Theoretical Analysis of Three Phase Flow Experiments in Porous Media
Authors: Sahni Akshay , Guzman Rafael , Blunt Martin 
Published: December 2007

Abstract
This paper presents a framework for analyzing three phase flow experiments in porous media. We analyze previously published data from a dynamic displacement experiment, where three phase relative permeabilities have been measured. The relative permeability of each phase is, to a good approximation, a polynomial function of the saturation of that phase. Then analytically, we calculate the saturation paths and recovery for the experiments using the method of characteristics. The predicted paths agree well with the experimental measurements, and with one-dimensional numerical solutions.

Introduction
The simultaneous flow of three immiscible fluids in porous media is an essential component of enhanced oil recovery and aquifer remediation processes. It is important to understand three phase flow for designing optimum methods for recovering oil by gas injection, gas gravity drainage, surfactant flooding and thermal recovery.
Although three phase flow experiments are hard to perform, there are many published studies. Grader and O'Meara performed dynamic displacement experiments using three immiscible fluids. Virnovsky and Grader and O'Meara developed a theory to obtain three phase relative permeability as a function of saturation by an extension of the Welge and JBN methods to three phases. Siddiqui et. al verified the theory using X-ray computerized tomography to obtain in-situ saturations for three phase dynamic displacement experiments. Sarem obtained three phase relative permeability by unsteady state displacement experiments assuming that relative permeability of each phase was a function of its own saturation. Oak et. al presented a steady-state study of three phase relative permeability using fired Berea cores. Minssieux and Duquerroix analyzed water alternating gas experiments in porous media with residual oil.
Three phase gravity drainage experiments have shown that it is possible to obtain very low residual oil saturations. Vizika and Lombard studied the effects of wettability and spreading characteristics of the fluid system in three phase gravity drainage. Skurdal et. al analyzed gas gravity drainage experiments using spreading and non-spreading systems under oil wet, water wet and mixed wet conditions. Naylor et. al performed gravity drainage experiments by measuring insitu oil and brine saturations using a radioactive tracer technique. Chalier et. al used a gamma-ray absorption technique to obtain three phase relative permeability for tertiary gas gravity drainage experiments. Skauge et. al summarized results from gas gravity drainage experiments at different water saturations. Espie et. al established and interpreted laboratory data quantifying the dynamics of oil bank growth during the waterflood/gravity drainage interaction in Prudhoe Bay cores.
In three phase flow, oil may form layers in crevices and roughness of the pore space, between water and gas. It is the drainage of these layers that is responsible for the good recoveries observed in gravity drainage experiments (Blunt et. al). Although the residual oil saturation is very low, the relative permeability at low oil saturations may also be very small, making gas injection schemes uneconomic over any reasonable time scale. It is therefore important to have a good understanding of the three phase relative permeability, especially at low oil saturation.
Most numerical models of three phase flow in porous media use empirical relationships for capillary pressure and relative permeability. Delshad and Pope, Oak and Fayers and Matthews compared empirical models to published experimental data and showed that in most cases the empirical models fail to match the measurements.
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