Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4256079, 12 pages

http://dx.doi.org/10.1155/2016/4256079

## Free Finite Element Approach for Saturated Porous Media: Consolidation

ETSI Caminos, Technical University of Madrid, Profesor Aranguren s/n, 28040 Madrid, Spain

Received 4 March 2016; Accepted 8 May 2016

Academic Editor: Giovanni Garcea

Copyright © 2016 M. M. Stickle et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The free finite element approach is applied to the governing equations describing the consolidation process in saturated poroelastic medium with intrinsically incompressible solid and fluid phases. Under this approach, where Voigt notation is avoided, the finite element equilibrium equations and the linearization of the coupled governing equations are fully derived using tensor algebra. In order to assess the free approach for the consolidation equations, direct comparison with analytical solution of the response of a homogeneous and isotropic water-saturated poroelastic finite column under harmonic load is presented. The results illustrate the capability of this finite element approach of reproducing accurately the response of quasistatic phenomena in a saturated porous medium.

#### 1. Introduction

The range of interesting problems involving porous media has increased dramatically in recent years. These problems might arise from different environmental fields like agriculture, petroleum engineering, or natural hazards, to mention a few [1–3]. A key aspect in order to develop accurate models, capable of reproducing the principal features of this wide class of phenomena, is the adequate description of multiphase porous media. Such multiphase materials can be properly described within the thermodynamically consistent theory of porous media (TPM), where all constituents are assumed to occupy simultaneously the space [4].

In order to obtain accurate, robust, and efficient solutions of the coupled partial differential equations that come up from TPM, numerical techniques are usually required. In this context, the finite element method is one that has achieved significant results [5–11].

Due to the widespread application of the finite element technique in engineering modeling, it is of great relevance how this numerical procedure is taught in graduate and undergraduate courses. In this context, the free finite element approach recently proposed in [12] is of paramount significance. The main ideas behind the free finite element approach are the following ones:(1)Use of Voigt algebra is avoided within free approach. Under Voigt algebra, second-order symmetric tensors in 3 dimensions are represented as a -dimensional vectors and a fourth-order symmetric tensor is rewritten as a matrix. Instead of this vectorization approach, which might be error-prone [12], the so-called free approach is tensor based. Only vectors and second-order tensors and their natural operations are considered, at least in solid mechanics. Fourth-order tensors are employed through development of the finite element equations; however, the Cartesian components of such type of tensors will never appear within the implementation.(2)From the previous point, it can be inferred that the strain operator , which relates the strain vector (increment) to the nodal displacements (increment), is not required. In the same way, the constitutive matrix, appearing in the tangent stiffness matrix in the standard finite element implementation, is also not required.

The main contribution of the free finite element approach is that it allows a direct translation of the continuum mechanics operations into the finite element implementation, simplifying the standard approach.

To the best of the authors knowledge, free finite element approach has not been considered before within the context of the TPM, even though some tensorial based finite element approaches have been developed.

The free finite element approach is related to [13]. See also [14, 15]. In [13], the authors proposed an implementation where the use of and matrices is avoided. However, this work is restricted to isotropic and linear elastic nonporous material. Also, the fourth-order tensor of classical elasticity appears in the expression of the tangent stiffness matrix, making use of indicial notation to operate with it. However, the free approach can be employed irrespective of the material type, which applies to the computation of internal forces, and no fourth-order tensors ever appear within the implementation.

Regarding the numerical resolution of saturated porous media equations, it is worth mentioning the tensorial form of the finite element discretization of the Generalized Biot field equations [5] presented by Zienkiewicz and coworkers [7]. The tensorial approach considered by these authors follows [13]. However, as stated by the authors, Voigt notation is preferred for the finite element computation. In other words, Zienkiewicz and coworkers presented the tensorial form for the sake of completeness of the referred text, but it seems that this approach was avoided in the finite element implementation.

It is also worth mentioning [16]. In this work, the author proposed a tensorial form of the finite element method for the formulation within TPM framework. Again the tensorial approach presented in this work presents similarities with [13]. However, in [16], the tensorial form is extended to deal with unsaturated porous media. Also the material model considered for the solid skeleton was extended to deal with a Drucker-Prager elastic-perfectly plastic model. On the other hand, a fourth-order tensor (consistent operator) appears in the expression of the tangent stiffness matrix in [16], making use of indicial notation to operate with it. However, in the free approach, the Cartesian components of a fourth-order tensors will never appear explicitly, irrespective of the material type considered.

An important point is whether the free approach is more efficient than the standard one. In [12], the authors stated that this is true in certain problems. However, this claim is not easy to sustain as there are specific aspects of the implementation, sparsity of the matrices and parallelization of process are only two of them, that might complicate the task of efficiency assessment. In the present work, this type of assessment is not accomplished.

The paper is structured as follows. For the sake of completeness, in Section 2, the consolidation equations of a fluid saturated porous medium within the theory of porous media (TPM) [4, 17] will be derived. Assumptions considered to obtain the full set of equations will be stated first. Then, the balance equations of mass and momentum of each phase will be presented under the scope of the assumptions made. In Section 3, the variational formulation of the consolidation equations is presented. In Section 4, the tensor based free finite element approach is developed in detail for the present problem. Evaluation of the accuracy of numerical procedures is of paramount importance in practical engineering. Therefore, in Section 5, the free numerical technique is applied to investigate the response of a one-dimensional column of finite length, composed by a fluid saturated poroelastic material and subjected to a time dependent loading. In order to assess the accuracy of the free approach, a direct comparison with an analytical solution is presented.

#### 2. Consolidation Equations for Saturated Soils

The assumptions considered to obtain the full set of governing equations are the following ones:(1)Porous medium is fully saturated.(2)Solid and fluid phases are intrinsically incompressible.(3)Neither mass nor heat exchanges between the solid and the fluid phase are considered.(4)Viscous shear stress within the fluid is neglected.(5)Permeability is considered isotropic.(6)Porous solid is considered isotropic and linear elastic under small strain theory.(7)All dynamic terms, including convective terms, are neglected.(8)Only loading by external forces is considered. Body forces are excluded.

Within the framework of TPM, a fluid saturated porous medium is described by two superimposed but immiscible species or phases with particles . The superscript denotes pore fluid for and solid skeleton for .

The kinematics in TPM is based on the assumption [4] that the spatial position in the current configuration at time is simultaneously occupied by particles of both constituents . Each particle proceeds from its own reference position in the reference configuration at time ; therefore, each phase has its own motion and at the common spatial position the phases interact (Figure 1).