Rapid In Situ Neutron Tomography and X-ray Imaging of Vapor Condensation in Fractured Sandstone

Abstract

Despite their diverse applications, experimental studies on multi-phase flow with phase change in fractured porous media are rare in the literature. In this study, water condensation of vapor in a fractured sandstone is investigated by means of 3D rapid in situ neutron tomography (30 s per tomogram). A water vapor and air mixture is injected at a constant rate, and the accumulation of condensed water is characterized through neutron tomography. Both phase-contrast and absorption-contrast X-ray tomography are jointly used to study the microstructure of the sandstone and measure the morphology of the splitting crack. The interplay between the pore matrix and fissure on the spatiotemporal distribution of water is studied through a combination of these techniques. The condensed water accumulates at the inlet boundary and gradually diffuses into the sample. Once the extracted crack is correctly aligned with with the neutron tomographies (registered), a propensity is seen for water to accumulate in small crack openings close to the cracks and migrate into the porous matrix predominantly due to the capillary effect. When enough liquid water condenses beyond a critical content, the air pressure in the crack transfers it into the porous matrix. This results in higher water content away from the crack. The water front propagates both along and transversely to the crack at a linear rate, albeit at different velocities. The final water distribution is found to be the result of two competing processes: condensation, occurring predominantly in the crack, and diffusion toward the matrix due to the capillary effect, as well as the pressure acting on the bottom of the sample and the crack walls.

Appendices

Appendices

1.1 Appendix 1: Pore Network Modeling of the Fontainebleau Sample

Compared to direct numerical simulations, pore network modeling makes it possible to simulate large networks derived from tomographic images of representative samples at a much lower computational cost. The extraction of the pore network from the synchrotron microtomography images was performed using the SNOW algorithm (Sub-Network from an Oversegmented Watershed) (Gostick 2017). It is an efficient algorithm, applicable to a wide variety of porous media, that computes the peaks of the distance transform between various constituents and progressively reduces the number of local maxima by ruling out several classes of peaks. The implementation of the algorithm is available in the open-source python package of PoreSpy (Gostick et al. 2019). OpenPNM Python package was used to apply the flow simulation on the pore resulting network (Gostick et al. 2016). The network was assumed to be filled with a single-phase under incompressible steady-state laminar flow, with two constant pressure conditions applied to two opposite sides. Intertial effects were assumed to be negligible, and the flow is governed by Stokes equation. The flow rate between each pore pair of the network was treated as flow through a pipe with a specified hydraulic diameter and a Poiseuille parabolic velocity profile. The flow rate was related to the pressure in each pore pair and the hydraulic conductance of the pipe element. A linear system of equations for the whole network was solved, the total flow rate of the network was calculated, and the numerical permeability of the network was estimated using the Darcy law.

In order to find a representative volume, cubic subvolumes of the porous matrix with 100 pixels growth in each step were investigated. In total, 20 cubic subvolumes were chosen within the entire sample volume and were analyzed in various locations in the sample. After 10 steps of numerical progressive increasing of the analyzed volume size to 1000$^3$ pixels$^3$ - 6.5$^3$ mm$^3$, the average permeability was found to vary less than 1.2% with respect to the previous step and was considered to be within representative elementary volume of the entire sample. The permeabilities were calculated in three directions, and the average permeability was in agreement with the permeability-porosity correlation of the Bourbie and Zinszner (1985), characteristic for Fontainebleau. The convergence of the permeability calculation as the dimension of the sampling subvolume increases is presented in Fig. 13a for one of the subvolumes. The porosity and permeability were found to be increasing as the sampling height from the bottom of the sample is increased, as shown in Fig. 13b. Less permeability along the height of the sample was measured compared to the radial direction.

Fig. 13

a The porosity and permeability convergence in three directions for a cube of the Fontainebleau sample using pore network modeling. z and x are the coordinates along the height and normal to the crack surface, respectively, as in Fig. 4b. b the porosity and permeability of the sampling subvolumes (cubes of 1000 pixels) as a function of the height from the bottom of the sample

1.2 Appendix 2: Quantification of the Zone Near Fracture Using Synchrotron Phase-Contrast Microtomography

Thanks to the sufficiently small pixel size of 6.5 $\upmu$m of the synchrotron microtomography, the pores and sand grains are clearly distinguishable with a relatively sharp edge which stems from the phase-contrast modality and allows the phases to be easily thresholded (e.g., using Otsu criteria). Due to the limitation in beam size, a core cylinder of 30 mm diameter was imaged in 5 sub-scans. Each scan has a 2.5 mm overlap with the upper and lower scans, which was used to stitch them together. The stitching was done using the SPAM package in Python (Stamati et al. 2020) where each overlap section in subsequent scans was registered to be spatially aligned in one large stack of images.

Since the segmented crack region is interconnected to the global network of pores and micro-cracks, two levels of median filter were used in order to separate the two. First, a 3D median filter with a size of [20 20 3] ${\text{pixels}}^3$ (longer along the crack surface) was applied to cut off the large branches and pathways that are normal to the crack surface. Next, to remove the smaller branches, a 2D median filter of [10 10] ${\text{pixels}}^2$ was used in each horizontal slice. The small particles corresponding to the pores were removed from the images so that only the single splitting crack remained in all the slices. The surfaces of this crack on the two sides were identified. The corresponding porosity of the original image on each of these two surfaces was calculated step by step as getting distance from the crack. Figure 14 shows a sample of the identification of the crack edge on the original image and the porosity as a function of distance from the crack surface. An elevated porosity can be seen within 50 $\upmu$m from the crack, which corresponds to a layer with micro-cracks and branches.

Fig. 14

The segmentation of the main crack without the micro-branches from the synchrotron microtomography scan and the porosity field as a function of distance from the crack wall. The yellow area shows the crack, and the green areas show the areas attributed to the elevated porosity

1.3 Appendix 3: Crack Segmentation and Morphological Analysis Based on Laboratory X-ray Tomography

In order to take advantage of the multi-modal aspect of the imaging measurements (neutron and x-ray in this work), it is necessary to extract the crack from the X-ray tomography data. The X-ray tomography was performed just before the injection procedure, capturing the sample geometry as close as possible to the in situ conditions. The goal was to analyze the sample crack morphology representative of injection conditions and register it with the neutron tomography data as to follow the evolution of saturation within and around the fracture.

The applied steps for the crack segmentation process are visualized in Fig. 15. The slices of the sample along the flow direction were treated sequentially:

• The reconstructed image (Fig. 15i) was filtered for high-frequency noise and sharpened by applying edge-preserving noise-reducing filters, namely Gaussian unsharp mask (blur radius of 6 pixels and mask weight 0.8) and bilateral filter (spatial and range radius of 3 and 50 pixels) (Fig. 15ii).

• Then, a threshold was used to pass to a binarized image that contained the crack field (Fig. 15iii)

• The resulting particles were analyzed, and those with a surface area of more than 5000 and circularity of less than 0.2 were segmented in the next step (Fig. 15iv).

• The resulting redundant small branches were removed, considering their connection to the main branch by applying a binary open operation, and then the same particle analysis process as (Fig. 15iv) was used to converge to only having the central part of the main fracture (manual editing is necessary in some cases).

• The resulting 3D representation of the crack (e.g., Fig. 15v) was used throughout this study and as a mask after being aligned to the neutron data to extract the distribution of water in the crack.

• The segmented crack was skeletonized by measuring its average value in the x-direction to have a one-pixel wide crack. It is used later on to determine the distance of each pixel from the crack using successive dilations of Fig. 15vi, resulting in what can be seen in Fig. 8.

Fig. 15

The approach to segment the crack from the X-ray image. (i) the original image of a vertical slice of the sample, (ii) applying edge-preserving noise-reducing filters, namely unsharp mask and bilateral filter, (iii) threshold of the crack and the pores, (iv) keeping the largest and the least circular particle, (v) the final crack where the small branches are separated by an open binary operation, and (vi) the skeletonized crack by measuring the averaged value in the x-direction

1.4 Appendix 4: Local Thermal Equilibrium Criterion

To manifest the local thermal equilibrium between the fluid and solid phases in quantitative manner, it is possible to use the criterion suggested by Carbonell and Whitaker (1984):

$$\begin{aligned} \begin{aligned} \frac{\epsilon (\rho C_{p})_{f} d^{2}_{p}}{t} \left( \frac{1}{\lambda _{f}} + \frac{1}{\lambda _{s}}\right)<< 1 \end{aligned} \end{aligned}$$

where $\epsilon$, $\rho$, $C_{p}$, $d_{p}$, t, $\lambda _{f}$, and $\lambda _{s}$ are porosity, fluid density, fluid specific heat, characteristic length scale of pore size, time scale, fluid conductivity, and solid conductivity, respectively. Substituting relevant values for vapor, given the characteristic length scale to be 25 ${\upmu }\hbox {m}$ (corresponding to median pore size), even for a time scale of the order of milliseconds, yields a value far below one ($\approx$ 10$^{-3}$).