Brinkman porosity influence
The current Darcy-Brinkman implementation seems to make some assumptions on porosity that should be clarified. Maybe there are some adjustments needed. I think the basic governing equations should be something like (based on theory of porous media)
Fluid mass balance:
\begin{equation}
\frac{\partial \left( n_f \rho_f \right)}{\partial t} + \operatorname{div}{\left( \rho_f \vec{w}_f \right)} = 0,
\end{equation}
with porosity (fluid volume fraction) n_f
, Darcy velocity \vec{w}_f := n_f (\vec{v}_f - \vec{v}_s) = n_f\vec{v}_f
(rigid solid skeleton, \vec{v}_s \equiv 0
).
Fluid momentum balance:
\begin{align}
\frac{\partial \left( n_f \rho_f \vec{v}_f \right)}{\partial t}
+ \operatorname{div}{\left( n_f \rho_f \vec{v}_f \otimes \vec{v}_f \right)} &=\operatorname{div}{\underbrace{\left(2 \mu_f \vec{D}(\vec{v}_f) - n_f p \vec{I} \right)}_{\text{fluid stress,}\; \boldsymbol{T}_f}} + n_f\rho_f\vec{g} + \underbrace{p \nabla n_f - n_f^2 \mu_f \epsilon_B \vec{K}^{-1} \vec{v}_f}_{\text{momentum production,}\; \hat{p}_f},
\end{align}
with \vec{D}(\vec{v}_f) = \frac{1}{2}\left( \nabla{\vec{v}_f} + \nabla{\vec{v}_f}^T \right)
, \epsilon_B
some scaling factor for the permeability (0 in free-flow, 1 in porous medium, 0 < \epsilon < 1
in transition zone).
This reduces to Darcy's law when inertia and viscous stress can be neglected, and \epsilon_B=1
,
\begin{align}
\vec{0} &= -n_f \nabla{p} + n_f\rho_f\vec{g} - n_f \mu_f \vec{K}^{-1} \vec{w}_f \quad \longrightarrow \quad \vec{w}_f = - \mu_f^{-1}\vec{K} ( \nabla{p} - \rho_f \vec{g}),
\end{align}
But various terms containing n_f
don't easily simplify otherwise (transition zone and when n_f
is not constant).
Note: Discussion of the viscous term in https://doi.org/10.1017/S0022112005007998, which indicates \vec{D}(\vec{w}_f)
should be used instead of \vec{D}(\vec{v}_f)
. Then we can rewrite in terms of Darcy velocity
\begin{align}
\frac{\partial \left( \rho_f \vec{w}_f \right)}{\partial t}
+ \operatorname{div}{\left( n_f^{-1} \rho_f \vec{w}_f \otimes \vec{w}_f \right)} &=\operatorname{div}{\underbrace{\left(2 \mu_f \vec{D}(\vec{w}_f) - n_f p \vec{I} \right)}_{\text{fluid stress,}\; \boldsymbol{T}_f}} + n_f\rho_f\vec{g} + \underbrace{p \nabla n_f - n_f \mu_f \epsilon_B \vec{K}^{-1} \vec{w}_f}_{\text{momentum production,}\; \hat{p}_f},
\end{align}
Note: Darcy-Brinkman was introduced after release 3.8 so there is until release 3.9 to fix things if need be.
Notes: Brinkman.pdf