scholarly journals The Tidal Potential of a Homogeneous Torus with an Elliptical Section of the Sleeve

2016 ◽  
Vol 25 (3) ◽  
Author(s):  
B. P. Kondratyev ◽  
N. G. Trubitsyna
Keyword(s):  
Cross Section ◽  
Kuiper Belt ◽  
Analytical Form ◽  
Elliptical Cross Section ◽  
External Region ◽  
Elliptical Cross ◽  
Tidal Potential ◽  
Gravitational Model ◽  
Elliptical Section

AbstractIn this paper the problem of the tidal potential of a homogeneous gravitating torus with an elliptical cross-section sleeve is solved. In particular, the potentials in analytical form in the vicinity of the center of the torus and its external region are found. This torus can serve as a gravitational model of the Kuiper belt.

Saint–Venant torsion of anisotropic elliptical bar

2017 ◽  
Vol 45 (3) ◽  
pp. 286-294 ◽  
Author(s):  
István Ecsedi ◽  
Attila Baksa
Keyword(s):  
General Solution ◽  
Cross Section ◽  
Elliptical Cross Section ◽  
Elliptical Cross

The object of this article is the Saint–Venant torsion of anisotropic, homogeneous bar with solid elliptical cross section. A general solution of the Saint–Venant torsion for anisotropic elliptical cross section is presented and some known results are reformulated. The case of non-warping cross section is analysed.

Stresses and Deformations of Toroidal Shells of Elliptical Cross Section: With Applications to the Problems of Bending of Curved Tubes and of the Bourdon Gage

1952 ◽  
Vol 19 (1) ◽  
pp. 37-48
Author(s):  
R. A. Clark ◽  
T. I. Gilroy ◽  
E. Reissner
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Practical Interest ◽  
Closed Shell ◽  
Open Shell ◽  
Elliptical Cross Section ◽  
Axially Symmetric ◽  
Elliptical Cross ◽  
Two Parameters ◽  
Toroidal Shells

Abstract This paper is concerned with the application of the theory of thin shells to several problems for toroidal shells with elliptical cross section. These problems are as follows: (a) Closed shell subjected to uniform normal wall pressure. (b) Open shell subjected to end bending moments. (c) Combination of the results for the first and second problems in such a way as to obtain results for the stresses and deformations in Bourdon tubes. In all three problems the distribution of stresses is axially symmetric but only in the first problem are the displacements axially symmetric. The magnitude of stresses and deformations for given loads depends in all three problems on the magnitude of the two parameters bc/ah and b/c where b and c are the semiaxes of the elliptical section, a is the distance of the center of the section from the axis of revolution, and h is the thickness of the wall of the shell. For sufficiently small values of bc/ah trigonometric series solutions are obtained. For sufficiently large values of bc/ah asymptotic solutions are obtained. Numerical results are given for various quantities of practical interest as a function of bc/ah for the values 2, 1, 1/2, 1/4 of the semiaxes ratio b/c. It is suggested that the analysis be extended to still smaller values of b/c and to cross sections other than elliptical.

The Effect of Thermoelastic Stresses on Injection Well Fracturing

1985 ◽  
Vol 25 (01) ◽  
pp. 78-88 ◽  
Author(s):  
T.K. Perkins ◽  
J.A. Gonzalez
Keyword(s):  
Hydraulic Fracturing ◽  
Cross Section ◽  
Hydraulic Fracture ◽  
Outer Boundary ◽  
Elliptical Cross Section ◽  
Injection Well ◽  
Plan View ◽  
Elliptical Cross ◽  
Thermoelastic Stresses ◽  
Line Crack

Abstract When a cool fluid such as water is injected into a hot reservoir, a growing region of cooled rock is established around the injection well. The rock matrix within the cooled region contracts, and a thermoelastic stress field is induced around the well. For typical waterflooding of a moderately deep reservoir, horizontal earth stresses may be reduced by several hundred psi. If the injection pressure is too high or if suspended solids in the water plug the formation face at the perforations, the formation will be fractured hydraulically. As the fracture grows, the flow system evolves from an essentially circular geometry in the plan view to one characterized more nearly as elliptical. This paper considers thermoelastic stresses that would result from cooled regions of fixed thickness and of elliptical cross section. The stresses for an infinitely thick reservoir have been deduced from information available in public literature. A numerical method has been developed to calculate thermoelastic stresses induced within elliptically shaped regions of finite thickness. Results of these two approaches were combined, and empirical equations were developed to give an approximate but convenient, explicit method for estimating induced stresses. An example problem is given that shows how this theory can be applied to calculate the fracture lengths, bottomhole pressures (BHP's), and elliptical shapes of the flood front as the injection process progresses. Introduction When fluids are injected into a well, such as during waterflooding or other secondary or tertiary recovery processes, the temperatures of the injected fluids are typically cooler than the in-situ reservoir temperatures. A region of cooled rock forms around each injection well, and this region grows as additional fluid is injected. Formation rock within the cooled region contracts, and this leads to a decrease in horizontal earth stress near the injection well. In Ref. 1, the magnitude of the reduction in horizontal earth stress was given for the case of a radially symmetrical cooled region. Another factor, which may occur simultaneously, is the plugging of formation rock by injected solids. There is extensive literature indicating that waters normally available for injection contain suspended solids. Laboratory tests demonstrate that these waters, when injected into formation rocks, can plug the face of the rock or severely limit injectivity. In field operations, injection often simply continues at a BHP that is high enough to initiate and extend hydraulic fractures." The injected fluid then can leak off readily through the large fracture face area. Because of the lowering of horizontal earth stresses that results from cold fluid injection, hydraulic fracturing pressures can be much lower than would be expected for an ordinary low-leakoff hydraulic fracturing treatment. For this reason, the well operator may not be aware that injected fluid is being distributed through an extensive hydraulic fracture. If injection conditions are such that a hydraulic fracture is created, then the flow system will evolve from an essentially circular geometry in the plan view to one characterized more nearly as elliptical. In this paper, thermoelastic stresses for cooled regions of fixed thickness and of elliptical cross section are determined, and a theory of hydraulic fracturing of injection wells is developed. Conditions under which secondary fractures (perpendicular to the primary, main fracture) will open also are discussed. Finally, an example problem is given to illustrate how this theory can be applied to calculate fracture lengths, BHP'S, and elliptical shapes of the flood front as the injection process progresses. Thermoelastic Stresses in Regions of Elliptical Cross Section If fluid of constant viscosity is injected into a line crack (representing a two-wing, vertical hydraulic fracture), the flood front will progress outward. so its outer boundary at any time can be described approximately as an ellipse that is confocal with the line crack. If the injected fluid is at a temperature different from the formation temperature, a region of changed rock temperature with fairly sharply defined boundaries will progress outward from the injection well but lag behind the flood front. The outer boundary of the region of changed temperature also will be elliptical in its plan view and confocal with the line crack (see Fig. 1). Stresses within the region of altered temperature, as well as stress in the surrounding rock, which remains at its initial temperature, will be changed because of the expansion or contraction of the rock within the region of altered temperature. The thermoelastic stresses within an infinitely tall cylinder of elliptical cross section can be determined from information available in the literature. 10 The interior thermoelastic stresses perpendicular and parallel to the major axes of the ellipse are given by Eqs. 1 and 2, respectively. SPEJ P. 78^

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