The high energy consumption induces a scarcity of hydrocarbon resources, forcing us to work in extreme conditions with increasingly difficult operating conditions (difficult reservoirs, depth, etc.)

This requires more advanced recovery techniques in which complex fluids find their place.

In fact, the recovery of hydrocarbons uses several complex fluids that are the hydrocarbons themselves and the different types of fluids (liquid water and water vapor, gases such as carbon dioxide and dinitrogen) used in different methods of recovery.

Complex fluids are used during the different phases of the production process. The stimulation of the porous medium is done by injecting through a well into the subterranean formation under high pressure so as to overcome the main stress of the rock and cause a crack in the porous medium. When fracture pressure is exceeded, very penetrating cracks or drainage ditches are initiated and propagated [1].

An oil field is not, as is often interpreted by the public, an underground lake where it is enough to draw, but it is a rock which contains the oil in its pores: these being sufficiently small (capillaries). Extracting this viscous liquid is not easy in practice. From then on the exploitation of an oil field is carried out according to several stages that are indicated by an order (possibly) chronological:

1) Primary recovery:

When the oil spontaneously arrives at the surface thanks to the presence of compressed gas, or to the differences of hydrostatic pressure: the quantities during this phase are generally small (5% - 10%) compared to the potentialities of the deposit.

2) Secondary recovery:

It consists of using methods that do not modify the physico-chemical properties of the oil, but that replaces it with another fluid, necessarily cheaper. We distinguish between the use of submerged pumps, the injection of water and the injection of the formation gas or other miscible gases (nitrogen, CO₂) [2] [3], the latter two processes that we intend to address within this context research work.

3) Tertiary (or improved) recovery:

Here the physico-chemical properties of the oil will be modified, for example by lowering the viscosity of the oil or facilitating its mobility in the deposit. These results are obtained by injecting water vapor, heat, CO₂, immiscible gases, or even more surfactants and polymers. Other methods, such as the injection of microorganisms capable of reducing the length of hydrocarbon chains and of producing in situ surfactants and CO₂, have been proposed. Assisted Recovery is sometimes referred to as the combination of the last two steps. Be that as it may, it is the study of the reservoir and the oil and the natural means on the spot that will determine the nature of the actions to be carried out [4].

Consider the two-phase flow for these two immiscible fluids, water and oil.

The independent variables are:

—for both fluids: densities ρ w and ρ o , interfacial tension σ , the viscosity μ w and μ o , the wettability coefficient in the form cos ( θ ) , filtration speeds v w (water) and v o (oil) for the porous medium: the pore radius, the shape factor, the characteristic length L.

The porous medium is confined in a cylinder mounted horizontally, the external forces related to the gravity are negligible. Unlike monophasic flows, two-phase flows, when we neglect the effects related to gravity, we must take into account the phenomena complex physic-chemicals at the interfaces of the two fluids and in contact with the fluids with the matrix.

It does not exist in general, in the flow, a sharp front through which we pass abruptly from one fluid to another, but a transition zone, in which the two fluids move simultaneously and the capillary forces play a very large role.

We must therefore take this phenomenon into account in a global way by introducing a macroscopic capillary pressure. We start from the equation system representing two-phase displacement in a porous medium. This system consists of the equations of motion (generalized Darcy’s law) [5] :

v → w = − k k w μ w g r a d → ( p w + ρ w g z ) (1)

v → 0 = − k k 0 μ 0 g r a d → ( p 0 + ρ 0 g z ) (2)

where k w and k 0 are called the relative permeabilities of water and oil that we assume that they are only a function of the saturation S of one of the two fluids, water for example.

Consider the conservation equations of mass:

d i v ( v → ) = 0 (3)

Φ ∂ s ∂ t + d i v ( v → w ) = 0 (4)

With v → = v → w + v → o (5)

where v → is the total filtration rate and Φ porosity. The pressures P w and p o in the two phases are connected via the capillary pressure p c : P w = p o − p c where p c is assumed only depending on saturation S. by eliminating P w , p o , v → w , v → o and d i v ( v → w ) between Equations ((1)-(4) and (5)), we obtain:

Φ ∂ s ∂ t + d ξ d s ( u ∂ s ∂ x + v ∂ s ∂ y + ∂ s ∂ z ) + k μ o [ ∂ ∂ x ( η d p c d s ∂ s ∂ x ) + ∂ ∂ y ( η d p c d s ∂ s ∂ y ) + ∂ ∂ z ( η d p c d s ∂ s ∂ z ) ] − k μ o g Δ ρ d η d s ∂ s ∂ z = 0 (6)

where ξ = ( 1 + k o k w μ w μ o ) − 1 , Δ p = p w − p o , and η = k o ξ are dimensionless functions of saturation S.

Figure 2 shows that this pressure drop decreases after 300 seconds and then increases over time according to the number of water volume injected without stabilizing, see Figure 2(a) and Figure 2(b). In addition, the volumes of water and oil recovered at the outlet of the porous medium vary according to the size of the sand grains used. This suggests that the porous medium retains oil depending on the nature of the porous medium either by adsorption or by trapping by the pores [6] [7].

In these two-phase flows, the transport of oil-lubricated oils is based on the natural phenomenon in which a less viscous phase migrates to the high-shear region near the cylinder wall used at the laboratory scale, where it lubricates the flow less well for sand with a diameter of less than 2 mm whose friction forces are important (more oil trapping in the rock), than sand with a diameter greater than 2 mm whose friction forces become less important (less than trapping of oil in the rock), see Table 1. Once pumping pressures are balanced by wall shear, see Figure 3 whose pressure stabilizes from 1650 seconds, lubricated transport requires pressures comparable to pumping water alone, regardless of oil viscosity.

The phases of oil and water can be organized differently during pumping. The porous medium is placed horizontally, these flows are stratified based on the density of water and oil [8].

The physico-chemical methods of improved recovery of oil allow increasing the production of oil by adding agents (polymers, water, gases ...) acting on different physical parameters.

Figure 4 shows that a limiting factor concerns the nature and properties of the porous medium. Our experiments have shown that we need to use porous media with sand greater than 4 mm in diameter to reduce oil entrapment and efficiently move oil into the porous medium.

The perspectives of this study concern several points which it would be interesting to go into further.

Firstly, it would be judicious, experimentally, to inject polymers in the porous medium which will allow us to increase the viscosity of the displacing fluid, which would lead, during the injection, to a reduction of the instabilities of front, modifying the wettability of the rock to hope to decrease the water-oil interfacial tension and to remobilize oil drops trapped by the capillary effects [9] [10] [11] [12].

The authors declare no conflicts of interest regarding the publication of this paper.

Mohamed, M.S., Ndyaye, S., Talla, K., Mbow, C. and Beye, A.C. (2020) Displacement of Oil in a Porous Medium on Diphasic Flow by Water Injection. Open Journal of Fluid Dynamics, 10, 1-7. https://doi.org/10.4236/ojfd.2020.101001