Syringomyelia is a medical condition in which one or more fluid-filled cavities (syrinxes) form in the spinal cord. The syrinxes often form near locations where the spinal subarachnoid space (SSS; the fluid-filled annular region surrounding the spinal cord) is partially obstructed. Previous studies showed that nonlinear interactions between the pulsatile fluid flow in the SSS and the elastic deformation of the tissues surrounding it can generate a fluid pressure distribution that would tend to drive fluid from the SSS into the syrinx if the tissue separating the two regions was porous. This provides a potential explanation for why a partial occlusion of the SSS can induce the growth of an already existing nearby syrinx. We study this hypothesis by analysing the mass transfer between the SSS and the syrinx, using a poroelastic fluid–structure interaction model of the spinal cord that includes a representation of the partially obstructed SSS, the syrinx and the poroelastic tissues surrounding these fluid-filled cavities. Our numerical simulations show that poroelastic fluid–structure interaction can indeed cause an increase (albeit relatively small) in syrinx volume. We analyse the seepage flows and show that their structure can be captured by an analytical model which explains why the increase in syrinx volume tends to be relatively small.