Overview
The Buckley-Leverett (1942) theory describes one-dimensional, immiscible displacement of oil by water in a porous medium. It predicts the water saturation profile as a function of distance and time during waterflooding. Combined with the Welge (1952) tangent construction, it provides breakthrough time, oil recovery factor, and water-oil ratio at breakthrough.
Theory
The fractional flow equation describes the fraction of total flow that is water at any saturation:
fw = 1 / (1 + (kro/krw) * (μw/μo))
The Buckley-Leverett frontal advance equation gives the velocity of any saturation:
dx/dt = (qt / (A * φ)) * dfw/dSw
where qt = total flow rate, A = cross-sectional area, φ = porosity.
The saturation front moves as a shock (discontinuity) from Swi to Swf (front saturation), determined by the Welge tangent construction.
Formulas
Fractional Flow (with gravity and capillary pressure neglected)
fw = 1 / (1 + (kro/krw) * (μw/μo))
With Gravity Term
fw = (1 + (kro * A * Δρ * g * sin(α)) / (qt * μo)) / (1 + (kro/krw) * (μw/μo))
Welge Tangent Construction
Draw a tangent from (Swi, fwi=0) to the fw curve. The tangent point gives:
- Swf = water saturation at the front (breakthrough saturation)
- fw_bt = fractional flow at breakthrough
- The x-intercept of the tangent extended gives average saturation behind the front
Recovery at Breakthrough
RF_bt = (Sw_avg - Swi) / (1 - Swi - Sor)
where Sw_avg = average water saturation behind the front at breakthrough.
Dimensionless Time (Pore Volumes Injected)
tD = Wi / (A * L * φ) = 1 / (dfw/dSw)|_at_Swf
Water-Oil Ratio (WOR)
WOR = fw / (1 - fw)
Worked Example
Given: μo = 5 cp, μw = 1 cp, Swi = 0.20, Sor = 0.25.
Corey relative permeability: kro = kro_max(1-Sw-Sor)^no, krw = krw_max((Sw-Swi)/(1-Swi-Sor))^nw
With kro_max = 0.8, krw_max = 0.3, no = 2, nw = 2.
At Sw = 0.50:
Se = (0.50 - 0.20) / (1 - 0.20 - 0.25) = 0.30/0.55 = 0.545
kro = 0.8 * (1 - 0.545)^2 = 0.8 * 0.207 = 0.166
krw = 0.3 * 0.545^2 = 0.3 * 0.297 = 0.089
fw = 1 / (1 + (0.166/0.089)*(1/5)) = 1 / (1 + 0.373) = 0.728
Breakthrough saturation (from Welge tangent): Swf ≈ 0.48
PV injected at breakthrough: tD = 1/(dfw/dSw)|_Swf ≈ 0.35 PV
Valid Ranges
| Parameter | Typical Range | Notes |
|---|---|---|
| Swi | 0.10 – 0.40 | Irreducible water saturation |
| Sor | 0.15 – 0.35 | Residual oil saturation |
| μo/μw | 0.5 – 100 | Higher ratio → less favorable displacement |
| b (Corey exponent) | 1.5 – 4.0 | Controls relative permeability shape |
Assumptions and Limitations
- One-dimensional, linear flow only
- Incompressible fluids
- No capillary pressure (valid for high-rate floods)
- Homogeneous reservoir (no layering, no channeling)
- Immediate gravity segregation not modeled
- Buckley, S.E. & Leverett, M.C. (1942). "Mechanism of Fluid Displacement in Sands." Trans. AIME, 146, 107–116.
- Welge, H.J. (1952). "A Simplified Method for Computing Oil Recovery by Gas or Water Drive." Trans. AIME, 195, 91–98.
- Craig, F.F. (1971). The Reservoir Engineering Aspects of Waterflooding. SPE Monograph Vol. 3.
- PetroWiki — Buckley-Leverett: https://petrowiki.spe.org/Buckley-Leverett_theory