Multiphase Flow (Chp. 3)

7/31/2019 Multiphase Flow (Chp. 3)

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7232/dnr 1Multi-Phase Flow in Thermal Recovery

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CHAPTER 3

MULTIPHASE FLOW

IN HEAVY OIL RESERVOIRS


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MUTIPHASE FLOW IN HEAVY OIL

RESERVOIRS

In thermal recovery operations, the heat balance and heat

transfer considerations (discussed so far) are only a part of

the problem.

In this Chapter, let us consider the problem of the

displacement of oil by water, by hot water and by steamand the mechanisms governing this displacement.


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Darcys Law for Two-Phase Flow

Assignment #1 dealt with deriving various forms of Darcys lawfor single-phase flow in different units.

When two (or more) separate phases are flowing simultaneouslythrough a porous solid, then the flow of each phase is less thanthat which would be produced by the same pressure gradient if itwere the only phase present.

The individual fluids compete with each other as they flowthrough the medium and impede each others progress.

This is accounted for by introducing new variables, calledRelative permeabilities, into Darcys equation.

These empirical correction factors are obtained for any particularporous medium by experimental measurements.


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The Darcy equations for one-dimensional flow of oil and water in a bed

of cross-sectional area A that is inclined at an angle (see Figure 3.1) are:

3.1

( Note: Po is not same as Pw, because of _________________________ )

where k is the absolute permeability, kro and krw are relative permeability of oiland water respectively.

The product, k.kr, is the effective permeability of oil and similarly the effective

permeability to water is k.krw.

sin.

gx

PAkkq o

o

o

r

oo

sin.

gx

PAkkq ww

w

r

ww 3.2


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Figure 3.1: Darcys Law for Two - Phase Flow


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Absolute permeability

Permeability of a porous medium to a fluid when the pore

space is 100% saturated with the same fluid.

Effective permeability

It is the measured permeability of a porous medium to afluid when the saturation of that fluid is less than 100%.

Relative permeability

It is the ratio of effective permeability of a fluid at a givensaturation to the absolute permeability.


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NOTE

The base permeability used in defining relative permeability

varies in practice.

Some use the absolute permeability to water.

Some others prefer the effective permeability to oil at connate(or irreducible) water saturation while some others use

absolute air permeability.

Care should be taken to know which of these base permeabilities

was used in a particular set of relative permeability data.

Please refer to rel perm ppt in P-drive for additional information.


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Since the effective permeability of a phase is dependent on its

saturation, so is its relative permeability.

Thus relative permeabilities are plotted against phase saturations.

The shapes of these curves are a function of the distribution of fluids

in the porous medium which, in turn, depends on the saturation

history and on the wetting characteristics of the rock

Relative permeability is presented in a graphical form as a function

of saturation.

The data are often referred to as drainage or imbibition curves.


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Imbibition relative permeability

A displacement where the wetting phase saturation is increasing.An example is water flood of water-wet rock, or coring with awater base mud.

Drainage relative permeability

A displacement where the non-wetting phase saturation isincreasing (or the wetting phase saturation is decreasing).

An example is gas (non-wetting phase) displacing oil duringprimary depletion, or waterflooding in an oil-wet core, or gasexpelling fluid during core recovery.

Water displacement of oil differs from gas displacement of oil. Water

normally wets the rock and gas does not.

This difference in wetting characteristics results in different relativepermeability curves for the two displacements.


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Important Factors Affecting Displacement of

Oil in the Reservoir

Rock Properties: Permeability, homogeneity, capillarity and

wettability (Fracturing of rock is included within permeability and

homogeneity)

Oil properties:VISCOSITY!Interfacial tension against the displacing phase.

Displacing Fluid Properties: Viscosity, tendency to change phase

(condensation and evaporation), potential to transfer heat

Conditions and Geometry:Flood velocity (injection and production

rates), geometric parameters (dip, spacing and pattern)


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Relative Permeabilities

In most situations, the relative permeabilities are largely independent of

flow velocity and of the fluid viscosities.

Figure 3.2: Effect of Viscosity Ratio on Oil - Water Relative Permeabilities. (Leverette, 1939).


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Figure 3.3: Comparison of A vs. S Relations for Two Liquid Pairs of Widely Different Interfacial Tensions.

Leverette (1939) also reported relative permeabilities to be independent of

interfacial tension (over a limited range from 534 dynes/cm).

These dated results have been used by Butler (1998) to generate hypothetical

relative permeability curves shown in Figure 3.4.


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Figure 3.4: Hypothetical Relative Permeability Curves used by Butler (1998).

The very unfavorable mobility ratio involved in the displacement of heavy oils

could result in pronounced flow rate sensitivity of relative permeabilities as shown

by Maini (1998). Figure 3.5 and 3.6 are reproduced from Mainis paper. (JCPT,

April 98, p. 56).


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Figure 3.5: Effect of Flow Rate on Relative Permeability at Viscosity Ratio of 405.

Figure 3.6: Effect of Flow Rate on Relative Permeability at Viscosity Ratio of 43.


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Since, theoretically, the relative permeabilities should beindependent of flow rate, these results raise some troubling

questions as to their applicability in describing the field

behavior.

Considering the large uncertainties usually involved in other

data (such as oil viscosity and reservoir heterogeneity), the

relative-permeability-based description of flow in

reservoirs appears to be more than adequate for all practicalpurposes.


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Displacement Models

Muskats model:Displacement assuming the existence of a sharp

interface between the displacing fluid (such as water) and the oil.

As the water advances, the oil is displaced, and it is assumed thatthere is no mingling of oil and water.

Buckley-Leverette Model: In this model, allowance is made for thepossibility that there can be an intermingling of oil and displacingfluid, which changes the front of the Muskat model to an advancinginterfacial region with a thickness that increases as the processproceeds.

This intermingling can also be accomplished by large-scalefingering. (See handout paper by Leverette (1939),and Buckley andLeverette (1942)).


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Heavy oil displacement concept combines some of the characteristics of both

Muskat model and B-L model by assuming that the flow of the displacing

fluid occurs as numerous fingers.

It is assumed that at any section in the reservoir, the flowing fluid in some pores

consists entirely of oil and that in others it is entirely water or steam. This is

called the stratified flow assumption.

The 1-D displacement of oil by steam is a coupled process. Within the steamchambers there is an isothermal displacement of oil by steam.

Beyond the steam chamber, the oil and water that have left the steam chamber

flow through the reservoir. In this region, there is also some displacement of

oil by the flowing water.

Between these two regions lies the advancing condensation interface, the

position of which is determined largely by heat balance that was discussed

earlier.


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The Fractional Flow Equation

The fractional flow equation relates the composition of the flowing steam

to the saturation in the rock.

Derivation:

By equating oil flow to total flow minus waterflow and using

equation (3.1), we can write

sin.

gx

PAkkqqq o

o

o

r

wtoo

3.3

sin..

gx

PA

kk

q

kk

qo

o

r

ow

r

ot

oo

3.4


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By using a rearranged version of equation (3.2) and the definition of capillary

pressure (Pc = PoPw) and = w - o, we can get the following fraction waterflow equation:

3.5

This equation contains within it the effects of four different factors upon the fraction

flow:

1. The ratio of the viscosities of the two fluids.

2. The dependence of relative permeability upon saturation.

3. The effect of gravity. For upward displacement shown earlier (in Figure 3.1)of a light oil with water, is positive. Since sin is also positive, thegravity term has the effect of decreasing the water fractional flowwhich

is a desirable effect.

4. The effect of the capillary pressure.

w

o

o

r

rw

c

ot

r

t

ww

k

k

o

gxP

q

Akk

q

qf

1

sin

.

1


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x

S

S

P

x

P w

w

cc

3.6

The terms Pc/S

wand S

w/x are both negative for water displacing oil

in a water-wet system.

The Pc/Sw is negative because, for water-wet systems, Pc monotonically

decreases with increasing Sw as shown in a typical plot below.

The term Sw/x is negative when water is displacing oil from a core

having a high initial oil saturation.

Thus, it follows that the effect of the capillary pressure term, in this case,

will be to increase the water flow.

This means that water tends to be drawn ahead into the oil-rich zone by

capillary attraction.


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For the sake of simplicity, let us assume that

1. Gravity term is zero (either the system is horizontal, = 0 =sin , or the two phases have the same density, = 0)

2. Capillary pressure term can be neglected. Except in the

immediate vicinity of the displacement front, this is often a

reasonable assumption, since the magnitude of the saturation

gradient is small.

The fractional flow equation, with the above noted assumptions,

reduces to:

w

o

r

r

o

w

w

k

kf

1

1

3.7


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This shows that the fraction of water in the flowing stream (and

hence in the produced stream) depends on the relative permeability

ratio as well as the viscosity ratio.

For a fixed saturation (i.e., fixed value of kro/krw ratio), an increase

in oil viscosity results in increased water fraction in the flowing

stream.

This is the reason for the poor performance of waterflood in heavy

oil reservoirs.

These effects are demonstrated in Figure 3.8, which uses therelative permeability plot of Figure 3.4 for calculating fw.


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Effect of the Gravity Term on Fractional

Water Flow

The dimensionless gravity term in equation (3.5) is:

This term becomes more significant when the density difference is

higher, when the reservoir is steeply inclined, when the total flow is

low and when the ratio k/o is high.

The term can be interpreted as being proportional to the ratio ofgravity potential gradient (.g.sin) to the viscous potential gradient

Akk

q

g

or

ot

sin

(3.5)

Akk

q

or

ot.


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Figure 3.9 shows the effect of the gravity term on fractional water flow.

Figure 3.9: Effect of Gravity Term on Fractional Water Flow.

1.0

o

w


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High velocities (that is high viscous forces) tend to overcome gravity

effects. Gravity tends to stabilize the flood if the heavier fluid is

below the lighter fluid.

For waterflood of heavy oils where the densities of the two fluids are

very similar and when o is very large, gravity has little effect.

As increases, fw decreases markedly for any Sw.

The curve for = -5 in the figure corresponds to a case where thewater flow is downward; there is counter-current flow in the range

where fw is greater than 1.0.

This means the oil is rising through the falling water. Similarly, withpositive values, there is a range where fw is negative.

This means water is falling counter-currently through a rising oil

stream.


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The Buckley-Leverette Displacement Theory

In 1942 Buckley and Leverette developed a theory that

provides a quantitative description of the displacement ofone fluid from a porous medium by an immiscible flood.

They introduced the idea of a dispersed front (between the

displacing and the displaced fluids) which is essentially azone of varying water saturation.

This theory makes use of the fractional flow conceptdiscussed earlier.

Please refer to my 4083 notes and the B-L paper in the P-Drive for more on this.


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Velocity of the shock frontWhen oil is displaced from a porous medium by a waterflood, a front

advances through the reservoir, and across this front is an abrupt change

in the saturation profile leading to a discontinuity in Sw/x.

There may also be a discontinuity in Sw, but this is not necessary.

By striking a water material balance, the velocity of the front (dxf/dt) can be

written as below in terms of the fractional water flow behind the front and

the saturation behind the front:

The bracketed term on the right-hand side depends on the fractional flow

curve.

It is the slope of the straight line joining the point corresponding to the front

conditions to the point (Swi, 0) on the fractional flow graph -as shown in

Figure 3.10.

wiwf

wftf

SS

f

A

q

dt

dx

3.9


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The maximum value of this slope can be found by drawing

a tangent as shown in this figure.

When this maximum slope is used, the above equation

yields the maximum velocity at which the saturation shock

front can move.

A shock front at this condition will overrun any front

having a different saturation.


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Figure 3.10: Shock Front Conditions .


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The Saturation Behind the FrontBehind the front, the water saturation falls from 1Sor right at the point of

injection down to the saturation at the shock front, as found by the

construction in Figure 3.10.

Figure 3.11 below shows a differential reservoir element.

The saturation within this stationary element will, in general, tend to change

with time because the concentration of the stream flowing out of it is not the

same as that of the stream entering the element; the element is being

depleted of oil.

Figure 3.11: Conditions behind the Front. Sw Varies Continuously .


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The following continuity equation including the depletion (time

dependent term) describes the situation:

3.10

Where fw = qw/qt as before (fractional waterflow) which is assumed to

be a function of saturation alone.

Behind the front, Sw is a continuous function of x and t, so

3.11

= 0 at constant Sw.

From this the rate of advancement of the front, can be obtained

as:

3.12

t

wt

x

w

x

f

A

q

t

S

dtt

Sdx

x

SdS

x

w

t

ww

wSt

x

t

w

x

w

S

x

S

t

S

t

x

w


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which can also be written as:

3.13

where N is the number of pore volumes of injected water (vol. of

pores up to point x) required to bring the water saturation at position

x to the level corresponding to fw.

w

wt

SS dS

df

A

q

t

x

t

x

ww

'

11

w

w

wS

t

f

dS

dfAx

tqN

w

Note: fw = dfw/dSw

= slope of fractional

flow curve

h df d i i f hi h b iwf


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The term dfw/dSw originates from which can be written

as an ordinary differential, since fw is assumed to be a function of Sw

alone.

This equation (3.13) is known as the Buckley-Leverette equation.

It shows that the velocity at which a plane of a fixed saturation

advances is proportional to the average fluid velocity multiplied by afunction of saturation.

Note that Sw must be greater than Swf, the water saturation at the front.

If not, the assumption that Sw is a differential function of x and t,

which is implied by the use of equation 3.11, is not correct.

The term dfw/dSw is the slope of the fractional flow curve which is

plotted in Figure 3.12 below.

t

w

t

x

S

x

The tangent to the curve drawn from the point (S 0) determines the conditions


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Figure 3.12: Slope of Fractional-Flow Curve.

The tangent to the curve drawn from the point (Swi, 0) determines the conditions

at the shock front.

The conditions behind the shock front correspond to that part of the fw versus

Sw curve for which Sw > Swf, that is, the upper right-hand part of the saturation

curve. At the shock front, the saturation drops rapidly from Swfto Swi.

C diti t B kth h


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Breakthrough occurs when the front reaches the limit of the reservoir, or xf= L. Up

until this time, only oil is displaced from the reservoir. The quantity of water that has

been injected can be found as qtt as below

3.14

or the number of pore volumes to breakthrough can be calculated as

3.15

The above equation can be used also to calculate the number of pore volumesinjected at any time, t when the front is at x, by replacing L

with x and with .

fw

w

t

dS

df

ALtq

fw

w

t

ds

dfAL

tqN

1

fw

w

ds

df

xw

w

ds

df

Conditions at Breakthrough

Recovery at and after Breakthrough


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At some time after breakthrough, the saturation profile will be as in Figure 3.13

below.

The average saturation over the length of the reservoir will be given by

3.16

Combining this with the Buckley-Leverette equation (3.13) (with L substituted for x),

we can get

3.17

Figure 3.13: Average Water Saturation after Breakthrough.

Lt

wLw fAL

tq

SS 1.

'

1

L

LwLw

f

fSS

Recovery at and after Breakthrough

1 LL

ff

Sw - SwLOR


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This was first published by Welge (1952). The geometric significance of this

equation can be seen in Figure (3.14) below.

Figure 3.14: Relation between Outlet Conditions and Average Water Saturation.


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For any point on the fw curve, which lies at or above the point

corresponding to the shock front (Swf, fwf), a tangent drawn upward

intersects the line fw = 1 at a water saturation corresponding to the

average water saturation in the reservoir.

This is indeed a remarkably simple answeran almost magical result!

The Buckley-Leverette-Welge method is applied as follows:

Construct the fractional flow curve from relative permeability data.

Draw a tangent from the base point of the curve and find the

conditions at the shock front (Swf, fwf) and the average water

saturation at breakthrough Swf, from the intersection with the line

fw=1.


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3 Calculate the oil recovery at breakthrough from :

pore volume of oil 3.18

recovered at breakthrough

4 Calculate the time of breakthrough from the total injection volume (qtt)

from equation (3.14) and the injection rate.

5 For various arbitrary values of SwL, draw the tangents and calculate the

corresponding recoveries and injection volumes.

wiWf SS


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Analysis of Steam Flood Using the Buckley-Leverette

Theory

For the sake of simplicity we consider only a one-dimensional adiabatic

steam flood, an intermediate stage of which is shown in Figure 3.15 below.

Figure 3.15: Diagram of Adiabatic, One-Dimensional Steamflood.


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Steam is introduced at the left and sweeps through the steam-swept

region, where the temperature is constant at Ts.

(The small drop in steam temperature due to pressure drop is ignored).

At the condensation front, all steam condenses abruptly and the liberated

heat of condensation and the sensible heat of condensate are absorbed in

raising reservoir and its contents from the initial temperature TR to the

steam temperature Ts.

The assumption is made that there are no lateral heat losses.

(Methods of determining these lateral losses have been discussed in

section 2.1 and 3 before).

The fluid velocities are assumed to be uniform across the cross section.


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In the steam-swept region, steam flows at a relatively high velocity and

moves the oil forward at the saturation temperature.

The displacement of oil in this region can be described quantitativelyusing the Buckley-Leverette theory in the same manner as for

waterflooding.

[See (1)Shutler N.D., and Boberg T.C.,A One-Dimensional Analytic

Technique for Predicting Oil Recovery by Hot Water or Steam Flooding,SPEJ p.489-498, Dec1972; (2)Boberg, T.C., Thermal Methods of Oil

Recovery, J. Wiley, New York, 1987]

No condensation of steam occurs within the steam-swept region, since it

is at constant temperature and the heat losses are assumed to be absent.

The water saturation in the steam-zone is close to Swi, the irreducible

water saturation.


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If the steam is wet (quality less than 100%) or if there are heat losses, a

higher water saturation is required in order to move the liquid water to

the interface (discussed later).

Hence an inherent assumption in this particular treatment is of injecting

dry and saturated steam and absence of heat losses. (More steam has

flushed the left part of the steam-swept zone).

The oil saturation increases from left to right in the steam-swept regiondepicted in the above schematic diagram.

The position of the condensation front is determined by a heat balance,

by equating the heat liberated by the condensation of steam and by

cooling of the condensate from the start of the process to the time

corresponding to the situation in the above diagram to the sensible heat

within the steam-swept region.


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It is intuitively obvious that the volume of the steam-swept region

increases with the quantity of injected steam.

At the condensation front, steam saturation drops to zero, the watersaturation increases and the oil saturation rises abruptly.

The oil which is swept from the steam region, together with the

condensate, flows forwards from the condensation front into the

waterflood region.

In the waterflooding region, some oil is displaced by the flowing water.

The oil displaced from the steam chamber also flows through the

waterflood region.

Immediately beyond the condensation front, the saturation conditions

correspond to the fractional flow of oil and water that is leaving the front.


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Figure 3.16 below depicts the general distribution of saturations within

the steam-swept region.

The abscissa is the dimensionless distance and is equal to dfs/ds

s(the

slope of the fractional steam flow curve fs).

Figure 3.16: Saturations in the Steam Chamber.

Th i ith t t il t ti S l if th


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The region with constant oil saturation, Sor, occurs only if the

curve of fs is not tangent to the saturation axis at its upper

terminal point (otherwise, Sor=0).

The water saturation is constant at Swi throughout because there is

no water flow within the steamflood region.

As mentioned before, if the injected steam is wet (i.e., quality

Multiphase Flow (Chp. 3)

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