Linear Displacement Effficiency_In waterflooding [Compatibility Mode]

CHAPTER IIICHAPTER III

Macroscopic Displacement Efficiency

Reservoir Engineeringg g

Learning Objectives of Lecture 13Learning Objectives of Lecture 13

Macroscopic displacement efficiencyWaterflood efficiency equationDesign parameters and variablesObjectives of waterflooding

Macroscopic Displacement Efficiency

• Macroscopic efficiency is used to describe the

p p y

Macroscopic efficiency is used to describe the displacement efficiency of a waterflood in a specified volume of the rock.

• In other words the term refers to the portion of the oil initially in place that water displaces from a unit volume of the reservoir.

• To discuss and quantify how water displaces oil from a reservoir of complex permeability and porosity we must first understand the behavior of a linear reservoir segmentfirst understand the behavior of a linear reservoir segment

having uniform properties.

Fractional flow equation

The fractional flow of water is defined as

ow

w

t

ww qq

qqqf

+==

substituting the Darcy’s law

( )woco SingPAk αρρ ⎥

⎤⎢⎡

−+∂

+1 ( )

o

w

rw

ro

cwo

tow

kk

gxqf

μμ

ρρμ

+

⎥⎦

⎢⎣ ∂=

1

(Note: the dip angle is positive with flow moving up-dip)

orwk μ

Fractional flow equation

Note SPP wcc ∂∂∂

As we do not have a way to calculate. xSxw

w

cc

∂∂=

∂

wSx

∂∂

In practice the capillary pressure term is neglected but not forgotten.

Typical fractional flow curve

0.8

1

0.4

0.6

fw

0

0.2

Fig.1 Fractional flow curve

00.2 0.4 0.6 0.8Sw

Significance of fractional flow

• For a given set of rock, formation and flooding conditions the fractional flow of water f is aconditions the fractional flow of water, fw, is a function of water saturation alone.

• If both fluids are flowing the fraction of water in the total rate at any location is dictated by Sw at that location

• If Sw varies with location fw varies accordingly

• It is therefore possible to have only water flowing at some point and no water flowing at another.

Fractional Flow

Variation of fw with location

qo qo

Large fw

Small fw

Small Sw

qw qw

Large Sw

Fractional Flow

Water Injection

Variation of Sw vary with location

Small Sw Large Sw

Factors affecting fractional flow

For a given saturation Sw, fractional flow varies as:

• rock wettability changes• the density contrast between fluids for a given

dipping condition changes• the dip angle and direction of flow (i.e. flow being

up-dip, horizontal or down-dip) variesp p, p)• the total flow rate within the reservoir varies• viscosity contrast between the fluids changes• reservoir permeability changes• reservoir permeability changes

Influence of Wettability

1

06

0.8

1

Wettability increases

0.4

0.6

fw

0

0.2

0.2 0.4 0.6 0.8Sw

Effect of capillary pressure on fw

As the capillary pressure decreases with increasing water saturation i e in a Pc curveincreasing water saturation. i.e. in a Pc curve

0c

w

PS∂

<∂

Since Sw must decrease with distance in the direction of flow:

0wSx

∂<

∂Thus the capillary pressure gradient is always positive

x∂

0cP∂

and its effect is to increase the value of fw.

0cx

>∂

Modes of immiscible displacement

• When water invades pore space of rock containing oil some oil will be displaced by water causing an increase in Sw

• With continued flooding more oil will be displaced from any given portion of the rock

• Initial reduction in So (increase in Sw) depends onInitial reduction in So (increase in Sw) depends on the rock/fluid properties specially relative pore size and geometry, permeability, viscosity and IFT.

• In the extreme case, all mobile may be displaced initially leaving only residual oil. This ideal mode is called Piston-like displacement.


Piston-like displacement• For extremely water wet rock the capillary pressure is large

over most S rangeover most Sw range.

• When water is injected in such a rock water is imbibed into most oil containing pores. Only the largest pores (with themost oil containing pores. Only the largest pores (with the smallest capillary pressure ) are spared

• Such imbibition of water expels oil out of all invaded pores leaving only residual oil saturation.

• With more water injection only water advances into new area since oil is immobile in the invaded area

• This mode of displacement results in near total flushing of mobile oil within the area swept by the injected watermobile oil within the area swept by the injected water

Piston Like Displacement

Piston Like Displacement: Reservoir Conditions

Water Injection:qw

sw ≈1-Sor Sw≈ swi

qw ≈qt

qo ≈0

qw≈ 0

qo≈ qt Sw

Flood front

Modes of immiscible displacementFrontal advance displacement• For ordinary water wettability, Pc is large only at the lowest

Sw range.g• When water is injected into such a rock, water is imbibed into

the smaller pores only. Only some of oil is expelled leaving behind a significant amount of mobile oil.

• With continued water injection water imbibes into larger pores and larger pores within the invaded area. Thus in this

b th il d t fl i lt larea both oil and water flow simultaneously

• Sw within the invaded area decreases in the direction of flow, but increases with time at a specific locationbut increases with time at a specific location.

• This mode of displacement oil recovery depends on how much So is reduced in the swept area as a function of t and xmuch So is reduced in the swept area as a function of t and x

Frontal Advance Displacement

Frontal Displacement: Reservoir Conditions

Water Injection:qt

sw>swi

sw≈ swi

qw ≈ 0

qw<qt

qo>0

qw ≈ 0

qo≈ qt Sw

fw

invaded area uninvaded area Flood front


• In modeling the immiscible displacement bothIn modeling the immiscible displacement both modes are useful if care is exercised in regarding the range of applicability of each mode.

• In addition one has to be consistent in using a mode throughout the whole system. Since modeling is based on determining and combining efficiencies in 1D, areal and vertical to obtain volumetric efficiency whichever mode is selectedvolumetric efficiency whichever mode is selected it should be applied throughout


• For the pistonlike displacement two p palternatives are possible for residual oil saturation in an invaded area, namely Sor from an end point pemeability k =0 andan end point pemeability kro=0 and

from a fractional flow curve. Use of values are closer to the reality specially for

l M

or wfS S=wfS

low M cases.• In this work we will cover the more general

case of frontal advance displacement and will pbriefly refer to pistonlike displacement’s applicability for the considered case studies.

Frontal advance theory

Unsteady state flow

The immiscible displacement of one fluid by another is an unsteady state process, because thesaturations change with time. This causes changes in kr’s and either pressure or phase velocities.

Next Figure shows four representative stages of a linear waterflood at interstitial water saturation.


[After Whillhite]


Buckley-Leveret frontal advance model is used topredict unsteady displacement performancepredict unsteady displacement performance depicted in the previous figure.

It i l ti l d l d hi lIt is an analytical model and uses graphical techniques developed by Welge.

The model is a valuable tool for insight into displacement mechanism of oil by water in addition, it is proven to be sufficient both by lab and field data.it is proven to be sufficient both by lab and field data.


Using assumptions of (1) incompressible flow(2) fw is a function of saturations only and (3)(2) fw is a function of saturations only and (3)No mass transfer between phasesThe frontal advance equation is derived as:

w wtS dfqdxv == Eq.1wS

wSw dSAdt

vφ

==


The frontal advance equation states that the rate of d (i l it ) f l f ifi d tadvance (i.e. velocity) of a plane of a specified water

saturation is equal to the total fluid velocitymultiplied by the change in composition of thep y g pflowing stream caused by change in the saturation of the displacing fluid.


Within the invaded area by the injected water, thelocation x of a particular saturation S is found bylocation, xSw of a particular saturation Sw is found byintegrating Eq.1 with respect to time as follows:

wt dftqEq.2

ww Sw

wtS dS

dfA

tqxφ

=

In terms of dimensionless variables:

wdfQ Eq.3wSw

wiD dS

fQx =


For a given Sw, by computing we can determine the location of that Sw, and

wSw

w

dSdf

Swxhence the location of all saturations in the system (i.e the saturation profile in the system at a particulartime, ).time, ).

We can also determine saturation history at a particular distance as wellparticular distance as well.

Setting XD=1 in Eq3, states that the required pore volume injection to increase end point saturation to that particular saturation is equal to the inverse slope of fw curve at that saturationo w cu e at t at satu at o


Examining the following figure and frontal advance equation w

SwtS

Sdfqdx

v ==reveals: wS

wSw dSAdt

vφ

1. Intermediate saturationsf t th ll

78

are faster than small saturations2. Faster saturations

123456

fw'must catch up small

saturations and must overcome them

Fig. x. Derivatives of the fractional flow curve for example 3.5 in textbook

01

0 0.2 0.4 0.6 0.8 1

Sw

overcome them3. Result should be formation of a saturation di ti itdiscontinuity


Thus the expected saturation profile is as follows :

1-Sor

wfS

wSx

fx

[After Whillhite][ ]

Frontal advance theoryThe figure shows when water enters the reservoir water saturation in the invaded area builds up to Swfb f i f d I tbefore moving forward. In essence, water moves as a bank whose front is always maintained at Swf called as front saturation.

In the invaded area, water saturation is equal to (1-S ) at the inlet and decreases in flow direction(1 Sor) at the inlet and decreases in flow direction down to Swf at the front. The small and slows saturations ranging Swi to Swf merged into Swf .

The positions of saturations from Swf to (1-Sor) is determined from the frontal advance equation and fractional flow curve.

Frontal advance theoryHow to determine this Swf in the fractional flow curve?

The answer is much easier to describe mechanically but quiteThe answer is much easier to describe mechanically but quite involved to explain mathematically.

The mechanical description is that you have to draw a tangent to the fw curve through the point (Swi, fwi).

11

0 50.60.70.80.9

f w

fwf

(S i,f i)

0.10.20.30.40.5w

Swf

(Swi,fwi)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6500

0.70.24 S w

wf


An elegant method for the determination of the floodfront saturation and is based on the mathematicalconstruction of Welge which will not be repeatedhere. But several other useful relations from Welge’swork will be utilizedwork will be utilized.

The Welge’s equation for the average water saturation f th i t l ifor the interval x1<x<x2 is,

121122 fftqSxSxS wwtww −⎟⎞

⎜⎛−

=1212 xxAxx

S w −⎟⎠

⎜⎝

−−

=φ


Welge’s equation is general. But we mostly used itby setting x1=0 (inlet) and x2=L (outlet) unless

th i t t dotherwise stated.

( )22 1 wt

ww fA

tqSS −⎟⎠

⎞⎜⎝

⎛+=

φ 0222 fQSS iww +=

The Welge’s equation relates three factors of prime interest in waterflooding;

2Ax ⎠⎝ φ

interest in waterflooding;

1. The the average water saturation and hence the iloil recovery,

2. Cumulative injected pore volume, 3. Oil cut and hence producing WOR.


The two important contributions from Welge’s work are:are:

the flood front values ( )are determined by d i t t li t th f ti l th t

wfwf fS ;drawing a tangent line to the fractional curve that originates from the point ( ). wiwi fS ;

In addition intersection of that tangent line with axis will give us a very valuable parameter

namely, (i.e the average saturation within the1=wff

fSnamely, (i.e the average saturation within the invaded area by the injected water).

wfS


We can also show that:

The intersection of a tangent line of any point with axis will give us, (i.e the average

t ti b hi d th l ti f th t t ti )1=wff wS

;w wS f

saturation behind the location of that saturation).Also:

dfw

D i Sww

dfxx QL dS

= =w

Swi w

dfxQ L dS

=

The inverse of the slope gives the pore volume injection wrt pore volume of reservoir behind the location of that saturationlocation of that saturation.

Estimating the displacement performance for a linear waterflood at constant ratefor a linear waterflood at constant rate

In simulating the displacement performance we arei t t d i d t i iinterested in determining;1. Volume oil displaced , Np2. The rate of oil production, qop q3. The volume of water that must be handled per

volume of oil after water production begins, i.e. WOR;WOR;

4. And we want to determine them as a function of time.

U ll di id th l i i t BT dUsually we divide the analysis as prior to BT and post BT performance estimation.

Performance prior to BT

The computation of performance prior to BT is quite straight forward. Prior to BT all injected q g jwater is retained in the reservoir. Since the fluid are considered incompressible prior to BT the production of oil must equal injected water if the

iN W= 2 if f= 2 2 tq f q=

production of oil must equal injected water if the initial water is immobile:

p iN W 2w wif f 2 2w w tq f q=

2 1o wif f= − 2 (1 )o wi tq f q= − 2 2

2 21w w

o w

f fWORf f

= =−

' 't is wf wfwf

q t Wx f fA Aφ φ

= = 1wDSw

i w

dfx for any Sw between Swf and SorQ dS

= −

Example Project

A waterflood is under consideration for a “shoestring” reservoir that is 300 ft wide 20 ft thickshoestring reservoir that is 300 ft wide, 20 ft thick and 1000 ft long. The reservoir is horizontal and has a porosity of 0.15 and an initial water saturation of 0.363, which is considered immobile. It is proposed to drill a row of injection wells at the one end of the reservoir and flood the reservoir by injecting water at y j ga rate of 338 B/D. viscosities of oil and water are 2.0 and 1.0 cp respectively.

Example Project

Relative permeability data corresponding to the displacement of oil by water are given by the p y g yfollowing equations.

SS −

Th id l il t ti i 0 205 B

56.2)1( wDro Sk −= 72.3)(78.0 wDrw Sk =iwor

iwwwD SS

SSS−−

=1

The residual oil saturation is 0.205. Base permeability is the effective permeability to oil at interstitial saturation which is assumed to be equal qto the absolute permeability. Oil and water FVF’s are 1.0.

Fractional flow curve plotted0 7fS =

0.911

f 0 899

0.7wfS

0 60.70.8 fwf=0.899

f’ f=2 698

0 40.50.6

f w

f wf 2.698

0.20.30.4

Swf=0.665

0 3 0 35 0 4 0 45 0 5 0 55 0 6 0 65 0 7 0 75 0 800.1

0

wf

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.780.363 S w

Example project

From frontal advance equation:

' ' 40 3*5.615 2 967 740 4t iq t W ef f f

That means BT has not occurred yet, since fluid are assumed incompressible Np=Wp=40000 bbl

2.967 740.40.15*900000

t is wf wfwf

qx f f ftA Aφ φ

= = = =

assumed incompressible Np=Wp=40000 bbl

Verifying by:Oil displaced=V of invaded area*reduction in averageOil displaced Vp of invaded area reduction in average

So in invaded part

Reduction in average So=Increase in avearge Swg o g w0.70 0.363 0.337oi of wf wiS S S S− = − = − =

740.43*300*20*0.1540000 *(0.71 0.363)5 615p iN W bbl= = = −5.615p

Example project

For 80000 bbl injection BT occurs as calculated xSwfwould be twice of previous case which is greater th i l th f 1000 ftthan reservoir length of 1000 ft.

At the moment of BT the performance parameterswill jump to those of front saturation Swf.

And beyond BT S 2 begins to increase causing f 2And beyond BT Sw2 begins to increase causing fw2to increase as well. This means a corresponding increase in qw and decrease in qo.

The main problem is how to determine average saturations within the reservoir to calculate Np.

Post BT Performance AnalysisCumulative oil displaced, NpSince the fluids are considered to be incompressible, Np is equal to change in the volume of water in the system; and the

l ti di l t i l l t dcumulative displacement is calculated as:

or)( 2 wiwpp SSVN −= )( 2 wiwp SS

VN

−=

Where the average saturation value is obtained from the intersection of the tangent line to the fractional curve with

(again from Welge’s work).01=f

pV

(again from Welge s work).

Production ratesThe fractional flow of water is determined from the frontal

0.1=wf

The fractional flow of water is determined from the frontal advance solution for every value of Sw2. Thus qw2 and qo2 are given by

and2 2w w tq f q= 2 2(1 )o w tq f q= −

WOR the water oil ratio

WOR is a measure of the efficiency of the displacement process at a point in the process. It represents the volume of water that must berepresents, the volume of water that must be handled to produce a unit volume of oil and is expressed as:

2 2w wf fWOR

Time required for displacement

2 2

2 21w w

o w

f fWORf f

= =−

Time required for displacement

Since the injection rate is assumed constant, the time required to inject Q PV’s of fluid is:time required to inject Qi PV’s of fluid is:

LAqQ

tt

i φ=qt

Possible Questions Regarding Applications of F l d ThFrontal advance Theory

Consider fractional flow curve below

1

0.6

0.8

w

0.2

0.4

fw

Fig.1 Fractional flow curve

00.2 0.4 0.6 0.8Sw

Possible Questions Regarding Applications f F l d Thof Frontal advance Theory

1 What is the value of irreducible water saturation1. What is the value of irreducible water saturation

2. If the connate water saturation is 0.25 at the start f th fl d h t i th t t tiof the flood what is the average water saturation

at the breakthrough?

3. What will be the water saturation at the production end at the time of 1.5 PV injection

4. What will be the PV injected at the breakthrough and the average water saturation at the b kth h?breakthrough?

Possible Questions Regarding Applications f F l d Thof Frontal advance Theory

6. How many PV must be injected to recover 80% of OOIPOOIP

7. What is the distance of front at 0.5 PV injection j

8. Given PV and qt How long you need to inject to recover 0 6 OOIP?recover 0.6 OOIP?

9. When do you produce the water you inject for the fi t ti ?first time ?

10. What is the fractional flow of water at the breakthrough? and so on.

Example Project

You are asked to estimate the oil displacement rate and cumulative oil displaced as function of timeand cumulative oil displaced as function of time for the shostring reservoir described previously.

P f ti l fl d d t i tha. Prepare a fractional flow curve and determine the breakthrough saturation.

a. Prepare a graph of water saturation versus distance at the instant of time when the flood front (Swf) is 500 ft [152.4 m] from injection wells.front (Swf) is 500 ft [152.4 m] from injection wells.

Example Project

c Determine oil production rate as a function ofc. Determine oil production rate as a function ofPV’s injected to a WOR of 100:1. Plot the oil rate(bbl/D) versus PV’s injected.

d. Determine cumulative oil displaced (Np in bbl’s)as a function of PV’s injected (Qi) to a WOR ofi100:1. Plot Np vs Qi, with Np as the ordinate.

e. Plot the WOR vs N on semilog paper. Usee. Plot the WOR vs Np on semilog paper. Usesemilog scale for WOR on the ordinate.

SOLUTION: a. Prepare a fractional flow curve and determine the breakthrough

S w0.363

=

Sor 0.205:= Siw 0.363:=

curve and determine the breakthrough saturation. First set Sw values:

0.363

0.392

0.421

0 449

α1 1.0:= α2 0.78:=

0.449

0.478

0.507

0 536

n 3.72:= m 2.56:=

0.536

0.565

0.593

0 622

μw 1.0:= cp μo 2.0:= cp

0.622

0.651

0.68

0 709

B1

1 Sor− Siw−:=A

α1 μw⋅

α μ:=

0.709

0.737

0.766

0 7951 Sor Siwα2 μo⋅ 0.795

Relative permeability and fractional flow equations in l d f f ticlosed form functions.

SwDi

SwiSiw−

1 S S:=

i 1 Sor− Siw−

kroiα1 1 SwDi

−⎛⎝

⎞⎠

m⋅⎡⎢⎣

⎤⎥⎦:= krwi

α2 SwDi( )n⋅:=

fwi

SwDi( )n

( ) ⎛ ⎞m:=

i SwDi( )n A 1 SwDi−⎛

⎝⎞⎠

m⋅+

A B S( )n 1− 1 S⎛ ⎞mS( )n 1 S⎛ ⎞

m 1−⎡⎢

⎤⎥

f'wi

A B⋅ n SwDi( )⋅ 1 SwDi−⎛

⎝⎞⎠

⋅ m SwDi( )⋅ 1 SwDi−⎛

⎝⎞⎠

⋅+⎢⎣

⎥⎦

⋅

SwDi( )n A 1 SwDi−⎛

⎝⎞⎠

m⋅+⎡⎢⎣

⎤⎥⎦

2:=

i( ) i⎝ ⎠⎣ ⎦

Swi0.363

= SwDi0

= krwi0

= kroi1

= fwi0

= f'wi0

=

0.38

0.405

0.43

0.039

0.097

0.155

0

0

0.001

0.902

0.77

0.65

0

0

0.002

0.002

0.033

0.146

0.455

0.48

0.505

0 53

0.213

0.271

0.329

0 387

0.002

0.006

0.012

0 023

0.542

0.445

0.36

0 286

0.009

0.026

0.065

0 137

0.43

1.028

2.114

3 7770.53

0.555

0.58

0.605

0.387

0.444

0.502

0.56

0.023

0.038

0.06

0.09

0.286

0.222

0.168

0.122

0.137

0.256

0.418

0.597

3.777

5.721

7.068

6.942

0.63

0.655

0.68

0.618

0.676

0.734

0.13

0.182

0.247

0.085

0.056

0.034

0.754

0.867

0.936

5.466

3.584

2.039

0.705

0.73

0.755

0 78

0.792

0.85

0.907

0 965

0.327

0.425

0.543

0 684

0.018

0.008

0.002

0

0.973

0.991

0.998

1

1.026

0.448

0.152

0 0240.78 0.965 0.684 0 1 0.024

Fractional flow curve plotted

0.911

0 60.70.8

0.40.50.6

f w

0.20.3

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.800.1

0

0 780 363 S 0.780.363 S w

Determination the flood front mathematically not hi llgraphically

Swf 0.7:=

SwfDSwf Siw−

1 S Si:=

1 Sor− Siw−

k f α1 1 S fD−( )m⋅⎡⎣

⎤⎦:= k f α2 S fD( )n⋅:=krof α1 1 SwfD( )⎣ ⎦:= krwf α2 SwfD( ):=

S bt t th d i ti i f ti l f f th tSubtract the derivative in functional form from thatobtained by slope of the graph.

Solve by a root finding method for SWfD and then t S fcompute Swf from

b. Preparation of a graph of water saturation versus distance at the instant of time when the flood front (Swf) is 500 ft [152.4 m] from injection wells. Discretize the saturation range for plotting Swf to 1-Sor

Then calculate Qibt as follows:

Determine number of PV injected when dimensionless front distance is 500/1000

and position of each saturation value for Qi=0.185

Plot the saturations vs dimensionless distance

c. Determine oil production rate as a function of PV’s injectedto a WOR of 100:1. Plot the oil rate (bbl/D) versus PV’sinjected.

WORa 100:=

WORafwa

WORaWORa 1+

:=fwa 0.99=

From a root finding algorithm we obtain Swa=0.728

Th t f’ f i f t ti b tThen compute f’w for a series of saturations betweenSwf and Swa.

Plot the results

d. Determine cumulative oil displaced (Np in bbl’s) as afunction of PV’s injected (Qi) to a WOR of 100:1. Plot Np vspQi, with Np as the ordinate.

Sw_aviSw2i

Qii1 fw2i−( )⋅+:=

NpiSw_avi

Siw−:=

e. Plot the WOR vs Np on semilog paper. Use semilog scale for WOR on the ordinate.semilog scale for WOR on the ordinate.

WORi

f w2i1 f 2−

:=1 f w2i−

Appendix D- Derivation of frontal theory relations for applicationsrelations for applications

The fractional flow of water is defined as

ow

w

t

ww qq

qqqf

+==

substituting the Darcy’s law owt qqq

⎤⎡ ( )c

woc

to

o Singg

xP

qAk

fαρρ

μ ⎥⎦

⎤⎢⎣

⎡−+

∂∂

+=

1

o

w

rw

row

kkf

μμ

+

⎦⎣=1

orw μ

3.1 . Fractional flow equation

This gradient can be expressed as

xS

SP

xP w

w

cc

∂∂

∂∂

=∂∂

although the value of can be determined from the appropriate ( given the knowledge whether it is

w

w

c

SP

∂∂

pp p ( g gimbibition or drainage) oil water capillary pressure curve, the values are not available. Sw

∂∂

So in practice the capillary pressure term is neglected but not forgotten

x∂

but not forgotten.

3.1 . Fractional flow equation

As for the effect capillary pressure gradient, we know f h ill h i lP∂from the capillary pressure curves that is always negative. That is capillary pressure decreases with increasing water saturation

w

c

SP

∂∂

increasing water saturation.

Since the water saturation must decrease with distance in the direction of flow, we can deduce that is always negative as well.

xSw

∂∂

Thus the capillary pressure gradient has a positive sign and its effect is to increase the value of f

xPc

∂∂

sign and its effect is to increase the value of fw.

3.2 Frontal advance theory

Using the following assumptions:1 I ibl fl1. Incompressible flow2. fw is a function of saturations only3 No mass transfer between phases3. No mass transfer between phasesIn its differential form; the frontal advance equation is

derived as:

S dfqdxEq.3.4

w

wS

w

wtSSw dS

dfAq

dtv

φ==

wφ


The frontal advance equation states that the rate of d ( l ) f l f f dadvance (i.e. velocity) of a plane of a specified water

saturation is equal to the total fluid velocitythe total fluid velocitymultiplied by the change in composition of the flowing streamg p f f gcaused by a change in the saturation of the displacing fluid.


In other words the frontal advance equation states that h i S l h heach water saturation, Sw, travels through system at a

constant velocity that can be computed from the derivative of the fractional flow curve with respect toderivative of the fractional flow curve with respect to Sw.

It also states that as the total rate increases the velocity of the plane of saturation increases correspondingly and vice versa.


The location of a particular saturation is found by i i E 3 4 i h i f llintegrating Eq.3.4 with respect to time as follows:

dftqww S

w

wtS dS

dfA

tqxφ

=

It is more convenient to work with dimensionless variables hence dividing both sides by the total system length L;

ortS dftqx wdfQx =or w

wS

w

wtS

dSdf

ALtq

L φ= wS

wiD dS

Qx =


In the previous equation Qi ( or dimensionless time ) i d fi d l i j itD ) is defined as pore volume injection.

Provided that we a formula for or we canwSw

dSdf

Provided that we a formula for or we can determine it accurately from a plot of fw vs Sw; we can determine for each Sw, and hence the

wwdS

Swx w,location of all saturations in the system (i.e the saturation profile in the system at a particular time, )

Sw

We can also determine saturation history at a particular distance as well.


Since the term is difficult to evaluate, we expect h bl i d i i

xPc

∂∂

dfto have problems in determining .

However whenever then f can be computed0≅∂Pc

wSw

w

dSdf

However, whenever , then fw can be computed directly from the relative permeability data.

0≅∂x

The question is then, is there a water saturation range for which that assumption holds ?and Can we determine that portion of the saturation

values?values?


Examining the following figure and frontal advance equation w

wS

wtSSw dS

dfAq

dtdx

vφ

==qreveals several important points.

wwdSAdt φ

78

1. Intermediate saturationsare faster than small saturations

123456

fw'

saturations2. Faster saturations must catch up small

t ti d t

Fig. x. Derivatives of the fractional flow curve for example 3.5 in textbook

01

0 0.2 0.4 0.6 0.8 1

Sw

saturations and must overcome them3. Result should be formation of a saturation discontinuity


Is the previous conclusion supported by experiments?

Y ( t fi )Yes. ( see next figure )

The experimental observations of Terwilliger indicated that lower range of saturations all moved at the same velocity. i.e. The shape of the saturation distribution over this range of saturations was constant with time This zone was calledof saturations was constant with time. This zone was called STABILIZED ZONE.

Th l b d th t i th hi h f t tiThey also observed that in the higher range of saturations called NONSTABILIZED ZONE, saturations will move at different speeds and hence will continuously get separated apart.

3.2 Frontal advance theoryE i l b i f T illi FiExperimental observations of Terwilliger Fig.


Supported by the above experimental observations, it wasti li d th t th f t l drationalized that the frontal advance

solution is characterized by a saturation discontinuity at the flood front where the water saturation jumps from Swi, initial wisaturation to Swf, flood front saturation.

In conclusion we represented the stabilized zone whereIn conclusion we represented the stabilized zone whereall saturations moves at a constant speed as a line discontinuity i.e. the front

Thus the expected saturation profile is as follows from frontal advance equation.


In this figure, let’s represent the saturation at the front as S fSwf.

wfS

wSx

fx


Remember that frontal advance equation assumes Sw is ti d diff ti bl lcontinuous and differentiable along x.

Therefore, it is inappropriate to describe the situation at the front itself. Now we have an half of the answer for the appropriate portion of the fractional flow curve:

The saturation range from S i to S f of the fractionalThe saturation range from Swi to Swf of the fractionalflow curve constitutes the portion where frontal advanceequation does not apply,

dandthe saturations from Swf to (1-Sor) constitute the region of applicability for the frontal advance equation.

How to determine this Swf ?


The answer is much easier to describe mechanically but quite involved to explain mathematically.q p yThe mechanical description is that you have to draw a tangent to the fw curve through the point (Swi, fwi) such as the one in the following figurethe one in the following figure.

0.80.9

11fwf

0 30.40.50.60.7

f w

(Swi,fwi)

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.6500.10.20.3

0

S

Swf

0.70.24 S w


An elegant method for the determination of the floodf i d i b d h h i lfront saturation and is based on the mathematicalconstruction of Welge.

The Welge’s equation for the average water saturation for the interval x1<x<x2 is,1 2 ,

121122 ffA

tqSxSxS wwtwww

−⎟⎠

⎞⎜⎝

⎛−−

=φ 1212 xxAxxw −⎠⎝− φ


Welge’s equation is quite general and the point 2 can b di f h i l 0 d Q i hbe at any distance from the inlet, x=0 and Qi2 is the pore volume injection in the portion of the system from the the inlet to the point 2from the the inlet to the point 2.

( )1t ftqSS ⎟⎞

⎜⎛ ( )2

22 1 w

tww f

AxqSS −⎟

⎠⎜⎝

+=φ

0222 fQSS iww +=


Th W l ’ i l h f f iThe Welge’s equation relates three factors of prime interest in waterflooding;

1. The the average water saturation and hence the oil recovery,

2. Cumulative injected pore volume, 3. Oil cut and thus water cut and producing WOR.


Coming back to determining the flood front equation, l ’ id h f ll i h ilet’s consider the following three equations.

1 a material balance from the inlet to flood front1. a material balance from the inlet to flood frontIn-Out=Accumulation

Eq.3.9

SSfwiwftwit xASStqftq φ)( −=−

Eq.3.10wi

wiwf

f

t

fSS

xAtq

−

−=

1φ wif fφ


2. frontal advance solution written for flood front positionposition

t tq=

1φ

wfSw

wf

SfxA

∂∂φ

3. Welge’s equation written for the flood front position

Eq 3 12Eq.3.12f

twfwfwf xA

tqfSSφ

)1( −+=


Rearranging Eq 3.12

)1(wfwft

fSS

xAtq −

=φ

Then comparing the above three equations we see;

)1( wff fxA −φ

p g q ;

f fff −−∂ )1()1(

wiwf

wi

wfwf

wf

Sw

w

SSf

SSf

Sf

wf−

=−

=∂∂ )1()1(


Now, number 1:

wfwf

wf

Sw

w

SSf

Sf

−

−=

∂∂ )1(

Derivative means slope of the tangent line to the curve of the function at the point where derivative

wfwfSwwf

curve of the function at the point where derivative is calculated. The above relation dictates that the tangent line to the fractional curve at the flood gfront values ( ) must pass through the point wfwf fS ;( ); 1wf wS f =


Number 2:

if

wi

ff

wf

SSf

SSf

−−

=−− )1()1(

The above relation implies that the line connecting the

wiwfwfwf SSSS

p gpoints ( ) and ( ) will also have the same slope of the tangent at ( )

wiwi fS ; 1;wfS

wfwf fS ;


Number 3: If two lines are required 1. to pass from the same point and 2 t h th l2. to have the same slope

They have to be identical lines.

In other words you cannot pass two different lines from the same point and require them to have f p qdifferent slopes.

Therefore the line originating from ( ) and tangent to the fractional curve fixes the location of the front This is the mathematical proof

wiwi fS ;

the front. This is the mathematical proof.


As a result we can state that we can determine the fl d f l ( ) b dflood front values ( ) by drawing a tangent line to the fractional curve that originates from the point ( )

wfwf fS ;

fS ;point ( ).

In addition intersection of that tangent line with

wiwi fS ;

f gaxis will give us a very valuable parameter

namely, (i.e the average saturation behind the 1=wff

wfSfront).

3.3 Estimating the displacement performance for a linear

waterflood at constant rate

I i l i h di l fIn simulating the displacement performance we are interested in three quantities;

1. Volume oil displaced 2. The rate of oil production,p ,3. The volume of water that must be handled per

volume of oil after water production begins;4. And we want to determine them as a function of

time.

3.3.1 Cumulative oil displaced, Np

By definition,)( wiwpp SSVN −=

and again by definition the pore volume injection for h l l h ithe total length is

LAtqQ t

i φ=

Let’s revisit Welge’s equation expressed for the interval x=0 to x2 which is applicable at all times to 2 ppcalculate the average water saturation.

0222 fQSS i+= 0222 fQSS iww +


R i W l ’ iRearranging Welge’s equation

22 ww SSQ −

Also from the frontal advance equation2

2 1 wi f

Q−

=

q

12i f

Q⎞∂

=

2wSw

w

Sf

⎟⎟⎠

⎞∂∂


Equating the above two equations, one obtains

Eq.3.1922

21

2 ww

w

Sw

w

SSf

Sf

w−−

=⎟⎟⎠

⎞∂∂

Let’s also see the graphical form of the derivative

2w

g pIn the following figure

Fig (3.11) showstangent drawn to the fractional flow curvefractional flow curve at a saturation Sw2.

From (fig 3.11)

21 ww ff −=⎟⎟

⎞∂

Eq 3 20

22 weSw SSSw

−=⎟⎟

⎠∂

Eq.3.20


Comparing Eq 3.19 and 3.20, one can see thatSS

Hence, average water saturation in the region from

2we SS =

x=0 to x=xSw2 can be obtained by finding the intersection of the tangent to the fractional flow

ith li01fcurve with line.

Note XS 2 is the position of the saturation S 2

0.1=wf

Note XSw2 is the position of the saturation Sw2.


The above relation has an implication of great value.When XSw2=L, it allows us to determine:When XSw2 L, it allows us to determine:

the average water saturation, after the breakthrough.

wS

In other words, finding the intersection of the tangent t th f ti l fl t S lto the fractional flow curve at any Sw2 value between (Swf and (1-Sor) with line will give us the average saturation in the system when

0.1=wf

give us the average saturation in the system when that production end saturation is equal to that particular saturation where we draw the tangent. p g


Remember that Sw2 can also be Swf and for that special case we have arrived at the same intercept conclusion earlier about the intersection of tangent with 0.1=wf


Since the fluids are considered to be incompressible, Np is equal to change in the volume of water in the system; and the cumulative displacement is calculated as:the cumulative displacement is calculated as:

or)( 2 wiwpp SSVN −= )( 2 wiwp

p SSVN

−=

Where the average saturation value is obtained from the intersection of the tangent line to the fractional

pV

curve with3.3.2 Production ratesThe fractional flow of water is determined from the frontal

0.1=wf

The fractional flow of water is determined from the frontal advance solution for every value of Sw2. Thus qw2 and qo2 are given by andtw

w Bqfq 2

2 =tw

o Bqf

q)1( 2

2−

=wB oB

3.3.3 WOR the water oil ratio

WOR is a measure of the efficiency of the displacement process at a point in the process. It represents, the volume of water that must be handled to produce a unit volume of oil and is expressed as:

w

o

o

wwo B

BffF

2

2=

3.3.4 Time required for displacement

Si th i j ti t i d t t th ti i dSince the injection rate is assumed constant, the time required to inject Qi PV’s of fluid is

LAqQ

t i φ=qt

Linear Displacement Effficiency_In waterflooding [Compatibility Mode]

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