Fundamentals of Fluid Flow in Porous Media
Chatenever and Calhoun (1952) found that in a system at steady state with two immiscible fluids that flow simultaneously in a porous medium will establish their own pathways. In their work the simultaneous injection of oil and water into a porous medium resulted in the creation of pore pathways fully occupied by one fluid and fluid-fluid interfaces such that the flow channels followed by one fluid were not influenced by the flow behaviour of the other fluid. This was used as the basis for extending Darcy’s Law to apply for each of the immiscible fluids:
ke1 = effective permeability of fluid phase 1
ke2 = effective permeability of fluid phase 2
V1 = Darcy velocity of fluid phase 1, as defined by Q1 / A (linear flow conditions)
V2 = Darcy velocity of fluid phase 2, as defined by Q2 / A (linear flow conditions)
μ1 = viscosity of fluid phase 1
μ2 = viscosity of fluid phase 2
ρ1 = density of fluid phase 1
ρ2 = density of fluid phase 2
The effective permeability is a function of the saturation and the saturation history of the sample being examined. It is customary to express effective permeability as a fraction of some base permeability; such as the single phase (absolute) permeability, k, of the medium. These fractional permeabilities are also known as relative permeabilities, kr1, and kr2, and are defined by:
Using the relative permeabilities, the individual phase velocities can then be written as follows:
The pressures values (Pi, where i = 1, 2) of each of the phases at any macroscopic point in the porous medium are assumed to be related to each other via the capillary pressure, Pc. Let P1 denote the pressure of the wetting phase and P2 denote the pressure of the non-wetting phase. The gradients ∇P1 and ∇P2, are related to each other by the capillary pressure gradient ∇Pc.
In the case of uniform saturation along the length of a core sample, ∇Pc = 0.
Some behaviours to note regarding relative permeability include:
- For a given water saturation ( Sw ) value under water-wet conditions, ( kw / ko ) is higher for a higher permeability sample in comparison to ( kw / ko ) in a low permeability sample.
- For a given Sw value under oil-wet conditions, the ( kw / ko ) ratio is higher for low permeability samples in comparison to high permeability samples.
- The relative conductivity of large pores in a low permeability system is larger than that in a high permeability system.