Effect of Pore Pressure on Strain Rate-Dependency of Coal

Coal seams are a great source of natural gas supply to fulfill the current energy demands to a large extent. The optimisation of gas production from coal seams, however, offers significant challenges due to the complex structure of coal and coupled physical processes involved in gas production (Perera and Sampath 2020; Van Eeckhout 1976). Gas production from coal seam is a strong function of coal geomechanical behaviour thus accurate prediction of coal effective stress is important. The effective stress behaviour of coal is often described through dual continuum poroelasticity (Aghighi et al. 2021, 2022; Espinoza et al. 2016; Lv et al. 2021) because of its distinct fracture and matrix pore network. Previous experimental studies have, however, shown apparent viscoelastic deformation of coal under different loading strain rates (Cao et al. 2019; Song et al. 2021; Zhao et al. 2016); the effect of which can significantly influence the stress evolution in coal. Mechanical experiments such as uniaxial and triaxial compressive tests have been used to investigate the strain rate dependency of coal mechanical properties, especially Young’s modulus and uniaxial compressive strength (UCS) (Fan et al. 2020; Xiao et al. 2020; Khan et al. 2022; Zhong et al. 2021). For instance, Fan et al. (2020) used a split Hopkinson pressure bar (SHPB) to conduct uniaxial compression tests on coal to study its dynamic mechanical response under different loading strain rates. Their results showed that UCS and Young’s modulus have apparent strain rate dependency (the faster the loading strain rate, the higher the UCS and Young’s modulus of coal specimens). These observations are consistent with results from Xiao et al. (2020), who performed uniaxial compression tests to study the influence of strain rate on coal mechanical properties and failure process. In their study, Young’s modulus showed a strong strain rate dependency, especially at high strain rates. Recently, Li et al. (2022) carried out triaxial compression tests on coal specimens to study their mechanical properties and deformation under different strain rates and confining stresses. Their results indicated that the stress–strain response and peak strength of coal samples have a clear strain rate dependency. Also, it was observed that an increase in confining stress intensifies the strain rate dependency of peak strength, but it has no consistent influence on the stress–strain response. It is noted that the above-discussed rate dependency experiments were conducted on dry coal specimens.

Despite the research conducted to date on the viscous behaviour of dry coal specimens, the effect of gas pressure on the coal viscoelastic response has received little attention. The presence of gas is ubiquitous in coal and has an important influence on many engineering problems, such as well completion, wellbore stability, and hydraulic fracturing amongst others (Detournay and Atkinson 2000; Helmons et al. 2016). The rock pore pressure variation during loading can affect the pore and fracture system, and in turn affect the strain rate dependency (Al-Bazali et al. 2008; Zhong et al. 2022). The effect of pore pressure on strain rate-dependent behaviour of rocks is in general, however, complex because of the interactions between boundary conditions of the loading process and the hydromechanical response as observed in some fractured shale rocks (Helmons et al. 2016; Swan et al. 1989). The authors were able to find only a few studies in the literature investigating the effect of pore pressure on strain rate dependency of rock samples where the micro-scale physical processes causing macro-scale experimental observations were not studied. Swan et al. (1989) conducted triaxial compression tests on shale specimens with water to investigate their strain rate-dependent behaviour with water pore pressure. Under undrained condition, almost no strain rate dependency on the mechanical properties of shale was observed, especially for strain rates below 0.1 min⁻¹. This behaviour was linked to pore pressure build-up in the specimen due to undrained testing conditions stiffening the specimens and reducing the viscous effect. In drained testing conditions, their results also showed no rate dependency. It was postulated that water does not have enough time to drain during deviatoric loading due to the low permeability of shale and is trapped in the pores causing localised pore pressure build-up (localised undrained response) which leads to the insensitivity of specimen to different strain rates. For the strain rates above 0.1 min⁻¹, the shale strength, however, increased with the strain rate. A similar observation was made by Brace and Martin (1968), who conducted triaxial compression tests on crystalline silicate rocks with low porosity at different strain rates and pore pressures. Their results showed that specimens have trivial rate dependency for the strain rates lower than a critical value; once the strain rate surpasses this critical value, the rock becomes more competent compared to zero pore pressure. Again, the localised pore pressure increase was used as the hypothesis for such strengthening behaviour (Al-Bazali et al. 2008; Brace and Martin 1968). Al-Bazali et al. (2008) conducted triaxial compression tests on shale specimens and developed a theoretical model to study the impact of strain rate on specimens’ mechanical behaviour. In their study, axial loading caused pore pressure build-up under drained conditions due to low shale permeability thus reducing rate dependency of mechanical response. Helmons et al. (2016) used the discrete element method to simulate the rock failure process under different strain rates. In their study, both drained and undrained boundary conditions were considered. Their results indicated that the strength of rock has no rate dependency under undrained conditions when the strain rate is lower than a critical value. These numerical results supported the experimental results of Swan et al. (1989) and Brace and Martin (1968).

The above studies reveal that the rate-dependent behaviour of dry and gas-saturated specimens is relatively different. However, the micro-processes controlling such macro-scale behaviours have not been fully understood nor systematically analysed. This study, therefore, aims to assess the coal rate dependency at dry and inert gas-saturated conditions systematically and shed light on the microscale processes controlling the coal rate dependency especially with gas pore pressure without sorption. To achieve this, we perform a set of triaxial compressive experiments on dry and gas-saturated specimens under different loading strain rates. To exclude the gas adsorption effect, non-sorbing gas (Helium) was used in the experiments. We further design a set of experiments combining X-ray micro-Computed Tomography (XRCT) scanning with hydromechanical testing to investigate the gas trapping in coal specimens under different loading rates leading to localised undrained conditions.

In order to analyse the viscoelastic response of the specimens, the effective stress values should be obtained (\(\sigma^{\prime} = \sigma_{t} - \alpha p\)); the values of which are required for calculating the effective stress coefficient (Biot 1941). Figure 2 shows the results of the solid bulk modulus (K_s) of specimens S1, S2 and S3 (Adelina et al. 2019). Having the solid bulk modulus and bulk modulus, the effective stress coefficient can be calculated using Eq. 2 which in turn enables calculating the effective stress values. It is noted that the bulk modulus of each strain rate and solid bulk modulus of each specimen were used to calculate the effective stress coefficient of the specimen at every test.

The results of rate-dependent experiments on specimen S1 under dry and different pore pressures (0.5, 1, 1.5 MPa) are shown in Fig. 3. Figure 3(a) shows the evolution of effective isotropic stress (at zero pore pressure) versus volumetric strain for different strain rates. It is seen from this figure that stress–strain relations are identical for low strain rates of \({4} \times {10}^{{ - 5}}\) and \({6} \times {10}^{{ - 5}}\) s⁻¹ meaning a negligible strain rate dependency under these strain rates. The slope of effective isotropic stress versus volumetric strain curve represents the bulk modulus of the specimen under different loading strain rates. When the strain rate reaches a threshold (approximately \({6} \times {10}^{{ - 5}}\) s⁻¹) for this specimen (S1), the stress–strain relation starts to show clear rate-dependent behaviour leading to bulk modulus increase with an increase in strain rates. These experimental results on coal specimens under dry conditions are consistent with previous studies (Fan et al. 2020; Xiao et al. 2020).

Figure 3(b–d) further present results of effective isotropic stress versus volumetric strain of specimen S1 at different strain rates for pore pressure of 0.5, 1.0 and 1.5 MPa, respectively. The effect of pore pressure on strain rate-dependent behaviour can be seen by comparing Fig. 3(a–d). The results evidently show that under the same effective stress, the strain rate dependency decreases significantly with the introduction of pore pressure. The specimen under a higher strain rate of \({2} \times {10}^{{ - 4}}\) s⁻¹ still shows some level of rate dependency at 0.5 MPa pressure but such rate dependency vanishes toward higher pore pressures e.g., no strain rate dependency is observed at 1.5 MPa pore pressure. These results confirm that an increase in pore pressure reduces the strain rate dependency of coal specimen, S1.

Figures 4 and 5 show the results of strain rate-dependent experiments on coal specimens, S2 and S3, respectively. For dry conditions, both specimens show similar rate-dependent stress–strain relations where a higher strain rate increases the rate dependency and the bulk modulus of coal specimens. Similar to specimen S1, specimen S3 shows a clear rate-dependency at strain rates above approximately \({5} \times {10}^{{ - 5}} {\text{s}}^{{ - 1}}\). The specimen S2 shows rate-dependency at all axial strain rates under dry conditions. The rate-dependent behaviour of specimens S2 and S3, similar to specimen S1, reduces significantly with an increase in pore pressure. At a pore pressure of 1.0 MPa, no rate dependency is observed for specimens S2 nor S3.

The strain rate-dependent testing of specimens S1, S2 and S3 individually and collectively show that pore pressure at drained condition reduces the rate dependency. Specialized experimental design is required to investigate the physical mechanisms responsible for such behaviour, especially at micro-scale. We thus designed a set of XRCT experiments on specimen, S4 as previously explained. Figure 6 shows a grayscale 3D tomograph of specimen S4 at the initial condition with Krypton gas pressure of 1 MPa and isotropic stress of 1.5 MPa. The lighter gray represents the areas with a higher concentration of Krypton. In the XRCT experiments, the specimen, S4 was fully saturated with Krypton gas at the initial condition and then loading was applied at two different loading speeds (fast and slow) as previously explained.

Figure 7 shows the XRCT results of the experiments on coal specimen (S4) under different loading rates with 1 MPa Krypton gas pressure and 1.5 MPa isotropic stress where (a) represents the difference of grayscale intensity of specimen between images captured at faster loading test and initial condition, (b) the difference of grayscale intensity of specimen between images captured at slower loading test and initial condition and (c) the difference of grayscale intensity of specimen between images captured at faster and slower loading tests. The XRCT images from each two condition were registered to ensure the images can be compared before subtracting the grayscale intensity.

Figure 7 interestingly shows that although the drained condition was maintained throughout the fast and slow tests (isotropic loading), some gas was trapped inside the specimen and caused an increase in grayscale intensity (representing higher gas concentration). From this figure, it is seen that the gas trapped in the specimen has localised higher pressures (higher intensity thus concentration and pressure) when the load is applied to the specimen. Such gas trapping is considerably higher when the loading rate is fast [Fig. 7(a)]. It is intuitive that higher loading rates will close micro-fractures and pores and compress the trapped gas with an increase in volumetric strain. This compression of the trapped gas leads to an increase in pore pressure locally causing localised undrained response. Such an undrained response strengthens the specimen thus reducing the rate-dependent behaviour. Previous studies on sedimentary rocks indicated similar behaviour. For example, Swan et al. (1989) conducted triaxial tests on shale specimens under both drained and undrained conditions. They hypothesized that pore pressure inside the specimen can be higher than the measured pore pressure values during compression even under drained conditions leading to reduced rate-dependency of shale specimens which is supported by our observations.

Our experimental results show apparent rate-dependency of dry coal specimens where the higher strain rates increase the specimen bulk modulus. Previous studies postulated different mechanisms for such rate dependency. One mechanism of strain rate-dependent mechanical properties of rock, for example, is related to the compaction-induced pre-existing micro-fractures closure during the isotropic loading process (Helmons et al. 2016; Liang et al. 2011). The faster the isotropic loading, the faster the pre-existing micro-factures closure causing a strengthening effect on the specimen and increasing its bulk modulus. Another proposed mechanism for such a hardening effect induced by high strain rates is linked to fracture development changing from macroscopic fracturing at low strain rates to micro-cracking at high strain rates (Hou et al. 2019). It is noted that the proposed mechanisms of rate dependency are often related to fracture development in specimens under different loading rates. In this study, 3 MPa effective isotropic (hydrostatic) stress was applied in all tests which are relatively low compared to UCS of specimens (7–8 MPa). In addition, the unloading was followed by each loading stage to monitor any permanent deformation (damage) and ensure only an elastic deformation (no damage) is experienced. This in turn rolls out the effect of micro-fracture generation on strain-rate dependency. Therefore, only the pre-existing fractures are likely to affect the strain-rate dependency.

It can be seen from the experimental results (Figs. 3, 4 and 5) that specimens S1 and S3 show relatively similar rate-dependent behaviour which is different from specimen S2. When looking at the stress–strain results of these specimens, it is observed that specimens S1 and S3 reach around 6.5 × 10^–3 volumetric strain value at 3 MPa effective stress while specimen S2 experiences only 4.5 × 10^–3 volumetric strain value at 3 MPa effective stress magnitude. Knowing that (i) the compressibility of fractures is much higher than matrix, (ii) the matrix compressibility of the three specimens are similar (sister cores) and (iii) the coal specimens are cleated and are under low stresses (< 3 MPa), it can be concluded that the specimens S1 and S3 have similar fracture systems. The porosity measurement of the specimens S1 and S3, however, returned different porosities of 2.57% and 5.46%, respectively, yet these two specimens show similar volumetric strain responses. Considering that the matrix porosity across specimens is similar, the difference in porosity indicates different fracture porosity e.g., specimen S3 has higher fracture porosity than specimen S1. The same volumetric response across two specimens with different fracture porosities, therefore indicates that fracture stiffness should dominate the volumetric strain development. Where the fracture stiffness is a dominant factor controlling the strain, one expects that it should also control the strain evolution with external loading rates.

The results of rate dependency of different coal specimens under dry condition and different pore pressures are evident for this observation (Figs. 3, 4 and 5). This observation cannot be linked to poroelastic response as all experiments were conducted at the same effective stress. Thus, other processes should be involved to observe less rate dependency with higher pore pressure. Al-Bazali et al. (2008) conducted experimental and theoretical studies to investigate the effect of strain rate on the failure characteristics of shale. They postulated that pore pressure build-up can occur in the specimen under drained conditions where the rate of pore pressure build-up depends on the strain rate. Higher the strain rate, the higher the pore pressure build-up both in its rate and eventual magnitude. Other numerical studies on intact (Roshan and Aghighi 2012) and fractured shale (Roshan and Fahad 2012) also showed that localised undrained response can occur in low permeable formations under drained conditions.

Although coal is a fractured sediment and different from shale in terms of permeability, the small-scale cleats/fractures can act as gas-trapping sites under different loading rates. We proved this postulation using our XRCT experiments (Fig. 7) where gas can be trapped inside the specimen during isotropic loading, especially at a faster loading rate. The trapped gas and its associated pressure increment leads to localised undrained response under drained testing conditions. This in turn causes the specimen to strengthen thus reducing the rate dependency. We, therefore, conclude that the interaction of gas and macropores–microfractures is the process controlling the rate dependency in coal samples.

This study investigates the effect of non-sorptive gas pore pressure on strain rate-dependent mechanical behaviour of coal specimens systematically. A series of triaxial experiments with different strain rates were conducted on both dry specimens and specimens saturated with pressured Helium. Another set of experiments combined with the X-ray micro-CT (XRCT) scanning technique was conducted to obtain the gas distribution in the specimen under different loading rates and to assist in understanding the effect of pore pressure on the strain rate-dependent behaviour of the specimens. The experimental results show clear rate-dependent stress and strain relation of coal specimens under dry conditions where the rate-dependency increases with strain rates. The higher strain rates cause an apparent stiffening effect on specimens and increase their bulk modulus. This rate-dependency, however, decreases significantly with the increase of pore pressure. The results of XRCT experiments demonstrate that gas can be trapped inside the coal specimen during isotropic loading, especially at a higher loading speed. This in turn leads to a localised undrained response in an apparently drained experiment causing the specimen strengthening effect and reducing the rate-dependency of coal. The coupled effect of sorption on the viscoelastic response of coal is more complicated and is a focus of our future investigations.