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Geomechanical Modeling of In-Situ Stresses Around a Borehole

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Geomechanical Modeling of In-Situ Stresses Around a Borehole Geomechanical Modeling of In-Situ Stresses Around a Borehole Samantha Grandi, Rama Rao, and M. Nafi Toks¨oz Earth Resources Laboratory Dept. of Earth, Atmospheric, and Planetary Sciences Massachusetts Institute of Technology Cambridge, MA 02139 Abstract In this paper, we present a modelling of the in-situ stress state associated with the severe hole enlargement of a wellbore. Geomechanical information is relevant to assure wellbore stability, i.e., to prevent damages in the formation and later on, the casing. Many of the drilling parameters, as mud weight or the optimal orientation of the borehole, require some knowledge of the mechanical behaviour of the rock. The lack of these kind of data in exploratory areas, where there are usually insufficient constraints for the geological model, increases even more the risk, hence the costs. The present model uses the concepts of poroelasticity theory to compute the stationary 2D, brittle response of the formation around a borehole that is submitted to effective compressive horizontal stresses. The numerical solution is obtained using a finite element approximation. The initial stress state at the far field was estimated combining a frictional-failure theory with the observations of dipmeter caliper in a particular borehole that presents elongations in a preferential di- rection. The direction and relative extension of the observed breakouts at a particular depth are modelled successfully using formation realistic parameters and dimensions, although the exact shape of the borehole (at all angles) was unknown. For the particular case study, the orientation of the breakout is NE-SW, at about 82 degrees azimuth. Therefore, the maximum horizontal stress lies at approximately 350 degrees azimuth. The ratios of horizontal principal stresses to vertical stress that best honor the observations are SHmax = 2.3Sv and Shmin = 1.7Sv. The compressive strength necessary for the rock to fail, as indicated by the caliper data under this stress field, is about 140 MPa. 1 Introduction In this paper, we present a numerical modelling of the geomechanical conditions of a wellbore where an heterogeneous stress field is responsible for numerous breakouts. The governing equations are the equations of motion (Navier’s equations) for perfectly elastic materials, therefore, only a brittle behavior is assumed and no plasticity is included. A numerical solution computed using the finite element method is presented below. The modelling carried out is a mechanical one and thermal stresses are disregarded. The displacements are assumed invariables with time. To have the ability of predicting the geomechanical behaviour of the column rock that is to be drillled is a dream-like scenario for the oil industry. In exploration, it is specially relevant, because the uncertainties of the area difficult the selection of drilling parameters as mud weigth, liners depths and so forth. The issue of wellbore stability has gained particular importance in the last 10 years as a response to the increasing of exploration of complex areas which represent major engineering challenges in drilling and production (Charlez and Onaisi, 2001). Severe hole enlargement has a long list of consequences, from reducing the drill bit life to causing cleaning hole problems, stuck pipe, poor cementing, logging problems, and often the need to sidetrack (Last, 2001). There are several problems in engineering to which wellbore stability refers. The one addressed in this project, is related to drilling performance where instability is mainly driven by stress concentrations around the cavity. 1 During drawdown, depletion of fluid could alter pressures significantly. Sometimes, and very much associated with the kind of lithology supporting the reservoir, the casing could suffer important damage and even the total loss of the wellbore (Frederick et al., 1998). A third problem considered as wellbore instability, is sanding production in the completion stage. During drilling, there are other sources which can also contribute to the development of stability problems. In particular, the orientation of the well has great relevance. Selecting a drilling orientation that does not agree with the natural weak orientation of the formation could introduce difficulties that translate in time and money. On the other hand, there are some laboratory experiments that indicate the damaging effect of some mud components reactions (Charlez and Onaisi, 2001), however, these could only be responsible for washouts of uniform-radius occurrence and not for breakout areas with preferential direction. There are well established methods to derive the in-situ stress state in a borehole. Among them, breakout analysis, hydraulic fracturing experiments and earthquake focal plane mechanisms, are of wide used (Zoback, 1989). Barton and Zoback (1988) described another stress measurement technique based on stress-induced polarization of tube waves (Stoneley). Sinha et al. (1994) introduced a method that uses the flexural mode recorded in cross-dipole data to distinguish induced stresses in the borehole. Finally, the resolution improvement of borehole imaging logs, have made the in-situ stress derivation another application of this tool, overcoming the limitation of four-arms caliper data to provide all-azimuth measurements (Brudy and Kjorholt, 2000). Each method has its own limitations. In all of these techniques (with the exception of focal mechanisms), the stress state inverted includes the effects of any drill-induced stress in the wellbore superimposed to the tectonic field. If the formation is anisotropic, the stress anisotropy would amplify or alter by the presence of the bore (Zoback, 1989), being these effects sometimes larger than the in-situ stresses existing before drilling the well. Knowledge of the ambient stress field is important in elucidating tectonic processes. To understand the behaviour of a wellbore in a tectonically pre-stressed region, forward modelling is a necessary procedure. In this project we are willing to quantify the tectonic stress effect in the formation around a well that does not present any significant anisotropy, at least at the depth of study. However, the heterogeneties of the far field stresses are responsible for important geomechanical damages. The paper is structured in the following way: first, a background theory is presented to facilitate the description of the numerical modelling. Second, the problem is described along with the derivation of the model parameters. Finally, the results are presented and discussed. In brief, the formation parameters are computed from the well data available and the initial range of values of the principal stresses is obtained by a standard analysis of breakouts inferred from the dipmeter caliper data. 2 Theoretical Background 2.1 Structural Mechanics When a layer of rock, buried at certain depth, is submitted to a stress field, the displacements and displace- ment gradients are sufficiently small for the theory of linear elasticity to be valid. The equations of motion, or equilibrium, can be considered to be satisfied in the undeformed reference configuration. We chose the Lagrangian description of motion to express the global conservation laws. Because of the fixed material viewpoint, the conservation of mass is automatically satisfied. Using conservation of momentum, the Cauchy’s equations of motion are obtained (Malvern, 1969). In vectorial notation, ∂u ∇T + f = ρ , (1) ∂t where u is the velocity, ρ is the density of the material, f represents the body forces per unit volume and T is the stress tensor defined as: σx τxy τzx T = τxy σy τyz (2) τzx τyz σz σi represent the normal stresses and τij the shear stresses. 2 ∂u In the special case of static equilibrium, the acceleration ∂t is zero and equation (1) reduces to: ∇T + f = 0 (3) In general, Tij = Tji, and there are only six independent unknown stress components instead of nine. However, the 3 equations (1) are still not enough to solve for the stresses, and constitutive equations have to be used. The stress-strain relations describe the ideal macroscopic behavior of the material under consideration. For an ideal elastic solid, the constitutive equations are known as Hooke’s law : T = C, (4) where C is a fourth-rank tensor with 36 independent coefficients (in the most general case), and is the strain tensor x γxy γzx = γxy y γyz (5) γzx γyz z By definition, the strains are related to the displacements,u ¯ = (u,v,w), as: γ 1 ∂u ∂v = ∂u ; = xy = + x ∂x xy 2 2 ∂y ∂x γ 1 ∂v ∂w = ∂v ; = yz = + (6) y ∂y yz 2 2 ∂z ∂y γ 1 ∂w ∂u = ∂w ; = zx = + z ∂z zx 2 2 ∂x ∂z For an elastically isotropic material, the coefficients of C can be expressed in terms of only two elastic modulii. Usually, these are either the Lame constants, λ and µ, the Young’s modulus (E) and Poisson ratio ν, or the bulk modulus K and the shear modulus (G). The bulk modulus is a measure of the fractional volume change when the pressure is increased by δP . It is obtained through isotropic compression experiments. Most frequently, uniaxial compression is simpler to perform, in which case, it is the Young’s modulus that it is measured. By definition, σ E = zz (7) zz On the other hand, the deformations that take place in the transversal direction are described by the Poisson ratio: ν = − yy (8) zz (Gueguen, 1994). Using the Young’s modulus (E), and the Poisson’s ration (ν), Hooke’s law reduces to: E ν σx = 1+ν x + 1−2ν (x + y + z) ; τxy = Gγxy E ν σy = 1+ν y + 1−2ν (x + y + z) ; τyz = Gγyz (9) E ν σz = 1+ν z + 1−2ν (x + y + z) ; τzx = Gγzx where G is defined as: E G = (10) 2(1 + ν) Finally, substituting equation (6) in (9) and equation (9) in (3), Navier’s displacement equations of motion are: −∇C∇u = f, (11) 3 where, 2G + µ 0 0 0 µ 0 0 0 µ 0 G 0 G 0 0 0 0 0 0 0 G 0 0 0 G 0 0 0 G 0 G 0 0 0 0 0 C = µ 0 0 0 2G + µ 0 0 0 µ (12) 0 0 0 0 0 G 0 G 0 0 0 G 0 0 0 G 0 0 0 0 0 0 0 G 0 G 0 µ 0 0 0 µ 0 0 0 2G + µ Eν and µ = (1+ν)(1−2ν) Once the derivatives of the displacements have been obtained, all strain and stress components can be derived.
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