Relative permeability
For two-phase flow in porous media given steady-state conditions, we can write where
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.
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 has been observed in heavy oil petroleum reservoirs when the gas phase flows as bubbles or patches (disconnected).
[1] 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.
Based on data from special core analysis laboratory (SCAL) experiments,[2] simplified models of relative permeability as a function of saturation (e.g. water saturation) can be constructed.
is the fraction of the pore volume that is filled with water, and similarly for the oil saturation
This gives the constraint The model functions or correlations for relative permeabilities in an oil-water system are therefore usually written as functions of only water saturation, and this makes it natural to select water saturation as the horizontal axis in graphical presentations.
be the residual (minimal) oil saturation after water flooding (imbibition).
), we get an endpoint parameter for both oil and water relative permeability.
oil permeability with irreducible water saturation present,
It occurs at irreducible water saturation, and it is the largest value of
If the permeability basis is oil with irreducible water present, then
The oil and water relative permeability models are then written as The functions
are called normalised relative permeabilities or shape functions for oil and water, respectively.
A number of busy core analysts, reservoir engineers and scientists often skip using tedious and time-consuming subscripts, and write e.g. Krow instead of
A variety of symbols are therefore to be expected, and accepted as long as they are explained or defined.
The effects that slip or no-slip boundary conditions in pore flow have on endpoint parameters, are discussed by Berg et alios.
[3][4] An often used approximation of relative permeability is the Corey correlation [5] [6] [7] which is a power law in saturation.
are called curve shape parameters or simply shape parameters, and they can be obtained from measured data either by analytical interpretation of measured data, or by optimization using a core flow numerical simulator to match the experiment (often called history matching).
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[8] [9] adds more degrees of freedom in order to accommodate the shape of relative permeability curves in SCAL experiments[2] and in 3D reservoir models that are adjusted to match historic production.
These adjustments frequently includes relative permeability curves and endpoints.
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.
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.
[11] Moghadasi et alios[10] evaluated 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 (three) of uncertain parameters, was clearly the best one for both oil and water relative permeability.
Sakhaei et alios[11] 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.
