PSI - Issue 13

A. Kostina et al. / Procedia Structural Integrity 13 (2018) 1159–1164 Author name / Structural Integrity Procedia 00 (2018) 000–000

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increase. Therefore, in this work we will focus on the temperature effect on the stress strain state of the reservoir and its properties. Early works was devoted to the simulation of thermoelastic media subjected to the injection of the hot steam into the reservoir. Aktan and Farouq Ali (1978) have shown that thermal stresses are the key factor defining the stress strain state of the reservoir overwhelming the influence of the pressure and gravitational forces. Botao et al. (2017) proposed methodology for mechanical response of the reservoir and dependence of permeability on mechanical deformation. As a result, it has been shown that the considered conditions induce only elastic strains. Rahmati et al. (2017) investigated effect of the anisotropy and natural fractures on strength of the caprock. They concluded that maximal operating pressure is sensitive to the density, orientation and height of the defects. In this work the damage evolution in the reservoir is described by the statistical model proposed by Naimark (2003). According to this model, volumetric and shear defects are described by the macroscopic internal variable which has a physical meaning of the additional (structural) strain induced by the initiation and evolution of the defects. This parameter is introduced as the statistical averaging of the symmetrical tensor characterizing a unit defect (microcrack or microshear). Constitutive equation for the structural variable is derived under an assumption of the local thermodynamic equilibrium. The three-dimensional numerical simulation of the mechanical response of the sandstone reservoir demonstrates application of the developed approach during the steam injection process. Experimental data on triaxial loading of the cubic sample were used for the identification of material parameters. The proposed model has been used for the investigation of the damage evolution in the reservoir and effect of the volumetric damage on the permeability and caprock integrity. 2. Thermo-mechanical model of damage evolution in quasi-brittle materials In case of a small deformations and absence of the heat supply, the first and second thermodynamics laws involve the following thermodynamic quantities: density ρ , strain and stress tensors σ and ε , absolute temperature T , heat flux vector q , Helmholtz free energy F , entropy S . These laws can be written in the form (Murakami (2012)): 1 : e     σ ε q   , (1)

1

: T       σ ε q    , 0 F TS T

(2)

where e F TS   - internal energy, the upper dote denotes time derivative,  - Nabla-operator. The symmetrical tensor p which has a meaning of the additional strain induced by the initiation and coalescence of the defects is used as an internal variable describing structural evolution in the material. Therefore, the kinematic relation for the full strain rate has the form (Naimark (2003), Plekhov et al. (2009)):

0 e d d      ε ε p ε p ε       , 0 e T

(3)

0 p - volumetric part of the structural strain tensor, e d ε -

where 0

e ε - volumetric part of the elastic strain tensor,

deviatoric part of the elastic strain tensor, d p - deviatoric part of the structural strain tensor, T  - thermal expansion coefficient, E - unit tensor. Let us assume that the free energy is a function of elastic strain and structural strain, i.e. ( , ) e F F  ε p . Taking into account this hypothesis, (3) and the standard thermodynamic formalism, we can obtain the constitutive equations in the form: ' k T    q , (4) T   ε E  - thermal strain,

0 0 σ ε e K    ,

(5)

2 d d G  σ ε   , e

(6)