Line source in a poroelastic layer bounded by an elastic space

The fundamental solution of a continuous line source, injecting fluid at a constant rate over the thickness of a poroelastic reservoir bounded by infinite impermeable elastic layers, is derived in this paper. This idealized problem has applications in hydrogeology and in petroleum engineering, as it can be used to assess the mechanical perturbations caused by injection or withdrawal of fluid in the subsurface through a vertical well. Construction of the solution takes advantage of the uncoupling of the pore pressure field, which, in this particular case, is given by the classical singular solution of the diffusion equation for an infinite line source. The mechanical fields then are determined by solving an elasticity-like problem with a body force field proportional to the time-dependent pore pressure gradient. On account of the problem symmetries, the Navier equations of elasticity reduce to two uncoupled partial differential equations for the radial and vertical (axial) displacement components, which are solved by a twofold application of Fourier and Hankel transforms. The solution exhibits different regimes at small, intermediate, and large times. When the diffusion radius, proportional to the square root of time, is smaller than or comparable to the thickness of the permeable layer, the induced deformation is confined to a region with a characteristic dimension of the same order as the diffusion radius. At large time, when the diffusion radius is large compared with the permeable layer thickness, the deformation rate in the reservoir is essentially oedometric (uniaxial). The different regimes of solutions are justified with a conceptual model based on identifying the evolving characteristics of complementary interior and exterior domain problems. The derived solution can serve as a valuable benchmark for coupled reservoir simulators. It also provides insights in to such problems as waterflooding, shearing at reservoir/cap rock interfaces, and stress redistribution around producing wells.