Time‐And‐Space Averaging Applied to Intermittent Multiphase Flow Experiments

Various researchers have studied fluctuations in pore‐scale phase occupancy during multiphase flow in porous media using synchrotron‐based X‐ray microcomputed tomography (micro‐CT). However, the impact of these fluctuations on the concept of a representative volume is not yet fully understood. In this study, we performed spatial and temporal averaging of multiphase flow experiments visualized with synchrotron‐based micro‐CT, focusing on oil saturation as the key parameter to determine a representative time‐and‐space average. Our findings revealed that a saturation value representative of both time and space was achieved during fractional flow experiments in drainage mode with fractional flows of 0.8, 0.5, and 0.3. Furthermore, we computed a range of relative permeabilities on the basis of whether momentaneous saturation or time‐and‐space averaged saturation was utilized for direct simulation. Our results highlighted the importance of time‐and‐space averaging in determining a representative relative permeability and indicated that the temporal and spatial scales covered in a typical micro‐CT flow experiment were sufficient to obtain a representative saturation value for sandstone rock under intermittent flow conditions.


Introduction
Multiphase flow in porous media occurs in many natural and engineered systems, such as petroleum reservoirs (Blunt, 2001), soil remediation (Van Dijke et al., 1995), and hydrogen fuel cells (You & Liu, 2002).In petroleum engineering, Special Core Analysis (SCAL) tests are commonly performed to measure the relative permeability of a rock (Ebeltoft et al., 2014).Similar tests are conducted in other engineering fields to measure the mobility (or relative permeability) of a given fluid in a porous system (Francisca & Montoro, 2015;Greenkorn, 1981;Stylianou & DeVantier, 1995).Such tests are commonly conducted under the steady co-injection of two immiscible phases where the total pressure drop is measured at different fractional flows (McPhee et al., 2015).Fluctuations in pressure and/or fluid saturation are commonly observed, but are regarded as noise, and thus average values are commonly reported with no theoretical basis for averaging the fluctuations, while the fluctuations are not necessarily random and do represent relevant flow physics (Rücker et al., 2021).Currently, with the development of synchrotron-based X-ray microcomputed tomography (micro-CT) technology, combined with image processing tools, we can observe inside the pore structure of a porous system and monitor the flow behavior of fluids (Berg et al., 2013;Blunt, 2001).
Recent literature refers to the phenomenon of fluctuations during multiphase flow as "intermittency," a term that is central to fully developed turbulence and has been extensively studied in classical works by Kolmogorov and Obukhov (Birnir, 2013).The concept of intermittency in turbulent systems has been applied analogously to the complex dynamics observed during multiphase flow in porous media (Gao et al., 2017(Gao et al., , 2019;;Spurin et al., 2021), and aligns with the broader application of the term used for nonlinear dynamics (Pomeau & Manneville, 1980).While intermittency is sometimes associated with a nonlinear flow regime at a high Capillary number near the onset of capillary desaturation (Zhang et al., 2022), intermittent behavior has also been observed for ganglion dynamics at low Capillary number, that is, less than 10 6 (Rücker et al., 2015).In recent porous media literature, intermittency was associated with the observation that pore occupancy by one immiscible fluid or the other fluctuates (possibly over periodic time scales) during the co-injection of immiscible fluids.
Fluctuations during multiphase flow can be caused by a variety of factors, such as the rapid and discontinuous movement of interfaces caused by pore-scale displacement events (Haines jumps, snap-off), and associated changes in flow velocity, pressure, and pore occupancy.Fluctuations have been shown to influence the stability and efficiency of flow paths of multiphase systems (Spurin et al., 2020).Intermittent phenomena in multiphase flow appear to be common, such as when a pore alternates between a nonwetting phase and a wetting phase occupancy.Various researchers have observed the existence of intermittency on the pore scale and on the millimeter (mm) scale to the centimeter (cm) scale (Armstrong et al., 2016;Gao et al., 2019;Lin et al., 2016;Rücker et al., 2021;Spurin et al., 2019a).For example, Armstrong et al. (2016) observed ganglion dynamics at the pore scale, with the snap off and coalescence of the fluids, and connectivity that changed continuously even at fixed bulk saturation.Gao et al. (2019) measured relative permeability by experimentation at the mm-scale, which compared well with the results of Lin et al. (2016) with measurements taken at the cm-scale.Gao et al. (2019) also observed that the occupancy of the pore space by oil and water was not constant; conversely, there were intermittent regions in which the occupancy alternated between the two fluids over a given time period.These regions appeared to be important for the nonwetting phase to flow even if connectivity was only present periodically.Furthermore, Spurin et al. (2019b) showed that intermittency was important for the flow of the non-wetting phase (NWP) because it provided additional flow pathways.In this way, the connectivity of the NWP was improved over a periodic intermittent time scale.Spurin et al. (2019b) also observed that intermittency occurred mainly in medium-sized pores and regions where the fluids are not well connected, or at least the NWP itself was not well connected.Lastly, Spurin et al. (2022) observed red noise during co-injection experiments, the red noise, also known as Brownian noise, indicated that the pressure fluctuation associated with intermittency occurred over a wide range of time scales, that is, from approximately milliseconds to minutes.
Since intermittency has been identified as having a physical origin linked to mechanisms that dissipate energy at a significant rate (Berg et al., 2013;Rücker et al., 2021) it poses a complication for the Darcy scale interpretation of experimental data.The aspects of discontinuous connectivity and nonergodicity (McClure et al., 2021a) have created a challenge for the correct treatment through the length scales.When upscaling from the pore scale to the Darcy scale, fluctuations are commonly assumed to average out, and thus a representative volume can be defined such that a single value of, for example, phase saturation is representative for a given fractional flow.However, information does not propagate instantaneously within the system: the time scale for mixing is controlled by diffusive mechanisms, which are slow.For this reason, we should not expect to identify a valid spatial Representative Elementary Volume (REV) without also considering the relevant temporal scale.Although intermittency may ultimately be caused by discontinuous events triggered by displacements on the pore scale, such as Haines jumps and snap-off, it appears to be a multiscale phenomenon observed on the pore scale, on the mesoscale as ganglion dynamics (Rücker et al., 2015), and on the Darcy scale as traveling waves (Rücker et al., 2021) with a continuous transition between the different scales.This poses a challenge to the traditional definition of a REV for multiphase flow, as there is a continuous transition between displacement physics clearly associated with the pore scale and fractional flow physics clearly associated with the Darcy scale (Rücker et al., 2021).
The analysis of a REV is typically based on a spatial analysis only, as the name implies, such as measuring porosity, permeability, or static phase saturation over different length scales.However, the work by Rücker et al. (2021) suggested that there was no such averaging scale for saturation and pressure where fluctuations would average out based on a spatial analysis alone.Alternatively, McClure et al. (2021b) proposed time-andspace averaging to account for energy dynamics in fluctuating systems.Relative permeability can be derived based on the conservation of energy equation, with explicit criteria to define stationary conditions (McClure et al., 2022).The question that arises is whether the observed fluctuations result in any net accumulation of energy.Simulation work has shown that the sum of energy fluctuations over a long enough period of time is zero for steady co-injection conditions (McClure et al., 2022).However, there is no experiment work that explores the energy fluctuations using the proposed time-and-space averaging approach, which is a key aim of the current work.
Herein, we study intermittency during steady fractional flow using the time-and-space averaging approach proposed by McClure et al. (2021b) to provide new information on the temporal and spatial scales required for representative relative permeability values.We explore fractional flow experimental images collected using synchrotron-based X-ray micro-CT.Pore-scale simulations using the Lattice-Boltzmann method (LBM) (McClure et al., 2021c) are used to determine the relative permeability for a range of observed (momentaneous) and time-and-space averaged fluid saturations.Various works have used the LBM for relative permeability and capillary pressure simulations, for example, (Ahrenholz et al., 2008;Armstrong et al., 2016;Zhao et al., 2017).Recent works have also shown that imaged fluid distributions under connected pathway flow provide predictive relative permeability, for example, (Berg et al., 2016;Hussain et al., 2014;Wang et al., 2020).For intermittent flow, these connected pathways vary depending on the moment at which they are observed (Reynolds et al., 2017).Then the question arises: To what extent is relative permeability impacted when the flow pathways change during intermittent behavior, and on what temporal and spatial scales do the fluctuations average out?
In the following, we first explain the experimental images collected using synchrotron-based micro-CT followed by an explanation of the temporal REV analysis versus the spatial REV analysis used.Our work focuses on oil saturation as a target parameter for time-and-space averaging.Multiphase flow simulations are then conducted using the LBM to determine relative permeability under different fluctuating (momentaneous) states versus relative permeability for a time-and-space averaged saturation.Lastly, we provide practical guidance for the study of intermittent multiphase flow systems using time-and-space averaging.

Experimental Data
Bentheimer sandstone (ϕ = 23.8%,K = 1.3 Darcy) was used for the presented fractional flow experiments.The sample size was 5 mm in diameter and 10 mm in length, and imaged at a resolution of 3.5 µm.The scan was taken from the middle of the core sample.The sample was cleaned with ethanol followed by DI water and then dried in a vacuum oven until a constant weight was achieved.Following this drying process, a dry scan was conducted.Brine was used as the wetting phase, it was prepared using a weight ratio of potassium iodide to water at 1 to 6.The NWP used was decane.The sample was initially saturated with brine, followed by co-injection of brine and decane in a drainage-mode fractional flow experiment.The fractional flow is defined as F w = q w /(q w + q o ), where q o and q w are the Darcy flux for oil and water.In our study, the calculation of the Capillary number was based on the definition provided by Foster (1973).The Capillary number (Ca) was calculated using the formula: where v represents the velocity of the fluid, μ is the viscosity of the fluid, σ is the interfacial tension between the brine and the decane and ϕ is the porosity of the sample.Consequently, the total flow rate of 100 μL/min in our experiments corresponds to a Capillary number of 1.19 × 10 5 .The fractional flow sequence was F w = 0.8, 0.5, and 0.3.Dynamic 3D images were obtained from GeoSoilEnviroCARS Sector 13 at Argonne National Laboratory Advanced Photon Source (APS).Pink beam was used for imaging and images were collected in 20 s with a 20 s delay between sequential scans.Each time the fractional flow was changed, we waited for at least three pore volumes before capturing the images.The spatial and temporal data collected are summarized in Table 1.In total, 43 3D images were captured at F w = 0.8, spanning a cumulative duration of 860 s.Furthermore, 48 3D images were obtained at F w = 0.5, totaling 960 s, while 8 3D images were acquired at F w = 0.3, representing a combined duration of 160 s.Total Pore Volumes Injected during imaging is also reported in Table 1.
For image processing reasons, we obtained a single dry image of the sample in addition to the dynamic images.All images were registered to the dry image.After registration, we denoised the image using a non-local means Water Resources Research 10.1029/2023WR036577 filter, similar to the approach used by Higgs et al. (2024).Next, the dry image was segmented into two phases, solid and pore.Dynamic images were first segmented into two phases, the NWP phase and all other phases.Then combining the segmented dry image and segmented dynamic images resulted in a three-phase segmented image.However, in this way, the segmentation result was not as good as we expected; as shown in Figure 1d, the NWP does not align well with the solid grains creating an artificial gap.To alleviate this problem, we dilated the oil phase by 3 voxels, while keeping the solid unchanged, then located the overlapping voxels and set them to grain phase.The final result is shown in Figure 1e.For each fractional flow data set, we performed the same image processing workflow, and the resulting images are 1340 × 1325 × 1184 voxels in size.

Time-And-Space Analysis
In a fractional flow experiment for a mostly homogeneous domain, in absence of spatial gradients, each position in the flow direction is, in principle, equivalent and can therefore be used to obtain statistics of the system.Therefore, fluctuations in saturation, that is, intermittency, can be studied over different time and length scales.A single image slice could be subject to large temporal fluctuations because spatial information is limited.On the contrary, an entire image domain over two sequential images could result in no observed fluctuations because the temporal information is limited.Clearly, these two extremes are not representative of the flow.However, one  To determine a representative time-and-space domain for oil saturation, we conducted two types of REV analysis -"spatial" and "temporal," as shown in Figure 2.Although the experimental data include both temporal and spatial information, how the information was combined to generate subsets defines the REV analysis used.A spatial REV analysis was performed by generating a subset of images at different times.This subset is then sampled with increasing spatial information.A temporal REV analysis was performed by generating a subset of images with different volumes.Then, this subset was sampled with increasing temporal information.Both approaches consider time and space, and thus should converge to the same representative value.However, the convergence of each method is expected to occur in different combinations of temporal and spatial information.
Spatial REV analysis was performed by sampling subvolumes of varying size from a given subset of images.First, we need to define a subset of images.For example, a single fractional flow image represents 20 s in time.
Each additional fractional flow image added to the subset increased the sampled time scale by 20 s.As temporal information is added, the size of the subset increases by N × (1340 × 1325 × 1184) voxels, where N is the number of fractional flow images added.Oil saturation was then measured by sampling subvolumes of varying size from the subset; hence the name, "spatial REV" analysis.The sampling subvolumes ranged from 100 3 to 1340 × 1325 × 1184 voxels.For each subvolume, we randomly perform 50 calculations of oil saturation from the subset and represent the individual measurements using scatter plots.The "average oil saturation" for each subvolume size was also computed by using all 50 measurements.
Temporal REV analysis was performed by sampling a subset of images at different time scales.First, we need to define a subset of images.For example, a single image slice (S = 1) analyzed for N fractional flow images provides (1340 × 1325 × S) × N voxels of information.We created subsets with S = 1, 500, and 1100.The oil saturation was then measured by sampling the subset over different time scales; hence the name, "temporal REV" analysis.The time scales sampled ranged from 20s to the total duration of the given fractional flow.For each sampled time scale, we randomly performed oil saturation calculations for 50 possible sequential sets of images that represent the given time scale.The "average oil saturation" for each sampled time scale was also computed by using all 50 measurements.
We used the following criteria to determine data convergence, that is, when a representative time-and-space average was obtained.(a) The average oil saturation for a given time-and-space sampling is required to be no more than 3% different from the total oil saturation.(b) The fluctuation range of the maximum and minimum oil saturation for a given time-and-space sampling must fall within a 3% difference.In this study, "total oil saturation" is defined as the average oil saturation across all images for a given fractional flow experiment.The "average oil saturation" is determined by conducting 50 random measurements and then calculating their average oil saturation for each subvolume of information.

2-Phase Flow Simulations
Relative permeabilities were determined by LBM simulations (Da Wang et al., 2021;McClure et al., 2021c;Wang et al., 2020) using the time-and-space average saturation in addition to the momentaneous minimum and maximum saturation from each fractional flow experiment as an initial condition.The color gradient method was used to simulate multiphase flow of decane and water in the pore space of the rock, and a body force was applied to the fluids with periodic boundary conditions to emulate a co-injection scheme under steady state conditions at a Capillary number of 10 5 .LBM simulation are limited in their ability to reach very low Capillary numbers within the capillary-dominated flow regime due to spurious currents in the simulation (Wang et al., 2020).The noise floor of these simulations is approximately 10 5 , which represents the value of the Capillary number where the capillary-dominated flow begins for Bentheimer sandstone.In our experiments, the measured Capillary number is 1.19 × 10 5 , which indicates that both our experiments and simulations are within the capillary-dominated flow regime.The periodic boundary conditions also eliminated the possibility of saturation fluctuations, and thus provided relative permeability for a user-defined saturation.The inlet and outlet boundaries were aligned with 16 2D slices of a transition porous structure that was a wall-distance weighted average of the inlet and outlet faces.The simulations were initialized with a segmented fluid distribution at the saturation (or fractional flow) of interest and run to steady state under multiphase flow conditions.The simulation domain measured 1340 × 1325 × 1166, and utilized 80 Nvidia V100 GPUs each from the Gadi supercomputer (Australian Government, National Computational Infrastructure (NCI)).The simulations were run at each fractional flow, with the aforementioned boundary conditions until the measured relative permeability stabilized, that is, the values remained equal to 4 significant figures over 1,000,000 time steps.Full details of the steady-state simulation protocol are reported in McClure et al. (2021c).

Results and Discussions
Figure 3 demonstrates the observed variations in the NWP occupancy during steady fractional flow.The first row of images illustrates the changes in NWP when F w was 0.8, while the second and third rows show the changes when F w was 0.5 and 0.3, respectively.It is noteworthy that intermittency was observed in these fractional flows.Observations of intermittency in our experiments with a Capillary number of 1.19 × 10 5 are consistent with the phenomena described in the work of Gao et al. (2020).
For F w = 0.8, Figure 3a represents the initial stage with a time step of 220 s, Figure 3b represents the intermediate stage with a time step of 400 s, and Figure 3c illustrates the final stage of this fractional flow with a time step of 860 s.This pattern is consistent in the other two fractional flows as well.For F w = 0.5, Figures 3d and 3f represent the first and last stages, respectively, with the final time step being 960 s.Similarly, for F w = 0.3, Figures 3g and  3i represent the first and final stages, respectively, with a final time step of 160 s.Figures 3e and 3h represent the intermediate stages with a time step of 400 and 100 s, respectively.Observations from Figure 3g, where the pores within the square are colored red, indicated that these pores are filled with decane at this stage.Subsequently, the decane was displaced by water in the intermediate stage and then decane reoccupied the pores in the final stage.
On the contrary, during the other two fractional flows, F w = 0.5 and F w = 0.8, the observed sequence was water being displaced by decane and subsequently replaced by water again.Therefore, intermittency was an observable phenomenon during these fractional flows.
From the spatial analysis (Figure 4), it was observed that all fractional flows exhibited more pronounced fluctuations when there was a limitation in the temporal information provided.For F w = 0.8, fluctuations were more obvious at 20 s than those observed under other temporal conditions.An increase in time information to 500 s resulted in our convergence criteria being met at a subvolume size of 1000 3 voxels.Furthermore, for F w = 0.5, when only 20 s were considered, we do not ever meet our convergence criteria even if we consider the largest volume possible.However, we meet our convergence criteria when more temporal information was considered.At 500 s, the data demonstrated convergence at a subvolume size of 1260 × 1260 × 1184 voxels.In the case of F w = 0.3, significant fluctuations were observed at 20 s.When more temporal information was added to 100 s,

Water Resources Research
10.1029/2023WR036577 WANG ET AL.
convergence was achieved when the volume size reached 1184 3 voxels.Therefore, it can be concluded that F w = 0.5 exhibits more fluctuations compared to the other two fractional flows.However, all three fractional flows demonstrated convergence within the experimental time frame for the rock volume imaged.This observation aligns with the findings reported by Rücker et al. (2021), which suggest that fluctuations in fractional flow may follow non-obvious trends, that is, some fractional flows exhibit more fluctuations than others.
The temporal analysis (Figure 5) reveals that for F w = 0.8, the initial results obtained from S = 1 (1340 × 1325 × 43 voxels) provided no convergence.However, convergence was observed as volume increased, as demonstrated in S = 500 and S = 1100, respectively.Convergence was reached at 400 s for S = 500, and 200 s for S = 1100.In the case of Fw = 0.5, the initial calculation based on S = 1 (1340 × 1325 × 48 voxels) deviated significantly from the total average oil saturation.However, as more volumes were integrated, convergence was reached in 900 s when S = 500 was considered and in 260 s when S = 1100 was considered.Under the condition of F w = 0.3, the average oil saturation of S = 1 (1340 × 1325 × 8 voxels) closely approximated the total oil saturation; however, the fluctuation range was quite large.As more volumes were incorporated, the system reached our convergence criteria at 160 s for S = 500 and 100 s for S = 1100.
In the formation of fluid interfaces, a larger frequency and magnitude of fluctuations lead to more energy consumption.This phenomenon is especially pronounced in the case of F w = 0.5, where a larger time-and-space average is required.These fluctuations can be attributed to the dynamic competition between the two immiscible fluids as they compete for the pore space.Gao et al. (2017) also mentioned that for the case where F w = 0.5, the percentage of intermittency phase is 0.328, a value that surpasses those observed under other conditions.For such cases, it is clear that a larger spatial volume and temporal average would be required to determine a representative value of saturation, which is well-aligned with our results presented herein.
Figure 6 reveals the specific combinations of time and space in which convergence was achieved in our experiments, based on the convergence criteria provided in Section 2.2.For example, for Fw = 0.8, the system meets convergence at a volume size of 8.88 × 10 8 voxels at 400 s, which corresponds to a physical size of approximately 800 mm 3 .Additionally, for Fw = 0.5, convergence is achieved for a volume size of 1.95 × 10 9 voxels at 260 s, equivalent to roughly 1,000 mm 3 .With the fractional flow of Fw = 0.3, the system reaches convergence faster than the other two fractional flows.This is evident as convergence is reached at 160 s for a volume size of 8.88 × 10 8 voxels or around 300 mm 3 in physical terms.
In Figure 7, we present the relative permeabilities derived from the simulations, illustrating the expected results for each fractional flow.The plot draws attention to the range in relative permeability values for both water and decane under each specific fractional flow condition, providing the relative permeabilities at time-space averaged saturations, along with relative permeability values at the maximum and minimum momentaneous saturations observed for each fractional flow.By means of the error bars, it is apparent that if we focused only on a single image (momentaneous saturation), without taking the time-space averages into account, the resulting relative permeability could arbitrarily fall anywhere within the maximum and minimum error bar range.For example, the maximum value might represent a high-permeability phase morphology that forms intermittently during flow, whereas the minimum value represents a lower-permeability phase morphology.Therefore, these error bars highlight the potential variability that may occur if time-and-space averages are ignored.We also know that the heterogeneity of the pore space has an influence on the relative permeability.Therefore, incorporation of timeand-space averaging provides a thorough and logical approach to characterizing fluid saturation, thus enabling the acquisition of representative relative permeability curves.

Conclusion
We performed a representative analysis in a way that differs from conventional REV methods by studying the significance of time-and-space averages using core flooding experiments imaged with synchrotron-based micro-CT.We observed intermittency under fractional flow conditions, where certain pore regions were first occupied by water, then displaced by decane and eventually reoccupied by water.With time-and-space averaging, we observed convergence for a typical synchrotron-based micro-CT fractional flow experiment.This suggests that the temporal and spatial scales sampled were sufficient to provide a representative saturation value for the sandstone rock under intermittent flow.The system reached convergence within the experimental time frame for all fractional flows, while F w = 0.5 showed more fluctuations and a prolonged convergence time.Overall, timeand-space averaging provided a means to determine an invariant saturation for the tested fractional flow conditions.
Our study's application of a 20-s temporal resolution with synchrotron-based micro-CT imaging captures the intermittency phenomena of multiphase flow dynamics, particularly emphasizing the importance of time-andspace averages in understanding intermittent flow behaviors under different fractional flow conditions.
Although our analysis provides insight into these phenomena within the specified temporal resolution, it is recognized that flow dynamics occurring over significantly shorter or longer timescales also play a role in the broader understanding of multiphase flows.Previous studies (Spurin et al., 2019a(Spurin et al., , 2020) ) have shown the importance of intermittency in multiphase flows on shorter and longer timescales.Yet, the key objective is to find a separation in the time scales to time-and-space average the relevant phenomena; the focus herein was on intermittent behavior.In addition, to address the complexities observed in core-scale experiments, future studies can specifically investigate the impact of physical length scales on flow behavior, considering both pore geometry and a broader spectrum of space and time scales.
For steady flow to occur (no intermittency), both phases would need to form stable connected pathways.Intermittency appears to occur when the connectivity of one phase is occluded by the other phase resulting in fluctuations between different geometric states.We observed that such behavior is most prevalent at intermediate fractional flows, which corresponds to the minimum total phase mobility, that is, when the obstruction of one phase's mobility to the other and vice versa is a maximum.Such conditions are likely to favor intermittency over stable connected pathway flow.
We speculate that percolation-like critical points could be identified near intermediate fractional flows.Prolonged fluctuations on significant time scales could be an indication of critical behavior.The noise signature of pressure transducer readings during fractional flow (McClure et al., 2021b;Spurin et al., 2022) also suggests the development of critical behavior (Bak et al., 1987).While we cannot provide further evidence herein in regard to critical behavior, our current work demonstrates that time-and-space averaging provides a practical way to deal with intermittency.Future research could focus on the critical behavior of fractional flow with inspiration from the classical works of Kolmogorov and Obukhov (Birnir, 2013).From a practical perspective, an invariant saturation is important for applying the two-phase extension of Darcy's law, in addition to other upscaling approaches.With LBM simulations, we compared relative permeabilities determined from momentaneous saturations with the time-and-space averaged saturation.We observed a range of momentaneous relative permeabilities since a range of momentaneous saturations were observed for any given fractional flow.Time-and-space averaging provided a means to average the momentaneous saturations over long enough time scales to provide an invariant saturation for a given fractional flow.Future research should focus on the time correlation of the intermittent states, given that highly correlated states will contribute toward the timeand-space averaged saturation more so than longer time scale oscillations.For experimental measurements of relative permeability, we suggest that averaging of state variables should be performed in both time and space.
Disregarding the time-and-space average could lead to inaccuracies in determining representative relative permeabilities, since either a momentaneous state or some combination of momentaneous states that do not necessarily represent the full fractional flow condition could be used.

Figure 1 .
Figure 1.An example of the image processing workflow.(a) Is the dry image; (b) is one of the wet images; (c) is the segmented dry image (pore is black and grain is white); (d) is the initial segmented image; and (e) is the image after dilation.The red square in panel (a) highlights the region that is enlarged in panels (c), (d), and (e).

Figure 2 .
Figure 2. Methodologies for analyzing Representative Elementary Volumes (REVs) are depicted.(a) focuses on Temporal REV by observing the region of interest over various time points.(b) Explores Spatial REV by analyzing ROIs of increasing size.Panel (c) synthesizes these information in a combined Spatial and Temporal REV analysis.(d) and (e) Illustrate examples of oil saturation convergence over time and space, respectively.

Figure 4 .
Figure 4. Spatial Representative Elementary Volume analysis of different fractional flows.(a), (b), (c) are for F w = 0.8 for subsets of 20, 500, and 860s, respectively; (d),(e), (f) are for F w = 0.5 for subsets of 20, 500, and 960s, respectively; (g), (h), (i) are for F w = 0.3 for subsets of 20, 100, and 160s, respectively.Each data point represents the oil saturation for a subvolume of a given size taken from a given subset.

Figure 6 .
Figure 6.Time and space requirement for different fractional flows, (a) is for Fw = 0.8, (b) is for Fw = 0.5, and (c) is for Fw = 0.3.Green circles (a), squares (b), and triangles (c) represent points that have reached convergence, whereas red circles (a), squares (b), and triangles (c) represent points that have not.

Figure 7 .
Figure 7. Relative permeability differences between water (Krw) and decane (Kro) at various fractional flows, shown on a linear scale (a) and a logarithmic scale (b).

Table 1
Spatial and Temporal Data Collected During Synchrotron-Based Micro-CT Imaging of Fractional Flow Experiments