Wellbore stability analysis in porous carbonate rocks using cap models

Wellbore stability analysis in porous carbonate rocks using cap models L. C. Coelho 1, A. C. Soares 2, N. F. F. Ebecken 1, J. L. D. Alves 1 & L. Landau 1 1 COPPE/Federal University of Rio de Janeiro, Brazil 2 Petrobras Research Center, Brazil Abstract It is well established that hydrocarbon wellbores fail under tension mode or shear compressive mode. Although there are sophisticated constitutive models that may represent the behavior of the rock, the lack of data available lead to the use of simple models such as elastic models with a failure criteria, and the Drucker- Prager or Mohr-Coulomb elastic-plastic models with associated or nonassociated flow rules. Recently, many studies have been published on the behavior of porous rocks. Mechanisms of pure compression failure (no shear) have been described. Cap plasticity models may represent the material pure compression failure. An experimental program and numerical simulations on porous carbonate rock from Campos Basin, offshore Brazil, carried out investigation on this phenomenon. The experimental program consisted on defining the hydrostatic strength of the rock through laboratory tests for stress paths that varied from hydrostatic to uniaxial. All tests followed paths below the shear line. Data from these tests was fitted to a cap plasticity model. Parameters obtained were used on stability analysis of a horizontal borehole drilled in the same field. A comparative study of the Mohr-Coulomb shear model and the cap model was carried out. The Mohr-Coulomb shear model pointed to stability of the borehole in the compressive mode, while the cap model indicated damage to the rock in the vicinity of the well. Keywords: compression failure, cap models, wellbore stability analysis.

266 Damage and Fracture Mechanics VIII 1 Introduction Efforts introduced in the rock caused by drilling and production may reach the strength of the material, leading to its failure. Many constitutive models were proposed to evaluate wellbore stability, with linear or non-linear failure criteria [1]. Although there are sophisticated constitutive models that may represent the behavior of the rock, lack of data available lead to the use of simple models such as elastic models with a failure criteria, and Drucker-Prager or Mohr-Coulomb elastic-plastic models with associated or non-associated flow rules. It is well established that hydrocarbon wellbores fail under tension mode or shear compressive mode [2]. During drilling, tension failure occurs when the fluid pressure on the borehole wall is large such that tensile strength is overcome. This causes hydraulic fracturing (figure 1a). Compressive failure occurs when the fluid pressure magnitude is not enough to avoid the development of deviator stresses higher than the shear strength of the rock. Under this condition, breakouts (figure 1b) may occur in brittle formations. In rocks with plastic behavior, yielding may lead to hole tightening. During production, a too low wellbore pressure may induce shear failure. High production rates may trigger tension failure [2]. Regard to wellbore compressive mode of failure, Haimson et al. [3], [4] have studied the compressive behavior for a number of boreholes drilled in blocks of rock with different porosity levels subjected to a true far-field triaxial stress. Dog-ear breakouts (figure 1b) were observed in granite, limestone and low porosity sandstone along the minimum in situ stress σ h direction. High porosity sandstone has shown an entirely different pattern: breakouts were thin, tabular and very long, extending in the direction perpendicular to the maximum principal horizontal stress σ H (figure 1c). A narrow zone ahead of the breakout tip showed lower porosity and compact grains. The mechanism of this breakout consists of grain debonding and crushing caused by compressive stress concentration. The circulating fluid carries out the loose grains. Further stress concentrations are generated in the adjacent region, consequently more grain is debonded, and the length of the fracture is increased. Haimson suggests that these types of breakout are empty compaction bands. Prediction of this failure mode depends on defining the hydrostatic strength of porous rocks. Most of data available on compacting behavior of rocks are based on triaxial or hydrostatic tests. Soares [5] has done an experimental study to define hydrostatic strength in a porous carbonate rock from a reservoir in Campos Basin, offshore Brazil. In this work, the hydrostatic strength was defined through proportional loading tests (constant K = σ 3 σ 1 ) for stress paths below shear line, associated to hydrostatic and uniaxial tests, in order to assure that material would not fail under shear. The stress paths varied from hydrostatic (K=1) to uniaxial (K=0.3). These paths are closer to the path followed by a reservoir under production.

Damage and Fracture Mechanics VIII 267 σ H Compaction bands σ h Breakout a) Hydraulic fracturing b) Shear failure a) Compaction bands Figure 1: Wellbore failure modes. σ 3 Von Mises Drucker-Prager Mohr-Coulomb Triaxial Extension Tests σ 1 - b + σ 2 Rigid Plastic A Plane Strain Pure Shear line (p=0) Triaxial Compression Tests Cap model in the I 1 x J 2 plane Failure surface on principal stresses plane Figure 2: Cap model. In this study, the cap model was implemented in an in-house elastic-plastic finite element code for wellbore stability analysis. The failure surface parameters were fitted to data available from Soares [5]. A horizontal wellbore drilled in this field was used to study the near borehole stability. The Mohr-Coulomb shear model was compared to the cap model. Stability during drilling and production was evaluated. As the hydrostatic effective stress increased due to fluid withdrawal, compressive strength was reached during production, leading to compaction failure. The most usual methods for wellbore mechanical stability analysis do not take this kind of failure into account. 2 Cap model Cap model yield surface consists of a shear-type failure cone limited in the direction of hydrostatic stress by an elliptical cap for volumetric compaction, as shown in figure 2.

268 Damage and Fracture Mechanics VIII Schwer & Murray [6] proposed a generalized complete failure surface for cap model. Its functional form is given by: f g = H ( I1, J 2, β ) = J 2 R( β ) h( I1 ) = 0, I1 < L( κ ) ( I1, J 2, β, κ ) = J 2 R( β ) F( I1, κ ), I1 L( κ ) (1) where h( I 1 ) is the Drucker-Prager perfect plastic failure surface, taken as the circumscribed circle fitted to compressive apex of Mohr-Coulomb hexagon, given by: h ( I1 ) k αi1 Parameters k and α are given by: = (2) α = k = 3c cosφ 3 3 ( 3 senφ ) 2senφ ( 3 senφ ) (3) (4) R ( β ) is a function that fits the Drucker-Prager function ( ) h I 1 to the meridians of classical Mohr-Coulomb surface (figure 2). It is defined by: R ( β ) 3 senφ 3 3 + senφ = 3 senφ 3 senφ 1+ cos β 3 1 senβ 3 + senφ 3 + senφ (5) The elliptical cap, is given by: H ( I1, J 2, β, κ ) J 2 R( β ) F( I1, κ ) = (6) where κ is a hardening parameter. This function F ( I 1,κ ) for cap model has the form proposed by Sandler et al [7]: F 1 2 { 1 } 2 2 ( I, κ ) [ X ( κ ) L( κ )] [ I L( κ )] 1 1 = (7) R

Damage and Fracture Mechanics VIII 269 R is a shape factor, given by the ratio between major axis and minor axis of I axis and the ellipse, X ( κ ) is the value of I 1 at the intersection of the cap with 1 L ( κ ) is the value of I 1 at the intersection of the cap with shear line ( ) h I1. It is assumed that material is subjected to strain hardening during compaction. The hardening law is a function of plastic strains and is determined through experimental data. Hardening law was assumed as the linear relation, obtained from the tests: p kk ( ) X ) ε = Aw *( X κ o (8) p where ε kk is the volumetric plastic strain and X o is the initial position of the cap. Hardening parameter is considered to be: ( κ ), if L( κ ) L( κ o ) L( κ ), otherwise L > κ = (9) o The hardening law is related to the hardening parameter through the relation that couples the shear surface to the ellipse arc: X 3 Borehole modeling ( κ ) L( κ ) + R h( L( κ )) = (10) Borehole stability model assumes plane strain, isotropic material, linear elasticity, rate-independent plasticity and associative flow. The principle of effective stresses and poroplasticity are considered valid. During drilling, the initial stress state is altered by the borehole and the fluid pressure inside it. The first effect is represented in the finite element analysis by superposition of two loading applied on borehole wall. The first load is the differential pressure between drilling fluid pressure and reservoir pressure. Second load is a tension with magnitude equivalent to initial stresses within the rock. During fluid production, differential pressure between the fluid in the borehole and the formation fluid pressure lead to a pore pressure field, which is a function of the distance from borehole axis and time dependent. The constitutive equation that relates the effective stress and rock matrix is considered to be independent of pore pressure field. Incremental stress vs. strain relation is given by: [ ε ij ( kk ) ] σ (11) ij = Cijkl ε whereσ ij is the effective stress rate, ε ij is the total strain rate, ( ε kk ) p is the volumetric compression caused by fluid withdrawal and C ijkl is the constitutive tensor. So, during production, the stress state is altered due to the effect of the p

270 Damage and Fracture Mechanics VIII kk ) p fluid pressure on the borehole associated to the volumetric compression ( ε, which is given by: where ε kk p = δ ij p p ks (12) k is given in terms of ( ) p p is pore pressure rate and the compressibility s the Young modulus E, Poisson coefficient ν and Biot s coefficient ρ : k s = ( 1+ν)( 1 ρ ) E (13) 200 150 Legend Stress Path- 20% Stress Path - 24% Stress Path - 27% Stress Path - 31% Pore Collapse Stress - 20% Pore Collapse Stress - 24% Pore Collapse Stress - 27% Pore Collapse Stress - 31% Shear Envelope (20%porosity) Shear Failure Stress Shear Stress - q (MPa) 100 Original In-Situ Stress 20% 50 24% 20% 31% 27% 4 Case study 31% 27 % 24% 0 0 20 40 60 80 100 120 140 Hydrostatic Pressure - p (MPa) Figure 3: Curve fitting for the 30% porosity tests. A horizontal wellbore drilled in limestone reservoir rock from Campos Basin, offshore Brazil, was studied. Ductile heterogeneous porous limestone (20% to 35%) composed this reservoir. Samples obtained from this material were tested for stress paths varying from hydrostatic compression to uniaxial strain, all below shear failure line. Among these paths, proportional loading tests in conventional triaxial cell were done to evaluate hydrostatic strength. Details on these tests may be found in Soares [5], Soares & Ferreira [8] and Coelho [9]. Figure 3 shows the hydrostatic strength obtained from the tests grouped by porosity. It may be seen that this strength is a function of porosity: the higher the porosity, lower the strength. Data from 30% medium porosity was used to fit the tests. The average parameters obtained (table 1) were used to a borehole stability analysis of an 8.5 open-hole horizontal wellbore drilled at production region. It was assumed that the formation around the borehole was homogeneous and

Damage and Fracture Mechanics VIII 271 presents isotropic behavior. It was assumed that initial stress state was the same around the interest region. The external boundary was defined at 20 meters from borehole axis, assuming that at that point the stress state is not affected by drilling and production operations. Young Modulus (MPa) ν Table 1: Material parameters. cohesion (MPa) friction angle shape factor R X o (MPa) A w (MPa) -1 2100 0.1 8,5 42 o 2.0 108 4 x10-6 It was assumed isotropic stress state in the horizontal plane. The initial effective horizontal stresses σ h were 9 MPa, and the effective initial vertical stress σ v was 32.1 MPa. The fluid pressure in the formation was 32.4 MPa and fluid pressure at the borehole wall for drilling was 34 MPa. Conventional triaxial tests obtained the shear strength parameters. Simulation of drilling and production phases was done incrementally. First increment corresponds to the drilling phase. It assumes that the well is impermeable and evaluates the changes on stresses around the well caused by drilling together with the application of fluid pressure. The second increment considers radial steady state fluid flow through the well, assuming a 200 meters drainage ratio. In lack of further data about transient pore pressure around the borehole, transient effects were neglected. Pore pressure profile was given by: pw pe re P() r = ln (14) re r ln rw where pw is the fluid pressure on borehole wall, p e is the reservoir pressure, r e is the drainage radius, r is the distance from borehole axis to the point and P() r is the differential pressure, function of position. At the borehole wall the differential pressure p is 14.9 MPa. Results are presented in terms of plastic ratio maps and comparative graphics representing radial and tangential stresses as function of the distance from r = J R β h for shear failure or borehole axis. Plastic ratio is described by 2 ( ) ( I 1 ) r J R( β ) F(, κ ) = for compacting failure. It represents how far is the actual 2 I 1 stress state from the failure surface. Its value ranges from 0 to 1.0 (damaged). Figure 4a and 4b show the damaged regions for Mohr-Coulomb shear model analysis. No plastic damage occurred during drilling phase (fig 4a). During production (fig 4b), the borehole vicinities became more stable. Figure 4c and 4d show the Mohr-Coulomb cap model. It may be seen that very close to the borehole, plastic damage occurred during drilling in the direction perpendicular to maximum horizontal stress. Damaged region increased during production due

272 Damage and Fracture Mechanics VIII to the increase of the effective stress during fluid withdrawal. Damaged region corroborates Haimson s study: compaction failure occurred in the region perpendicular to the highest principal horizontal stress. (a) Plastic ratio during drilling Mohr-Coulomb shear model (b) Plastic ratio during production Mohr-Coulomb shear model (c) Plastic ratio during drilling Mohr-Coulomb cap model (d) Plastic ratio during production Mohr-Coulomb cap model Figure 4: Plastic ratio around the borehole. Figure 5 presents circumferential and radial stresses, respectively, along the direction of σ h during drilling (where damage occurs for the cap model) for both models, while figure 6 presents the stresses around the borehole for the production phase, also comparing both models. These figures show that the stress state was little affected by compaction due to the hardening effect. However, the numerical analysis pointed that material is damaged, and this damage possibly includes debonding, grain crushing and pore collapse, leading to a reduction in permeability and porosity.

Damage and Fracture Mechanics VIII 273 100 80 Drilling Phase Stresses along σ h direction Circumferential Stress - Mohr-Coulomb Shear Model Circumferential Stress - Mohr-Coulomb Cap Model Radial stress - Mohr-Coulomb shear model Circumferential stress - Mohr-Coulomb cap model Stresses due to drilling (MPa) 60 40 20 0 0 0.1 0.2 0.3 0.4 0.5-20 Distance from Borehole Axis (meters) Figure 5: Stress state in σ h direction during drilling. 120 Production Phase Stresses along σ h direction Circumferential stress - Mohr-Coulomb shear model Circumferential stress Mohr-Coulomb cap model Radial stress - Mohr-Coulomb shear model Radial stress - Mohr-Coulomb cap model 80 Circumferential Stresses (MPa) 40 0 0 0.1 0.2 0.3 0.4 0.5-40 Distance from Borehole Axis (meters) 5 Conclusions Figure 6: Stress state in σ h direction during production. Porous rock may fail under a compaction mode under confining pressures that overcome its hydrostatic strength. In this study, data from porous limestone was fitted to the cap model for stress paths below shear line. Stability analysis of a wellbore drilled in the same reservoir was done. Mohr-Coulomb shear model presented no failure. During production, analysis with these models showed stability around the borehole because they followed stress paths below shear line.

274 Damage and Fracture Mechanics VIII Mohr-Coulomb cap model presented failure caused by compaction in drilling phase and an increase in the compacted area during production. This plastic failure pointed by the cap models is caused by the decrease in effective stresses during fluid withdrawal. The final stress state is little affected by the compaction due to hardening effects. However, there is a permanent damage in the material. This damage is associated to pore collapse, with grain crushing and debonding, leading to porosity and permeability reduction. Around boreholes, massive sand production may occur in open hole completions. If obstacles to particle flux are present, compacted regions may act as barriers to fluid flow. Results presented herein show that wellbore stability models must be reviewed for drilling and production in high porosity rocks. Compaction behavior of these materials needs to be taken into account for prediction of borehole stability and sand production. Further studies on this failure mechanism and on the compressive strength characterisation of reservoir rocks is recommended. References [1] McLean, M.R. and Addis, M.A., Wellbore Stability Analysis: a Review of Current Methods of Analysis and Their Field Applications. IADC/SPE Drilling Conference, Houston, Texas, SPE paper 19941, 1990. [2] Fjaer, E., Holt, R.M., Horsrud, P., Raaen and A.M., Risnes, R., Petroleum Related Rock Mechanics. Elsevier Science, Amsterdam, The Netherlands, 1992. [3] Haimson, B., Fracture-Like Borehole Breakouts in High-Porosity Sandstone: Are They Caused by Compaction Bands?, Physics and Chemistry of Earth (A) 26, 15-20, 2001. [4] Haimson, B.C. and Song, I., Borehole Breakouts in Berea Sandstone: Two Porosity-Dependent Distinct Shapes and Mechanisms of Formation, 1996. SPE/ISRM Eurorock 96, Trondheim, Norway, SPE paper 47249,1996. [5] Soares, A.C., Um Estudo Experimental para Definição de Colapso de Poros em Rochas Carbonáticas. MS. Thesis, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil (in Portuguese, with English Abstract), 2000. [6] Schwer, L. and Murray, Y.D., A Three-Invariant Smooth Cap Model with Mixed Hardening, International Journal for Numerical and Analytical Methods in Geomechanics, 18, 657-688, 1994. [7] Sandler, I.S., DiMaggio, F.L., Baladi, G.Y., Generalized Cap Model for Geological Materials. Journal of the Geotechnical Engineering Division, ASCE, 102 (7), 683-699, 1976. [8] Soares, A.C. and Ferreira, F.H., An Experimental Study for Mechanical Formation Damage. 2002 SPE International Symposium and Exhibition on Formation Damage Control, Lafayette, Louisiana, SPE paper 73734, 2002. [9] Coelho, L.C., Modelos de Ruptura de Poços de Petróleo. D. Sc. Dissertation, COPPE, Federal University of Rio de Janeiro, Brazil (in Portuguese, with English Abstract), 2001.