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## DESCRIPTION OF A SATURATED POROUS MEDIUM

Let us assume a porous medium, composed of a solid matrix, and a porous space in which the pores are interconnected. It is through this connected porous space that the transport of fluid mass occurs. Any two points in its domain can be connected by a generic arc totally contained in it, so that the fluid phase in that space can be treated as a continuum. There may also be closed pores included in the solid matrix, in which the occurrence of flow is not considered, at least not in the timescale considered in this theory. Hence, from this point of the text, the term pore is applied to the effective pores of the connected space, while the disconnected pores will be treated as part of the solid matrix. Ph.D.

Therefore, it is understood that the saturated porous medium is described by the superposition, temporal and spatial, of two continuous media: The first represents the solid skeleton and the second, the fluid phase. Usually, the deformation of the porous media is described in relation to the skeleton deformation, which can actually be observed and shows a more accessible physical meaning.

An infinitesimal volume of porous medium can be represented by the composition of two elementary material particles (Figure 2.1), one that is solid which also contains occlusions and disconnected pores and one that is fluid. Considering that the porous medium is heterogeneous at a microscopic level, its treatment as a continuous medium requires the choice of a macroscopic scale, in which the internal constitution of the material can be neglected, when analyzing the physical phenomenon of interest. Therefore, the continuity hypothesis admits the existence of an infinitesimal control volume of representative dimensions at a macroscopic scale, in the study of all phenomena involved in the intended application.

### BEHAVIOR OF THE SKELETON

If there are external forces or pressure variations in the fluid, the solid skeleton deforms. This deformation is analyzed according to the classical theory foreseen in the continuum mechanics, whose main concepts are briefly described below.

#### Motion of a Continuum. Displacement. Deformation Gradient

Consider a solid body occupying a determined region of space, at a time t 0 . In this initial configuration, a particle is represented by its position vector X of components i X , in a Cartesian coordinate system, of orthonormal basis i e (i 1,2,3) . After deforming, in time t , the body is in a current configuration, with its reference particle represented by the position vector x of components i j x (X , t) as shown in Figure 2.2. One can then write: skeleton particle fluid particle infinitesimal volume of porous medium X Xiei ; x xi (Xj, t)ei (2.1).

the displacement vector u of a particle is defined, from its initial position X to the current position x as: x Xu (2.2) Supposing two particles, positions X and X+dX in the initial configuration. After the deformation, the infinitesimal material vector dX becomes dx , and connects the two particles in their current positions x and x+dx . Any vector material dX is transported to its corresponding deformed dx by a linear application called the deformation gradient F , as follows. dx FdX.

**BEHAVIOR OF THE FLUID PHASE**

In the development of constitutive equations for a porous medium, the description of the fluid motion in relation to the initial configuration of the skeleton is necessary.

**Particle Derivative**

As aforementioned in the previous section, the description of the skeletons deformation can be done as a function of time t and the position vector X , both referenced in the initial configuration of the particle. In this Lagrangian description, the skeletons strain kinematics is formulated by the derivatives in total time. In some cases, it may be of interest to formulate the problem according to an Eulerian description, taking into account only the current configuration of the skeleton at a given time instant. In this type of approximation, it is necessary to define a velocity field ( t) V x, of the particle, which can be either a fluid particle or a skeleton particle (indicated by f or s , respectively). The particle derivative concept is shown below.

**MOMENTUM BALANCE**

Now we formulate the momentum balance for a porous medium, still according to the hypothesis adopted in the preceding paragraphs, which is treated as a superposition of two continuous media, interacting with each other. The momentum balance concept is important to obtain the total stress tensors.

**The Hypothesis of Local Forces**

Any material domain t can be subject to two types of external forces: the body forces and the surface forces. Generally, the body forces solicit the skeleton and the fluid in the same way. This is the case, for example, of forces due to gravity. An infinitesimal force f acting on the elementary volume t d is defined by a volume force density per unit mass f : f f (x, t)dt.

**Table of contents :**

**1. INTRODUCTION **

1.1. OVERALL CONSIDERATIONS AND OBJECTIVES

1.2. METHODOLOGY

1.3. BRIEF LITERATURE REVIEW

1.3.1. Poromechanics and Linear Poroelasticity

1.3.2. Strain Localization and Continuum Damage Mechanics

1.3.3. Porous Media Subjected to Damage

1.3.4. Integral Equations and BEM Applied to Poroelasticity and to Damage Mechanics

1.4. THESIS STRUCTURE

**2. ASPECTS ON POROMECHANICS AND CONTINUUM DAMAGE MECHANICS **

2.1. OVERALL CONSIDERATIONS

2.2. DESCRIPTION OF A SATURATED POROUS MEDIUM

2.3. BEHAVIOR OF THE SKELETON

2.3.1. Motion of a Continuum. Displacement. Deformation Gradient

2.3.2. Porosity. Void Ratio

2.3.3. Strain Tensor

2.4. BEHAVIOR OF THE FLUID PHASE

2.4.1. Particle Derivative

2.4.2. Relative Flow Vector of a Fluid Mass. Filtering Vector. Fluid Mass Content

2.5. MASS BALANCE

2.5.1. Eulerian Continuity Equations

2.5.2. Lagrangean Continuity Equations

2.6. MOMENTUM BALANCE

2.6.1. The Hypothesis of Local Forces

2.6.2. Momentum Balance

2.6.3. The Dynamic Theorem

2.7. STRESS TENSOR

2.8. EQUILIBRIUM EQUATION

2.9. PARTIAL STRESS TENSOR

2.10. ASPECTS ON THE CONTINUUM DAMAGE MECHANICS

2.10.1. Damage Variable and Effective Stress

2.10.2. Isotropic Local Damage Model (Marigo, 1981)

2.10.3 Comments on Strain Localization

**3. PORO-DAMAGE FORMULATION AND BEM IMPLEMENTATION **

3.1. OVERALL CONSIDERATIONS

3.2. PORO-DAMAGE FORMULATION

3.2.1. Constitutive Laws

3.2.2. Fluid Transport Law

3.2.3. Fluid Continuity Equation

3.2.4. Equilibrium Equation

3.2.5. Rates of the Variables

3.3. INFLUENCE OF DAMAGE ON THE POROELASTIC PARAMETERS

3.4. ASPECTS ON THE BOUNDARY ELEMENT METHOD

3.4.1. Boundary Elements and Discretization

3.4.2. Domain Discretization

3.5. BEM FORMULATION

3.5.1. Integral Formulation for the Solid Phase

3.5.2. Integral Formulation for the Fluid Phas

3.5.3. Time-dependent Integral Formulation

3.5.4. Algebraic System

3.5.5. Solution Procedure

3.5.6. Algorithm to Evaluate the Damage Level

**4. NUMERICAL EXAMPLES **

4.1. OVERALL CONSIDERATIONS

4.2. LINEAR POROELASTICITY EXAMPLES

4.2.1. One-dimensional Consolidation

4.2.2. Plane Consolidation

4.2.3. Poroelastic Response under Different Loading Conditions

4.3. EXAMPLES ON THE ADOPTED DAMAGE MODEL

4.3.1. Characterization and Parametric Analysis of the Model

4.3.2. Solid under Cyclic Loading

4.3.3. Solid with Defect under Uniaxial Tension

4.4. EXAMPLES ON POROELASTICITY COUPLED TO DAMAGE

4.4.1. Poroelastic Column Subject to Damage

4.4.2. Poroelastic Plane Domain subjected to Damage

4.4.3. Brief Comments on the Shallow and Deep Foundation Structures

CONCLUSION AND PERSPECTIVES

**BIBLIOGRAPHY**