Inthe single well system, oil and associated fluids move from the reservoir tothe tank. Energy losses must be overcome in order for fluids to flow throughvarious interconnected components from the reservoir to the stock tank. Figure 12 below shows the locations ofcommonly used nodes.The Systems Analysis or NODAL analysis concept:inflow involves the components (inside the reservoir) = outflow all of thecomponents (from intake point (6) and up word to point (1)). Inthe gas lifted well, generally the solution node is selected at the midperforations depth.
Atthis location, IPR (inflow performance relation) = VLP (vertical liftperformance).as shows in the figure -13 below.Proper design and analysis of an oil wellrequires knowledge of reservoir flow rates into the wellbore at current as wellas future conditions. Minimally, the pressure at the bottom of the well and thecorresponding liquid production rate is needed for design and analysis. Therelationship between the liquid influx into the wellbore and the driving force– caused by the difference between the average reservoir pressure and thebottom hole flowing pressure – is called the Inflow Performance Relationship orIPR.
The simplest IPR representation is astraight line wherein the flow rate is directly proportional to the drivingforce or the pressure differential between the average reservoir pressure PRand the bottom hole flowing pressure PWFTheproportionality constant is referred to as the Productivity Index or PI orJ. The flow rate is given by the following expression: Inthe English units, q is flow rate in STB/day, and the pressures – PR andPWF – are in psig, resulting in the units of PI to beSTB/day/psi. Aproper production well-test would provide values for the bottom hole flowingpressure and the corresponding flow rate.
The average reservoir pressure can beeither inferred from shut-in pressures or reservoir simulation techniques. This IPRrelationship can also be derived from the Darcy equation on flow in porousmedia under simplified assumptions of radial, single-phase (liquid) flow in ahomogeneous reservoir, whereby: Where,k is effective permeability in mD, h is pay thickness in ft, ?is liquid viscosity in cP, B is liquid formation volume factor inbbl/STB, re is well drainage radius in ft, and rw is wellbore radiusin ft. Forthe cases where this relationship holds, mainly where the PWF is abovethe bubble point pressure, PB, the Productivity Index will be theinverse of the slope of the IPR line.