1337 Words6 Pages

3 Analytical Solution

We try to simplify the above model and solve it analytically. Since the micro-bubble fluid consist of liquid and bubbles, so it can be assumed as semi-incompressible fluid. Assume that the liquid phase (water) is incompressible in wide ranges of the pressure. i.e. ρ_w=constant . Expanding equation of 19 and rewrite it, we have:

{█(ϕ ∂(S_w )/∂t+∇.((u_w ) ⃗ )=0 @ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ )+1/ρ_m (Dρ_m)/Dt=0 @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+ϕS_m ∂n/∂t =ϕS_m [K_g (n_∞-n)-K_d n] )┤ 20

where the (Dρ_m)/Dt (material derivative of the density) is defined as below:

(Dρ_m)/Dt=ϕS_m (∂ρ_m)/∂t+(u_m*…show more content…*

Equation 23 indicated that the semi-incompressible micro-bubble fluid arise when its pressure remain constant or pressure variation be small.

{█(ϕ ∂(S_i )/∂t+∇.((u_i ) ⃗ )=0 @(u_i ) ⃗= -λ_i ∇P_i @ϕS_m ∂n/∂t+(u_m ) ⃗.∇n=ϕS_m [K_g (n_∞-n)-K_d n] )┤

24

Now, the fractional flow analysis is introduced. The water fractional flow function fw, defined as ratio of water velocity to total velocity. So the equation 17 become:

{█(ϕ (∂S_w)/∂t+(u ⃗ ).∇f_w+f_w ∇.u ⃗=0 @ @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+Dn/Dt ==ϕS_m [K_g (n_∞-n)-K_d n] )┤*…show more content…*

The micro-bubble generation and destruction coefficients i.e. K_g and K_d were assume constant; this assumption is not so far from the reality [20]. Mechanism of the micro-bubble generation depends on the liquid and gas interstitial velocities; micro-bubble fluid apparent viscosity and the bubble density, this is a known concept from the literature, but here, we can see that, bubble density controls the micro-bubble yield stress.

A typical saturation and bubble density profiles are shown in the Figure 1. It shows that the micro-bubble fluid front, flows with constant speed. This is agree with literature experiments, except when the capillary effects occurs. The Figure 2 shows the bubble density versus bubble generation parameter. It indicates that, any increasing in bubble generation, increase the bubble density at specified

We try to simplify the above model and solve it analytically. Since the micro-bubble fluid consist of liquid and bubbles, so it can be assumed as semi-incompressible fluid. Assume that the liquid phase (water) is incompressible in wide ranges of the pressure. i.e. ρ_w=constant . Expanding equation of 19 and rewrite it, we have:

{█(ϕ ∂(S_w )/∂t+∇.((u_w ) ⃗ )=0 @ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ )+1/ρ_m (Dρ_m)/Dt=0 @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+ϕS_m ∂n/∂t =ϕS_m [K_g (n_∞-n)-K_d n] )┤ 20

where the (Dρ_m)/Dt (material derivative of the density) is defined as below:

(Dρ_m)/Dt=ϕS_m (∂ρ_m)/∂t+(u_m

Equation 23 indicated that the semi-incompressible micro-bubble fluid arise when its pressure remain constant or pressure variation be small.

{█(ϕ ∂(S_i )/∂t+∇.((u_i ) ⃗ )=0 @(u_i ) ⃗= -λ_i ∇P_i @ϕS_m ∂n/∂t+(u_m ) ⃗.∇n=ϕS_m [K_g (n_∞-n)-K_d n] )┤

24

Now, the fractional flow analysis is introduced. The water fractional flow function fw, defined as ratio of water velocity to total velocity. So the equation 17 become:

{█(ϕ (∂S_w)/∂t+(u ⃗ ).∇f_w+f_w ∇.u ⃗=0 @ @n(ϕ ∂(S_m )/∂t+∇.((u_m ) ⃗ ))+Dn/Dt ==ϕS_m [K_g (n_∞-n)-K_d n] )┤

The micro-bubble generation and destruction coefficients i.e. K_g and K_d were assume constant; this assumption is not so far from the reality [20]. Mechanism of the micro-bubble generation depends on the liquid and gas interstitial velocities; micro-bubble fluid apparent viscosity and the bubble density, this is a known concept from the literature, but here, we can see that, bubble density controls the micro-bubble yield stress.

A typical saturation and bubble density profiles are shown in the Figure 1. It shows that the micro-bubble fluid front, flows with constant speed. This is agree with literature experiments, except when the capillary effects occurs. The Figure 2 shows the bubble density versus bubble generation parameter. It indicates that, any increasing in bubble generation, increase the bubble density at specified

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