Selecting the Right Fluid Flow Interface for Porous Media
To model fluid flows in porous media, use either Darcy’s Law or the Brinkman Equations interface. Darcy’s law is preferable when the major driving force for the flow is the pressure gradient, and the momentum transfer by shear stresses within the fluid is negligible. This is usually true in the case of low permeability in the porous medium.
The corresponding physics interfaces combine arbitrarily and link to other physics including solute transport, heat transfer, electric potentials, magnetic potentials, and structural deformation, to name a few. Because any expression can be entered for coefficients like permeability or density, the equations set up in this module are especially flexible.
Figure 3-1 illustrates typical settings where the various flow regimes apply.
Figure 3-1: Vertical cross section through ground near a river. Labels indicate flow regimes. The triangle denotes the water table.
The Single-Phase Flow, Laminar Flow interface uses Navier-Stokes equations for flows of liquids and gases moving freely in pipes, channels, caves, and rivers. These equations detail fluid movements through the intricate networks of pores and cracks in porous media. The scale of interest in most geologic problems, however, makes solving for velocity profiles within pores unfeasible owing to the sheer volume of the data required and machine constraints on computational effort.
The Porous Media and Subsurface Flow interfaces — Darcy’s Law, Brinkman Equation, and Richards’ Equation — circumvent detailing flow in each pore. Instead these equations estimate flow rates by lumping the properties of the solid grains and the spaces between them in representative volumes.
The Fluid Flow interfaces in this module can be grouped according to driving forces. Consider Bernoulli’s equation for mechanical energy along a streamline in inviscid fluids:
Here s represents a location on the streamline, u denotes the fluid velocity, ρf is the fluid density, p refers to the fluid pressure, g represents gravity, and D is the elevation. The Bernoulli equation states that the total mechanical energy is constant along the fluid trajectory in a steady flow system. Moreover, the energy can shift between velocity, pressure, and elevation along the streamline.
With the extended Laminar Flow interface and the Brinkman Equations, all the driving forces are nonnegligible. When solved, the equations give the directional fluid velocities and the pressure; elevation is a coordinate in the model. With Darcy’s Law and Richards’ Equation, the momentum is so small it can be neglected. Pressure alone drives the flow for these physics interfaces.