To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:34, paper No. 072.
Copyright © 1997 Elsevier Science Ltd
Copyright © 1997 Elsevier Science Ltd Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 34, 1997 ISSN 01489062 To cite this paper: Int. J. RockMech. &Min. Sci. 34:34, Paper No. 072
Modeling Wellbore Breakouts D. Elata
Faculty of Mechanical Engineering Technion  Israel Institute of Technology, Haifa 32000, Israel ABSTRACT
In this work a constitutive model of hard sandstone is developed in order to study the development and stabilization of wellbore breakouts. The rock is modeled as a brittle, porous, hyperelastic material in which the solid phase is a linear elastic material. The effects of porosity and damage are described phenomenologically, and the resulting constitutive model describes a nonlinear damagable poroelastic material in which damage continuously reduces the resistance to distortion. Breakout geometry predicted by preliminary simulations using a simple material are in agreement with field data. The results also predict an inhomogeneous stiffness within the damaged zone, that may be verified by acoustic measurements. Copyright © 1997 Elsevier Science Ltd KEYWORDS Boreholes
• Brittle Failure • Constitutive
Relations
• Continuum
• Mechanical
Properties
• Models
INTRODUCTION The objective of this work is to study the process of breakouts development and stabilization in hard sandstone. A vertical wellbore is considered and the farfield stress boundary condition is assumed to have one principal stress parallel to the well axis. In a recent study (Cheatham 1993), the rock was modeled as a linear elastic material with two semicircular regions at the breakout zones, in which the material was softer than the surrounding rock. The softer material reduces the stress intensity at the well face and leads to stress redistribution that stabilizes the growth of the breakouts. In the present study the process of breakouts development and stabilization is simulated by using a general constitutive model in which gradual and inhomogeneous degradation of the elastic moduli is allowed. The size and shape of the breakouts zones in which the rock is fully damaged is to be calculated rather than predetermined. The rock is modeled as a brittle, porous, hyperelastic material in which the solid phase is an isotropic linear elastic material. The effects of porosity are described phenomenologically by relating global strain invariant to the same invariants within the solid phase. In addition, a simple damage modeled that describes degradation of the overall shear stiffness is considered. The resulting constitutive model describes a nonlinear damagable poroelastic material in which damage continuously reduces the resistance to distortion. Also, the phenomenological model of porosity couples volumetric and distortion effects. Breakout shapes resulting from preliminary simulations are in agreement with field data. The results also predict an inhomogeneous stiffness within the damaged zone, that can be verified by acoustic measurements. By varying the farfield stress, wellbore pressure, and material parameters, the effect of these variables on breakout development and stabilization may be studied.
ISSN 01489062
To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:34, paper No. 072.
Copyright © 1997 Elsevier Science Ltd
C O N T I N U U M M O D E L OF R O C K The mechanical response of a porous material depends on the response of its solid constituent and the geometry of its pore space. The effect of pore space geometry is so crucial that the properties of a porous material of a given solid phase and given porosity may vary widely. In natural materials it is difficult to measure the pore space geometry and any such measurement describes only a specific material sample. Nevertheless, the mechanical effects of porosity are similar in different samples of the same material even though their geometrical details vary. In contrast to the pore space, the properties of the solid phase can be readily measured and are valid anywhere within the material. It is therefore logical to base the constitutive model of a porous material on the constitutive model of its solid constituent, and to suggest a phenomenological model to account for the homogeneous effect of the pore space. Rubin et al. 1996 suggested a model that related the local volumetric deformation within the solid phase to the global volumetric deformation of the porous material. The resulting constitutive model of a porous rock had excellent predictive capabilities even though very few material parameters were used. In a current study (Elata 1997) this approach is generalized by relating local distortion within the solid phase to the global distortion of the porous material. Also the model is generalized to include damage. The porous rock considered in this work is a hard and well consolidated. The kinematic domain of interest is limited to small strain in which the rock demonstrates nonlinear response with negligible residual strain in the stressfree state. However, within this kinematic domain, the mechanical response is characterized by damage evolution that degrades the deviatoric stiffness of the rock. The local stress experienced by the solid constituent is small and therefore the solid is modeled as a linear elastic material. By way of background, let x be the position vector of a material point, u be the displacement of x, H=0u/0x be the displacementgradient tensor, E=H+H T be the infinitesimal strain tensor, (EoI) and E'=E1/3(EoI)I be the spherical and deviatoric parts of E, respectively, and let J=(I+EoI) be the volumetric deformation. Here a superposed 'T' denotes transposition, '.' denotes scalar product, and I is the secondorder unit tensor. Also, let the porosity ~)be defined as the volume fraction of pore space within the material, and let qb denote the initial value of ~). Let T be the Cauchy stress, p=l/3(ToI) be the pressure and T'T+pI be the deviatoric part of T. The constitutive model is a smallstrain version of a general finite deformation model currently developed (Elata 1997). The stressstrain relations of the porous rock are given by p=k
s JfL~i
T' = ( l  ~ ) 2 j  I g s
1¢~J
J
2 °¢'l~~'JoJ
'
~'t + ( Z l O a l )
(i)
(2)
where k s and gs are the bulk and shear moduli of the solid constituent, ~Zl=2(E'oE'), ~)=p(J) and
~,=q(~),(Zl,O) ) are functions that relate global and local kinematics Js = 1+ E ' s ' I = J 1  ~
'
(3)
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To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:34, paper No. 072.
C(I s  2(E
Copyright © 1997 Elsevier Science Ltd
(4)
,o , )V2 s E s) = ot[ ,
where E'sOI and E' s are the volumetric and deviatoric strains within the solid constituent, and co is a damage variable that is limited by the constraint g = )(J,(E"E'))
(5)
 e0 _< 0
Within the context of isothermal processes, the bulk and shear moduli are given by
k=J
(E'=o) = ( 1  ¢ ) k 1 1 } , aJ
lOT'
gffj~
(1~
(a
+ Ps(2}~ +  a j 2 '
(6a)
'
l)
(6b)
) ~ l + a l ~   7 Ms
These functions together with Eqns. (1) and (2) can be used to determine the functional form and parameters of p and q from experimental data. NUMERICAL SIMULATION The experimental data required to determine the functional form and parameters of p, q, and u, has not been collected yet. Therefore, the simulations are calculated for a linear elastic rock (~) and X are constant). The material parameters of Massillion sandstone are used [Winkler, Liu 1996] =0.21,
k=6.1GPa,
Ix=6.3GPa
(7)
,
and the damage constraint function (5) is postulated in the form }=1exp[(E~~
I 1
(8)
The material parameters sco and Pco are arbitrarily chosen s m = 1.05E3 ,
(9)
Pm= 6 . 0 .
The principal stress components at far field are the overburden (Yz and the horizontal stresses (Yxand (yy ~z =  6 . 0 M P a ,
~x =  2 . 0 M P a ,
(10)
~y = 6.0 M P a ,
and the well face is assumed to be a free boundary. The equilibrium equations were solved by a finite element code and the resulting damage distribution is described in Figure 1., where the zones of high damage intensity represent the breakouts.
ISSN 01489062
To cite this paper: Int. J. R o c k M e c h . & Min. Sci. 34:34, p a p e r N o . 072.
C o p y r i g h t © 1997 E l s e v i e r S c i e n c e Ltd
FIGURES Paper 072, Figure 1.
5f i
•.
1 5
.'
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~ " , \ ; < , ' X ?~'~ ", "~'"''. "~2~\\ "~. ':" ". ", ...X"" ', x.
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'~':,~.' •,?..?,....v°, '". ., ' •.,.,.,.,.'~ v '(~ %~ .
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1
...
 "L . ",, "" ' .,
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i
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.5
1. Damage concentration at the breakout zone.
References
References Elata D. 1997. A model of Nonlinear poroelasticity with application to brittle rock. (work in progress). Cheatham J. B. 1993. A new hypothesis to explain stability ofborehole breakouts. International Journal o f Rock Mechanic's and Mining Sciences, 30, 10951101. Rubin M. B., Elata D., Attia A. V. 1996. Modeling added compressibility of porosity and the thermomechanical response of wet porous rock with application to Mt. Helen tuff. Int. J, Solids' Structures, 33, 761793. Winkler K. W., Liu X. 1996. Measurements of thirdorder elastic constants in rocks', J. Acous. Soc. Am., 100, 3, 13921398.
ISSN 01489062