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The effect of near-wellbore yield on elastic wave velocities in sandstones Colin M. Sayers*, José Adachi, Schlumberger, Arash Dahi Taleghani, The University of Texas at Austin Summary During oilfield drilling operations, the rock originally in the volume occupied by the wellbore is replaced by drilling fluid that exerts pressure on the borehole wall. This leads to a redistribution of stress in the vicinity of the wellbore, and may lead to yield of rock close to the borehole. This results in a decrease in stress near the borehole wall. This redistribution in stress is studied in this paper for a vertical well using a computational model that accounts for rock deformation and plastic strain in the near-wellbore region. The stress changes around the borehole lead to changes in elastic wave velocities that may be used to monitor the changes in stress that occur. The change in elastic wave velocities are sensitive to the mechanical properties of the rock, and may therefore be used to calibrate mechanical earth models used to predict rock failure due to production.
In figure 1 the maximum horizontal stress SH is assumed to act in the x1 direction, while the minimum horizontal stress Sh acts in the x2 direction. It is seen that the radial and hoop stress components vary strongly as a function of radius, r, and azimuth φ measured from the x1 direction. The hoop stress is seen to be strongly compressive at azimuth
φ = 90o , corresponding to the azimuth of the far-field minimum horizontal stress. For wells drilled in sandstone, the changes in stress shown are expected to lead to changes in elastic wave velocities due to the presence of stresssensitive grain boundaries within the rock. However the stress changes may also lead to yield of the rock, and the purpose of this paper is to investigate the effect of this on the wave velocities in the vicinity of the borehole.
Introduction The production of hydrocarbons leads to changes in rock stress that may result in damage or failure of the rock. Knowledge of the geomechanical properties of the reservoir is necessary, therefore, to optimize production. Current methods for determining rock strength include laboratory measurements of geomechanical properties, and the use of correlations between properties derived from logs such as dynamic elastic moduli, porosity, clay content, etc., and rock mechanical properties such as unconfined compressive strength, friction angle, etc. However, the drilling of a well can be considered as a rock mechanical test, since the stress acting in the vicinity of the wellbore is perturbed by the replacement of the rock originally in the volume occupied by the borehole by the drilling fluid, which exerts a pressure on the borehole wall. This redistribution in stresses is illustrated in figure 1, which shows the variation in effective radial stress σ rr = S rr − Pp and effective hoop stress σ φφ = Sφφ − Pp for a vertical borehole for an elastic calculation (i.e. assuming no rock yield) assuming plane strain (Jaeger and Cook, 1979). The case in which the pore pressure Pp, the wellbore pressure Pw, the vertical stress SV, the maximum horizontal stress SH, and the minimum horizontal stress Sh take the values given in Table 1 is assumed. Table 1: Stress, pore pressure and wellbore pressure for the examples in this paper. 1 MPa is equal to 145.0378 psi. SV 100 MPa
SH 90 MPa
Sh 80 MPa
Pp 60 MPa
Pw 60 MPa
Figure 1. Variation in effective radial stress (top) and effective hoop stress (bottom) in the vicinity of a borehole (dark blue) of radius a plotted for the stress state in Table 1 and a Poisson’s ratio of 0.15. The color scale is in MPa.
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Effects of rock yield on wave velocities
Rock yield The redistribution in stress shown in figure 1 may cause the rock to yield. Yield of the rock leads to further changes in the velocity of compressional and shear waves in the vicinity of the wellbore that vary with position relative to the borehole. Recent developments in sonic logging allow such changes in wave velocity to be monitored by making it possible to map the 3D variation in elastic wave velocities around the borehole (Pistre et al., 2005). Determination of the variation in velocity as a function of azimuth and radial distance from the wellbore therefore has the potential of allowing the strength characteristics of the formation to be determined. In this paper, the effect of the stress redistribution resulting from rock yield on the velocity of vertically propagating elastic waves is investigated. To allow for the effects of rock yield, the rock is assumed to be linearly elastic, perfectly plastic, with a failure surface defined by the Mohr-Coulomb criterion with cohesion S0, friction angle θ, and dilation angle ψ. For the purpose of illustration, the case of a clean gas sand with a porosity ϕ = 0.2 and clay content C=0 is considered with P- and Swave velocity assumed to be given by the relations: v P (km/s) = 5.41 − 6.35ϕ − 2.87C
(1)
v S (km/s) = 3.57 − 4.57ϕ − 1.83C
(2)
Computations were performed using FLAC, a Lagrangian finite difference model that can incorporate large deformations (Cundall and Board, 1988). Figure 2 shows the variation in radial, hoop and vertical effective stress as a function of radius r from the borehole divided by the borehole radius a for the stress state given in Table 1. Results for two azimuths, φ = 0o corresponding to the azimuth of the far-field maximum horizontal stress, and φ = 90o corresponding to the azimuth of the far-field minimum horizontal stress are shown for two cases: (i) an elastic calculation (i.e. assuming no rock yield) and (ii) an elastoplastic calculation assuming a cohesion of 10 MPa, a friction angle of 30 degrees, and a dilation angle of 15 degrees. The corresponding value of the unconfined compressive strength C0, given by C0 = 2 S0 tan γ where γ = 45o + θ / 2 , is then 34.64 MPa. It is seen in Figure 2 that the redistribution of stress predicted by the elastic calculation causes the rock to yield close to the borehole. As a result, the stress in the yielded zone close to the borehole is reduced. However, the radius of the yielded zone is fairly small for this case.
obtained by Han (1986) for dry sandstones. These relations give a dynamic Poisson ratio of 0.15 for ϕ = 0.2 and C=0. The static elastic moduli were obtained by assuming that the static Poisson’s ratio is equal to the dynamic Poisson’s ratio, and that the static Young’s modulus is given by the correlation of Wang (2000) for soft rocks: Estatic = 0.4145Edynamic − 1.0593
(3)
with Estatic and Edynamic in units of GPa, where the dynamic Young’s modulus is obtained from the compressional and shear wave velocities using the relation: Edynamic = 9 K µ /(3K + µ ) .
(4)
Here the dynamic bulk modulus K and shear modulus µ are related to the compressional-wave (P-wave) velocity vP and shear-wave (S-wave) velocity vS by: vP =
( K + 4 µ / 3) / ρ ,
vS = µ / ρ ,
where ρ is density.
(5)
Figure 2. Stress distribution in the vicinity of a borehole of radius 0.1 m for the stress state in Table 1, calculated by assuming (i) an elastic calculation (dashed curves) and (ii) an elastoplastic calculation using a cohesion of 10 MPa, a friction angle of 30 degrees, and a dilation angle of 15 degrees (full curves). For comparison, the results for the same friction angle and dilation angle as in figure 2, but for a cohesion of 5 MPa and 2 MPa are shown in figure 3. The radius of the yielded zone is seen to increase with decrease in cohesion.
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Effects of rock yield on wave velocities
0 + ∆Sijkl , Sijkl = Sijkl
(6)
0 is the compliance that the rock would have if where Sijkl
the rock grains formed a continuous framework, and ∆Sijkl is the excess compliance due to the presence of the grain boundaries in the rock. The term ∆Sijkl can be written as ∆Sijkl =
1 δ ikα jl + δ ilα jk + δ jkα il + δ jlα ik + β ijkl , 4
(
)
(7)
where, for sandstones, the second-rank tensor αij and the fourth-rank tensor βijkl may be written in the form π / 2 2π
∫ ∫Z
α ij =
T (θ ,φ ) ni n j sin θ dθ dφ
θ =0 φ =0
(8)
π / 2 2π
β ijkl =
Z ∫ ∫ θ φ
N (θ ,φ ) − ZT (θ ,φ ) ni n j nk nl sin θ dθ dφ
=0 =0
where ni represents the ith component of the normal to the grain boundary (Sayers, 2007). In these equations, Z N (θ ,φ )sin θ dθ dφ and ZT (θ , φ )sin θ dθ dφ represent the normal and shear compliance of all grain boundaries with normals in the angular range between θ and θ + dθ and φ and φ + dφ in a reference frame X1, X2, and X3, with axis X3 aligned with the normal to the grain boundary.
Figure 3. Stress distribution in the vicinity of a borehole of radius 0.1 m for the stress state in Table 1, calculated by assuming (i) an elastic calculation (dashed curves) and (ii) an elastoplastic calculation using a friction angle of 30 degrees, and a dilation angle of 15 degrees (full curves) for a cohesion of 5 MPa (top) and 2 MPa (bottom).
Stress-dependence of elastic wave velocities Key to the determination of the mechanical properties from the variation in velocity in the vicinity of the borehole is the ability to calculate wave velocities as a function of stress. Elastic wave velocities in sandstones vary with changes in effective stress due to the presence of stresssensitive grain boundaries within the rock. Sayers and Kachanov (1995) show that the elastic compliance tensor Sijkl of a sandstone may be written in the form
If the normal and shear compliance of the grain boundaries are approximately equal, it follows that the fourth-rank tensor βijkl will be small, and the elastic stiffness tensor will be determined primarily by αij. Inversion of velocity data for αij and βijkl has shown that this is a reasonable approximation for the grain contacts in sandstones (Sayers, 2002). Henceforth, the contribution of βijkl will be neglected. Following Schoenberg (2002), it is assumed that the normal and shear compliance of a grain boundary are functions only of the component of the effective stress σ n = niσ ij n j acting normal to the plane of the boundary, and that the compliance of the grain boundaries decreases exponentially with increasing stress applied normal to the grain boundaries as follows:
Z N = Z T = Z 0 e −σ n / σ c
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(9)
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Effects of rock yield on wave velocities
Here σc is a characteristic stress that determines the rate of decrease in grain boundary compliance with increasing compressive stress acting normal to the boundary. A comparison of the predictions of the theory with the measurements of the stress dependence of the P- and Swave velocities for the 24 room dry Gulf of Mexico sandstone studied by Han (1986) showed good agreement, the average values of Z0 and σc being σ c = 10.13 MPa and
µ Z 0 = 0.2583 , where µ is the shear modulus under high compressive stress. In the following, the values σ c = 10 MPa and µ Z 0 = 0.25 are used, and the multiple integral required to compute αij was evaluated numerically using a 128-point 15th-degree Gauss integration formula (Sayers, 2007). Figure 4 shows the variation in shear wave velocity with normalized radius predicted for the distribution of stress shown in Figures 2 and 3. It is seen that the yielded zone predicted by the elastoplastic theory grows as the unconfined compressive strength decreases, and this causes an increased reduction in the shear wave velocity near the wellbore. This reduction in shear wave velocity may be used to characterize the extent of the yielded zone and so determine the rock strength. The optimum value of the unconfined compressive strength can be obtained by computing the variation in shear-wave velocity for a range of possible values of unconfined compressive strength, C0, friction angle and other parameters related to rock yield and failure, and choosing the vector of parameters that gives the best agreement with the observed shear-wave velocity profile.
Conclusion The stress redistribution that occurs in the vicinity of a borehole may lead to yield of the rock. Recent developments in sonic logging allow the variation in elastic wave velocities with radius and azimuth around a borehole to be characterized. Since elastic wave velocities in sandstones are sensitive to changes in stress, owing to the presence of stress-sensitive grain boundaries within the rock, this enables a determination of the mechanical properties of the rock using the variation of elastic wave velocities in the vicinity of the wellbore. These properties are essential for drilling future wells, and designing the completion based on logs and drilling data together with the acoustic radial profiles obtained from a multi-pole sonic tool. It is expected that the techniques presented will lead to better choice of completion and field development in reservoirs subject to geomechanical problems due to depletion.
Figure 4. Shear-wave velocity calculated using the stress distribution shown in figures 2 and 3 (full curves) for shearwave polarization parallel ( φ = 0o ) and perpendicular ( φ = 90o ) to the maximum horizontal stress. The calculation assumes a friction angle of 30 degrees, a dilation angle of 15 degrees for a cohesion of 10 MPa (top), 5 MPa (middle) and 2 MPa (bottom). The dashed curves show the velocities calculated using the elastic solution.
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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2008 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Cundall, P. A., and M. Board, 1988, A microcomputer program for modeling large-strain plasticity problems: in G. Swoboda, ed., Numerical methods in geomechanics. Han, D. H., 1986, Effects of porosity and clay content on acoustic properties of sandstones and unconsolidated sediments: Ph.D. thesis, Stanford University. Jaeger, J. C., and N. G. W. Cook, 1979, Fundamentals of rock mechanics: Chapman and Hall. Pistre, V. et al., 2005, A new modular sonic tool provides complete acoustic formation characterization: Presented at the 75th Annual International Meeting, SEG. Sayers, C. M., 2002, Stress-dependent elastic anisotropy of sandstones: Geophysical Prospecting, 50, 85–95. ———2007, Effects of borehole stress concentration on elastic wave velocities in sandstones: International Journal of Rock Mechanics and Mining Science, 44, 1045–1052. Sayers, C. M., and M. Kachanov, 1995, Microcrack-induced elastic wave anisotropy of brittle rocks: Journal of Geophysical Research, 100, 4149–4156. Schoenberg, M., 2002, Time-dependent anisotropy induced by pore pressure variation in fractured rock: Journal of Seismic Exploration, 11, 83–105. Wang, Z., 2000, Dynamic versus static elastic properties of reservoir rocks: in Z. Wang and A. Nur, eds., Seismic and acoustic velocities in reservoir rocks, 3: SEG.
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