THE LATTICE BOLTZMANN BRINKMAN FLOW MODEL: UNIFYING DARCY AND NAVIER-STOKES EQUATIONS

Although Darcy~{!/~}s law was first established empirically, it has been subsequently derived rigorously from the Stokes or Navier-Stokes equation. While the Navier-Stokes equation is capable of describing flow in detailed geometries at the pore scale, Darcy~{!/~}s law averages the flow over a representative volume. The Brinkman equation includes a viscous resistance term in the Darcy equation to account for solid/fluid interface effects. On the basis of a transient version of the Brinkman equation, we developed a unified lattice Boltzmann (LB) model for flow in multiscale porous media. This model can simulate not only flow in porous media of various length scales but also flow in porous systems where multiple length scales coexist. It has been found that the model recovers Darcy~{!/~}s law when the effects of inertial forces and the Brinkman correction are negligible and it reduces to the Navier-Stokes equation by specifying zero resistance in the void spaces and infinite resistance on solid walls. This LB model is particularly suitable for simultaneously simulating flow in the fractures and the matrix of fractured porous media. Here, both the Navier-Stokes flow in the fractures and the averaged Darcy flow in the matrix are solved from the same equation with the interface handled internally. This, together with other advantages of the LB method, leads to an efficient approach for simulating flow in multiscale porous media. A number of examples are used to illustrate this LB Brinkman flow model.