Overview
Capillary pressure (Pc) is the pressure difference across a curved fluid interface in a porous medium. It controls initial fluid distribution, transition zones, irreducible water saturation, and imbibition/drainage behavior. The Leverett J-function (1941) normalizes capillary pressure data to account for variations in porosity, permeability, and interfacial tension, enabling correlation of Pc data from different rock types or wells.
Theory
Capillary pressure in a tube is described by the Young-Laplace equation:
Pc = 2σ cos(θ) / r
In porous media, the pore radius is replaced by a characteristic length related to permeability and porosity. Leverett proposed normalizing Pc data as a dimensionless function J(Sw).
Formulas
Capillary Pressure (Porous Media)
Pc = Po - Pw (for oil-water system)
Pc = Pg - Po (for gas-oil system)
Leverett J-Function
J(Sw) = Pc * sqrt(k/φ) / (σ * cos(θ))
where:
| Symbol | Description | Units |
|---|---|---|
| Pc | Capillary pressure | psi (convert to dynes/cm² if using CGS) |
| k | Permeability | md (convert to cm²: 1 md = 9.869e-12 cm²) |
| φ | Porosity | fraction |
| σ | Interfacial tension | dynes/cm |
| θ | Contact angle | degrees |
In Oilfield Units
J(Sw) = 0.2166 * Pc * sqrt(k/φ) / (σ * cos(θ))
where Pc in psi, k in md, σ in dynes/cm.
Brooks-Corey Model
Pc = Pd * (Se)^(-1/λ)
Se = (Sw - Swi) / (1 - Swi)
where Pd = displacement (entry) pressure, λ = pore size distribution index (typically 0.5–4.0).
Height Above Free Water Level (HAFWL)
h = Pc / (Δρ * g) = Pc * 144 / (Δρ_water-oil)
In oilfield units:
h (ft) = Pc (psi) / (0.433 * ΔSG)
Worked Example
Given: Pc = 5 psi, k = 100 md, φ = 0.20, σ = 26 dynes/cm, θ = 0° (water-wet).
J-function value:
J = 0.2166 * 5 * sqrt(100/0.20) / (26 * cos(0°))
= 0.2166 * 5 * sqrt(500) / 26
= 0.2166 * 5 * 22.36 / 26
= 24.21 / 26
= 0.931
Height above FWL (oil-water, ΔSG = 0.25):
h = 5 / (0.433 * 0.25) = 5 / 0.108 = 46.2 ft
If a nearby well has k = 50 md, φ = 0.15, same J-function, what is Pc?
Pc = J * σ * cos(θ) / (0.2166 * sqrt(k/φ))
= 0.931 * 26 / (0.2166 * sqrt(50/0.15))
= 24.2 / (0.2166 * 18.26)
= 24.2 / 3.96
= 6.11 psi
Valid Ranges
| Parameter | Typical Range | Notes |
|---|---|---|
| Pc (drainage) | 0 – 200 psi | Depends on pore throat size |
| J(Sw) | 0.1 – 10 | Dimensionless |
| λ (Brooks-Corey) | 0.5 – 4.0 | Higher = more uniform pore size |
| σ (oil-water) | 15 – 35 dynes/cm | Decreases with pressure/temperature |
| θ (water-wet) | 0 – 50° | Mixed-wet: 50–120°; oil-wet: 120–180° |
References
- Leverett, M.C. (1941). "Capillary Behavior in Porous Solids." Trans. AIME, 142, 152–169.
- Brooks, R.H. & Corey, A.T. (1964). "Hydraulic Properties of Porous Media." Hydrology Papers, Colorado State University.
- Dake, L.P. (1978). Fundamentals of Reservoir Engineering. Elsevier. Chapter 4.
- PetroWiki — Capillary pressure: https://petrowiki.spe.org/Capillary_pressure