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With selecting lower-convected Maxwell constitutive equation, polymer solution flow equation in eccentric annulus is established. And the distribution rule of the force acting on the sucker rod by polymer solution has been gained. The result shows the eccentric ratio and the movement velocity of the sucker rod are major influence factors on the radial force. The greater the ratio is, the greater the force of sucker rod is. Then the eccentric wear is easier to appear.

With the development of the enhanced oil recovery technology, Polymer flooding has been widely used. But because of the viscous-elasticity of polyer flooding produced fluid, there would be a radial force acting on the sucker rod, which was vertical to axis. The combination of radial force, inertial force and resistance makes sucker rod bent. Thus, the phenomenon of eccentric wear will be heavier and rupture of sucker rod will always happen, which has become the major problem in Polymer flooding Wells production. So, it has practical significance to carry out the study of viscous-elastic fluid in eccentric annulus.

Before there is few study about radial force due to the viscous-elasticity of polymer flooding produced fluid. In this article, on the base of the steady flow velocity profile in the viscous-elastic fluid in eccentric annulus with the internal rod moving axially, the lower-convected Maxwell constitutive equation model is used, the stress distribution of the flow field and the radial force acting on the sucker rod are calculated, the influence factors of radial force and eccentric wear of sucker rod are analyzed, which provides theoretical basis for the prevention of eccentric wear.

The velocity of eccentric annulus flow of viscous-elastic fluid can be solved by using control volume method and the ADI method [

The lower-convected Maxwell constitutive equation [

where:

—viscosity.

Under the condition of flowing in the paper, each component of the lower-convected Maxwell constitutive equation in the bipolar reference frame can be expressed as:

Flow field velocity distribution isolines

After the velocity distribution of flow field is got, the deformation rate distribution and the Ericksen tensor of flow field can be calculated further. And then the stress distribution of the flow field can be calculated by using the constitution equation. The normal stress S_{rr} acting on the inner tube in the flow field is the most interested. For the resultant force is the major factor of sucker rod eccentric wear. The normal stress on the surface of the inner tube can be expressed as

The unit length rod radial force F can be calculated by integrating S_{rr} along the inner tube by angle, namely

Because the flow field is symmetrical when the sucker rod and tubing is concentric (Namely the various angles of normal stress is equal), the resultant force is zero. But, in the engineering practice, it is difficult to guarantee the sucker rod and tubing concentric, velocity field is asymmetric and the normal stress field is asymmetric, too. Then the resultant force F of the normal stress is no longer zero.

In engineering practice, the flow in the wellbore is combined by the flow produced by the pressure gradient and the flow produced by sucker rod. The velocity distribution curves of the sucker rod at the certain speed U of down run and up run are shown in

The major factors of the radial force in the sucker rod are the eccentricity and its velocity. The greater the ec-

The velocity distribution of sucker rod down run in the wellbore. (a) The velocity distribution when the sucker rod is rest; (b) The velocity distribution when the sucker rod moves; (c) The velocity distribution when the sucker rod moves and the fluid flows at the same time

The velocity distribution of sucker rod up run in the wellbore. (a) The velocity distribution when the sucker rod is rest; (b) The velocity distribution when the sucker rod moves; (c) The velocity distribution when the sucker rod moves and the fluid flows at the same time

Normal stress S_{rr} distribution on the surface of the inner tube

The relationship between the radial force and the eccentricity

centricity is, the bigger the radial force is, so the velocity of the sucker rod is, which provides theoretical basis for the further analysis of the cause of the sucker rod eccentric wear and the method of preventing eccentric wear.