Simultaneous Estimation of Relative Permeability and Capillary Pressure for Non-Darcy Flow-Steady State
Authors: Ucan Sezai , Civan Faruk 
Published: December 2007

Abstract
An inverse interpretation method has been developed to determine the relative permeability and capillary pressure curves from steady-state core flood experiments with or without the non-Darcy effect included. The Forchheimer equation has been incorporated for multiphase non-Darcy flow and the forward model has been solved numerically by the finite difference method. The simulated annealing method has been applied as the non-linear global optimization technique for history matching. The flow functions were generated by automatic history matching of the saturation and pressure profiles along the core.
Different steady state experimental methods have been introduced in the literature to eliminate or minimize the influence of capillary end effects. However, in this study, capillary end effect has been used as an asset to determine the relative permeability and capillary pressure curves. Because the capillary end effect is strongly related to the properties of the multiphase fluid in porous core, it can be best incorporated by using the saturation profiles as the in-situ measurement.
This interpretation method also reduces the required time frame to conduct tests. Thus, instead of repeating various tests at many different gas/water ratio and for flow rates for covering a full saturation range, experiments for two or three different gas/water ratio, which give different saturation ranges, will be sufficient to capture the characteristic flow functions. The main difficulty in this method is the requirement of accurate saturation profiles by high resolution X-ray computer tomography, specifically at the ends of the core.
Relative permeability of gas for non-Darcy flow has a higher curvature than Darcy's flow. Due to greater pressure dissipation, the gas saturation develops more rapidly under non-Darcy flow conditions. Results indicate that non-Darcy flow is not negligible for estimation of relative permeability and capillary pressure curves, particularly for the gas/brine system.

Introduction
Traditional methods for interpretation of the simultaneous immiscible fluid flow data to generate relative permeability and capillary pressure curves facilitate the external core data such as the production/injection rates and pressure drop histories. Steady-state or unsteady-state techniques may be used to obtain such information. Both methods have some advantages and disadvantages. The steady-state method usually requires a long time to obtain stabilization, but it is the simplest method to determine relative permeability and capillary pressure. Whereas, the unsteady-state method requires numerical or graphical differentiation of experimental data which amplifies the errors when inaccurate measurements are used. In this method, it is usually difficult to describe the flow functions over the entire saturation range of interest for low mobility ratio cases. The majority of the previous interpretation methods for both assume incompressible fluids and Darcy flow, and no capillary pressure.
Numerous investigations have been conducted to study these assumptions and estimate the capillary pressure and relative permeability simultaneously. Efforts have been made to gather internal core data such as the saturation and pressure history profiles which can be measured along the core by the gamma-ray attenuation technique, CT scanning (Eleri, et al, Satik et al), or NMR imaging (Enwere and Archer) in addition to the overall pressure drop and production data. Various functional representations of relative permeability and capillary pressure curves with a variety of interpretation methods and optimization techniques have also been investigated.
Different experimental and interpretation methods have been introduced in the literature to eliminate or reduce the influence of capillary forces and other simplifying assumptions. These efforts can be classified in two categories as the direct (forward) and indirect (inverse) methods. In direct methods, special experimental set-ups are designed.
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