Relative Permeability at Near-Critical Conditions
Authors: Blom S.M.P. , Hagoort J. , Soetekouw D.P.N. 
Published: December 2007

Abstract
We have measured a series of two-phase relative permeability curves at near-critical conditions by means of the displacement method. As a fluid system we have used the model system methanol/n-hexane that exhibits a critical point at ambient conditions. In the measurements we have varied the interfacial tension and the flow rate.
Our results show a clear trend from immiscible relative permeability functions to miscible relative permeability lines with with decreasing interfacial tension and increasing Darcy velocity. The relative permeability measurements show that the controlling parameter is the ratio of viscous to capillary forces on a pore scale, denoted by the capillary number Nc = k p/.
To demonstrate the significance of using the proper relative permeability functions, we present an example calculation of well impairment due to drop out of liquid hydrocarbon in gas condensate wells. The calculations show that near-miscible relative permeability functions come into play in the vicinity of the well bore. This is contrary to the implications of the relative permeability model of Coats. Furthermore, we show that when dependency of relative permeability on the capillary number is ignored, well impairment in highly permeable reservoirs may be overestimated.

Introduction
As hydrocarbon exploration moves to deeper geological formations, volatile oil and gas condensate reservoirs become increasingly important. At initial reservoir conditions the hydrocarbon fluids in these reservoirs are often found at near-critical conditions. As a consequence, the physical properties of the oil phase and the gas phase are very similar and the interfacial tension between oil and gas is very low. The latter may have an important bearing on the multi-phase flow characteristics in the reservoir during the production phase. An example of an important multi-phase fluid problem at near-critical conditions is condensate drop out in the vicinity of wells in a gas condensate reservoir.1 This drop out causes an apparent skin resistance at the well bore that impairs the production capacity of the well.
Traditionally, multi-phase flow in porous media is described by means of the concept of relative permeability functions, empirical relationships for the decrease in effective permeability to a flowing fluid phase as a function of the fluid saturation. At conditions far from the critical point multi-phase flow is, when viewed on a pore scale, dominated by capillary forces relative to viscous and gravitational forces. Hence relative permeability functions may be considered constant, i.e., independent of flow rate and interfacial tension. The constant functions are commonly referred to as immiscible relative permeability functions. At the other extreme, in the limit of zero interfacial tension, relative permeability curves reduce to linear functions of the fluid saturation. These straight lines are referred to as miscible relative permeability functions.
The effect of near-criticality on relative permeability is still an unsolved issue in reservoir engineering. Experimental studies published in the literature do indicate a trend from immiscible to miscible relative permeability curves as the interfacial tension approaches zero, but are non-conclusive on how the miscible curves are approached. Lacking firm experimental data, relative permeability curves at near-critical conditions are usually represented by heuristic models, notably the Coats model. In this model the relative permeability is a weighted linear function of immiscible and miscible relative permeability curves, where the weighting factor is an exponential function of the interfacial tension.
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