We develop a model for the determination of gas and water relative permeabilities from pore size distribution functions. The model originates from percolation ideas, and considers two situations:

  • gas and liquid without surfactant, and

  • gas and liquid with a surfactant.

In the first case, we show that the relative permeabilities are coherent with those obtained in past studies. This proves the consistency of our model.

In the second case, the surfactant leads to the dispersion of gas in the liquid phase (foam). We determine the ensuing partitioning of mobile and immobile gas. Then we show that foam relative permeability increases with the permeability. We argue that the decline of foam mobility with increasing permeability is probably due to dominant increase of foam effective viscosity. However, at the present stage, the model could not confirm the dependence of foam mobility on a (-1/2) power of the permeability.


The removal of near wellbore formation damage by acid wash is an important issue in oil recovery.1 Efficient damage treatment mostly needs an agent to divert the acid from undamaged (high permeability) to damaged (low permeability) zones.1,2 Foams are such diversion agents.

The permeability is a decisive parameter in the control of foam generation and stability in heterogeneous porous media. Yet, its role is poorly known. Kibodeaux et al.3,4 performed foam flow experiments at various velocities, using nitrogen (N2) foam of fairly high quality, and Berea sandstone cores of permeability 90 and 800 mD. They found that foam mobility decreases when the permeability increases. This shows that foam is stronger in high permeability rocks than in low permeability ones. Dixit et al.5 observed the same behavior during the flow of carbon dioxide (CO2) foam in Berea sandstone and Baker dolomite cores. In the former studies foam mobility decreased by orders of magnitude allowing us to expect sufficient diversion.

However, this prediction is not entirely consistent with other studies. Parlar et al.6 investigated the flow of N2 foam in Berea sandstone cores with a permeability range between 50 and 1200 mD. They found that foam mobility ?f varies with the permeability k according to

Equation (1)

in the permeability interval 400 mD to 1200 mD. Below 400 mD, foam mobility was less sensitive to permeability variations. Zerhboub et al.,7 in turn, performed core flow tests of N2 foams in sand packs covering a wide permeability range (200 mD to 40 D). They showed that the dependence of foam mobility on permeability is in fair agreement with equation (1). Note that, since sand has a wide grain size distribution, the experiments might have been influenced by the migration of fine particles.

The decrease of foam mobility with increasing permeability is consistent with the idea of critical capillary pressure introduced by Khatib et al.8 However, it can be shown that the decrease of foam mobility given by equation (1) is insufficient to produce diversion. Moreover, the physical significance of the exponent 1/2 in equation (1) is unclear.

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