Brooks-Corey RelPerm Models

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Brooks-Corey and related models

In 1954, Corey[1] 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[2] 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[3] 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[3] 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[4] to explain the relative permeability behavior of vuggy and fractured samples.

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  Fig.1 – Predictions of Eqs. 7 through 10 for flow parallel and perpendicular to laminations.
  
  
  
  
• 
  
  
  
  Fig.2– Observed relative permeabilities parallel and perpendicular to stratification for a sample of Berea outcrop (from Corey and Rathjens[3]).
  
  
  
  

Chierici model

Chierici[5] 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.[6]

Correlations of Honarpour et al. and Ibrahim

Other researchers have offered correlations for laboratory measurements of relative permeabilities. Honarpour et al.[7] suggest correlations for two sets of rock samples with varying wettabilities:

• Sandstones and conglomerates
• Limestones and dolomites
  

Ibrahim[8] 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[9] and Carlson[10] are used in some commercial simulation software. Fayers et al.[6] 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.

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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.[12]

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[13] 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[14] 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[15] proposed the following expression for Som:

....................(23)

with α = 1 - Sg/(1 - Swc - Sorg) . Aleman, as reported by Baker,[13] 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[16] 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:

........(25)

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

Baker[13] 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[17] 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.[6] 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.[15] 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.

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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)

References

• ↑ Corey, A.T. 1954. The interrelation between gas and oil relative permeabilities. Producers Monthly 19 (November): 38–41.
• ↑ Brooks, R.H. and Corey, A.T. 1964. Hydraulic Properties of Porous Media. Hydrology Papers, No. 3, Colorado State U., Fort Collins, Colorado.
• ↑ Jump up to:3.0 3.1 3.2 Corey, A.T. and Rathjens, C.H. 1956. Effect of Stratification on Relative Permeability. J Pet Technol 8 (12): 69-71. http://dx.doi.org/10.2118/744-G.
• ↑ Ehrlich, R. 1971. Relative Permeability Characteristics of Vugular Cores - Their Measurement and Significance. Presented at the Fall Meeting of the Society of Petroleum Engineers of AIME, New Orleans, Louisiana, 3-6 October 1971. SPE-3553-MS. http://dx.doi.org/10.2118/3553-MS
• ↑ Chierici, G.L. 1984. Novel Relations for Drainage and Imbibition Relative Permeabilities. SPE J. 24 (3): 275-276. http://dx.doi.org/10.2118/10165-PA
• ↑ Jump up to:6.0 6.1 6.2 Fayers, F.J., Foakes, A.P., Lin, C.Y. et al. 2000. An Improved Three Phase Flow Model Incorporating Compositional Variance. Presented at the SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 3-5 April 2000. SPE-59313-MS. http://dx.doi.org/10.2118/59313-MS
• ↑ Honarpour, M., Koederitz, L., and Harvey, A.H. 1986. Relative Permeability of Petroleum Reservoirs, 16-41. Boca Raton, Florida: CRC Press.
• ↑ Ibrahim, M.N.M. 1999. Two-Phase Relative Permeability Prediction Using a Linear Regression Model. PhD thesis, University of Missouri-Rolla.
• ↑ Killough, J.E. 1976. Reservoir Simulation with History-Dependent Saturation Functions. SPE J. 16 (1): 37–48. SPE-5106-PA. http://dx.doi.org/10.2118/5106-PA
• ↑ Carlson, F.M. 1981. Simulation of Relative Permeability Hysteresis to the Nonwetting Phase. Presented at the SPE Annual Fall Technical Conference and Exhibition, San Antonio, Texas, USA, 4–7 October. SPE-10157-MS. http://dx.doi.org/10.2118/10157-MS
• ↑ Jump up to:11.0 11.1 11.2 11.3 11.4 Alpak, F.O., Lake, L.W., and Embid, S.M. 1999. Validation of a Modified Carman-Kozeny Equation To Model Two-Phase Relative Permeabilities. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October 1999. SPE-56479-MS. http://dx.doi.org/10.2118/56479-MS
• ↑ Fenwick, D.H. and Blunt, M.J. 1998. Network Modeling of Three-Phase Flow in Porous Media. SPE J. 3 (1): 86-96. SPE-38881-PA. http://dx.doi.org/10.2118/38881-PA.
• ↑ Jump up to:13.0 13.1 13.2 Baker, L.E. 1988. Three-Phase Relative Permeability Correlations. Presented at the SPE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma, 16-21 April 1988. SPE-17369-MS. http://dx.doi.org/10.2118/17369-MS
• ↑ Stone, H.L. 1970. Probability Model for Estimating Three-Phase Relative Permeability. J Pet Technol 22 (2): 214–218. SPE-2116-PA. http://dx.doi.org/10.2118/2116-PA
• ↑ Jump up to:15.0 15.1 Fayers, F.J. and Matthews, J.D. 1984. Evaluation of Normalized Stone's Methods for Estimating Three-Phase Relative Permeabilities. Society of Petroleum Engineers Journal 24 (2): 224-232. SPE-11277-PA. http://dx.doi.org/10.2118/11277-PA
• ↑ Stone, H.L. 1973. Estimation of Three-Phase Relative Permeability and Residual Oil Data. J Can Pet Technol 12 (4): 53–61. http://dx.doi.org/10.2118/73-04-06
• ↑ Blunt, M.J. 1999. An Empirical Model for Three-Phase Relative Permeability. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October 1999. SPE-56474-MS. http://dx.doi.org/10.2118/56474-MS