Uniqueness And Simultaneous Predictability of Relative Permeability And Capillary Pressure By Discrete And Continuous Means
Authors: Ucan S. , Civan F. , Evans R.D. 
Published: March 2010

Abstract
Requirements for unique estimation of two-phase flow functions are investigated. This is achieved by increasing the number of observable parameters by using both external and internal data, by representing the flow functions via global empirical functions, discrete values, and piece-wise local functions and by implementing the imulated annealing method for non-linear global optimization. Applications involving unsteady-state drainage and imbibition unsteady-state core flood tests are presented. It is shown that global functional representations of the relative permeability and capillary pressure data are not sufficient and a discrete representation of flow functions leads to non-smooth functions. However, a piece-wise functional representation of relative permeability and capillary pressure data can be determined uniquely when the transient-state internal saturation profiles and the overall pressure differentials are used together for history matching.

Introduction
Petrophysical properties of multiphase flow systems in porous rock are complex functions of the morphology and topology of the porous medium, interactions between rock and fluids, phase distribution and flow patterns and regimes. It is impractical to deconvolute the effect of the individual factors and forces from the macroscopic petrophysical properties. Therefore, the effect of these properties on the flow behaviour is lumped in the form of empirically determined relative permeability and capillary pressure functions which are used as the primary flow parameters for the macroscopic description of multiphase flow in porous media. However, development of reliable methods for the simultaneous determination of relative permeability and capillary pressure data from laboratory core flood tests is a challenging task and is of continuing interest to the oil and gas industry.
This paper addresses the issue of uniqueness in the determination of relative permeability and capillary pressure functions by means of the history matching of unsteady-state displacement data obtained from laboratory core flow tests. History matching (the inverse problem) requires a reliable porous media averaged, macroscopic flow description model (the forward problem) to predict the values of the observable parameters such as cumulative production, pressure drop histories and saturation history profiles so that the best estimates of the relative permeabilities and capillary pressure functions can be determined. However, some model parameters may be overdetermined while leaving the others underdetermined, unless the interpretation method is designed to assimilate a proper set of experimental data. Lack of intrinsic data, experimental uncertainties, and an accurate physical representation of the complex flow affect the reliability of the predictions.
When a problem is ill-posed, its solution may not necessarily be unique and, therefore, perturbation of any model parameters may lead to arbitrary variations of the solutions. As demonstrated in the present study, observed values are quite sensitive to the flow functions. In reservoir simulation, the flow functions are usually chosen as the first model parameters to be tuned for history matching. Presently, there are no satisfactory general theoretical flow functions available. For the prediction of two phase relative permeabilities alone, over thirty different empirical models have been proposed (Honarpoor et al.(1), and Siddiqui et al.(2)). Although these empirical models are applicable only to specific conditions, in many applications, they have been selected arbitrarily without any particular basis.