  Models for branched networks are often expressed as the minimization of an
energy $M^\alpha$ over vector measures concentrated on $1$-dimensional
rectifiable sets with a divergence constraint. We study a Modica-Mortola type
approximation $M^\alpha_\varepsilon$, introduced by Edouard Oudet and Filippo
Santambrogio, which is defined over $H^1$ vector measures. These energies
induce some pseudo-distances between $L^2$ functions obtained through the
minimization problem $\min \{M^\alpha_\varepsilon (u)\;:\;\nabla\cdot
u=f^+-f^-\}$. We prove some uniform estimates on these pseudo-distances which
allow us to establish a $\Gamma$-convergence result for these energies with a
divergence constraint.
