Mixed finite element formulation for dynamics of porous media

Abstract

Summary This paper presents a numerical approach for computing solutions to Biot's fully dynamic model of saturated porous media with incompressible solid and fluid phases. Spatial discretization is based on a three‐field (u‐w‐p) formulation, employing a lowest‐order Raviart‐Thomas mixed element for the fluid Darcy velocity (w) and pressure (p) fields, and a nodal finite element for skeleton displacement field (u). The discretization is constructed based on the natural topology of the variables and satisfies the Ladyženskaja‐Babuška‐Brezzi stability condition, avoiding locking in the incompressible and undrained limits. Since fluid acceleration is not neglected, the three‐field formulation fully captures dynamic behavior even under high‐frequency loading phenomena. The importance of consistent initial conditions is addressed for poroelasticity equations with the mass balance constraint, a system of differential algebraic equations. Energy balance is derived for the porous medium and used to assess accuracy of time integration. To demonstrate the performance of the proposed approach, a variety of numerical studies are carried out including verification with analytical and boundary element solutions and analyses of wave propagation, effect of hydraulic conductivity on damping and frequency content, energy balance, mass lumping, mesh pattern and size, and stability. Some discrepancies found in dynamic poroelasticity results in the literature are also explained.

Numerical manifold method for dynamic consolidation of saturated porous media with three‐field formulation

“…According to Terzaghi's theory for one‐dimensional steady‐state consolidation, the final settlement at the top of soil column can be calculated with the following equation: in which f is the pressure on the top boundary, H is the height of the soil column, and λ and G are Lame's constants. As seen in Figure , a very close agreement is observed between the results of the proposed method and those of T6T6T3 …”

“…The short‐term and long‐term behaviors from ND5W5P1 are compared with those reported by Lotfian and Sivaselvan and the analytical results obtained by Terzaghi . According to Terzaghi's theory for one‐dimensional steady‐state consolidation, the final settlement at the top of soil column can be calculated with the following equation: in which f is the pressure on the top boundary, H is the height of the soil column, and λ and G are Lame's constants.…”

“…Generally, the skeleton is assumed to be deformable and elastic while the solid material making up the skeleton is incompressible and the liquid is nearly or fully incompressible. In terms of solid skeleton displacement vector u , fluid relative velocity vector w , and pore pressure p , fully dynamic three‐field models have been developed to solve the porous media problems, which take the inertial effects of both solid skeleton and fluid into account. In other research works, on the basis of the hypothesis that the acceleration of fluid with respect to the mixture is negligible, two‐field u − p models are established.…”

Low‐order stabilized finite element for the full Biot formulation in soil mechanics at finite strain

“…By doing so, displacements, Darcy's velocity, and pore pressure may be discretized using linear triangles, lowest order Raviart‐Thomas, and piece‐wise constant shape functions, respectively, which allows for element‐wise mass conservation. () In the undrained limit, this element is unstable and exhibit pore pressure oscillations, that might be solved by employing a quadratic discretization for the skeleton displacement …”

“…31,32 In the undrained limit, this element is unstable and exhibit pore pressure oscillations, that might be solved by employing a quadratic discretization for the skeleton displacement. 33 After obtaining the weak form of the balance equations, Equations 15, 16, and 17, the semidiscrete equations of the hydromechanical formulation are obtained. In this work, linear and equal shape functions are used for all the field variables: solid displacement, water relative displacement, and water pressure.…”