Modeling of the Stability of Highly Inclined Boreholes in Anisotropic Rock Formations (includes associated papers 19213 and 19886 )

Abstract

Summary. Simulators have been developed to study the fracture and collapsebehavior of boreholes. To take into account the directional properties ofreal rocks, an anisotropic stress model is used. The model takes intoaccount anisotropic elastic properties, directional shear, and directionaltensile strengths. The orientation of the borehole, the in-situ stresses, and the bedding plane can all be arbitrarily related to each other to modelactual field situations. This paper presents some of the results of themodel. It is shown that neglecting the anisotropic effects introduces anerror. Also, studying the anisotropic model yields further insight intothe behavior of the borehole. Introduction The integrity of the borehole plays an important role in many well operations. During drilling. lost circulation and borehole collapsecause economic losses, and in production operations, controlledfracturing and sand control are of utmost importance for the economyof an oil field. Therefore, a better understanding of the rockmechanics is necessary to improve the economy of the well operations. In the past, accurate methods to predict critical fracturing orcollapse pressures have been unavailable. Simple isotropic stressequations have been used to some extent, but these fail to take intoaccount real rock properties that are clearly anisotropic. Sedimentaryrocks have a laminated structure, with directional elastic properties as well as directional shear and tensile strengths. To understandfield situations better, a complete mathematical model was developedthat takes into account all directional properties. Fig. 1 gives an overview of typical borehole problems. Shownare the only two cases covered in this paper-i.e., borehole collapseat low borehole pressures and fractures initiating along the boreholeaxis at high pressures. Our analysis is limited to wells great than 2,000ft [greater than 610 mi] deep. The model is based on linear elasticity.and neglects plastic or time-dependent effects. Fig. 1 shows theversatility of the simulators. The borehole, the in-situ stresses, and therock-property reference frame may assume any orientation. This givesus a toot to model any field situation. Mathematical Model The core of the simulators is the mathematical model for stresses around boreholes in anisotropic materials. The complicated modelis derived in detail by Aadnoy and is based on a generalizedplane-strain concept and linear elasticity. It results in two coupledpartial-differential equations to determine the parameters of the stressfunctions. Aadnoy applies the model with a transversely isotropictype of anisotropy. This is a good description for sedimentary rocks;all parameters of elasticity are equal in a horizontal plane, but differvertically, The appendix briefly outlines the model. Table 1 is takenfrom Chenevert's work and gives a summary of the elastic parameters asapplied to different rock types. It is observed that the degree of anisotropy, Kani, is lithology-dependent. Therefore, the sandstone can be classified as very anisotropic, the shales as moderately anisotropic, and the limestone as isotropic for this particular data set. The two simulators, ANISFRAC for borehole fracture analysis and ANISCOLL for collapse analysis, also perform all transformationsnecessary to implement randomly oriented boreholes, in-situstresses, and bedding planes, and thus are able to model any fieldsituation. The introduction of anisotropy vastly complicates themathematics involved, but it is believed that it describes real fieldbehavior better. From Chenevert's work, a set of directional tensile-strength datfor common sedimentary rocks is also obtained. Table 2summarizes these data. It is observed that the tensile strength is 20 to35 % lower parallel to the bedding plane than perpendicular to it. This, of course, will affect the behavior of inclined boreholes. Finally, the simulators use shear data also obtained by Chenevert for the samerock types. An extended Mohr-Coulomb criterion with a plane ofweakness is introduced. allowing both the cohesive strength andthe angle of internal friction to vary. The results are summarizedin Table 3. Note that the limestone is isotropic in both its elasticproperties (Table 1) and its shear properties(Table 3). Some ofthe resultant shear envelopes are plotted in Fig. 2. For beta = 15 degrees[0.26 rad], the material is weakest because it fails along the beddingplane. At 0 or 90 degrees [0 or 1.6 rad], it fails across the beddingplane and is therefore strongest for these directions. Applications In Borehole Fracturing The simulator ANISFRAC will be used to study a few field cases. The failure criteria applied are that when the least effective principalstress exceeds the strength of the rock in tension, a tensile failureoccurs. Two different tensile strengths will be used. One is themeasured values given in Table 2 and the other is zero tensile strengthassuming that the rock contains cracks that are merely reopened. One shale and one sandstone are studied. The shales are generallyalmost nonpermeable but porous and contain pore fluid at a givenpressure. Therefore, these will be modeled with a pore pressureinside the porous rock with an instantaneous rise (a step function)to the borehole pressure outside the borehole wall. The shales can also be abnormally pressurized. Thus, the porepressure is also a very important parameter in shales because thefracture gradient is strongly sensitive to the magnitude of the porepressure. The sandstone is somewhat different from the shalesbecause it is permeable. Two different situations will be studied. Inthe first one, a perfect mudcake is assumed, and we have ananalogous situation for the shales. Inside the borehole is the boreholepressure, and immediately inside the rock wall is the pore pressure, the transition being a step function. The second situation assumes no mudcake; therefore. fluid communications between the formationand the borehole are allowed. This means that during fracturingoperations. the pore pressure immediately inside the porous rockwall is equal to the borehole pressure. Stress contributions causedby the flowing fluid are neglected. The results of the calculations for the Permian shale are shownin Fig. 3. The parameters used are typical for an intermediate deepwell and are given in the figure. The bedding plane and the in-situstresses are assumed to lie in the horizontal plane. The numericalvalues used in the following simulations are not chosen to simulateactual oil fields, but to study different parameters. The hatched band defines the two limits of tensile strength. SPEDE P. 259^