The Effect of Capillary Pressure in a Multilayer Model of Porous Media
Authors: Hawthorne R.G. 
Published: December 2007

Abstract
This paper shows how the Hiatt-Hearn method for computing pseudo-relative permeability functions is modified by capillary pressure. The model is one-dimensional and without gravity forces. It provides another limiting case to computed pseudo-relative permeability functions. pseudo-relative permeability functions.The model was used for theoretical analysis and numerical experiments on water drives. The study shows that capillary forces promote a sharper front and a higher recovery at breakthrough than would be predicted for a noncapillary case.

Introduction
Reservoir performance prediction by analytical methods is based on some definition of fluid permeability as a function of saturation. In the permeability as a function of saturation. In the past, we have tried to use laboratory tests on past, we have tried to use laboratory tests on cores (relative permeabilities) to predict reservoir performance. Not only do laboratory data on performance. Not only do laboratory data on different cores from the same field vary widely, but data obtained from the same core by different methods are not always the same.
Extrapolation of small-scale core measurements to the reservoir scale is questionable. Various averaging and combining schemes are tried, but only sometimes do laboratory flow tests relate to actual field performance.
There is extensive literature on attempts to define relative permeability from basic physical principles. A review of this literature is beyond principles. A review of this literature is beyond the scope of this paper; for further information, see Refs. 1 and 2.
Alternatively, pseudo-relative permeability curves calculated for layered models have been tried. In practice, the layer methods sometimes work, but practice, the layer methods sometimes work, but often the laboratory core analysis is just as good. The technique presently used to avoid these difficulties is history matching. This is basically extrapolating a curve and, unfortunately, the method depends on knowing a good part of the performance we wish to predict.
Two basic layered models are used in petroleum engineering. The Dykstra-Parsons method assumes the layers are isolated except for common pressure at the input and output ends. The Hiatt method assumes crossflow between layers such that the layers are always in pressure equilibrium at any given distance. Both methods assume a sharp displacement front between displacing and displaced fluids. Recently, Hearn showed how to compute pseudo-relative permeability curves based on the pseudo-relative permeability curves based on the Hiatt model. Carrying the idea even further, jacks et al. showed how to compute pseudo-relative permeability functions under dynamic conditions permeability functions under dynamic conditions using a reservoir simulator.
It seems possible that the failure of layered models was partly caused by neglecting capillary forces in computing fluid displacement. This study was made to determining the part that capillary forces could play in the Hiatt-Hearn model. The results show how capillary forces change pseudo-relative permeability curves. While thick layers in the permeability curves. While thick layers in the reservoir probably could not reach the pressure equilibrium postulated by this model, the method does impose a limiting condition. With this and other models, it should be possible to bracket extremes of relative permeability (as caused by permeability variation, capillary imbibition, permeability variation, capillary imbibition, crossflows, viscous fingering, etc.) and to limit the range of performance predictions.
In the following development of theory, I emphasize the water-drive casethat is, the case for a wetting fluid displacing a nonwetting fluid from the reservoir. Flow is treated as one-dimensional and horizontal so that there are no gravity effects. First, the formulas are given for two parallel capillaries, and then the results are generalized for any number of capillary channels.
Sample calculations were made to illustrate the effects of flow parameters on the shape of the displacement front. In the dynamic system, these sample calculations become numerical experiments in fluid flow. Moreover, these numerical experiments did not give entirely the expected results for water drive at an unfavorable mobility ratio.
The gas-dove case (nonwetting fluid displacing a wetting fluid) has been studied in an analogous manner.
SPEJ
P. 467