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# 6 Deviation from linear Inflow performance

By Edwin Carter,2014-06-18 08:30
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6 Deviation from linear Inflow performance

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

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

sand

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

fields.

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

conditions:

- 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

permeabilities

- 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

non-dimensional relationship

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

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

relationship

*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

References

Asheim, H.: "Potential flow principles extended to simultaneous

flow and gas and liquid in Porous Media"

Recent advances and challenges in oil recovery, Moscow, May 13-23, 2003

Brooks, RH, and Corey, AT: "Properties of Porous Media affecting Fluid Flow",

J. of Irrigation and Drainage Division. Proc. Ascea, vol. 92, no. IR2, 1966, 61

Burdine, NT: "Relative Perm Ability Calculations from pore Size Distribution Data", Trans. AIME 198, 1953, 71

Fetkovich, MJ: "The Isochronal Testing of Oil Wells", 48 Annual Fall Meeting of SPE, Las Vegas, Sept. 30-Oct. 3, 1973

Forchheim, Ph. "Wasserbevegung durch Boden"

Z. Ver. Deutsh. Ing., Vol. 45, 1901, p.1781-1788

Geertsma, J.: "Estimating the coefficient of Inertial Resistance in Fluid Flow Through Porous Media" SPEJ, Oct. 1974, 415

Golan, M., and Whitson, C.H.: Well Performance

Prentice Hall, N.J., 1986.

Standing, MB: Notes on Relative Perm Ability Relationships Department of petroleum, NTH 1974.

Tek, M.R., Coats, K.H. and Katz, DL: The Effect of Turbulence on Flow of Natural Gas Through the Porous Reservoir. Journal of P. Tech., July 1962, 799; Trans AIME, 225

Unneland, T.: Performance of high-rate gravel-packed oil wells Dr. thesis, NTNU, 2001

Vogel, JV: "Inflow Performance Relationships for Solution - Gas Drive wells",

Petr. Trans. of AIME, Journal of P. Tech. Jan. 1968, 83

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