Poromechanical Properties of a Sandstone Under Different Stress States

Abstract

Poromechanical properties of a sandstone from an underground gas storage site are investigated with the use of a neutral gas to control the pore pressure. The poroelastic theory of anisotropic medium is used to evaluate elastic properties and coupling coefficients under different stress states. In-situ CT observations have been made under uniaxial compression test and they are used to underline some effects due to the occurrence of damage and microcraks. Under hydrostatic loading, pores and micro-cracks are gradually compressed. The initial state of the sandstone is “slightly” transversely isotropic but in a first approach the Biot’s tensor reduces to a scalar. This coefficient decreases with the increase in confining pressure that can be attributed to the closure of micro-cracks. The effect of damage and cracking is investigated with two series of conventional triaxial tests conducted at different confining pressures, either with an increase in pore pressure (series 1) or with a decrease in pore pressure (series 2), to evaluate the coupling coefficients. The results, derived from these two test series, are consistent as regards the effects of cracking on the material behavior. There is an obvious damage due to the triaxial loading which induces a parallel decrease in the ratio E₃/(1 − ν₃) and in the modulus H₃. Axial compaction results in the continual increase in the H₁ modulus observed for every sample whatever the measurement technique was.

1 Introduction

Underground gas storage plays a vital role in competitive natural-gas markets and is a beneficial economic method to compensate the imbalance between supply and demand (Arfaee and Sola 2014; Azin et al. 2014; De Jong 2015).

The injection (respective withdrawal) of natural gas into a depleted reservoir or an aquifer results in the increase (respective decrease) in pore pressure. This cycling injection and withdrawal of gas pressure modifies the stress field in the reservoir, especially close to the borehole. These stress variations are suspected to damage the rock vicinity of the borehole and to induce sand production, which can degrade well performances. Understanding the poromechanical behavior of porous rock under stress changes is thus an essential for the storage operator.

We focus in this study on the evolution of coupling poromechanical properties of a sandstone under hydrostatic and deviatoric loading. The theory of linear isotropic poroelasticity was first introduced by Biot (1941), Cheng (1997) and Coussy (2004). The theoretical foundation of anisotropic poromechanical properties has also been widely developed and has also been investigated by the help of numerical simulations (Aichi and Tokunaga 2011; Carroll 1979; Cheng 1997; Shao 1998; Tan and Konietzky 2014; Tan et al. 2015). Anisotropy will be induced by variation of ambient stress (Liu and Shao 2017; Liu et al. 2015). However, direct laboratory measurements of anisotropic poromechanical properties, especially their evolution with deviatoric stresses, are still very few (Hu et al. 2010; Lockner and Beeler 2003; Suarez-Rivera and Fjær 2013). In this study, a series of laboratory tests were made. The poroelastic behavior of the material is determined in the vicinity of a given stress state, which can be hydrostatic or deviatoric.

Computed tomography (CT) has emerged as a powerful nondestructive scanning technique that can yield high-resolution-3D renderings of a scanned rock (Alshibli and Alramahi 2006). In-situ compression CT, which can scan the internal structures of rocks under progressive loadings is now available and has been employed in this study in parallel with triaxial compression tests. Observations with the in-situ axial compression CT test reveal damage, in particular the micro-cracks evolution in axial and lateral directions with deviatoric stress. In-situ compression CT results will well explain the evolution of the macroscopic poromechanical elastic properties under deviatoric loading, at least under low confining pressure.

2 Experimental Material and Apparatus

2.1 Material Used for the Experiments

The sandstone used in this study is a Vosges sandstone. Samples are 69.5 ± 0.5 mm in height and their diameter is 37.5 ± 0.5 mm. The initial porosity measured with distilled water is around 19.4%. All the samples used are in dry condition at a room temperature of 20 ± 2 °C. The micro-structure of this sandstone obtained with optical microscope and X-ray tomography can be observed in Fig. 1. The grain size is mainly in the range 250–300 µm. The resolution of CT scanning is 30 µm (note that for the in-situ compression CT test a smaller sample is drilled with 12.14 mm in height and 5.31 mm in diameter).

Fig. 1

Micro-structure of the sandstone with optical microscope and X-ray tomography

2.2 Experimental Techniques

We combine triaxial compression tests and in-situ compression CT observations to characterize the sandstone.

2.2.1 Triaxial Compression Cell

Stress loading and pressure loading can be imposed independently to estimate the material poromechanical properties. The choice of the same gas pressure, applied at one end or both ends of the sample has been developed in our lab; such experimental setup makes it easier to estimate not only elastic behavior, but also both permeability and poromechanical properties under loading. Moreover, using gas, instead of liquid (generally water) for the pore pressure control, involves numerous advantages such as the chemical neutrality (argon in the present case) especially for reactive materials (clay rocks, concrete, etc) and the ability to simultaneously measure the gas permeability and the porosity under stresses. It has been successfully used for argillite, claystone and for cementitious material (Pei et al. 2017; Song et al. 2017; Yuan et al. 2017). Moreover, the use of gas makes experiments closer to the in-situ conditions that we are interested (i.e., gas pressure variations in a sandstone reservoir).

A triaxial compression device, shown in Fig. 2, was used; it is composed of an autonomous, self-compensated, triaxial cell, and a gas device which allows gas permeability, gas porosity, and poromechanical measurements to be performed. Two cross gauges were glued at the mid-height of the sample with 180° to each other. Each gauge can monitor the strains in two directions (axial and lateral directions). The final axial strain (ɛ_a) and lateral strain (ɛ_l) are the average of two measured values, respectively. The volumetric strain (ɛ_v) is calculated by ɛ_a + 2ɛ_l. Before gluing gauges, a thin layer of fast-cured epoxy was applied to the sample to seal its porous surface and then polished. Strain values are recorded with a LabView System. The sample is confined with oil pressure (σ₂ = σ₃ = P_c) and is sealed by a rubber membrane. The axial force is applied by a Zwick mechanical press to obtain the usual deviatoric stress (σ₁ − σ₃). The fluid used to apply the relative pore pressure (P_i) is pure argon. At each step of loading (or unloading) of confining pressure and/or of pore pressure, it was necessary to wait for 10 min for strain stability. The coordinate system is also shown in Fig. 2.

Fig. 2

Schematic illustration of the triaxial compression apparatus (1–4 valves, 5 manometer, 6 accumulator, 7 gas reservoir, 8 sandstone sample, 9 pump, 10 cross gauge, 11 aluminum cylinder, 12 axial strain gauge)

To perform the poromechanical experiment, it is necessary to change internal pore pressure. As discussed later in Sect. 5.4, this operation can lead by itself to additional sample damage. To assess the effect of damage on the poromechanical properties, with a better objective, it has been decided to perform two different series of triaxial tests. In the first one, pore pressure is increased (loading pore pressure), whereas in the second one, pore pressure is decreased (unloading pore pressure). As it is shown in Fig. 2, a pressure sensor (an aluminum cylinder equipped with a axial strain gauge) was specially designed for the second series of tests to obtain the real axial stress on the sample, i.e., corrected for the friction effects between the cell and the axial piston. This led to a cell calibration, which was used for the first test series.

2.2.2 In-Situ Compression CT Apparatus

Figure 3 shows the in-situ compression CT test apparatus. To get high-quality tomography images, a high resolution (5 µm) is chosen. The acceleration voltage is 110 kV. A press machine (Fig. 3a) is directly mounted on the rotating stage of the tomograph. The maximum force the machine can apply is 5000 N. A Quartz tube is used as part of the loading machine, and the tube allows a 360° rotation without hiding the sample, and gives a negligible attenuation of the X-ray photos (Li 2016). The axial loading speed was 0.12 mm/min which was also used for the macroscopic tests. As shown in Fig. 3b, four scans were made at different loading stages (0.45, 22.50, 31.5, and 36.00 MPa). The sample failed at 39.95 MPa. This critical axial stress is close to the value (42.7 MPa) recorded with a bigger sample (37.45 mm in diameter and 69.63 mm in height). The difference between both critical stresses may be attributed to the material natural dispersion or to an effect of coring. As an initial state, a small force (10 N) is applied to prevent any sample move during the press machine rotation. 2742 projections were taken at each loading step for a complete 360° sample rotation. The total scan time was about 80 min for each step. There was also a usual small stress relaxation during this process (Baud et al. 2012).

Fig. 3

In-situ compression CT apparatus and deviatoric stress loading stage. a Press machine and the small sample. b Evolution of deviatoric stress with time

3 Micro-structure Evolution with Deviatoric Stress

The 2D and 3D images were processed and analyzed by the AVIZO FIRE 9.0 software. Due to the sample compression and rotation during the scanning, the direct images obtained from CT scanning-at different loading-change and it is difficult to locate and compare the same cross section with the original images during the loading process. There is, however, a ‘register’ process in AVIZO FIRE 9.0 that can help to find the same cross sections at different loading stages. This process chooses the middle horizontal and vertical cross sections of the first scan as the reference. After ‘register’, one can get one horizontal cross section and two vertical cross sections in the middle position of the sample for the different loading stages which are always the same. Figure 4 shows the 3D rock digital structure and the three reference cross sections.

Fig. 4

3D rock digital structure with three reference cross sections

Figure 5 shows the same grains or micro-cracks in horizontal and vertical directions at different levels of axial stress. The size of each figure is 0.5 mm × 0.5 mm. A dotted white line is used to highlight the micro-cracks. ‘a_i to d_i’ are the same horizontal or vertical cross section at the four different loading stages. Two main phenomena can be easily observed on either horizontal or vertical (axial) micro-cracks. In Fig. 5(a₁-h)–(d₁-h), the occurrence of a new microcrack inside a grain [dotted line in (d₁-h)] which is perpendicular to the horizontal plane (i.e., axial) can be located. In Fig. 5(a₂-v)–(d₂-v), there is a clear compaction of a horizontal microcrack (between two grains), the same phenomenon is also visible in Fig. 5(a₃-v)–(d₃-v), whereas a new vertical microcrack occurred at the end of the loading. As it will be seen in the following, this phenomena is useful to explain some changes observed in the sample poromechanical behavior.

Fig. 5

Deformation of three different grains and micro-cracks in the same middle horizontal cross section (h) and the same middle vertical cross section (v) with increase of deviatoric stress a 0.45 MPa, b 22.50 MPa, c 31.51 MPa, d 36.00 MPa

Figure 6 shows five failed samples under different confining pressures. It can be noticed that the failure mode depends on the level of confining pressure. This means that conclusions about the micro-structure evolution v.s. deviatoric stress can only be drawn for uniaxial test. For other conditions (with confining pressure), it can just be assessed whether the mechanical property evolutions are consistent (or not) with the micro-CT test.

Fig. 6

Failed samples under different confining pressures

4 Poromechanical Properties Under Hydrostatic Stress

4.1 Isotropic Behavior

Hydrostatic compression tests are used to evaluate the poromechanical property evolution with confining pressure.

Under hydrostatic stress, the material is assumed to be elastic isotropic. Skeleton bulk modulus (K_b) is measured from an unloading in confining pressure (∆P_c). The variation of volumetric strain (∆ɛ_v) is determined at different levels of confining pressure (P_c):

$${K_{\text{b}}}=\frac{{\Delta {P_{\text{c}}}}}{{\Delta {\varepsilon _{\text{v}}}}}.$$

(1)

H, the coupling coefficient, is measured by an increase in pore pressure (∆P_i). 1/H is defined as the poroelastic expansion coefficient (Wang 2000). At 5 MPa of confining pressure, the pore pressure is increased by 2 MPa with two steps of 1 MPa. Fot the other confining pressures, the pore pressure is increased by 3 MPa with the same 1 MPa steps:

$$H=\frac{{\Delta {P_{\text{i}}}}}{{\Delta {\varepsilon _{\text{v}}}}}.$$

(2)

K _b and H variations with confining pressure are shown in Figs. 7 and 8. The variation of volumetric strain with confining pressure and pore pressure is also included in the two figures. These two properties increase with confining pressure. Between 5 and 20 MPa, the increase in K_b and H is significant, while from 20 to 30 MPa, this increase is comparatively smaller. This material hardening is mainly attributed to crack closure. It can be then assumed that the cracks are almost closed at 20 MPa of confining pressure.

Fig. 7

Skeleton bulk modulus (K_b) as a function of effective confining pressure

Fig. 8

Coupling coefficient (H) as a function of effective confining pressure

Biot’s coefficient (b) can be obtained by the ratio of K_b and H (Chen et al. 2009; Coussy 2004; Lion et al. 2004):

$$b=\frac{{{K_{\text{b}}}}}{H}.$$

(3)

Biot’s coefficient estimates are given in Table 1. From 5 to 20 MPa, the decrease in Biot’s coefficient is significant. This result is consistent with the closure of cracks mentioned before. As the cracks are almost closed after 20 MPa, the Biot’s coefficient remains constant. Compared to the behavior of a tight sandstone studied in our laboratory (Wang et al. 2017), the Biot’s coefficient decrease remains weak (Biot’s coefficient of the tight sandstone can vary by example from 0.9 to 0.6 between 3 and 40 MPa of confining pressure).

Table 1 Biot’s coefficient under different confining pressures

When the variation of confining pressure is equal to that of pore pressure, i.e., ∆P_c = ∆P_i, the measured volumetric strain is equal to the solid matrix volumetric strain. The solid matrix bulk modulus K_s is, therefore, deduced from (Chen et al. 2009; Coussy 2004; Lion et al. 2004):

$${K_{\text{s}}}=\frac{{\Delta {P_{\text{c}}}=\Delta {P_{\text{i}}}}}{{\Delta {\varepsilon _{\text{v}}}}}.$$

(4)

The results, obtained with another sample, are shown in Fig. 9. The K_s values, derived from relation (4), reveal to be too high (81 GPa) at the highest confining pressure especially for a rock matrix which is mainly composed of pure silica grains. This can be attributed to the low variation in confining pressure used to do the measurements (i.e., ΔP_c = ΔP_i = 3 MPa), therefore, leading to very small resulting strains (around 12 µm/m). As a consequence, there is a high sensitivity of the K_s value towards any small variation in the strains measured. Such small variations can be due to slight temperature change into the cell, quality of the wire welding that can be sensitive to the confinement, etc. Hence, only the tendency of the K_s variation can be clearly analyzed. The constant increase in this modulus with increasing P_c can be attributed to the closure of micro-cracks or grain joints that are not connected to the pore network (Duan et al. 2014; Wang et al. 2017).

Fig. 9

Relationship between volumetric strain and confining pressure (pore pressure) for determination of K_s

In addition, the value of K_s can also be evaluated with the properties K_b and H (1/K_s = 1/K_b − 1/H). The resulting values are 35, 39, 41, and 44 GPa, respectively, for 5, 10, 20, and 30 MPa confining pressures. These values are more realistic for this kind of rock.

4.2 Remarks About the Material Anisotropy

Anisotropy can be observed from the ratio “lateral strain/axial strain” obtained for a hydrostatic loading and/or an increase in pore pressure. This ratio should be equal to 1 for an isotropic material. This anisotropy is also tracked from the “H_i” modulus which is calculated as

$$\Delta {P_{\text{i}}}={H_i}\Delta {\varepsilon _i}.$$

(5)

Experimental results (Table 2) indicate indeed that the material is less deformable in the axial direction than in the lateral one, whatever the loading is (pore pressure or confining pressure). This anisotropy could be due either to pre-existing cracks or to an intrinsic anisotropy related to the rock’s structure, as X-ray tomography (Fig. 1) shows horizontal layers of various density. However, the sensitivity of the material to confining pressure (Figs. 7, 8, 9) and the CT data support an interpretation in term of cracks closure and pore compression.

Table 2 Initial anisotropic parameters under hydrostatic stress

An interesting case for anisotropic material arises when b₁ = b₃ = b, as observed for clayed rocks (Yuan et al. 2017). In this specific situation, the estimation of b reduces to

$$b={b_i}=\frac{{\Delta {\varepsilon _i}\left( {\Delta {P_{\text{i}}}=\Delta {P_{\text{c}}}} \right)}}{{\Delta {\varepsilon _i}\left( {\Delta {P_{\text{c}}}} \right)}}.$$

(6)

It can be seen in Table 2 that this approximation is valid for the sandstone studied and that the Biot’s tensor is nearly isotropic under hydrostatic loading.

There is, however, a 10% difference in the Biot’s coefficient which only occur for 20 MPa of confining pressure. For other values of this pressure, the difference is less than 5%, so it can be assumed that the difference at 20 MPa should not be considered as a physical phenomenon but as an experimental artefact. We have thus performed a new experiment and found that the values of b₁ and b₃ at 20 MPa of confining pressure were 0.689 and 0.704, respectively.

On another hand, the decrease in the Biot’s coefficients with the increase in confining pressure is systematic, so it can be supposed to be a physical phenomenon attributed to the closure of micro-cracks. The material anisotropy is evidenced by the ratio ∆ɛ_l^{c or g}/∆ɛ_a^{c or g} being always different from 1. b₁ = b₂ = b₃ just means that the ratio ∆ɛ_l^g/∆ɛ_l^c is almost equal to ∆ɛ_a^g/∆ɛ_a^c.

This kind of observation was also made on the CO_x clay rock (Yuan et al. 2017) which exhibited a stronger anisotropy than this sandstone and for which the Biot’s coefficients were virtually identical.

The purpose of the next section is to investigate the poroelastic properties under non-hydrostatic triaxial loading.

5 Poromechanical Properties Evolution with Deviatoric Stress

In this section, the poroelastic properties and Biot’s tensor in the vicinity of a given deviatoric stress state are investigated. Some numerical simulations had already been done on the Biot’s coefficient evolution under deviatoric stress (Shao 1998; Tan and Konietzky 2014; Tan et al. 2015), but direct experimental results are still very few and are not always in good agreement with simulated results (Al-Tahini and Abousleiman 2010; Aoki et al. 1993; Cheng 1997; Hu et al. 2010; Lockner and Beeler 2003). We focus mainly in this study on the evolution of the coupling poromechanical properties.

5.1 Theoretical Background

Let us consider the case of transversely isotropic poroelastic material. The poroelastic behavior can be described using the following relationship (Hu et al. 2010; Lings et al. 2000):

$$\left[ {\begin{array}{*{20}{c}} {\Delta {\varepsilon _1}} \\ {\Delta {\varepsilon _2}} \\ {\Delta {\varepsilon _3}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{1}{{{E_1}}}}&{\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{{ - {\nu _1}}}{{{E_1}}}} \\ {\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{1}{{{E_3}}}}&{\frac{{ - {\nu _3}}}{{{E_3}}}} \\ {\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{{ - {\nu _3}}}{{{E_3}}}}&{\frac{1}{{{E_3}}}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\Delta {\sigma _1} - {b_1}\Delta {P_{\text{i}}}} \\ {\Delta {\sigma _2} - {b_2}\Delta {P_{\text{i}}}} \\ {\Delta {\sigma _3} - {b_3}\Delta {P_{\text{i}}}} \end{array}} \right],$$

(7)

where E₁ and E₃ are the Young’s modulus in vertical (axial) and horizontal (lateral) directions, respectively. ν₁ is the Poisson’s ratio for horizontal strain due to vertical strain, and ν₃ represents Poisson’s ratio in the isotropic plane. b₁ is the Biot’s coefficient in axial direction and b₂ = b₃ is the Biot’s coefficient in lateral direction.

If confining pressure and deviatoric stresses are kept constant (∆σ₁ = ∆σ₂ = ∆σ₃ = 0), the variation of strains can also be expressed as a function of variation in pore pressure:

$$\left[ {\begin{array}{*{20}{c}} {\Delta {\varepsilon _1}} \\ {\Delta {\varepsilon _2}} \\ {\Delta {\varepsilon _3}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{1 \mathord{\left/ {\vphantom {1 {{H_1}}}} \right. \kern-0pt} {{H_1}}}} \\ {{1 \mathord{\left/ {\vphantom {1 {{H_2}}}} \right. \kern-0pt} {{H_2}}}} \\ {{1 \mathord{\left/ {\vphantom {1 {{H_3}}}} \right. \kern-0pt} {{H_3}}}} \end{array}} \right] \cdot \Delta {P_{\text{i}}}.$$

(8)

The stress–strain relation becomes

$$\left[ {\begin{array}{*{20}{c}} {\Delta {\varepsilon _1}} \\ {\Delta {\varepsilon _2}} \\ {\Delta {\varepsilon _3}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {\frac{1}{{{E_1}}}}&{\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{{ - {\nu _1}}}{{{E_1}}}} \\ {\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{1}{{{E_3}}}}&{\frac{{ - {\nu _3}}}{{{E_3}}}} \\ {\frac{{ - {\nu _1}}}{{{E_1}}}}&{\frac{{ - {\nu _3}}}{{{E_3}}}}&{\frac{1}{{{E_3}}}} \end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}} {\Delta {\sigma _1}} \\ {\Delta {\sigma _2}} \\ {\Delta {\sigma _3}} \end{array}} \right]+\left[ {\begin{array}{*{20}{c}} {\frac{{\Delta {P_{\text{i}}}}}{{{H_1}}}} \\ {\frac{{\Delta {P_i}}}{{{H_3}}}} \\ {\frac{{\Delta {P_{\text{i}}}}}{{H{}_{3}}}} \end{array}} \right].$$

(9)

The two poroelastic expansion moduli are related to the Biot’s coefficients:

$$\frac{1}{{{H_1}}}=\frac{{{b_1}}}{{{E_1}}} - \frac{{2{\nu _1}}}{{{E_1}}}{b_3}$$

(10)

$$\frac{1}{{{H_2}}}=\frac{1}{{{H_3}}}= - \frac{{{\nu _1}}}{{{E_1}}}{b_1}+\frac{{1 - {\nu _3}}}{{{E_3}}}{b_3}.$$

(11)

Thus, the Biot’s coefficient in axial and lateral direction can be estimated from the two previous equations. It becomes

$${b_1}=\frac{{{E_1}}}{{{H_1}}}+\left( {\frac{1}{{{H_3}}}+\frac{{{\nu _1}}}{{{H_1}}}} \right) \cdot \frac{{2{\nu _1}}}{{\frac{1}{L} - \frac{{2\nu _{1}^{2}}}{{{E_1}}}}}$$

(12)

$${b_2}={b_3}=\left( {\frac{1}{{{H_3}}}+\frac{{{\nu _1}}}{{{H_1}}}} \right) \cdot \frac{1}{{\frac{1}{L} - \frac{{2\nu _{1}^{2}}}{{{E_1}}}}}$$

(13)

with $L=\frac{{{E_3}}}{{1 - {\nu _3}}}$.

5.2 Experimental Procedure for the First Series of Tests

According to the relations (12) and (13), five properties need to be identified to estimate the Biot’s coefficients: the Young’s modulus E₁, the Poisson’s ratio ν₁, two poroelastic expansion coefficients 1/H₁ and 1/H₃ (or coupling coefficients H₁ and H₃), and L. All these five properties can be obtained during a triaxial compression test. The testing procedure is as follows:

• 1. The confining pressure is applied up to a fixed value under drained conditions (∆σ₁ = ∆σ₂ = ∆σ₃ = P_c, then P_c remains constant). A deviatoric loading is applied.

• 2. An unloading–reloading deviatoric cycle is made (∆σ₁ = − c, ∆σ₂ = ∆σ₃ = 0), which allows the measurement of ∆ɛ₁ = ∆σ₁/E₁ and ∆ɛ₂ = ∆ɛ₃ = (− ν₁∆σ₁)/E₁, based on relationship (7). Hence, the value of E₁ and ν₁ can be determined.

• 3. The confining pressure is now increased up to a constant value (∆σ₁ = ∆σ₂ = ∆σ₃ = c) and the lateral strain is monitored. It becomes ∆ɛ₂ = ∆ɛ₃ = (− ν₁/E₁ + (1 − ν₃)/E₃) ∆σ₂, based on Eq. (7). Confining pressure is unloaded to its initial value. E₃/(1 − ν₃) is evaluated.

• 4. For this last step, a pore pressure is applied by injecting gas into the sample (∆P_i = c). The variation of axial and lateral strains can be given by ∆ɛ₁ = ∆P_i/H₁ and ∆ɛ₃ = ∆P_i/H₃: the coupling coefficients (H₁ and H₃) are estimated.

Step (1)–step (4) were repeated gradually along with the increase in deviatoric stress. The variations of deviatoric stress, confining pressure and pore pressure for unloading–loading cycles in each test are shown in Table 3.

Table 3 Variation of deviatoric stress, confining pressure and pore pressure for the first series of tests

5.3 Experimental Results (the First Series of Tests)

The five measured elastic properties are listed in Table 4. The triaxial stress–strain curves are shown in Fig. 10. At each confining pressure, five unloading–reloading cycles were made.

Table 4 Evolution of the five elastic properties and Biot’s coefficient for the first series of tests

Fig. 10

Stress–strain curves under different confining pressures for the first series of tests

The variations of lateral strain in hydrostatic loading step are shown in Fig. 11. There is a continuous decrease of the slopes (Fig. 11) with the increase in deviatoric stress: E₃/(1 − ν₃) is decreasing. This phenomenon can be related to the opening and occurrence of axial micro-cracks as we observed in Fig. 5. Despite this damage there is no strong variation in E₁ (Table 4) that either slightly increases at the beginning of the axial loading or exhibits a weak decrease close to the stress peak. Even if conducted only under axial loading the in-situ micro-CT gave results, which are in accordance with a compaction effect leading to the increase in E₁ visible at the beginning of the triaxial test. However, the increase in Poisson ratio is observed close to the failure, i.e., when the dilatancy obviously occurs as the material is damaging.

Fig. 11

Variation of lateral strain due to hydrostatic loading at different levels of confining pressure

Figures 12 and 13 show the variation in lateral and volumetric strains due a pore pressure loading (increase in pore pressure). The axial strain is smaller, by nearly one order of magnitude, than the lateral strain. Moreover, note that there is a clear loss of linearity for the latter with the increase in deviatoric stress (i.e., see the number of loading–unloading cycles). These observations can be interpreted in term of cracks: at higher deviatoric stresses, that is above the onset of dilatancy, vertical cracks have been nucleated/propagated which make the effect of pore pressure more pronounced in the lateral direction. This phenomenon induces a strong decrease in modulus H₃ (see Table 4): an expansion of the sample due to the occurrence of micro-cracks induced by deviatoric stress, as confirmed by the increase in the volumetric strains (Fig. 13). The evolution of two coupling moduli variations (H₁ and H₃) is given in Fig. 14. The coefficient in axial direction (H₁) increases continuously, while H₃ decreases with deviatoric stress. This evolution of H_i is consistent with the material compaction which was observed with the in-situ micro-CT experiments under axial loading (Fig. 5). It will be confirmed in the following with the second test series.

Fig. 12

Variation in lateral strain due to the increase in pore pressure at different levels of confining pressure

Fig. 13

Variation of volumetric strain due to the increase of pore pressure at different levels of confining pressure

Fig. 14

Evolution of the two coupling coefficients (H₁ and H₃) with deviatoric stress

5.4 Remarks About the Measurements Methods and Experimental Procedure for the Second Series of Tests

At this stage, it is useful to, once again, underline the fact that the Biot’s coefficients (and all the properties derived from these tests) are involved in a poroelastic theory and should be estimated under elastic path of loading (or unloading). The case of triaxial loading is particular as, if there is damage, only a deviatoric stress unloading step is likely to be elastic. Any change in the effective confining pressure is indeed a hybrid case as, even if the deviatoric stress remains constant, such a change will modify the reference stress state and the sample behavior (Fig. 15). As a consequence, the strains measured are composed of both the effect of loading (or unloading) and the effect of a different behavior. This will induce some inaccuracy concerning the Biot’s coefficient calculation, since it will be seen in the following that they are very sensitive to tiny variations of the other properties from which they are deduced. It is the reason why the direct measurements of the moduli H_i are preferable to link more accurately the strains to the stress variations (relation (9)). On another hand, it has been underlined below that an increase in pore pressure, necessary to evaluate the H_i moduli, may also induce damage. This potential damage would lead to some bias in the result interpretation. To assess this point and as a purpose of comparison, it has been decided to change the experimental method as follow and to carry out a new test series. Most parts of the experimental procedure of the new test series were similar to the first series. These complementary experiments are now conducted with an internal gas pressure of 2 MPa (or 1.5 MPa for the lowest confining pressure) which is brought back to 0 to measure the H_i moduli. Four new complete tests were then carried out with a confining pressure being: 6.5, 12, 22, and 32 MPa as we have chosen to perform both series of triaxial tests under the same level of effective Terzaghi confining pressure. The different loadings that were used are given in Table 5, they are similar to those carried out in the first case except at the lowest confining pressure (4 unloading–loading steps instead of 5). It has to be emphasized here that, to obtain a better accuracy in the calculation of the elastic properties E₁ and ν_1, the strains used were the average of those obtained with two consecutive unloading steps from the same deviatoric stress (σ₁ − σ₃). The unloading was of 20%, i.e., Δ(σ₁ − σ₃) = 0.2(σ₁ − σ₃).

Fig. 15

Modification of sample behavior by changing confining pressure or pore pressure at constant deviatoric stress

Table 5 Variation of deviatoric stress, confining pressure, and pore pressure for the second series of tests

5.5 Experimental Results Obtained (the Second Series of Tests)

The new stress–strain curves are given in Fig. 16 and the whole results on the elastic and poroelastic properties can be obtained in Table 6. The analysis of Fig. 10 (first test series) and Fig. 16 shows that no significant differences can be observed in the stress–strain behavior of the samples as regards the strength and the occurrence of dilatancy. The E₁ modulus variations are similar for both cases in the sense that they do not reveal particular damage (i.e., decrease in E₁). Similar variations of the Poisson ratio ν₁ can also be noticed even if its final increase, occurring close to the failure, is now less pronounced. As for in the first case the mechanical damage is mainly visible with the variation of the ratio E₃/(1 − ν₃) which is closely linked to the crack opening (or closure) due to changes in confining pressure. A strong decrease is also present at the end of the triaxial test after dilatancy and close to failure. This can be seen in Fig. 17 in which are plotted the normalized values of the ratio E₃/(1 − ν₃) divided by its initial value at the beginning of the test. This graphical representation is interesting as it allows the whole set of results for cases 1 and 2 to be compared and to extract the global tendency followed by this property when there is damage.

Fig. 16

Stress–strain curves under different confining pressures for the second series of tests

Table 6 Evolution of the five elastic properties and Biot’s coefficient for the second series of tests

Fig. 17

Normalized values of the ratio E₃/(1 − ν₃)—i.e., divided by its value at the beginning of the triaxial loading vs normalized values of deviaotric stress—i.e., divided by its peak value

There are strong similarities between Figs. 17 and 18 that plots the normalized values of modulus H₃. As this modulus is linked to the sample expansion due to a change in pore pressure, its decrease indicates the effect of axial cracks due to the deviatoric stress (see Sect. 5.3). Obviously the decreases in E₃/(1 − ν₃) and in H₃ come from this phenomenon. The purpose of the second test series (case 2) was also to identify whether an increase or a decrease in pore pressure would influence by itself this behavior observed. As no systematic effect of the pore pressure loading path can be underlined in Fig. 18, we can now assume that the cracks are mainly produced by the triaxial loading and not by the way they are detected (i.e., pore pressure increase or decrease). The variations in H₁ and H₃ are given in Fig. 19 that can be compared to the Fig. 14 (first series of tests). There is an obvious increase in H₁ modulus related to the material compaction previously mentioned so as the difference between H₁ and H₃ increases constantly when the damage occurs. As for the previous properties, it is very useful to examine the variations of the modulus H₁ normalized by its initial value (after the first step of loading). They are presented in Fig. 20 in which we have removed some results obtained at the very end of the test, i.e., close to the failure. In this figure, the results for the two test series can be compared and showing that, for samples tested at the same effective confining pressure (Terzaghi pressure), the normalized variations in H₁ are virtually identical even if the absolute values are different.

Fig. 18

Normalized values of H₃—i.e., divided by its value at the beginning of the triaxial loading vs normalized values of deviatoric stress—i.e., divided by its peak value

Fig. 19

Variations in H₁ and H₃ for the second series of tests

Fig. 20

Normalized values of H₁—i.e., divided by its value at the beginning of the triaxial loading vs normalized values of deviatoric stress—i.e., divided by its peak value

The normalized values of H₁ and H₃ are also represented vs the deviatoric shear strain γ that is often used for damage modelling. As H₁ was found to be mainly related to the material compaction, this representation is more useful to describe the H₃ variation. It can be observed in Fig. 21 that there is a constant decrease of H₃ with the shear strain, but that this phenomenon is obviously linked to the level of confining pressure as the latter clearly reduces, for a given value of shear strain, the material expansion to due the internal pore pressure (and/or the micro-cracking). As a consequence, the poromechanical property damage cannot be simply modelled with the sole shear strain and should involved another state variables related with the confining pressure, i.e., volumetric strain for example. This is beyond the scope of the present experimental study.

Fig. 21

Normalized values of H₁ and H₃ vs deviatoric shear strain. a, b Are, respectively, for the first and second series of tests

5.6 Results Obtained for the Biot’s Coefficients

As mentioned before and indicated by relation (9), the knowledge of the elastic properties measured by the two series of tests is sufficient to describe the elastic part of the material behavior. On another hand, the relation (9) is theoretically equivalent to relation (7) in which the Biot’s coefficient are involved. The physical effects of the material damage and the associated cracking are mainly described with the variation of the normalized values of E₃/(1 − ν₃) and H₃.

When calculated with the relations (12) and (13), the cumulated error on coefficient b₁ and b₃ can be very high and as large as 20 for 5% uncertainty on the other properties. These coefficients are very sensitive to small changes in the properties from which they are obtained. It is thus very difficult to draw conclusions and analyses on these coefficient variations due to the material damage. They can, however, be found in Tables 4 and 5.

6 Conclusions

Poromechanical properties of a sandstone from an underground porous gas storage have been investigated with the use of a neutral gas to control the pore pressure. The poroelastic theory of anisotropic medium has been used to evaluate the poromechanical elastic parameters including Biot’s tensor, elastic properties and coupling coefficients under different stress states. In-situ CT observations has been made but under uniaxial compression test only (no confining pressure). Despite this restriction, they can be used to underline some effects due to the occurrence of damage and microcraks.

Under hydrostatic loading, pores and micro-cracks are gradually compressed. The initial state of the sandstone is “slightly” transversely isotropic, as shown by the ratio “lateral strain/axial strain” obtained for a hydrostatic loading and/or an increase in pore pressure. Despite this anisotropy, the Biot’s tensor reduces to a scalar. The skeleton bulk modulus, the solid matrix bulk modulus, and the coupling coefficients (H₁, H₃) increase gradually, whereas Biot’s coefficient decreases with the increase in confining pressure. This can be attributed to the closure of micro-cracks.

The effect of damage and cracking has been investigated with two series of conventional triaxial tests conducted at different confining pressures. These tests are difficult to perform as the poroelastic property measurements require changes in the pore pressure that result in variations in effective confining pressure. As a consequence the measurement methodology can produce additional perturbative damage and it has been decided to carry out two series of tests, either with an increase in pore pressure (series 1) or with a decrease in pore pressure (series 2), to evaluate the coupling coefficients. The results, derived from these two test series, are consistent as regards the effects of cracking on the material behavior. The main concluding remarks can then be drawn: there is an obvious damage due to the triaxial loading which induces a parallel decrease in the ratio E₃/(1 − ν₃) and in the modulus H₃. Despite the damage the modulus E₁ is almost constant or slightly increases, a slight decrease is nevertheless observed close to the failure. This phenomenon is consistent with the observations of the in-situ CT test and is attributed to an axial compaction. On a poromechanical point of view this compaction results in the continual increase in the H₁ modulus observed for every sample whatever the measurement technique was (series 1 or 2). All these measures can be used to calculate the Biot’s coefficient in the anisotropic case that is, however, not essential to describe the poroelastic part of the material behavior.