From arteries to boreholes: transient response of a poroelastic cylinder to fluid injection

Abstract

The radially outward flow of fluid through a porous medium occurs in many practical problems, from transport across vascular walls to the pressurization of boreholes in the subsurface. When the driving pressure is non-negligible relative to the stiffness of the solid structure, the poromechanical coupling between the fluid and the solid can control both the steady state and the transient mechanics of the system. Very large pressures or very soft materials lead to large deformations of the solid skeleton, which introduce kinematic and constitutive nonlinearity that can have a non-trivial impact on these mechanics. Here, we study the transient response of a poroelastic cylinder to sudden fluid injection. We consider the impacts of kinematic and constitutive nonlinearity, both separately and in combination, and we highlight the central role of driving method in the evolution of the response. We show that the various facets of nonlinearity may either accelerate or decelerate the transient response relative to linear poroelasticity, depending on the boundary conditions and the initial geometry, and that an imposed fluid pressure leads to a much faster response than an imposed fluid flux.

Large poroelasto-plastic deformations due to radially outward fluid injection

“…For comparison with our poroelasto-plastic models, we additionally extend the purely poroelastic models from Auton and MacMinn [2,3] To investigate the impact of fluid injection into the pre-stressed subsurface, we wish to isolate the additional stress and deformation due to fluid injection from those due to background compression. We approximate the disturbance in all quantities due to fluid injection by subtracting their values in the compressed initial state (III in Figure 1) from their values in the final steady state (V in Figure 1).…”

“…For a given set of parameters, our poroelasto-plastic models will not be well posed for all values of σ a . Specifically, the material will remain elastic if σ a is sufficiently small; this purely poroelastic problem was studied extensively by Auton and MacMinn [2,3] for fully permeable materials (ζ ≡ 1). Conversely, the material will yield completely if σ a is sufficiently large.…”

“…In the plastic region, the governing ODE (Equation 38a) is first order and thus must be enforced at exactly N − 1 nodes. We enforce this for the first N − 1 nodes of the discretised domain The main structural difference between the models presented here and the models presented in Auton and MacMinn [2,3] is the coupling of an elastic domain with a plastic domain at a free (but non-material) boundary. As a result, the derivation of an exact analytical Jacobian matrix is nontrivial and we instead approximate the Jacobian matrix numerically.…”

“…For a more detailed derivation and discussion of the above aspects of the model, see Auton and MacMinn [2,3].…”

“…For the LL, QL, and QQ models, the governing ODE in the plastic region (Equation 37) can also be solved analytically. However, doing so leads to a strongly nonlinear root-finding problem for {a, s, b}, in which we do not have a good initial guess for s. We avoid this by instead solving all of these models and the NQ model numerically by adapting the Chebyshev spectral collocation method of Auton and MacMinn [2,3].…”