Relative permeability
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, 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, owing to capillary hysteresis one often resorts to a function or curve measured under drainage and another 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. 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.
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 saturation can be constructed. If is the irreducible water saturation, and is the residual oil saturation after water flooding, we can define a normalized water saturation valueand a normalized oil saturation value
We cannot normalize both oil relative permeability and water relative permeability, but we can normalize oil relative permeability by selecting oil phase permeability, with present, as absolute permeability. This gives the desired properties
Corey-type
An often used approximation of relative permeability is the Corey correlationwhich is a power law in saturation. The Corey correlations of the relative permeability for oil and water are then
when the permeability basis is oil with irreducible water present.
The empirical parameters and are called curve shape parameters or simply shape parameters, and they 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. = 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-correlation or Corey model has only one degree of freedom for the shape of each relative permeability curve, the shape parameter N.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 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 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 the following reasonable ranges for the parameters L, E, and T: L ≥ 0.1, E > 0 and T ≥ 0.1.
In case of gas-water system or gas-oil system there are LET correlations similar to the oil-water relative permeabilities correlations shown above.
Evaluations
After Morris Muskat et alios established the concept of relative permeability in late 1930'ies, the number of correlations, i.e. models, for relative permeability has steadily increased. This creates a need for evaluation of the most common correlations at the current time. Two of the latest and most thorough evaluations are done by Moghadasi et alios and by Sakhaei et alios. Moghadasi et aliosevaluated Corey, Chierici and LET correlations for oil/water relative permeability using a sophisticated method that takes into account the number of uncertain model parameters. They found that LET, with the largest number of uncertain parameters, was clearly the best one for both oil and water relative permeability. Sakhaei et alios
evaluated 10 common and widely used relative permeability correlations for gas/oil and gas/condensate systems, and found that LET showed best agreement with experimental values for both gas and oil/condensate relative permeability.
Relative permeability versus [TEM-function]
Relative permeability is just one of the factors that affect fluid flow dynamics, and therefore can’t fully capture dynamic flow behavior of porous media. A criterion/metric has been established to characterize dynamic characteristics of rocks, known as True Effective Mobility or TEM-function. TEM-function is a function of Relative permeability, Porosity, permeability and fluid Viscosity, and can be determined for each fluid phase separately. TEM-function has been derived from Darcy's law for multiphase flow.in which k is the permeability, kr is the Relative permeability, φ is the Porosity, and μ is the fluid Viscosity.
Rocks with better fluid dynamics have higher TEM versus saturation curves. Rocks with lower TEM versus saturation curves resemble low quality systems.
While TEM-function controls the dynamic behavior of a system, the Relative permeability alone has conventionally been used to classify different fluid flow systems. Despite Relative permeability is itself a function of several parameters including permeability, Porosity and Viscosity, the dynamic behavior of systems may not necessarily be fully captured by this single source of information and, if used, it can even result in misleading interpretations.
TEM-function in analyzing Relative permeability data is analogous with Leverett J-function in analyzing Capillary pressure data.