Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium

Abstract

Exact analytic solutions are derived for the time-dependent deformation of a poroelastic medium within a two-dimensional finite domain. Solutions are given with a specific set of boundary conditions for the case of a source of fluid at an arbitrary point and for an applied pressure on the boundary. These solutions are ideal for testing numerical schemes for poroelastic flow and deformations due to their relative simplicity.

A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity system

“…Figure 5 shows for this setting the computed displacement and pressure solution at timet = =2. The solution resembles the exact solution in Reference [23] very well, see also [3]. O(h 2 + t 2 ) accuracy is observed for the displacements, and, asymptotically, for the pressure too (despite the occurrence of the delta function which usually in uences the numerical accuracy negatively) [3].…”

“…By choosing a unit squared domain, a source term Q = 2 sint · 0:25;0:25 (t = ( + 2 )at, is the Kronecker delta function), the following boundary and initial conditions: at y = {0; 1}; u= 0; @v=@y = 0 at x = {0; 1}; v= 0; @u=@x = 0 and pressure p = 0 at the boundaries, we can mimic the dimensionless situation. In this case, the solution can be written as an inÿnite series [23]. An interesting feature is that this solution is independent of the Lamà e coe cients.…”

“…Some analytical reference solutions are known in the literature [23] for (1) in dimensionless form, where scaling has taken place with respect to a characteristic length of the medium ', Lamà e constants + 2 , time scale t 0 and a. By choosing a unit squared domain, a source term Q = 2 sint · 0:25;0:25 (t = ( + 2 )at, is the Kronecker delta function), the following boundary and initial conditions: at y = {0; 1}; u= 0; @v=@y = 0 at x = {0; 1}; v= 0; @u=@x = 0 and pressure p = 0 at the boundaries, we can mimic the dimensionless situation.…”

An efficient multigrid solver for a reformulated version of the poroelasticity system

“…In this case, the solution can be written as an infinite series [1], see also [7]. An interesting feature is that this solution is independent of the Lamé coefficients.…”

“…Some analytical reference solutions are known in the literature [1] for (7) in dimensionless form, where scaling has taken place with respect to a characteristic length of the medium ', Lamé constants k + 2l, time scale t 0 and a (7).…”