Temperature-Induced Ductile–Brittle Transition in Porous Carbonates and Change in Compaction Band Growth Revealed by 4-D X-Ray Tomography

Abstract

Deformation bands featuring localised material failure are ubiquitous in nature. They form important flow barriers and reduce/compartmentalise fluid flow in oil/gas/water reservoir rocks. Moderate temperature changes have been observed to play a fundamental role in the formation and style of these bands, but the mechanisms underpinning these changes are often obscure. Here, we show compaction experiments of highly porous limestone from Mt Gambier, Australia, with chemically non-reacting gaseous (helium) and fluid (kerosene) pore fluids. Gas-filled limestones showed a lower static elastic stiffness than fluid-filled specimens. The discrepancy in elastic modulus is particularly noted at high temperature. This indicates the important effect of temperature-sensitive compressibility of gas-filled pores versus nominally incompressible fluid-filled pores. A moderate temperature rise from 25 to 80 °C also led to a sharp transition in compaction band growth from dominantly ductile diffuse band growth at low temperatures to prevailing brittle growth at higher temperatures. We attribute this change to a temperature-sensitive change in micro-mechanism from rate-sensitive calcite twinning at room temperature to activation of a near-ideal plastic Peierls mechanism at 80 °C. The inverse-to-normal brittle–ductile transition is documented by time-lapse X-ray CT micro-tomographic images and associated digital image and volume image correlation techniques.

Highlights

• 4D time-lapse triaxial experiments on highly porous carbonates reveal thermomechanical and thermohydromechanical couplings.

• Gas saturated specimens show higher yield stress and lower elastic modulus when compared to fluid-saturated specimens.

• A changeover from ostensibly ductile to a dominantly brittle micro-deformation mechanisms is encountered when raising the temperature from 25 to 80 °C.

Appendices

Appendix 1

A series of pre-tests were conducted at a constant water flow rate to warm up the specimen to the desired temperature. The temperature of the specimen was measured with a thermocouple probe made of Cu–CuNi wires which was placed inside the specimen. The thermocouple was connected to ALMEMO 2590 data logger, which has a resolution of 0.1 °C. Figure 15 shows the desired temperature of 80 °C requiring circulating hot water (95 °C) at 90 ml/min for 60 min and temperature of 50 °C requiring circulating hot water (95 °C) at 90 ml/min for 8 min and then reduced to 24 ml/min for another 30 min until the temperature stabilises. The temperature fluctuation is ± 0.5 °C throughout.

Fig. 15

The relationship between desired elevated temperature versus time at different flow rates

Appendix 2

2.1 Digital Image Processing

Each scan might have slightly different greyscale intensity because of the filament lifetime changes. Differences in greyscale intensity can result in bias when obtaining the accurate registration and thresholding segmentation process across all the images. The greyscale intensity was carefully calibrated according to selected homogeneous regions: air, kerosene, triaxial cell body, top and bottom platens, and rubber sleeve. All differences were corrected using an obtained linear function.

When the X-ray beam passes through a cylindrical specimen, the outer regions absorb and scatter the lower energies in the X-ray spectrum, which results in the exterior regions being brighter than the inner regions. Beam-hardening correlation thus applied a Gaussian smoothing kernel on specified regions of the data to reduce the beam hardening effect (Ketcham and Hanna 2014). Noise is another artifact that makes it challenging to differentiate low-density areas, thereby reducing the ability to segment effectively (Nagarajappa et al. 2015). A nonlinear anisotropic diffusion filter denoising the images and preserving the edges using a similar method by Frangakis and Hegerl (2001) was used in our study.

Image registration is a crucial step in time-lapse experiments. A 3D registration algorithm developed by Latham et al. (2008) was used for image registration. This technique brings two or more images into geometric alignment for further digital image correlation analysis. In our study, each scan was carefully registered to a previous scan to ensure that undeformed parts are overlapped.

The histogram of all the image slices can be exported before image segmentation (Fig. 1). The greyscale values of the tomographs correspond to the X-ray attenuation produced by material, i.e., for example, lower for air than kerosene and the calcite grains. The histogram shows two peaks corresponding to the pores (air, low X-ray attenuation) and matrix (mainly calcite, high X-ray attenuation). With compression of the specimen, the volume of the pore phase reduces, and the matrix phase increases. Two dashed lines indicate the greyscale value of the pore phase and the matrix phase. The converging active contours’ (CAC) method was used for the segmentation of these images (Sheppard et al. 2014; Sheppard et al. 2004). This method uses a combination of the watershed and active contour methods to segment the greyscale data. The real boundary of the pore and matrix can be determined by this method. Later on, the layer-by-layer porosity profile can be extracted from the segmented images along with Z-axis.

2.2 Digital Image Correlation (DIC) and Digital Volume Correlation (DVC) Analysis

Since the 1980s, the 2D digital image correlation (DIC) technique has been widely used to measure real-time full-field data of displacements and strains (Chu et al. 1985; Vendroux and Knauss 1998). This non-destructive testing method can measure the deformation behaviour of a material over a wide area in exceptional detail. DIC uses image registration algorithms to track the relative displacement of material points between a reference image and a deformed image. Our study uses an open-source 2D subset-based DIC software package: Ncorr (Blaber et al. 2015). Ncorr uses the reliability guided (RG-DIC) method to obtain displacement values for a subset. The shear and compressive strain field can be extracted based on the displacement field using the Green–Lagrangian strain tensor, which can be used to determine the nucleation and propagation of the compaction bands. This DIC code has been shown to work well for tracking the materials displacements (Caselle et al. 2019; Lv et al. 2019; Siddiqui et al. 2021; Stanier et al. 2016). With the wide use of X-ray CT and synchrotron 3D images, the digital volume correlation (DVC) technique has become popular (Bay et al. 1999). This technique can be considered as the extended version of the 2D-DIC method to the 3D-DVC domain in conjunction with 3D images, which effectively determines the internal volumetric deformation behaviours of solid materials. Avizo (Thermofisher Scientific) digital volume correlation (DVC) was used for 3D-DVC analysis in our study. A subset-based (local) approach is used to capture the large displacements on a coarse, regular grid.

Appendix 3

The radial deformations for both helium and kerosene saturated specimens at low and high temperatures were extracted from every X-ray CT scan. The radial and axial strain versus axial stress is plotted in Fig. 16.

Fig. 16

The axial and radial strain versus axial stress of a specimen MG-D25 (dry 25 °C) and MG-D80 (dry 80 °C) and b specimen MG-K25 (kerosene 25 °C) and MG-K80 (kerosene 80 °C)

Appendix 4

The contact angles of kerosene–calcite–air and water–calcite–air were measured by viewing the drop profile (Link and Schlünder 1996; Siddiqui et al. 2019) and are shown in Fig. 17.

Fig. 17

Contact angle measurement using distilled water and purified kerosene with crushed powder of Mt Gambier limestone. Because of the highly permeable sample powder, the contact angle of two different fluids was measured using a high-speed camera. The results were plotted by contact angle versus time. Two dashed lines show the power-law fit of the kerosene and water contact angles, respectively

The equilibrium contact angle (${\theta }_{\mathrm{C}}$) is determined from Young equation (Butt et al. 2013)

$${\gamma }_{\mathrm{SG}}+{\gamma }_{\mathrm{SL}}+{\gamma }_{\mathrm{LG}}\mathrm{cos}({\theta }_{\mathrm{C}})=0,$$

where, $\gamma$ is the surface energy, and SG, SL, and LG represent the solid–gas, solid–liquid, and liquid–gas, respectively. We write the contact angle equations for water–calcite–air and kerosene–calcite–air system as

$${\gamma }_{\mathrm{Sa}}+{\gamma }_{\mathrm{Sw}}+{\gamma }_{\mathrm{wa}}\mathrm{cos}({\theta }_{\mathrm{w}})=0,$$

$${\gamma }_{\mathrm{Sa}}+{\gamma }_{\mathrm{Sk}}+{\gamma }_{\mathrm{ka}}\mathrm{cos}({\theta }_{\mathrm{k}})=0,$$

where Sa, Sw, wa, and w represent calcite–air, calcite–water, water–air, and water, and Sk, ka, and k represent calcite–kerosene, kerosene–air, and kerosene, respectively. The difference between ${\gamma }_{\mathrm{Sw}}$ and ${\gamma }_{\mathrm{Sk}}$ is

$$\Delta \gamma ={(\gamma }_{\mathrm{Sk}}-{\gamma }_{\mathrm{Sw}}{)= \gamma }_{\mathrm{ka}}\mathrm{cos}\left({\theta }_{\mathrm{k}}\right)-{\gamma }_{\mathrm{wa}}\mathrm{cos}\left({\theta }_{\mathrm{w}}\right),$$

where ${\gamma }_{\mathrm{wa}}$ is 0.0728 N/m at 25 °C (Lange and Dean 1967) and ${\gamma }_{\mathrm{ka}}$ is 0.0267 N/m at 25 °C (Landry et al. 2011). Also, ${\theta }_{\mathrm{k}}$ and ${\theta }_{\mathrm{w}}$ are 5° and 20°, respectively, leading to

$$\Delta \gamma 0.042\mathrm{ N}/\mathrm{m}.$$