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Derivation of an Enhanced Pressure Differential Expression, for a Penetration Injection with Back Pressure

In real-world water injection applications, an in-line injection facilitates a pressure differential that boosts the current flow. A pressure differential created by the injection of a pressurized flow into the mainline of flow is derived from the momentum transfer equation. Heat loss is disregarded, and such empirical equations provide a ballpark value to these pressure differentials during the injection. In industrial applications, injection of the fluid is done on the surface, due to weld and other constraints where losses due to friction and eddy current formation are imminent. On the other hand, penetration injection provides a far more augmented pressure differential that has a polynomial impact based on the mainline flow rate and the injection flow rate. This paper aims to derive an accurate representation of the pressure differential values obtained from a penetration injection through experimentation and compare it against a surface injection or empirical calculation. The paper concludes by indicating that the penetration injection augments the pressure differential with a new empirical formula for the derived pressure differential as a polynomial equation for this apparatus and can be extended across different sizes of the mainline and injection line diameters. This work provides a precise formula that can be used to derive pressure differential and estimate the flow and pressure rates. The formula also provides a platform for further utility in the fracturing operations where fracture flow from the well upstream presents multiple injection fractures to the mainline through fracture pores.