1 Introduction

Transport properties of underground reservoirs depend largely on the permeability of geo-formations, which has been receiving much attention in sequestration of CO2, extraction of fluids and isolation of hazardous and radioactive waste. In these applications the host rock is typically characterised by fissures and pores, thus geo-formation is often classified as fissured porous media. The permeability of fissured porous media is generally regarded as a function of the in situ stress, the internal microstructures of the solid matrix, and the complex interactions of fluid flow and volume change of the solid skeleton. Thus, understanding the effect of stress on permeability plays a significant role in quantifying fluid flow in geo-formations.

Permeability is generally defined as the ability of a medium to allow the passage of fluid (Friedman 1977). Experimental investigation is the main avenue for examining the permeability evolution in rock under hydro-mechanical conditions. Current work in this area is primarily confined to the laboratory experimentation, with a few investigations conducted on-site. In laboratory, triaxial compression, extension and hybrid tests are conducted to study the effects of hydrostatic and deviatoric stress and the damage effect on rock mass (David et al. 1994; Zhu et al. 2007). These works reveal that the changes of permeability and porosity strongly coincide with each other. Friedman (1977) and Zhu et al. (2007) highlighted that under mechanical conditions, only the variation of the effective “inter-connected” porosity rather than the total porosity should be taken into account in the evolution of permeability with loading. Nevertheless, it is difficult to address such effective porosity based on current technology, thus the total porosity is still preferred for convenience.

To predict permeability evolution of rock subject to different loading scenarios, many permeability evolution models have been developed. Notable contributions include the network model (Bernabé et al. 2010; Pan et al. 2010; Zhu et al. 1996; Zhu and Wong 1999), the discrete model (Cheng-Haw et al. 2007) and the constitutive model (Bouteca et al. 2000; Crawford and Yale 2002; Gessner 2009; Morris et al. 2003; Pride and Berryman 1998; Rudnicki 2001; Yale 2002). Although these models are proposed taking into account either the microstructures of solid matrix, or the in situ stress, or the variation of porosity, most of these models are not complete, further development is required to precisely predict permeability evolution. For instance, in the network model developed by Zhu et al. (1995) and Bernabé (1991), the shape and size of the fracture space is idealised to the extent that it may not capture the true response of the fracture to fluid flow in large-scale fields. Among these remarkable contributions, one of the most widely accepted constitutive models is the generalised power law, which is formulated in the permeability–porosity space, the log (permeability)–log (porosity) space and the semi-log space (Bernabé et al. 2003; Morris et al. 2003; Zhu et al. 2007). These models are simple and enable engineers to predict the main trend of permeability in general applications. However, they are unable to accurately capture permeability evolution in various loading paths. For example, under low to medium confining pressure, these models are inadequate for modelling permeability evolution where the rock sample undergoes shear-enhanced dilatancy after peak strength point. The possible reason, as pointed out by Morris et al. (2003), is that permeability evolution during shear-enhanced dilatancy results from increased tortuosity as some pore spaces have been increased but also have been isolated. However, the way to directly estimate tortuosity has not been found from current literature. Thus, further modifications to constitutive models are highly required. Some investigators (Vajdova et al. 2004) have only considered the effects of inelastic strain, whereas others (Main et al. 2000; Ngwenya et al. 2003) have only taken into account the axial strain. Although these models make some acceptable predictions, the basic mechanisms of these models are not reasonable or rigorous. Specifically, some deformations are recoverable in general; thus part of the reduced permeability may be recovered after unloading, which has been proved by the experimental work on Westerly Granite (Zoback and Byerlee 1975). Furthermore, shear deformation is regarded as the main role in the evolution of tortuosity; thus it is not accurate to consider the inelastic strain or the axial strain only.

In this paper, a unified and complete constitutive model is proposed to address the stress-induced permeability evolution in fissured porous media subject to various loading paths. This is done by introducing a permeability resistance parameter, which gives more considerations to the effects of tortuosity and fissures’ closure. The proposed model is applied to fissured porous sandstones under a wide range of confining pressure. Comparison between model simulation and experimental results demonstrates that the characteristics of permeability evolution in fissured porous media are captured appropriately in all cases considered.

2 Permeability Evolution Model

It is widely accepted that only the effective porosity (e.g. connected pores and fissures) could provide channels for fluid flow. However, in reality, it is difficult to identify the effective porosity during various deformation processes. Therefore, a simple, but effective, way to address permeability evolution is to analyse the relationship between permeability and the overall porosity. However, there is no universal permeability–porosity relationship that is valid for all fissured porous media because the closure of fissures and the development of tortuosity induced by stress would affect the permeability during deformation. To address the effect of these factors, a unified permeability resistance parameter (\(\tau\)) is introduced to capture the effects of increased fissures’ closure and tortuosity on permeability.

2.1 Permeability Resistance Parameter

For a reference configuration (state A), two configurations, state B and C would be induced corresponding to shear deformation and compaction, respectively. Within this context, τ c and τ t are introduced to quantify the effects of increased resistance to fluid flow due to fissures’ closure (state C) and tortuosity (state B), respectively. Thus, the total permeability resistance parameter is expressed in a simple form:

$$\tau = \tau_{c} + \tau_{t}$$
(1)

Notice that τ c and τ t are merely permeability resistance parameters relative to a reference state (initial permeability \(k_{\text{o}}\) and initial porosity \(\varphi_{\text{o}}\)) and the corresponding changes in porosity and deformation.

2.1.1 Increased Fissures’ Closure

In compression tests, the effective mean stress has been found to be a significant role in controlling permeability evolution. Mean effective stress renders increased closure of fissures, which consequently causes a progressive reduction in the permeability. The consequence of compaction is characterised by dead ends and isolated pore/fissure spaces. Similarly, a certain amount of these fissures would be closed during both shear dilatancy and compaction. The only difference between shear dilatancy and compaction is that shear compaction may be characterised by more closure of fissures than its counterpart. The dilatancy-induced closure of micro-cracks also has been investigated by experimental study and theoretical analysis on Westerly granite (Zoback and Byerlee 1975). The change in the fluid permeability resistance factor due to compaction is in turn related to mean effective stress as:

$$\tau_{\text{c}} = \alpha \left( {\frac{{\sigma^{\prime}_{\text{o}} }}{{\sigma^{\prime}_{\text{oi}} }}} \right)^{\chi }$$
(2)

where \(\tau_{\text{c}}\) is value of the permeability resistance factor due to increased fissures’ closure; \(\alpha\) is the permeability resistance parameter for fissures’ closure; \(\chi\) is the mean effective stress parameter, which is a constant for a certain material; \(\sigma^{\prime}_{\text{o}}\) is the current mean effective stress and \(\sigma^{\prime}_{\text{oi}}\) is the reference mean effective stress.

It is noted that, all the variables in Eq. (2) are non-negative, thus \(\tau_{\text{c}}\) is proportional to the mean effective stress and is inversely proportional to fluid pressure. This in turn captures the enhancement of permeability due to increasing in fluid pressure when maintaining a constant total stress, i.e. fluid injection in deep reservoir.

2.1.2 Increased Tortuosity

In the context of fissured porous media, tortuosity is traditionally defined as the ratio of effective path length through which fluid flows to the shortest straight-line and straight-forward distance, from the ‘flow in point’ to the ‘flow out point’ (Scott 2001). Therefore, tortuosity serves as an additional barrier within the fissured porous media that prevents direct fluid flow along a straight-forward linear path. On a microscopic level, the value of tortuosity depends on the fabric of the fissured porous media. However, it is impossible to identify such fabric in rock materials soundly, even if advanced CT or SEM (scanning electron microscopy) is applied. For macroscopic deformation, a model is generally applied to address the tortuosity based on experimental observations (Zhu et al. 2007).

In triaxial compression tests, when the deviatoric stress increases, shear-enhanced compaction or dilatancy may occur and continue until failure. From a microscopic view, the current fabric within fissured porous media changes significantly upon shear deformation. Specifically, shear deformation may distort current flow paths, thus increasing the barriers to fluid flow. For a macro-scale solution, without taking into account the increased tortuosity of every single possible flow path, an average permeability resistance \(\tau_{\text{t}}\) is introduced to address the effect of the increased overall tortuosity induced by shear deformation. Based on experimental observations (David et al. 1994; Zhu et al. 2007), it is reasonable to define the increased permeability resistance \(\tau_{\text{t}}\) as a function of the shear strain.

$$\tau_{\text{t}} = \beta (\varepsilon_{\text{q}} )^{\varsigma }$$
(3)

where \(\beta\) is the permeability resistance parameter for tortuosity, \(\varepsilon_{\text{q}}\) is the total shear strain, and \(\varsigma\) is the shear strain parameter, which is a constant for certain materials. Shear strain in conventional triaxial tests is defined as \(\varepsilon_{\text{q}} = 2\;\left( {\varepsilon_{1} - \varepsilon_{3} } \right)/3\), which is back calculated through axial strain and volumetric strain; where \(\varepsilon_{1}\) and \(\varepsilon_{3}\) are axial and lateral strains, respectively. All the variables in Eq. (3) are non-negative.

2.1.3 Permeability Resistance Function

To elucidate the effects of the mean effective stress and shear deformation on the permeability evolution, permeability resistance function is developed to estimate the contribution of increased fissures’ closure and tortuosity to the evolution of permeability.

$$\tau = \alpha \left( {\frac{{\sigma^{\prime}_{\text{o}} }}{{\sigma^{\prime}_{\text{oi}} }}} \right)^{\chi } + \beta \left( {\varepsilon_{\text{q}} } \right)^{\varsigma }$$
(4)

2.2 Permeability Evolution Model

A complete permeability evolution model is developed, which introduces a permeability-resistant parameter (\(\tau\)) to quantify the effects of fissures’ closure and tortuosity on permeability.

$$k = k_{\text{o}} \left( {\frac{\varphi }{{\varphi_{\text{o}} }}} \right)^{Z} \left( {\frac{1}{\tau }} \right)^{\xi }$$
(5)

where \(k\) and \(k_{\text{o}}\) are the current and reference permeabilities, respectively; \(\varphi\) is the current porosity and \(\varphi_{\text{o}}\) is the reference porosity; \(Z\) and \(\xi\) are the parameters of permeability and permeability resistance, respectively. The second term of Eq. (5), \(\left( {{\varphi \mathord{\left/ {\vphantom {\varphi {\varphi_{\text{o}} }}} \right. \kern-0pt} {\varphi_{\text{o}} }}} \right)^{Z}\), along with the conventional permeability–porosity approach (Bernabé et al. 2003; Morris et al. 2003; Zhu et al. 2007), addresses the effect of the total porosity on permeability, which means that the change in porosity includes the increased porosity induced by the fracturing of new micro-cracks, as well as the decreased porosity caused by pore collapse. The last term, \(\left( {{1 \mathord{\left/ {\vphantom {1 \tau }} \right. \kern-0pt} \tau }} \right)^{\xi }\), captures the effects of fissures’ closure (\(\tau_{\text{c}}\)) and tortuosity (\(\tau_{\text{t}}\)) on permeability evolution.

3 Parameter Identification

The main advantage of this model is that all model parameters can be calibrated based on experimental data. In the equation of permeability resistance (Eq. 4), \(\alpha\) and \(\beta\) are material constants, \(\chi\) and \(\varsigma\) are the mean effective stress and the shear strain parameters, respectively. The permeability resistance parameter \(\xi\) is a material constant, can be calibrated based on experimental observation. The permeability exponential parameter \(z\) is suggested to be 3 to maintain a unified relationship between permeability and porosity.

In the permeability evolution equation (Eq. 5), reference porosity (\(\phi_{\text{o}} = {{e_{\text{o}} } \mathord{\left/ {\vphantom {{e_{\text{o}} } {\left( {1 + e_{\text{o}} } \right)}}} \right. \kern-0pt} {\left( {1 + e_{\text{o}} } \right)}}\) or initial void ratio \(e_{\text{o}}\)) is obtained through some physical testing methods, i.e. grinding powder and liquid infiltration tests. Current porosity (\(\phi = {e \mathord{\left/ {\vphantom {e {\left( {1 + e} \right)}}} \right. \kern-0pt} {\left( {1 + e} \right)}}\)) can be calculated based on volumetric strain (\(\varepsilon_{\text{v}} = \varepsilon_{1} + 2\varepsilon_{3}\) in triaxial test), which can be predicted by some constitutive models, i.e. bounding surface model (Ma 2014) and single hardening model (Lade and Kim 1995). The solids in fissured porous media is assumed to be non-deformable in general, and the volumetric change mainly comes from the change in pore space: \(\varDelta e = \left( {1 + e_{\text{o}} } \right)\varepsilon_{\text{v}}\), thus current porosity can be expressed as:

$$\phi = \frac{e}{1 + e} = \frac{{e_{\text{o}} - \left( {1 + e_{\text{o}} } \right)\varepsilon_{\text{v}} }}{{1 + \left[ {e_{\text{o}} - \left( {1 + e_{\text{o}} } \right)\varepsilon_{\text{v}} } \right]}}$$
(6)

Thus, the shear strain in Eq. (4) also can be back calculated with available data of volumetric and axial strains predicted some constitutive models (Lade and Kim 1995; Ma 2014).

In isotropic compression tests, there is no shear deformation, thus the permeability resistance parameter (\(\tau\)) is contributed by \(\tau_{\text{c}}\) (Eq. 2) only, the value of which is not less than 1 due to the nature of the multiplicative function. Thus, in the initial stage, \(\tau_{\text{c}} = 1\), and \(\sigma^{\prime}_{\text{o}}\) (the current mean effective stress) is equal to \(\sigma^{\prime}_{\text{oi}}\) (the reference mean effective stress), yielding the material parameter \(\alpha = 1\). The value of exponential parameter \(\chi\) is recommended to be 1 for the purpose of simplicity, which is also consistent with the conventional approach of permeability–effective mean stress relationship in literature, such as Rice (1992) and David et al. (1994).

In direct shear test, the main contribution of permeability resistance comes from the increased tortuosity. Similarly, the expression of tortuosity is a multiplicative function; the value of Eq. (3) is no less than 1 as defined. Thus, \(\tau_{\text{t}}\) is 1 at the initial stage and may be extreme large in some ultimate cases. For the lower band, tortuosity is likely to increase when the current shear strain \(\varepsilon_{\text{q}}\) goes beyond a threshold value \(\varepsilon_{\text{qt}}\), which meets the initial state condition as:

$$\tau_{\text{t}} = \beta (\varepsilon_{\text{qt}} )^{\varsigma } = 1$$
(7)

For ultimate conditions, the tortuosity values may be on the order of 100–1000, according to the numerical results on general porous media (Scott 2001). However, these extreme conditions may not be the representative of all fissured porous media, as some mediums with high porosities would have lower average tortuosity. Thus, based on the numerical investigation of general porous media conducted by Scott (2001), the upper limit of tortuosity for fissured porous media is assumed to be 90 when an ultimate shear strain \(\varepsilon_{\text{qu}}\) is approached.

$$\tau_{\text{t}} = \beta (\varepsilon_{\text{qu}} )^{\varsigma } = 90$$
(8)

Considering Eqs. (7) and (8),

$$\left( {\frac{{\varepsilon_{\text{qu}} }}{{\varepsilon_{\text{qt}} }}} \right)^{\varsigma } = 90$$
(9)

After studying some experimental observations (Zhu et al. 2007), it is reasonable to assume that the value of ultimate shear strain is around 20 times the shear strain threshold value, indicating \(\varsigma = 1.5\) for sandstones. Also, the threshold value of shear strain for most fissured porous rocks is around 0.6 %, thus, the material parameter for tortuosity \(\beta\) is back calculated to be 2151 for sandstones through Eq. (7). For the purpose of simplicity, \(\beta\) is suggested to be 2000 in this study. These model parameters analysed above are listed in Table 1.

Table 1 Model parameters in this study
Full size table

4 Model Validation

To examine the permeability evolution model, simulation result is compared with experimental data for sandstones under both hydrostatic and triaxial compressive tests from Zhu and Wong (1997). Darley Dale and Berea sandstone samples were cored perpendicular to the bedding, and Adamswiller sandstone samples were cored parallel to the bedding. The shape of the sandstone samples is cylindrical, with diameter being 18.4 mm and length of 38.1 mm. All rock samples were fully saturated. A fixed pore water pressure of 10 MPa was applied, and the strain rate was controlled at a constant value of \(5\; \times \;10^{ - 5}\) (Zhu and Wong 1997). The permeability along the loading axial was measured, and permeability versus porosity/deviatoric stress was plotted together to examine the proposed model. The model simulation results are represented by lines, whereas the experimental data are plotted by discrete symbols.

4.1 Isotropic Tests

The initial porosities (under 3 MPa effective confining pressure) for Berea, Darley Dale and Adamswiller sandstones are 21, 14 and 23 %, respectively. All experiment results, including axial strain, volumetric strain, mean effective stress and permeability were reported in the literature (Zhu and Wong 1997). The permeability resistance parameter \(\xi = 2/3\) is adopted for all sandstone samples.

Figure 1 shows that the model simulations agree very well with experimental results. Under isotropic compressive loading, the existing fissures tend to close, and this process is also characterised by the reduction of both porosity and permeability (Fig. 1a). As a result, some equivalent channels may be blocked by these closures. Accordingly, the permeability decreases as the porosity decreases (Fig. 1b). As aforementioned, there is no unique relationship that can be identified for permeability–porosity, which is due to the effects of closure of fissures on the permeability evolution. The proposed model takes into account such effects at macroscopic level, thus the model is able to capture the stress-induced permeability evolution of fissured porous media under isotropic test.

Fig. 1
figure 1

Comparison of the model simulation and experimental data for hydrostatic tests for Berea, Darley Dale and Adamswiller sandstones: a permeability–mean effective stress, and b permeability–porosity

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4.2 Triaxial Tests

Figures 3, 4, 6 and 7 illustrate the comparison between the model simulation and experimental data in triaxial tests from the literature: the variation of permeability–deviatoric stress (Figs. 3, 6) and the relationship between permeability–porosity (Figs. 4, 7). All experiment results, including axial strain, volumetric strain, shear stress, mean effective stress and permeability were reported in the work of Zhu and Wong (1997). The conventional porosity–deviatoric stress and shear strain–deviatoric stress were also plotted to help understand the behaviours of rocks, see Figs. 2 and 5. The permeability resistance parameters \(\xi\) used in this analysis are: 0.75 for Darley Dale sandstone, and 1 for Berea sandstone.

Fig. 2
figure 2

Experimental data: porosity–deviatoric stress and shear strain–deviatoric stress plots for Darley Dale sandstone under triaxial test [after Zhu and Wong (1997)]

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4.2.1 Darley Dale Sandstone

Figure 2 shows the experimental results of Darley Dale sandstone in triaxial tests: porosity–deviatoric stress and shear strain–deviatoric stress plots. Rock sample experienced shear-enhanced compaction under high confining pressures (100–200 MPa) or dilatancy under low confining pressures (5–20 MPa). Figures 3 and 4 demonstrate that permeability evolution is reasonably captured by the presented model. Specifically, under low confining pressures (Fig. 3a: 5–20 MPa), rock samples undergo shear-enhanced compaction in the first stage, where permeability reduces along with reduction of porosity (Fig. 4a). Then rock samples enter into the second-stage shear-enhanced dilatancy when continuing loading, this stage is typically characterised by increasing porosity. This increase in porosity, against the general law of permeability–porosity relationship, does produce decrease instead of increase in permeability, which has not been captured by most permeability evolution models. The continuing reduction in permeability is captured by the proposed model through taking tortuosity and fissures’ closure into consideration when shear deformation occurs. Under high confining pressures (Fig. 3b: 100–200 MPa) deformation is featured with shear-enhanced compaction, the permeability keeps decreasing with the reduction of porosity (Fig. 4b). A unique relationship between permeability and porosity/deviatoric stress is obtained through considering the shear deformation-induced tortuosity and compaction-induced closure of fissures.

Fig. 3
figure 3

Comparison of model simulation and experimental data for Darley Dale sandstone, permeability versus deviatoric stress: a p c = 5, 10, 20 MPa and b p c = 100, 200 MPa

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Fig. 4
figure 4

Comparison of model simulation and experimental data for Darley Dale sandstone, permeability versus porosity: a p c = 5, 10, 20 MPa and b p c = 100, 200 MPa

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4.2.2 Berea Sandstone

Berea sandstone samples were tested under a wide range of confining pressures: low pressures (p c = 5, 10 MPa), a medium pressure (p c = 40 MPa) and high pressures (p c = 160, 250 MPa); the corresponding deformations are brittle, brittle–ductile transition and ductile, respectively (Fig. 5). Applying deviatoric stress, initially porosities in all rock samples dropped, then reduction in porosity had been observed for rocks under medium to high pressure, while dilatancy for low-pressure cases. Model simulations match the experimental data appropriately for all cases (Figs. 6, 7). Similar to the case of Darley Dale sandstone in Sect. “Darley Dale sandstone”, a unique relationship between permeability and porosity/deviatoric stress is obtained for rock samples under different confining pressures. With increasing in shear deformation-induced tortuosity and compaction-induced closure of fissures, permeability decreases with both decreasing of porosity under shear-enhanced compaction and increasing of porosity under shear-enhanced dilatancy.

Fig. 5
figure 5

Experimental data: porosity–deviatoric stress and shear strain–deviatoric stress plots for Berea sandstone under triaxial test [after Zhu and Wong (1997)]

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Fig. 6
figure 6

Comparison of model simulation and experimental data for Berea sandstone, permeability versus deviatoric stress: a p c = 5, 10 MPa and b p c = 40, 160, 250 MPa

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Fig. 7
figure 7

Comparison of model simulation and experimental data for Berea sandstone, permeability versus porosity: a p c = 5, 10 MPa and b p c = 40, 160, 250 MPa

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Note that, for Berea sandstone under lower confining pressures (Fig. 7a), there is some deviation from experimental date in the shear compaction stage. In general, a dramatic reduction of permeability has been observed soon after increasing effective mean stress, which is mainly due to the closure of fissures. However, Berea sandstone samples under low confining pressures (p c = 5, 10 MPa) behave differently as if the fissures did not close that much compared with model prediction. This slight overestimation mainly comes from the permeability resistance parameter for fissures’ closure (\(\alpha\)) and the mean effective stress parameter (\(\chi\)). Thus, in further study, a large set of experimental data is required to investigate the influence of each model parameter on the behaviour of the model for specific loading conditions. Nevertheless, the main trend of permeability evolution in both shear-induced compaction and dilatancy has been captured. For rocks under a wide range of confining pressures, a unified, small number of model parameters have been applied, which also demonstrates the capability of the proposed model.

5 Conclusions

A unified permeability evolution model is formulated within a conventional permeability–porosity power function to describe the development of permeability induced by stress. The effects of shear deformation-induced tortuosity and compaction-induced closure of fissures have been captured by a permeability resistance parameter. This model contains a small amount number of parameters, which can be calibrated based on conventional experiments. The proposed model is validated through comparison between model simulation and experimental results for sandstones under a wide range of confining pressures. Good performance demonstrates the capability of the proposed constitutive permeability evolution model.