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

Hypothesis: We define contact angles, theta, 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. Theory: For three phases labelled 1, 2 and 3, and solid, s, using conservation of energy ignoring viscous dissipation (Delta alpha(1s) cos theta(12) - Delta alpha(12) phi kappa(12)Delta S-1)sigma(12) = (Delta alpha(3s) cos theta(23) Delta alpha(23) phi kappa(23)Delta S-3)sigma(23) Delta alpha(13)sigma(13), where phi is the porosity, sigma is the interfacial tension, alpha is the specific interfacial area, S is the saturation, and kappa is the fluid-fluid interfacial curvature. A represents the change during a displacement. The third contact angle, theta(13) can be found using the Bartell-Osterhof relationship. The energy balance is also extended to an arbitrary number of phases. Findings: 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 theta(12) approximate to 48 degrees and theta(13) 44 degrees, while theta(23) approximate to 0 since oil is always present in spreading layers. In contrast, for the alteredwettability case, oil is most wetting, gas is intermediate-wet, while water is most non-wetting with contact angles of theta(12) approximate to 134 degrees +/- similar to 10 degrees; theta(13) =119 degrees +/-similar to 10 degrees, and theta(23) = 66 degrees +/-similar to 10 degrees. (C) 2020 The Author(s). Published by Elsevier Inc.