Quantification of Uncertainty and Best Practice in Computing Interfacial Curvature from Complex Pore Space Images

Operando Liquid Pressure Determination in Polymer Electrolyte Fuel Cells

Extending the operating range of fuel cells to higher current densities is limited by the ability of the cell to remove the water produced by the electrochemical reaction, avoiding flooding of the gas diffusion layers. It is therefore of great interest to understand the complex and dynamic mechanisms of water cluster formation in an operando fuel cell setting as this can elucidate necessary changes to the gas diffusion layer properties with the goal of minimizing the number, size, and instability of the water clusters formed. In this study, we investigate the cluster formation process using X-ray tomographic microscopy at 1 Hz frequency combined with interfacial curvature analysis and volume-of-fluid simulations to assess the pressure evolution in the water phase. This made it possible to observe the increase in capillary pressure when the advancing water front had to overcome a throat between two neighboring pores and the nuanced interactions of volume and pressure evolution during the droplet formation and its feeding path instability. A 2 kPa higher breakthrough pressure compared to static ex situ capillary pressure versus saturation evaluations was observed, which suggests a rethinking of the dynamic liquid water invasion process in polymer electrolyte fuel cell gas diffusion layers.

In-situ capillary pressure and wettability in natural porous media: Multi-scale experimentation and automated characterization using X-ray images

HYPOTHESIS

Geometrical analyses of pore-scale fluid-fluid-rock interfaces have recently been used for in-situ characterization of capillary pressure and wettability in natural porous media. Nevertheless, more robust techniques and multi-scale, well-characterized experimental data are needed to rigorously validate these techniques and enhance their efficacy when applied to saturated porous media.

EXPERIMENTS AND IMAGE ANALYSIS

We present two new techniques for automated measurements of in-situ capillary pressure and contact angle, which offer several advancements over previous methodologies. These approaches are methodically validated using synthetic data and X-ray images of capillary rise experiments, and subsequently, applied on pore-scale fluid occupancy maps of a miniature Berea sandstone sample obtained during steady-state drainage and imbibition flow experiments.

FINDINGS

The results show encouraging agreement between the image-based capillary pressure-saturation function and its macroscopic counterpart obtained from a porous membrane experiment. However, unlike the macroscopic behavior, the micro-scale measurements demonstrate a nonmonotonic increase with saturation due to the intermittency of the pore-scale displacement events controlling the overall flow behavior. This is further explained using the pertinent micro-scale mechanisms such as Haines jumps. The new methods also enable one to generate in-situ contact angle distributions and distinguish between the advancing and receding values while automatically excluding invalid measurements.

Determination of contact angles for three-phase flow in porous media using an energy balance

HYPOTHESIS

We define contact angles, θ, during displacement of three fluid phases in a porous medium using energy balance, extending previous work on two-phase flow. We test if this theory can be applied to quantify the three contact angles and wettability order in pore-scale images of three-phase displacement.

THEORY

For three phases labelled 1, 2 and 3, and solid, s, using conservation of energy ignoring viscous dissipation (Δa_1scosθ₁₂-Δa₁₂-ϕκ₁₂ΔS₁)σ₁₂=(Δa_3scosθ₂₃+Δa₂₃-ϕκ₂₃ΔS₃)σ₂₃+Δa₁₃σ₁₃, where ϕ is the porosity, σ is the interfacial tension, a is the specific interfacial area, S is the saturation, and κ is the fluid-fluid interfacial curvature. Δ represents the change during a displacement. The third contact angle, θ₁₃ can be found using the Bartell-Osterhof relationship. The energy balance is also extended to an arbitrary number of phases.

FINDINGS

X-ray imaging of porous media and the fluids within them, at pore-scale resolution, allows the difference terms in the energy balance equation to be measured. This enables wettability, the contact angles, to be determined for complex displacements, to characterize the behaviour, and for input into pore-scale models. Two synchrotron imaging datasets are used to illustrate the approach, comparing the flow of oil, water and gas in a water-wet and an altered-wettability limestone rock sample. We show that in the water-wet case, as expected, water (phase 1) is the most wetting phase, oil (phase 2) is intermediate wet, while gas (phase 3) is most non-wetting with effective contact angles of θ₁₂≈48° and θ₁₃≈44°, while θ₂₃=0 since oil is always present in spreading layers. In contrast, for the altered-wettability case, oil is most wetting, gas is intermediate-wet, while water is most non-wetting with contact angles of θ₁₂=134°±~10°,θ₁₃=119°±~10°, and θ₂₃=66°±~10°.

Three-phase flow displacement dynamics and Haines jumps in a hydrophobic porous medium

We use synchrotron X-ray micro-tomography to investigate the displacement dynamics during three-phase—oil, water and gas—flow in a hydrophobic porous medium. We observe a distinct gas invasion pattern, where gas progresses through the pore space in the form of disconnected clusters mediated by double and multiple displacement events. Gas advances in a process we name three-phase Haines jumps, during which gas re-arranges its configuration in the pore space, retracting from some regions to enable the rapid filling of multiple pores. The gas retraction leads to a permanent disconnection of gas ganglia, which do not reconnect as gas injection proceeds. We observe, in situ, the direct displacement of oil and water by gas as well as gas–oil–water double displacement. The use of local in situ measurements and an energy balance approach to determine fluid–fluid contact angles alongside the quantification of capillary pressures and pore occupancy indicate that the wettability order is oil–gas–water from most to least wetting. Furthermore, quantifying the evolution of Minkowski functionals implied well-connected oil and water, while the gas connectivity decreased as gas was broken up into discrete clusters during injection. This work can be used to design CO₂ storage, improved oil recovery and microfluidic devices.

Contact Angles in Two-Phase Flow Images

In this work, we calculate contact angles in X-ray tomography images of two-phase flow in order to investigate the wettability. Triangulated surfaces, generated using the images, are smoothed to calculate the contact angles. As expected, the angles have a spread rather than being a constant value. We attempt to shed light on sources of the spread by addressing the overlooked mesh corrections prior to smoothing, poorly resolved image features, cluster-based analysis, and local variations of contact angles. We verify the smoothing algorithm by analytical examples with known contact angle and curvature. According to the analytical cases, point-wise and average contact angles, average mean curvature and surface area converge to the analytical values with increased voxel grid resolution. Analytical examples show that these parameters can reliably be calculated for fluid–fluid surfaces composed of roughly 3000 vertices or more equivalent to 1000 pixel2. In an experimental image, by looking into individual interfaces and clusters, we show that contact angles are underestimated for wetting fluid clusters where the fluid–fluid surface is resolved with less than roughly 500 vertices. However, for the fluid–fluid surfaces with at least a few thousand vertices, the mean and standard deviation of angles converge to similar values. Further investigation of local variations of angles along three-phase lines for large clusters revealed that a source of angle variations is anomalies in the solid surface. However, in the places least influenced by such noise, we observed that angles tend to be larger when the line is convex and smaller when the line is concave. We believe this pattern may indicate the significance of line energy in the free energy of the two-phase flow systems.

Dynamics of water injection in an oil-wet reservoir rock at subsurface conditions: Invasion patterns and pore-filling events

We use fast synchrotron x-ray microtomography to investigate the pore-scale dynamics of water injection in an oil-wet carbonate reservoir rock at subsurface conditions. We measure, in situ, the geometric contact angles to confirm the oil-wet nature of the rock and define the displacement contact angles using an energy-balance-based approach. We observe that the displacement of oil by water is a drainagelike process, where water advances as a connected front displacing oil in the center of the pores, confining the oil to wetting layers. The displacement is an invasion percolation process, where throats, the restrictions between pores, fill in order of size, with the largest available throats filled first. In our heterogeneous carbonate rock, the displacement is predominantly size controlled; wettability has a smaller effect, due to the wide range of pore and throat sizes, as well as largely oil-wet surfaces. Wettability only has an impact early in the displacement, where the less oil-wet pores fill by water first. We observe drainage associated pore-filling dynamics including Haines jumps and snap-off events. Haines jumps occur on single- and/or multiple-pore levels accompanied by the rearrangement of water in the pore space to allow the rapid filling. Snap-off events are observed both locally and distally and the capillary pressure of the trapped water ganglia is shown to reach a new capillary equilibrium state. We measure the curvature of the oil-water interface. We find that the total curvature, the sum of the curvatures in orthogonal directions, is negative, giving a negative capillary pressure, consistent with oil-wet conditions, where displacement occurs as the water pressure exceeds that of the oil. However, the product of the principal curvatures, the Gaussian curvature, is generally negative, meaning that water bulges into oil in one direction, while oil bulges into water in the other. A negative Gaussian curvature provides a topological quantification of the good connectivity of the phases throughout the displacement.

Using energy balance to determine pore-scale wettability

HYPOTHESIS

Based on energy balance during two-phase displacement in porous media, it has recently been shown that a thermodynamically consistent contact angle can be determined from micro-tomography images. However, the impact of viscous dissipation on the energy balance has not been fully understood. Furthermore, it is of great importance to determine the spatial distribution of wettability. We use direct numerical simulation to validate the determination of the thermodynamic contact angle both in an entire domain and on a pore-by-pore basis.

SIMULATIONS

Two-phase direct numerical simulations are performed on complex 3D porous media with three wettability states: uniformly water-wet, uniformly oil-wet, and non-uniform mixed-wet. Using the simulated fluid configurations, the thermodynamic contact angle is computed, then compared with the input contact angles.

FINDINGS

The impact of viscous dissipation on the energy balance is quantified; it is insignificant for water flooding in water-wet and mixed-wet media, resulting in an accurate estimation of a representative contact angle for the entire domain even if viscous effects are ignored. An increasing trend in the computed thermodynamic contact angle during water injection is shown to be a manifestation of the displacement sequence. Furthermore, the spatial distribution of wettability can be represented by the thermodynamic contact angle computed on a pore-by-pore basis.

Evaluation of methods using topology and integral geometry to assess wettability

HYPOTHESIS

The development of high-resolution in situ imaging has allowed contact angles to be measured directly inside porous materials. We evaluate the use of concepts in integral geometry to determine contact angle. Specifically, we test the hypothesis that it is possible to determine an average contact angle from measurements of the Gaussian curvature of the fluid/fluid meniscus using the Gauss-Bonnet theorem.

THEORY AND SIMULATION

We show that it is not possible to unambiguously determine an average contact angle from the Gauss-Bonnet theorem. We instead present an approximate relationship: 2πn(1-cosθ)=4π-∫κ_G12dS₁₂, where n is the number of closed loops of the three-phase contact line where phases 1 and 2 contact the surface, θ is the average contact angle, while κ_G12 is the Gaussian curvature of the fluid meniscus which is integrated over its surface S₁₂. We then use the results of pore-scale lattice Boltzmann simulations to assess the accuracy of this approach to determine a representative contact angle for two-phase flow in porous media.

FINDINGS

We show that in simple cases with a flat solid surface, the approximate expression works well. When applied to simulations on pore space images, the equation provides a robust estimate of contact angle, accurate to within 3°, when averaged over many fluid clusters, although individual values can have significant errors because of the approximations used in the calculation.

Dynamics of fluid displacement in mixed-wet porous media

We identify a distinct two-phase flow invasion pattern in a mixed-wet porous medium. Time-resolved high-resolution synchrotron X-ray imaging is used to study the invasion of water through a small rock sample filled with oil, characterized by a wide non-uniform distribution of local contact angles both above and below 90°. The water advances in a connected front, but throats are not invaded in decreasing order of size, as predicted by invasion percolation theory for uniformly hydrophobic systems. Instead, we observe pinning of the three-phase contact between the fluids and the solid, manifested as contact angle hysteresis, which prevents snap-off and interface retraction. In the absence of viscous dissipation, we use an energy balance to find an effective, thermodynamic, contact angle for displacement and show that this angle increases during the displacement. Displacement occurs when the local contact angles overcome the advancing contact angles at a pinned interface: it is wettability which controls the filling sequence. The product of the principal interfacial curvatures, the Gaussian curvature, is negative, implying well-connected phases which is consistent with pinning at the contact line while providing a topological explanation for the high displacement efficiencies in mixed-wet media.

Pore-scale numerical simulation of low salinity water flooding using the lattice Boltzmann method

HYPOTHESIS

The change of wettability toward more water-wet by the injection of low salinity water can improve oil recovery from porous rocks, which is known as low salinity water flooding. To simulate this process at the pore-scale, we propose that the alteration in surface wettability mediated by thin water films which are below the resolution of simulation grid blocks has to be considered, as observed in experiments. This is modeled by a wettability alteration model based on rate-limited adsorption of ions onto the rock surface.

SIMULATIONS

The wettability alteration model is developed and incorporated into a lattice Boltzmann simulator which solves both the Navier-Stokes equation for oil/water two-phase flow and the advection-diffusion equation for ion transport. The model is validated against two experiments in the literature, then applied to 3D micro-CT images of a rock.

FINDINGS

Our model correctly simulated the experimental observations caused by the slow wettability alteration driven by the development of water films. In the simulations on the 3D rock pore structure, a distinct difference in the mixing of high and low salinity water is observed between secondary and tertiary low salinity flooding, resulting in different oil recoveries.