Relative permeability
Encyclopedia
In multiphase flow
in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be viewed as an adaptation of Darcy's law
to multiphase flow.
For two-phase flow in porous media given steady-state conditions, we can write
where
is the flux, 
is the pressure drop, 
is the viscosity. The subscript 
indicates that the parameters are for phase 
.
is here the phase permeability (i.e., the effective permeability of phase 
), as observed through the equation above.
Relative permeability,
, for phase 
is then defined from 
as
where
is the permeability
of the porous medium in single-phase flow, i.e., the absolute permeability
. Relative permeability must be between zero and one.
In applications, relative permeability is often represented as a function of water saturation
, however due to capillary hysteresis
, one often resorts to one function or curve measured under drainage
and one measured under imbibition
.
Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).
Each of the phase saturation must be larger than the irreducible saturation, and each phase is assumed continuous within the porous medium.
in the water saturation
. If 
(also denoted 
, or 
, or 
) is the irreducible (minimal) water saturation, and 
is the residual (minimal) oil saturation after water flooding, we can define a normalized (or scaled) water saturation value
The Corey correlations of the relative permeability of oil and water are then
and
when the permeability basis is oil with irreducible water present.
We note the desired properties
The empirical parameters
and 
can be obtained from measured data either by optimizing to analytical interpretation of measured data, or by optimizing using a core flow numerical simulator to match the experiment(often called history matching). 
= 
is sometimes appropriate.The physical property 
is called the end point of the water relative permeability, and it is obtained either before or together with the optimizing of 
and 
.
In case of gas-water system or gas-oil system there are Corey correlations similar to the oil-water relative permeabilities correlations shown above.
).
The LET-correlation adds more degrees of freedom in order to accommodate the shape of measured relative permeability curves in SCAL experiments.
The LET-type approximation is described by 3 parameters L, E, T. The correlation for water and oil relative permeability with water injection is thus
and
written using the same
normalization as for Corey.
Only
, 
and 
have direct physical meaning, while the parameters L, E and T are empirical. The parameter L describes the lower part of the curve, and by similarity and experience the L-values are comparable to the appropriate Corey parameter. The parameter T describes the upper part (or the top part) of the curve in a similar way that the L-parameter describes the lower part of the curve. The parameter E describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the L- and T-parameters. Increasing the value of the E-parameter pushes the slope towards the high end of the curve. Decreasing the value of the E-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates that the parameter, L ≥ 0.3, E > 0.3 and T ≥ 0.3.
In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.
Multiphase flow
In fluid mechanics, multiphase flow is a generalisation of the modelling used in two-phase flow to cases where the two phases are not chemically related or where more than two phases are present In fluid mechanics, multiphase flow is a generalisation of the modelling used in two-phase flow to...
in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability. It can be viewed as an adaptation of Darcy's law
Darcy's law
Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...
to multiphase flow.
For two-phase flow in porous media given steady-state conditions, we can write
where
is the flux, is the pressure drop, is the viscosity. The subscript indicates that the parameters are for phase . is here the phase permeability (i.e., the effective permeability of phase ), as observed through the equation above.Relative permeability,
, for phase is then defined from aswhere
is the permeabilityPermeability (fluid)
Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it.- Units :...
of the porous medium in single-phase flow, i.e., the absolute permeability
Permeability (fluid)
Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it.- Units :...
. Relative permeability must be between zero and one.
In applications, relative permeability is often represented as a function of water saturation
Water content
Water content or moisture content is the quantity of water contained in a material, such as soil , rock, ceramics, fruit, or wood. Water content is used in a wide range of scientific and technical areas, and is expressed as a ratio, which can range from 0 to the value of the materials' porosity at...
, however due to capillary hysteresis
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
, one often resorts to one function or curve measured under drainage
Drainage
Drainage is the natural or artificial removal of surface and sub-surface water from an area. Many agricultural soils need drainage to improve production or to manage water supplies.-Early history:...
and one measured under imbibition
Imbibition
'Imbibition' is defined as the displacement of one fluid by another immiscible fluid. This process is controlled and affected by a variety of factors...
.
Under this approach, the flow of each phase is inhibited by the presence of the other phases. Thus the sum of relative permeabilities over all phases is less than 1. However, apparent relative permeabilities larger than 1 have been obtained since the Darcean approach disregards the viscous coupling effects derived from momentum transfer between the phases (see assumptions below). This coupling could enhance the flow instead of inhibit it. This has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).
Assumptions
The above form for Darcy's law is sometimes also called Darcy's extended law, formulated for horizontal, one-dimensional, immiscible multiphase flow in homogeneous and isotropic porous media. The interactions between the fluids are neglected, so this model assumes that the solid porous media and the other fluids form a new porous matrix through which a phase can flow, implying that the fluid-fluid interfaces remain static in steady-state flow, which is not true, but this approximation has proven useful anyway.Each of the phase saturation must be larger than the irreducible saturation, and each phase is assumed continuous within the porous medium.
Approximations
Based on experimental data, simplified models of relative permeability as a function of water saturation can be constructed.Corey-type
An often used approximation of relative permeability is the Corey correlation which is power lawPower law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
in the water saturation
. If (also denoted , or , or ) is the irreducible (minimal) water saturation, and is the residual (minimal) oil saturation after water flooding, we can define a normalized (or scaled) water saturation valueThe Corey correlations of the relative permeability of oil and water are then
andwhen the permeability basis is oil with irreducible water present.
We note the desired properties
The empirical parameters
and can be obtained from measured data either by optimizing to analytical interpretation of measured data, or by optimizing using a core flow numerical simulator to match the experiment(often called history matching). = is sometimes appropriate.The physical property is called the end point of the water relative permeability, and it is obtained either before or together with the optimizing of and .In case of gas-water system or gas-oil system there are Corey correlations similar to the oil-water relative permeabilities correlations shown above.
LET-type
The Corey approximation only has one degree of freedom for the oil relative permeability and two degrees of freedom for the water permeability (in).The LET-correlation adds more degrees of freedom in order to accommodate the shape of measured relative permeability curves in SCAL experiments.
The LET-type approximation is described by 3 parameters L, E, T. The correlation for water and oil relative permeability with water injection is thus
and
written using the same
normalization as for Corey.Only
, and have direct physical meaning, while the parameters L, E and T are empirical. The parameter L describes the lower part of the curve, and by similarity and experience the L-values are comparable to the appropriate Corey parameter. The parameter T describes the upper part (or the top part) of the curve in a similar way that the L-parameter describes the lower part of the curve. The parameter E describes the position of the slope (or the elevation) of the curve. A value of one is a neutral value, and the position of the slope is governed by the L- and T-parameters. Increasing the value of the E-parameter pushes the slope towards the high end of the curve. Decreasing the value of the E-parameter pushes the slope towards the lower end of the curve. Experience using the LET correlation indicates that the parameter, L ≥ 0.3, E > 0.3 and T ≥ 0.3.In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.
See also
- Permeability (earth sciences)
- Capillary pressure
- ImbibitionImbibition'Imbibition' is defined as the displacement of one fluid by another immiscible fluid. This process is controlled and affected by a variety of factors...
- DrainageDrainageDrainage is the natural or artificial removal of surface and sub-surface water from an area. Many agricultural soils need drainage to improve production or to manage water supplies.-Early history:...
- Buckley–Leverett equationBuckley–Leverett equationIn fluid dynamics, the Buckley–Leverett equation is a transport equation used to model two-phase flow in porous media. The Buckley–Leverett equation or the Buckley–Leverett displacement can be interpreted as a way of incorporating the microscopic effects to due capillary pressure in two-phase flow...