The Relation Between Recovery Deformation and In-Situ Stress Magnitudes

Abstract

Abstract In this paper a theoretical basis is developed for calculating in situ stress magnitudes from recovery deformations measured in rock. The constitutive theory used assumes that the rock is linearly viscoelastic, homogeneous, and non-aging. These assumptions yield explicit integral equations relating strain as a function of time to stress history, which is taken to be instantaneous unloading of the rock. Solutions are obtained for recovery of isotropic and transversely isotropic-core in which time-dependent behavior is contained in creep compliance terms and Poisson's ratio terms are constant. The main results for both cases are equations that allow calculation of horizontal principal stresses from vertical stress, principal recovery principal stresses from vertical stress, principal recovery strains, and material properties that do not depend on time. For the isotropic case, the only material property required is Poisson's ratio. For the transversely isotropic case, two Poisson's ratio. For the transversely isotropic case, two Poisson's ratio terms as well as the ratio of creep compliances are Poisson's ratio terms as well as the ratio of creep compliances are required. This last solution assumes that the vertical stress is parallel to the axis of material property symmetry. parallel to the axis of material property symmetry Introduction Recent improvements in the technique of measuring recovery deformations offer promise of this method developing into an effective stress-measuring tool. Results published to date, however, have been primarily directed at relating recovery stresses to stress orientations. In this paper the method is extended to include calculation of stress magnitudes with the idea that stress recovery measurements may become a useful alternative to other stress-measuring techniques. The practical significance of knowing in situ stresses results mainly from their overriding influence on the orientation and geometry of hydraulic fractures. The ability to predict these fracture characteristics is often critical to the design of optimal stimulation treatments. INTEGRAL REPRESENTATIONS OF VISCOELASTIC BEHAVIOR Time-dependent recovery strains in rock are the result of a viscoelastic response to removal of in situ stresses. In developing relations between measured strains and in situ stresses, a viscoelastic constitutive theory based on hereditary integrals will be used. This approach is particularly convenient because it does not require any knowledge of the physical mechanisms involved and it allows treatment of a broader range of materials than mechanical models based on springs and dashpots. The gain in generality does not cause a loss in utility, and, in fact, this method is often easier to use in applications. The only restriction placed on behavior by this type of integral representation is that the system be linear. Viscoelastic systems in general involve history-dependent behavior and thus are described mathematically by functionals, in this case a relation that maps an entire function onto a point. For example, the strain at a particular time depends not just on the stress at that time but on the entire stress-history up to and including that time. For a functional to be linear it must satisfy the following conditions:Homogeneity: { } = { }Superposition: { } = { } + { } where: : is strain: are different stress histories{ } : are used to denote a functionalc : is a constant p. 213