The simulation of the nonlinear consolidation of porous media

In this paper, the general formulation of the problem of coupled deformation of a porous deformable medium with a fluid flowing through the pores is formulated, mathematically investigated and numerically implemented within the framework of physical and geometric nonlinearity. We present the formulation of the problem in velocities of solid phase displacements and the rate of pore pressure change in differential and variational forms. A phenomenological approach was used to formulate the mechanical model. The equations of the coupled consolidation model were derived from the general conservation laws of continuum mechanics using spatial averaging over a representative volume element. The consolidation model took into account the change in the porosity and permeability of the medium during deformation. The equations of filtration and porosity change, originally presented in Euler approach, were reformulated in Lagrangian coordinates of the solid phase using the relative fluid velocity according to ALE (Arbitrary Lagrangian - Eulerian) approach. The Gâteaux differentiation technique was used to linearize the variational equilibrium equations. For spatial discretization of the saddle system of equations, the finite element method (FEM) was used: quadratic serendipity elements for approximating the equilibrium equations and Brick type elements for approximating the filtration equation. To solve the system of equilibrium and filtration equations, a generalization of the implicit scheme with internal iterations at each time step by the Uzawa method was used. The results of numerical simulation of elastoplastic deformation of a water-saturated soil under load with fluid outflow are presented. To simulate the constitutive relations of elastoplastic deformation of soil under short-term loads, a generalization of S.S. Grigoryan's model to large deformations is proposed. The calculations were carried out in our own program code. The developed consolidation model can be used to simulate the formation of tracking ruts and unevenness of natural roads, as well as to calculate the uneven settlement of engineering structures.