Multi-String Casing Design with Wellhead Movement
Authors: Halal A.S. , Mitchell R.F. , Wagner R.R. 
Published: December 2007

Abstract
Multi-string axial interactions at points of joined tubular movement such as the wellhead are necessary for accurate determination of axial loads and displacements. In conventional single-string stress analysis, a zero-displacement boundary condition is usually assumed at end points. In the actual well completion, these boundary conditions are not fixed, but instead represent connections between multiple strings or with the formation, which are all elastic to some degree. In addition, single-string predictions of burst and collapse loads generated by thermal expansion or contraction of trapped annular fluids can be exceedingly conservative. This is so because the radial interaction across uncemented intervals and the composite elasticity of cemented sections are not considered. This paper presents the required theoretical framework for addressing the multi-string problem, together with an efficient solution technique for solving the governing non-linear equations. Application of the technique to sample wells shows that while conventional design may be adequate for casing strings landed in tension, it could result in substantial error for free-standing structural casings.

Introduction
Multi-string stress analysis can be classified as axial, radial, or coupled axial-radial according to the well completion type.
In onshore and platform wells, the readily accessible wellhead allows all annuli to be monitored and bled-off to relieve any annular heat-up pressures. However, the wellhead, downhole hangers, and packers represent points of joined tubular movement and the conventional single-string assumption of fixed ends may be inadequate for determining axial loads and displacements. This problem is considered a multi-string axial one where the radial interaction is not of significant concern.
However, this is not the case with subsea wells where limited access to the wellhead requires careful consideration of annular heat-up and cool down pressures dining casing design. Subsea high-pressure, high-temperature wells can experience significant casing heat-up, not only during production, but also during testing and even during drilling. Tubulars will elongate and trapped annular fluids will expand causing severe loads that must be considered for casing design. Burst and collapse are not the only concern. Casing axial loads due to constrained thermal elongation together with "reverse ballooning" from high annular pressures can generate sufficient compression to relieve all hanging weight and cause upward forces at the mudline hanger. For simple subsea completions with uniform casing weights, good conductor cement bond, and packer set in a cemented interval, the problem may be considered a multi-string radial one. This is so because the effect of the pressure induced change in axial stress (through Poisson's ratio) on the armulus volume change is small compared to that resulting from the corresponding hoop strain. However, for most practical applications, the interaction between axial stress and annular pressures may be significant which requires a simultaneous solution to the coupled axial and radial problems.
The methodology presented in this paper addresses the multi-string problem by decomposing it into two elastic problems:the radial stresses and displacements, andthe axial stresses and displacements.
These are solved successively and iterated until converged.
The radial stresses and displacements are determined through applications of Hook's law and Lame's thick-wall cylinder equations. For composite systems, such as cemented concentric strings, boundary conditions of stress and displacement continuity are applied between different materials. For sealed annuli, fluid volume changes are obtained by numerical integration of fluid elements using constitutive PVT relations so as to preserve variations in fluid compressibility and thermal expansion with temperature and pressure. This is particularly important in the presence of annular gas.
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