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# Relative permeability and capillary pressure

Reservoir engineers use relative permeability and capillary pressure relationships for estimating the amount of oil and gas in a reservoir and for predicting the capacity for flow of oil, water, and gas throughout the life of the reservoir. Relative permeabilities and capillary pressure are complex functions of the structure and chemistry of the fluids and solids in a producing reservoir.[1] As a result, they can vary from place to place in a reservoir. Most often, these relationships are obtained by measurements, but network models are emerging as viable routes for estimating capillary pressure and relative permeability functions.

## Permeability

Before defining relative permeability and capillary pressure, let us briefly review the definition of permeability. Permeability represents the capacity for flow through porous material. It is defined by Darcy’s law (without gravitational effects) as

....................(1)

Darcy’s law relates the flow rate q to the permeability k, cross-sectional area A, viscosity μ, pressure drop ΔP, and length L of the material. High permeability corresponds to increased capacity for flow. The dimensions of permeability are length squared, often expressed as darcies (1 darcy = 0.987×10–8 cm2), millidarcies, or micrometers squared. Some writers use "absolute permeability" or "intrinsic permeability" in place of permeability.

For multiple-phase flow, the following expressions define relative permeabilities, specifically written for oil and water flow (without gravitational effects) in the x direction:

....................(2)

and

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

where kro and krw are the relative permeabilities of oil and water, respectively. Relative permeabilities are dimensionless functions that usually range between 0 and 1. Eqs. 2 and 3 allow for differences in the pressure in the oil and water phases. The difference in pressure between the two phases is the capillary pressure:

....................(4)

### Capillary pressure and permeability relationship

Capillary pressure relationships are dimensional functions that range from large negative to large positive values. (Capillary pressure is often defined as the pressure of the less-dense phase minus the pressure of the more-dense phase.) Relative permeabilities and capillary pressures are usually viewed as functions of the saturation of phases in the porous sample—so, for oil/water flow in the absence of a gas phase, we have:

• kro(Sw)
• krw(Sw)
• Pc(Sw)

Saturation is the fraction of pore space that is occupied by a phase. In the present example of oil/water flow, Sw +So =1.

### Effective permeability

In some discussions, the products of permeability and relative permeability (e.g., kkro and kkrw in Eqs. 2 and 3) are termed the effective permeabilities. Effective permeability of oil at irreducible water saturation, or ko(Swi), is sometimes used to normalize relative permeabilities in place of absolute permeability. With this normalization, kro(Swi ) equals 1. It is possible for water relative permeability to exceed 1 when ko(Swi) is the normalizing factor. One must be very careful when using data to note whether absolute permeability or an effective permeability is used for normalizing.

## Trends and cautions in use of rock/fluid properties

Five important trends and cautions are worth emphasizing.

1. The need for accurate measurement of capillary pressure and relative permeability functions increases with the resolution of reservoir models. With low-resolution models, there is a need for algorithms to "upscale" permeabilities, relative permeabilities, and capillary pressures from the scale of measurement on a small sample of rock to the relatively huge size of blocks in reservoir models. The results of the averaging processes of upscaling are insensitive to the quality of measurements on small samples. The need for upscaling should diminish as increases in computer power permit higher-resolution models.
2. To obtain accurate measurements of capillary pressure and relative permeabilities, tests with representative samples at representative conditions are critical. Much of the available data in our industry do not pass this standard.
3. Capillary end effects and viscous fingering have corrupted a significant portion of relative permeability data. If capillary pressure and relative permeabilities are available, the extent of this corruption for a sample can be assessed and sometimes corrected.
4. We often interpret water/oil wettabilities from the shape of relative permeabilities. Such interpretations are particularly susceptible to error caused by the heterogeneity of the sample used for measurements. This susceptibility was conceded in the original literature on wettability interpretation, but it is not widely acknowledged.
5. The quality of estimates of capillary pressure and relative permeability with network models is increasing. These models offer the hope of providing estimates for a large set of rock samples for any particular reservoir while avoiding the costs of measuring capillary pressure and relative permeability.

## Nomenclature

 A = area perpendicular to flow, L2 k = permeability, L2, md kro = relative permeability for oil krw = relative permeability for water L = length Pc = capillary pressure, m/Lt2, psi Pcow = capillary pressure between oil and water phases, m/Lt2, 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 So = saturation of oil Sw = saturation of water Swi = irreducible or residual saturation of water x = position in the x direction, L μ = viscosity, m/Lt, cp μo = viscosity of oil, m/Lt, cp μw = viscosity of water, m/Lt, cp

## References

1. Content of this page relies heavily on Christiansen, R.L. 2001. Two-Phase Flow through Porous Media, Ch. 1, 4, and 5. Littleton, Colorado: KNQ Engineering.