Influence of subsurface injection on time-lapse seismic: laboratory studies at seismic and ultrasonic frequencies

2.1 Pierre shale

Laboratory tests were conducted on a Pierre shale I core plug with a diameter of 1 in. and a length of 2 in. Pierre shale I used in this work is an outcrop shale from North Dakota; it exhibits transverse isotropy (TI) (e.g., Islam and Skalle 2013) and is considered as an analogue for some overburden formations. Tested core plugs were drilled from outcrop material preserved in oil at 0^o angle of inclination with respect to the symmetry axis. X-ray diffraction (XRD) data show that the studied batch contains about 48 wt% clays, 44 wt% silicates (27 wt% quartz), and around 6 wt% of carbonates. Due to the short sample consolidation time (strain measurements shows that, while exposing the core plugs to brine under stress, the strains stabilize within ̴ 53 hours), as-received material is considered as nearly fully saturated. The porosity of the tested sample was not determined directly; however, based on the measured bulk density of as-received material and mineral density of its composites (taken from XRD test), we have estimated a porosity of 16% ± 1%. This result is in good agreement with data reported in the literature ( ̴ 10%–25% porosity) (e.g., Schultz et al. 1980; Olgaard, Nuesch and Urai 1995; Holt et al. 2015).

2.2 Experimental set-up

The experiments were carried out in a low frequency apparatus at the Formation Physics Laboratory of SINTEF Petroleum Research. The apparatus combines three measuring techniques that allow for determination of ultrasonic velocities, quasistatic rock deformations, and dynamic elastic stiffness at seismic frequencies under deviatoric stresses, within a single test. Design of the apparatus, together with an in-house-developed strain gauge fitting/sample mounting procedure, allows for sealing of the core plugs within ̴ 1 hour after removing them from preservation oil, thus most likely providing maintenance of their initial state. A schematic illustration of the experimental set-up is shown in Fig. 1.

The sample stack consists of a loading piston, two endcaps with the sample placed in between them, a holder with a set of three linear variable displacement transducers (LVDTs) measuring displacements in the vertical direction, an aluminium piece with attached semiconductor strain gauges, a piezoelectric force sensor, a piezoelectric actuator, and an internal load cell. The entire stack is placed on the bottom flange of a compaction cell equipped with electronic feed-throughs and fluid channels allowing for independent control of pore pressure and confining pressure. The compaction cell is placed inside the mechanical loading frame that exerts an axial force onto the sample. Measurements at ultrasonic frequencies are performed by the pulse-transmission technique. Two pairs of in-house-built compressional (P) and shear (S) wave transducers are embedded in the top and bottom endcaps allowing for ultrasonic velocity measurements along the sample axis. The excitation frequency used during the experiments for both P-wave and S-wave transducers was 500 kHz (centre frequency of the piezoelectric crystals). The received P-wave velocity signal had an average frequency of about 450 kHz, whereas received S-wave velocity signal had an average frequency of about 320 kHz (Fig. 2(a)–(h)). The picking of the arrivals includes (i) comparing the actual waveforms with the waveforms recorded with aluminium standard, (ii) tracking frequency-related arrival time shifts obtained from waveforms acquired at several frequencies lower than 500 kHz (data recorded immediately after 500 kHz measurements), and (iii) comparing the ultrasonic data with seismic measurements. Quasistatic rock deformations were determined with the use of an internal load cell (measuring deviatoric stress), a set of three LVDTs distributed equally around the circumference of a sample, and a set of four sealed foil strain gauges attached to the sample (measuring axial strain locally). Redundancy in the determination of axial deformation is used for verification purposes or identification of large heterogeneities in the sample. Finally, rock mechanical properties (Young's modulus and Poisson's ratio) at seismic frequencies (1 Hz–155 Hz) are measured with the use of the forced-deformation method described by Spencer (1981). An amplified sinusoidal electric signal is applied to the piezoelectric displacement actuator, which results in oscillatory axial force modulations measured by the piezoelectric force sensor. In order not to exceed the elastic regime of the tested specimen, the sample strains are kept below 10⁻⁶ (e.g., Gordon and Davis 1968; Iwasaki, Tatsuoka and Takagi 1978). Strains are measured locally with the use of eight sealed strain gauges attached to the sample (four measuring strains in axial direction and four measuring strains in radial direction). From the force, F, and strain, ε, modulation amplitudes, both Young's modulus, E, and Poisson's ratios, ν, are calculated according to the following equations:

(1)

where A is the cross-section area of the sample and the indices ax and rad indicate axial and radial direction, respectively. The low frequency experimental errors were quantified with the help of standard materials (aluminium and polyetheretherketone) and are smaller than 5%. More details about experimental set-up and strain gauge fitting/sample-mounting procedures are given by Szewczyk, Bauer and Holt (2016).

2.3 Experimental conditions

According to the linear elastic nucleus of strain model introduced by Geertsma (1973), for an isotropic homogenous sub-surface, the mean stress remains constant in the overburden during reservoir depletion or inflation (constant mean stress (CMS) loading). This implies that the overburden above the centre of the reservoir will experience a decrease in horizontal stress and increase in vertical stress, as a response to injection into a reservoir. At the edge of the storage site, the response is opposite, i.e., an increase in horizontal stress and a decrease in vertical stress. In order to simulate those effects, we have performed an experiment in which a Pierre shale core plug was loaded according to the idealized stress path characteristic for caprock above the centre of a reservoir (Fig. 2(i)). First, the sample was loaded hydrostatically to a 5 MPa confining pressure. Next, confining pressure, axial stress, and pore pressure were increased simultaneously to 19.5 MPa, 21 MPa, and 2 MPa, respectively. Subsequently, after sample has stabilized (which took around 53 hours), a CMS loading cycle was performed to simulate the effect of reservoir inflation. Here, the axial stress was increased by to 5 MPa, and the confining pressure was reduced by 2.5 MPa to keep the mean stress constant. It is known that properties measured during first loading differ from those obtained during subsequent loading cycles (e.g., Walsh 1965). For this reason, a second CMS cycle was performed. After each loading/unloading step, we have allowed the sample to stabilize before conducting low-frequency and ultrasonic measurements (the stabilization time was varying between 4 to 10 hours). After the second CMS loading cycle, the axial stress was gradually reduced to 19 MPa while holding the confining pressure constant at 17 MPa (triaxial unloading). This phase of the experiment was used to determine quasistatic properties of Pierre shale. Finally, to assure that the experiments were not executed close to the material failure, the axial load was increased with simultaneous reduction of the confining pressure until the stress state of 31 MPa axial load and 14.5 MPa confining pressure were reached (not shown in Fig. 2(i)). During this step, the shale did not fail. For all loading/unloading stages, the applied loading rate was equal to 5 MPa/h for all stresses and pressures, which corresponds to axial strain rates between and . All segments of the experiments were conducted under drained conditions (assured by a metal mesh wrapped around the sample and covering about 60%–70% of the sample's lateral surface), with 3.5% NaCl brine as a pore fluid, and at constant pore pressure of 2 MPa. It should be noted, that at the time scale of typical 4D seismic surveys, shale formations in the field are usually considered undrained, which implies pore pressure change. However, during CMS loading, the pore pressure change is usually small, and for the Pierre shale, we can roughly estimate the pore pressure change to be between 10% and 25% of the axial stress change (Skempton 1954; Holt et al. 2015).

2.4 Conversion between properties measured at seismic and ultrasonic frequencies

While velocities measured at ultrasonic frequencies are functions of P-wave and S-wave moduli, the low frequency technique provides a direct measurement of Young's modulus, E, and Poisson's ratio, ν, for a given loading direction. Therefore, in order to compare results obtained at seismic and ultrasonic frequencies, conversion between those different types of moduli is required. A transversely isotropic (TI) material is characterized by five independent coefficients forming the stiffness tensor, C_ijkl, that relates the stress tensor, , and the strain tensor, , of any elastic material (Hooke's law), as follows:

(2)

where Einstein's summation convention applies. The elastic stiffness tensor (assuming that the z-axis is the symmetry axis), can be written as a matrix using Voigt notation (Voigt 1928; Nye 1985):

(3)

with .

The anisotropy of a TI material can be characterized by three parameters introduced by Thomsen (1986):

(4)

(5)

(6)

In conventional compaction cells, where only biaxial stresses can be applied (axial stress and confining pressure) and only axial velocities measured, determination of all five independent stiffness coefficients (C₁₁, C₃₃, C₄₄, C₆₆, C₁₃) requires the repetition of experiments on three differently oriented samples: (i) specimens drilled at 0^o angle of inclination with respect to the symmetry axis, (ii) samples drilled at 90^o with respect to the axis of symmetry, and (iii) samples cut under oblique angle with respect to the symmetry axis. The equations that form the basis for inversion of laboratory data for the five independent stiffness coefficients are as follows (e.g., Helbig 1994; Mavko, Mukerji and Dvorkin 2009):

• a)

  For properties acquired at ultrasonic frequencies:

(7)

(8)

(9)

(10)

(11)

(12)

where ρ denotes bulk density, P and S stands for compressional and shear waves, respectively, and and represent phase velocities of “quasi P-waves” and “quasi S-waves” with wave-fronts normals oriented at angle θ with respect to the symmetry axis. Subscript V denotes the direction of the symmetry axis, whereas subscript H denotes any direction within the symmetry plane.

• b)

  For properties acquired at seismic frequencies:

(13)

(14)

(15)

(16)

(17)

(18)

In the present work, since the tested core plug was oriented at 0^O with respect to the symmetry axis, only vertical (quasistatic, low frequency, and ultrasonic) responses to the applied stress path were measured. However, if the Thomsen parameters and vertical properties of tested specimens are known, equations 4–6, together with equations 7–8 for ultrasonic measurements, and equations 13–14 for seismic frequency measurements can be inverted for the five independent stiffness coefficients characteristic for a given frequency. This procedure leads to two independent solutions. The choice of the physically valid solution may be achieved from TI material energy considerations, which implies the following inequalities (Nye 1985):

(19)

In the following, values of Thomsen parameters representative for our specimen at seismic frequencies were assumed to be identical with the one obtained from Pierre shale field measurements at 950-ft depth performed by White, Martineau-Nicoletis and Monash (1983): , , . At ultrasonic frequencies, Thomsen's parameters were previously measured on samples cut from the same batch of material by Holt et al. (2015): , , . Possible errors associated with this approach will be discussed later.

3.1 Rock mechanical properties

Measured static and dynamic rock mechanical properties of Pierre shale I (vertical Young's modulus, E_V, and vertical Poisson's ratio, ν_VH) obtained during both constant mean stress (CMS) cycles, as well as during triaxial unloading, are shown in Fig. 3 and summarized in Table 1. Data points in the seismic range were measured directly, whereas ultrasonic parameters were calculated from measured P-wave and S-wave velocities. Solid lines in the figure represent a manual Cole–Cole fits (phenomenological model originally developed for the frequency dependence of the complex dielectric constant that has also been used to describe relaxation in a visco-elastic materials) (Cole and Cole 1941). The Cole-Cole model may be written in the following form:

(20)

(21)

where are high and low frequency moduli, τ₀is the relaxation time, and stand for real and imaginary part of complex modulus, ω is angular frequency, and α is a parameter controlling the width of the relaxation time distribution. Open symbols at ultrasonic frequencies represent isotropic conversion from velocities to rock mechanical parameters and were added to illustrate the effect of anisotropy and the error associated with neglecting material anisotropy.

Table 1. List of vertical Young's modulus, Poisson's ratio, and vertical P-wave and S-wave velocities together with their dispersions, V_PV/ V_SV ratios, stress sensitivity and strain sensitivity factors obtained during both CMS loading cycles

	Reference stress		1st CMS loading		Reference stress (repetition)		2nd CMS loading
	Seismic (1 Hz)	ultrasonic	Seismic (1 Hz)	ultrasonic	Seismic (1 Hz)	ultrasonic	Seismic (1 Hz)	ultrasonic
Vertical Young's modulus, EV (GPa)	7.03	9.54	7.25	9.65	7.34	9.62	7.62	9.74
EV dispersion	36%		33%		31%		28%
Vertical Poisson's ratio, (-)	0.394	0.373	0.404	0.373	0.394	0.373	0.4	0.373
Vertical P-wave velocity, VPV (m/s)	2443	2981	2578	3014	2496	2999	2599	3028
VPV dispersion	22.0%		16.9%		20.2%		16.5%
Vertical S-wave velocity, VSV (m/s)	1087	1254	1105	1260	1114	1258	1132	1265
VSV dispersion	15.4%		14.0%		12.9%		11.8%
VPV/ VSV (-)	2.248	2.377	2.333	2.392	2.241	2.384	2.296	2.394
Stress-sensitivity factor, (MPa−1)	−	−	11.5·10−3	2.4·10−3	−	−	8.4·10−3	2·10−3
Strain-sensitivity factor, (-)	−	−	23	4.5	−	−	25	6

Figure 3 (a) and (b) shows data obtained during initial CMS loading (indicated with numbers 1 and 2 in Fig. 2(i)). Rather large seismic dispersion (about 36%) is observed for vertical Young's modulus at the reference stress state. In case of shales, large dispersion of Young's moduli is not unusual and, in some cases, it may exceed 50% (e.g. Suarez-Rivera et al. 2001; Duranti et al., 2005, Szewczyk et al. 2016), which indicates the importance of accounting for dispersion effects during seismic data interpretation. Simulated reservoir inflation leads to the stiffening of the sample in both frequency regimes. Note, however, that the responses are different at seismic and ultrasonic frequencies. While at 1 Hz Young's modulus increases by around 0.2 GPa, the change at ultrasonic frequency is about two times smaller (see Table 1). Frequency-dependent stress sensitivity of Young's modulus results, during loading, in a decrease of dispersion. Different stress sensitivities at seismic and ultrasonic frequencies were observed before in relatively dry Pierre shale (Holt et al. 2016) and Mancos shale subjected to isotropic loading (Szewczyk, Bauer and Holt 2018a).

The quasistatic vertical Young's modulus (equal to E_static = 5.5 GPa) shown in Fig. 3(a) was obtained from the average slope of the stress–strain curve during triaxial unloading (indicated with number 5 in Fig. 2(i)). A clear difference of around 27% between quasistatic and low frequency (1 Hz) dynamic rock stiffness is observed. The static stiffness is generally smaller than the dynamic stiffness due to several factors including strain amplitude, dispersion, saturation, anisotropy, and relevant rock volume. As pointed out by Fjær, Stroisz and Holt (2013), careful design of experiments may eliminate most of the factors except the strain amplitude effect and dispersion. As observed for several rock types, the difference between static and dynamic mechanical compliance increases nearly linearly with stress amplitude during triaxial unloading. This allows for simple linear extrapolation of the measured compliance to zero stress (the point where unloading started), where the extrapolated value corresponds to zero-strain amplitude. Any remaining difference between the zero-strain static compliance and the dynamic compliance is believed to reflect dispersion between the frequencies corresponding to the strain rates of a static and a dynamic measurements, i.e.:

(22)

where f is the frequency, ε₀ is the strain amplitude associated with an elastic wave, and .

The result of applying this method is shown in Fig. 3(e). The compliance, dε_ax/dσ_ax, is plotted as a function of unloading stress, Δσ_ax, and extrapolated towards zero stress leading to the zero-strain limit of Young's modulus E_zero-strain = [dε_ax/dσ_ax(Δσ_ax = 0)]⁻¹ = 6.9 GPa. This zero-strain-extrapolated Young's modulus agrees well with the dynamic Young's modulus measured at 1 Hz. Since the strain rate used during triaxial unloading phase of our experiment () roughly corresponds to that of a 1 Hz elastic wave in our test (), this result suggests that, indeed, the difference between static and dynamic Young's modulus at a given frequency originates from both strain effect and dispersion effects (Holt et al. 2015). It also confirms that, in some cases, dynamic stiffness can be obtained from quasistatic measurements.

The vertical Poisson's ratio () shown in Fig. 3(b) does not exhibit large stress sensitivity. For seismic frequencies, it changes by around 0.01 under CMS loading. At ultrasonic frequencies, of Pierre shale exhibits even smaller stress sensitivity. Measured values are slightly lower than for seismic frequencies and stable at both stress states. Usually, an increase in Poisson's ratio from seismic to ultrasonic frequencies is observed (e.g., Hofmann 2006). It should be noted, however, that the values of Thomsen parameters used for anisotropic corrections were obtained during tests performed at different stresses, which may introduce error (see Discussion section).

Figure 3 (c) and (d) shows data obtained during second CMS loading (indicated with numbers 3 and 4 in Fig. 2(i)). The values of Young's modulus at seismic and ultrasonic frequencies are systematically higher than during the first CMS cycle (see Table 1), which is expected to be seen due to the sample over-consolidation. Note that during second CMS loading cycle, at both seismic and ultrasonic frequencies, increase of the vertical Young's modulus is similar to the case of the first loading cycle. This indicates that the measured stress sensitivity of Pierre shale (in absolute values) during CMS loading is not affected by material consolidation. Changes in vertical Poisson's ratio are also similar to the one obtained during the first loading cycle (see Table 1). After first CMS unloading, the acquired values did not show any considerable change. Slightly smaller increase of Poisson's ratio was observed after the second CMS loading; however, differences are within the experimental errors.

3.2 Wave velocities

Vertical P-wave and S-wave velocities obtained during both loading cycles are shown in Fig. 4 and summarized in Table 1. Ultrasonic velocities were measured directly, whereas seismic velocities were calculated from the measured Young's moduli and Poisson's ratios, assuming stress-independent Thomsen parameters (, , ; White et al. 1983).

Measurements performed at the reference stress exhibit around 22% dispersion in vertical P-wave velocity. First CMS loading shows that the P-wave stress sensitivity is different at low and high frequency regimes. While a large increase of V_PV was observed at seismic frequencies, ultrasonic measurements revealed only a small velocity increase upon loading (see Table 1). We define a stress sensitivity factor and strain sensitivity factor as follows:

(23)

(24)

with being the axial strain and being the axial stress. Both stress sensitivity and strain sensitivity factors are about five times higher at seismic frequencies compared with ultrasonic frequencies. The S-wave velocity exhibits smaller stress sensitivity than a P-wave. At the reference stress, vertical S-wave velocities exhibit around 15% dispersion. Again, CMS loading caused different increase of S-wave velocities at seismic and ultrasonic frequencies. As a consequence, dispersion decreased slightly. Different stress sensitivity of P-wave and S-wave results in a distinct V_P/V_S evolution at various frequencies (see Fig. 4(e)). In the seismic regime, the V_P/V_S changes are about five to six times larger than at ultrasonic frequencies (see Fig. 4(f)). Note that unloading in the first CMS cycle reduces the V_P/V_S ratio almost to its original values despite rather large irreversible changes in V_P and V_S. At seismic frequencies, the CMS unloading to the reference stress, induces a reduction of V_PV and V_SV. The measured velocities are, however, higher than those measured at the reference stress before the first CMS cycle. At ultrasonic frequencies, we also observe a small increase in absolute values of P-wave and S-wave velocities after CMS unloading, but the effect is about four to five times smaller than for seismic frequencies (see Table 1). During the second CMS cycle, S-wave velocities show a stress sensitivity comparable to the one measured during the first loading cycle, for both seismic and ultrasonic frequencies. As a consequence, dispersion does not change significantly. Also, P-wave velocity measurements roughly reproduce values obtained after the first CMS loading. During the second CMS loading, we again observe rather large stress sensitivity at seismic frequencies and a much smaller sensitivity at ultrasonic frequencies, leading to reduced dispersion. The average velocity change with stress () during second loading decreased with respect to those of the first loading cycle for both frequency ranges. The strain sensitivity factors, on the other hand, slightly increased when compared with the previous loading. Finally, the V_P/V_S changes during second loading cycle are comparable with the one obtained during initial loading, and their absolute values are reproduced with an accuracy of ±0.05.

4.1 Effect of frequency-dependent stress sensitivity

Presented experimental studies aim to demonstrate how the stress changes caused by an injection into a reservoir influence the mechanical properties of overburden shales for monitoring purposes. It has been observed that velocities and stiffness of shales exhibit significant seismic dispersion, and available rock physics models are not always capable of capturing the observed phenomena (Müller, Gurevich and Lebedev 2010). Therefore, the applicability of ultrasonic laboratory tests to seismic field measurements is questionable. For a better understanding of dispersion effects, we have performed experiments at seismic and ultrasonic frequencies. The obtained results indicate that also stress sensitivity of velocities in clay-rich materials strongly depend on frequency. In situations where pore-pressure changes in a reservoir influence a thick zone above the reservoir, the effect of dispersive stress sensitivity on the interpretation of 4D seismic data would be significant. Numerical simulations (Kenter et al. 2004) indicate that, in case of a 100-m thick reservoir, the influence of a pore-pressure change may affect a zone up to 1,000 m above the injection site. In such case, based on measurements performed during first CMS loading, the modelled change in P-wave two-way travel time would differ by a factor of five depending on whether stress sensitivities obtained from seismic or ultrasonic measurements are taken into account. Data acquired during the second CMS cycle exhibits smaller difference between seismic and ultrasonic stress sensitivities; however, the changes in travel time would still differ by a factor of ̴ 4.

4.2 Representativeness of laboratory tests for in situ studies

Presented experiments were conducted on Pierre shale core plugs drilled from outcrop material. Even though Pierre shale is considered as an analogue for overburden rocks, such materials are obviously not fully representative for in situ situations. Coring the samples and associated stress release could influence their mechanical properties (Holt, Brignoli and Kenter 2000) and make measured responses to differ from in situ behaviour. Also, in contrast to the present experiments, in a real case, there is usually a stiffness contrast between reservoir and overburden, which results in overburden stress paths that differ from a CMS loading. It is known that velocities are not only stress dependent but also stress path dependent. We have recently showed (Holt et al. 2016) that velocities in shales near in situ stresses are characterized by nearly monotonous (linear) dependence on stress path. In a real field case, inhomogeneities in rock elastic properties, rock plasticity, and the real geometry of reservoir and overburden formations should be taken into account, and the appropriate stress path should be determined by numerical modelling.

Finally, Thomsen's parameters that were used to convert between velocities and rock mechanical parameters were not measured directly during our experiments, which obviously introduces an additional error. To this end, it should be noted that (i) properties of Pierre shale I, including Thomsen's parameters differ depending on the shale origin (e.g., Schultz et al. 1980) and that (ii) Thomsen's parameters are dispersive and may be affected, among others, by stress applied to the sample, loading path, saturation, and stress history. Ultrasonic Thomsen's parameters used in this paper were obtained with core plugs drilled from the same batch of material as used in the present work (Holt et al. 2015). However, measurements performed by Holt et al. (2015) were conducted at slightly higher stresses (25 MPa confining pressure, 30 MPa axial stress, and about 12 MPa pore pressure) than those used in our experiments, which could lead to a change of the Thomsen's parameter value. Thomsen's parameters at seismic frequencies were taken from Pierre shale field measurements performed by White et al. (1983) at 950-ft depth. White et al. (1983) do not give direct information about saturation, mechanical parameters, in situ stresses, or density of their shale. However, suggested by White et al. (1983), density of 2.25 g/cm³ is smaller than the density of Pierre shale used in the present study (2.39 g/cm³). Based on the velocity data published by White et al. (1983), the vertical Poisson's ratio, ν_VH, of the field shale was 0.397, which is in good agreement with our measurements. Taking into account the large saturation sensitivity of Poisson's ratio in shales (e.g., Vales et al. 2004; Szewczyk, Bauer and Holt 2018b), it is reasonable to assume that the field shale was fully saturated, as the sample used in the present study. Also, based on the velocity data, the dynamic vertical Young's modulus, E_V, of the Pierre field shale was 5.3 GPa, about 1.7 GPa smaller than E_V of our sample. Young's modulus in Pierre shale, however, strongly increases with increasing stress experienced by the material (e.g., Szewczyk et al. 2018a). Bearing in mind that under normal pressure conditions, effective stress used in our experiment corresponds to a depth of about 1,000 m, which is significantly deeper than the shale formation studied by White et al. (1983), the discrepancy between Young's modulus values can be attributed to the difference in stress. In addition, taking into account that the Thomsen's parameters are stress dependent, the fact that Thomsen's parameters obtained by White et al. (1983) correspond to a lower stress could mean that they are not fully representativeness for our case. Qualitatively, the dependence of Thomsen's parameters on stress and stress path can be deduced from Szewczyk et al. (2018a,b) and Holt et al. (2016). The sensitivity of conversion between seismic and ultrasonic parameters measured in this work to the value of Thomsen's parameters ε, γ, and δ is illustrated in Fig. 5.

Figure 5 shows that the conversion from ultrasonically measured vertical velocities to rock mechanical parameters is more sensitive to Thomsen's parameters than conversion from low frequency laboratory measurements of E and ν to velocities. The converted data are particularly sensitive to ε and δ. In the vicinity of the values used in this work, while converting from velocities to mechanical parameters, 0.02 variation in ε value causes around 5% difference in the vertical Young's modulus, E_V, and about 4% difference in the vertical Poisson's ratio, . Similar variation of the δ parameter creates around 6% difference in E_V and about 2% difference in . The influence of the variations of the γ parameter is below 1% for both Young's modulus and Poisson's ratio. While converting from rock mechanical parameters to velocities, the effects of similar variation of Thomsen's parameters are on average two to three times smaller.

4.3 Possible leakage from reservoir

Pierre shale used in our experiment was tested in fully brine-saturated conditions. In the field case, there could be some gas partially saturating the caprock (e.g., due to a leakage from the reservoir). In case of CO₂ storage, leakage from the storage site is a concern, and suitable and reliable monitoring methods are required. Here, we present possible effects of CO₂ leakage on 4D seismic attributes.

A common way used in geophysics for identifying effects of fluid substitution involves use of the Gassmann model (Gassmann 1951). Many of the assumptions underlying the Gassmann model are violated in shales during any field or laboratory investigations. Real shales are heterogeneous and composed of several minerals, and the pore connectivity is rather poor. Increased volume of the pore fluid at seismic frequencies causes strong softening of the shale and changes the shear moduli (Szewczyk et al. 2018b). Water near the clay surfaces influence the rock physical properties of shales (Holt and Kolstø 2017). Moreover, due to observed dispersion (Figs. 3 and 4), the shale is often unrelaxed or only partially relaxed even at seismic frequencies. Nevertheless, as shown elsewhere (Szewczyk et al. 2018b), in some cases the anisotropic Gassmann model may still be applicable for seismic velocities in shales if modifications accounting for the rock–fluid interaction effects, such as adsorption effects leading to frame weakening (e.g., Spencer 1981; Murphy, Winkler and Kleinberg 1984; Moerig et al. 1996) and capillary pressure effects, are introduced into the model. The equations of such modified anisotropic Gassmann model accounting for the adsorption driven frame weakening are as follows:

(25)

(26)

(27)

(28)

(29)

(30)

where the superscript fr denotes the properties of frame; K_f and stands for the bulk modulus of pore fluid and solid phase, respectively; φ indicates the porosity; represent the independent coefficient of stiffness elastic tensor; is the wetting phase saturation; and is the fitting parameter accounting for the fluid-phase driven frame weakening through adsorption effects, which, in general, may be different for each of the independent ’s. The capillary pressure effects at a microscopic scale and at low frequency limit, on the other hand, may be accounted for by calculating the effective fluid modulus through Brie's mixing law (e.g., Santos, Corberó and Douglas 1990; Papageorgiou, Amalokwu and Chapman 2016), despite the typical patchy saturation interpretation of this mixing law. The Brie's empirical equation (Brie et al. 1995) is as follows:

(31)

where stands for non-wetting phase saturation, and are bulk moduli of wetting and non-wetting phases, respectively, and e is an adjustable parameter, which, in case of high values, would move the effective fluid modulus towards Reuss average and, in case of low values, towards Voigt average. The use of the Gassmann model requires anisotropic frame properties. In the present work, the frame properties were estimated in such a way that the modified Gassmann model reproduces values measured for full brine saturation. The effective bulk moduli of solid phase (), Brie's adjustable parameter (), and parameter accounting for frame weakening ( between 0.42 and 0.45) of Pierre shale were taken from Szewczyk et al. 2018b). We have considered a two-phase fluid system filling up the pore space (brine and CO₂). The bulk modulus and the density of the brine were and , respectively, (e.g., Batzle and Wang 1992), whereas bulk modulus and density of CO₂ were and (e.g., Burke 2011). Being aware that CO₂ properties are affected by pressure and temperature, we decided to keep them constant, since bulk modulus and density of CO₂ are low compared with those of brine; thus, accounting for pressure and temperature dependence would have only a small impact on the final results.

Figure 6(a) shows vertical P-wave velocities as a function of CO₂ saturation calculated based on seismic (1 Hz) measurements performed at all examined stress states.

The modelled effect of gas saturation (in relative terms) is quite similar during the first and second CMS cycles. The modelling of the fluid substitution indicates that small leakages of the CO₂ into the overburden formation would create noticeable travel time shifts, under the assumption that the overburden formation was initially fully brine saturated. Ten percent CO₂ saturation in the overburden due to possible leakage could create up to 3% P-wave velocity change. Obviously, the possibility of detecting such events during 4D surveys depends on the related changes of the two-way travel times and, thus, on the length of CO₂ penetration into the shale formation. Assuming good quality data, current 4D detection limit for changes of the two-way travel time is below 1 ms (e.g., Landrø and Amundsen 2017). Using vertical P-wave velocities from our 1 Hz data (about 2,500 m/s) and assuming 3% velocity change due to the CO₂ leakage, this threshold would be reached after the CO₂ has propagated for about 40 m into the overburden formation, which is quite likely in the case of the over-pressured reservoirs (e.g., during CO₂ injection for storage or enhanced recovery). The fluid substitution effects are visible up to about 50%–60% gas saturation. Afterwards, the changes of the velocities are mainly controlled by the adsorption effects present in shales and leading to the weakening of anisotropic frame moduli (those effects may have magnitude comparable with the fluid substitution effects). This can be seen by the drop of seismic velocities while going from fully gas-saturated shale up to about 30%–40% fluid saturation and increase of the velocities afterwards (similar behaviour was observed experimentally in Szewczyk et al. 2018). Another interesting observation is that the CMS loading seems to decrease the range in which fluid substitution effects are dominant (compare black and grey curves in Fig. 6(a). Note also that the P-wave velocity changes associated with fluid substitution may be of comparable magnitude as one associated with a stress change: based on seismic data obtained at reference stress, ten percent CO₂ saturation would lead to a velocity change comparable to that obtained by about 1.9 MPa shear stress change during CMS loading.

Velocity changes associated with fluid substitution are clearly stress dependent (additional load decreases fluid phase sensitivity). Moreover, fluid substitution effects depend on the material hysteresis. This is further shown in Fig. 6(b) where the shear stress sensitivity of vertical P-wave velocities () is plotted as a function of CO₂ saturation. In absolute values, during initial CMS loading, the seismic stress sensitivity reduces from 115 m/s MPa⁻¹ to 38 m/s MPa⁻¹ when going from full CO₂ saturation to full brine saturation. For the second CMS cycle, the seismic stress sensitivity of the P-wave velocity decreases from 95 m/s MPa⁻¹ to 28 m/s MPa⁻¹. Note, however, that in the relative values, the stress sensitivity at seismic frequencies is about three times higher in case of full CO₂ saturation than in case of full brine saturation regardless of the loading cycle (during first CMS loading stress sensitivity reduces by ̴ 67%, whereas during second CMS loading, it reduced by ̴ 70%).