## 2 Factors Impacting the Intensity of Hydroabrasive Wear of the Pumping Stages of the ESP
This value can be calculated using Equation (4):  
\[\Delta=\frac{10^{-4}L^{2}}{2E}\Big{[}\gamma-2\gamma_{f}(1-\mu)\Big{]}+\frac{(1 -2\mu)F_{t}LP_{w}}{0.1 EF}, \tag{4}\]  
where \(\Delta\)--deformation value of the oil-well tubing, m; \(L\)--the length of the oil-well tubing, m; \(E\)--modulus of elasticity of tubing material, MPa; \(\mu\)--Poisson's ratio of tubing material; \(\gamma\)--specific weight of tubing, N/m\({}^{3}\); \(\gamma_{f}\)--specific weight of fluid in tubing, N/m\({}^{3}\); \(F_{t}\)--area of tubing passage, m\({}^{2}\); \(F\)--area of annular cross-section of tubing, m\({}^{2}\); \(P_{w}\)--pressure at the wellhead, atm [35]. The left side of Equation (4) is a constant when the wellhead pressure varies with the filled oil-well tubing, i.e.,  
\[\frac{10^{-4}L^{2}}{2E}\Big{[}\gamma-2\gamma_{f}(1-\mu)\Big{]}=\text{const.} \tag{5}\]  
Therefore, the deformation value \(\Delta\) of the oil-well tubing when changing \(P_{w}\) is determined by Equation (6):  
\[\Delta=\frac{(1-2\mu)F_{t}LP_{w}}{0.1 EF}. \tag{6}\]  
The following conditions were considered for execution of calculations using Equation (6): the oil-well tubing consists of tubes with an outside diameter of 73 mm and wall thickness of 5.5 mm; tubing material--30 CrMA steel (foreign analogues are 25 CrMo4, 34 CrMo4); modulus of elasticity of tubing material \(E\) = 2.12\(\cdot 10^{5}\) MPa; [PERSON]'s ratio of tubing material \(\mu\) = 0.3; tubing length \(L\) = 1000 m; area of tubing passage \(F_{t}\) = 30.2\(\cdot 10^{-6}\) m\({}^{2}\); area of annular  
Figure 4: Scheme of the borehole slotted self-cleaning filter with deformable filter element. cross-section of tubing \(F=11.7\cdot 10^{-6}\) m\({}^{2}\). If excess pressure \(P_{w}\) changes from 0 to 50 atm at the wellhead, the deformation value of the oil-well tubing is \(\Delta=0.024\) m.  
Therefore, the submersible pump unit together with slotted filter is lowered along the production well by the value \(\Delta=0.024\) m when the wellhead pressure increases to \(P_{w}=50\) atm and the length of the tubing is \(L=1000\) m. Switching off the pump unit and releasing the pressure at the wellhead to \(P_{w}=0\) atm lifts the pump unit along the production well by the same value \(\Delta=0.024\) m. At the same time, the cuff _13_ is in contact with the walls of the well _15_ (Figure 4). The aforementioned factors cause short-term stretching of deformable filter element 7, increasing of the size of slots 9, destruction of \"bridges\" of deposits of mechanical impurities, and restoration of the permeability of the filter element 7. Subsequently, the spring _16_ compresses the stop _10_ with the cuff _13_, which leads to the restoration of the original dimensions of slots 9 and the filter element 7. The use of slotted filters with rigid, non-deformable filter elements equipped with movable cleaning devices is possible when developing ESPs installed on a load-bearing cable when working with complex wells. Figure 5 shows a slotted borehole self-cleaning filter lowered along the oil-well tubing as a part of an ESP unit on a load-bearing cable [36]. The slotted filter contains a tube \(1\), which is perforated with through holes 2. Upper 3 and lower 4 end rings are mounted on the ends around the tube \(1\). Filter element 5 is concentrically placed around the tube \(1\). The filter element 5 is made in the form of longitudinal bars 6, between which slots 7 are formed. Transversal bars \(8\), connected to the longitudinal bars 6, are provided in order to make the structure of filter element 5 rigid. The ends of the filter element 5 are placed inside the annular gaps of the end rings 3 and 4. The movable device 9 for cleaning from mechanical impurities, made in the form of a ring _10_ with a fixed resilient element _11_ with through holes _12_, is concentrically installed on the outer surface of the filter element 5. The resilient element _11_ is in contact with the inner walls of the tubing _13_. There are support-cleaning elements _14_ on the inner side of the ring _10_.