STUDY OF STRESSES IN MODELS OF ANISOTROPIC ROCK BY THE PHOTOELASTICITY METHOD

Yu. L. Boyarko

(Presented by the scientific seminar of the Department of Exploration Engineering of the MPI, Tomsk Polytechnic Institute)

The destruction of rocks during mechanical drilling is preceded by the appearance of elastic stresses in the rock. Knowing how stresses are distributed in the medium under investigation, and relying on work in this field [1, 2], we can judge with a sufficient degree of accuracy in what directions failure will occur and what the volume of failure will be. These judgments subsequently help in selecting the optimal shape of the destructive elements for drilling the corresponding rocks and, in particular, may shed light on the influence of the shape of the destructive elements of bits, crowns, and other tools on the process of borehole deviation when drilling in anisotropic rocks.

The theory of elasticity and plasticity gives clear and concrete concepts of how stresses propagate in an isotropic medium (Fig. 1). Most rocks, however, possess anisotropic mechanical properties to one degree or another, and therefore this classical theory can be applied to explaining the causes of rock failure only with considerable assumptions.

To clarify the qualitative aspect of the problem of stress distribution in anisotropic rocks under the action of a concentrated load, we carried out corresponding investigations. The photoelasticity method was adopted as the basis, since it makes it possible to obtain the required results relatively simply in cases where analytical investigation is seriously difficult. The physical essence of the photoelasticity method is described in detail in the monograph by M. Frocht [3], and therefore there is no need to dwell on this question here.

In the investigations a plane polariscope was used, making it possible to observe both isochromatic and isoclinic patterns. Rectangular parallelepipeds measuring \(22 \times 22 \times 5\) mm were used as models of anisotropic rock; they were glued together from thin plates of organic glass according to A. V. Stepanov [2], and corresponded to a two-dimensional anisotropic medium characterized by the alternation of homogeneous parallel layers of two isotropic materials (Plexiglas and glue) with different elastic properties. The thickness of the organic-glass plates was 1.2 mm. The thickness of the glue layer was within the range \(0.1—0.15\) mm.

Such models reproduce, with sufficient reliability, layered and schistose anisotropic rocks and, to a lesser degree, rocks with cataclastic and similar structure [2]. A load applied to the model with a constant force of 50 kg was transmitted through a wedge-shaped indenter by means of a special loading frame. The angles of inclination

Fig. 1. Patterns of isochromatics (a), isoclinics (b), trajectories of principal stresses (c), and maximum shear stresses (d) under loading by a static concentrated force of an elastic half-space (after M. Frocht)

plates imitating bedding in the models, relative to the faces to which the concentrated static force was applied, were varied in steps of \(10^\circ\) over the range from 0 to \(90^\circ\). The observed fringe patterns were photographed and sketched. For comparison, a model of isotropic rock, cut from monolithic Plexiglas and having the same dimensions, was investigated.

The studies showed that the patterns of stress distribution in an anisotropic medium differ sharply from the known patterns of stress distribution in an isotropic medium and depend to a great extent on the angle of inclination of the layers in the model.

Figure 1a shows an isochromatic pattern reflecting the distribution of the greatest shear and total stresses in an isotropic semi-infinite plate under a concentrated load. Figure 2 presents isochromatic patterns appearing in anisotropic models when they are loaded by a concentrated force. In this case the stresses develop fully only in one layer, passing into the next only when plastic deformation occurs in the original layer and the contour of the deformed core touches the next layer. However, before this moment the stresses partly spread into the subsequent layers owing to the elastic deformation of the overlying layers. This principle of stress distribution in the layers of an anisotropic rock model leads to a sharply different appearance of the fringes. The isochromatics seem to spread out along the most heavily loaded layer, while the center or axis of nucleation—

Fig. 2. Isochrome patterns in models of anisotropic rock at different angles of incidence

Their concentration is found on the line of action of the concentrated load only in the first layer beneath the indenter, except in the case when the force is applied perpendicular to the simulated bedding. In the other cases, the deformation core, having a spherical surface, causes the initiation of stress at the point of contact of the next layer, and the latter is located on the line perpendicular to the bedding and lowered from the point of application of the concentrated force.

It is also characteristic that the number of bands of one color in the most heavily loaded layer decreases as the angle of incidence between the direction of application of the concentrated force and the plane of bedding decreases, at one and the same pressure. This can be explained by an increase in the expenditure of energy on the elastic displacement of one layer relative to another as the angle of incidence decreases. The elastic displacement is clearly reflected, for example, in the exposure with an angle of incidence of 40°: a black, long band directed along the plane of cohesion shows that the lower layer has shifted downward relative to the upper one and has created noticeable bending stresses in the upper layer; under unconfined conditions such bending leads to easy splitting of the rock.

The isocline patterns in anisotropic models also differ from the trajectories of isoclines in an isotropic medium (Fig. 3 and Fig. 1, b). The zero isocline, which in an isotropic plate passes along the line of application of the concentrated force, coincides with this direction in the case of loading anisotropic models only in two exposures, namely at incidence angles of 90° and 0°. In the other cases it is deflected toward the dip of the simulated bedding. This deflection occurs smoothly as the angle of incidence decreases and reaches a maximum when the latter is within the range of 40–50°.

The isocline of any parameter is the geometric locus of points at which the directions of the principal stresses are parallel. In this connection, on the basis of the patterns of their trajectories it becomes possible to reconstruct the trajectories of the principal stresses (Fig. 4) and to analyze them qualitatively.

Fig. 3. Isochromatics in models of anisotropic rock at different angles of intersection

Fig. 4. Trajectories of principal stresses in models of anisotropic rock at different angles of intersection

The principal stresses $\sigma_1$ (shown by dashed lines in Fig. 4) are a family of concave curves, widely spread out and gradually straightening as the distance from the point of application of the concentrated force increases in exposures with a large angle of intersection. As the angle of intersection decreases, the concavity of these trajectories increases, and the branches close together more and more. The lock of the fold of the trajectories at first gradually shifts toward the dip of the simulated bedding, and then, after the angle of intersection decreases to $45^\circ$, again returns to a symmetrical position. The trajectories of the principal stresses $\sigma_2$, whose families are orthogonal to the trajectories of the principal stresses $\sigma_1$, also gradually change their shape, especially near the application of the concentrated forces, passing from convex upward at large angles of intersection to concave downward at small angles of intersection. In this case the concavity of the trajectories of the principal stresses $\sigma_2$ appears only at angles of intersection of $25$–$20^\circ$ and less.

At these same angles the isoclines at first partly, and then almost all, curve inward into the models being studied (Fig. 3, the last two exposures). At angles of intersection greater than $20$–$25^\circ$ all the isoclines tend to emerge at the surface and close at the point of application of the concentrated force.

Joint consideration of the patterns of isochromatics, isoclines, and trajectories of the principal stresses makes it possible to draw the essential conclusion that, at large angles of intersection, bending stresses are created in the upper layer (and, with thin stratification, in the upper layers), which, in the absence of clamping, greatly facilitate destruction of the rock. In Figs. 2, 3, 4, and 5, in the exposure with an angle of intersection of $90^\circ$, the upper layer has patterns of fringes, isoclines, and trajectories of the principal stresses approximately the same as those of a beam bent at the middle by a concentrated load [3]. In our case the upper layer of anisotropic rock may be regarded as a beam on an elastic foundation. The lower layers are an elastic bed on which the upper layer, under the action of the concentrated force, bends to a certain limit. Then spalling occurs of a certain volume of rock located around the penetrating indenter and subjected to bending stresses. Spalling is also assisted by the bursting forces of the volume of rock compressed beneath the indenter. Separation of the peripheral volume of destruction takes place along the surface in which the maximum shear stresses are most strongly developed.

This effect explains the relatively easy drillability of anisotropic rocks when the drill bit is positioned in the direction across the schistosity or bedding. The volume of rock located directly beneath the teeth of roller bits or cutters is destroyed by compression (destruction proceeds along the trajectories of greatest shear stresses), while the peripheral part is torn off by tension during its bending. As the angle of intersection decreases, the layers on one side become clamped, and the destruction becomes asymmetric and less effective. When drilling along the schistosity, only destruction by compression takes place, and spalling of the destroyed volume of rock occurs along the trajectories of the maximum shear stresses directly beneath the indenter and at a small distance from it, since these trajectories tend to emerge at the surface.

Considering the trajectories of the maximum shear stresses (Fig. 1, г and Fig. 5) and comparing them, we see that, as the angle of intersection decreases, the trajectories gradually increase their steepness, and at angles of intersection of $10$–$20^\circ$ there is also an asymmetry of the trajectories relative to the line of application of the concentrated load. The latter conclusion helps explain the fact of the regular curvature of boreholes in

toward the dip of the bedding or schistosity at small angles of intersection (on the order of \(25\)—\(0^\circ\)), since at these angles the clamping of the layers does not permit failure by bending, while the asymmetry of the trajectories of the shear stresses leads to one-sided failure of the rock volume. The greater chipping occurs on the side where the isoclines still tend to emerge at the surface, while the trajectories of the principal stresses \(\sigma_2\) tend to go deeper.

In addition to theoretical conclusions, which make it possible to understand the mechanics of failure of anisotropic rocks under static indentation of indenters and to understand the causes of borehole curvature when drilling them in anisotropic rocks, practical recommendations can also be made here for directional drilling of boreholes in anisotropic rocks. On the basis of the fact that, at different angles of intersection of the line of action of the destructive forces with the plane of schistosity of the rock, stresses of different configuration arise beneath the cutters, it is possible, by changing the geometry of the cutting edges of downhole bits, to control the intensity of failure of individual regions of the bottomhole and thereby control the direction of advance of boreholes in masses of anisotropic rocks.

Fig. 5. Trajectories of maximum shear stresses in models of anisotropic rock at different angles of intersection

Thus, if the face of a core bit or chisel bit is made conical so that the cutting edges of the cutters or other breaking elements form, during rotation, an internal cone (Fig. 6), then when anisotropic rocks are intersected at an acute angle to the axis of the tool, nonuniform failure of the bottomhole will occur. Those cutting edges that pass perpendicular to the schistosity encounter much greater resistance than the cutting edges passing parallel to the schistosity. As a result, unequal reactions \(R_1\) and \(R_2\) arise on the cutting edges under consideration, with \(R_1 > R_2\). In addition, a reaction \(R\) arises on the lateral surface of the drilling bit. The reactions \(R_1\) and \(R_2\) are directed parallel to the axis of the drilling tool, and therefore an overturning moment \(M\) arises on the tool, deflecting the upper-

Fig. 6. Diagram of the forces on a crown bit forming a conical shape of the bottomhole and adapted to deflect boreholes in the direction of the strike of schistosity

the lower part of the tool toward the dip of the plane of anisotropy of the rocks, and this leads to curvature of the borehole in the direction of the strike of schistosity with a definite intensity. The action of the overturning moment \(M\) increases the reaction \(R\), which deflects the drill bit in the same direction. Laboratory tests of the described crown bit confirmed the theoretical conclusions, and this method of directed borehole curvature is recommended for practical introduction instead of the use of single-hinge deflectors.

REFERENCES

1. B. V. Baidyuk, N. N. Pavlova. Mechanism of deformation and failure of rocks under indentation by a punch. In: Mechanical Properties of Rocks under Indentation and Their Practical Application. “Nedra,” 1967.

2. A. V. Stepanov. Causes of the peculiarities of failure of elastic-anisotropic bodies. Izv. AN SSSR. Physical Series, vol. XIV, No. 1, 1950.

3. M. Frocht. Photoelasticity. Vols. 1 and 2. Translated from English, GITTL, 1948.