Modeling of composite coupling technology for oil-gas pipeline section resource-saving repair

The article presents a variant of modeling and calculation of a main pipeline repair section with a composite coupling installation. This section is presented in a shape of a composite cylindrical shell. The aim of this work is mathematical modeling and study of main pipeline reconstruction section stress-strain state (SSS). There has been given a description of a structure deformation mathematical model. Based on physical relations of elasticity, integral characteristics of rigidity for each layer of a two-layer pipe section have been obtained. With the help of the systems of forces and moments which affect the layers differential equations for the first and second layer (pipeline and coupling) have been obtained. The study of the SSS has been conducted using the statements and hypotheses of the composite structures deformation theory with consideration of interlayer joint stresses. The relations to describe the work of the joint have been stated. Boundary conditions for each layer have been formulated. To describe the deformation of the composite coupling with consideration of the composite cylindrical shells theory a mathematical model in the form of a system of differential equations in displacements and boundary conditions has been obtained. Calculation of a two-layer cylindrical shell under the action of an axisymmetric load has been accomplished.


Introduction
The growth of fuel and energy resource scarcity and environmental concerns make energy-saving technologies one of the priorities of the oil industry. Main pipeline transport is the most common and efficient type of petroleum products transportation. Pipelines long-term operation leads to a decrease in bearing capacity and increased risk of accidents and failures. To improve the reliability of pipelines a selective repair of defective sections is carried out. Among the existing technologies of main and technological pipeline repair an installation of reinforcing couplings (coupling technology) is one of the promising methods [1,2]. In contrast with the overhaul repair methods use of repair structures in the form of a series of couplings is a resource-saving and environmentally friendly method, as it is performed without stopping oil pumping and pipe replacement. For example, the use of such couplings as composite and clamp weld-on ones allow restoring a pipeline defective section bearing capacity to a failsafe level during further operation. During the repair coupling installation, the SSS of the recovered pipeline section significantly changes which causes the need for strength calculations.
A pipeline section which is repaired with coupling installation can be regarded as a composite structure, consisting of two layers in the form of cylindrical shells and the joint. Connection between the layers provides consistency of the layers work. The connection of layers depends on a method of fixing the coupling. Paper [1] states that when repairing pipelines they use the following couplings: welded reinforcing and sealing ones. Reinforcing couplings cover the defect area without sealing. For sealing couplings sealing the ends with "girth side fillet weld" [1] is provided for. The boundary conditions which meet the specified conditions of fixing the couplings can be considered as a hinged support for reinforcing couplings and as rigid fixing for sealing ones.

Materials and Methods
To describe the mathematical model and to study the SSS of a composite coupling which is used to restore a pipeline defect section the composite structure theory statements are applied [2 -5]. Each shell is considered as the th i  layer ( (the thickness of pipe walls and couplings). The SSS of each layer of the shell is described by a system of forces, moments, strains, and displacements, which is applied in the classical theory using Kirchhoff-Love's hypotheses (L. I. Balabuha -I. V. Novozhilov variant). To describe the structure deformation the curvilinear, orthogonal right-handed coordinate system is used: The shell thickness is measured from the middle surface of the i-th layer in the direction of coordinate w, according to [2,3] is adopted as similar for each layer.
For a circular cylindrical shell we consider the case when boundary and surface loads do not depend on angle . Then the deformation of the shell will not depend on this angle, i.e. will be axisymmetric.
Here tangential deformation parameters  Physical correlations of the structure material are defined according to generalized Hooke's law. Thus, for the stresses components in layers [3,4] are used: Here the coefficients with account of material orthotropy of the structure layers are determined by formulas [1,2].
For the case of isotropic properties of the layers material the coefficients are equal to [1 -4].
We can go to line forces that act at the level of the middle surface of the th i  layer from the stresses at an arbitrary point of the th i  layer by means of integrating by thickness: where To calculate the line forces under axisymmetric load, it is necessary to integrate the expressions for strains at an arbitrary point of this layer. After the integration, we obtain the following elasticity relations: Here the rigidity parameters Differential equations of equilibrium for the th i  layer subject to shear stresses in interlayer joint  will be presented in accordance with the methodology [1 -5]:  Differential equations of equilibrium in displacements for the th i  structure layer are presented in papers [1 -5]. The order of the system of differential equations determines the number of boundary conditions.

Results
Composite structures in the form of a circular cylindrical shell have been regarded. Along coordinate х on each of the two transverse layers boundary conditions have been specified. Along coordinate  periodicity conditions of the required function must be met.
In the case of axisymmetric deformation of a two-layer cylindrical shell, the system of differential equations in displacements will be written as [3,5]:   Technical description and diagrams that correspond to the boundary conditions are given in [1].
Thus, the mathematical model of the SSS of the recovered section of a main pipeline under axisymmetric load is a system of differential equations (12) and boundary conditions. For example, if the boundary conditions correspond to hinged support of the ends (case 4), then the system of differential equations can be solved by decomposition of the displacement functions and load vector in trigonometric series [7]:

Discussion
The solution of the problem of bending of a closed two-layer cylindrical shell (composite coupling) with steel layers under axisymmetric load showed that with stiffness exceeding mm N 3 5 10 the structure can be regarded as single, without taking into account slippage between layers. The load distribution variation for a closed composite cylindrical shell showed that the interlayer connection stiffness under evenly distributed load significantly affects the stressstrain state at the nearanchorage area, and under local load at the central area. To assess the authenticity of the calculation (14) Here, w is a deflection function, x is the distance along the generatrix,  is Young's modulus, r is the cylindrical shell radius of curvature, h is the cylindrical shell thickness. General solution of differential equation (14)  where constant Ci and particular solution w ch are determined from the boundary conditions.

Conclusions
Thus, a mathematical model has been obtained in the form of differential equations which allows calculating the values of deformations, stresses and forces in layers under different methods of coupling installation, assessing the structural reliability of the pipeline, and determining the influence of geometrical and mechanical structure parameters on the strength of the repair pipeline section. Reconstruction of damaged pipeline sections using coupling technology leads to a decrease in stress and unloading the main pipe, which improves resistance to damage and increases the service life of pipelines due to an additional layer. Modeling and calculation of an operating pipeline without stopping oil pumping allow using energy resources efficiently and environmentally friendly; therefore, are a promising direction for further scientific and technical studies.