A Methodological Determination of Sorptive Poromechanical Constitutive Coefficients for Double Porosity Shales

Abstract

Double porosity constitutive models are necessary components in mechanical analysis of fractured rocks. To solve the field constitutive equations, values of constitutive coefficients are required. Several theoretical expressions have been proposed to calculate the double porosity constitutive coefficients as their physical estimation is challenging due to a lack of efficiently designed experiments. Yet experimental determination of constitutive coefficients provides confidence in the accuracy of the models. Here, we present a methodological framework to determine new constitutive coefficients arising by coupling of gas sorption kinetics in the sorptive poromechanical model developed by the authors for double porosity media (Algazlan et al. 2022a). To physically characterise the new constitutive coefficients, we designed an innovative methodological process consisting of (i) obtaining gas diffusivities and Langmuir parameters by measuring gas uptake on different shales, (ii) computing the time evolutions of the free and adsorbed matrix gas concentrations, and (iii) optical imaging of temporal changes in the porosity of fractures. A novel shear-microscopy cell was used allowing stress application to the samples whilst being exposed to surficial fluid flow coupled with optical monitoring. The methodological framework resulted in the efficient determination of the new sorptive constitutive coefficients paving way for further advanced validation studies of multiphysics constitutive models.

Appendices

Appendix A

Samples Description and Experimental Procedures

Shale samples from three different local Australian basins were used for the gas uptake and diffusivity measurements and the microscopy-based investigations. Using a RockLab benchtop ring mill, samples from the three different basins were ground down to an average particle size of ~ 200 µm, to be used for the gas uptake measurements. The powdered samples were then left overnight in a vacuum oven to remove moisture. Another set of samples from the MVBF was cut into cuboids using an electrical circular saw cutting machine. Samples were secured and hold in place firmly during cutting to minimise/eliminate damage. Air was used instead of water to clean/cool the blade whilst cutting the samples. The surfaces of the cut samples were then polished using a sandpaper with 1000 p grit-size maintaining a constant polishing time for consistency across all samples.

The gas uptake measurements were conducted using a system developed by CSIRO Energy North Ryde laboratories. This system was designed based on a volumetric method, whereby the mass of adsorbed gas is calculated from direct measurements of gas pressures in various parts of the system. It allows simultaneous measurements of four samples. The samples are generally crushed to less than 200 µm particle size. The system requires feeding samples of ~100 g mass and can operate at temperatures of up to 70 °C and pressures of up to 15 MPa. To increase the gas pressure, the adsorbent gas is first fed from a set of cylinders into a dual ISCO pumps system. Once the gas pressure reached a pre-set level, it is injected into a reference cell. This reference cell is connected to 4 sample cells via shut-off valves. All cells are mounted in a temperature-controlled water bath. A thermocline unistat heater circulator is used to heat the water bath to the desired temperature and a thermocouple connected to a computer records the temperature values in real-time. During measurements, gas pressure values in reference and sample cells are recorded continuously through pressure transducers connected to a computer. The pressure sensors have a 0–15 MPa range and accuracy of ±0.04%. The valves into and between the reference cell and the four sample cells are controlled pneumatically. A vacuum pump is connected to the system and used to keep the cells under vacuum at the start of the experiment. The volume of the reference cell and the empty sample cells is accurately measured and used in data processing.

The prepared cuboid shale samples were hosted in the high-pressure shear cell, which allows for stress application to the samples whilst exposing them to fluid flow and monitoring their surface changes through a scratch resistant sapphire glass window that is fitted and screwed on top of the housed sample with a flat O-ring. Two Teledyne ISCO pumps filled with hydraulic oil were connected to the cell’s two hydraulic rams that push steel platens against the housed shale sample creating the desired stresses, provided that the sides opposite to the rams are fixed. Another ISCO pump filled with injection gas (i.e. either helium or CO₂) is connected to the upstream port and is used to circulate the gas and control its pressure, with the downstream port being fitted with a Swagelok high-pressure valve. The gas enters from the upstream port and is introduced first to the top face of the sample. Applied gas pressures and confining stresses were monitored using high-precision digital pressure transducers connected to a computer for data acquisition and recording. The changes in fractures’ apertures were captured using ISCapture, an image acquisition software, at a magnification value of 90x with an image pixel resolution of 0.5 μm and a capture rate of one frame/second.

Appendix B

Numerical Simulations

Two physics interfaces were used to model the gas diffusion and adsorption, a built-in transport of diluted species in porous media interface for CO₂ diffusion utilising the equation:

$$\frac{{\partial (\phi c_{i} )}}{\partial t} + {\triangledown }. J_{i} + u.{\triangledown }c_{i} = - R$$

(B.1)

where ${c}_{i}$ is the free gas concentrations, and $J_{i} = - D_{i} .\nabla c_{i}$ with ${D}_{i}$ being the diffusion coefficient. A general form PDE interface was utilised for modelling CO₂ adsorption using the following equation:

$$\frac{{\partial c_{s} }}{\partial t} + {\triangledown }. \Gamma = R$$

(B.2)

where ${c}_{s}$ is the adsorbed gas concentrations, and R is the source/sink term and is defined as follows:

$$R = k_{ads}\cdot\, c_{i} ( c_{\max } - c_{s} )-k_{des}\cdot\, c_{s}$$

(B.3)

where ${k}_{ads}$ is the adsorption rate, ${k}_{des}$ is the desorption rate, and ${c}_{max}$ is the maximum surface sorption concentrations. The initial/boundary gas concentrations were calculated from the CO₂ injection pressure value of 30 bar, used in the fracture investigation, and the void space between the shear cell’s closed chamber and the hosted sample at the experiment’s temperature, i.e. 25 °C. The values used in the model for CO₂ diffusivity coefficient and CO₂ Langmuir parameters were the same values obtained from the CO₂ gas uptake measurements introduced earlier, at the same temperature and pressure conditions. Finally, a source/sink term was used to couple gas diffusion with gas sorption.