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# Relative permeability models

Relative permeability has important implications for flow of reservoir fluids. A number of models have been developed to relate relative permeability to other reservoir properties. This page provides an overview of those models.

## Brooks-Corey and related models

In 1954, Corey combined predictions of a tube-bundle model with his empirical expression for capillary pressure to obtain expressions for gas and oil relative permeabilities. In 1964, Brooks and Corey extended Corey’s results using Eq. 1 for capillary pressure to obtain the following expressions for oil and gas relative permeabilities: ....................(2)

and ....................(3)

Eqs. 2 and 3 apply to a porous material that is initially fully saturated with oil and then invaded by gas. These equations do not allow for nonzero critical gas saturation.

• For λ=2, Eqs. 2 and 3 reduce to the 1954 Corey expressions.
• Brooks and Corey related the parameter λ to the distribution of pore sizes.
• For narrow distributions, λ is greater than 2
• For wide distributions, λ is less than 2.
• λ =7.30 for an unconsolidated pack of glass beads of uniform diameter.
• For sandpacks with broader distributions of particle sizes, λ ranged from 1.8 to 3.7.
• For a particularly homogeneous consolidated sandstone, they reported λ =4.17.

The following "power-law" relationships are often used to describe oil, water, and gas relative permeabilities, respectively: ....................(4) ....................(5) ....................(6)

• The exponents no, nw, and ng range from 1 to 6
• The maximum relative permeabilities, kro max, krw,max, and krg,max, are between 0 and 1

These expressions are often referred to as modified Brooks-Corey relations, reflecting their similarity to the Brooks-Corey expression for oil relative permeability.

## Model for heterogeneous rock

In 1956, Corey and Rathjens extended Corey’s earlier work to explain the effect of laminations on gas/oil relative permeability relations. They considered flow parallel and flow perpendicular to N laminations. For flow perpendicular to laminations, ....................(7)

and ....................(8)

For flow parallel to laminations, ....................(9)

and ....................(10)

In these expressions,

• ki is the permeability of each layer
• kroi and krgi are oil and gas relative permeabilities for each layer, respectively
• fi is the fraction of total sample volume for each layer
• kT is the total permeability of the laminated materials
• kroT and krgT are the resulting total relative permeabilities of the laminated materials

The predictions of Eqs. 7 through 10 for N = 2 are shown in Fig.1. The permeability of one region is one-fourth that of the other region. The relative permeabilities are given by Eqs. 2 and 3 with λ = 2 and with Sor = Sgc = 0.20. Corey and Rathjens noted two consequences of laminations: bumps in the relative permeability relationships and a shifting of the critical gas saturation. As shown in Fig.1, the critical gas saturation shifts from 0.20 for the uniform sample to 0.10 for flow parallel to strata, and to 0.33 for flow perpendicular to laminations. Also, the relative permeability for flow perpendicular to laminations is much less than that parallel to laminations. As shown in Fig.2, measured relative permeabilities for a sample of Berea outcrop sandstone with visually apparent laminations show the predicted trends. The Corey-Rathjens observations were used by Ehrlich to explain the relative permeability behavior of vuggy and fractured samples.

## Chierici model

Chierici proposed exponential expressions for fitting gas/oil relative permeabilities: ....................(11)

and ....................(12)

with gas saturation normalized as follows: . His expressions for oil/water relative permeabilities are ....................(13)

and ....................(14)

with water saturation normalized as follows: . The Chierici expressions were used in a recent discussion of three-phase relative permeabilities.

## Correlations of Honarpour et al. and Ibrahim

Other researchers have offered correlations for laboratory measurements of relative permeabilities. Honarpour et al. suggest correlations for two sets of rock samples with varying wettabilities:

• Sandstones and conglomerates
• Limestones and dolomites

Ibrahim reported a more extensive set of correlations. Such correlations are useful for understanding general trends and for preliminary estimates; however, they can be far from correct when applied to a specific formation.

## Hysteresis models

The effect of relative permeability hysteresis on reservoir performance can be significant for processes with variable directions of saturation change. For example, during coning of water toward an oil-producing well, the water saturation is increasing; however, if the production rate is decreased or set to zero, the water saturation can decrease. Hysteresis also can affect the performance of a waterflood if the relative production rate of a well in a pattern of producers is changed. To facilitate simulation of these processes, a number of models have been proposed for representing hysteresis effects on relative permeability and capillary pressure. The hysteresis models of Killough and Carlson are used in some commercial simulation software. Fayers et al. proposed another model for incorporating hysteresis effects in reservoir simulation. These models extrapolate and interpolate from measured drainage and imbibition curves to generate "reasonable" estimates of relative permeabilities. Although these models may be satisfactory for preliminary estimates, many more experimental data on hysteresis of relative permeability and capillary pressure are needed.

## Carman-Kozeny models

Alpak et al. used concepts from a model for capillary pressure to build a relative permeability model with the form of the Carman-Kozeny expression for permeability. (See Alpak et al. for references to other related models.) The Alpak et al. model relates relative permeabilities to the total surface area of the solid AT, the oil/water interfacial area Aow, the water/solid interfacial area Aws, and the oil/solid interfacial area Aos: ....................(15)

and ....................(16)

This relative permeability model includes tortuosity relationships, τw/τ and τo/τ , that do not arise in the capillary pressure model of Alpak et al. This interesting difference suggests that relative permeabilities cannot be estimated from capillary pressure information alone. Alpak et al. suggested relationships for the tortuosity and the area functions. They used the model to fit relative permeability data for unconsolidated and consolidated media. Future research will test the merit of this approach to modeling relative permeability.

## Network models

The advent of computers led to the development of models that represent porous structure as 2D and 3D networks of flow channels. Analysis of these network models leads to capillary pressure and relative permeability relationships. The capacity of these models to represent real behavior has increased with improved descriptions of the displacement mechanisms.

## Models for three-phase relative permeabilities

Many reservoir processes, such as waterflooding below the bubblepoint pressure of the oil in place, involve simultaneous flow of three phases. To model these processes, three-phase relative permeabilities are mandatory. Measurements of three-phase relative permeabilities are much rarer than those for two-phase relative permeabilities, and there is more uncertainty in the reported three-phase data, as noted by Baker in his 1988 review of three-phase correlations.

Current efforts in three-phase relative permeability studies are weighted toward identification of models for extrapolating two-phase relative permeability data to three-phase applications. Stone started this trend in 1970 with a model that is now known as the Stone I model. In this model for water-wet porous media, the three-phase water relative permeability krw,wog depends only on water saturation and is identical to krw,wo measured in water/oil displacements: ....................(17)

Similarly, the three-phase gas relative permeability krg,wog depends only on gas saturation and is identical to krg,go measured in gas/oil displacements at irreducible water saturation. ....................(18)

The equality of water and gas relative permeabilities in two- and three-phase flow, as described by Eqs. 17 and 18, is supported by much of the three-phase data in the literature for water-wet media. On the other hand, the three-phase oil relative permeability kro,wog depends nonlinearly on water and gas saturations (see note below): ....................(19)

with the oil, water, and gas saturations scaled, respectively, as follows: ....................(20) ....................(21)

and ....................(22)

According to Stone, the minimum oil saturation Som:

• Should be in the range of ¼ Swc to ½ Swc
• Should be less than or equal to the smaller of Sorw or Sorg, the residual oil saturations for waterflooding and gasflooding, respectively

Fayers and Matthews proposed the following expression for Som: ....................(23)

with α = 1 - Sg/(1 - Swc - Sorg) . Aleman, as reported by Baker, proposed an alternative expression: ....................(24)

To account for hysteresis effects in three-phase flow, Stone recommended use of the appropriate two-phase relative permeabilities. For example, in a water-wet system, if oil saturation is decreasing and gas and water saturations are increasing in the three-phase setting, then the following two-phase relative permeabilities should be used:

• kro,wo for decreasing oil saturation
• krw,wo for increasing water saturation
• krg,go for increasing gas saturation

In 1973, Stone proposed another model, which has become known as the Stone II model. In this model, the three-phase water and gas relative permeabilities are again equal to those measured in two-phase flow, as expressed by Eqs. 17 and 18, while the oil relative permeability is as follows:

The water/oil and the gas/oil relative permeabilities in Eq. 25 are functions of water saturation and gas saturation, respectively.

Baker compared four versions of the Stone models and various other models to available data for three-phase relative permeabilities. He concluded that models based on linear interpolation from the two-phase relative permeabilities perform as well as the other models. As an example of linear interpolation models, Baker suggested the following saturation weighting of two-phase relative permeabilities: ....................(26) ....................(27)

and ....................(28)

Blunt showed how to extend this model to describe relative permeability of oil at very low saturations, often termed the "layer drainage" regime.

Another model for three-phase relative permeabilities and capillary pressures was proposed recently by Fayers et al. The lasting value of this or any of the above-mentioned models, will be known only when more data become available for comparisons.

Note: The factors that contain kro,wo(Swc) in Eqs. 19 and 25 reflect normalization of relative permeabilities with respect to absolute permeability. For a comparison of these normalized equations to data, see Fayers and Matthews. Originally, Stone normalized with respect to oil permeability at irreducible water saturation. The Stone versions of Eqs. 19 and 25 can be recovered exactly by omitting the extra factors from Eqs. 19 and 25.

## Nomenclature

 A = area perpendicular to flow, L2 A, B = parameters in Chierici functions AT = total surface area of a porous sample, L2 Aow = area of oil/water interface in a porous sample, L2 Aos = area of oil/solid interface in a porous sample, L2 Aws = area of water/solid interface in a porous sample, L2 C = parameter in the Land function e = basis of natural logarithm (2.718...) fi = fraction of total sample volume occupied by layer i in a laminated sample G = pore geometric factor in Thomeer function j ( Sw) = Leverett j-function k = permeability, L2, md ki = permeability of layer i in a laminated sample, L, md kT = total permeability in a laminated sample, L2, md krg = relative permeability for gas krgi = relative permeability for gas in layer i of a laminated sample krg, max = maximum relative permeability for gas in modified Brooks-Corey functions krg,go = relative permeability for gas in gas/oil flow for three-phase models krg,gw = relative permeability for gas in gas/water flow for three-phase models krg,wog = relative permeability for gas in oil/water/gas flow for three-phase models krgo = relative permeability for oil in layer i of a laminated sample krgT = total relative permeability for gas in a laminated sample kro = relative permeability for oil kroi = relative permeability for oil in layer i of a laminated sample kro, max = maximum relative permeability for oil in modified Brooks-Corey functions kro,og = relative permeability for oil in oil/gas flow for three-phase models kro,wo = relative permeability for oil in oil/water flow for three-phase models kro,wog = relative permeability for oil in oil/water/gas flow for three-phase models kroT = total relative permeability for oil in a laminated sample krw = relative permeability for water krw,max = maximum relative permeability for water in modified Brooks-Corey functions krw,gw = relative permeability for water in gas/water flow for three-phase models krw,wo = relative permeability for water in oil/water flow for three-phase models krw,wog = relative permeability for water in oil/water/gas flow for three-phase models L = length L, M = parameters in Chierici functions ng = gas exponent for modified Brooks-Corey functions no = oil exponent for modified Brooks-Corey functions nw = water exponent for modified Brooks-Corey functions N = number of laminations Nc = capillary number Pc = capillary pressure, m/Lt2, psi Pcgo = capillary pressure between gas and oil phases, m/Lt2, psi Pcow = capillary pressure between oil and water phases, m/Lt2, psi Pcgw = capillary pressure between gas and water phases, m/Lt2, psi Pcgwt = threshold capillary pressure between gas and water phases, m/Lt2, psi Pci = capillary pressure at inside face of sample in centrifuge test, m/Lt 2, psi Pct = threshold capillary pressure, m/Lt 2, psi Pcs = shape parameter in Bentsen-Anli function , m/Lt 2, psi pg = pressure in the gas phase, m/Lt 2, psi po = pressure in the oil phase, m/Lt 2, psi pw = pressure in the water phase, m/Lt 2, psi ΔP = pressure drop, m/Lt 2, psi q = flow rate, L3/t qo = flow rate of oil, L3/t qw = flow rate of water, L3/t r = radius to a point in sample in centrifuge test, L ri = radius to inside face of sample in centrifuge test, L ro = radius to outside face of sample in centrifuge test, L R1, R2 = principal radii of curvature, L Scc = critical saturation of condensate Sg = saturation of gas Sgc = critical saturation of gas Sgi = initial saturation of gas SgN = normalized gas saturation Sgr = residual saturation of gas Sgr,max = maximum residual or trapped gas saturation SgS = scaled saturation of gas for Stone I model Sgt = trapped saturation of gas SHg = saturation of mercury Snwr = residual saturation of nonwetting phase So = saturation of oil Som = minimum saturation of oil for three-phase models Sor = residual saturation of oil Sorg = residual saturation of oil for a gas/oil displacement Sorw = residual saturation of oil for a water/oil displacement SoS = scaled saturation of oil for Stone I model Sw = saturation of water Swc = critical saturation of water Swi = irreducible or residual saturation of water Swn = normalized water saturation Swr = residual saturation of water or wetting phase SwS = scaled saturation of water for Stone I model v = velocity, L/t, cm/s x = position in the x direction, L α = parameter in Eqs. 23 and 24 β = parameter in Eq. 24 θ = contact angle, degrees λ = pore-size-distribution parameter in Corey functions μ = viscosity, m/Lt, cp μo = viscosity of oil, m/Lt, cp μw = viscosity of water, m/Lt, cp Δρ = density difference for fluids in centrifuge tests, m/L3 σgw = gas/water interfacial tension, m/t 2, dyne/cm σow = oil/water interfacial tension, m/t 2, dyne/cm τ = tortuosity of porous sample τo = tortuosity of oil phase in porous sample τw = tortuosity of water phase in porous sample Φ = porosity ω = spin rate for centrifuge tests, 1/t (ω=2π×RPM/60)