1. Introduction

This chapter presents a continuous Galerkin FE formulation for linear isotropic elasticity. It covers in detail how to derive such formulation beginning with the equilibrium equation and the virtual work statement. It also discretizes the continuity equation for slightly compressible single-phase flow to show how to couple different physics with elasticity. It discusses several coupling approaches such as the monolithic and iterative ones, i.e., loosely coupled. This chapter also mentions the affinity of the poroelastic case with the thermoelastic one. It thus also includes thermoelasticity in the treatment herein. It shows concrete numerical examples

© 2018 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

covering two- and three-dimensional problems of practical interest in thermo-poroelasticity. The sample problems employ triangular, quadrilateral, and hexahedral meshes and include comments about implementing boundary conditions (BCS). An introduction to domain decomposition ideas such as iterative coupling by the BCS, i.e., Dirichlet-Neumann domain decomposition and mortar methods for non-matching interfaces is included.

while f<sup>0</sup> and p<sup>0</sup> define for a reference or initial state. The common BCS for the pressure equation

one should also consider an initial or reference pressure distribution in the whole domain. Sources and sinks simulate injector and producer wells, respectively. Herein <sup>b</sup><sup>n</sup> is the outer unitary normal vector as usual. For the mechanics part, one begins from the equilibrium equation for a quasi-steady process, i.e., Newton second law, which means that one discards

�<sup>∇</sup> � <sup>σ</sup> <sup>¼</sup> <sup>f</sup> in <sup>Ω</sup> ; <sup>Γ</sup> <sup>¼</sup> <sup>Γ</sup><sup>u</sup>

<sup>t</sup> <sup>¼</sup> <sup>σ</sup> � <sup>b</sup><sup>n</sup> on <sup>Γ</sup><sup>u</sup>

where σ is the stress tensor, f corresponds to the vector of body forces, such as gravity and electromagnetic effects, for instance. One can decompose BCS in Dirichlet type, i.e., Γ<sup>u</sup>

where T ¼ T xð Þ ; t is the temperature, C is the elastic moduli, β corresponds to the coefficient of thermal dilatation while K is the bulk modulus. The Kronecker delta becomes δ while λ, and G, are the Lamé constants, and I represents the fourth-order identity tensor. The strain tensor ε is

One can derive a weak form by substituting Eq. (2) into Eq. (1) and then multiplying by a test

K � ∇pð Þ ∇v

v 1 μ

ð

∂Ω<sup>p</sup> N

A weak form for the equilibrium Eq. (4) can be derived in a similar way, by testing against a

<sup>2</sup> <sup>∇</sup><sup>u</sup> <sup>þ</sup> ð Þ <sup>∇</sup><sup>u</sup>

<sup>0</sup>ð Þ Ω and integrating over Ω and using the Gauss-divergence theorem, this

T

� dx ¼ ð

<sup>K</sup>ð Þ� <sup>∇</sup><sup>p</sup> � <sup>r</sup>g∇<sup>z</sup> <sup>b</sup>nTds:

Ω

D

N

<sup>u</sup> <sup>¼</sup> 0 on <sup>Γ</sup><sup>u</sup>

law combined with Biot's poroelastic theory defines σ by the following expression:

<sup>ε</sup> <sup>¼</sup> <sup>∇</sup><sup>s</sup>

v þ αv∇ � u\_ þ

K � ∇zð Þ ∇v

<sup>u</sup> <sup>¼</sup> <sup>1</sup>

1 μ

� �

T � �dx <sup>þ</sup>

<sup>∇</sup><sup>p</sup> � <sup>b</sup><sup>n</sup> <sup>¼</sup> 0 on <sup>Γ</sup>, (3)

Linear Thermo-Poroelasticity and Geomechanics http://dx.doi.org/10.5772/intechopen.71873

(4)

225

<sup>D</sup>, and

(7)

<sup>D</sup> ∪ Γ<sup>u</sup> N

<sup>N</sup>, where the external tractions are known or prescribed. Hooke's

<sup>T</sup> h i: (6)

q � vdxþ

<sup>σ</sup> <sup>¼</sup> <sup>C</sup> : <sup>ε</sup> � <sup>α</sup> <sup>p</sup> � <sup>p</sup><sup>0</sup> � � <sup>þ</sup> <sup>3</sup>K<sup>β</sup> <sup>T</sup> � <sup>T</sup><sup>0</sup> � � � � <sup>δ</sup> ; <sup>C</sup> <sup>¼</sup> λδ <sup>⊗</sup> <sup>δ</sup> <sup>þ</sup> <sup>2</sup>GI, (5)

imply Neumann or no-flow namely:

the acceleration term:

Neumann type BCS, i.e., Γ<sup>u</sup>

given by:

function v∈ H<sup>1</sup>

ð

1 M ∂p ∂t

> rg μ

Ω

ð

Ω

given virtual displacement, χ. One arrives at:

yields:

The treatment herein demonstrates that the continuous Galerkin formulation for linear isotropic elasticity is the foundation to develop codes for mechanics. Indeed, after discretizing linear elasticity is straightforward to extend the implementation to nonlinear mechanics such as rateindependent plasticity. It thus provides some comments about such extension. Applications of practical interest show that industrial size problems will require domain decomposition techniques to handle such simulations in a timely fashion. Unquestionably, domain decomposition techniques can exploit current parallel machines architectures which brings high-performance computing into the picture. For instance, recently the author showed that the Dirichlet-Neumann scheme could handle problems at the reservoir field-level as well as the mortar method decoupled by this last one. Its current results are backed up by papers published in peer-reviewed journals and conferences thus this book chapter summarizes that effort.
