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.

Relating Hydraulic–Electrical–Elastic Properties of Natural Rock Fractures at Elevated Stress and Associated Transient Changes of Fracture Flow

Monitoring the hydraulic properties within subsurface fractures is vitally important in the contexts of geoengineering developments and seismicity. Geophysical observations are promising tools for remote determination of subsurface hydraulic properties; however, quantitative interpretations are hampered by the paucity of relevant geophysical data for fractured rock masses. This study explores simultaneous changes in hydraulic and geophysical properties of natural rock fractures with increasing normal stress and correlates these property changes through coupling experiments and digital fracture simulations. Our lattice Boltzmann simulation reveals transitions in three-dimensional flow paths, and finite-element modeling enables us to investigate the corresponding evolution of geophysical properties. We show that electrical resistivity is linked with permeability and flow area regardless of fracture roughness, whereas elastic wave velocity is roughness-dependent. This discrepancy arises from the different sensitivities of these quantities to microstructure: velocity is sensitive to the spatial distribution of asperity contacts, whereas permeability and resistivity are insensitive to contact distribution, but instead are controlled by fluid connectivity. We also are able to categorize fracture flow patterns as aperture-dependent, aperture-independent, or disconnected flows, with transitions at specific stress levels. Elastic wave velocity offers potential for detecting the transition between aperture-dependent flow and aperture-independent flow, and resistivity is sensitive to the state of connection of the fracture flow. The hydraulic-electrical-elastic relationships reported here may be beneficial for improving geophysical interpretations and may find applications in studies of seismogenic zones and geothermal reservoirs.