}c} {\left( {\left( {\frac{{p_{{\text{c}}} }}{{p_{{{\text{entry}}}} }}} \right)^{{\text{n}}} } \right)^{{\frac{{1 - n}}{n}}} ,} & {p_{{\text{c}}} > 0} \\ {1,} & {p_{{\text{c}}} \le 0} \\ \end{array} } \right.$$
(6)

where pentry is the gas entry pressure and n is an empirical material parameter. However, since the porous medium exhibits some residual saturation and a maximum saturation at all times, the effective saturation of the wetting liquid Se is calculated as follows:

$${S}_{{\text{e}}}=\frac{{S}_{{\text{w}}} - {S}_{{\text{r}}}}{{S}_{{\text{max}}} - {S}_{{\text{r}}}}$$
(7)

where Sr is the residual saturation of the wetting fluid and Smax is the maximum saturation of the wetting fluid. The relative permeability in the porous medium of the two fluid phases depends on the effective saturation. The relative permeability for the wetting fluid is according to the Mualem analytical function (Mualem 1976) as follows:

$${k}_{{\text{w}}}^{{\text{r}}}={S}_{{\text{e}}}^\frac{1}{2}{\left[1-{\left({S}_{{\text{e}}}^{\frac{n}{n-1}}\right)}^{\left(1-\frac{1}{n}\right)}\right]}^{2}$$
(8)

The relative permeability of the non-wetting liquid is described with a cubic function, which is represented as follows:

$${k}_{{\text{g}}}^{{\text{r}}}={100 \left(1-{{\text{S}}}_{{\text{w}}}\right)}^{3}$$
(9)

The cubic function is multiplied by a factor of 100 to account for measurements that the intrinsic permeability in the COx is about 100 times higher for the non-wetting fluid than for the wetting fluid (Amri et al. 2022; Zhang and Rothfuchs 2004).

The Darcy law (Darcy 1856) calculates the flow of both fluid phases (q) based on the relative permeabilities and fluid pressures as follows:

$${\mathbf{q}}_{\xi }=-{\rho }_{\xi }\frac{\mathbf{k}{k}_{\xi }^{{\text{r}}}}{{\eta }_{\xi }}\left(\nabla {p}_{\xi }-{\rho }_{\xi }\mathbf{g}\right)$$
(10)

where the subscript ξ stands for both fluid phases (w: wetting fluid, g: non-wetting fluid).

2.3 Mechanically-Coupled Permeability Constitutive Equations

After drilling the boreholes, two zones can be identified in the resulting EDZ: a continuous fracture zone (CFZ) with extensional and shear fractures and a discrete fracture zone (DFZ) with only shear fractures (Armand et al. 2014). The CFZ is located around the borehole, while the DFZ is generally more extensive. The overall BDZ is likely to have an ellipsoidal shape, and its structure depends on the orientation of the borehole axis with respect to the principal stresses and bedding (Armand et al. 2013, 2014). A schematic representation of the EDZ with division into CFZ and DFZ can be seen in Fig. 1.

Fig. 1
figure 1

Schematic representation of the excavation damage zone with measured values from Armand et al. (2014)

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In this work, the H2M-coupled behaviour in the COx during excavation and under a high-gas-injection rate is simulated by two constitutive equation sets. The combination of the two equation sets is based on the different fracture modes of the investigated processes (Table 1). The excavation-induced permeability increase in the vicinity of the borehole is assumed to be triggered by unloading tensile fractures (mode I) and shear fractures due to stress redistribution (Armand et al. 2013). In this work it is assumed that the dimension of the EDZ is equal to the dimension of the DFZ and is determined by shear fractures. Tensile fractures in the CFZ region are implicitly represented by high permeability values near the borehole. In the case of a shear-stress induced yield, the permeability variation is determined by the failure-index permeability model (Wang et al. 2021). During the subsequent gas injection, diffusion and conventional two-phase flow initially occur in a negligible proportion (Senger et al. 2018). Should the gas pressure increase to a point where it corresponds approximately to the minor principal stress, dilatancy-controlled gas propagation occurs in the claystone and the permeability will increase. Dilatancy is defined in this context as inelastic volumetric change of the rock. This process is associated with a local volume reduction in the case of compressive stresses or with pathway dilation in the case of tensile stresses (Horseman 1996). The numerical implementation of this process is carried out with the strain-dependent permeability model (Xu et al. 2013).

Table 1 Application areas in the mechanical stress state of the constitutive equation sets and their effects on the permeability
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To prevent the two constitutive sets of equations from overlapping and multiplying each other in the area of the shear failure (during excavation), the respective sets are only activated in the corresponding stress states. To implement this numerically, the failure index f is calculated for the shear failure (Wang et al. 2021):

$$f=\frac{\left|{\tau }_{{\text{m}}}\right|}{\text{cos}\left(\varphi \right){\tau }_{f}\left({\sigma }_{{\text{m}}}\right)}$$
(11)

where \({\tau }_{{\text{m}}}=({\sigma }_{1}-{\sigma }_{3})/2\) is the maximum shear stress, calculated with the major and minor principal stresses σ1 and σ3, \(\varphi\) is the inner friction angle and \({\tau }_{f}({\sigma }_{{\text{m}}})\) is the shear strength dependent on the mean stress \({\sigma }_{{\text{m}}}=({\sigma }_{1}+{\sigma }_{3})/2\). The shear strength is calculated by the Mohr–Coulomb failure criterion (Labuz & Zang 2012):

$${\tau }_{{\text{f}}}\left({\sigma }_{{\text{m}}}\right)=c-{\sigma }_{{\text{m}}} {\text{tan}}(\varphi)$$
(12)

where c is the cohesion. The Mohr–Coulomb equation is used here as the failure criterion, since the failure-index model is formulated on its basis, even though it is only a rough approximation of the yield strength and other failure criteria such as Drucker-Prager (Chiarelli et al. 2003) or Hoek–Brown (Souley et al. 2017) provide a better approximation of the yield strength in COx. The constitutive set of equations is determined using the failure index, whereby the following applies:

$${\text{constitutive equation}} = \left\{ {\begin{array}{*{20}c} {failure~index~dependent~permeability} & {\text{if}~f \ge 1} \\ {dilatancy~induced~permeability} & {\text{if}~f < 1} \\ \end{array} } \right.$$
(13)

2.3.1 Failure Index-Dependent Permeability

The failure index-dependent permeability model was developed by Wang et al. (2021) and is implemented in OGS-5.

$$\mathbf{k}={\mathbf{k}}_{0}+H\left(f-1\right){k}_{{\text{r}}}{e}^{bf}\mathbf{I}$$
(14)

where k0 is the initial intrinsic permeability, H is the Heaviside step function, kr is a reference permeability, b is a fitting parameter. The model is originally defined for both shear and tensile fractures, but is only applied here for shear fractures. Since shear fractures from the drilling process affect a larger area, this assumption is made for simplification, even though it does not completely correspond to the observed in-situ processes. The failure index f used in the permeability model was defined in Eq. (11) and can be calculated with the Mohr–Coulomb failure criterion Eq. (12). Wang et al. (2021) have previously calibrated the empirical values of the equation on COx claystone.

2.3.2 Dilatancy-Induced Permeability

The permeability increase in the process of dilatant deformations is simulated in the model by the strain-dependent permeability model, developed by Xu et al. (2013). This is a hydraulic-mechanical-coupled empirical model based on volumetric strain and equivalent plastic strain. The parameters are supplemented with empirical values depending on the prevailing stress state. As a result, permeability decreases in the compressed stress state and increases in the extensional stress state. These functions are regulated by the volumetric deformation (Braun et al. 2020). The equivalent plastic deformation may exclusively contribute to an exponential increase in permeability.

$$\mathbf{k}={f}_{sdp}\left({\varepsilon }_{{\text{vol}}}\right){e}^{{b}_{1}{\overline{\varepsilon }}_{{\text{pl}}}}{\mathbf{k}}_{0}$$
(15)

where εvol is the volumetric strain, b1 is an empirical value related with the permeability increase due to tensile plastic deformation, \({\overline{\varepsilon }}_{{\text{pl}}}\) is the equivalent plastic strain. The function fsdp is dependent on the behaviour of the volumetric strain as follows:

$$f_{{sdp}} \left( {\varepsilon _{{{\text{vol}}}} } \right) = \left\{ {\begin{array}{*{20}c} {10^{{b_{2} \varepsilon _{{{\text{vol}}}} }} ,} & {~~~~~ \varepsilon _{{{\text{vol}}}} \le 0{~}\left( {{\text{compaction}}} \right)} \\ {10^{{b_{3} \varepsilon _{{{\text{vol}}}} }} ,} & {~ \varepsilon _{{{\text{vol}}}} > 0~\left( {{\text{extension}}} \right)} \\ \end{array} } \right.$$
(16)

where b2 and b3 are empirical values related to the decrease and increase of the permeability due to dilatancy, respectively. To account for microfractures due to tensile forces, a tension cut-off is added to the yield criterion. The tension cut-off is directly related to the equivalent plastic strain and is formulated based on Eq. (12). A description of the non-associated yield criterion and the implementation to OGS can be found in Nagel et al. (2017).

2.4 Mechanical Constitutive Equations

The mechanical regime is based on elasto-plasticity for the COx as previously shown in Eqs. (11) and (12), to account for permeability caused by failure or plastic dilatancy. The elastic behaviour of the COx is orthotropic. The stress state can be calculated using the effective stress law, noting that the tensile stress is defined as positive, as follows:

$${\varvec{\upsigma}}={{\varvec{\upsigma}}}{\prime}-\alpha {p}_{{\text{p}}}\mathbf{I}$$
(17)

where σ is the total stress. The effective stress can be used to calculate the elastic strain, by using the Hooke’s law, which can be expressed in a generalized form as follows:

$${\varvec{\upsigma}}=\mathbf{C}{\varvec{\upvarepsilon}}$$
(18)

where C is the stiffness tensor and ε is the strain tensor. Since plane strain conditions apply in the simulated models (\({\varepsilon }_{zz}={\varepsilon }_{zx}={\varepsilon }_{yz}=0)\), Hooke’s law for an orthotropic material can be expressed as follows:

$$\left[\begin{array}{c}{\varepsilon }_{xx}\\ {\varepsilon }_{yy}\\ 2{\varepsilon }_{xy}\end{array}\right]=\left[\begin{array}{ccc}\frac{1-{\nu }_{xy}{\nu }_{yx}}{{E}_{x}}& -\frac{{\nu }_{yx}-{\nu }_{xy}{\nu }_{yx}}{{E}_{y}}& 0\\ -\frac{{\nu }_{xy}-{\nu }_{xy}{\nu }_{yx}}{{E}_{x}}& \frac{1-{\nu }_{xy}{\nu }_{yx}}{{E}_{y}}& 0\\ 0& 0& \frac{1}{{G}_{xy}}\end{array}\right]\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\tau }_{xy}\end{array}\right]$$
(19)

where ν is the Poisson’s ratio and G is the shear modulus. The strain can be divided into elastic and plastic strain as follows:

$${{\varvec{\upvarepsilon}}}_{{\text{t}}}={{\varvec{\upvarepsilon}}}_{{\text{el}}}+{{\varvec{\upvarepsilon}}}_{{\text{pl}}}$$
(20)

where εt is the total strain, εel is the elastic strain and εpl is the plastic strain.

3 Modelling the PGZ Experiment

3.1 PGZ Experiment

The PGZ experiment was conducted in the Meuse/Haute-Marne Underground Research Laboratory (MHM URL), France (Fig. 2). The MHM URL has been in operation since 2000. All relevant aspects of the final disposal of radioactive materials in the connection with argillaceous rocks are investigated there (Plúa et al. 2021). Among other experiments, the PGZ experiment was carried out by ANDRA at the MHM URL to investigate gas flow processes through claystone (de La Vaissière et al. 2014). The PGZ experiment started in 2009 with the experimental characterisation of hydraulically-induced fractures and the influence of the EDZ. Since 2020, a second phase of the experiment has been conducted, this time focusing on the effects of stress orientation on fracture development. The numerical calculation of the PGZ experiment is carried out in the framework of the current phase of the DECOVALEX project (DECOVALEX 2022). For the numerical simulations, the dimensions of the borehole, the gas-injection rate and most of the rock properties were provided by the DECOVALEX project.

Fig. 2
figure 2

Layout of the MHM URL with the area highlighted where the PGZ experiment was carried out (edited after Seyedi et al. 2017)

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Several boreholes were drilled in the GEX drift for the PGZ experiment (de La Vaissière et al. 2019). In the following, two of them are examined: one parallel to the minor horizontal stress (PGZ1002) and one parallel to the major horizontal stress (PGZ1003). The dimensions of the EDZ, created in the process of drilling can be estimated by in-situ investigations (Armand et al. 2014; de La Vaissière et al. 2015; Guayacán-Carrillo et al. 2015). The length of the borehole is 35 m and the radius of the borehole is rb = 0.038 m. After the excavation process, a filter was inserted to which a Neumann boundary condition is attached to simulate the gas inflow. The inner radius of the filter is ri = 0.033 m. A schematic representation of the processes taking place and the estimation of the dimension of the EDZ are shown in Fig. 3. This shows the drilling process and the following gas injection in three stages for both boreholes. PGZ1002 is drilled in parallel with the minor horizontal stress (σh) and PGZ1003 in parallel with the major horizontal stress (σH). Based on various in-situ investigations, the EDZ is expected to form as shown in the figure.

Fig. 3
figure 3

Schematic drawing of the processes taking place for (PGZ1002) the borehole in the direction of the minor horizontal stress and (PGZ1003) the borehole in the direction of the major horizontal stress

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3.2 Model Parameter

The numerical model is based on the finite element method. The domain has a side length of 5 m. The 2D mesh has 3420 nodes and 3149 quadratic elements. At the beginning of the simulation, the COx claystone is fully saturated with a pore pressure of pp = 4.7 MPa. The pore pressure is calculated from gas pressure and capillary pressure according to Eq. (5). Since the COx is fully saturated, the initial gas pressure is pg = 0.1 MPa and the initial capillary pressure is pc = − 4.7 MPa. A Dirichlet condition with no displacement applies to all boundaries, except for the two boundaries where the stress is applied. Furthermore, on the far boundaries there are Dirichlet boundaries that keep the gas pressure at pg = 0 Pa and the capillary pressure at pc = − 4.7 MPa. The initial conditions in the filter are determined by measurements during the experiment. A gas pressure of pg = 4.7 MPa was built up and a saturation of approx. Sw = 0.6 was estimated, which corresponds to a capillary pressure of pc = 9.0 kPa. Most of the rock and fluid parameter values were provided in the DECOVALEX project description and can be found in Table 2. The values for the failure-index permeability model and the Mohr–Coulomb failure criterion are from Wang et al. (2021) and are already calibrated to COx claystone.

Table 2 Model parameter from DECOVALEX (2022) and Wang et al. (2021)
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During the gas-injection test, the gas flow rate was measured in standard temperature and pressure (STP). The presented gas flow rate was used in the in-situ experiments and are applied in the respective models (Fig. 4). The standard Galerkin finite element approach is applied for spatial discretisation of the domain. Linear shape functions are employed for gas pressure and capillary pressure, while quadratic shape functions are employed for the displacements. The time discretisation is carried out according to the implicit Euler method with a fixed time step size. The time increment varies between 60 s during the injection phase and 500 s during the decay phase. The set of equations is solved by a staggered scheme and linked with a coupling loop for each time step until convergence is reached. The biconjugate gradient stabilized method (BiCGSTAB) is used as a linear solver with an overall error tolerance of 10–16. The Newton–Raphson iteration scheme is used as a nonlinear solver with a relative tolerance of 10–6.

Fig. 4
figure 4

Measured gas flow rate in the in-situ gas-injection test

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The parameter b1, b2 and b3 of the dilatancy-dependent permeability model were calibrated so that the calculated gas pressure in the injection filter matches the measured gas pressure during the gas-injection test. The permeability model with failure index was applied as well, to achieve the closest possible similarity to the experimental gas injection test. Figure 5 shows the calibrated values and their effect in the H2M coupling regarding the permeability.

Fig. 5
figure 5

Dilatancy-dependent permeability model with calibrated values

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4 Results

4.1 Failure-Index Permeability for PGZ1002 and PGZ1003

The first step of the simulation is the excavation of the borehole. The failure-index permeability based on the Mohr–Coulomb model calculates the permeability around the borehole (Fig. 6). The calculated range of the EDZ is approximately 0.03–0.05 m into the claystone. This means that the calculation is within the range of measured dimensions of the EDZ in the vicinity of boreholes for COx (Armand et al. 2014). It can be observed that the influence of the stress distribution has a greater influence in the simulation on the formation of the EDZ than the transversal isotropic behaviour of the COx. The structure in the PGZ1002 model corresponds in shape to the measurements, schematically shown in Fig. 1. The calculated EDZ in the PGZ1003 model does not completely correspond in shape to the measurements. This is because in the failure-index model, the stress distribution is mainly responsible for the development of the EDZ, and the transversal isotropic behaviour of the COx has less influence. The calculated EDZ of borehole PGZ1002 extends approximately 0.05 m vertically and 0.025 m horizontally into the rock. Whereas the calculated EDZ of borehole PGZ1003 extends approximately 0.02 m vertically and only 0.03 m horizontally into the rock.

Fig. 6
figure 6

Excavation-induced permeability in log-scale for PGZ1002 and PGZ1003

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4.2 Dilatancy-Induced Gas Flow

To better analyze the influence of the respective constitutive models on permeability and pressures, base tests for both models and four model configurations (cases) are presented below:

Base test: PGZ1002 & 1003 without failure-index permeability without dilatancy-controlled permeability

Case A: PGZ1002 with failure-index permeability without dilatancy-controlled permeability

Case B: PGZ1002 with failure-index permeability and dilatancy-controlled permeability

Case C: PGZ1003 with failure-index permeability without dilatancy-controlled permeability

Case D: PGZ1003 with failure-index permeability and dilatancy-controlled permeability

The validation of the approach and the comparison between the respective model cases is performed with the gas pressure and the water saturation in the filter. The measured values for the gas pressure in the filter are included as a black dashed line for comparison. The pressure and saturation development during gas injection and the decay phase of PGZ1002 is shown in Fig. 7 and for PGZ1003 in Fig. 8.

Fig. 7
figure 7

Temporal evolution of gas pressure and water saturation during the injection and decay phase for PGZ1002. The section with gas injection are marked in grey

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Fig. 8
figure 8

Development of gas pressure and water saturation during the injection and decay phase for PGZ1003 the measured gas pressure. The sections with gas injection are marked in grey

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The figures show that the calculation without an additional constitutive equation (base test) describing the hydro-mechanical coupling misses the measured values by far. The gas pressure in the injection filter is overestimated by 1.5 MPa (PGZ1002) and 4 MPa (PGZ1003). By applying the failure-index permeability model, a better agreement with the measured gas pressure can be achieved. Only at very high pressures (above ~ 12 MPa) does an overestimation of the gas pressure of 1—2 MPa in the injection filter become apparent in the two models. The pressure corresponds approximately to the minimum principal stress (11.7 MPa) beyond which dilatant micro tensile cracks occur. The additional application of the dilatancy-dependent permeability model demonstrates that the maximum gas pressures in the filter can be better predicted with this additional constitutive model.

The increased gas pressure in the filter affects the liquid pressure near the borehole, since water is displaced from the injection filter into the COx. Figure 9 shows the fluid pressure for selected points near the borehole. The figure shows that the high liquid pressure quickly transfers from the injection filter into the COx and levels off after a few hours at close range. The effect is more obvious in case B and D where the dilatancy-controlled permeability model was also used. The orthotropic behaviour of the COx is minimized in case C and D by the shape of the EDZ.

Fig. 9
figure 9

Liquid pressure for all four cases at different points in the injection filter and COx. The measured liquid pressure in the filter is added for comparison. The sections with gas injection are marked in grey

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Figure 10 shows the gas pressure for selected points near the borehole. The gas pressure evolution is similar to the liquid pressure, but the increase is stronger and the gas pressure in the COx claystone is 0.1 MPa at the beginning of the calculation. This corresponds to the initial condition for the gas pressure, which was selected here according to the atmospheric pressure, since there is no gas phase in the claystone at the beginning of the calculation, since the claystone is fully saturated. It can also be observed that the calculation can be simulated better at lower pressures where a microfacture-induced gas flow at lower gas pressures occurs (PGZ1002) than at higher gas pressures (PGZ1003).

Fig. 10
figure 10

Gas pressure for all four cases at different points in the injection filter and COx. The measured gas pressure in the filter is added for comparison. The sections with gas injection are marked in grey

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5 Conclusion

The approach to combine two independent constitutive models regarding the hydro-mechanically coupled processes during an excavation and gas-injection test was demonstrated on a number of models. The approach uses the dominant mechanical stress states of the analyzed processes to combine the sets of equations without creating a reciprocal coupling. Thus, only one permeability model is used at a time for a specific stress state and never both at the same time and stress state. The equation sets are thereby adjusted to the extent that they merge continuously into one another. To be precise, the transition occurs at compressive stress, namely when the yield strength is exceeded the index failure criterion formulates an increasing permeability, below the yield strength the compression leads to a decrease in permeability, simulated by the strain-dependent permeability model. The application was carried out in a well-studied area of the COx argillaceous rock formation. The drilling of two boreholes was simulated and subsequent gas-injection tests were calculated. The simulation was carried out in a coupled H2M finite element model, calculated with the open-access software OGS. The evaluation of the hydraulic and mechanical parameters was conducted with measured data recorded during the gas-injection test for the PGZ experiment. The comparison with the measured data and between the test cases indicates the following:

  • The failure-index permeability model has calculated plausible results regarding the extent and magnitude of the EDZ, which are also within the range of published data. However, when applying the Mohr–Coulomb flow criterion, the applied model cannot accurately reproduce the structure of the BDZ in the PGZ1003 model, which has an assumed ellipsoidal shape due to the transversely isotropic behavior of the COx.

  • The combination of both constitutive equations can reflect the development of the gas pressure in the filter better than one equation alone could.

  • The decay phases are well reproduced for the combined approach in cases B and D.

The developed approach has been successfully applied to the modelling of gas percolation in claystone and could prove to be a valuable tool for the safety analysis of a future nuclear waste repository.