Representation Of Hysteresis In Capillary Pressure For Reservoir Simulation Models
Authors: Tan Thomas 
Published: March 2010

Abstract
A new formulation to represent capillary pressure hysteresis for use in numerical reservoir simulation is described. Unlike previous methods, the new method reproduces the experimental bounding drainage and imbibition curves. It also provides for a smooth transition between the drainage and imbibition curves at the maximum wetting saturation.
After implementation in a fully implicit numerical simulator, radial model coning cases and cartesian water drive cases were run. The results show capillary pressure hysteresis has a noticeable influence on water coning behaviour, but has a negligible effect on prediction of performance for water drive reservoirs.

Introduction
The hysteresis of capillary pressure is a well-known phenomenon that has been reproduced in many laboratory experiments. Many paper(1,2) have presented data on both drainage and imbibition capillary pressure curves. It is of interest to be able to represent the hysteretic behaviour in reservoir numerical simulation models so that more accurate or realistic predictions or reservoir behaviour may be made. Other authors(3) have commented that capillary pressure hysteresis may affect well coning behaviour.
As there is no theoretical analysis available to predict the values of the drainage and imbibition curves at any saturation value, empirical methods have been applied. The most commonly used one is that presented by Killough(4) in a classic paper on reservoir simulation with history-dependent saturation functions. His model uses interpolation with a regression parameter to generate intermediate scanning curves that are the hysteresis curves to be used.
Our attempt to implement Killough's method into our numerical simulators, however, revealed certain deficiencies. The subject or this paper is a new formulation that resolves these difficulties. Following the implementation of the new algorithm in the simulation model, several runs were made to investigate the effect of capillary pressure hysteresis on recovery.

Discussion of Killough's Formulation for Capillary Pressure
In the algorithm presented by Killough, experimental data are required only for the bounding imbibition and drainage curves as the method generates intermediate scanning curves using an interpolative scheme and a regression parameter that may be varied to allow a closer fit of experimental scanning curves, if available, The discussion that follows uses water as the wetting phase.
For a drainage imbibition curve, using Killough's notation, it' Sw is the current saturation, SWHyx the departure point on the drainage curve, and e is a given regression parameter, then the capillary pressure on the scanning curve is given by:
Equation (Available In Full Paper)
and Swmax is the maximum wetting phase saturations (= Swor). If this formulation is consistent, then it should reproduce the behaviour of the experimentally measured imbibition curve if the departure point on the drainage curve is the residual water saturation. This is the obvious boundary condition. In such a case, F should be equal to I for all values of Sw, Substituting
Equation (Available In Full Paper)
and it is clear that F is equal to I only when Sw = Swmax. Thus, the scanning curve produced does not reflect the behaviour of the experimentally measured imbibition curve.