In
the single well system, oil and associated fluids move from the reservoir to
the tank. Energy losses must be overcome in order for fluids to flow through
various interconnected components from the reservoir to the stock tank.

Figure 12 below shows the locations of
commonly used nodes.The Systems Analysis or NODAL analysis concept:
inflow involves the components (inside the reservoir) = outflow all of the
components (from intake point (6) and up word to point (1)).  In
the gas lifted well, generally the solution node is selected at the mid
perforations depth.

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At
this location, IPR (inflow performance relation) = VLP (vertical lift
performance).as shows in the figure -13 below.Proper design and analysis of an oil well
requires knowledge of reservoir flow rates into the wellbore at current as well
as future conditions. Minimally, the pressure at the bottom of the well and the
corresponding liquid production rate is needed for design and analysis. The
relationship between the liquid influx into the wellbore and the driving force
– caused by the difference between the average reservoir pressure and the
bottom hole flowing pressure – is called the Inflow Performance Relationship or
IPR.The simplest IPR representation is a
straight line wherein the flow rate is directly proportional to the driving
force or the pressure differential between the average reservoir pressure PR
and the bottom hole flowing pressure PWFThe
proportionality constant is referred to as the Productivity Index or PI or
J. The flow rate is given by the following expression:

In
the English units, q is flow rate in STB/day, and the pressures – PR and
PWF – are in psig, resulting in the units of PI to be
STB/day/psi.

A
proper production well-test would provide values for the bottom hole flowing
pressure and the corresponding flow rate. The average reservoir pressure can be
either inferred from shut-in pressures or reservoir simulation techniques.

This IPR
relationship can also be derived from the Darcy equation on flow in porous
media under simplified assumptions of radial, single-phase (liquid) flow in a
homogeneous 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 in
bbl/STB, re is well drainage radius in ft, and rw is wellbore radius
in ft.

For
the cases where this relationship holds, mainly where the PWF is above
the bubble point pressure, PB, the Productivity Index will be the
inverse of the slope of the IPR line.

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