Capillary Pressure Calculator (Young-Laplace)

An engineering tool to calculate capillary pressure based on fluid properties and pore geometry.

Capillary Pressure (P_c)

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Results Table & Chart

The table and chart below demonstrate how capillary pressure changes as the pore radius varies, while holding interfacial tension and contact angle constant. This illustrates the inverse relationship: as pores get smaller, the capillary pressure required to displace the wetting fluid increases significantly.

Chart: Capillary Pressure vs. Pore Radius

Pore Radius ()	Capillary Pressure (kPa)	Capillary Pressure (psi)

Table: Capillary pressure at various pore radii.

What is Capillary Pressure?

Capillary pressure (P_c) is the pressure difference that exists across the interface separating two immiscible fluids (like oil and water, or air and water) in a narrow tube or a porous medium. This phenomenon is a direct consequence of surface tension. The Young-Laplace equation provides the mathematical framework to calculate capillary pressure. It’s a fundamental concept in soil mechanics, petroleum engineering, and material science.

Essentially, it’s the amount of extra pressure required for a non-wetting fluid to displace a wetting fluid in a tiny space. For example, in a water-wet rock, significant pressure is needed to force oil (the non-wetting phase) into the small pores already occupied by water. Understanding how to calculate capillary pressure using Young-Laplace is crucial for predicting fluid behavior in reservoirs and other porous systems. You can learn more about the properties of porous media in our guide on geotechnical engineering formulas.

Capillary Pressure Formula and Explanation

The calculator uses the Young-Laplace equation for a cylindrical capillary tube, which is the most common application. The formula is:

P_c = (2 * γ * cos(θ)) / r

This equation elegantly connects the material properties of the fluids and solid with the geometry of the pore.

Variables in the Young-Laplace Equation

Variable	Meaning	Common Unit	Typical Range
Pc	Capillary Pressure	Pascals (Pa), psi	0.1 – 10,000+ psi
γ (gamma)	Interfacial Tension	N/m or dyn/cm	0.02 – 0.075 N/m
θ (theta)	Contact Angle	Degrees (°)	0° – 180°
r	Pore Radius	micrometers (µm)	0.1 – 1,000 µm

Practical Examples

Example 1: Water in a Hydrophilic Glass Capillary

Consider a scenario where water is rising in a very clean, narrow glass tube exposed to air. This represents a strongly water-wet system.

• Inputs:
  • Interfacial Tension (γ): 0.072 N/m (water-air)
  • Contact Angle (θ): 10° (highly wettable)
  • Pore Radius (r): 5 micrometers (µm)
• Calculation:
  • P_c = (2 * 0.072 N/m * cos(10°)) / 0.000005 m
  • P_c ≈ 28,362 Pa or 28.36 kPa
• Result: The capillary pressure is approximately 4.11 psi. This is the pressure drawing the water up into the tube.

Example 2: Oil Displacing Brine in a Reservoir Rock

In petroleum engineering, you might want to calculate the pressure needed for oil to enter a water-filled pore in a reservoir rock. The wettability might be less ideal. For more details on this topic, see our article on reservoir rock properties.

• Inputs:
  • Interfacial Tension (γ): 0.035 N/m (oil-brine)
  • Contact Angle (θ): 45° (moderately water-wet)
  • Pore Radius (r): 2 micrometers (µm)
• Calculation:
  • P_c = (2 * 0.035 N/m * cos(45°)) / 0.000002 m
  • P_c ≈ 24,749 Pa or 24.75 kPa
• Result: The entry pressure required is approximately 3.59 psi. Oil will not enter these pores until the pressure exceeds this value.

How to Use This Capillary Pressure Calculator

1. Enter Interfacial Tension (γ): Input the interfacial tension between your two fluids. Ensure you select the correct units, either N/m or mN/m (which is equivalent to dynes/cm).
2. Set the Contact Angle (θ): Input the contact angle in degrees. Values less than 90° indicate a wetting surface, while values greater than 90° indicate a non-wetting surface.
3. Provide the Pore Radius (r): Enter the radius of the pore or capillary. This value is critical, so be sure to select the appropriate unit: meters, millimeters, or micrometers.
4. Interpret the Results: The calculator instantly provides the capillary pressure in Pascals (Pa), kilopascals (kPa), and pounds per square inch (psi). The intermediate values used in the Young-Laplace equation are also shown for transparency.
5. Analyze the Chart and Table: Use the dynamic chart and table to visualize how capillary pressure is affected by changes in pore radius, a key factor in many applications.

Key Factors That Affect Capillary Pressure

Pore Radius (r)

This is the most sensitive parameter. Capillary pressure is inversely proportional to the pore radius. Smaller pores lead to exponentially higher capillary pressures.

Interfacial Tension (γ)

A higher interfacial tension between the two fluids results in a proportionally higher capillary pressure. This can be influenced by temperature and chemical composition. Our surface tension calculator can provide more insight.

Wettability (Contact Angle, θ)

Wettability determines the sign and magnitude of the cosine term. For wetting fluids (θ < 90°), cos(θ) is positive, leading to capillary rise. For non-wetting fluids (θ > 90°), cos(θ) is negative, leading to capillary depression.

Pore Geometry

The Young-Laplace equation assumes a perfect cylindrical pore. In reality, pores are irregular. This irregularity (e.g., shape, roughness) complicates the pressure profile but the cylindrical model remains a vital first approximation.

Fluid Densities

While not directly in this form of the equation, the density difference between fluids is what allows the capillary pressure to support a column of fluid against gravity, leading to capillary rise or fall. This is explained further in the meniscus height formula.

Temperature

Temperature primarily affects capillary pressure by altering the interfacial tension of the fluids. Generally, IFT decreases as temperature increases, which in turn lowers the capillary pressure.

Frequently Asked Questions (FAQ)

1. What does a negative capillary pressure mean?

A negative capillary pressure occurs when the contact angle is greater than 90 degrees. This happens in non-wetting systems (e.g., mercury in a glass tube). It means that external pressure must be applied to force the fluid *into* the capillary, and the meniscus will be depressed relative to the free surface.

2. Why is the unit mN/m the same as dyn/cm for interfacial tension?

This is a unit conversion. 1 Newton = 100,000 dynes, and 1 meter = 100 centimeters. Therefore, 1 N/m = (100,000 dyn/m) / 100 = 1000 dyn/cm. A milliNewton (mN) is 1/1000 of a Newton, so 1 mN/m = 1 dyn/cm.

3. How do I find the correct interfacial tension for my fluids?

Interfacial tension values are determined experimentally. For common fluid pairs like water-air, oil-water, or mercury-air, these values can be found in engineering handbooks or scientific literature. They are temperature-dependent.

4. What is the difference between pore radius and pore diameter?

The radius is half the diameter. The Young-Laplace equation specifically uses the radius (r) of the pore. If you have a measurement of the pore diameter, be sure to divide it by two before using it in this calculator.

5. Can this calculator be used for non-cylindrical pores?

This calculator is based on a cylindrical model. For different geometries (e.g., slits, square pores), the `2` in the numerator of the Young-Laplace equation is replaced by a different geometric factor. However, the cylindrical model provides a very useful and common estimate for general porous media fluid flow.

6. What is wettability and why is it important?

Wettability is the preference of a solid to be in contact with one fluid over another, quantified by the contact angle. It’s critical because it determines whether a fluid will spontaneously enter a pore or need to be forced in. A wettability index calculator can further explore this concept.

7. What is the ‘entry pressure’ in reservoir engineering?

Entry pressure is the minimum capillary pressure required for a non-wetting phase (like oil or gas) to start displacing a wetting phase (like water) from the largest pores in a rock. This calculator directly computes that pressure for a given pore size.

8. Does this calculation work for gravity?

This calculator determines the pressure difference across the meniscus itself. To find the equilibrium height a fluid will rise or fall in a tube against gravity, you must balance this capillary pressure against the hydrostatic pressure of the fluid column (P = ρgh).