Research Papers

Early-Age Stress and Pressure Developments in a Wellbore Cement Liner: Application to Eccentric Geometries

[+] Author and Article Information
Thomas Petersen, Franz-Josef Ulm

Department of Civil and
Environmental Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 13, 2016; final manuscript received June 22, 2016; published online July 13, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(9), 091012 (Jul 13, 2016) (11 pages) Paper No: JAM-16-1181; doi: 10.1115/1.4034013 History: Received April 13, 2016; Revised June 22, 2016

This paper introduces a predictive model for the stress and pressure evolutions in a wellbore cement liner at early ages. A pressure state equation is derived that observes the coupling of the elastic changes of the solid matrix, the eigenstress developments in the solid and porespaces, and the mass consumption of water in course of the reaction. Here, the transient constitution of the solid volume necessitates advancing the mechanical state of the poroelastic cement skeleton incrementally and at constant hydration degree. Next, analytic function theory is employed to assess the localization of stresses along the steel–cement (SC) and rock–cement (RC) interfaces by placing the casing eccentrically with respect to the wellbore hole. Though the energy release rate due to complete debonding of either interface is only marginally influenced by the eccentricity, the risk of evolving a microcrack along the thick portion of the sheath is substantially increased. Additionally, it is observed that the risk of microannulus formation is principally affected by the pressure rebound, which is engendered by the slowing reaction rate and amplified for rock boundaries with low permeability.

Copyright © 2016 by ASME
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Fig. 1

Diagram of a wellbore cement liner

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Fig. 2

The cement poroelastic constants in function of thehydration degree (G = 11.1 GPa and N = 174.3 GPa; see Table 2 in Appendix C for the calculation of the volume fractions and the upscaling of the poroelastic constants). The inset shows the evolution of the primary cement phases.

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Fig. 3

Plots depicting the (a) driving forces of the bulk stress developments and (b) the pressure evolution p̂=p/p0 for low and high permeability formations. Herein, the solid eigenstress σ* was assumed to evolve linearly with ξ. As a validation of concept, the black contours in (b) show smoothed, nondimensionalized pressure-log data for a well of undisclosed identity provided by Schlumberger (solid) and the pressure evolution simulated by our model (dashed) for a cement with w/c = 0.45.

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Fig. 4

Bulk effective radial ((a) and (b)) and tangential ((c) and (d)) stress development along the SC interface ((a) and (c)) and the RC interface ((b) and (d)). At complete hydration, a residual Δp remains in the system. Thus, the dashed contours show the asymptotic values of the effective stress, once the pressure has fully recovered to the reference value p = p0.

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Fig. 5

Contour plots of the influence of the shear modulus ratio between rock and cement Gr/Gc and the Newton coefficient λfl on the effective radial stress along SC (left column of panel) and RC (right column) at complete hydration (top row) and once the pressure has completely recovered (bottom row). The casing placement is assumed concentric with the borehole.

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Fig. 6

Diagram of the bilinear transformation that maps the eccentric contours SC and RC in the z-plane to concentric contours in the s-plane

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Fig. 7

(a) The evolution of the energy release rate for a vertically propagating microannulus in a concentric geometry for λfl = 1 × 10−5 s−1; values normalized by Ĝ=GGc∞/2πriΣrr∞2, where i = 1 along the SC and i = 2 along RC. The dashed contours show the energy release rate once the pressure has fully recovered. (b) The normalized energy release rate at t in function of the rock-to-cement shear modulus ratio.

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Fig. 8

The top panels show the distribution of work done along (a) SC and (b) RC to produce a microannulus along the respective interface for different degrees of casing eccentricity. The bottom panels display the energy release rate for a microcrack Gi, i.e., the risk of crack initiation, along (c) SC and (d) RC. dG¯/dθ and G¯i have been normalized with respect to the uniform values in a concentric geometry.

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Fig. 9

The energy release rate for a microcrack along SC and RC G¯i(θ)=Giecc(θ)/Gicent in function of different Gr/Gc




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