Triaxial Deformation of the Goldwyer Gas Shale at In Situ Stress Conditions—Part II: Viscoelastic Creep/Relaxation and Frictional Failure

Mandal et al. (2022) provide a detailed geological setting of the Goldwyer formation. The following can be summarised from that study:

Mandal et al. (2022a) reported the composition of the various samples of the Goldwyer formation available for this study (Fig. 2). The key aspects of interest here can be summarised as follows:

The low-pressure nitrogen gas adsorption (LPNA) technique at room pressure and a temperature of 77 K were used to characterize the pore space in the available suite of Goldwyer shale specimens (Mandal et al. 2021a). Applied to degassed powder samples of shale, this technique yields the mesopore volume, the pore size distribution (PSD), and the specific surface area value (S_N2) of pores in the size range of 2–100 nm (Kuila 2013). A Micromeritics^(R) TriStar II PLUS instrument (Fig. 3b) is available to conduct these measurements on powdered specimens prepared following the ISO 9277:2010 standard, i.e., rock samples are crushed into powder with grain size less than 450 µm (< 40 µm mesh size). Prior to the measurement, 1 g of powder is degassed and dried at 105 °C over 12 h (degas setup in Fig. 3a) to remove adsorbed, capillary, and clay-bound water without modifying the internal clay structures (Kamruzzaman et al. 2019; Kuila and Prasad 2013; Yuan et al. 2019). Within the apparatus, the dry powdered shale sample is cooled down to 77 K, and the volume of N₂ gas adsorbed by the specimen is measured for a range of relative equilibrium pressures P/P₀ (0.1–0.99) along the corresponding isotherm, where P and P₀ are the absolute equilibrium pressure and the nitrogen condensation pressure, respectively (Zou et al. 2018).

In practice, the pressure in the test chamber is increased until the condensation peak pressure P₀ is reached. It is then gradually reduced to generate a desorption isotherm. Classification of isotherms (Type I–IV) according to the International Union of Pure and Applied Chemistry (IUPAC) definition is described in great length by Rouquerol et al. (1998). For a purely mesoporous material, a hysteresis loop is expected (Kuila and Prasad 2013), which is associated with capillary condensation and evaporation mechanism occurring within mesopores (equivalent to Type IV isotherm). An important feature occasionally observed in the recorded hysteretic sorption/desorption curves is the forced closure (discontinuities) of the desorption curve at a P/P₀ value of ~ 0.4 (Iqbal et al. 2021; Kamruzzaman et al. 2019; Kuila and Prasad 2013) which is due to the instability of the hemispherical meniscus of nitrogen vapor during capillary evaporation in pores. These closure features are indicative of the presence of pores with a size < 4 nm (Kuila 2013). The SSA and PSD are determined by inverting adsorption isotherm using the BET (Brunauer, Emmett, and Teller) theoretical model and BJH (Barrett, Joyner, and Halenda) model, respectively (refer to following papers to know more about the underlying principals and assumptions of BET and BJH inversion techniques—Brunauer et al. 1938; Rouquerol et al. 1998; Sing et al. 1985). The built-in software platform within Micromeritics^(R) TriStar II PLUS instrument inverts the adsorbed isotherm data to supply SSA and PSD.

The PSD is derived from the inversion of the LPNA adsorption curve by assuming cylindrical pore geometry. Based on the model’s assumption, pore size refers to the diameter of an equivalent cylindrical pore in this study. It is necessary to account for the dependency of adsorbed layer thickness on P/P₀ when inverting adsorbed data to obtain pore volumes. The algorithm inverts measured isotherm data using Kelvin’s equation. The technique cannot evaluate pore diameter ranges below 2 nm since Kelvin’s formula is invalid in micropores (Kuila and Prasad 2013). Since PSD is confined within mesopores range as per IUPAC definition (Micropore: pores < 2 nm; mesopores: pore between 2 and 50 nm; macropores: pore > 50 nm), it will be interesting to analyse what percentage of mesopores contribute to total porosity of the examined gas shale samples.

The semi-quantitative assessment of SSA derived from the LPNA corresponds to the amount of nitrogen molecules required to cover the specimen’s mineral surfaces, including its external surfaces, and the walls of the macro- to mesopores (Kuila 2013; Zou 2019).

Creep tests were carried out on cylindrical shale samples during the multistage triaxial tests (MSTs) using CSIRO’s Autonomous Triaxial cell (Fig. 4a) at confining pressure (Table 2) consistent with in situ stress at the samples’ native depth (Mandal et al. 2022a). The available samples were unpreserved (dry) since their recovery from the Theia-1 well. The MSTs were conducted at room temperature and in dry/drained conditions. No re-saturation of the samples or control of the pore pressure was attempted to (i) avoid shale swelling/shrinking associated with clay hydration and chemoporoelastic effects; (ii) avoid pore pressure build-up and (extremely) slow equilibration/drainage times during triaxial loading; and (iii) focus on gas shales, which are often only partially saturated with formation water (as opposed to fully saturated sealing shales/caprocks). Following the recommendations of the International Society of Rock Mechanics (ISRM), cylindrical shale specimens were prepared with a length-to-diameter ratio of two and flat and parallel end faces. They were cored either parallel (horizontal, 25 mm in diameter) or perpendicular (vertical, 38 mm in diameter) to the visible bedding plane (Fig. 4b).

Creep deformation was recorded during the fourth loading stage of each MST (Fig. 5), while the shale sample was held under a pre-defined and constant triaxial stress, i.e., differential stress of 30–40 MPa, depending on the native depth of each sample, which corresponds to the addition of mean effective stress p_c defined as p_c = (S_v + S_hmin + S_Hmax)/3 − P_p calculated from a 1-D geomechanical model of Theia-1 (Table 2) (Mandal et al. 2020b) and 40% value of peak stress. This triaxial state of stress was achieved by increasing the differential axial stress at a rate of 2 MPa/min while maintaining the in situ confining pressure constant. In all creep tests, we observe that most of the strain occurs instantaneously upon application of the differential stress (elastic response). Once the target differential stress is achieved, it is maintained constant, while creep strains (axial and radial) are recorded for about 6 h (primary creep stage). The limitations of creep deformation are covered in great length by Mandal et al. (2021a).

As observed in Fig. 5b, the radial strain remains virtually unaffected during the axial creep of the sample over the six hours of monitoring. In other words, time-dependent deformation occurs in a direction orthogonal to the maximum principal stress direction, i.e., triaxial loading direction. This is observed for all tested samples, regardless of their orientation with respect to bedding, i.e., vertical, and horizontal.

For a few samples, we also conducted a stress relaxation test at the pre-defined triaxial stress, i.e., time-dependent stress change at zero displacement/constant strain. The aim is to assess the reciprocity between creep and relaxation parameters and validate the use of linear viscoelasticity for our shale samples. The relaxation test is achieved in a similar way to the creep test, except that once the target differential stress is achieved, the loading actuator is halted and locked in place so that the axial displacement is zero, while the evolution of the axial stress is recorded.

At the last (fifth) loading stage of each MST (confining pressure stage—1.25 × p_c as outlined in Table 2), the sample is brought to failure (peak stress), and beyond to record the post-failure response up to a total axial strain of about 4% (Fig. 5a). The combined stress–strain data from the last stage of each MST was used to characterize the frictional failure properties of each sample, i.e., shear stress, effective normal stress, residual strength, and sliding friction coefficient µ_s (covered in Appendix 3).

The N₂ sorption isotherm curves obtained for our suite Goldwyer shale samples are shown in Fig. 6 as the volume of adsorbed nitrogen per unit mass of test material versus the relative pressure P/P₀ in the test chamber. Overall, the clay and organic-rich samples with the highest TOC (sub-group 1: samples Th7, Th9, and Th10) (Fig. 6b) exhibit larger amounts of adsorbed gas than the carbonate-rich samples (sub-group 2: samples Th4, Th5, and Th8) (Fig. 6a). We observe a so-called forced-closure discontinuity mostly in the desorption isotherms of the clay and organic-rich samples at around 0.4 P/P₀ (Fig. 6b). This behaviour suggests that in these samples, the volume of pores < 4 nm is significant. At high relative pressures (P/P₀) in the range of 0.98–1, the sorption curves of clay and organic-rich samples indicate a higher volume of larger mesopores.

The overall outcomes of the LPNA technique applied to our Goldwyer shale samples are shown in Fig. 7 and summarised in Table 3 in terms of total porosity, pore size distribution, and specific surface area S_N2, where the pore diameter D is comprised in the range 2–100 nm. Total porosity is computed from the density porosity equation with grain density coming from laboratory helium gas pycnometer measurement, while bulk density from the volume-mass relationship of cylindrical sample and  with an assumed fluid density of 1 g/cm³.

The pore size distribution is defined as the pore volume per unit mass of material versus (the decimal logarithm of) the pore diameter D (Fig. 7a). Overall, the pores exhibit a bi-modal size distribution with two noticeable peaks: (i) smaller mesopores narrowly centred around 2–3 nm; and larger mesopores more broadly centred around 20–30 nm. The total volume of pores in the investigated range of 2–100 nm is essentially composed of the larger type of mesopores (20–30 nm). As expected, we also observe higher overall porosity values and higher fractions of the smaller mesopores in the organic-rich samples (Th7, Th9, and Th10) compared to the organic-lean samples (Th4, Th5, and Th8). In Fig. 7b, the correlation of total porosity and mesopore volume indicates a linear positive trend. It is evaluated from Table 3 that on average more than 50% contribution in total pore space comes from interpreted mesopores in the Goldwyer gas shale formation.

Figure 8a shows a representative set of creep curves recorded for vertical and horizontal samples of the Goldwyer shale for six hours after the application of an average differential stress of 40% of peak stress at a constant confining pressure in the range of 14–17 MPa replicating the conditions at depth (Table 2).

Laboratory creep strain data were fitted with a linear regression function in the log₁₀(t)–log₁₀(J) space, corresponding to a power-law of time with the compliance B (y-intercept) and time-dependent exponent n (slope) as the constitutive parameters of creep to be identified (see example in Fig. 8b). The resulting creep parameters for all Goldwyer shale samples are shown in Fig. 9 and summarised in Table 4. Figure 9a plots B against n, for both vertical and horizontal samples, color-coded with the weak fraction, i.e., clay content, total organic carbon (TOC), and porosity fractions summed. Figure 9b plots the comparison (cross-plot) between the static Young's modulus (tangent from axial stress–strain curve in the range of 40–60% of differential peak stress) derived from the triaxial loading part of the test and 1/B derived from the fitting of the subsequent creep data. These results suggest that: (i) weaker samples (higher weak fraction, clay- and organic-rich—sub-group 1) tend to display higher values of B and n than carbonate-dominated samples (sub-group 2); and (ii) E and 1/B are nearly equal, with a high confidence coefficient of R² = 0.93. It is noticeable that vertical samples deformed orthogonal to bedding tend to display higher values of B and n compared to horizontal samples. Also, based on additional stress relaxation tests on a few samples, the viscoelastic parameters B and n derived independently from creep and relaxation data vary within 2%, which is consistent with prior results published on US gas shale formations by Sone (2012), and supports the hypothesis that the Goldwyer shale samples can be treated as viscoelastic under the in situ testing conditions replicated in the laboratory. These samples also exhibit anisotropic nature in their viscoelastic responses (Th5 sample in Fig. 8a; Table 4 and Fig. 9a vertical samples have relatively higher values of B and compared to horizontal samples,) equivalent to their observed elastic anisotropy (Mandal et al. 2021a, b).

Existing methods for estimating the horizontal stress components S_hmin and S_Hmax at depth often refer to Eaton’s approach (Eaton 1969; Thiercelin and Plumb 1994). This method predicts the horizontal stress components within a subsurface porous formation, a 3D volume with vertical boundaries subjected to a horizontal tectonic loading and vertical gravity loading. It accounts for the possible vertical transverse isotropy (VTI) of the formation, and poroelastic effects, i.e.,where E_v (E₃₃) and E_h (= E₁₁) are the vertical and horizontal static young’s modulus, respectively; ν_v (= ν₃₁ = ν₃₂) and ν_h (= ν₁₂ = ν₂₁) are the vertical and horizontal static Poison’s ratio, respectively. Here the vertical direction is defined as X3, while the plane of symmetry (bedding) is horizontal (X1, X2). S_v is the gravity-driven vertical stress; α is considered as isotropic Biot’s consolidation coefficient for anisotropic material; P_p is the formation pore pressure; ε_h and ε_H are the tectonic strains in the direction of S_hmin and S_Hmax, respectively. In the right-hand side of each equation (Eq. 6), the first term describes the part of the horizontal stress generated by gravitational loading in the anisotropic formation assuming vertical transverse isotropy, as well as accounting for poroelastic effects. The second term describes the part of the horizontal stress generated by tectonic loading at the vertical and horizontal boundaries of the domain.

The viscoelastic parameters of the Goldwyer shale derived from laboratory triaxial stress–strain data are used to estimate present-day stress state at depth, accounting for (i) the short-term elastic response; (ii) the primary time-dependent stress and strain response over approximately 6 h; (iii) available geological information about the basin, i.e., interpreted strain rates and duration of tectonic loading). However, we do not explicitly account for complex depositional processes, such as burial, diagenesis, and organic matter maturation.

Considering a constant-rate tectonic strain loading \(\dot{\varepsilon }_{\text{o}}\) at the vertical boundaries of the subsurface domain of interest, the evolution of the stress with time is given by Eq. 19 (Appendix 1), which, in this specific scenario becomes

Given the viscoelastic creep constitutive parameters B and n obtained from laboratory triaxial data, at a prescribed loading rate, the accumulated differential horizontal stress can be calculated over a given time duration t [s], within the described model assumptions, i.e., neglecting complex processes such as uplift and erosion.

For the Goldwyer formation, we use as input a tectonic loading duration of approximately 300 million years (Ma), which is two-third of the total geological age of the formation (440 Ma), during which maturation and diagenesis occurred, i.e., prior to any uplift or erosion event (Ghori and Haines 2006). It is indeed reasonable to assume that the processes governing the present-day mechanical properties of the shale result primarily from the maturation and diagenesis stages, rather than from the uplift and/or erosions episodes. Different gas shale formations may have different geological ages. However, Sone and Zoback (2014a) and Yang and Zoback (2016) have shown that the duration of the tectonic loading in these circumstances is irrelevant unless the geological timeframe of tectonic loading varies by an order of magnitude between reservoirs.

We focus here on gas shale prospects located within an intraplate region (Delle Piane et al. 2015; Ghori and Haines 2006). Relying on the rigidity of the plates over geological time scales provides an approximate upper bound of the intraplate tectonic strain rate of the order of 10^–18 s⁻¹ (Haines 2004; Zoback and Townend 2001). The tectonic strain rate in the Goldwyer formation is derived from the computed 15 MPa horizontal stress difference (S_Hmax‐S_hmin) value of the vertical Theia-1 well in the Goldwyer gas shale (Mandal et al. 2020a). The value of S_Hmax‐S_hmin at a specific depth point of G-III unit is derived from the field measured Leak-off test (LOT) and modelled S_Hmax magnitude via the stress polygon method (Mandal et al. 2020a; Zoback 2010). Laboratory triaxial deformation experiments yield a range of horizontal static Young’s modulus of Goldwyer gas shale between 20 and 35 GPa. The process required an average elastic strain rate of 4–7.5 × 10^–4 s⁻¹ over the inferred geological time of 300 Ma to accommodate the evaluated horizontal stress difference if only elastic deformation is accounted for. Following this computation, we come up with a lower bound of strain rate which varies between 1 and 4 × 10^–19 s⁻¹. Given the bounds of intraplate strain rate, we used a strain rate of 2 x 10^–19 s⁻¹ in the viscoelastic rheology-driven differential stress estimation, which is similar to outcomes from previous studies conducted on Barnett shale by Sone and Zoback (2014b) and Yang et al. (2015).

The horizontal differential stress (S_Hmax–S_hmin) at depth accumulated over 300 Ma at an average intraplate tectonic strain rate of 2 x 10^–19 s⁻¹ is shown in Fig. 10 as iso-stress contours in the (B, n) parametric space. The viscoelastic parameters B and n derived from the triaxial creep experiments on the vertical and horizontal Goldwyer shale samples are also displayed.

The viscoelastic rheology model predicts horizontal differential stress that reaches up to 15 MPa, which is in a close match with the observed value in the Goldwyer gas shale (Mandal et al. 2020a). The coloured contour lines represent the stress relaxation predictions after the application of a strain step perturbation.

The values shown on the top of the contours of Fig. 10 refer to pure elastic response of the gas shale when intersected horizontal axes at n = 0. It is resulting from the strain step perturbation. A higher magnitude elastic stress is expected for shale with lower values of elastic compliance B. The role of viscous stress relaxation comes into effect with the increment of n values along y-axis i.e., stress relaxation dominance of shale rock relative to the instantaneous elastic response.

It can be noticed that contours are closer to horizontal than the engineering timescale predicted values shown in Appendix 2. This emphasises the influence of creep constitutive component n on calculating current differential in situ stress magnitudes. In the case of n > 0.04, the amount of differential stress retained by rock is between 2 and 5 MPa irrespective of their elastic response.

Since obtaining a continuous profile of mechanical and elastic properties of rocks beyond reservoir intervals largely depend on sonic logs (compressional wave velocity V_p and shear wave velocity V_s), quality control of these logs has utmost importance in mechanical characterization of subsurface rocks.

These logs are required to extrapolate stress magnitudes, mechanical and elastic properties in carbonate-dominated Nita, Goldwyer-II, and Willara formations to observe their variations across lithologies. However, the experimental viscoelastic fitting parameters are mostly confined within G-I and G-III units. To overcome this limitation, the composition of each lithology obtained from Fourier Transform Infrared Spectroscopy (FTIR) technique (Mandal et al. 2020b) was plotted in a ternary diagram (Fig. 11a) along with investigated Goldwyer shale sample’s mineral composition and organic quantity from pyrolysis and XRD analysis.

Lithologies of G-I and G-III units vary across wide mineral compositions spanning quartz, calcite, and clay with more than 50 vol% of clay-rich minerals. Moreover, the composition of the laboratory samples does not effectively cover the low-carbonate to carbonate-rich shale intervals of the Goldwyer formation. To fill that data gap, compositional representative US gas shale dataset is included for which creep constitutive parameters are acquired at similar experimental conditions and their mineralogy dataset are readily available in the literature (Sone 2012; Sone and Zoback 2014b). This additional data aided in obtaining a continuous profile of creep exponent parameter n where continuous depth profile of creep elastic compliance constant B is built from the inverse of advanced sonic log derived static Young’s modulus.

Compressional wave and shear wave velocities were plotted from each lithological unit and laboratory-measured ultrasonic data at in situ condition of Goldwyer shale samples (Fig. 11b). The acoustic velocity ranges of the three limestone formations are within the velocity range of the Goldwyer shale formation. The laboratory ultrasonic velocity covers the whole spectrum of sonic logs. As a result, the mechanical and elastic properties of those limestone formations are within the studied samples from Goldwyer shale, which provides confidence for extrapolation beyond the reservoir intervals. Most of the velocity data clusters are lies around and within brine saturated carbonate and shale empirical trend as reported by (Castagna and Backus 1993; Omovie and Castagna 2019). Datapoints lie in the left section of shale empirical trend line are relevant to the observed organic-rich gas shale trend. No obvious outlier was identified from the sonic log cross-plot, which gives further confidence in its usability for deriving continuous profiles of elastic and mechanical properties beyond the target zone.

The study conducted by Mandal et al. (2020b) reported that the stress state at the Goldwyer shale formation is hybrid (S_v ≈ S_Hmax > S_hmin) with a combination of normal faulting (NF)—strike-slip faulting (SSF) from conventional stress estimation method, while Bailey et al. (2021) proposed a depth-dependent stress regime in the Canning Basin. The differential horizontal stress accumulation can be predicted from Eq. 7 using a constant tectonic strain loading rate over geological timeframe along different lithological intervals of Goldwyer shale formations, although a constant strain rate may be questionable because deformation mostly happened at a non-constant strain rate over geologic time. However, Sone and Zoback (2014a) and Yang et al. (2015) have concluded that total strain (ε₀) is more important compared to loading history when evaluating current differential stress magnitudes.

To proceed, viscous stress relaxation and dynamic elastic properties (Young’s modulus and Poisson’s ratio) were evaluated from the quality-controlled cross-dipole sonic logs (Fig. 13b), followed by static-dynamic conversion available for Goldwyer shale (Mandal et al. 2020b) from the chosen vertical wells PE-1 and TH-1. The continuous profile of B was determined from the inverse of horizontal static Young’s modulus (1/E_h), considering the requirement of differential horizontal stress computation which arises from tectonic horizontal strain difference ε_diff = ε_H–ε_h. We have shown in Fig. 9b that static young’s modulus is inversely proportional to elastic compliance constant B of laboratory creep data. Building a continuous profile of n along the depth, extrapolating beyond clay-rich intervals, necessitates sufficient laboratory creep measurement from carbonate-rich intervals. To mitigate data gaps and broaden the statistical significance of creep database and their usability, US gas shales (Haynesville and Eagle Ford) with similar experimental conditions was also included (Fig. 11a). From the empirical relationship (Fig. 12a) between creep constitutive parameters B and n as described by Eq. 8, profile of n was obtained as:

Thereafter, the differential horizontal stress (S_Hmax–S_hmin) accumulated due to tectonic loading was computed from Eq. 7. Finally, the least principal stress magnitude along depth was defined based upon constraining relative variation of in situ stress magnitudes ϕ with depth, where ϕ is defined using Eq. 9 provided by (Angelier 1979)where S_max, S_int, and S_min are maximum, intermediate, and minimum principal stress magnitude, respectively, and ϕ is a constant that varies between 0 and 1 when intermediate principal stress magnitude S_int moves from S_int = S_min to S_int = S_max, respectively. In a physical sense, ϕ refers to the kinematics of slip along faults and can be used to define faulting categories (normal, strike-slip, combination of normal and strike-slip, and others). With the above assumption of a constant value of ϕ in a specific stress regime at depth, S_hmin and S_Hmax can be calculated from Eq. 10 with a viscous stress relaxation approach:

Contours of constant ϕ are presented in Fig. 12b through a stress polygon on the defined stress regime based upon the presence of drilling-induced tensile fracture (DITF), rock strength properties, drilling data, and other components of principal stress magnitudes. It would be fair to consider that kinematics of the sedimentary formation remained uniform since different lithological units are part of it. Uniformity of ϕ with depth is in view of confirming regional crustal strain constraints (Sone and Zoback 2014b; Zoback 2010). The vertical stress S_v is defined with a stress gradient of 23.1 MPa/km (Mandal et al. 2020b).

Despite our focus on viscous stress relaxation on two horizontal principal stress components, experimental creep data on vertical samples showed stress relaxation is also occurring under a vertical and a horizontal stress. The combined outcome of simultaneous differential stress relaxation among all principal stress components changes the stress state towards an isotropic stress condition (S_v = S_Hmax = S_hmin), which accords with a constant stress path shown in Fig. 12b. Since stress relaxation occurs along a constant ϕ, the effects of vertical stress relaxation do not impact spatial constraint on ϕ. From the constrained stress magnitudes at 1500 m depth through the stress polygon, we confine ϕ between 0.5 and 1 in a NF/SSF regime. Given the limitation of rock frictional strength, ϕ = 0.6 honours the upper limit of effective horizontal stress ratio within 3.1–4.3 for frictional coefficients ranging between 0.6 and 0.8, assuming a strike-slip faulting scenario. The viscous stress relaxation model predicted stress profiles are displayed in Figs. 13 and 14 where S_hmin profile is calibrated with regional S_hmin gradient of 18.5 MPa/km in onshore Canning basin and a nearby well leak-off test (LOT) data. A pretty good match is observed at depth point and stable regional gradient trend within clay-rich interval. The continuous profile of B and n honour the stratigraphic layering (track 2 in Figs. 13 and 14 for PE-1 and TH-1 well) and provides more confidence to the predicted model in shale intervals despite limited field measurements. The S_Hmax profile is also confined within the pre-defined range of S_Hmax magnitude obtained from the stress polygon.

Sone and Zoback (2014a) demonstrated that the poroelastic effects had minimal contribution to the creep behaviour of studied gas shales when comparing time-dependent deformation of room-dry and oven-dried Haynesville samples under constant axial stress. Li and Ghassemi (2012), Sone and Zoback (2014a), and Rassouli and Zoback (2018) did not find any dependence of creep on in situ confining pressure ranges. Rybacki et al. (2017) computed primary creep strain response of US shale under 30 MPa differential stress after 3 years using 1-D power-law equation and noticed that the compositional influence on primary creep is of similar order of magnitude as from the temperature and confining pressure influence on Posidonia shale. It is noteworthy that temperature and confining pressure conditions were tested beyond the Posidonia sample’s original in situ stress conditions. The presence of water possibly enhances the amount of creep in the gas shale referred from the studies conducted by Sone and Zoback (2014a) and Almasoodi et al. (2014). The water effects could be effective through induced poroelasticity from pore pressure perturbation, capillary suction, swelling, and microcrack enhancement in highly stressed clay-rich rock with dominant swelling clay materials, such as Smectite and Illite–Smectite (I–S) mixture. In fine-grained mudrocks, the grain size is an order of magnitude smaller and the surface area is an order of magnitude larger compared to siliciclastic rocks (Passey et al. 2010). In the studied clay-rich rocks, which contains mostly Illite, the water retention capacity of clay surfaces played an important role in creep deformation.

To understand the impact of grain rearrangement through pore collapse and the relative motion of grains, additional creep experiments were conducted using Berea sandstone and Savonnerie limestone as a proxy for strong and intermediately strong phase components. The experiment was conducted at room temperature and dry conditions at varying confining pressure conditions. The creep responses are presented in Fig. 15. Previously Sone and Zoback (2014a) and Rassouli and Zoback (2018) showed first-order influence of differential stress on axial creep strain. Our observations are no different from those of additional sample creep responses (Fig. 15a), as well as that from a clay-rich shale sample (Th10). In Fig. 15b, the creep compliance function of Savonnerie limestone can be described by 1-D power-law formulation with relatively small slope in log₁₀J(t)–log₁₀t space, i.e., lower value of power-law exponent n. Before the creep test and post creep deformation, porosity and permeability were also measured to observe the change in grain readjustment due to compaction of pore network of the sample from creep. Presence of large pore network in Savonnerie limestone allowed compaction during creep through reduction of pore volume by ~ 9%. Since the time-dependent behaviour of our studied samples mostly occurred in the axial direction followed by slight compaction, we expect the amount of pore volume (Micro to Meso pores) within and in between clay minerals to directly influence creep response through grain alternation and reduction of pore space contained inside weak phase component of gas shale. As reported in Table 3, most of the mesopore volume is confined within clay-rich samples and have contributed 30–70% of total pore volume. Further, we investigated SSA value S_N2 and its correlation with weak mineral phase and mesopore volume. Most clay-rich rocks have higher SSA which is confirmed through its strong linear relationship (Eq. 11 and Fig. 16a) with ClayTocPHI and therefore indirectly contributing to the interpreted mesopore volume (Fig. 16b). The correlation between S_N2 and weak phase volume ClayTocPHI reads:

Mesopore volume correlates with S_N2 reflecting a higher correlation coefficient follows

Sone and Zoback (2013b) and Kohli and Zoback (2013) emphasized load-bearing framework moved from clastic and carbonate supported to clay and organic matter supported when the amount of clay and TOC content reaches above ~ 30–40 vol%. Relying on that investigation, load-bearing capacity of most of the current investigated shale samples is expected to be controlled by weak phase constituents. Hence, the amount of weak phase becomes a proxy for deformation, especially the time-dependent component.

The surface of the grain is where most chemical reactions and rock mechanical deformations happen over depositional history. Hussaini and Dvorkin (2021) found a complex relationship between specific surface area and porosity in low to medium porosity sandstone and carbonate rocks. Also, Mavko et al. (2020) found the influence of SSA and porosity on the geometric effects of permeability as illustrated by the Kozeny-Carman equation. Although we are investigating the creep and failure frictional behaviour of ultra-low permeable gas shale, we believe SSA may act as a proxy to find any possible inter-relationship with mechanical deformation. The gas shales reservoir rocks are predominantly clay rich and have a large specific surface area (Passey et al. 2010) depending upon clay type (Smectite has the largest total surface area of ~ 800 m²/g while Illite and Kaolinite have surface areas of the order of ~ 15–30 m²/g). We have tried to understand if indirectly derived S_N2 can provide any empirical correlation (Eqs. 13–14) with creep constitutive parameters as follows (see Fig. 16c, d).

It is clear from Fig. 16c, d and Eq. 13–14 that the power-law exponent n and elastic compliance constant B are well  correlated with S_N2.

Given the established relationship between S_N2 and weak phase volume ClayTocPHI, followed by a strong correlation between creep parameters B, n with S_N2, it is evident to recognize how mineral composition and microstructure (here SSA describe the semi-quantitative measure of microstructure) possibly influence the creep deformation of clay and organic-rich shales. These interrelationships testify to the significance of the specific surface area of the studied gas shale reservoir rocks. Therefore, creep parameters n and B can be alternatively estimated from the studied rock formation’s SSA without going through expensive experimental laboratory triaxial tests.

Creep deformation reported in this study both for bedding parallel and perpendicular shale samples of Goldwyer formation was acquired under 1-D consideration, neither reflecting actual far-field boundaries nor the formation pore pressure conditions. It is an indicator of the assumption of a drained creep measurement over a geological time frame owing to small tectonic loading rates. Gas shale reservoir rocks are mostly located under higher temperature and pressure conditions. Samples were received from the core library and then tested in the laboratory. Therefore, exact field conditions cannot be replicated. For example, (i) triaxial tests were conducted at room temperature, whereas reservoir temperature can be above ~ 100 °C; (ii) stress field at depth is anisotropic, whereas tested samples are exposed to axi-symmetry conditions in which two horizontal stress components are equal (S_Hmax = S_hmin = S_h) and vertical stress is always greater than or equal to the horizontal stress (S_v ≥ S_h); (iii) samples were tested in as-received condition at ambient pressure, while gas shales are partially saturated with water. Temperature largely influenced the creep behaviour of gas shales in a similar order as mineral compositions of gas shales reported in the literature (Herrmann et al. 2020; Rybacki et al. 2017). Further, creep constitutive parameters were empirically correlated with quantitatively measured petrophysical and elastic properties. In wellbore stability problems, subsurface rock strength properties, UCS, and friction coefficient (µ) are correlated with V_p and porosity (Chang et al. 2006; Dewhurst et al. 2015). Although there is no physical resemblance between mechanical strength and compressional wave velocity, a strong correlation was reported. Here, we also refer to that strong correlation to help field engineers build subsurface stress profiles from data-driven correlation function as a proxy approach. For example, we found a relationship between creep constitutive parameters n and B with the semi-quantitative assessment of SSA and weak mineral phase fraction ClayTocPHI. To provide statistical significance of the proposed empirical-based equations, we have tested the model’s applicability by analysing r squared (R²) value together with another statistical variable such as rank correlation coefficient and p-value. Spearman’s rank correlation was reported along with their R² value and correlation category in Table 5.

To make a comparison of the viscous stress relaxation model predicted horizontal stress profiles (S_hmin, S_Hmax), two horizontal stress profiles (Figs. 18, 19) of PE-1 and TH-1 wells were computed from Eaton’s modified model (Eq. 6). Anisotropic elastic properties (a vertical and horizontal component of Young’s modulus and Poisson’s ratio) were obtained from cross-dipole sonic logs (Fig. 17) with standard methods (covered in great length by Mandal et al. (2020b) in Appendix 1). We assume an isotropic value of Biot’s coefficient as 1 for simplicity, which may not be an actual representation of the studied gas shale sample. Pore pressure was defined from the organic matter corrected total porosity log (Mandal et al. 2020b) with identified overpressure zones at G-I and G-III units. Horizontal tectonic strain components were kept constant with a magnitude of ε_h = 3E−04 and ε_H = 9E−04 adhering to the regional Canning basin stress study by Bailey et al. (2021).

Two stress profiles derived from viscous stress relaxation and Eaton’s modified model, respectively, are compared in Figs. 18 and 19. Comparison of S_hmin profiles from these two methods produced completely different stress profiles along depth and thus expected a variation of fracture gradient along lithological units as well. Eaton’s modified model S_hmin is mostly uniform while the other model showed a variable S_hmin profile with depth from clay-rich to carbonate-rich formations. S_hmin trends from both models show a relatively small difference in fracture gradients at the boundary between the Nita formation and G-I unit and a significant contrast in fracture gradients when the wells enter Willara limestone from the organic-rich G-III unit. The viscous stress relaxation model predicts a decrease in S_hmin magnitude. This observation indicates that Willara limestone may allow downward fracture propagation, even though this would be less likely using S_hmin calculated from Eaton’s modified model. To demonstrate the influence of stress layering on hydraulic fracture growth, we simulated the hydraulic fracturing process using a commercial software package (MFrac Suite, Baker Hughes).

A 3D planar hydraulic fracture model was used in the prospective G-III unit to quantify the impact of stress layering on its spatial propagation. Mandal et al (2021a, b) demonstrated the application of the viscous stress relaxation model in Perth Basin where direct field data followed lithology controlled S_hmin profile. From the hydraulic fracturing simulation, the authors further showed that the downward propagation of simulated fracture from the perforated shale interval into the shaly-sandstone formation.

The key geomechanical parameters were derived from the available wireline logs, field data, and viscoelastic stress relaxation modelling reported earlier. A constant stress gradient was assumed in each layer of a four-layer geomechanical model (Fig. 20a). Average elastic properties such as Young’s modulus and Poisson’s ratio were computed from sonic measurement followed by dynamic to static conversion through empirical equations. Assuming a homogeneous rock property in each layer (Table 6), we simulated a vertical well configuration at the centre of the lower G-III unit with the following configurations: (i) slick water injection at a rate of 50 bbl/min for 2 h (ii) single perforation cluster with perforation diameter of 0.3 inch and 12 perforations per cluster (iii) proppant concentration of 1 ppg with 40/70 mesh size. Figure 20b displays the fracture geometry and proppant distribution 1 day after the shut-in of the hydraulic fracturing operation.

There are no continuous direct measurements available to validate the stress models expect a single depth LOT data in the Goldwyer shale formation. For US Barnett shale, it has been reported that the hydraulic fracture in the main reservoir interval propagates downwards into a limestone formation (Sone and Zoback 2014b; Yang and Zoback 2016). Propagation of a single fracture in an isotropic stress field shows uniform elliptical growth (Salimzadeh et al. 2019), while the application of stress layering controls vertical and lateral propagation of the fracture depending upon S_hmin contrast at layer boundaries of the perforated interval (Singh et al. 2019). Our previous study in gas shale formations of Perth Basin (Mandal et al. 2021b) reiterated the fact that the fracture propagation and proppant distribution at subsurface intervals are driven by variation of least principal stress at depth. Our simulation shows that lateral hydraulic fracture growth is several times that of its vertical growth (Fig. 20). The fracture propagates in the upward direction where the least principal stress magnitude is lower. The lower layer does not act as a full fracture barrier; it is rather a soft barrier, i.e., allows slight downward growth.

Yang et al. (2015) showed that the viscous stress relaxation model derived S_hmin profile can explain out-of-zone microseismicity, while Eaton’s modified model failed to characterise the observed microseismicity beyond the perforated geologic formation. The uncertainty associated with Eaton’s modified model is substantial, considering the contribution of elastic anisotropy parameters, Biot’s coefficient, pore pressure, and arbitrarily defined horizontal tectonic strains. On the other hand, the advantages of the viscous stress relaxation model are several such as (i) direct utilization of laboratory creep deformation data, (ii) usage of physical rock properties from wireline logs, and (iii) honouring the kinematics condition of stress state within a lithologic interval. Further, Eaton’s modified model suggests a hybrid-faulting regime, while the viscoelastic model confirms a consistent strike-slip faulting regime within the Goldwyer shale formation, similar to the present-day in situ stress variation study conducted in the onshore Canning Basin by Bailey et al. (2021).

We already covered how composition and microstructure can control overall mechanical deformation including creep. Further, we established a reliable correlation between S_N2 and creep constitutive parameters. It was demonstrated that production from ultra-low permeable gas shale is feasible through inducing slip along pre-existing fracture networks by slick water injection. Favourably orientated natural fractures and faults experience instantaneous slip during hydraulic fracturing, but slow slip emanates from long-duration and long-period slip events from regional fault networks (Das and Zoback 2011; Kohli and Zoback 2013). Therefore, it is important to dig further to understand the influence of mineral composition and specific surface area on fault stability and frictional strength. Kohli and Zoback (2013) reported a strong influence of clay framework on the slip stability of several gas shale reservoirs (e.g., Barnett, Haynesville, and Eagle ford). Figure 21a shows a moderate negative correlation between failure frictional coefficient µ_s and weak phase ClayTocPHI as shown in Eq. 15:

However, a rapid decrease in µ_s (from an average of 0.8 to 0.6) value was observed when the weak phase fraction reached above 40 vol%. This behaviour indicates a possible shift from carbonate to a clay-rich load-bearing framework. The correlation between µ_s and S_N2 in Fig. 21b reads a negative trend with a correlation coefficient of 0.66. Equation 16 presents the correlation as

Again, this relationship points to the importance of the specific surface area in analysing intact mechanical properties as well as failure frictional properties of gas shale. A strong positive relationship between µ_s and static Young’s modulus (E) is reported separately for Goldwyer and US gas shales (Eq. 17), equivalent to the observed correlation between mechanical compressive strength and Young’s modulus (E) of intact global gas shales (Mandal et al. 2022a).

These empirically derived equations for Goldwyer and US gas shales allow direct prediction of continuous sliding friction coefficient profiles from cross-dipole sonic logs. Therefore, µ_s profile indirectly helps in identifying possible intervals that are prone to shear slip following hydraulic fracturing operation.