PSI - Issue 6

I. Bazyrov et al. / Procedia Structural Integrity 6 (2017) 228–235 Author name / Structural Integrity Procedia 00 (2017) 000–000

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3 6 7 8 9

  =  − /  +    .

Flow discharge per unit area can be described by Darcy’s law [5]:  = −   grad ,  – permeability tensor,  – fluid dynamic viscosity. Now, we determine the mass of the liquid [5, 6]:  =   +  .  = 1/ –is the Biot modulus (fluid compressibility according to porous medium deformation). The flow rate in porous medium is a continuity equation:     +  ∙  =  ,  - filtration rate,  – flow rate intensity. To rearrange equations (1)–(9) the governing equations can be obtained: div −  grad = 0 ,       − div    grad =  , where ,   ,   are the bulk moduli of porous medium, soil and fluid respectively,  is the porosity. So, rock mechanics is coupled with hydrodynamics. The change of porosity is the result of the rock matrix deformation which is a function of both pressure and stress, permeability of the medium is also a function of effective stresses in the reservoir. Also we need to specify initial and boundary conditions. To simplify the approach let’s consider homogeneous conditions of the 1 st and 2 d kind. The coupled (4D) model consists of geomechanical and hydrodynamical domains ( Ω  and Ω  respectively , Ω  ⊂ Ω  ) with boundaries (figure 1)   =   =   +   +   and   =   + Σ     . (10) (11)       +  = 1 −    ,   =    +     , (12)

Figure1. Coupled models’ domains.

The geomechanical model has the following boundary conditions:  = 0,  ∈  Boundary conditions for hydrodynamic model ( Γ   :

 ,   = 0,  ∈   ,  =   ,  ∈   .     = 0,  ∈   , ,  =    ,  ∈    . , 0 =   ,  ∈ \⋃      .

13 14 15

Initial conditions for pressure:

Initial conditions for displacement follows equation (11) according to (15):