Fractional Flow & Buckley-Leverett Calculator

The Buckley-Leverett equation is one of the most important analytical tools in reservoir engineering for predicting the performance of immiscible displacement processes such as waterflooding. Developed by S.E. Buckley and M.C. Leverett in 1942, it describes the movement of a saturation front through a porous medium by combining the fractional flow equation with mass conservation. The theory assumes one-dimensional, incompressible, immiscible displacement with no capillary pressure or gravity effects — simplifications that nonetheless provide remarkably useful predictions for screening studies and initial waterflood design.

The fractional flow of water (f_w) represents the fraction of the total flowing stream that is water at any given water saturation (S_w). It is computed from relative permeability curves and fluid viscosities: f_w = 1 / (1 + (k_ro/k_rw) × (μ_w/μ_o)). The shape of the fractional flow curve is governed by the relative permeability characteristics of the rock-fluid system and the viscosity ratio of the displacing and displaced fluids. A favorable mobility ratio (M < 1) yields a more piston-like displacement, while unfavorable ratios (M > 1) lead to early water breakthrough and lower sweep efficiency.

The Welge tangent construction is a graphical method applied to the fractional flow curve to determine key waterflood performance parameters. By drawing a tangent line from the point (S_wi, 0) to the fractional flow curve, one can identify the breakthrough saturation (S_w,bt) — the water saturation at the producing well when water first arrives — and the average water saturation behind the front (S̅_w) at breakthrough. The recovery factor at breakthrough is then RF = (S̅_w − S_wi) / (1 − S_wi).

This calculator uses the Corey relative permeability model, which parameterizes the water and oil relative permeabilities as power functions of normalized saturation. The Corey model is widely used in reservoir simulation and provides a good fit to laboratory-measured relative permeability data for many rock-fluid systems. The exponents n_w and n_o control the curvature of the relative permeability curves — higher exponents produce more concave curves, indicating stronger wettability preference.

After breakthrough, continued water injection recovers additional oil, but at increasingly higher water cuts. The post-breakthrough recovery table shows how recovery factor improves with additional pore volumes of water injected, helping engineers evaluate the economic limit of a waterflood project. The mobility ratio M = (k_rw,max/μ_w) / (k_ro,max/μ_o) is a key indicator of displacement efficiency: values near or below unity suggest a stable, efficient flood, while high mobility ratios indicate viscous fingering and poor sweep.

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