# Measurements of the Effective Stress Coefficient for Elastic Moduli of Sandstone in Quasi-Static Regime Using Semiconductor Strain Gauges

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## Abstract

**:**

^{−6}. The study was performed using a forced-oscillation laboratory apparatus utilizing the stress–strain relationship. The dynamic elastic moduli were measured in two sets of experiments: at constant pore pressures of 0, 1, and 5 MPa and differential pressure (defined as a difference between confining and pore pressures) that varied from 3 to 19 MPa; and at a constant confining pressure of 20 MPa and pore pressure that varied from 1 to 17 MP. It was shown that the elastic moduli obtained in the measurements were in good agreement with the Gassmann moduli calculated for the range of differential pressures used in our experiments, which corresponds to the effective stress coefficient equal to unity.

## 1. Introduction

^{−6}. Such a strain amplitude limit better corresponds to the real conditions of seismological measurements and imposes important restrictions on experimental measurements of the elastic/anelastic properties of rocks in the laboratory. However, due to the complexity of the technical implementation of the strain-level control in ultrasonic systems, most ultrasonic measurements are performed without strain-amplitude control, which often leads to inadequate values of the measured elastic moduli [36].

^{−6}was conducted at a low frequency on Savonnieres limestone for the effective pressure coefficient for porosity by Tan et al. [23]. Analyzing the drained-to-undrained transition of bulk modulus in n-decane-saturated Savonnieres limestone observed in forced-oscillation experiments with varying dead volume, Tan et al. [23]. demonstrated that the prediction of the poroelastic model is highly consistent with the experimental data if the effective stress coefficient for porosity is equal to unity.

^{−6}. In addition to the predominant mineral presented by quartz, the sample selected for our measurements (Mungaroo Formation, Western Australia) contains a number of other minerals, including a significant amount of clay minerals. We present the results of the laboratory measurements of the elastic properties of rock performed using a forced-oscillation apparatus at a frequency of 2 Hz. Using the standard low-frequency forced-oscillation method (a comprehensive review of the forced-oscillation method is presented in [32]), we examined the dependences of the elastic moduli of the sandstone and extensional attenuation on the differential pressure, which is defined as the difference between confining and pore pressures, and determined the effective stress coefficient for elastic moduli. To avoid lacerations of strain gauges due to the high compressibility of the sample, we had to limit our measurements to an upper confining pressure of 20 MPa. The measurements were carried out in two sets. The first set of the measurements was conducted at pore pressures equal to 0, 1, and 5 MPa and a differential pressure gradually increasing from 3 MPa to 19 MPa, and the second set was performed at a constant confining pressure of 20 MPa and a pore pressure varying from 1 to 17 MPa; in both sets of the measurements, n-decane was used as the pore fluid. In all measurements, the dynamic strains did not exceed 10

^{−6}.

## 2. Experimental Setup

^{−8}to 10

^{−6}. The apparatus measures the elastic parameters of rock samples at confining or uniaxial pressures from 0 to 70 MPa.

## 3. Sample Description and Experimental Procedure

^{3}, respectively. The difference in density of the sample in dry and wet states corresponds to a pore space of 11.5 cm

^{3}or porosity of 13.7%. The permeability of the sandstone is equal to 1.9 mD.

^{3}/s (1.2 cm

^{3}/min).

Mineral | Content (Volume), % | Bulk Modulus, GPa |
---|---|---|

Quartz | 72 | 36.6 [39] |

K-feldspar | 2 | 57 [40] |

Micrite | 4 | 71 [41] |

Illite | 5 | 21 [42] |

Kaolinite | 12 | 11 [43] |

Calcite | 4 | 76.8 [39] |

## 4. Results and Discussion

#### 4.1. Measurements with Variable Confining Pressure

#### 4.2. Measurements with Variable Pore Pressure

#### 4.3. Discussion

## 5. Conclusions

^{−6}. The dynamic elastic moduli were measured in the drained regime with zero pore pressure, as well as in the experiments with non-zero constant pore pressures of 1 and 5 MPa at a differential pressure that varied from 3 to 19 MPa, and with pore pressure changed from 1 to 17 MP at a constant confining pressure of 20 MPa. It was shown that the measurement results of elastic moduli were in good agreement with the Gassmann moduli calculated for the range of differential pressure used in this study, which corresponds to the effective stress coefficient equal to unity. It was also demonstrated that the results of the measurements of extensional attenuation obtained at non-zero pore pressures are in good agreement with each other and are uniquely determined as functions of differential pressure.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The mechanical assembly and electrical schematics of the low-frequency laboratory apparatus (

**a**); the measured sample in the Hoek cell with two strain gauges mounted along the axis and circumference of the sample (

**b**).

**Figure 2.**Two-dimensional mutually perpendicular X-ray cross-sections of the Mungaroo sandstone sample.

**Figure 3.**The dependences of static strains in the sample and standard on confining pressure measured in the drained regime.

**Figure 4.**The dependences of Young’s modulus (

**a**) and Poisson’s ratio (

**b**) on differential pressure, measured on the Mungaroo sandstone sample at a frequency of 2 Hz.

**Figure 5.**The dependence of the extensional attenuation on differential pressure, measured on the Mungaroo sandstone sample at a frequency of 2 Hz.

**Figure 6.**The dependences of the dynamic strains measured on the sample and standard on differential pressure, obtained for the Mungaroo sandstone sample at a frequency of 2 Hz.

**Figure 7.**The dependences of the bulk modulus (

**a**) and shear modulus (

**b**) on differential pressure, computed using Equations (4) and (5).

**Figure 8.**The experimental dependences of the bulk moduli on differential pressure, obtained using Equation (4), and the pressure dependence of the bulk modulus, computed using modified Gassmann Equation (11).

**Figure 9.**The dependences of the Young’s modulus (

**a**) and Poisson’s ratio (

**b**) on differential pressure, measured on the n-decane-saturated Mungaroo sandstone sample at a frequency of 2 Hz under a confining pressure of 20 MPa and a pore pressure varied that from 1 to 17 MPa.

**Figure 10.**The dependence of the extensional attenuation on differential pressure, measured on the n-decane-saturated Mungaroo sandstone sample at a frequency of 2 Hz and presented with the attenuation obtained in the drained mode.

**Figure 11.**The differential pressure dependences of the dynamic strains measured on the n-decane-saturated Mungaroo sandstone sample and standard at a frequency of 2 Hz.

**Figure 12.**The dependences of the bulk modulus (

**a**) and shear modulus (

**b**) on differential pressure, calculated from measured Young’s modulus and Posson’s ratio.

**Figure 13.**The dependences on differential pressure obtained for the bulk modulus (

**a**) and shear modulus (

**b**) in the measurements with variable and constant pore pressures.

**Figure 14.**The dependences on differential pressure obtained for Poisson’s ratio in the measurements with variable and constant pore pressures.

**Figure 15.**The dependences on differential pressure obtained for the extensional attenuation in the experiments with variable and constant pore pressures.

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**MDPI and ACS Style**

Mikhaltsevitch, V.; Lebedev, M.
Measurements of the Effective Stress Coefficient for Elastic Moduli of Sandstone in Quasi-Static Regime Using Semiconductor Strain Gauges. *Sensors* **2024**, *24*, 1122.
https://doi.org/10.3390/s24041122

**AMA Style**

Mikhaltsevitch V, Lebedev M.
Measurements of the Effective Stress Coefficient for Elastic Moduli of Sandstone in Quasi-Static Regime Using Semiconductor Strain Gauges. *Sensors*. 2024; 24(4):1122.
https://doi.org/10.3390/s24041122

**Chicago/Turabian Style**

Mikhaltsevitch, Vassily, and Maxim Lebedev.
2024. "Measurements of the Effective Stress Coefficient for Elastic Moduli of Sandstone in Quasi-Static Regime Using Semiconductor Strain Gauges" *Sensors* 24, no. 4: 1122.
https://doi.org/10.3390/s24041122