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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.
Modes of immiscible 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
Modes of immiscible displacement
• 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
Modes of immiscible displacement
• 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.
Frontal advance theory
[After Whillhite]
Frontal advance theory
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.
Frontal advance theory
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
==
Frontal advance theory
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.
Frontal advance theory
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 =
Frontal advance theory
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
Frontal advance theory
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
Frontal advance theory
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
Frontal advance theory
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
Frontal advance theory
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.
Frontal advance theory
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
Frontal advance theory
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
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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 =
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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?
3.2 Frontal advance theory
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
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
In this figure, let’s represent the saturation at the front as S fSwf.
wfS
wSx
fx
3.2 Frontal advance theory
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 ?
3.2 Frontal advance theory
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
3.2 Frontal advance theory
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
3.2 Frontal advance theory
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 +=
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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
3.2 Frontal advance theory
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( −+=
3.2 Frontal advance theory
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(
3.2 Frontal advance theory
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 =
3.2 Frontal advance theory
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 ;
3.2 Frontal advance theory
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.
3.2 Frontal advance theory
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 +
3.3.1 Cumulative oil displaced, Np
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
⎟⎟⎠
⎞∂∂
3.3.1 Cumulative oil displaced, Np
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
3.3.1 Cumulative oil displaced, Np
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.
3.3.1 Cumulative oil displaced, Np
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
3.3.1 Cumulative oil displaced, Np
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
3.3.1 Cumulative oil displaced, Np
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