Fact-checked by Grok 2 weeks ago

Capillary pressure

Capillary pressure is the pressure difference across the interface between two immiscible fluids confined within a narrow space, such as a capillary tube or pore in a porous medium, resulting from the effects of surface tension and interfacial curvature.^[1] This fundamental phenomenon, governed by the Young-Laplace equation \Delta p = \sigma \nabla \cdot \mathbf{n} (where \Delta p is the pressure jump, \sigma is the surface tension, and \nabla \cdot \mathbf{n} is the mean curvature of the interface), determines the equilibrium shape of the fluid meniscus and drives processes like fluid rise or depression against gravity.^[1] For a cylindrical capillary, the simplified form is \Delta p = \frac{2\sigma \cos \theta}{r}, with \theta as the contact angle between the fluid and the solid surface and r as the tube radius, highlighting how wettability influences the direction and magnitude of the pressure.^[2] In isolated capillary tubes, capillary pressure manifests as capillary action, where a wetting fluid (e.g., water in glass, \theta < 90^\circ) rises to a height h = \frac{2\sigma \cos \theta}{\rho g r} due to the balance between surface tension forces and hydrostatic pressure, with \rho as fluid density and g as gravitational acceleration; conversely, non-wetting fluids exhibit depression.^[2] The contact angle \theta quantifies wettability: \theta = 0^\circ for complete wetting (e.g., water on clean glass) and \theta = 180^\circ for complete non-wetting (e.g., mercury on glass), affecting meniscus concavity or convexity.^[1] This basic behavior scales with the capillary length l_c = \sqrt{\sigma / (\rho g)}, approximately 2.7 mm for water at 20°C, beyond which gravitational effects dominate over surface tension.^[1] In complex porous media, such as soils, rocks, or reservoir formations, capillary pressure controls the distribution and saturation of multiple fluid phases, dictating how non-wetting fluids (e.g., oil or air) displace wetting ones (e.g., water) during drainage or vice versa in imbibition.^[2] Capillary pressure curves, plotting \Delta p versus wetting-phase saturation, reveal key parameters like displacement pressure (threshold for invasion of larger pores) and irreducible saturation (residual fluid trapped after displacement), which exhibit hysteresis due to contact angle variations and pore geometry trapping.^[2] In petroleum engineering, these curves are essential for identifying fluid contacts (e.g., oil-water contact depth) and predicting recovery efficiency, as higher capillary pressures in finer pores retain more wetting phase.^[2] Beyond reservoirs, capillary pressure underpins natural processes like soil moisture retention and upward water transport in plant xylem via cohesive and adhesive forces, as well as engineering applications including oil extraction, irrigation systems, biomedical wicking in diagnostic devices, and anti-fouling surface designs.^[3] In hydrology, it influences groundwater recharge and contaminant transport, while in materials science, it affects ink absorption in porous substrates and heat pipe efficiency in thermal management.^[3] Overall, capillary pressure bridges microscopic interfacial physics with macroscopic fluid behavior in multiphase systems across disciplines.

Basic Concepts

Definition

Capillary pressure is defined as the pressure difference across the curved interface, or meniscus, separating two immiscible fluids within a narrow confinement, such as a capillary tube or pore space. This differential arises from the tendency of the fluid interface to minimize its surface free energy, leading to a curvature that sustains the pressure imbalance between the phases. The generation of capillary pressure is fundamentally driven by surface tension, which quantifies the contractile force at the fluid-fluid interface due to molecular cohesive interactions, and by interfacial energy, the excess energy required to maintain the interface relative to the bulk phases.^[4] In configurations involving a solid boundary, the interfacial energies between the solid and each fluid contribute to the overall equilibrium, influencing the meniscus shape and the resulting pressure difference.^[4] The distinction between wetting and non-wetting phases is central to capillary pressure, with the wetting phase being the fluid that spreads more readily on the solid surface and the non-wetting phase exhibiting higher repellency.^[5] This behavior is characterized by the contact angle θ, defined as the angle formed between the solid surface and the tangent to the fluid interface at the three-phase contact line, measured through the wetting phase; θ < 90° indicates wetting, while θ > 90° indicates non-wetting.^[6] The foundational description of capillary pressure traces back to the early 19th century, when Thomas Young qualitatively explained surface tension and capillary action in his 1805 work on fluid cohesion, and Pierre-Simon Laplace provided a rigorous mathematical treatment in 1806 as part of his celestial mechanics analysis.^[7] In porous media, capillary pressure governs the spatial distribution of immiscible fluids, affecting saturation and displacement processes.^[4]

Physical Principles

Capillary pressure arises from the interplay of intermolecular forces at the interface between a liquid, a solid, and a gas, primarily governed by adhesion and cohesion. Adhesion refers to the attractive forces between molecules of the liquid and the solid surface, while cohesion describes the attractions within the liquid molecules themselves. When adhesion exceeds cohesion, the liquid spreads along the solid surface, forming a concave meniscus; conversely, when cohesion dominates, the liquid contracts, resulting in a convex meniscus.^[8]^[9] The behavior of the meniscus is closely tied to wetting regimes, determined by the contact angle θ that the liquid-vapor interface makes with the solid surface. In complete wetting (θ < 90°), the liquid strongly adheres to the solid, promoting capillary rise, as seen with water in clean glass where θ ≈ 0° due to hydrogen bonding. Partial wetting occurs around θ = 90°, with balanced forces leading to minimal curvature. In non-wetting regimes (θ > 90°), cohesion prevails, causing capillary depression, exemplified by mercury in glass where θ ≈ 140° because of weak adhesion and strong metallic bonding within the liquid.^[10]^[11]^[12]^[13] Geometric factors, such as the radius of a tube or pore size, significantly influence meniscus curvature and the resulting fluid behavior. Smaller radii increase the curvature of the meniscus, amplifying the effects of surface forces relative to bulk forces, which leads to greater liquid rise in wetting systems or depression in non-wetting ones. For instance, in narrow capillaries, water rises higher than in wider tubes due to the heightened relative strength of adhesive forces. This curvature generates a pressure difference across the interface, as explored in quantitative models.^[14]^[15] At its core, capillary action represents an energy minimization principle where the system seeks to reduce total surface free energy. This involves a balance among surface tension forces at the interfaces, gravitational potential energy, and viscous dissipation during flow. In equilibrium, the upward pull from surface forces is counteracted by the downward gravitational force on the liquid column, stabilizing the height of rise or depression. Viscous forces play a role in the dynamics of fluid movement but are secondary to the static balance in many cases.^[16]^[17]^[10]

Theoretical Framework

Governing Equations

The Young-Laplace equation provides the fundamental relationship between capillary pressure and the curvature of the fluid interface in a capillary tube. For a spherical meniscus in a narrow cylindrical tube of radius r, the capillary pressure P_c, defined as the pressure difference across the interface, is given by

P_c = \frac{2\gamma \cos \theta}{r},

where \gamma is the interfacial tension, and \theta is the contact angle between the liquid and the solid surface. This equation arises from a force balance on the meniscus: the pressure difference exerts a force P_c \pi r^2 downward on the liquid column, balanced by the upward surface tension force $2\pi r \gamma \cos \theta acting along the contact line. Equating these forces yields the expression above, assuming hydrostatic equilibrium and a hemispherical meniscus shape.^[1] For more general meniscus shapes, the Young-Laplace equation extends to arbitrary interfaces using the principal radii of curvature R_1 and R_2, yielding

P_c = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right).

This form derives from the normal stress balance across the interface, where the pressure jump is proportional to the mean curvature \kappa = \frac{1}{R_1} + \frac{1}{R_2}, obtained through differential geometry of the surface. In the cylindrical capillary case with a spherical meniscus, R_1 = R_2 = \frac{r}{\cos \theta}, yielding P_c = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right) = \frac{2 \gamma \cos \theta}{r}, recovering the earlier equation.^[18] This generalization applies to non-spherical menisci, such as those in wider tubes or irregular geometries, while neglecting viscous effects and assuming constant \gamma.^[19] The dynamics of capillary rise in a tube are described by the Washburn equation, which models the time-dependent height h(t) of the liquid meniscus:

h(t) = \sqrt{\frac{r \gamma \cos \theta}{2 \eta} t},

where \eta is the liquid viscosity. This is derived by balancing the capillary driving force \frac{2\gamma \cos \theta}{r} with the Poiseuille flow resistance in the tube, leading to the differential equation \frac{dh}{dt} = \frac{r \gamma \cos \theta}{4 \eta h}, integrated under the assumption of negligible inertia and gravity.^[20] Key assumptions include laminar flow, constant contact angle, and a fully developed meniscus; limitations arise from inertial effects at short times, gravity at long times, and deviations in non-circular pores, which can cause overestimation of rise rates. From a thermodynamic perspective, capillary pressure can be interpreted as the derivative of the Gibbs free energy with respect to the interfacial area. In multiphase systems, the change in total free energy dG = -S dT + V dP + \gamma dA implies that at constant temperature and pressure, P_c = \left( \frac{\partial G}{\partial V} \right)_{T,P} relates to interfacial contributions via P_c = \gamma \frac{dA}{dV}, where A is the interface area and V is the volume. This basis underscores capillary pressure as a manifestation of interfacial energy minimization in porous media.^[21]

Hysteresis and Saturation Relationships

In porous media, the relationship between capillary pressure (Pc) and wetting-phase saturation (Sw) is typically represented by Pc-Sw curves, which exhibit distinct behaviors during drainage and imbibition processes due to hysteresis effects. During primary drainage, where a non-wetting phase displaces the wetting phase, the Pc-Sw curve is generally monotonic, with Pc increasing as Sw decreases, reflecting the progressive invasion of larger pores as pressure overcomes capillary entry thresholds.^[22] In contrast, during imbibition, where the wetting phase displaces the non-wetting phase, the curve is non-monotonic, often showing lower Pc values at intermediate saturations compared to drainage, primarily attributed to contact angle hysteresis that alters meniscus configurations and interfacial energies.^[23] Hysteresis arises from the sequence-dependent nature of fluid displacement in pore networks, encompassing primary drainage (initial non-wetting phase invasion into a fully wetting-saturated medium), primary imbibition (subsequent wetting phase re-entry), and secondary drainage (further non-wetting phase advancement). These processes involve competing mechanisms: piston-like displacement, where the meniscus advances uniformly across a pore throat like a piston, dominates in drainage and favors efficient sweeping in uniform pores; whereas snap-off, the fragmentation and trapping of non-wetting phase ganglia in converging-diverging pore geometries, prevails during imbibition and leads to higher residual non-wetting phase saturations. The resulting scanning curves between main drainage and imbibition paths form a hysteresis loop, with the area of the loop quantifying energy dissipation from trapped fluids and contact angle variations.^[22] To compare Pc-Sw relationships across heterogeneous media, normalized saturation data—often expressed as effective saturation (Se = (Sw - Swr)/(1 - Swr - Snwr), where Swr and Snwr are residual wetting and non-wetting saturations)—enable scaling that accounts for variations in pore size distribution and connectivity. In heterogeneous systems, such scaling reveals that Pc scales inversely with a characteristic length (e.g., pore throat radius), allowing upscaled curves for layered or stochastic media while preserving hysteresis features like residual trapping.^[24] Recent advances highlight environmental influences on these relationships. Thermodynamic models demonstrate that rising temperature decreases drainage Pc at a fractional rate of approximately -0.008 K⁻¹, driven by reductions in interfacial tension and entropic contributions to the Gibbs free energy of the meniscus, though imbibition Pc may increase slightly due to differential phase density changes.^[25] In mixed-wet systems, fractional wettability—where a fraction of the surface is preferentially wet by one phase—alters Pc-Sw curves by introducing intermediate wetting states; for instance, increasing the oil-wet fraction reduces Pc at fixed Sw by promoting partial pore occupancy and reduced snap-off efficiency.^[26]

Measurement Techniques

Laboratory Methods

Mercury injection porosimetry (MIP) is a widely used laboratory technique to measure capillary pressure curves by forcing non-wetting mercury into the pore network of a rock sample under increasing pressure. The method relies on the Washburn equation, which relates the applied pressure P to the pore throat diameter d via P = \frac{-4 \gamma \cos \theta}{d}, where \gamma is the interfacial tension of mercury (typically 485 dynes/cm) and \theta is the contact angle (assumed 140° for mercury-air-rock systems).^[27] In the procedure, a dried core sample is placed in a penetrometer, evacuated to remove air, and then subjected to stepwise pressure increments from atmospheric to up to 60,000 psia, with intrusion volumes measured via dilatometer capacitance changes.^[28] The cumulative intruded volume at each pressure step corresponds to the non-wetting phase saturation S_{nw}, yielding the capillary pressure P_c versus wetting phase saturation S_w curve after conversion using the pore size-pressure relationship. Calibration involves blank runs on non-porous quartz to correct for system compressibility and identification of closure pressure at the inflection where mercury enters the sample's inter-particle space.^[28] Limitations include overestimation of porosity and entry pressures in clay-rich samples due to loss of bound water during oven drying at 100°C, which artificially enlarges apparent pore volumes, and potential sample deformation under high pressures in compressible lithologies like shales.^[28] The porous plate method measures capillary pressure by achieving equilibrium between a fluid-saturated core sample and a surrounding non-wetting phase under controlled pressure differentials, leveraging the plate's uniform pore size to prevent premature intrusion. In the standard procedure, a brine-saturated plug (e.g., with 5,000 ppm NaCl) is placed on a wettable porous ceramic or fritted glass plate with pore throats rated to withstand up to 200-2,000 psig, then enclosed in a pressure cell where air or decane is applied incrementally (e.g., 100, 200, 400, 1,000 psig) to desaturate the sample.^[29] Equilibrium is confirmed when produced fluid volume stabilizes, typically over days per step, with S_w determined from weight loss or produced volume, and P_c equated to the applied pressure. Calibration ensures plate integrity by testing maximum pressure without breakthrough, and data are extended to higher P_c via vapor desorption methods around 2,000 psig.^[29] This technique provides accurate drainage curves for reservoir fluids but is limited by long equilibration times (weeks to months) and maximum P_c constrained by plate pore size, making it less suitable for tight rocks requiring pressures beyond 5,000 psig.^[29] The centrifuge method simulates capillary pressure by applying centrifugal force to a core sample to displace wetting fluid with a denser non-wetting phase, generating an effective pressure gradient across the sample. Core plugs are saturated with brine, mounted in a rotor (typically 1,500-10,000 rpm), and spun at stepwise increasing speeds while monitoring produced effluent volumes to construct the P_c - S_w curve.^[30] The effective capillary pressure at the outflow endface is calculated as P_c = \frac{1}{2} \Delta \rho \omega^2 (R_b^2 - R_e^2), where \Delta \rho is the density difference between phases, \omega is the angular velocity, R_b is the bucket outer radius, and R_e is the endface radius; saturation profiles are inferred assuming quasi-equilibrium and negligible capillary end effects except near equilibrium.^[30] Data interpretation, often via the Hagoort method, corrects for capillary pressure gradients by integrating Darcy's law for each phase, assuming uniform acceleration, high non-wetting phase mobility, and Corey-type relative permeability forms to derive average saturations from production data.^[30] This approach accelerates measurements compared to porous plate but may underestimate low S_w due to non-equilibrium transients and requires corrections for capillary back-pressure in imbibition modes. To scale capillary pressure data across samples of varying porosity and permeability while assuming similar wettability and pore geometry, the Leverett J-function is applied as J(S_w) = \frac{P_c \sqrt{k}}{\sigma \cos \theta}, where k is permeability, \sigma is interfacial tension, and \theta is contact angle. This dimensionless function collapses multiple P_c - S_w curves onto a universal form when plotted against S_w, facilitating reservoir-scale correlations under the assumptions of uniform wettability and homologous rock types. Hysteresis in these curves arises from contact angle variations during drainage and imbibition but is not derived here.

Advanced Characterization

Nuclear Magnetic Resonance (NMR) relaxometry provides a non-invasive approach to characterize capillary pressure by leveraging transverse relaxation time (T2) distributions, which correlate with pore sizes and fluid saturation in porous media. The principle relies on the surface relaxivity of pore walls, where shorter T2 times indicate smaller pores filled with wetting fluids at higher capillary pressures, allowing inference of pore size distributions and capillary pressure-saturation (Pc-Sw) relationships without direct pressure application. In shales, recent applications have demonstrated NMR's utility in quantifying capillary pressure in organic-rich nanopores, where T2 shifts during drainage reveal entry pressures as low as 1-10 MPa, aiding evaluation of fluid trapping in tight formations. X-ray microtomography (μCT) enables three-dimensional imaging of fluid distributions within pores, directly visualizing mechanisms like snap-off and capillary trapping that govern hysteresis in Pc-Sw curves. With resolutions down to 1-5 microns, μCT captures in situ fluid interfaces during imbibition or drainage, allowing computation of local capillary pressures via curvature measurements from segmented images using tools like maximal inscribed spheres or level-set methods.^[31] Data processing involves image segmentation to identify phases, followed by interface tracking to extract pore-by-pore entry pressures and trapping efficiencies; for instance, studies on sandstones have shown snap-off events trapping up to 30% non-wetting phase saturation at capillary numbers below 10^{-6}.^[32] Advanced characterization techniques also include cryo-porosimetry, which measures pore sizes and capillary pressures in frozen, wet samples by analyzing melting point depressions in confined water, offering advantages over gas sorption methods by preserving native hydration states.^[33] Environmental scanning electron microscopy (ESEM) facilitates in-situ wettability assessment by observing fluid imbibition dynamics under controlled humidity, revealing contact angles and capillary rise in real-time without sample drying, which is critical for understanding mixed-wettability effects on Pc.^[34] Integration of machine learning enhances curve fitting by clustering μCT or NMR data to identify flow regimes and predict full Pc-Sw profiles, with unsupervised algorithms detecting heterogeneity in trapping that traditional parametric fits overlook.^[35] As of 2025, recent developments include AI-driven methods such as Bayesian optimization for capillary pressure estimation from NMR data and microfluidic experiments combined with pore-network modeling to quantify corner-bridge flow effects on Pc curves.^[36]^[37] Challenges in advanced characterization persist for nanoscale pores in nanomaterials, where continuum assumptions like the Young-Laplace equation break down below 5 nm due to molecular layering and disjoining pressures, complicating Pc estimation.^[38] Temperature-dependent effects further exacerbate this, as interfacial tension variations can alter Pc by 20-50% per 50°C rise, requiring coupled thermal-fluid models for accurate interpretation in high-temperature reservoirs or nanomaterials.^[39]

Engineering Applications

Petroleum and Reservoir Engineering

In petroleum and reservoir engineering, capillary pressure plays a critical role in governing multiphase flow dynamics within porous media, directly influencing relative permeability curves and residual phase saturations.^[40] Relative permeability, which quantifies the simultaneous flow of oil, water, and gas phases, is modulated by capillary forces that create pressure gradients across phase interfaces, thereby affecting fluid distribution and mobility during production.^[41] For instance, higher capillary pressures in water-wet reservoirs promote imbibition, leading to residual oil saturation levels that can exceed 30% in heterogeneous formations, while in oil-wet systems, they hinder water invasion and reduce sweep efficiency.^[42] This interplay is particularly evident in waterflooding operations, where capillary end effects at core boundaries or reservoir heterogeneities can lower recovery factors by 10-20% if not accounted for in simulations, emphasizing the need for integrated capillary-relative permeability models to optimize injection strategies.^[43] To model these relationships, empirical correlations such as the Brooks-Corey and van Genuchten functions are widely adopted for relating capillary pressure (P_c) to water saturation (S_w) in reservoir simulations. The Brooks-Corey model, originally derived for drainage processes in porous media, expresses capillary pressure as P_c = P_e (S_w^*)^{-1/\lambda}, where P_e is the entry pressure marking the threshold for non-wetting phase invasion, S_w^* is the normalized effective water saturation (S_w^* = (S_w - S_{wr}) / (1 - S_{wr}) with S_{wr} as residual water saturation), and \lambda is a pore-size distribution index typically ranging from 1 to 3 for sandstones, reflecting heterogeneity. Parameters are fitted to laboratory data using nonlinear regression, often yielding \lambda values around 2 for North Sea chalks to capture sharp drainage-imbibition transitions.^[44] The van Genuchten model, suited for smoother saturation profiles, expresses effective saturation as S_e = \left[1 + (\alpha P_c)^n \right]^{-m}, inverted to P_c = \frac{1}{\alpha} \left[ (S_e)^{-1/m} - 1 \right]^{1/n}, with \alpha (inverse of air-entry pressure, ≈0.01–0.1 cm⁻¹), n (>1, shape parameter), and m = 1 - 1/n fitted via least-squares to match hysteresis in mixed-wet reservoirs. These models enable upscaling from core-scale data to field-level predictions, improving volumetric estimates by incorporating capillary trapping.^[45] In enhanced oil recovery (EOR), capillary pressure is leveraged through wettability alteration techniques to boost displacement efficiency. Surfactant or low-salinity flooding alters rock wettability from oil-wet to water-wet, reducing residual oil saturation by increasing favorable capillary pressures that drive spontaneous imbibition, potentially recovering an additional 10-15% of original oil in place (OOIP).^[46] For CO2 sequestration in depleted reservoirs, capillary forces provide residual trapping by immobilizing CO2 ganglia in pore throats, with trapping indices up to 25% of injected volume at saturations below 0.6, as modeled by van Genuchten relations under reservoir conditions (5-20 MPa, 25-50°C).^[47] This mechanism enhances long-term storage security by countering buoyancy-driven migration.^[48] Case studies from North Sea reservoirs, such as the South Arne chalk field, demonstrate capillary pressure's practical impact, where Brooks-Corey fitting to centrifuge data revealed entry pressures of 0.1-1 bar, informing waterflood designs that mitigated early breakthrough and improved recovery by 5-10%.^[49] Recent advances in digital rock physics further refine predictions by simulating P_c-saturation curves from micro-CT images of carbonate plugs, achieving relative permeability matches within 15% of experimental values and enabling rapid assessment of heterogeneity effects without extensive core testing.^[50] These tools have been applied to Middle Eastern analogs, predicting capillary trapping for CO2-EOR hybrids with uncertainties reduced to under 10%.^[51]

Microfluidics and Nanotechnology

In microfluidics, capillary pressure plays a pivotal role in driving passive fluid flow within lab-on-a-chip devices, enabling the design of channels that facilitate autonomous pumping without external pressure sources. This capillary-driven mechanism relies on surface tension forces to propel liquids through microchannels, often enhanced by evaporation or wicking structures at channel ends, allowing precise control of flow rates for applications such as diagnostics and chemical analysis.^[52]^[53] For instance, hydrophilic microstructures generate sufficient capillary pressure gradients to sustain steady flow, with typical pressures on the order of 10-100 kPa in channels of 10-100 μm width, eliminating the need for bulky pumps and simplifying device portability.^[54] Key applications leverage this principle for targeted functionalities. In drug delivery systems, capillary bursts—sudden pressure releases from meniscus instability—enable controlled release of therapeutics from microreservoirs, where capillary action transports drugs through porous matrices to achieve sustained delivery profiles over hours.^[55] Similarly, in inkjet printing, capillary pressure governs ink meniscus dynamics within nozzles, ensuring stable droplet ejection by balancing viscous and surface tension forces, with optimized channel geometries preventing wetting instabilities that could disrupt print resolution.^[56] Superhydrophobic surfaces, engineered with nanoscale roughness to yield contact angles exceeding 150°, modulate capillary pressure to promote self-cleaning; here, low adhesion from Cassie-Baxter states allows water droplets to roll off, carrying contaminants while minimizing capillary retention forces.^[57] Post-2020 innovations have advanced these systems further. Capillary-drop platforms, such as the self-powered Cap-Drop device, utilize pre-programmed capillary forces to generate and immobilize uniform droplets for 3D cell culture, achieving sub-nanoliter volumes without external actuation and enabling high-throughput biological assays.^[58] In nanotechnology, nanotextured surfaces with light-controlled modulation of nanopixel heights—varying from 5-50 nm—allow tunable capillary pressure via nanocapillarity, facilitating grayscale nanopatterning with sub-10 nm vertical resolution for optical and sensing applications. These approaches draw on the Young-Laplace equation to predict pressure differences across curved interfaces in confined spaces. Despite these advances, challenges persist in scaling from micro- to nano-regimes. Scaling laws reveal that capillary pressure intensifies inversely with pore radius (ΔP ∝ 1/r), amplifying surface effects over bulk inertia at nanoscale dimensions below 100 nm, which complicates flow predictability and device fabrication due to enhanced stiction and wetting variations.^[59] In ultra-small pores (sub-10 nm), quantum effects further disrupt classical models; for example, quantized energy levels in confined fluids alter adsorption and capillary condensation, leading to anomalous flow rates up to four to five orders of magnitude faster than continuum predictions, necessitating hybrid quantum-classical simulations for accurate design.^[60]^[61]

Environmental and Natural Occurrences

In Soil and Groundwater Systems

Capillary pressure plays a central role in the soil water characteristic curve (SWCC), which describes the relationship between soil water content and matric potential in unsaturated soils, thereby governing water retention and availability in the vadose zone.^[62] The SWCC determines the amount of plant-available water by delineating the range of capillary pressures where water is held against gravity but accessible to roots, typically between field capacity (around -33 kPa) and permanent wilting point (around -1500 kPa), influencing hydrological processes such as drainage and transpiration in unsaturated zones.^[63] In groundwater systems, capillary pressure creates the capillary fringe, a transitional zone immediately above the water table where pore water is held by surface tension, extending upward from a few centimeters in coarse sands to several meters in fine clays.^[64] This fringe maintains near-saturation conditions, facilitating contaminant transport and microbial activity near the interface between unsaturated and saturated zones.^[65] Another phenomenon driven by capillary pressure is needle ice formation, where freezing temperatures induce upward migration of soil water through capillary action, leading to frost heave in cold climates as ice needles grow and lift the soil surface.^[66] Capillary pressure is critical in environmental applications like soil remediation, particularly for removing non-aqueous phase liquids (NAPLs) such as petroleum hydrocarbons, where high capillary entry pressures in fine soils trap residual NAPL, complicating extraction but enabling strategies like surfactant flushing to reduce interfacial tension.^[67] Climate change exacerbates these dynamics through soil drying, which increases capillary pressures and induces hysteresis in the SWCC, as demonstrated in recent studies showing drought conditions can shift soils to alternative stable low-moisture states with persistent high suction.^[68] Such hysteresis, where drainage and wetting paths diverge due to air entrapment and contact angle variations, amplifies water scarcity in aridifying regions.^[69] In the vadose zone, capillary pressure interacts with evaporation by generating upward gradients that sustain flux from deeper layers until critical suction thresholds are reached, limiting surface drying rates in fine-textured soils.^[70] During infiltration events, dynamic capillary pressures at the wetting front enhance or impede water movement, with higher pressures in drier soils promoting preferential flow paths and faster recharge compared to equilibrium conditions.^[71]

In Biological Systems

In plant physiology, capillary pressure facilitates water ascent in the xylem through tracheids and vessels, driven by cohesion-tension mechanisms where negative pressures (as low as -13.1 MPa) pull water upward against gravity via capillary wicking along cell walls.^[72] Air-seeding occurs at pit membranes between tracheids, where capillary failure allows air bubbles to enter under tension, causing embolism that blocks water transport; pore sizes in these membranes determine vulnerability, with larger pores increasing risk.^[72] Embolism repair relies on capillary forces to refill embolized protoxylem vessels, as seen in cushion plants like Azorella macquariensis, where water rapidly enters pit channels (within seconds) and dissolves bubbles (in under 8 minutes) due to hydrophilic walls and helical thickenings that enhance meniscus curvature, independent of root pressure.^[73] Bio-inspired designs draw from xylem capillary principles, such as evaporation-driven suction pressures up to -1.0 MPa in leaves, to develop nanofluidic systems with hierarchical nanopores and microchannels that mimic tracheid wettability for efficient, low-resistance fluid transport.^[74]

References

[1]
[PDF] LECTURE 3: Fluid Statics - MIT
The normal force balance is expressed by the Young-Laplace equation, where now ρ = ρw − ρa ≈ ρw is the density difference between water and air. We define the ...
[2]
None
### Definition and Context of Capillary Pressure
[3]
[PDF] Investigation of Engineering Applications of Capillary Effect
Nov 7, 2020 · Capillary action is used in irrigation, oil extraction, biomedical devices, and blood testing, and aids in soil water retention and plant water ...
[4]
Capillary pressure - MOOSE framework
The capillary pressure is the pressure difference between two fluid phases in a porous medium that arises due to the interfacial tension between the fluid ...Missing: physics | Show results with:physics
[5]
The Defining Series: Wettability - SLB
Wettability describes a preference of a solid surface to be in contact with one fluid rather than another.
[6]
Microfluidics-based analysis of dynamic contact angles relevant for ...
The contact angle is defined as the angle that a two-fluid interface makes with the solid surface. A contact angle bigger than 90° defines the non-wetting phase ...
[7]
https://doi.org/10.1016/j.micromeso.2011.09.022
[8]
Cohesion and adhesion of water (article) - Khan Academy
This upward motion against gravity, known as capillary action, depends on the attraction between water molecules and the glass walls of the tube (adhesion), as ...
[9]
Cohesion and Adhesion in Liquids: Surface Tension and Capillary ...
Cohesive forces between molecules cause the surface of a liquid to contract to the smallest possible surface area. This general effect is called surface ...
[10]
7.1: Surface Tension, Viscosity, and Capillary Action
Jun 5, 2019 · Mercury therefore does not wet glass, and it forms a convex meniscus when confined in a tube because the cohesive forces within the mercury tend ...Missing: regimes | Show results with:regimes
[11]
Wettability - an overview | ScienceDirect Topics
A wetting liquid has a contact angle with a solid of less than 90°, while a non-wetting liquid has a contact angle with a solid between 90° and 180° [5].Thin Liquid Films In Wetting... · Rp-Hplc Analytical Columns · A Critical Review On Recent...Missing: action | Show results with:action<|control11|><|separator|>
[12]
11.2 Properties of Liquids – Chemistry Fundamentals
Mercury therefore does not wet glass, and it forms a convex meniscus when confined in a tube because the cohesive forces within the mercury tend to draw it into ...Missing: regimes | Show results with:regimes
[13]
Capillary Action - Chemistry LibreTexts
Jan 29, 2023 · Capillary action can be defined as the ascension of liquids through slim tube, cylinder or permeable substance due to adhesive and cohesive forces.
[14]
11.8 Cohesion and Adhesion in Liquids: Surface Tension and ...
(a) Mercury is suppressed in a glass tube because its contact angle is greater than 900. Surface tension exerts a downward force as it flattens the mercury, ...Missing: regimes | Show results with:regimes
[15]
Capillary action – how contact angle and surface tension are related?
Apr 23, 2020 · Capillary action is driven by surface tension. Lower contact angles (more hydrophilic) increase capillary pressure, while high surface tension ...
[16]
[PDF] Mechanical approach to surface tension and capillary phenomena
May 7, 2020 · Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) ...
[17]
[PDF] Derivations of the Young-Laplace equation - Capillarity
Apr 28, 2021 · The classical Young-Laplace equation relates capillary pressure to surface tension and the principal radii of curvature of the interface ...
[18]
[PDF] Thermodynamic Basis of Capillary Pressure in Porous Media
According to this definition, macroscopic apillary pressure is related to the change in the free energy of phases and interfaces as a result of change in the ...
[19]
Reservoir Simulation With History-Dependent Saturation Functions
Feb 1, 1976 · THE HYSTERESIS MODEL. The hysteresis model allows both capillary pressures and relative permeabilities to range pressures and relative ...<|control11|><|separator|>
[20]
Relationship between capillary pressure, saturation, and interfacial ...
Nov 30, 2007 · Approximate correlations for interfacial area as a function of saturation are suggested for the subsequent imbibition and drainage processes.
[21]
Upscaling Capillary Pressure‐Saturation Curves in Heterogeneous ...
This study investigates the upscaling of capillary curves under static conditions of capillary-gravity equilibrium. The averaging regions are horizontal layers, ...
[22]
A Thermodynamic Analysis of the Impact of Temperature on the ...
Jul 31, 2021 · They concluded that the macroscopic capillary pressure is related to the free energy of the two fluid phases and all three interfaces that ...
[23]
Fractional wettability effects on two-and three-fluid capillary pressure ...
The oil-water and air-water capillary pressures decreased, at a particular water saturation, as the fraction of oil-wet sand increased. The water pressure is ...
[24]
Note on a Method of Determining the Distribution of Pore Sizes in a Porous Material | PNAS
### Publication Details
[25]
[PDF] 2000: Mercury Porosimetry Protocol for Rapid Determination of ...
The protocol uses high-pressure mercury injection to quickly determine drainage capillary pressure, quantifying pore systems and pore aperture size. Mercury is ...Missing: seminal | Show results with:seminal
[26]
[PDF] A Comparative Study of Laboratory Techniques for Measuring ...
In this section, we review the three primary capillary pressure measurement techniques used by the petroleum industry, including the porous plate, centrifuge, ...
[27]
[PDF] 1997: Analysis of Centrifuge Relative Permeability Data
The Hagoort method is based on three assumptions: that the effects of capillary pressure are negligible except near the equilibrium point (a correction ...
[28]
Pore‐by‐pore capillary pressure measurements using X‐ray ...
Sep 12, 2014 · In this paper, we use in situ X-ray microtomographic imaging coupled with new interface curvature measurement tools to examine the capillary ...
[29]
Design considerations for dynamic fluid flow in porous media ...
We present the state-of-the-art in design of X-ray transparent flow systems and discuss the technical challenges around performing µCT-based fluid flow ...
[30]
Studies of freezing–melting hysteresis in cryoporometry scanning ...
Cryoporometry provides an alternative to gas sorption and mercury intrusion, and offers advantages over them, since wet samples can be used, and pore shape can ...
[31]
Visualization of imbibition in porous media by environmental ...
ESEM allows visualization of fluid capillary rise in rock samples. It allows the observation of dynamic experiments, for instance, wetting fluid imbibition in ...
[32]
Using Unsupervised Machine Learning to Characterize Capillary ...
Jun 29, 2020 · An automated unsupervised machine learning method is developed to differentiate flow regimes and identify capillary heterogeneity trapping ...
[33]
Effectiveness of the Young-Laplace equation at nanoscale - Nature
Apr 1, 2016 · Currently, the Y-L equation is also widely used to describe the capillary pressure when the channel/pore size is down to nanoscale.
[34]
[PDF] Conceptual modeling of temperature effects on capillary pressure in ...
Apr 20, 2019 · The effect of temperature on the liquid–gas interface and consequently on the capillary pressure in unsaturated dead-end pores is conceptually ...
[35]
Chapter 15: Relative Permeability and Capillary Pressure - OnePetro
Reservoir engineers use relative permeability and capillary pressure relationships for estimating the amount of oil and gas in a reservoir and for predicti.
[36]
Influence of Capillarity on Relative Permeability in Fractional Flows
Oct 18, 2020 · The capillary pressure gradient was stronger in the mixed-wet case. It is likely that the sharper gradient was not accurately captured by ...
[37]
[PDF] reservoir engineering implications of capillary pressure and relative
Capillary pressure and relative permeability data are very important to reser- voir engineering. Capillary pressure data are used in fluid flow calcula- tions, ...
[38]
Capillary Pressure Effects on Estimating the Enhanced-Oil-Recovery ...
We present a mathematical model representing coreflooding that accounts for wettability changes caused by changes in the injected composition. For purpose of ...
[39]
Capillary pressure and relative permeability correlations for ...
Aug 20, 2017 · Brooks and Corey (1966) proposed a capillary pressure correlation method comprising two fitting parameters to represent the pore size ...
[40]
[PDF] with Brooks-Corey and Van Genuchten Type Capillary Pressure
Brooks-Corey models entry pressure heterogeneity, leading to capillary pinning. Van Genuchten models pore networks, causing capillary retardation. These ...
[41]
Wettability Alteration Mechanisms in Enhanced Oil Recovery ... - MDPI
When the capillary pressure is higher, the water imbibes the oil from small pores to the main channels, increasing the oil recovery. In contrast, when the ...
[42]
[PDF] Capillary trapping for geologic carbon dioxide storage
The impact of reservoir conditions was evaluated directly in. Niu et al. (2015). It was found that varying pressure 5–20MPa, temperature 25–50◦C, and brine ...
[43]
[PDF] A parametric analysis of capillary pressure effects during geologic ...
Simulation results show that capillary pressure strongly controls the spatial distribution of gas phase CO2 as it flows away from the injection well into the ...
[44]
A New Capillary Curve Modelling Method For North Sea Chalk And ...
Jun 22, 2003 · A new method for modelling of capillary pressure curves is presented with the South Arne Chalk field as a case story.
[45]
An Evaluation of Digital Rock Physics Technology for the Prediction ...
Jan 19, 2014 · Capillary Pressure Prediction from Rock Models Reconstructed Using Well Log Data. 12ATCE. Evaluation of Relative Permeability of a Tight Oil ...
[46]
Digital Rock Physics for Fast and Accurate Special Core Analysis in ...
Oct 31, 2012 · These numerical simulations require input parameters such as relative permeabilities, capillary pressures, and other rock and fluid porosity ...
[47]
Passive microfluidic pumping using coupled capillary/evaporation ...
Here a passive pumping approach is presented in which a large pressure differential arising from a small, curved meniscus situated along the bottom corners of ...
[48]
Wicking pumps for microfluidics | Biomicrofluidics - AIP Publishing
Nov 1, 2024 · This review describes mechanisms for pulling fluids through microfluidic devices using hydrophilic structures at the downstream end of the device.
[49]
Capillary microfluidics for diagnostic applications - Frontiers
This review examines the fundamental concepts of capillary-driven microfluidics, emphasizing significant progress in the design of capillary pumps and valves.
[50]
Applications of capillary action in drug delivery - PMC - NIH
Jul 23, 2021 · Capillary action transfers the liquid to deliver and release drugs. Capillary action transfers the liquid through the materials. For example, ...
[51]
[PDF] Fundamental Fluid Dynamics Challenges in Inkjet Printing
Oct 20, 2021 · the unsteady pressure, pc ≈ 0.05 bar for the capillary pressure, and pv ≈ 0.7 bar for the viscous pressure. Together the nozzle pressure ...
[52]
When and how self-cleaning of superhydrophobic surfaces works
Jan 17, 2020 · We monitor the self-cleaning process on a single-particle level and quantify the involved forces.
[53]
Cap‐Drop: A Pre‐Programmed, Self‐Powered Capillary Microfluidic ...
May 22, 2025 · A novel capillary‐driven platform that generates and immobilizes droplets with precision, eliminating the need for external pumps or intricate setups.
[54]
Tackling Multi-Physics Nano-Scale Phenomena in Capillary Force ...
Oct 26, 2023 · This paper uses known physical principles alongside artificial intelligence to uncover the unknown physical principles responsible for the sudden jump ...
[55]
https://pmc.ncbi.nlm.nih.gov/articles/PMC8282986/
[56]
Role of Quantum Effect for Nano-confined Substance Ultrafast Flow
Many researchers, however, found that (1) the flow of both liquid and gas through nanoscale pores is one to even seven orders of magnitude faster than that ...
[57]
Equations for the soil-water characteristic curve
The soil-water characteristic curve can be used to estimate various parameters used to describe unsaturated soil behaviour. A general equation for the ...
[58]
Analytical prediction of soil–water characteristics curves for granular ...
Sep 3, 2025 · The soil–water characteristic curve (SWCC) describes the ... 2, based on the Young-Laplace equation [36], the capillary pressure or ...
[59]
General Facts and Concepts about Ground Water
Between the unsaturated zone and the water table is a transition zone, the capillary fringe. In this zone, the voids are saturated or almost saturated with ...
[60]
R.W. Gillham and the Role of Capillary Fringe Processes in Shallow ...
Feb 5, 2025 · The capillary fringe exerts an important influence on the relation between precipitation and aquifer recharge, and groundwater discharge to ...
[61]
Formation of ice lenses and frost heave - Rempel - AGU Journals
Jun 21, 2007 · The maximum frost heave pressure depends only on the temperature at the lens boundary and not the details of the temperature profile beneath.
[62]
[PDF] Ground Water Issue: Light Nonaqueous Phase Liquids
Capillary conditions affect the configuration and magnitude of trapped residual NAPL. Field observations of the effects of capillary pressure include.
[63]
(PDF) Experimental evidence for drought induced alternative stable ...
Aug 6, 2025 · Here, we provide long-term manipulation and observation data supporting the existence of drought induced alternative stable soil moisture states ...
[64]
Modeling the hysteretic behavior of the capillary pressure in partially ...
Apr 18, 2014 · Modeling the hysteretic behavior of the capillary pressure in partially saturated porous media: a review ... soil-water characteristic curve.
[65]
Experimental and numerical studies of dynamic variation of liquid ...
As evaporation progresses, the water content near the surface decreases and water supplied from the lower part of the vadose zone through capillary rise cannot ...
[66]
Dynamic capillary pressure during water infiltration: Experiments ...
Jun 13, 2012 · A Green-Ampt model with a velocity-dependent capillary pressure is tested The modified Green-Ampt model describes downward infiltration ...Missing: evaporation | Show results with:evaporation
[67]
Physiology, Edema - StatPearls - NCBI Bookshelf
May 1, 2023 · Capillary pressure forces fluid from the capillaries into the ... Raised capillary pressure is a common cause of edema including ...
[68]
Interaction of Capillary and Interstitial Forces - NCBI - NIH
Increments in capillary pressure in excess of 15 mmHg lead to unopposed transcapillary fluid filtration, severe edema, and eventually exudation of fluid across ...
[69]
None
### Summary of Hydraulics of Vascular Water Transport, Air-Seeding via Pit Membranes, and Capillary Failure in Xylem Embolism
[70]
Easy Come, Easy Go: Capillary Forces Refill Embolized Xylem
In this study, embolism repair was investigated in relation to xylem structure in two cushion plant species, Azorella macquariensis and Colobanthus muscoides, ...
[71]
Podocytes are sensitive to fluid shear stress in vitro
Glomerular capillary pressure generates wall tension in glomerular capillaries that needs to be counterbalanced by endothelial cells (ECs) and podocytes.
[72]
The Life of a Kidney Podocyte - PMC - PubMed Central
Jul 23, 2025 · Podocytes physiologically react to changes in glomerular capillary pressure (mechanical stress) and increased glomerular filtration rate by ...
[73]
Fabrication of Artificial Leaf to Develop Fluid Pump Driven ... - Nature
Nov 7, 2017 · By mimicking a plant leaf, a leaf inspired pump was produced with cryo-agarose gel, which consists of hierarchical regions of random ...
[74]
Systemic and Cardiac Microvascular Dysfunction in Hypertension
This pathology is initiated by prolonged elevated blood pressure, which imposes sustained mechanical stress on vascular walls, leading to damage of vascular ...
[75]
Alterations in capillary morphology are found in mild blood pressure ...
Disruption of capillary integrity could potentially result in interstitial edema, extravasation of plasma proteins and blood cells, and inflammation triggered ...