Pore-Scale Fluid Dynamics Resolved in Pressure Fluctuations at the Darcy Scale

Complex pore-scale dynamics have been observed during multiphase flow through porous rocks. These dynamics are not incorporated in large scale models for the migration and trapping of subsurface fluids such as CO 2 or hydrogen. We show that fluctuations in pressure measured at the core-scale (centimeters) can reflect fluid displacements at the pore-scale (millimeters). The spectral characteristics of pressure data are shown to depend on the flow dynamics, the size of the rock sample, and the heterogeneity of the pore space. These results show that pressure data, transformed into the time-frequency domain using wavelets, provides information about flow dynamics, across scales, that are otherwise challenging to acquire.

• Pore-scale dynamics can be linked to pressure fluctuations measured across the core • The spectral signature of the pressure fluctuations can be used to classify the dominant flow regime • By exploring the spectral signature of pressure fluctuations at the larger scale, we are able to infer the underlying pore-scale dynamics

Supporting Information:
Supporting Information may be found in the online version of this article.
Micro-computed tomography (Micro-CT) experiments provide pore-scale observations of fluid-fluid interfaces in situ at resolutions of a few microns.However, experimental limitations including temporal resolution, expense, and management of the vast quantities of data produced, mean it is currently infeasible to observe fluid-fluid interfaces at the centimeter to meter scale (the core-scale).Instead, medical CT scanners are used to measure saturation distributions (Akin & Kovscek, 2003;Krevor et al., 2012;Pini & Madonna, 2016).If observations of flow from pore-scale experiments are representative of flow at larger scales they can, in principle, be used to understand results from core-scale experiments.However, it is unclear if information about flow dynamics is being lost due to the limited spatial and temporal scales of the pore-scale experiments, or if pore-scale dynamics differ when sample size is increased.For example, viscous and gravity forces may become more important at larger scales, even in capillary-dominated regimes.
Pore-scale and core-scale experiments have two overlapping quantities that are measured: saturation and pressure.Saturation is important, as it can indicate the amount of trapping, but without a measure of connectivity, it gives no indication of the underlying dynamics.However, pressure fluctuations have been related to pore-scale dynamics, and energy dissipation in the pore space through the creation and destruction of interfaces (Rücker et al., 2021;Spurin et al., 2022).In this work, we explore how pressure fluctuations measured during core-scale experiments can be used to provide insight into the underlying flow dynamics by using continuous wavelet transforms to map the spectral power of pressure data.We identify sources of spectral power as a function of time and frequency.The merits of using pressure data to obtain information about multiphase flow in porous media, including possible scaling relationships, are assessed.

Experimental Procedure
The experiments in this work were conducted at two different scales: the pore-scale and the core-scale.For the pore-scale investigation, the sample was a carbonate rock, 5 mm in diameter and 20 mm long.These experiments were conducted at a synchrotron facility, so fluid interfaces could be resolved in real time (Spurin et al., 2020).
There are two experiments in the pore-scale investigation, which both explored the transition to steady-state dynamics.One experiment observes intermittent pathway flow through the co-injection of gas and water, while the other observes connected pathway flow through the co-injection of oil and water (Spurin et al., 2020(Spurin et al., , 2021)).The capillary number, defined as Ca = q/σλ where q is the flow rate, σ is the interfacial tension and λ is the mobility of the fluids was 1.6 × 10 −7 for the gas/water experiments and 2.2 × 10 −6 for the oil/water experiments.See Spurin et al. (2020) for a full experimental description.
For the core-scale investigation, the sample was a carbonate rock, 5 cm in diameter and 12 cm long.The experiments were conducted in a medical CT scanner, so the fluid interfaces themselves cannot be resolved, but the saturation across many pores is measured (see Figure 1 for the difference in imaging resolution at the different scales).Three experiments were performed to explore the transition to steady-state dynamics; two explore the co-injection of gas and water.The same sample was used for both these experiments, but the sample orientation was reversed between experiments, to explore the role of heterogeneity of the pore space on flow dynamics.For the third experiment, oil and water were co-injected.The sample orientation was not reversed in this experiment as oil is difficult to remove from a sample, which would have influenced the observations.The capillary number was 2.0 × 10 −8 for the gas/water experiments and 5.4 × 10 −7 for the oil/water experiment.With similar capillary numbers, we aimed to observe the same manifestation of the pore-scale dynamics in the pressure data as the pore-scale experiments.The full experimental procedure is provided in the Supplementary Information.
An example of the images taken during an experiment at each scale is shown in Figure 1, which highlights the impact of the different imaging resolutions for the experiments.It demonstrates how connectivity of the fluid phases cannot be calculated from traditional medical CT imaging.Thus, due to these imaging constraints, the only parameters that are constant across the experimental scales are saturation and pressure.The pressure drop across the sample with time is the parameter of interest in this work, and was recorded for all experiments using a differential pressure transducer connected to the inlet line for the water, and the outlet line for both phases. 10.1029/2023GL104473 3 of 11

Spectral Analysis Using Wavelet Transformation
The spectral content of the pressure data was investigated by transforming it into the time-scale (and equivalent time-frequency) domain using a continuous wavelet transformation (CWT).This differs from previous work using Fourier transformation of pressure data that revealed a cascade of timescales for steady-state multiphase flow, with lower frequency events having larger amplitudes (Spurin et al., 2022).While insightful, Fourier transforms have some limitations that make further analysis difficult.These include significant power spectral leakage, noisy calculated power spectra, and the fact that stationary functions are unlikely to reflect changes in pressure as a result of flow, especially during transient flow, when average pressure is a function of time.Mapping spectral power as a function of frequency and time might provide additional insight into the dynamics of the system.
We use a transform that convolves a uniformly sampled pressure data time series, p t , with a mother wavelet, ψ.
Pressure time series, with constant sampling intervals of either δ t = 1.29 s (pore-scale experiments; Figure 2) or 9.3 s (core-scale experiments; Figure 3), were mirrored 7 times before being transformed to ameliorate edge effects (Roberts et al., 2019).The Derivative-of-Gaussian (DoG) wavelet, with derivative m = 6, was used as the mother wavelet in this study (see Supplementary Information for insight on choice of wavelet and derivative).It was scaled and translated along the time series by t′ to reveal variations in amplitude as a function of scale, s, and time, t.The wavelet transform, W t (s), has the form: where ψ* denotes the complex conjugate of the mother wavelet.N is the number of discrete measurements of pressure.In this study, N = 385 for the gas/water core-scale experiments (total sampling duration ≈ 1 hr), for the oil/water core experiment N = 577 (≈1.5 hr).For the pore-scale experiments N = 24,991 (≈9 hr) and The wavelet transform can be converted into power, ϕ, such that ϕ(t, s) = |W t (s)| 2 .The time-averaged power spectrum is thus: (2) Following Liu et al. (2007), power is rectified by scale.Minimum scales, s • = 2δ t , scales   = •2   , where j = 0, 1, …, J, and d j = 0.1.The largest scales, J, are determined by the length of the time series.Scales are converted  into equivalent Fourier frequencies,   = √ ( + 0.5)∕2 for the DoG mother wavelet used in this study (Torrence & Compo, 1998).It is now straightforward to compare calculated spectra to spectra generated using different mother wavelets, and to Fourier transformed series.Rectified time-averaged power spectra, ϕ r = ϕ(s) s −1 (dropping the j notation) are consistent with results obtained from Fourier transformation of the time series.
Relationships between power spectral amplitudes and frequencies, f, provide insight into the scaling regimes and dynamics of many physical systems (Fernandes et al., 2022;Moura et al., 2017;Rudnick & Davis, 2003;Spurin et al., 2022;Van der Schaaf et al., 2002).Many geophysical time series are characterized by:  ∝   . (3) Determining the value(s) of α from the power spectra of time series can be a way to identify scaling For example, α = −2 indicates that a time series can be characterized by red noise.If pressure time series exhibit red noise characteristics, it implies that the amplitudes of the pressure perturbations are directly proportional to their duration.White noise, α = 0, indicates that the amplitudes of pressure perturbations are roughly evenly distributed across all frequencies.A variety of other noise distributions and changing patterns of spectral content can be straightforwardly identified by plotting power as a function of frequency in log-log space.For example, black, pink, and blue noise have spectral slopes, α, of −3, −1, and 1, respectively.

Sources of Spectral Power
There are many different potential sources for the spectral power in pressure time series during multiphase flow.The main ones identified here are (a) flow mechanisms, such as intermittent or connected pathway flow, (b) heterogeneity of the pore space, and (c) the ratio of capillary to viscous forces.
In this paper we focus on flow mechanisms, and their manifestation in pressure signals in pore-scale experiments.We explore links between spectral power and different flow regimes.We then explore if pore-scale spectral scalings can be applied to core-scale experiments, which would allow flow regimes to be deduced without pore-scale imaging.With the larger samples, we explore the role of heterogeneity on fluid flow by repeating the experiment with the sample orientation reversed, so that the direction of flow relative to the heterogeneity is reversed.Note that the heterogeneity is linked to the flow mechanisms (Spurin et al., 2019a), so it is not trivial to isolate them.With larger cores, viscous forces may become significant, even at low capillary numbers (Wang et al., 2023).This may result in different spectral scaling relationships even at equivalent capillary numbers.

Pore-Scale Results
The results for the pore-scale experiments are shown in Figure 2, with panels a-d showing the results for the gas/ water experiment and panels e-h showing the results for the oil/water experiment.Panels a and e in Figure 2 show the pressure drop across the sample recorded during an experiment for gas/water and oil/water, respectively.The shaded green strips correspond to the time intervals for the time-averaged power spectra shown in panels d and h, with a later time denoted by a darker shade.Note these panels indicate ∼1 hr intervals for the gas/water experiment, and ∼30 min intervals for the oil/water experiment because steady-state was reached quicker during the oil/ water experiment.Figures 2b and 2f show power spectra of pressure data with time for gas/water and oil/water, respectively.Here, the dashed lines correspond to the shaded green strips in panels a and e. Figure 2c-d and g-h show time-averaged power against frequency for gas/water and oil/water, respectively.Spectra are shown for the full recording window, and the first and second half of the pressure time series in Figures 2c and 2g, which can be compared to the evolution of the power spectra for shorter intervals in Figures 2d and 2h.
With the pore-scale experiments, we can relate power spectra to different flow regimes observed during the experiments.For both experiments, the sample is initially saturated with water.First, the non-wetting phase (the gas or oil) percolates the sample, resulting in purely drainage events (gas or oil displacing the water).At approximately 20,000 s for the gas/water experiment (Figure 2a) and 3,000 s for the oil/water experiment (Figure 2e) the pressure plateaus, marking the transition to steady-state flow.For the gas/water experiment this leads to intermittent pathway flow, where gas flow pathways repeatedly connect and disconnect (Spurin et al., 2020).For the oil/water experiment no further displacement events occur during steady-state flow; the fluids flow in their own separate pathways that are connected across the pore space (Spurin et al., 2020).

Intermittent Pathway Versus Connected Pathway Flow
For the gas/water experiment fluid rearrangement events were larger and occurred even during steady-state flow, while the oil/water experiment had little to no fluid rearrangement once oil had percolated the sample (Spurin et al., 2020(Spurin et al., , 2022)).The different flow regimes are evident in the pressure in Figures 2a and 2e.First, in the oil/ water experiment the pressure overshoots the stabilization pressure (at around 3,000 s in Figure 2e), but then relaxes to approximately 65 kPa for the rest of the experiment.In the gas/water experiment, the pressure builds more and then plateaus at approximately 20,000 s in Figure 2a.There are significantly more fluctuations during the gas/water experiment, even if the mean pressure remains constant, due to intermittent gas pathways periodically connecting and disconnecting.
Variations in the pressure time series are highlighted by the power spectra, produced by the CWT, shown in Figures 2b and 2f.Several results are evident from the wavelet power spectra, which were not immediately obvious from inspection of the pressure time series alone.First, pressure at the longer periods/lower frequencies increases in power as the system transitions to steady-state for the experiment with intermittency (shown by the increase in power at periods of approximately 10 3 s in Figure 2b).This observation corresponds to the approximately 10 min cycles observed in the pressure data in Figure 2a).These cycles were linked to disconnection and re-connection events in a key location, controlling flow across the sample (Spurin et al., 2020).
The contribution of certain frequencies to the total power can be calculated using the mean error (%) between the pressure time series (p t ) and the filtered pressure time series p tf : where    is the mean pressure of the unfiltered series.The mean error for frequencies <10 −3 Hz is 3% for the gas/ water experiment and 2% for the oil/water experiment (Figures 2b and 2f: gray arrow heads).For frequencies <10 −4 Hz, the mean error is 5% for the gas/water experiment, and 8% for the oil/water experiment (Figures 2b  and 2f: white arrow heads).Thus, pressure at higher frequencies/shorter periods contributes less to total power than the longer period fluctuations.
While higher frequency variations in pressure (<10 −3 Hz), provide relatively little overall power, they account for a greater proportion of the total power in the gas/water experiments (3%) compared to the oil/water experiment (2%).In contrast, lower frequencies (<10 −4 Hz) contribute a larger proportion of the total power for the oil/ water experiment (8%).Thus pressure fluctuations at higher frequencies play a greater role in larger scale flow properties in the gas/water experiment.This observation is indicative of the role that pore-scale intermittency has in enabling flow at relatively little energy cost (Spurin et al., 2021).
The time-averaged power spectra (Figures 2c and 2d, 2g and 2h) show that, for the gas/water experiment, the spectral slope steepens for frequencies <10 −2 Hz, as the system approaches steady-state, whilst a roughly constant spectral slope exists at all times and across all timescales during the oil/water experiment.This observation highlights the complexity of intermittent pathway flow, with events occurring over a wide range of frequencies, length-scales, and being non-local in nature (Spurin et al., 2020).At steady-state, we observe, for intermittent pathway flow, a spectral slope of −2 (red noise) for intermediate frequencies (10 −3 -10 −4 Hz) that flattens at both higher and lower frequencies, while connected pathway flow could be characterized by a single spectral slope of −3.A slope of −3 is typical for pseudo-turbulent flows (Mendez-Diaz et al., 2013;Mercado et al., 2010;Roghair et al., 2011).These are flows that appear turbulent but are in fact the result of the complex interaction of fluids with the surrounding space (other fluids, and in this case, potentially the rock grains) instead of inertial forces (Mercado et al., 2010).Further research, including velocity measurements are required to determine if pseudo-turbulence is occurring in multiphase flow through porous media.

Possibility of Upscaling
For the oil/water experiment, where both fluids flowed in continuously connected pathways (as assumed in the multiphase extension of Darcy's law), a broadly constant spectral slope of −3 exists for all frequencies during transient and steady-state flow (Figure 2h).These observations imply that there is limited temporal evolution during connected pathway flow, creating less uncertainty in predictions made for periods outside the experimental observation window.For the gas/water experiment, spectral slopes depend on frequency and time (Figure 2d), which makes extrapolating observations to larger spatial or temporal scales difficult.Thus, the success of upscaling efforts depends on how the dynamics present manifest in larger samples.

Core-Scale Results
The core-scale experiments follow the same procedure as the pore-scale experiments, with the same fluid pairings.This allows us to establish if the pore-scale observations can be upscaled to the core-scale experiments typically used for subsurface characterization (Perrin et al., 2009;Pini & Benson, 2013;Ruprecht et al., 2014).Figures 3a-3d shows the results for the gas/water experiment, e-h shows the results for the oil/water experiment and i-l shows the results for the gas/water experiment in which sample orientation was reversed.

Gas/Water Versus Oil/Water
The pressure response for the gas/water experiment and the oil/water experiment shown in Figures 3a and 3e appear similar in nature; in the first 500 s there is a steep increase in pressure as the gas or oil percolates the sample, then there is a gradual increase in pressure with time, with pressure fluctuations of a similar magnitude (around 5 kPa).The spectral power provides additional insight into evolution of pressure, and reveals subtle differences between the experiments.First, power at longer periods (>10 3 s) decreases with time, as shown by the reduction in red colors with increasing time in Figures 3b and 3f.Pressure at periods > 10 3 s contributes a similar proportion of power (mean error ∼2%) in both the gas/water and oil/water experiments (white arrow heads in Figures 3b  and 3f).This implies that pressure at periods > 10 3 s contributes ∼98% of total power.Power at frequencies of 3 × 10 −2 to 2 × 10 −3 Hz also decrease with time (Figures 3d and 3h).Spectral slopes at these frequencies are observed to flatten (i.e., whiten).While longer periods contain less power at later experimental times, they continue to contribute significantly to the total power, shown by the spectral slope steepening as the system evolves in time.Pressure at higher frequencies (<10 −3 Hz) contributes far less (∼2%) to total power in both the oil/water and gas/water experiments.
When averaged over the entire experimental run time, as shown in Figures 3c and 3g, spectral power is broadly consistent with a single spectral slope of −2 that is, red noise for both experiments.However, at the end of both experiments (the darkest green lines in Figures 3d and 3h) the time-averaged power requires two spectral slopes, and differ for gas/water and oil/water, with a steeper spectral slope for frequencies >10 −3 Hz for the latter.

The Role of Heterogeneity
The role of heterogeneity is evident in Figures 3a-3d and 3i-3l.In the experiment where the section of the sample with a lower porosity was closest to the inlet, there is an overshoot in pressure prior to stabilization (seen in Figure 3i) at approximately 200 s.Whereas, when flow direction is reversed, there is a gradual increase in pressure, and no marked overshoot, as shown in Figure 3a.Both experiments reach the same differential pressure of approximately 200 kPa within 15 min of injection.
For both experiments, pressure at lower frequencies decreases in power with time (shown in Figure 3d and l).Power is increasingly concentrated at lower frequencies during an experiment, regardless of the orientation of the sample.During steady-state flow, the spectral slope for higher frequencies (<10 −3 Hz) resembles white noise that is, the amplitude is independent of frequency, while it is steeper for the lower frequencies.The spectral slope is broadly −2 (red noise) when averaged over the whole time series.However, the evolution of the flow properties is dependent on the heterogeneity of the pore space and its orientation with respect to flow.This result reinforces the importance of considering the orientation of heterogeneities, and not only variations in porosity and permeability, on flow properties (Li & Benson, 2015;Ni et al., 2019).

Core-Scale Versus Pore-Scale Results
The magnitude of pressure fluctuations (∼1-5 kPa) is similar for the core-and pore-scale experiments.The total pressure drop is significantly higher for the core-scale experiments, meaning the pressure fluctuations are much smaller relative to the total pressure drop across the sample.
While the magnitude of pressure fluctuations are similar, spectral analysis reveals differences between the poreand core-scale experiments.For the oil/water core-scale experiments, the spectral slope is approximately −3 for 10.1029/2023GL104473 9 of 11 frequencies <10 −3 Hz during steady-state flow but this it flattens at higher frequencies (Figure 3h).The spectral slope also flattens at higher frequencies in the core-scale gas/water experiments (Figures 3d and 3l).This suggests that higher frequency events are less significant during core-scale steady-state flow.This could be due to viscous dampening, but the exact mechanisms for this will be explored in future work.
For the core-scale gas/water experiments, power decreases across almost all frequencies (down to ∼0.1 Hz in Figures 3d and 3l), while power increases at frequencies 10 −2 to 10 −4 Hz in the pore-scale gas/water experiments.Thus the evolution of the time-averaged power spectra is dependent on scale.While similar spectral slopes are observable (between −1 and −3), these slopes are dependent on frequency and time.
For the larger, core-scale experiments, a single spectral slope, attributed to connected pathway flow/linear flow dynamics in the pore-scale experiments, is not observed under any of the experimental conditions explored in this work.This result suggests that the onset of non-linear flow dynamics may occur at lower capillary numbers in larger samples.This assertion agrees with dynamic pore network modeling observations showing that the onset of non-linear flow regimes start at lower flow rates as system size increases (Hansen et al., 2023;Pedersen & Hansen, 2023;Roy et al., 2019).

Conclusions
In this work we used CWT to investigate sources of spectral power in the pressure time series of multiphase flow experiments.We showed that spectral power is dependent on frequency, sample size, and the heterogeneity present.Since pressure series spectral slopes are dependent on frequency and time, it is challenging to extrapolate flow dynamics to larger spatial scales and longer temporal scales in a straightforward way.However, we can relate spectral signals to dynamics in the pore space.Thus pressure time series can provide useful information about underlying pore-scale dynamics at larger scales and an analysis of the pressure fluctuations is an important step toward understanding larger scale flow processes.We showed it is possible to gain new insights into the underlying flow regimes without recourse to novel experimental techniques, or increased imaging capabilities.
Further work is needed to fully characterize the impact of heterogeneity, for example, layering or different lithologies, on spectral power and associated scaling regimes.Experiments should be conducted at lower flow rates, and different fractional flows, to ascertain whether the connected pathway flow signal observed for the pore-scale results can arise in core-scale experiments.Further work could also include generation of analytical spectra from physical models, to increase the applicability of the findings made in this work to other flow regimes and different samples.

Figure 1 .
Figure 1.(a) The core-scale (large) sample shown alongside the pore-scale sample to highlight the difference in scale.(b) A CT slice through the pore-scale sample, where fluid interfaces are resolvable.(c) A CT slice through the core-scale sample, where fluid interfaces are not resolvable but grayscale values are proportional to saturation.

Figure 2 .
Figure 2. Spectral analysis of pressure time series from pore-scale experiments.(a) Black curve = pressure from the gas/water experiment.Green strips and dotted line = time intervals indicated in panels b-d.(b) Power spectrum calculated by transforming the black curve in panel (a).Dashed and dotted lines correspond to time intervals indicated in panel (a).Gray/white arrow heads indicate limit on low pass filters (at periods, P ≈ 10 3 and ≈10 4 s) discussed in body text.(c) Thick black curve = time-averaged, rectified, power spectrum for entire series.Dotted and dashed curves = time-averaged power for first and second half of the time series, respectively (separated by dotted line in panels a and b).(d) Time-averaged power spectra for intervals indicated by green strips in panel (a).Note graticule indicating red (−2), pink (−1) and white (0; flat) spectral slopes.(e-h) Results for the oil/water experiment.

Figure 3 .
Figure 3. Spectral analysis of pressure time series from core-scale experiments.(a-d) Gas/water experiment initial sample orientation.(e-h) Oil/water experiment.(i-l) Gas/water experiment with reversed sample orientation.Annotation is the same as Figure 2.