Issue 49

A. Kostina et alii, Frattura ed Integrità Strutturale, 49 (2019) 302-313; DOI: 10.3221/IGF-ESIS.49.30 305     vp F F A b ε σ , (11)       1 2 T ε u u , (12)               , , sat s s sat sat s w sat w w sat sat T T rnS T T T q q T T rnS T T T , (13) where subscript i takes values s , w , o , r and denotes steam, water, oil and solid skeleton respectively; n is the porosity;  i is the density; t is the time; w S is the water saturation; s S is the steam saturation; o S is the oil saturation; i v is Darcy’s velocity; K is the absolute permeability; ri k is the relative permeability;  i is the dynamic viscosity; p is the pressure; g is the gravity; T is the temperature; i c is the heat capacity;           , , 1 eff i i r i w o s nS n is the effective thermal conductivity;  w Q Lq is the heat source due to the phase change; L is the latent heat; σ is the stress tensor;           , , 1 eff i i r i w o s nS n is the effective density; C is the stiffness tensor which has two components in case of linear isotropic elasticity (Young’s modulus and Poisson’s ratio); ε is the strain tensor;  T is the thermal expansion coefficient; 0 T is the initial temperature; E is the unity tensor; vp ε is the viscoplastic strain tensor;  B is the Biot coefficient; A is the viscoplastic rate coefficient;  denotes McCayley brackets;    2 1 F J aI b ; 2 J is the second invariant of the deviatoric part of the stress tensor; 1 I is the first invariant of the stress tensor; a , b are material parameters; u is the displacement vector; r is the mass transfer intensity factor; sat T is the phase change temperature. The following functions were used to define relative phase permeabilities [28]:                   1 1 , 1 1 0, m w rw rw w rw rw ro w rw S S a S S k S S S S , (14)                    2 2 , 1 1 0, m s rs rs s rs rs rw ro s rs S S a S S k S S S S S , (15)                   3 3 , 1 1 0, m o ro ro o ro rw ro o ro S S a S S k S S S S , (16) where 1 a , 2 a , 3 a , 1 m , 2 m , 3 m are the empirical parameters; rw S , rs S , ro S are the residual values of water steam and oil saturations. In this work the effect of volumetric strains and effective stresses on the porosity evolution of the reservoir is investigated. For this purpose two qualitatively and quantitatively different models are considered. The first model [10] relates porosity to mean effective stresses by means of the coefficient which is linearly dependent on the current value of porosity: