  In this paper, we consider the numerical approximation of the quasi-static,
linear Biot model in a 3D domain $\Omega$ when the right-hand side of the mass
balance equation is concentrated on a 1D line source $\delta_{\Lambda}$. This
model is of interest in the context of medicine, where it can be used to model
flow and deformation through vascularized tissue. The model itself is
challenging to approximate as the line source induces the pressure and flux
solution to be singular. To overcome this, we here combine two methods: (i) a
fixed-stress splitting scheme to decouple the flow and mechanics equations and
(ii) a singularity removal method for the pressure and flux variables. The
singularity removal is based on a splitting of the solution into a lower
regularity term capturing the solution singularities, and a higher regularity
term denoted the remainder. With this in hand, the flow equations can now be
reformulated so that they are posed with respect to the remainder terms. The
reformulated system is then approximated using the fixed-stress splitting
scheme. Finally, we conclude by showing the results for a test case relevant
for flow through vascularized tissue. This numerical method is found to
converge optimally.
