Hey! Im 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).(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Two phase darcy flow

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