Representation of Capillary Pressure Hysteresis in Reservoir Simulation
Authors: Kleppe J. , Delaplace P. , Lenormand R. , Hamon G. , Chaput E. 
Published: December 2007

Abstract
A new method for construction of hysteresis capillary pressure relationships for use in reservoir simulation models is presented. The method is based on the experimental drainage-imbibition bounding curves and the history of the saturation changes. By scaling of the measured drainage and imbibition bounding curves into the saturation ranges in question for hysteresis scanning loops, any family of hysteresis curves may be constructed. The method is well suited for use in reservoir simulation models.
Results of highly accurate laboratory measurements of capillary pressures on a gas-oil system including re-imbibition and re-drainage are presented. Predictions of hysteresis behavior of the laboratory system by the new method show satisfactory agreement with the experiments, while prediction by the much used Killough's method fail to match the experiment, primarily because it does not scale saturations.
It is also observed that the commonly used Land equation for prediction of residual saturations is not representative for the system under investigation. Since the Land equation do not distinguish between rock types, it is not recommended used unless experimental support of its applicability exists. Our results show that the relationship between the residual saturation and the initial saturation for a hysteresis imbibition process is approximately linear.

Introduction
In a paper published in 1965, Morrow and Harris presented experimental results and a comprehensive discussion of capillary behavior of porous materials. They show that a hysteresis curve departing from one of the bounding drainage or imbibition curves is uniquely defined by the departing point on the curve. By the same token, virtually an infinite number of families of hysteresis curves may result from saturation reversals, and each branch is defined by the departing point and the history of saturation reversals.
In order to define an imbibition hysteresis curve, the residual saturation in addition to the departing point must be known. Commonly used for prediction of residual saturation is the semi-empirical relation presented by Land based on matching of experimental data. He found that for a given sand the difference in reciprocals of residual and initial saturations remains constant.
Several authors have discussed hysteresis behavior of porous media. Of particular interest is representation of hysteresis in reservoir simulation. Model input data normally includes complete drainage and imbibition curves. The simulation model then applies some method to predict hysteresis residual saturations and hysteresis capillary pressures and relative permeabilities. Both Killough and Carlson presented methods for predicting hysteresis in relative permeability. For prediction of hysteresis in capillary pressures, the method presented by Killough is frequently employed. His method computes hysteresis capillary pressures by weighting of the complete drainage and imbibition curves. However, as pointed out by Tan, Killough's method was specially formulated for the case where the drainage and imbibition curves meet at the residual saturation. Because of that, the method is often inadequate.
Recently, very accurate laboratory measurements of gas-oil capillary hysteresis have been made in the laboratories of IFP in a cooperation with Total and Elf Aquitaine. Measurements of capillary pressures including complete drainage and imbibition curves and intermediate drainage-imbibition and drainage-imbibition-drainage loops were made. The results are presented in this paper and used for evaluation of hysteresis prediction methods.

Experiment
Gas-oil drainage and imbibition capillary pressure cycles were measured using the Porous Plate Method. A schematic of the laboratory setup is shown in Fig. 1. The laboratory setup includes:
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