We examine a physical problem concerning the injection of an aquablock liquid into the interior of telecommunications cables for the purposes of waterproofing. We model the inside of the cable as a porous medium and use Darcy's law for the flow problem. Mathematically, we formulate a Laplace moving boundary problem, which by the technique of using a Baiocchi transformation we reduce to a variational inequality formulation. Numerical solutions of the reformulated problem are obtained via finite element and finite difference discretizations and the results are compared to solutions obtained using a quasi-static formulation. In addition, asymptotic solutions are obtained which are consistent with the numerical solutions and which furthermore' provide bounds for the time taken for the injected liquid to reach the centre of the cable.