🛢️ Oil Reservoir Waterflood Simulator: Buckley-Leverett Fractional Flow and Layered Sweep (2D Cross-Section)

Two-phase Darcy flow

Each phase moves according to Darcy's law, uw = -(k krw/μw) ∂p/∂x and likewise for oil. Relative permeabilities follow Corey curves krw = krw* Snnw, kro = (1 - Sn)no with normalised saturation Sn = (Sw - Swc)/(1 - Swc - Sor). The share of the flowing stream that is water is the fractional flow fw = 1 / (1 + (kro/μo)/(krw/μw)).

Buckley-Leverett and the Welge shock

With incompressible fluids, 1D flow conserves water: ∂Sw/∂tD + ∂fw/∂xD = 0, where tD counts injected pore volumes. The solution is a shock at the saturation Sf where the tangent from (Swc, 0) touches the fw curve, fw'(Sf) = fw(Sf) / (Sf - Swc), followed by a smooth rarefaction. Water breaks through at tD = 1/fw'(Sf). A high mobility ratio M = (krw*/μw)/(1/μo) (viscous oil) gives early breakthrough and poor sweep.

Layers

The slice is eight non-communicating layers sharing the same pressure drop, so layer flow is proportional to its permeability: the high-k layers flood first (thief zone) and the producer water cut rises before low-k layers are swept. Recovery is expressed as a fraction of the original oil in place.

Honest simplifications

No gravity, capillary pressure, compressibility, crossflow between layers, wells' pressure drop or 3D areal sweep; one-dimensional first-order upwind finite volumes (numerical diffusion smears the shock slightly). The slice is a teaching model of a waterflood, not a field simulator.

• Raise the oil viscosity: the Welge tangent moves left, breakthrough comes earlier and the recovery curve flattens.
• Increase layer heterogeneity: the top flow layers reach the producer long before the rest.
• Choose "Homogeneous" and compare the green numerical profile with the analytic one in the last graph.
• Change the Corey exponents to mimic oil-wet rock.