A consistent u‐p formulation for porous media with hysteresis

Abstract

SUMMARYThis paper presents a continuum formulation based on the theory of porous media for the mechanics of liquid unsaturated porous media. The hysteresis of the liquid retention model is carefully modelled, including the derivation of the corresponding consistent tangent moduli. The quadratic convergence of Newton's method for solving the highly nonlinear system with an implicit finite element code is demonstrated. A u-p formulation is proposed where the time discretisation is carried out prior to the space discretisation. In this way, the derivation of all consistent moduli is fairly straightforward. Time integration is approximated with the Theta and Newmark's methods, and hence the fully coupled nonlinear dynamics of porous media is considered. It is shown that the liquid retention model requires also the consistent second-order derivative for quadratic convergence. Some predictive simulations are presented illustrating the capabilities of the formulation, in particular to the modelling of complex porous media behaviour.

A solution to transient seepage in unsaturated porous media

“…The solution is then obtained with the finite element method [3,23,24]. Further details, especially on the derivation of all equations for porous media, are found in the works by de Boer, Ehlers, Lewis, Schrefler and Pedroso [13,14,25,22].…”

“…It is observed that this order (time first, space later) yields great convenience when dealing with complex liquid retention models (e.g. with hysteresis) [22]. As shown in the following, the Newton-Raphson method can then be directly applied after deducing the derivatives related with the liquid retention model.…”

“…The solution of this expression can be obtained as in Computational Plasticity. In the present paper, an implicit backward Euler method is employed [22].…”

“…One common observed characteristics of the LRM is the hysteresis and scanning curves that appear during experiments with cycles of drainage and imbibition. These may cause difficulties in the numerical analyses; however the presented formulation can easily accommodate hysteretic models [22]. This paper is organised as follows.…”

FDM solutions to linear dynamics of porous media: Efficiency, stability, and parallel solution strategy

“…The TPM can be consistently derived from continuum mechanics, thermodynamics, the theory of mixtures, and the concept of volume fractions . For the static and dynamic behaviour of porous media, simulations based on the TPM (and Biot theory) have generated a great deal of advances; see, eg, previous studies …”