The Impact of Constitutive Laws on Wellbore Stability: A General Review
- P.A. Charlez (Total Oil Marine)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling & Completion
- Publication Date
- June 1997
- Document Type
- Journal Paper
- 119 - 128
- 1997. Society of Petroleum Engineers
- 1.14 Casing and Cementing, 4.3.4 Scale, 1.11 Drilling Fluids and Materials, 2.4.3 Sand/Solids Control, 1.6 Drilling Operations, 1.11.2 Drilling Fluid Selection and Formulation (Chemistry, Properties), 1.2.2 Geomechanics, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment
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This paper is focused on soft deep rocks that can induce strong wellbore-stability problems. In the first part, it is shown that these rocks generally exhibit plastic-type behaviors. In the second part two elastoplastic models (Cam-Clay and Laderock) are briefly presented. In the third part, these constitutive laws are applied to wellbore stability through analytical solutions or finite element codes. Finally, in the fourth part, a case history is discussed in detail. We insist on the difficulty to properly define the boundary conditions of the problem (in-situ stresses and virgin pore pressure) and on the subjectivity of the criterion translating the boundary wellbore problem in stability recommendations. Some keys for future research are proposed.
Damaging and Ductile Geomaterials
Materials for which macroscopic rupture1,2,3 (under deviatoric loading) is preceded by a large linear elastic zone, then by a nonlinear damage zone (Fig. 1), are generally considered to be damaging.
The damage phase can be considered a preparation to macroscopic failure, during which progressive development of microcracks parallel to the major stress induces a strong anisotropy4. The damage phase is associated with a volumetric dilancy and a reduction of the elastic modulus5-7 (clearly observed if loading/unloading cycles are performed).
Susceptibility of damaging geomaterials to microcracking and their small capability to strain are mainly caused by a strong cohesion. When the rock is loaded, stresses are concentrated either into the intergranular cement (generally the case for sedimentary rocks) or into the grains themselves, if the mechanical resistance of the cement is higher than that of the grains. The structure will be progressively destroyed by opening and propagation of flaws. The coalescence of these microcracks will finally lead to the collapse of the structure (often a shear band).8
Under hydrostatic state of stress, damaging geomaterials do not generally exhibit any type of irreversibility; the behavior remains elastic contractant over a large range of confining pressures. This second property is caused by the good mechanical stability of the porous space, which is again related to the strong cohesion between grains. Consequently, the intrinsic curve of damaging materials is always open upon hydrostatic loading. It is called an open form.
By contrast to damaging materials, ductile rocks9 can often support large plastic strains (several percent) without any macroscopic failure. From a structural viewpoint, two main petrographic properties differentiate damaging and ductile rocks: a small cohesion and a high porosity. These two properties have determinant consequences on the rheological behavior of the material.
On one part, the high porosity allows ductile materials to be strained irreversibly under hydrostatic loading. The purely contractant-associated plastic mechanism called collapse corresponds to an irreversible reduction of the porosity by implosion of the material.
The plastic collapse mechanism is schematically described in Fig. 2. The grains are initially bonded by weak links. Under the effect of an increasing loading, three different phases can be observed. For moderate values of the mean stress (or of the confining pressure), the porous structure remains stable (phase 1) and strain energy (mainly elastic) is stored into the bonds. With the progressive rupture of the bonds, the capability of the material to strain increases and the concavity of the stress/strain curve points downwards. Once free from its bonds, the porous structure collapses, exhibiting very large deformations for only a small increase of the mean stress (phase 2 in Fig. 2). In a third phase, the grains contact with respect to each other. The number of contacts increases, the compressibility of the material decreases, and consequently, the concavity of the stress/strain curve is inverted. In fact, phase 3 corresponds to the common consolidation process that is well-known in soil mechanics. A clear example of such a plastic behavior is presented in Fig. 3a (Gorm chalk). For this material, the collapse stress is equal to 20 MPa.
All ductile geomaterials do not necessarily exhibit the three phases described above. For cohesionless materials (no initial bonding between the grains), only the consolidation phase exists. This is often the case for unconsolidated sands (Fig.3b), and also sometimes for shales and claystones.
Under deviatoric loading, and because of small cohesion, these materials are rather insensitive to fissuration. The damage mechanism is replaced by a sliding elastoplastic mechanism between grains mainly governed by internal friction. It very often takes place at constant plastic volume as shown in Fig. 4. Of course this second mechanism is coupled with the previous one because contacts between grains resulting from consolidation strongly affect shear failure.
In contrast to damaging materials, for which the appearance of a shear band remains today a poorly understood phenomenon,10 catastrophic rupture of plastic specimens is well explained by the localization (or bifurcation) theory11-13, as shown in the remarkable analogic14 experiment of Fig. 5.
The analogic material consists in a pack of rolls, the axes of which are perpendicular to the plane. On each roll a T is drawn pointing initially upwards. The structure is then biaxially loaded. As shown in Fig. 5, localization takes place by rotation of the rolls along the two diagonals. Elsewhere rotations remain small. Furthermore, this localization induces a strong dilancy, well marked along the lateral faces of the analogic sample. With the number of contacts decreasing during the localization, the hardening energy blocked up during the loading phase is released. By contrast to damaging geomaterials, for ductile rocks, dilancy appears in a later phase corresponding to the collapse of the global structure and not to a volumetric damage.
Examples of Elastoplastic Laws
This section succinctly presents two classical elastoplastic models commonly used in geomechanics: the Cam-Clay and the Laderock. They are both written in terms of Terzaghi effective stress.
Modified Cam-Clay Model.
The Cam-Clay model15,16 introduced during the 1960's by Roscoe and Burland is based on the existence of a single plastic mechanism with an associated plastic flow rule. In the P-Q diagram (mean effective stress, stress deviator), the yield locus is an ellipse (Fig. 6), the size of which evolves exponentially with the volumetric plastic strain. Physically, this model only reproduces the consolidation mechanism (phase 3 of Fig. 3).
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