Relative-Permeability Measurements: An Overview
Authors: Honarpour M. , Mahmood S.M. 
Published: December 2007

Introduction
Fluid transport through reservoir rocks is complex and cannot be described by theory alone. Darcy's law, an empirical equation describing the laminar flow of incompressible fluids, is largely used for calculation of fluid flow through porous media. It relates the macroscopic velocity (flux) of a fluid of known viscosity to the pressure gradient by a proportionality factor called absolute permeability, expressed in darcies. Permeability is a measure of the ability of porous materials to Permeability is a measure of the ability of porous materials to conduct flow and is dictated by the geometry of the pore network. Generally, the fluid flow in hydrocarbon reservoirs involves more than one fluid, in which case the ability of each fluid to flow is reduced by the presence of other fluids. Darcy's equation has been extended to such situations using the concept of effective permeability, which is the apparent permeability of a fluid at a given saturation. The sum of the permeability of a fluid at a given saturation. The sum of the effective permeabilities for all phases is less than the absolute permeability because of the interference between fluids that permeability because of the interference between fluids that share the same channels. The effective permeability to a fluid becomes zero while its saturation is finite because the fluids become discontinuous at low saturations.
Another useful concept in describing the flow of multiphase systems is relative permeability, which is defined as the ratio of the effective permeability of a fluid to the absolute permeability of the rock. Relative permeability has a first-order permeability of the rock. Relative permeability has a first-order dependency on saturation level. However, many interstitial fluid distributions are possible for each level of saturation, depending on the direction of saturation changes. Thus, values of relative permeability vs. saturation obtained for drainage (reduction of wetting-phase saturation) may be different from those for imbibition (increase in wetting-phase saturation). This phenomenon is called hysteresis. phenomenon is called hysteresis. Fig. 1 shows a typical plot of two-phase relative permeability vs. saturation. It is also helpful to present such permeability vs. saturation. It is also helpful to present such plots on a semilog scale to expand the relative-permeability plots on a semilog scale to expand the relative-permeability characteristics near the endpoint saturations.
Relative-permeability data are essential for almost all calculations of fluid flow in hydrocarbon reservoirs. The data are used in making engineering estimates of productivity, injectivity, and ultimate recovery from reservoirs for evaluation and planning of production operations and also can be used to diagnose formation damage expected under various operational conditions. These data are unquestionably one of the most important data sets required in reservoir simulation studies.
Laboratory Determination of Effective Permeability and Relative Permeability Permeability and Relative Permeability Steady-state methods for determining permeabilities have the widest application and greatest reliability because the capillary equilibrium prevails, the saturation is measured directly, and the calculation scheme is based on Darcy's law. Unsteady-state techniques present many uncertainties in calculation schemes. Operational constraints connected with use of viscous ohs and high injection rates diminish the role of capillarity such that the influence of wettability cannot always be manifested. Following is a description of both methods.

Steady-State Techniques.
The most reliable relative-permeability data are obtained by steady-state methods in which two or three fluids are injected simultaneously at constant rates or pressure for extended durations to reach equilibrium. The saturations, flow rates, and pressure gradients are measured and used in Darcy's law to obtain the effective permeability for each phase. Conventionally, curves of relative permeability for each phase. Conventionally, curves of relative permeability vs. saturation are obtained, in a stepwise fashion, permeability vs. saturation are obtained, in a stepwise fashion, by changing the ratio of injection rates and repeating the measurements as equilibrium is attained. Saturation changes are controlled to be unidirectional (i.e., imbibition or drainage) to avoid hysteresis.
The steady-state methods are inherently time-consuming because equilibrium attainment may require several hours or days at each saturation level. In addition, these methods require independent measurement of fluid saturations in the core. Their advantages are greater reliability and the ability to determine relative permeability for a wider range of saturation levels. The steady-state methods include the Hassler method, single-sample dynamic, stationary phase, Penn State, and modified Penn State. They vary in the method of establishing capillary equilibrium between fluids and reducing or eliminating end effects. Further details of these methods are provided in subsequent sections. provided in subsequent sections. Unsteady-State Techniques. The quickest laboratory methods of obtaining relative-permeability data are unsteady-state techniques. In these techniques, saturation equilibrium is not attained; thus, an entire set of relative-permeability vs. saturation curves can be obtained in a few hours. A typical run involves displacing in-situ fluids by constant-rate (or constant-pressure) injection of a driving fluid while monitoring the effluent volumes continuously. The production data are analyzed, and a set of relative-permeability curves is obtained using various mathematical methods.
The Buckley-Leverett equation for linear displacement of immiscible and incompressible fluids is the basis for all analyses. This equation relates the saturation levels, at each point and time, to capillary pressure, the ratio of fluid point and time, to capillary pressure, the ratio of fluid viscosities, the flow rates, and the relative permeabilities. The Welge, Johnson-Bossler-Naumann, and Jones-Roszelle methods are most commonly used for analysis.
Many difficulties are inherent in unsteady-state methods. Operational problems such as capillary end effects, viscous fingering, and channeling in heterogeneous cores are difficult to monitor and to account for properly.
JPT
P. 963