Capillary pressure, abbreviated as \( P_c \), is the pressure difference that occurs across the interface between two immiscible fluids (like water and oil, or air and water) when they’re in contact within a narrow space, such as the tiny pores of a rock or soil. It’s caused by the interplay of surface tension, the force that holds the surface of a liquid together, and the geometry of the space, like how curved the interface becomes in those tight confines.
Capillary Pressure Formula
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\( P_c \;=\; \dfrac{ 2 \cdot \sigma \cdot cos(\theta) }{ r } \) (Capillary Pressure) \( \sigma \;=\; \dfrac{ P_c \cdot r }{ 2 \cdot cos(\theta) }\) \( cos(\theta) \;=\; \dfrac{ P_c \cdot r }{ 2 \cdot \sigma } \) \( r \;=\; \dfrac{ 2 \cdot \sigma \cdot cos(\theta) }{ P_c } \) |
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| Symbol | English | Metric |
| \( P_c \) = Capillary Pressure | \(lbf \;/\; in^2\) | \(Pa\) |
| \( \sigma \) (Greek symbol sigma) = Interfacial Tension | \(lbf \;/\; in\) | \(dyn \;/\; cm\) |
| \( \theta \) = Angle of Wettability | \(deg\) | \(rad\) |
| \( r \) = Radius of the Pore | \(in\) | \(cm\) |
Surface Tension - It’s the cohesive force at the surface of a liquid that makes it act like a stretched membrane. When two fluids, like water and oil meet, surface tension at their interface resists mixing and creates a pressure difference. The stronger the surface tension, the higher the capillary pressure.
Wettability - This is about how much one fluid prefers to stick to a solid surface over the other. If a surface (like a rock pore) is water-wet, water spreads out and hugs it, while oil gets pushed aside. If it’s oil-wet, the reverse happens. Wettability shapes the curvature of the fluid interface, which directly affects the pressure difference.
Pore Size and Geometry - The smaller the space, like a tiny pore or capillary tube, the more pronounced the effect. When the interface between the fluids curves sharply in a narrow space, the pressure difference spikes. This is why capillary pressure is a big deal in fine-grained materials like clay or tight rocks, but less so in wide-open spaces.
Fluid Properties - The difference in densities or viscosities between the two fluids can influence how they behave under capillary forces. For instance, a denser fluid might resist displacement more, tweaking the pressure balance.
Interfacial Curvature - Tied to all the above, the shape of the boundary between the fluids matters. In a capillary tube or pore, this boundary (called the meniscus) curves based on wettability and pore size. The tighter the curve, the greater the capillary pressure, as described by the Young-Laplace equation: pressure scales inversely with the radius of curvature.
