Production from Two-phase Reservoirs:
-
Phase diagram
Gas oil ratio in saturated reservoirs
Properties of two phase fluids, relative permeability
Comparing single and two phase flows
Steady state
Pseudo steady state
- Inflow performance relationship (IPR)
Single phase flow
Two phase flow
Vogel’s correlations
Generalized Vogel’s correlations
Fetokovich’s approximation

Earlier flow relationships have considered only single-phase flow of oil. The effect
of simultaneously producing liquid (oil) and gas on the liquid flow rate will now be
considered.
(From, Dake, Fundamentals of Reservoir Engineering)
If the reservoir is below bubble point pressure, as depicted in fig. 2.1(b), the situation is
more complicated. Now there are two hydrocarbon phases in the reservoir, gas saturated
oil and liberated solution gas. During production to the surface, solution gas will be
evolved from the oil phase and the total surface gas production will have two
components; the gas which was free in the reservoir and the gas liberated from the oil
during production. These separate components are indistinguishable at the surface and the
problem is, therefore, how to divide the observed surface gas production into liberated
and dissolved gas volumes in the reservoir.
In a saturated reservoir each stock tank barrel of oil is produced in conjunction with R scf
of gas, where R (scf/stb) is called the instantaneous or producing gas oil ratio and is
measured daily (see fig 2.3). As already noted, some of this gas is dissolved in the oil in
the reservoir and is released during production through the separator, while the remainder
consists of gas which is already free in the reservoir. Furthermore, the value of R can
greatly exceed Rsi, the original solution gas oil ratio, since, due to the high velocity of
gas flow in comparison to oil, it is quite normal to produce a disproportionate amount of
gas. This results from an effective stealing of liberated gas from all over the reservoir and
its production through the relatively isolated offtake points, the wells. A typical plot of R
as a function of reservoir pressure is shown as fig. 2.4.
The producing gas oil ratio can be split into two components as shown in fig. 2.3, i.e.
R = Rs +(R-Rs)
The first of these, Rs scf/stb, when taken down to the reservoir with the one stb of oil,
will dissolve in the oil at the prevailing reservoir pressure to give Bo rb of oil plus
dissolved gas. The remainder, (R occupy a volume
and therefore, the total underground withdrawal of hydrocarbons associated with the
production of one stb of oil is
The shapes of the Bo and Rs curves below the bubble point, shown in fig. 2.5(a) and
(b), are easily explained. As the pressure declines below pb, more and more gas is
liberated from the saturated oil and thus Rs, which represents the amount of gas dissolved
in a stb at the current reservoir pressure, continually decreases. Similarly, since each
reservoir volume of oil contains a smaller amount of dissolved gas as the pressure
declines, one stb of oil will be obtained from progressively smaller volumes of reservoir
oil and Bo steadily declines with the pressure.

The presence of a gas phase reduces the relative permeability of oil.
Relative permeabilities are laboratory-derived relationships, are functions of fluid
saturations and functions of specific reservoir rock.

The expansion of free gas creates an effective mechanism for the production of fluids.
The reservoir pressure, which is the driving force for the flow, decreases rapidly with
production from a reservoir containing liquid only.
Comparing the Flow Equations for Single and Two-phase Flows:
STEADY STATE INFLOW:
qo 
qo 
qo 
hk
( pe  pwf )
141.2 B[ln( re / rw )  s ]
hko
( pe  pwf ) o
141.2 Bo  o [ln( re / rw )  s ]
pe
k ro
hk
dp
141.2[ln( re / rw )  s ] pwf  o Bo
PSEUDO-STEADY STATE FLOW:
qo 
qo 
hk
3
141.2 B[ln( re / rw )   s ]
4
( p  pwf )
hk
3
141.2[ln( re / rw )   s  Dq ]
4

p
p wf
k ro
dp
o Bo
INFLOW PERFORMANCE RELATIONSHIP (IPR):
All well deliverability equations relate the well production rate and the driving force in
the reservoir, that is, the pressure difference between the initial, outer boundary or
average reservoir pressure and the flowing bottomhole pressure.
If the bottomhole pressure is given, the production rate can be obtained readily.
However, the bottomhole pressure is a function of the wellhead pressure, which, in turn,
depends on production engineering decisions, separator or pipeline pressures, etc.
Therefore, what a well will actually produce must be the combination of what the
reservoir can deliver and what the imposed wellbore hydraulics would allow.
It is then useful to present the well production rate as a function of the bottomhole
pressure. This type of presentation is known as an "inflow performance relationship"
(IPR) curve. Usually, the bottomhole pressure, pwf, is graphed on the ordinate and the
production rate, q, is graphed on the abscissa.
Pseudo steady state IPR calculation is the most useful and most commonly done
for the forecast of well performance. Each IPR curve reflects a “snapshot” of well
performance at a given reservoir pressure. This is time dependent calculation done in
discrete intervals. In combination with “volumetric material balances” it will allow the
forecast of rate and cumulative production versus time.
pi  pwf 
162.6qBo
k
[log( t )  log(
)  3.23  0.87 S ]
kh
ct rw2
pe  pwf 
r
141.2qB
[ln( e )  s ]
hk
rw
p  pwf 

141.2qB 1
4A
[ ln
 s]
kh
2 C A rw2
The complex analytical solutions to the two-phase flow equations will not be
considered in this course. The two-phase correlations of Vogel and Fetkovitch will
be presented.
Vogel's Correlations:
Vogel developed a set of inflow performance relationship (IPR) correlations. The
particular correlation that is appropriate is dependent on the magnitude of the average
reservoir pressure,
P , and the wellbore pressure, Pwf, relative to the bubble-point
pressure, Pb. These correlations are valid for a wide range of reservoir and fluid
properties. Only the properties of the oil phase associated with the two-phase flow are
required for Vogel's correlations.
Case 1: Pwf < P Pb (original Vogel Correlation)
In this case there is two-phase flow throughout the reservoir:
o and B are evaluated at
P.
Case 2: P > Pb but
Pb Pwf (Vogel's Generalised Correlation)
A limiting volumetric flow, qb, is defined which represents the flow that occurs in the
specific case when the wellbore pressure is equal to the bubblepoint pressure (Pwf=Pb).
Fetkovitch Inflow Performance Relationship:

In some cases Vogel's correlations do not accurately represent well /reservoir
behavior. The correlation of Fetkovitch can also be applied to two-phase systems.

Fetkovitch developed an empirical equation based on two correlation parameters,
qo,max and n. To apply the correlation, well measurements must be performed during
at least two stable flow conditions. Fetkovich's equation is adjusted to fit to the data
using the parameters.
The following two equations are combined to give the final empirical equation: