Advective fluxes according to Forchheims's law (extends Darcy's law by an nonlinear drag) More...
#include <dumux/flux/forchheimerslaw_fwd.hh>
#include <dumux/flux/cctpfa/forchheimerslaw.hh>
#include <dumux/flux/box/forchheimerslaw.hh>
Go to the source code of this file.
Darcy’s law is linear in the seepage velocity v. As described in "darcyslaw.hh", this is valid for creeping flow (Re<<1). As v increases further, a nonlinear drag arises. The nonlinear drag arises, because the friction becomes comparable to surface drag due to friction. This additional friction is added as the forchheimer term to the Darcy's law. Resulting in the Forchheimer's law. see e.g. Nield & Bejan: Convection in Porous Media [56]
For multiphase flow, the relative passability \( \eta_r\) is the "Forchheimer-equivalent" to the relative permeability \( k_r\). We use the same function for \( \eta_r\) as for \( k_r \) (Van-Genuchten, Brooks-Corey, linear), other authors use a simple power law e.g.: \(\eta_{rw} = S_w^3\)
This leads to the equation in the form:
\[ \mathbf{v_\alpha} + c_F \sqrt{\mathbf{K}} \frac{\rho_\alpha}{\mu_\alpha } |\mathbf{v_\alpha}| \mathbf{v_\alpha} + \frac{k_{r \alpha}}{\mu_\alpha} \mathbf{K} \nabla \left(p_\alpha + \rho_\alpha g z \right)= 0 \]
This already includes the assumption \( k_r(S_w) = \eta_r(S_w)\):
This non-linear equations is solved for \(\mathbf{v_\alpha}\) using Newton's method and an analytical derivative w.r.t. \(\mathbf{v_\alpha}\).
The gradient of the Forchheimer relations looks as follows (mind that \(\sqrt{\mathbf{K}}\) is a tensor):
\[ f\left(\mathbf{v_\alpha}\right) = \left( \begin{array}{ccc} 1 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \\ \end{array} \right) + c_F \frac{\rho_\alpha}{\mu_\alpha} |\mathbf{v}_\alpha| \sqrt{\mathbf{K}} + c_F \frac{\rho_\alpha}{\mu_\alpha}\frac{1}{|\mathbf{v}_\alpha|} \sqrt{\mathbf{K}} \left( \begin{array}{ccc} v_x^2 & v_xv_y & v_xv_z \\ v_yv_x & v_{y}^2 & v_yv_z \\ v_zv_x & v_zv_y &v_{z}^2 \\ \end{array} \right) \]