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Permeability is one of the most fundamental reservoir rock properties required for modeling hydrocarbon production. However, shale permeability is not yet fully understood because of the high temperature of shale reservoirs. The third thermal stress that is caused by temperature change will decrease the permeability of shale. In this work, a theoretical model has been derived to describe the permeability of shale considering the third thermal stress; the principles of thermodynamics and the mechanics of elasticity have been employed to develop this model. The elastic modulus parameters of the shale were measured, along with Poisson’s ratio, as required. Lastly, the permeability of shale was tested by transient pulse-decay. Isothermal flow experiments were carried out at 303, 313, 323, and 333 K to assess the effects of shale expansion and deformation on shale permeability caused by the third thermal stress. The permeability of shale samples, as predicted by the model, was found to agree well with experimental observations. The model may provide useful descriptions of the gas flow in shale. The correction accuracy of the permeability was found to increase at lower permeability. However, the development of completely predictive models for shale permeability will require additional experimental data and further testing.

Shale gas is an unconventional but promising gas resource that has been used with significant success in the world. Permeability is one of the most fundamental properties of any reservoir rock, and it is required for modeling hydrocarbon production. Heller et al. [

The thickness of gas shales varies widely. For example, a study of five shale gas systems in the USA indicates a range in shale thicknesses of 2 to 700 m, and the maximum reported thickness of Chinese gas shales is 925 m [

Current studies mainly focus on the temperature effects on the permeability of rock. However, temperature effects on the permeability properties of rocks containing complicated components such as shale are comparatively unknown. Thermal stress can be divided into three categories [

In this paper, a model is proposed that is based on the principles of thermodynamics and mechanics of elasticity, to describe the temperature impacts of shale permeability under the third thermal stress conditions. The correction factor is tested by experimental measurements on the permeability of shale at 303, 313, 323, and 333 K.

Since shale is composed of different mineral components, the linear expansion coefficient of shale

The thermal stress caused by temperature is expressed as follows [

Thermal stress is a vector and its direction points from the region of largest expansion quantities to the region of smallest expansion, as shown in Figure

Schematic of the third thermal stress.

The direction of the thermal stress in bulk shale points from the organic matter to the minerals. Assuming that the minerals are fully mixed with the organic matter, the value of the thermal stress in small units can be calculated by the model of Figure

Temperature change in shale formations can be determined by the depth of the shale. A shale formation at the same depth can be considered to have isothermal conditions because the temperature would not change at the same depth of shale burial. Additionally, shale can be assumed to be an isotropic solid for calculation simplicity [

The thermal stress analysis diagram.

The equilibrium, geometric, and physical equations in Cartesian coordinates are as follows [

The equilibrium differential equation, using the displacement as the basic unknown function, is obtained through a transformation of Equation (

The solution to Equation (

A set of solutions to the equation of state of constant temperature, homogeneous plus, is the general solution of differential equations of equilibrium equations, boundary conditions to obtain simultaneous, displacement solution for determining:

The shale strain in the

The empirical formula

By introducing a modification coefficient

The shale porosity is not affected by the third thermal stress, so

The above equations are simplified by omitting higher order terms because the linear elastic deformation is small:

External factors that affect the correction coefficient of the shale permeability include the temperature difference, geothermal gradient, vertical stress, and horizontal stress. These factors are also related to the medium’s elastic modulus, Poisson’s ratio, coefficient of expansion, and the mass ratio of its mineral components.

Equation (

It is very difficult to measure shale permeability using conventional steady-state measuring methods because the shale permeability is very low (10-^{3} to 10^{-6} mD). So, an alternative method such as the transient pulse-decay (TPD) method [

In this work, the transient pulse-decay experiment was used to measure the permeability of shale samples affected by the temperature.

The shale samples used in the laboratory experiments were collected from the lower Silurian Longmaxi Formation in Yibin City, Sichuan Province. The total organic carbon (TOC) content of the samples is 7.88%, and the vitrinite reflectance (

The sample of shale.

The mineralogical compositions of shale samples were tested using X-ray diffraction. The X-ray diffraction data were obtained using a Siemens D5000 diffractometer, using CuK

X-ray diffraction results of shale samples.

Mineralogical composition (%) of shale samples | ||||||
---|---|---|---|---|---|---|

Quartz | Potash-feldspar | Plagioclase | Calcite | Dolomite | Pyrites | Clay |

53.5 | 1 | 3.1 | 20.3 | 8.7 | 3.6 | 9.8 |

The triaxial compression strength, elastic modulus, and Poisson’s ratio of the samples were tested using MTS815. The results are shown in Table

Mechanical indexes of shale under uniaxial compression.

Data | Average | |
---|---|---|

Uniaxial strength (MPa) | 158.88 | 158.65 |

154.95 | ||

160.14 | ||

Elastic modulus | 46.5 | 47.5 |

48.5 | ||

47.5 | ||

Poisson’s ratio | 0.324 | 0.298 |

0.289 | ||

0.281 |

Figure

The schematic of the experimental apparatus.

The maximum oil field thermostat system can be heated to 473 K, with a temperature control accuracy of 0.1 K. The servo loading system provides a maximum 3000 kN axial load to provide a maximum 200 MPa of confining pressure.

The experimental system consisted of a sample cell, two reference cell (5 mL), two pressure transducer (Rosemount, American, model 3051, range: 0 to 13.79 MPa, accuracy: 8 kPa), and a pressure difference sensor (Rosemount, American, model 3051, range: 0 to 0.68 MPa, accuracy: 0.68 kPa). The pore pressure is provided by the ISCO pump, with a maximum available gas pressure of about 5000PSI.

Prior to measuring the permeability of shale samples, the apparatus was checked for pressure leaks. The helium was injected into the reference cell. If the values of pressure collected from the pressure transducer decreased less than the accuracy of the pressure transducer (8 kPa) in 24 h, then it was no leaks in the system [

The sample was subsequently installed in the test cell. The system was allowed to reach thermal equilibrium at the target temperature, based on heating in the oil field, ensuring that any temperature-induced expansion took place. The swelling of the shale in response to increased temperature was considered to have reached its maximum when the strain gauges showed the strain of the sample had plateaued, at which point a vacuum was applied.

We apply a confining pressure of 9 MPa, then open the relevant valve of the pulse permeability instruments, and the

Kumar et al. [

The permeability was calculated as

The permeability of the sample.

Temperature (K) | 303 | 313 | 323 | 333 |
---|---|---|---|---|

Sample 1 | 0.278uD | 0.268uD | 0.250uD | 0.239uD |

Sample 2 | 0.256uD | 0.246uD | 0.234uD | 0.223uD |

Sample 3 | 0.233uD | 0.228uD | 0.216uD | 0.209uD |

In this work, we have established a model to predict shale permeability. Eight parameters are included in this model, i.e., temperature difference

Shale component parameter table.

Parameters | Data |
---|---|

Elastic modulus | 47.5 |

Poisson’s ratio | 0.29 |

Geothermal gradient ^{-1}) | 0.03 |

Axial stress | 30.0 |

Horizontal stress | 20.0 |

The linear expansion coefficient of organic matter (10^{-6}·K^{-1}) | 30.0 |

The linear expansion coefficient of minerals (10^{-6}·K^{-1}) | 3.0 |

The correction coefficient for the temperature difference was calculated using equation (

Correction factor tables.

Temperature difference (K) | Correction factor | Temperature difference (K) | Correction factor |
---|---|---|---|

5 | 0.9995 | 25 | 0.9410 |

10 | 0.9932 | 30 | 0.9125 |

15 | 0.9813 | 35 | 0.8785 |

20 | 0.9639 |

To eliminate the effect of the second thermal stress on the results, and based on the experimental conditions, the permeability at 303 K was considered the initial permeability of the model. Therefore, only the third thermal stress will affect the results. Figure

The relationship of temperature and permeability; the scatters were experiment data, while the curve was the model data.

Figure

This paper presents a theoretical model to describe the impacts of temperature on shale permeability under the third thermal stress condition. This model was built on the principles of thermodynamics and mechanics of elasticity. All parameters included in this model have physical meanings, which provide a predictive basis for constraining the shale permeability.

The main constituents in the shale sample are minerals and organic matter. A shale sample will produce uneven swelling because the thermal expansion coefficients of the minerals and organic matters are different as a result of the effects of temperature change. The third thermal stress is caused by uneven swelling. Furthermore, shale permeability will change under the effects of matrix swelling, which is caused by both ground and thermal stresses. We have also modified the formula for the correction coefficient of the shale permeability. The shale permeability predicted by the model coincides with the experimentally measured permeability, which decreases as the temperature rises. Additionally, the accuracy of the permeability corrections was found to be higher when the permeability was reduced. The shale permeability decreased more rapidly when the temperature increased, particularly while the difference in the shale component expansion coefficients was larger and the organic matter content was higher.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This study was funded by the National Key Basic Research Program of China (No. 2014CB239206), the Natural Science Foundation of Chongqing (Nos. cstc2019jcyj-msxmX0507 and cstc2019jcyj-zdxmX0024), and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJQN20200).