Determination of contact angles for three-phase flow in porous media using an energy balance

We define contact angles, θ, during displacement of three fluid phases in a porous medium using energy balance, extending previous work on two-phase flow. We test if this theory can be applied to quantify the three contact angles and wettability order in pore-scale images of three-phase displacement.

For three phases labelled 1, 2 and 3, and solid, s, using conservation of energy ignoring viscous dissipation (Δa_1scosθ₁₂-Δa₁₂-ϕκ₁₂ΔS₁)σ₁₂=(Δa_3scosθ₂₃+Δa₂₃-ϕκ₂₃ΔS₃)σ₂₃+Δa₁₃σ₁₃, where ϕ is the porosity, σ is the interfacial tension, a is the specific interfacial area, S is the saturation, and κ is the fluid-fluid interfacial curvature. Δ represents the change during a displacement. The third contact angle, θ₁₃ can be found using the Bartell-Osterhof relationship. The energy balance is also extended to an arbitrary number of phases.

X-ray imaging of porous media and the fluids within them, at pore-scale resolution, allows the difference terms in the energy balance equation to be measured. This enables wettability, the contact angles, to be determined for complex displacements, to characterize the behaviour, and for input into pore-scale models. Two synchrotron imaging datasets are used to illustrate the approach, comparing the flow of oil, water and gas in a water-wet and an altered-wettability limestone rock sample. We show that in the water-wet case, as expected, water (phase 1) is the most wetting phase, oil (phase 2) is intermediate wet, while gas (phase 3) is most non-wetting with effective contact angles of θ₁₂≈48° and θ₁₃≈44°, while θ₂₃=0 since oil is always present in spreading layers. In contrast, for the altered-wettability case, oil is most wetting, gas is intermediate-wet, while water is most non-wetting with contact angles of θ₁₂=134°±~10°,θ₁₃=119°±~10°, and θ₂₃=66°±~10°.