Two-Phase Darcy Flow: Modeling & Solving

[maka89]

Hey! I am currently working on modeling and solving a scenario with oil and water flow. The flow is supposed to be driven purely by the capillary pressure(Dont mind this too much).

The flow in the two phases are given by:
[itex]q_o = -\frac{k_o(\Delta P)}{\mu_o}\frac{\partial P_o}{\partial x}[/itex]
[itex]q_w = -\frac{k_w(\Delta P)}{\mu_w}\frac{\partial P_w}{\partial x}[/itex]

I ended up with these governing equations:
[itex]\frac{\partial q_o}{\partial x} = c(\Delta P)\frac{\partial \Delta P}{\partial t}[/itex] (Mass continuity oil phase)
[itex]\frac{\partial q_w}{\partial x} = -c(\Delta P)\frac{\partial \Delta P}{\partial t}[/itex] (Mass continuity water phase)

Now, you can see that I have two equations with two unknowns, [itex]P_o[/itex] and [itex]P_w[/itex]. But as you see, they have similar right hand sides, which makes little sense... What does it mean when I end up with these kinds of equations? How do I solve this problem?

While applying mass continuity i neglected the compressibility of the fluids as they should not be too big. If I were to include them my equations would become:

[itex]\frac{\partial q_o}{\partial x} = c(\Delta P)\frac{\partial \Delta P}{\partial t} + c_o\frac{\partial P_o}{\partial t}[/itex] (Mass continuity oil phase)
[itex]\frac{\partial q_w}{\partial x} = -c(\Delta P)\frac{\partial \Delta P}{\partial t} + c_w\frac{\partial P_w}{\partial t}[/itex] (Mass continuity water phase)

Do I have to do this? Is there a way to solve my original equations?

 

[Strum]

I would imagine you need some boundary condition between the two phases like ## v_n \propto [P]^{+}_{-} ## and Gibbs-Thompson conditions.