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Sand control treatments that use plastics to consolidate an incompetent formation are quite reliable when the principles of soil mechanics, engineering mechanics, reservoir engineering, and geology are combined to define the magnitude of the sanding condition. A concept based upon controlling reservoir strength and fluid flow velocity is advocated. Stress levels described by Mohr-Coulomb diagrams are used to limit and help understand the various pressure responses that take place while injecting fluids or slurries into the reservoir. Flow profiles and pressure transients provide the basis for determining how much plastic is required to go out a distance where the velocity of the produced fluids does not have the potential to dislodge formation sand grains, In practice, a preliminary injection test is used to confirm the well data and theory so that any discrepancies can be readily noticed and corrected by the use of supplementary chemicals or various mechanical methods. This concept has proved successful on hundreds of treatments in manly different geographical and geological areas.
The two-dimensional Mohr diagram is a convenient graphical method used to represent the relationship between stresses on an "elemental cube" within a body and the shearing and normal stresses within the same cube on planes inclined to the planes of principal stress (See Figures I and 2). Only the maximum and minimum principal stresses are used. The intermediate principal stress is neglected in two dimensional analysis; however, it is often equal to the minimum principal stress. The plot is made so that normal principal stresses are abscissas and shearing stresses are ordinates. (Ordinarily, no transformation of principal stresses to horizontal and vertical components will be necessary for earth stress calculations in unconsolidated sand areas. The overburden weight acts vertically and is thus the easily oriented major principal stress. Transformations should be considered, however, in wellbores which intersect the substrata obliquely or are in an active diastrophic area.) Now, if points are plotted to show all the combinations of shearing and normal stresses that occur within the intermediate principal plane (this is the plane of the paper) as the principal stresses are varied, it will be found that the locus of these points describes a circle upon this intermediate principal plane. The center of the circle lies along the abcissa, and the circle itself cuts the abscissa at points equal to the maximum and minimum principal stresses. Various circles will describe where the material has been made to rupture as the major and minor principal stresses are changed. In the case of sand, clay or rocks, which are the materials common to reservoirs, a curve drawn tangent to the top of all the circles (through the maximum obliquity points) that have rupture coordinates will describe the criteria for failure by forming what is known as an envelope for the particular material. For unconsolidated sands, usually the envelope starts out from the origin. But should the envelope intersect the shearing stress ordinate above the origin, the intercept value represents a shearing strength that is usually referred to as the cohesive strength. It is a relatively low, but nevertheless quite influential value when compared to formation loading stresses, because it helps to hold formations together in the absence of confinement. The slope of the envelope for these unconsolidated reservoirs will be remarkably constant (neglecting any geologic time-fluidity analogy or testing with intermittent applications of stress) for any formation in question. The variation will be between about 20 deg. for the most loosely packed sediments, often mixed with clayey material, to the usual values between 30 deg. and 35 deg. that represent the very compact or mildly-cemented formations. By way of comparison, consolidated reservoirs will normally vary between 30 deg. and 50 deg., and massive igneous rock, such as basalt or granite, might have a maximum slope approaching 75 deg. The more competent rocks, such as limestone, can have curved, often nearly parabolic, envelopes under very large stress differences. Their slopes are greater at the lower loadings. Every formation has its own envelope which is a specific description of its strength. And quite significantly, although the strength of the formation may have been surpassed previously during the long geologic time periods, resulting in some "flowing" or shifting of sediments, it will not have lost its essentially elastic properties in terms of the short duration testing which is representative of the operations considered in this paper. The upper half of the circles or envelope is all that is frequently shown because of symmetry, Any coordinates of stress that plot within the envelope are simply conditions of loading, and any above the envelope are failure conditions. Or restated, failure will be because of a critical combination of shearing and normal stress resulting from unequal principal stresses. All important stress relationships can be estimated from a Mohr diagram.