Empirical Capillary Pressure Relative Permeability Correlation

By

Cons

Po

Email:

James H. Schneider ulting Reservoir Engineer

P.O. Box 316 teet, Texas 78065-0316

jhschneider@jhschneider.com

Copyright 2003

Empirical Capillary Pressure Relative Permeability Correlation

When performing reservoir simulations capillary pressure data is rarely available and capillary pressure is one of the important variables in obtaining a history match especially when primary water production is present in the model study area. Other uses of capillary pressures is the data gives us the ability to predict the distance to the gas/water or the oil/water contact and to develop an relative permeability correlations. Hawkins, Luffel and Harris came up with an interesting capillary pressure correlation which appears to be reasonably accurate which is based on Water Saturation, Porosity and Permeability. The following equations are the ones that are necessary to compute the mercury capillary pressure: Pd = 937.8 / (ka

0.3406φ) Fg = [ ln(5.21ka

0.1254/φ)]2 / 2.303 Log PC = - Fg / ln(1-SW) + log Pd

Where: ka = Air permeability, md PC = Mercury capillary pressure, psi Pd = Mercury displacement pressure, psi SW = Water Saturation, fraction

φ = Porosity, % Example Calculations: Where : ka = 105 md φ = 26.7 % Pd = 937.8 / [( 1050.3406 )27.6 ] = 7.1975 Fg = [ ln(5.21(1050.1254)/27.6)]2 / 2.303 = 0.4792 Log PC = - 0.4792/ ln(1-SW) + log(7.195) = - 0.4792/ ln(1-SW) + 0.8572 PC = 10 - 0.4792/ ln(1-SW) + 0.8572

Or

SW = 1 – e[--Fg/log( Pc/Pd )] = 1 – e[--0.47925log( Pc/7.1975)]

Pc, psi Sw 10 0.965120 0.660330 0.538440 0.474550 0.434160 0.405770 0.384380 0.3676

100 0.3425140 0.3105160 0.2994180 0.2902200 0.2824

Converting Mercury/Air Capillary Data

To Oil Water System

Example: Mercury/air(Hg/a): σ = 480 dynes/cm θ = 140° (cos 140° = -.766) σ cos θ = 370 Oil/water(o/w): σ = 25 dynes/cm (range is typically 15 – 35) θ = 30° (cos 30° = 0.866) (range is typically 0 – 160°)

depending on wettability σ cos θ = 21.65 Po/w = PHg/a[(σo/w cos θo/w) / (σHg/a cos θHg/a)] Po/w = PHg/a [(21.65) / (370)] = 0.059PHg/a

The following table and graph shows the resultant values using the previous equation:

Sw PHg/a psi Po/w psi 0.9651 10 0.590.6603 20 1.180.5384 30 1.770.4745 40 2.360.4341 50 2.950.4057 60 3.540.3843 70 4.130.3676 80 4.720.3425 100 5.900.3105 140 8.260.2994 160 9.440.2902 180 10.620.2824 200 11.80

Height Above Free-Water To calculate the height above the free water level the following equation is used: h = (Po/w or Pg/w)/[.433(ρw - ρh)]

where: h = Height above free water level, ft. Po/w = oil/water capillary pressure, psi Pg/w = gas/water capillary pressure, psi ρw = Formation water density, g/ml ρh = Formation hydrocarbon density, g/ml Example: In this example we will use a gas/water system.

Mercury/air(Hg/a): σ = 480 dynes/cm θ = 140° (cos 140° = -.766) σ cos θ = 370

Gas/water(o/w): σ = 50 dynes/cm (range is typically 35 – 70) θ = 0° (cos 0° = 1.000) (usually assumed 0°)

depending on wettability σ cos θ = 50 Pg/w = PHg/a[(σo/w cos θo/w) / (σHg/a cos θHg/a)] Pg/w = PHg/a [(50) / (370)] = 0.135PHg/a

The following table shows the resultant values using the previous equation:

Sw PHg/a psi Pg/w psi 0.9651 10 1.350.6603 20 2.700.5384 30 4.050.4745 40 5.400.4341 50 6.750.4057 60 8.100.3843 70 9.450.3676 80 10.800.3425 100 13.500.3105 140 18.900.2994 160 21.600.2902 180 24.300.2824 200 27.00

Using the following values

ρw = 1.10 g/ml ρh = 0.24 g/ml Then h = (Pg/w)/[.433(1.10 – 0.24)] = (Pg/w)/(0.372) = 2.688Pg/w

The following table and graph shows the resultant values using the previous equation:

Sw PHg/a psi Pg/w psi h, ft 0.9651 10 1.35 3.600.6603 20 2.70 7.200.5384 30 4.05 10.810.4745 40 5.40 14.410.4341 50 6.75 18.010.4057 60 8.10 21.610.3843 70 9.45 25.210.3676 80 10.80 28.810.3425 100 13.50 36.020.3105 140 18.90 50.430.2994 160 21.60 57.630.2902 180 24.30 64.830.2824 200 27.00 72.04

Height Above Free-WaterPorosity = 26.70%, Permeability =105md

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Relative Permeability From Capillary Pressure Curves The computational technique most widely used in calculating relative permeability from capillary pressure curves is based on the Brooks and Corey Equations. These equations are: krw = [Sw*]( (2 + 3λ) / λ ) and krnw = [1-Sw*]2 [1-Sw* ( (2 + λ) / λ )] where Sw* = (Sw – Swr) / (1 – Swr) krw = wetting phase relative permeability krnw = non-wetting phase relative permeability λ = lithology factor obtained from capillary pressure data Brooks and Corey observed that a log-log plot of Sw* against PC results in a straight line with a slope -λ which is characteristic of the pore volume structure. Pe , the pore entry pressure is obtained from the intercept at Sw* = 1. The equation is defined as the following: Sw* = (Pe/PC)λ

Or Sw* = (PC/Pe)-λ

The following example shows this correlation:

Swir = 0.195

Sw PHg/a psi Sw* 0.9651 10 0.95660.6603 20 0.57800.5384 30 0.42660.4745 40 0.34720.4341 50 0.29700.4057 60 0.26170.3843 70 0.23520.3676 80 0.21440.3425 100 0.18320.3105 140 0.14350.2994 160 0.12970.2902 180 0.11830.2824 200 0.1086

Determine Lambda

y = 9.3284x-1.3887

R2 = 0.9999

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Sw*, (Sw-Swir)/(1-Swir)

Pc. p

si

In this example Swir was varied until the best straight line was generated, this seems to be the best estimate of the Swir. The slope of the line was determined to be –1.3887 and λ = 1.3887. For the residual oil saturation ( Sorw ) a good estimate is obtained at the point were PC = 1 psi. In this example the Sorw was estimated to be 0.25. Gas-Oil-Water Relative Permeability System The oil-water relationship can be defined with the following equations based on the Brooks and Corey Equations: krw = [Sw*]( (2 + 3λ) / λ ) For Sw<Swr krw =0.0

and kro = [1-Swo*]2 [1-Swo* ( (2 + λ) / λ )] For Sw<Swr (kro =1) and Sw>1-Sor (kro =0) where Sw* = (Sw – Swr) / (1 – Swr) Swo* = (Sw – Swr) / (1 – Swr - Sor) Sw = water saturation Swr = irreducible water saturation Sor = residual oil saturation to water krw = water phase relative permeability kro = oil phase relative permeability λ = lithology factor obtained from capillary pressure data Example:

Swir = 0.195Sor = 0.25λ = 1.3887

Sw kro krw 0.000 0.100 0.195 1.000000 0.000000 0.300 0.646108 0.000118 0.400 0.362695 0.002303 0.500 0.155823 0.013442 0.600 0.039186 0.047349 0.700 0.001670 0.126136 0.750 0.000000 0.191821 0.900 0.554900 1.000 1.000000

Oil-WaterRelative Permeability

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The gas-oil relationship can be defined with the following equations:

kro = [Swo*]( (2 + 3λ) / λ ) and krg = [1-Swg*]2 [1-Swg* ( (2 + λ) / λ )] where

Swo* = (SL – Sor - Swr) / (1 – Sor - Swr) Swg* = (SL – Swr) / (1 – Swr – Sgc) SL = Swr+So Swr = irreducible water saturation

So = oil saturation Sor = residual oil saturation to water

kro = oil phase relative permeability krg = gas phase relative permeability λ = lithology factor obtained from capillary pressure data Example:

Swir = 0.195Sor = 0.25λ = 1.3887Sgc = 0.02

So SL krg kro 0.000 0.195 1.000000 0.250 0.445 0.436013 0.0000000.300 0.495 0.345214 0.0000230.400 0.595 0.194121 0.0030000.500 0.695 0.087966 0.0289820.600 0.795 0.026714 0.1291110.700 0.895 0.002860 0.3940800.720 0.915 0.001304 0.4780130.805 1.000 0.000000 1.000000

Gas-OilRelative Permeability

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Gas-Water Relative Permeability System The oil-water relationship can be defined with the following equations based on the Brooks and Corey Equations: krw = [Sw*]( (2 + 3λ) / λ ) For Sw<Swr krw =0.0 and krg = [1-Swg*]2 [1-Swg* ( (2 + λ) / λ )] For Sw<Swr (krg =1) and Sw>1-Sgr (kro =0) where Sw* = (Sw – Swr) / (1 – Swr) Swg* = (Sw – Swr) / (1 – Swr – Sgr) Sw = water saturation Swr = irreducible water saturation Sor = residual gas saturation to water krw = water phase relative permeability krg = gas phase relative permeability λ = lithology factor obtained from capillary pressure data Example:

Swir = 0.195Sgr = 0.24λ = 1.3887

Sw krg krw 0.000 0.100 0.195 1.000000 0.000000 0.300 0.651941 0.000118 0.400 0.371777 0.002303 0.500 0.164720 0.013442 0.600 0.044606 0.047349 0.700 0.002702 0.126136 0.760 0.000000 0.207650 0.900 0.000000 0.554900 1.000 0.000000 1.000000

Gas-WaterRelative Permeability

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Sgr can be estimated from a set of empirical relationships. Narr-Henderson proposed the following Sgr correlation for consolidated sands: Sgr = 0.5Sgi Where Sgr = residual gas saturation Sgi = initial gas saturation Agarwal obtained the following correlations: Consolidated Sands Sgr = A1Sgi + A2Sgi2

Where Sgr = residual gas saturation, %

Sgi = initial gas saturation, % A1 = 0.80841168 A2 =-0.63869116x10-2

Unconsolidated Sand Sgr = A1Sgi + A2Sgiφ + A3φ + A4

Where

Sgr = residual gas saturation, % Sgi = initial gas saturation, % φ = Porosity, % A1 = -0.51255987 A2 =-0.26097212x10-1 A3 = -0.26769575 A4 = 0.14796539x102

Limestone Sgr = A1φ + A2logk + A3Sgi + A4

Where Sgr = residual gas saturation, %

Sgi = initial gas saturation, % φ = Porosity, % k = permeability, md A1 = -0.53482234 A2 = 0.33555165x10 A3 = 0.15458573 A4 = 0.14403977x102

Conclusions When measured data is limited empirical correlations can provide a reasonable estimate.

References Joseph M. Hawkins, Donald L.Luffel and Thomas G. Harris, “Capillary pressure model predicts distance to gas/water, oil/water contact”, Oil & Gas Journal, Jan 18 1993, pp39-43 Dan Smith, “How to predict down-dip water level”, World Oil, May 1993, pp 85-88 Naar, J. and Henderson, J.H., ”An Imbibition Model Its Application to Flow Behavior and The Prediction of Oil Recovery”, SPE J., June 1961 p61. Agarwal, R.G., “Unsteady State Performance of Water Drive Gas Reservoirs”, Ph.D. Dissertation, Texas A&M University, May 1967 Brooks, R.H. and Corey, A.T., “Hydraulic Properties of Porous Media.”, Hydrology Papers, No. 3, Colorado State University, Ft. Collins, Colo., 1964. Brooks, R.H. and Corey, A.T., “Properties of porous media affecting fluid flow”, J. Irrig. Drain. Div…6, p61, 1966