PSI - Issue 28

A. Kostina et al. / Procedia Structural Integrity 28 (2020) 675–683 Author name / Structural Integrity Procedia 00 (2019) 000–000

679

5

where τ p 0 is the relaxation time for volumetric structural strain,  denotes Macaulay brackets, 0 c σ is the threshold value of stress for activation of volumetric structural strains, ' K is the bulk modulus, b is non-dimensional parameter defining increase in the volumetric damage, c is non-dimensional parameter responsible for the coupling between volumetric and shear damage, τ pd is the relaxation time for deviatoric structural strain, σ :  σ σ is the stress intensity, dc σ is the threshold value of stress for the activation of deviatoric part of the structural strain tensor, G is the elastic shear modulus, : p  p p is the structural strain intensity, 2.4. Coupling equations As it is mentioned by Shafiei and Dusseault (2013) SAGD is accompanied by increase in porosity by 4%-6% due to the shear dilation with concomitant increase in permeability. To describe this effect we applied porosity equation which relates porosity with volumetric strains (Rahmati et al. (2017)):   0 1 vol vol n n      , (14)

where 0 n is the initial porosity, vol  is the volumetric strains. Absolute permeability was related to porosity according to the equation (Hu et al. (2013)):

3

n

,

(15)

K d 





2

1

n





 2 3

where 0 1 d K n n   ensures that initial value of absolute permeability is equal to 0 K . 3. Results of numerical simulation 0 0

The developed model (1)-(9), (12)-(15) was applied to the three-dimensional numerical simulation of reservoir’s area containing two pairs of SAGD wells. The reservoir properties correspond to Yarega oil deposit (Russian heavy oil deposit in Komi Republic). The reservoir has a height of 27 m, length of 21.5 and a width of 100 m. The injection and production wells have a radius of 0.178 m. Distance between the wells was equal to 5 m. Additional overburden layer of the soil with the thickness of 20 m was considered in mechanical problem while three-phase flow was considered only in the reservoir. The temperature and the pressure of the injected steam were equal to 495 K and 3.5 MPa respectively. The time of steam injection was equal to 400 days. Specific feature of the simulation is accounting for the non-uniform steam distribution along the horizontal wellbore which strongly affects the shape of the steam chamber. We carried out a comparative analysis of three cases: non uniform distribution of steam without structural strains (in this case only thermo-elastic strains were considered), uniform distribution of steam with accounting for structural strains and non-uniform steam distribution with structural strains. Fig. 1-2 demonstrate distribution of the volumetric structural strains for the last two cases. In case of the uniform steam distribution (Fig. 1) we observe steady increase of the volumetric structural strain during the whole injection time with the maximum concentration in a vicinity of the injection well. Non-uniform steam distribution induces non-uniform evolution of the structural strain (Fig. 2) and substantial qualitative difference in its spatial distribution compared to the previous case. Maximum value of the structural strain in this case is localized near the heel of the well. However, the magnitude of the maximum value in both cases are the same. Fig. 3 presents increase in porosity compared to the initial value in case of the absence and the presence of the structural strains. It can be seen that in case of thermo-elastic strains a non-uniformity produced by steam distribution is levelled off to the 400 th day of steam injection (Fig. 3(a)). As it can be seen from Fig. 1(c) and Fig. 2(c) volumetric part of the structural strain completely determines the shape of the porosity increment which is given by Fig. 3(a) - 3(b). Moreover, consideration of the structural strain allows one to obtain porosity increment one order of magnitude higher compared to the case of purely thermo-elastic strains. Similar results can be observed for the permeability (Fig.