PSI - Issue 18

A. Kostina et al. / Procedia Structural Integrity 18 (2019) 301–308 Author name / Structural Integrity Procedia 00 (2019) 000–000

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and Dusseault (2013), propagation of the thermal front within the reservoir leads to the rise in the horizontal compressive stresses and substantial reduction of the vertical stresses causing a shear failure at low confining pressures. In case of a high-porous media fragmentation of soil particles clogs pore space and leads to the inelastic compaction. Therefore, thermal methods of oil recovery are accompanied by substantial structural changes which affect porosity and permeability of the reservoir. Commonly, evolution of various defects in the material is described within the framework of continuum damage mechanics. In this case, damage state of the material is described by a kinetic equation for a scalar or tensorial variable characterizing damage accumulation. This concept was proposed by Kachanov (1958) who associated scalar damage variable with the effective reduction of the area due to the presence of microcracks. Kawamoto et al. (1988) applied Kachanov’s theory to the case of damage distribution in the rock mass using the concept of net stress which acts only on the undamaged area. Lubliner at al. (1989) have associated plastic-damage variable to the cohesion in such a way that macroscopic failure of the material corresponds to the vanishing of the cohesion. Lee and Fenves (1998) extended this model by including of two damage variables to account tensile and compressive damage. The model takes into account scalar degradation of the elastic properties and the recovery in case of the crack closure. The general case of anisotropic damage requires the use of tensorial varibles. Al-Shayea et al. (2003) considered softening behavior of dense soils as a combination of elasto-plastic and damage behavior. Therefore, the total strain tensor included elastic part, damage part due to the degradation of elastic properties and plastic part due to the dislocation and readjustment of soil particles. Shao and Rudnicki (2000) introduced a second-rank damage tensor to describe residual opening of microcracks after the unloading. The damage evolution is associated with propagation condition of microcracks. Vallapan et al. (1990) idealized damage state of the rock as an orthotropic and derived constitutive equations which can be conveniently applied for finite-element simulation due to the symmetry of the final constitutive matrix. Xu and Arson (2014) derived thermodynamically consistent anisotropic damage model using adaptation of Drucker-Prager yield criterion to represent damage surface and projection tensor to distinguish effects related to crack opening in tension and compression. Olsen-Kettle (2018) defined the damage evolution law for the fourth-rank damage tensor using the results of measurements of seismic wave velocities. He has shown that off-diagonal components of damage tensor can be important in case of the stress-induced anisotropy of initially isotropic solid. In this work, we restrict ourselves to the symmetric damage tensor and consider the case of randomly distributed defects in the reservoir. The statistical model proposed by Naimark (2003) is used to take into account effect of volumetric damage on porosity evolution in the reservoir. 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. The volumetric part of this tensor is associated with an increase in porosity due to thermal and mechanical loadings, which arises during crude oil recovery by thermal methods. The reservoir is considered as a multiphase system, which includes solid skeleton, pore water, oil and the steam which is injected in order to reduce oil viscosity. The specific feature of the proposed model is coupling between thermal, filtration and mechanical processes which let us to associate volumetric damage evolution caused by thermal expansion and pore pressure with the improvement in porosity and permeability of the reservoir. The three-dimensional numerical simulation of the mechanical response of the rectangular reservoir which physical and mechanical properties correspond to Yarega oil deposit (Russian crude oil deposit in Komi Republic) demonstrates application of the developed approach to the description of a vertical surface heave observed in the process of thermal oil recovery and associated with an increase in porosity (Shafiei and Dusseault (2013)). 2. Mathematical model Generally, porous media is considered as a multiphase system composed of several phases. In our case (crude oil deposit subjected to continuous steam injection) the fluid and the solid phase need to be considered. The fluid phase includes pore water, oil and the injected steam. Adequate description of each phase requires the development of a coupled model, which includes mass balance equation, energy balance equation, fluid filtration and constitutive equations relating stress-induced effects to the change in porosity and permeability of the reservoir.