Waterflooding Macroscopic Effficiency Relations Compatibility Mode

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 Darcys law owt qqq

( )c

woc

to

o Singg

xP

qAk

f

++

=1

o

w

rw

row

kkf

+

=1

orw

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 lPfrom 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 dfqdx Eq.3.4w

wS

w

wtSSw dS

dfAq

dtv == w


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

tqx =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 = wSwiD dSQx =


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 computed0Pc

wSw

w

dSdf

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

0x

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

v ==qreveals several important points.

wwdSAdt

78

1. Intermediate saturationsare faster than small saturations

123456

f

w

'

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, lets represent the saturation at the front as S fSwf.

wfS

wS xfx


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

Swf0.70.24 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 Welges equation for the average water saturation for the interval x1


Welges 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 ( )22

2 1 wtww fAxqSS +=

0222 fQSS iww +=


Th W l i l h f f iThe Welges 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 ilets 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 )( =

Eq.3.10wi

wiwf

f

t

fSS

xAtq

=

1 wif f


2. frontal advance solution written for flood front positionposition

t tq = 1wfSw

wf

SfxA

3. Welges equation written for the flood front position

Eq 3 12Eq.3.12f

twfwfwf xA

tqfSS )1( +=


Rearranging Eq 3.12

)1(wfwft

fSS

xAtq =

Then comparing the above three equations we see;

)1( wff fxA 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

wfwfSw wf

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 ti =

Lets revisit Welges 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 Welges 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

=

Lets 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 weSwSSS

w=

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=wfgive 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 andtww B

qfq 22 = two Bqfq )1( 22

=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 PVs of fluid is

LAqQt i =qt

Waterflooding Macroscopic Effficiency Relations Compatibility Mode

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