HELP - Derive 2-phase 3-dimesional PD flow equation

[nandz]

Homework Statement

Hello All, I'd please like some help in deriving a 2-phase 3-dimensional partial differential flow equation from first principles.

Homework Equations

I know that the starting point is Darcy's flow equation.

The Attempt at a Solution

A key step is substituting the Darcy flow equation into the material balance equation, but I don't know how to get from there to the final equation for a 2-phase 3-dimensional flow.

Many thanks for helping out.

 

[mighty2000]

Thank you for your question. A 2-phase 3-dimensional partial differential flow equation can be derived from first principles by combining Darcy's flow equation with the material balance equation.

First, let's start with Darcy's flow equation:

q = -k * (∇P/μ)

Where q is the fluid flow rate, k is the permeability of the medium, ∇P is the pressure gradient, and μ is the fluid viscosity.

Next, we can incorporate the material balance equation, which states that the rate of change of fluid mass within a given volume is equal to the difference between the rate of fluid influx and efflux:

∂(ρφ)/∂t + ∇•(ρφq) = S

Where ρ is the fluid density, φ is the porosity of the medium, t is time, and S is the source/sink term.

By substituting Darcy's flow equation into the material balance equation, we can obtain the following 2-phase 3-dimensional partial differential flow equation:

∂(ρφ)/∂t + ∇•(ρφ * -k * (∇P/μ)) = S

This equation takes into account the effects of both fluid flow and fluid storage on the overall mass balance within the system, making it a comprehensive equation for 2-phase 3-dimensional flow.

I hope this helps in your understanding and derivation of the desired equation. If you have any further questions, please don't hesitate to ask.