2D Phase-Field Modelling of Hydraulic Fracturing Affected by Cemented Natural Fractures Embedded in Saturated Poroelastic Rocks

Griffith’s hypothesis on the cracking phenomenon implies that new surfaces within the volume of a deforming body are formed as a result of the release of the stored strain energy due to the external loading. In fact, the released energy is consumed to form new surfaces, so the fracture propagation problem can be mathematically defined as searching for an equilibrium state of energy in a fracturing body (Griffith 1921). The energy contribution of the new surfaces (\({\Psi }_{\text{surface}}\)) within the volume of the body can be formulated as \({\int }_{\Gamma }{G}_{c} dA\), where \({G}_{c}\) is so-called the critical energy release rate. The domain of this integral (\(\Gamma\)) is mathematically unknown, because fractures form anonymously throughout the body. Therefore, Bourdin et al. (2000) re-formulated \({\Psi }_{\text{surface}}\) as an integral over the volume of the body (\(\Omega\)) by assuming the sharp discontinuity as a gradient-type diffusive damaged zone, represented by the phase-field parameter (\(d\)), and introducing the surface energy density functional \(\gamma \left(d,\nabla d\right)\) in the following manner (Ambrosio and Tortorelli 1990):

In the above equation, \(\nabla\) is the gradient operator in Cartesian coordinates, \(d\) is the so-called damage (phase-field) parameter, and \({l}_{0}\) is length-scale parameter which characterises the width of the diffusive fracture. The value of damage parameter varies between \(d\) = 0 for fully damaged region and \(d\)=1 associated with the intact region, which is governed by a function shown in Fig. 2.

The fracture propagation can be predicted by searching for the equilibrium in the internal potential energy of the body; the regularised functional which needs to be minimised for tracking the fracture propagation is formulated as (Bourdin et al. 2000)where \({\Psi }_{\text{bulk}}\left({\varvec{\varepsilon}}\right)\) is the strain energy stored in the bulk material (undamaged state), and \(g\left(d\right)={d}^{2}+{\kappa }_{\epsilon }\) is the degradation function based on the damage field which will be discussed later. In the process of energy minimisation, as the fractures propagate and new surfaces form throughout \(\Omega\), the first integral in the above equation decreases because of the effect of the development of damage throughout the body (i.e., decreasing effect of the degradation function when \(d\to 0\)), while the second integral grows constantly as the damaged zones spread in the continuum.

Consider a physical body \(\Omega\) in Euclidean space \({\mathbb{R}}^{3}\) as a saturated porous medium consisted of two phases, the porous solid skeleton with mass density \({\rho }_{\text{s}}\) and the fluid phase with mass density \({\rho }_{\text{f}}\) in the pores. The initial porosity for a volume element \(dV\) is defined aswhere \(d{V}_{\text{f}}=dV-d{V}_{\text{s}}\) is the volume of the fluid (equal to the volume of pores in the saturated condition). The initial mass of the unit volume \(dV\) is equal to \({m}_{0}={\rho }_{\text{s}}{\varphi }_{0}^{\text{s}}+{\rho }_{\text{f}}{\varphi }_{0}^{\text{f}}\). When the domain \(\Omega\) is subjected to the hydro-mechanical loading, including the fluid injection and the exertion of mechanical forces on the external boundaries, both fluid and solid undergo motion. Herein, the movement of the fluid phase is formulated with respect to the deformation of the solid skeleton, since the developed formulations are based upon a Lagrangian description of the continuum. In the process of deformation in time \(t\), the rate of the fluid mass \(m\) flowing into or out of the unit volume of the element is defined aswhere \({D}^{\text{s}}\left(\cdot \right)/Dt\) operator is the Lagrangian time derivative of the fluid mass with respect to the solid motion, and the vector \({{\varvec{m}}}^{\text{r}}\) is the fluid mass flow rate in the space moving throughout the porous medium, defined aswhere \({\varvec{q}}\) is the fluid volume rate affecting on the boundaries of the porous body, as shown in Fig. 3. Hydro-mechanical response of water-saturated porous rock is associated with two mechanisms: (i) the dilation of the solid skeleton due to an increase in the pore fluid pressure, and (ii) the increase in the pore fluid pressure when the rock is under compression (Detournay and Cheng 1993). For modelling such behaviour, Biot’s theory of poroelastic coupling in saturated porous rock is employed, assuming incompressible fluid and linear elastic behaviour for the solid skeleton (Biot and Willis 1957). The total Cauchy stress tensor \({\varvec{\sigma}}\), the strain energy stored in the solid skeleton \({\Psi }_{\text{eff}}\left({\varvec{\varepsilon}}\right)\), and the energy stored in the pore fluid phase \({\Psi }_{\text{fluid}}\left({\varvec{\varepsilon}},\xi \right)\) are defined, respectively, aswhere the strain tensor is calculated from the deformation field (\({\varvec{u}}\)) as \({\varvec{\varepsilon}}=1/2\left(\nabla {\varvec{u}}+{\nabla }^{\text{T}}{\varvec{u}}\right)\), and \({\varepsilon }_{\text{v}}=\text{tr}\left[{\varvec{\varepsilon}}\right]\). We have based our model on the infinitesimal strain theory; this assumption is valid when the deformations are very small due to a high stiffness in the rock and low confining stresses. Several studies have developed phase-field hydraulic fracture models in poroelastic media based on linear elasticity (Mauthe and Miehe 2017; Heider and Markert 2017; Zhou et al. 2018), similar to the model developed herein. In higher magnitudes of the confining pressure, rock’s mechanical response becomes nonlinear (Choo and Sun 2018), and deformations become significant when the stiffness decreases; therefore, employing the kinematics of large deformations (finite strains) would be essential (Miehe and Mauthe 2016). The tensor \({{\varvec{\sigma}}}{\prime}=\partial {\Psi }_{\text{eff}}\left({\varvec{\varepsilon}}\right)/\partial{\varvec{\varepsilon}}\) is so-called the effective Cauchy stress tensor is a part of \({\varvec{\sigma}}\) that only applies on the solid skeleton structure. The pore fluid pressure is defined as \(p=\partial {\Psi }_{\text{fluid}}\left({\varvec{\varepsilon}},\xi \right)/\partial \xi ={M}_{\text{b}}\left(\xi -\alpha {\varepsilon }_{\text{v}}\right)\), \(\xi\) is the volumetric fluid production term (or so-called fluid content), and \(d\xi =dm/{\rho }_{\text{f}}\) is the variation of the fluid volume per total unit volume \(dV\) due to fluid mass transport (Detournay and Cheng 1993). The pore fluid pressure (\(p\)) is considered as a positive value, which is incorporated in the total Cauchy stress tensor \({\varvec{\sigma}}\) (Eq. 8) in a way that negative value for the hydrostatic part of the stress would imply a compressive stress state. In the above equations, \(\alpha\) is the Biot’s coefficient defined as (Biot and Willis 1957)where \(K\) is the drained bulk modulus of the porous rock and \({K}_{\text{s}}\) is the bulk modulus of the solid constituents (i.e., grains). Several laboratorial methods can be found in the literature to measure Biot’s coefficient (Franquet and Abass 1999). Biot’s modulus \({M}_{\text{b}}\) is defined as \({M}_{\text{b}}={K}_{\text{s}}^{2}\left({K}_{\text{u}}-K\right)/{\left({K}_{\text{s}}-K\right)}^{2}\), where \({K}_{\text{u}}\) is the undrained bulk modulus of the porous rock and can be measured experimentally (Detournay and Cheng 1993).

The finite-element weak forms are derived by minimising the total energy with respect to arbitrary damage, displacements, and pore fluid pressure. The total potential energy functional \(\Pi \left({\varvec{\varepsilon}},\xi ,d\right)\) is consisted of the effective elastic strain energy of the fractured solid skeleton \({\Psi }_{\text{eff}}\left({\varvec{\varepsilon}},d\right)\), the energy of the pressurised fluid inside the pores \({\Psi }_{\text{fluid}}\left({\varvec{\varepsilon}},\xi \right)\), and the surface energy \({\Psi }_{\text{surface}}\left(d\right)\) which is the contribution of the newly propagated fractures within the volume (\(\Omega\)). Employing a continuous Galerkin method and applying the natural boundary conditions, shown in Fig. 3, the weak form of the balance of linear momentum is derived by finding a stationary position of \(\delta\Pi \left({\varvec{\varepsilon}},\xi ,\overline{d }\right)=0\) with respect to arbitrary displacement field \({\varvec{\eta}}\), while the damage field is assumed to be fixed (\(\overline{d }\)):

In the above equation, \({\mathbb{C}}\) is the fourth-order elastic stiffness tensor, \({{\varvec{T}}}^{*}\) is the vector of traction forces applied on the boundaries of the body (see Fig. 3), \(g\left(\overline{d }\right)={\overline{d} }^{2}+{\kappa }_{\epsilon }\) is the degradation function, as defined by (Bourdin et al. 2000; Miehe et al. 2010b), for a fixed state of damage in the body, and \({\kappa }_{\varepsilon }\ll 1\) is the regularisation parameter. The second weak form completing the u–p formulation is obtained from the total mass conservation law (Eq. 13) and applying the Galerkin weighting function \(\psi\) and natural boundary conditions shown in Fig. 3:

Obeying the staggered approach in solving for the damage field, both displacement and pore fluid pressure fields are assumed to be fixed (\(\overline{{\varvec{u}} },\overline{p }\)) throughout the domain, to eventually solve for the damage field (phase-field parameter \(d\)). The weak form of the damage evolution model is obtained by finding a stationary position of \(\delta\Pi \left(\overline{{\varvec{u}} },\overline{p },d\right)=0\) with respect to the arbitrary damage \(\omega\) aswhere \(\widehat{\mathcal{H}}\) is a history field, defined as the maximum energy acquired by a volume element during the history of deformation to satisfy the fracture irreversibility condition (Miehe et al. 2010b). The history field reads aswhere \({t}_{0}\) and \({t}_{\text{i}}\) are the initial and current times of the deformation history. The method for defining the contribution of the effective bulk energy in calculating the value of \(\widehat{\mathcal{H}}\) can be chosen differently, based on the physical failure behaviour of the material. The most widely used criteria on defining \(\widehat{\mathcal{H}}\) have been introduced by (i) Amor et al. (2009) based on splitting the strain tensor into volumetric and deviatoric parts, and (ii) Miehe et al. (2010b) based on the spectral decomposition of the strain tensor into tensile and compressive parts. These formulations are written in the following neglecting the operator \(\underset{t\in \left[{t}_{0},{t}_{\text{i}}\right]}{\text{max}}\left[\cdot \right]\) for the sake of simplicity:

In the above equations, \(\lambda ,\mu\) are the elastic Lame parameters and \(K\) is the bulk modulus of porous rock, \({\langle x\rangle }_{\pm }=\left(x\pm \left|x\right|\right)/2\), and \({{\varvec{\varepsilon}}}_{D}={\varvec{\varepsilon}}-\left(\text{tr}\left[{\varvec{\varepsilon}}\right]/3\right){\varvec{I}}\) is the deviatoric part of the strain. The positive part of the strain tensor \({\left[{\varvec{\varepsilon}}\right]}^{+}\) is calculated as stated in Eq. (33), based on the spectral decomposition of the strain tensor \({\varvec{\varepsilon}}\). It is worth mentioning that the method of defining \(\widehat{\mathcal{H}}\) also affects the calculation of the effective stresses within the solid skeleton, see Sect. 2.5. For the case of \({\widehat{\mathcal{H}}}_{\text{Miehe}}\), the degradation function \(g\left(d\right)\) only affects the elements under tension and not the ones under compression. For the case of \({\widehat{\mathcal{H}}}_{\text{Amor}}\), the negative part of the volumetric stress remains unaffected by the degradation function \(g\left(d\right)\). As a result, contacting compressive stresses remain in place within the damaged zone, which is a better physical representative of the fracturing phenomenon especially in mode-II failure (Amor et al. 2009).

For modelling the fluid flow within the HF, Poiseuille’s law representing the flow between parallel plates is postulated, so the fluid velocity field is formulated aswhere \(w\) is the fracture (HF) width. Since the HF is modelled as a gradient-type damaged zone over a continuous finite-element mesh, there will be no discrete boundary considered for calculating the HF’s width when employing the Poiseuille’s law. Equation (21) is only used to model the fluid flow inside the HF, so one needs to specify the elements which are guaranteed to be within the fully damaged zone. Herein, we recommend using a strain-based criterion, Eqs. (22) and (23), that can identify whether the critical strain state in the element has exceeded the critical tensile strain of the material or not; Poiseuille’s law is only employed over these elements. To illustrate these critical elements within the fully damaged zone, a typical phase-field fracture in mode-I opening status, including the relevant displacement and the maximum principal strain fields, is depicted in Fig. 4. It is shown that the fully damaged zone can be restricted to the critical elements for which the maximum principal strain \({\varepsilon }_{\text{p}1}\) is significantly higher compared to the adjacent elements. To maintain the consistency of the hydro-mechanical model (u–p formulation) for modelling the fluid flow in both HF and the intact porous rock, the permeability tensor \({\varvec{k}}\), used in Eq. (14), should be modified aswhere

In the above equations, \(H\left({\varvec{\varepsilon}},\boldsymbol{ }{\varepsilon }_{\text{t}}\right)\) is a jump function, \({\varepsilon }_{\text{i}}\) are the principal strains calculated from spectral decomposition of the strain tensor \({\varvec{\varepsilon}}\), refer to Eq. (33), and \({{\varvec{k}}}_{\text{intact}}=\left(\overline{k }/{\mu }_{\text{f}}\right){\varvec{I}}\) is defined for the intact porous rock with no damage. The critical tensile strain of the material is \({\varepsilon }_{\text{t}}={\sigma }_{\text{t}}/E\), which if exceeded, the element is encountered as “critical” and is ensured to be within the fully damaged zone.

Our method of capturing the critical elements based on exceeding the critical tensile strain limit \({\varepsilon }_{\text{t}}\), gives a physical meaning to the evolution of permeability for the elements inside the diffusive damaged zone. Calculation of the HF’s width (\(w\)) is affected by the element size (\({H}_{\text{el}}\)) and the strain tensor \(\left[{\varvec{\varepsilon}}\right]\), as stated in Eq. (24). According to Fig. 4, for a triangular mesh structure, the minimum \({H}_{\text{el}}\) in the vicinity of the fracture cannot be larger than \({l}_{0}/4\) to guarantee the existence of adequate elements capturing the fully damaged zone and to ensure a more accurate calculation of \(w\) (Sarmadi and Mousavi Nezhad 2023). Satisfying the condition \({H}_{\text{el}}\le {l}_{0}/4\) for the damaged zone is compatible with the recommendations of (Bourdin et al. 2000; Miehe et al. 2010b) on choosing the element size \({H}_{\text{el}}\) to ensure the mesh-independency of the answer. For calculating \(w\), we follow the procedure proposed by Mauthe and Miehe (2017), as mentioned below, albeit with further recommendations on choosing the value of \({L}_{\perp }\):

In the relationship above, \({L}_{\perp }\) is called the characteristic length which is measured in the direction perpendicular to the orientation of the diffusive fracture and is typically considered equal to \({H}_{\text{el}}\) (Mauthe and Miehe 2017). By comparing the distribution of the principal strain fields for two different mesh configurations in Fig. 4, we realise that having a finer mesh may result in appearing more than one element within the fracture walls. As a result, we recommend choosing \({L}_{\perp }\) equal to \({l}_{0}/4\) while keeping the element size around the HF-tip \({H}_{\text{el}}\le {l}_{0}/4\) by performing a simple h-refinement on the coarser elements. It is worth mentioning that the length-scale parameter \({l}_{0}\) can be chosen with regard to the physical parameters, such as the fracture toughness and tensile strength of the material (Amor et al. 2009; Miehe et al. 2015; Sarmadi et al. 2023).

In this section, we develop a mixed-mode formulation to calculate the crack driving force accounting for both tensile and shearing failure modes. This approach rests upon the Mohr–Coulomb–Griffith failure model (Jaeger et al. 2009) to account for tensile and compressive shear failure due to the accumulated stresses in the continuum. The final purpose is to use this formulation in modelling shear slippage of NFs which can affect the propagation path of the HF. The strong form of the damage evolution model can be derived from Eq. (17) aswhere \(\Delta\) is the Laplacian operator, and the term \(\widehat{\mathcal{H}}/{G}_{c}\), so-called the crack driving force, is specifically responsible for the evolution of damage throughout the solid skeleton. In geo-materials, the critical energy release rate for mode-II cracking is usually greater than the one for mode-I. Zhang et al. (2017) modified the crack driving force by introducing a hybrid format of Eq. (25) as follows:where \({G}_{c}^{I}\) and \({G}_{c}^{II}\) are the critical energy release rates for mode-I and mode-II fracture, respectively; also, \({\widehat{\mathcal{H}}}_{1}=\lambda /2{\left({\langle \text{tr}\left[{\varvec{\varepsilon}}\right]\rangle }_{+}\right)}^{2}\) and \({\widehat{\mathcal{H}}}_{2}=\mu \text{tr}\left[{\left({\left[{\varvec{\varepsilon}}\right]}^{+}\right)}^{2}\right]\) are defined based on the spectral decomposition of the strain tensor. This approach still accounts for the elements under tension only, while the shear cracking (mode-II) can happen for the elements under compression in rock-like materials. A comparison between the cracking behaviour of limestone and a type of glass (polymethyl methacrylate) in single-flaw square plates subjected to compression has been done by Ingraffea and Heuze (1980), see Fig. 5, which confirms having a different crack pattern in rocks. Calculating the crack driving force based on Miehe’s approach replicates a crack pattern similar to the one shown in Fig. 5c. Miehe’s approach on splitting the energy based on spectral decomposition of the strain tensor can predict the crack pattern in glass, since the brittle tensile failure is the dominant mode. A more advanced criterion based on a statistical damage model was developed by Zhou et al. (2019) to form the crack driving force (\(\widehat{\mathcal{H}}/{G}_{c}\)) based on the elastic parameters, internal angle of friction \(\varphi\), and the internal cohesion \(c\). The modification by Zhou et al. (2019) enforces the dominancy of the shearing mode in the damage evolution, while the work of Zhang et al. (2017) still tends to forecast tensile cracks. In modelling the cracking phenomenon in geo-materials, both tensile and shear failure might occur depending on the tensile and shear strength limits. To predict the correct crack patterns in geo-materials using the phase-field fracture model, which is essentially a continuum damage model (Marigo et al. 2016), we construct the crack driving force based on the idea of mixed-mode cracking, Eq. (26), but use two different formulations to calculate \({\widehat{\mathcal{H}}}_{1}\) and \({\widehat{\mathcal{H}}}_{2}\). The maximum principal stress criterion with the critical tensile strength threshold (\({\sigma }_{\text{t}}\)), introduced in (Miehe et al. 2015), is employed for calculating \({\widehat{\mathcal{H}}}_{1}\) and is in charge of driving the damaged zone in the opening mode-I:

In the above equation, \({\sigma }_{\text{i}}^{\prime}\) are the principal effective Cauchy stresses obtained via spectral decomposition of the undegraded Cauchy effective stress tensor, and \({E}^{\prime}\) is the effective elastic young modulus. For the compressive shear failure of geo-materials, Mohr–Coulomb–Griffith failure criterion (Jaeger et al. 2009) is implemented in the model to calculate the contribution of \({\widehat{\mathcal{H}}}_{2}\), see Eq. (28). If the stress state exceeds the Mohr–Coulomb shear failure envelope, \({\widehat{\mathcal{H}}}_{2}\) would take the value of the deviatoric part of the strain energy, defined by Amor et al. (2009):

Equation (29) introduces \(H\left({\sigma }_{\text{i}}^{\prime},c,\varphi \right)\), a jump function that depends on the principal stresses and checks if the Mohr–Coulomb shear failure condition (\(i,\text{j}=\text{1,2},3\)) is met. Parameters \(c\) and \(\varphi\) are cohesion and internal friction angle, and \({{\varvec{\varepsilon}}}_{D}\) is the deviatoric part of the strain tensor. Finally, the finite-element weak form of the damage evolution model, which was previously introduced in Eq. (17), is replaced by the following:

The effective stress tensor is defined in Eqs. (31) and (32), depending on whether the failure occurs in tensile or shear mode, respectively:

For the failed elements under compressive shearing, the procedure proposed by Amor et al. (2009) is followed using Eq. (32). The volumetric part of the effective stress remains unaffected by the degradation function \(g\left(d\right)\) to maintain the normal compressive stresses within the fully damaged zone. In tensile failure, however, the formulation of Miehe et al. (2010a) is employed for obtaining effective stresses, i.e., Eq. (31). Tensors \({\left[{\varvec{\varepsilon}}\right]}^{\pm }\) are obtained from the spectral decomposition of \({\varvec{\varepsilon}}\) aswhere \({\varepsilon }_{\text{i}}\) are the principal strains and \({{\varvec{n}}}_{\text{i}}\) are the relative principal directions. In the next section, the presented model will be verified by comparing the results to the KGD analytical solution of HF propagation and experimental data on the shear failure in rock-like materials.

The Newton–Raphson method is employed to solve for the displacement and fluid pressure fields in an iterative manner. After discretising Eq. (16) in time domain, the weak form of the mass conservation law reads aswhere \(\Upsilon\) is the time-integration parameter, \(\Delta t={t}_{\text{n}+1}-{t}_{\text{n}}\), \(\delta p={p}_{\text{n}+1}-{p}_{\text{n}}\) and \(\delta {\varvec{u}}={{\varvec{u}}}_{\text{n}+1}-{{\varvec{u}}}_{\text{n}}\). Note that for calculating the velocity fields \({{\varvec{v}}}_{\text{n}+1}^{r}\) and \({{\varvec{v}}}_{\text{n}}^{r}\), Eq. (14) is employed, albeit with incorporating the correction of the permeability tensor \({\varvec{k}}\) mentioned in Eq. (22) into the model. The weak form of the balance of linear momentum reads aswhere the degraded elasticity tensor \({\mathbb{C}}\left({{\varvec{\sigma}}}_{\text{n}}^{\prime},{\overline{d} }_{\text{n}}\right)\) is updated based on the current state of damage in the body at time-step \({t}_{\text{n}}\). Notice that the effective stress tensor \({{\varvec{\sigma}}}_{\text{n}}^{\prime}\) is calculated using Eqs. (31) or (32) depending on the failure’s mode. The numerical algorithm, elaborated in Fig. 6, to solve for the variables using finite-element method in an iterative manner is implemented in a MATLAB-based code for the case of 2D plane–strain configurations. We choose an implicit approach in all the simulations by setting the time integration parameter as \(\Upsilon\)=1. A simple h-refinement strategy is employed to re-fine first-order 3-noded elements by adding an extra node on the longest boundary of the element using the built-in MATLAB function “refineMesh” in the PDE toolbox. The algorithm goes through several cycles of refinement until the condition \({H}_{\text{el}}<{l}_{0}/4\) is satisfied for the elements in which the nodal values of damage are (\(d<0.7\)). It is noticed that the coupled u–p formulation and the damage evolution model are solved after each cycle of performing mesh refinement over the domain.

To simplify the inherent complexities of the HF propagation in reservoirs, the analytical KGD solution was proposed and developed in (Zheltov 1955; Geertsma and De Klerk 1969) to calculate the fluid pressure, width, and length of the HF based on the equilibrium of viscous forces of a Newtonian fluid between two parallel surfaces. The KGD solution proposes the following equations, assuming an elliptical cavity within an elastic continuum in plane–strain configuration:

In the above solution, \({l}_{\text{HF}}\left(t\right)\) is the HF’s half-length, and \({P}_{\text{HF}}\left(t\right)\) is the fluid pressure inside the HF during the propagation with respect to time, and the HF's width \({w}_{\text{HF}}\left(t\right)\) is the ellipse’s diameter on the axis perpendicular to the direction of propagation. The analytical KGD solution has been commonly used to validate the numerical simulation of the HF propagation using discrete fracture methods (Dahi Taleghani 2009; Xie et al. 2016). To compare the numerical results of our phase-field HF model to the KGD solution, we use the material properties in Table 1, similar to the values chosen in (Xie et al. 2016), and the boundary value problem is shown in Fig. 7. The critical energy release rate in our simulations is chosen based on LEFM-based relationship \({G}_{c}={K}_{\text{IC}}^{2}/E^{\prime}\).

To highlight the dependency of the results on the length-scale parameter (\({l}_{0}\)), simulations are done for three different values of \({l}_{0}\) equal to 0.02, 0.03, and 0.04 m, and the results are compared to the KGD solution in Figs. 8 and 9. The validity of the phase-field method for the HF simulation can be seen in Fig. 9a, where the propagation speed is compatible with the solution obtained from the analytical KGD solution. The reduction of fluid pressure inside the HF with the propagation, shown in Fig. 8b, is reasonably matched with the analytical KGD solution, albeit with significant discrepancies for larger values of \({l}_{0}\). According to Fig. 8a, the HF’s width is acceptably in accordance with the KGD solution. Considering the results in Figs. 8 and 9, it can be concluded that reducing the length-scale parameter \({l}_{0}\) (e.g., \({l}_{0}=0.02\)) results in outputting the numerical results that better match the KGD solution. This conclusion is compatible with the theoretical proof made by (Bellettini and Coscia 1994), implying that when \({l}_{0}\to 0\), the elasticity solution of sharp fracture is retrieved. In modelling HF using the phase-field method, the choice of element size also affects the characteristics of the HF, specifically the fracture width (w). Increasing the element size (\({H}_{\text{el}}\)) can result in an overestimate in the calculated value of \(w\) using Eq. (24), which can be inferred from Fig. 10. To clarify the effect of \({H}_{\text{el}}\) on the calculation of \(w\), three simulations are borrowed from Sect. 3.2 and run by setting the element size to \({H}_{\text{el}}\) equal to \({l}_{0}/6\), \(\frac{{l}_{0}}{4},\) and \({l}_{0}/2\). The calculated values of \(w\) for the critical (fully damaged) elements throughout the domain are shown in Fig. 10 for three cases of \({H}_{\text{el}}\) in one specific snapshot of the simulations at \(t\)=1 s. It is seen in Fig. 10c that the calculated values of \(w\) in some elements for the case of \({H}_{\text{el}}={l}_{0}/2\) are significantly higher than the plotted values in the other two cases in Fig. 10a, b; the expected HF’s width at \(t\)=1 s can be seen in Fig. 8a, which matches the majority of \(w\) values plotted in the histograms related to the cases of \({H}_{\text{el}}\) equal to \({l}_{0}/6\), and \({l}_{0}/4\) (Fig. 10a, b). Thus, setting the \({H}_{\text{el}}\) larger than \({l}_{0}/4\) can potentially cause an overestimate in calculating the \(w\) in some of the critical elements.

The results of the formulations presented in Sect. 2.4 to model shear failure is tested by comparing the outputs of this model to the experimental data and the numerical results of the particle flow code of the failure of single-flaw concrete specimens under compression, as documented in (Jin et al. 2017). The boundary value problem is identical to Fig. 5a with \(L=50\text{mm}\), \(H=100\text{mm}\), \(a=10\text{mm}\), and \(b=0.5\text{mm}\). Material properties are the same as those presented in (Jin et al. 2017) which are listed in Table 2. The tensile strength \({\sigma }_{\text{t}}\) is calculated with regard to the compressive strength \({\sigma }_{\text{c}}\) and the internal friction angle \(\varphi\) using the relationship \({\sigma }_{\text{t}}=\left(1-\text{sin}\varphi \right){\sigma }_{\text{c}} /\left(1+\text{sin}\varphi \right)\), recommended by Labuz and Zang (2012). The length-scale parameter is chosen as \({l}_{0}=1 \text{mm}\) by calibrating the model for the case \(\beta =60^\circ\), and it is kept the same for other cases of \(\beta\). The crack patterns taken from our analyses are compared to the experimental observations for two geometries with \(\beta =0^\circ\) and \(\beta =60^\circ\) in Fig. 11. The deformed geometries in Fig. 11c, d, g, and h show that compressive shear failure has been modelled in a realistic way without any interpenetration of two pieces of the broken samples along the diffusive fractures. In fact, the developed model can effectively capture the existing compressive stresses that have developed within the damaged zones. The strain–stress response of the single-flaw specimens with different inclination angles taken from our numerical simulations show a strong compatibility with the experimental data and the results of the discrete modelling of fracture using the particle-flow code (Jin et al. 2017), see Fig. 12a. However, the experimental strain–stress curves show significant discrepancies from the numerical ones which can be related to the potential heterogeneities and the existence of micro-cracks in the concrete specimens. The effect of micro-cracks can be seen in the experimental stress–strain curve with a low tangent stiffness at the beginning of loading, and its gradual increase by the application of the load. The inherent heterogeneity and existence of micro-cracks cannot be captured neither in our numerical model nor in the numerical analysis of Jin et al. (2017). Experimental data indicates that the captured maximum uniaxial compressive stress (UCS) grows as the inclination angle of the embedded flaw in the concrete specimen increases (Jin et al. 2017). Our numerical results in Fig. 12b show the same trend and confirm the reliability of our developed phase-field model in modelling shear failure in geo-materials.

A series of semi-circular bending (SCB) tests have been done by Wang et al. (2018) on hydrostone samples to mimic the propagation of the HF intersected by a cemented NF filled by plaster. The material properties of hydrostone and plaster are detailed in Table 3. In this section, we try to examine the capacity of our model in predicting the mixed-mode tensile and shear failure inside the NF by comparing our results to the data provided in (Wang et al. 2018). We model the interaction between the propagating fracture and the cemented NF without the effect of fluid and compare the outcome to the experimental observations. The SCB test has been introduced by Chong and Kuruppu (1984) as a versatile method to measure mode-I fracture toughness (\({K}_{IC}\)) in rocks based on the maximum load capacity (\({P}_{\text{max}}\)) of the specimen using the relationship:where \(B\) and \(R\) are the thickness and radius of the semi-circular specimen, \(a\) is the length of the initial notch in the centre. The value \({Y}_{I}^{\prime}\) is the non-dimensional stress intensity factor, calculated based on the relationships proposed by Lim et al. (1994) and Kuruppu et al. (2014) depending on the geometry. Having chosen the same geometry (a specific \({Y}_{I}^{\prime}\)) for both analytical solutions and the numerical simulations, numerically predicted values of (\({P}_{\text{max}}/B\)) with respect to (\(a/R\)) are compared to the analytical answers of (Lim et al. 1994; Kuruppu et al. 2014) in Fig. 13a. This comparison is done for three different values of \({l}_{0}\) due to the inherent sensitivity of the phase-field method to this parameter (Gironacci et al. 2018). The strong agreement between the predictions of the developed phase-field model and the analytical solutions ensures the reliability of the model in predicting the fracture propagation and the brittle behaviour of the hydrostone in the first place.

Thereafter, our developed model, presented in Sect. 2.5, is tested by modelling the hydrostone samples containing plaster inclusions representing a cemented NF. Using the geometry shown in Fig. 14a with the dimensions \(R\)=20mm, \(S\)=32mm, \(a\)=4mm, \({w}_{\text{NF}}\)=0.4mm, and \({L}_{\text{NF}}\)=16mm, the material properties presented in Table 3, and assuming \(\varphi =\left({{\sigma }_{\text{c}}-\sigma }_{\text{t}}\right)/\left({{\sigma }_{\text{c}}+\sigma }_{\text{t}}\right)=40^\circ\) and \(c\)=800kPa for the plaster inclusion (cemented NF), the interaction between the propagating vertical fracture and the pre-existing NF are simulated. Increasing the width and the inclination of the embedded NF reduces the value of the fracture toughness obtained from the simulations, according to Fig. 14b, since the overall stiffness of the sample has decreased. This conclusion, based on the ensuing numerical results, is in line with the experimental data. According to Fig. 15, it is found out that reducing the angle of approach (increasing the inclination of the NF) reduces is not in favour of the crossing outcome. The dominant scenarios seen for three approaching angles \(\beta\)=\(90^\circ\), \(\beta\)=\(60^\circ\), and \(\beta\)=\(30^\circ\) are crossing, offset crossing, and diversion, respectively, which are in line with the experimental observations in (Wang et al. 2018). As the vertically propagating fracture approaches an inclined NF, more elements inside the NF region undergo shear failure, because the shearing stresses exceed Mohr–Coulomb strength envelope. As the approaching angle decreases, the shearing mode becomes dominant and causes the development of damage along the NF.

It can be attested from the past studies (Xie et al. 2016; Zhang et al. 2020; Sun et al. 2022) that the approaching angle is the most important factor controlling the interaction outcome, as the crossing outcome usually occurs for nearly orthogonal intersections (Warpinski and Teufel 1987) and diversion is usually the case when the HF approaches a highly inclined NF. According to Fig. 18a, the interaction outcome is always diversion when \(\beta \le 30^\circ\) regardless of the NF’s strength (i.e., \({c}_{\text{int}}\) and \({\sigma }_{\text{t}}^{\text{int}}\)). For intermediate approaching angles (\(40^\circ <\beta <70^\circ\)), increasing the differential stress (\({\sigma }_{\text{H}}-{\sigma }_{\text{h}}\)) would be in favour of the crossing outcome. The interaction outcomes for the case of having high cohesion and high tensile strength of the NF for two different values of NF permeability (\({\overline{k} }_{\text{int}}\)=0.1 mD and \({\overline{k} }_{\text{int}}\)=100 mD) are shown in scatter plots in Fig. 18b, c. Our results are compared to the analytical criterion of (Blanton 1986) on specifying the interaction outcome based on the approaching angle and the differential stress. In this criterion, according to Eq. (1), parameter \(b\) depends on the hydro-mechanical characteristics of the NF, the amount of fluid penetrated the NF. It can be concluded that \(b\to \infty\) if no slippage is expected to happen, in which case the crossing outcome would only occur (Blanton 1986). By increasing the NF permeability (\({\overline{k} }_{\text{int}}\)), a larger area of the NF become pressurised by the fracturing fluid which makes the shear slippage more likely to occur because of reducing the effective resisting shear stresses inside the NF. Therefore, by increasing the NF permeability (\({\overline{k} }_{\text{int}}\)), it would make sense to choose a smaller value for \(b\) to be used in Blanton’s criterion for the sake of comparing our numerical results to this analytical reference. For relatively low values of the differential stress (e.g., 0.5 MPa and 1 MPa), the numbers of the crossing outcomes reduce significantly, especially in the case of having a high-permeable NF. It is also understood that the numerical predictions of crossing/diversion outcome better match the analytical predictions for the higher values of differential stress \({\sigma }_{\text{H}}-{\sigma }_{\text{h}}\). It is evident from Fig. 18b, c that increasing both the approaching angle and the differential stress would help the HF to cross the NF.

The NF permeability (\({\overline{k} }_{\text{int}}\)) is dependent on the hydraulic aperture, the degree of cementation inside the NF, and the confining in-situ stresses (Kear et al. 2017). Having a small hydraulic aperture for the NF is in favour of the HF crossing the NF (Xie et al. 2016), although other factors such as differential stresses and approaching angle have a bigger impact on the interaction outcome. We identify the permeability NF region to be the third important factor affecting the HF–NF interaction outcome. Capturing the crossing outcome becomes more likely by reducing the NF permeability, because the fracturing fluid gets less chance to affect a large area around the NF; therefore, the NF is less vulnerable to shear slippage, which is the main cause for the HF to divert into the NF’s direction. A comparison between Fig. 17a and b confirms that a lower value of \({\overline{k} }_{\text{int}}\) keeps the fluid pressurised region locally around the HF tip, avoiding the diversion. Figure 19 depicts the positive effect of lowering the NF permeability on increasing the possibility of crossing for different values of the approaching angle and the differential stress. Our results are also compatible with the experimental data of (Zhou et al. 2008; Kear et al. 2017) affirming that a low value of the NF’s aperture (NF’s permeability) makes the crossing scenario the dominant mode. The effect of NF permeability appears to play a more significant role when the differential stress is low.

A high degree of cementation in the NF region increases the tensile and shear strength of the NF, which is expected to be in favour of the crossing outcome. In Fig. 20, Crossing/No-Crossing is plotted with respect to different values of the approaching angle and the tensile strength \({\sigma }_{t}^{\text{int}}\) for various cases of the differential stress while keeping a constant value for the NF permeability (\({\overline{k} }_{\text{int}}\)=25 mD). For the case of (\({\sigma }_{\text{H}}-{\sigma }_{\text{h}}\)=2 MPa), increasing the NF strength has no effect on the crossing outcome, and the approaching angle remains the dominant factor. However, the NF’s tensile strength (\({\sigma }_{t}^{\text{int}}\)) comes into play as an important factor as the differential stress increases. The reason of capturing this behaviour can be related to the vulnerability of the NF against shear slippage. The NF tends to slip when touched by the HF, so a lower tensile strength of the cemented NF will worsen the situation and leads to a full diversion of the HF into the NF. Having a mechanically stronger NF is in favour of the crossing outcome in intermediate approaching angles especially when the differential stress (\({\sigma }_{\text{H}}-{\sigma }_{\text{h}}\)) is high.

The parametric studies presented in previous sections were all conducted under a constant fluid injection rate of \({Q}_{\text{inj}}\)=200 m³/s. However, the combination of the fluid injection rate and viscosity can affect the HF–NF interaction and create complex crack-patterns in naturally fractured reservoirs (Chuprakov et al. 2014). Here, we repeat the simulations with a different injection rate of \({Q}_{\text{inj}}\)=500 m³/s, and the relevant results are shown in Fig. 21. Increasing the fluid injection rate seems not to be in favour of HF crossing a high-permeable NF. Increasing \({Q}_{\text{inj}}\) only helps the HF crossing when the NF permeability is low enough, so that the NF will not be affected by the shear slippage mechanism. In more permeable and low-cohesive NFs, increasing the fluid injection rate will significantly reduce the possibility of crossing outcome by causing shear slippage along the NF and changing the stress field around the NF especially in intermediate approaching angles (\(40^\circ <\beta <70^\circ\)). As a result, the effect of fluid injection rate on the interaction outcome must be evaluated with respect to other parameters of the NF, such as the degree of cementation, hydraulic aperture, and the tensile and shear strength of the cemented NF.