Effect of Flow Rate on Imbibition Three-Phase Relative Permeabilities and Capillary Pressures
Authors: Akin Serhat , Demiral M.R. Birol 
Published: December 2007

Abstract
The effects of displacement pressure, pressure gradient, and flow rate on the shape of relative permeability curves have long been a controversial subject in petroleum literature. For drainage experiments it has been reported that the relative permeabilities are independent of flow rate. However, for imbibition experiments the rare literature, mainly concerned with oil-water phases, does not agree on this point.
Three phase, unsteady state CT scanned displacement tests were conducted using a fired Berea sandstone to obtain relative permeability and capillary pressure data. 8% Potassium Bromide doped brine, hexane and nitrogen gas was used. Relative permeabilities and capillary pressures were then estimated simultaneously after minimizing a least squares objective function containing all available and reliable experimental data obtained from three phase imbibition experiments using an automated history matching code where simulated annealing was utilized.
It has been found that brine and hexane relative permeability curves were affected much more compared to the gas relative permeability curve especially near the end points. Moreover, gas relative permeabilities decreased with increase in flow rate. Capillary pressure curves were affected in a similar manner. Finally, in order to confirm the above results an approach consisting of matching, at the same time, the fastest, the slowest and medium rate experimental data was tested. The algorithm failed to find a set of flow function curves which could fit both experimental data; therefore the conclusion was that for three phase imbibition the flow functions depend on the flow rate.

Introduction
Reservoir engineering calculations frequently require consideration of coexisting oil, water and gas phases. Such three phase flow occurs when oil is displaced by simultaneous gas/water flow as in carbon dioxide, water alternating gas flooding, steam flooding. For this reason, reservoir simulators generally include three phase relative permeability prediction methods. The effects of displacement pressure, pressure gradient, and flow rate on the shape of relative permeability curves have long been a controversial subject in petroleum literature. Leverett et al. reported, then disproved, the influence of flow rate upon relative permeability. They eventually assigned the observed deviations in their results to an end effect which was previously described by Hassler. Crowell et al. and Geffen et al. found that injection rate had no affect within the limits of viscous flow of water and gas. However, Henderson and Yuster, Morse et al., and Caudle et al. found that relative permeability curves were affected such that relative permeability decreased with increase in flow rate. The effect of flow rate on drainage relative permeability curves were investigated by Richardson et al., Osaba et al., and Leas et al. They found that drainage relative permeability was independent of flow rate as long as a saturation gradient was not introduced in the core by inertial forces.
The effect of flow rate on imbibition two phase relative permeability curves was addressed by Labastie et al. and Heaviside et al. Labastie et al. reported that relative permeabilities were independent of flow rate except near residual oil saturation. Capillary pressure data however depended on flow rate and porous medium wettability Moreover, they found that, imbibition capillary pressure changed very little with the flow rate on sandstones. For the carbonates, the capillary pressure, which was generally positive during the initial oil drainage phase, became negative immediately behind the front Heaviside et al. concluded that, numerical simulation incorporating capillary pressure could not explain the rate dependencies unless different relative permeability and capillary pressure data were used for different flow rates.
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