6 Deviation from linear Inflow performance
At sufficiently high flow rates, the inflow characteristics are
not linear. The pressure loss will almost always be greater than
predicted from a constant productivity index. Below, we quantify
flow mechanisms that provide such a discrepancy.
6.1 Second order pressure loss
6.1.1 Forchheimer equation
Darcy’s equation implies proportionality between flow velocity
and pressure gradient. Similar relations are used for other
engineering applications, such as: Ohm's law of electric
resistance, Hooks elasticity law, Fourier's heat flow law.
However such relations are often not applicable at high rate, or
Forchheimer (1901) proposed a relationship that can be seen as
Darcy’s equation with a 2.order adjustment
?dpo (6-1) ??v???vvdrk
-1???: Second-order constant, "turbulence factor" (m)
From Forchheimer equation (6-1), we can predict the relationship
between pressure loss and rate. By assuming radial and steady-
state influx, we get the following solution
2????BrB11??ooeoo?? ????? (6-2) pr?pr?lnq??qq??eooo22????2khr4hrr??e??The first part of (6-2) corresponds to Darcy’s law. The second part is independent of viscosity and increases with the square
of flow rate, corresponding to turbulent flow resistance in
pipes. Truly turbulent flow does not occur in porous media under
normal circumstances; the pore channels are too narrow and flow
speed too low. But the flow changes velocity and direction
through the pores. For complicated pore geometry we can expect
?-factor. For relatively straight and smooth relatively large
flow channels, we can expect little ?-factor.
Figure 6-1 illustrates 2.order pressure profile with low
viscosity and high production. The figure shows significant
pressure loss near the well. Further out the flow speed is so
small that 2.order pressure loss is negligible.
Figure 6.1 Pressure profile according to Forchheimer equation.
6.1.2 Flow characteristics
Figure 6.1 shows that the deviation from Darcy’s equation occurs
close to the well. Flow will hear be very close to steady-state.
there (near the wellbore). This will hold even if the conditions
further out is pseudo-steady-state, or even transient. Thus, the
well pressure can be expressed as
12 (6-3) p?p?q?FqwRooJ
By matching the (6-2) and (6-3), we get the following relation
for 2.order pressure loss parameter
2?B1oo (6-4) ?F?224?hrw
Figure 6.2 illustrates 2.order inflow characteristics, estimated
from (6-3), for the same parameters which formed the basis for
.-8 3 .9 32Figure 6-1: [J=1.01 10Sm/s/Pa, F=1.14 10Pa/(Sm/s) ] .
Figure 6.2 second order inflow characteristics
6.1.3 Estimate of 2.order inflow performance from measured data.
We can estimate 2.order inflow performance from measured well
pressure and production rates. For graphical estimation it is
practical to express (6-3) as
p?p1Rw (6-5) y???FqoqJo
Figure 6.3 shows pressures and production rates plotted
according to (6-5). If a straight line can be drawn through the
data points, the slope is equal to: F and the intersection with
the y-axis corresponds to: 1 / J
Figure 6.3 Graphical estimate of 2.order flow parameters
The points shown in Figure 6.3 are "simulated measurements" and
the straight line is adapted visually. The intersection with the
y-axis provides the estimate: 1 / J = 0.0115, productivity index:
J = 87 Sm3/d/bar (Estimation of "F " is left to the reader)
6.1.3 Turbulence factor correlated to the rock parameters
The ?-factor is often correlated to porosity and permeability. The correlation by Tek & al (1962), Katz & Coats (1968) is
illustrated in Figure 6.4
9???5.5/0.304810?? (6-6) 1.250.75k?
k : permeability (mD)
? : porosity ( - )
-1???????????????? : turbulence factor (m)
Figure 6.4 Correlation of --factor in natural rock
In order to have consistency dimension wise, you must have
permeability exponent "0.5". It is not the case in (6-6). Golan
& Whitson /1991 / presented a dimension terms satisfactory
correlation of ?-factor in the dry sand. This is given in (6-7)
illustrated in Figure 6.5.
7???1.746/0.304810?? (6-7) 0.51.5k?
Figure 6.5. Turbulence Factor Correlation for unconsolidated
6 6.1.4 Second-order pressure losses in gravel packs
By gravel packing, we understand that the granular material is
placed between the rock and well to prevent flow of reservoir
fines into the well. The flow rate through such gravel packs is
often so high that it may give significantly pressure loss.
Experience shows that -factor for gravel pack corresponds better to the permeability of the reservoir rock than the permeability
of the gravel. This is explained by that the production will
transport fines from the formation into the sand pack. Unneland
(2001) provides an overview of different correlations and -
factor for gravel pack and comparison with data for North Sea
6.2 Flow below saturation pressure.
When the pressure falls below the saturation, gas is released.
This leads to a number of changes in fluid behavior and flow
- The total flow volume gets larger
- Viscosity of oil becomes larger because of less dissolved gas
- gas and oil fill pores and will therefore have different
- Density difference may make gas will filter upwards
Such changes affect the pressure loss.
6.2.1 Vogel’s inflow performance
Vogel (1968) performed numerical simulations with pressures
below saturation. By systematizing numerical results he found
that the inflow performance could be expressed by the following
2??????qppoww?????? (6-8) ?1?0.2?0.8??????qppmaxss??????q: Extrapolated production capacity at zero pressure well max
p: Reservoir pressure at saturation s
Vogel’s characteristics curve (6-8) shows that the productivity
index decreases with decreasing well pressure.
The comparable dimensionless inflow performance for constant
productivity index reads
qpow?1? (6-9) *pqs
From (6-9) follows that the comparable production capacity at
* zero well pressure: q=Jp RMarshall B. Standing observed that at saturation pressure (6-8)
and (6-9) should provide the same productivity index. Thus, the
gradients of (6-8) and (6-9) should be equal at saturation
pressure, as illustrated in Figure 6.6. This gives the following
*q?1.8q (6-10) max
Figure 6.6 Comparison of inflow performance relations
Combining (6-9) and (6-10), the Vogel parameter may be related
to productivity index estimated above the saturation pressure
Jp?sq? (6-11) max1.8
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